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King-TransmissionLineTheory2

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Graduate textbook by Ronold W. P. King of Harvard, developed from his lecture notes. The text shown covers the preface and contents. Chapters treat infinite lines and their parameters, terminated lines, impedance and admittance, matching sections, baluns and hybrid junctions, current and voltage distributions, discontinuities, and oscillators, coupling and radiation. This is a published book by someone else, not Phil's own work.

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TranmlI~nn Theory BYRONOLD WPoKING DOVERBOOKS ONENGINEERING ANDENGINEERING PHYSICS TheoryofWingSections, JroH.AbbottandAlbertE.vonDoenhoff. $3.25 DeReMelallieD, Georgius Agricola. Clothbound $10.00 Chorlos Bobbogo ondHisColculating Engines, editedbyPhilipMorrison andEmilyMorrison. $2.00 Treulise onHydrodynamics, A.IJ.Bosset.Twovolumeset$3.50 Traveling WavesonTransmission Systems, L.V.Bowloy. $3.00 Two-Dimensional FieldsinEleclricol Engineering, L.V.Bowley. $1.50 FluxLinkages andElcclromognclic Induction, L.V.Bewlcy. $:1.25 ThoMeasurement ofPowerSpectru fromthePointofViewofCom munications Engineering, RnlphB.Blachmon ondJohnW.Tukey. $1.85 TheoryofShipMotions, S.N.Blogoveshchensky. 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Published intheUnitedKingdom byConstable andCompany Limited, 10Orange Strcet,London W.C.2. ThisDo\'eredition, firstpUblished in1965,is anunabridged andcorrected republication of!JIC workfirstpublished bytheMcGraw· HillBook Company, Inc.,in1955,towhichhasbeenadded .l newIndexofSymbols, LibraryofCongress CalalogCardNumbu: M-8269 Manufactured intileUnitedStatesofAmerica DoverPublications, 1m:. 180VarickStreet NewYork14,N.Y. Theoryassuch i~ofnouseexceptinsofaras itmakesusbelieveinthecoherence ofphenomena. Gmml.;, AIaximsandReflections Fe Ea Lone MeeR4 iNe sie 3zt 7b y ’= BY 2% CAR, Co <ssinay PREFACE TOTHEDOVER EDITION Although adecadehaspassedsince1'ransmission-line TheonJwas firstpublished, thematerial itcontains isstillmodernandcomplete. Thischaracterization islikelytoa.pplyformanyyearstocome.Indeed, astechnology continues torequireevergreaterprecision, thescienti ficallyaccurate treatment provided inTranamission-line TheON)may findincreasing appreciation anduaebyengineers. Inthissecondeditiollanumborofmisprints andminorerrorshave beencorrected. Inaddition, anindexofsymbols hasbeenprovided. l{ONOLDW.P.KUIG GordonA[oKayLaboratory ofAppliedScience Harvard University Cambridge, Mass.,August, 1964 PREFi\CE Asaconsequence oftheever-increasing preoccupation ofphysicists withproblems innuclearandsolid·state physics,thescientific andtech nicaladvance ofelectrical scienceisrapidlybecoming theconcernand responsibility oftheengineer. Astheresearch scientist knowsonlytoo well,anunderstanding ofreecntprogressinappliedelectricity, especially inthegltneratioll, transmission, andradiation ofelectromagnetic energy ateverhigherfrequencies, rcq\lircsadeeperappreciation ofphysical and mathematical fundamentals thancanbeprovided byevenamostthor oughknowledge ofelectric-nctWOI'k t.hcoryasappliedtolumpedelcmcnt8. Thisdeeperandmorefundamcntltl approach involves aknowledge of generalelectromagnctic theory. Perhaps themostinteresting bridgebetween thespecialized pointof viewoflumped-eonst.ant electriccircuitsandthegeneralandfundamental approach ofelectromagnetic theoryistheconventional transmission line. Sinceitstransverse dimensions Sfttisfytheconditions oflumpcd-eonstant circuits, whcreas itsIcngthisunrestricted, theelectromagnetic aspect involving theconceptofpropagation isolle-dimensional. Moreo\'er, by demanding thatequalandopposite currcnts andchargesbesufficicntly closetogether, thesmoothlinemaybeapproximatcd byarecurrent network oflumpedelements formostpur·poses. Thisisfoundtobea consequence oftheapplication ofgeneralelectromagnetic theoryandits specialization, subjecttoappropriate restrictions, totheboundaries of conventional transmission lines.Significantly itispossibletointroduce someofthemostfundamental concepts ofelectromagnetic theorywith· outbecoming involved inaUthecomplications ofvectorfieldtheory. [nChap.Ithcwell-known transmission-line equations foraninfinite linearedcduecd intheconventional manneran.dfromelectromagnetic fundamentals forvariousimportant crosssections. InChap.IIthe derivation oftheC<luations issl>ccializcd tolinesoffinitelength,andthe basicmethodoftreating terminated linesisformulated. Chapter III isconcerned withtheimpcdanec ofsectionsoftransmission lineAndtheir liseU8shuntandseriesclements ininsulators, tmllsformers, matching lIetworks, unbalanced loads,baluns,shielded loops,andhybridjUlIctions. Animportant featureofthetrel~tmCl1t ofterminal impedance andinput impedancc istheuseofcomplex terminal functions. Thesearcvery closelyrelatedtoexperimentally measured quantities, andtheyare viil viii PlIEt'ACE moreconvenient inanalyzing mllllYtypesofproblem;; thanarcthe reflection coefficients. Chapter IVi.lldevoted tothestudyofdistribu t-ionsofcurrentandvoltage, thet.ransferofpower,andananalysis of distribution andresonance curvesandtheirapplication intransmisaion linemeasurements. Chapter Vinvolves theanalysis ofdiscontinuities andnonunllormities alongsmoothlines.Included arediscussions ofthe Weissfloch tangentrelationandDeschamps's methodfordetermining the properties ofjunctions. Alsotreatedaredielectric slabsasdiscon tinuities, themeasurement ofdielectric constants andpermeabilities, changesincrossscction,terminations, bends,Tjunctions, andendcor rections, including thrnICfortransmission linesusedtodriveantennas. Chapter VIiseoncemed withtransmission-line oscillators, coupled circuitphenomena onlincs,andradiation. Mostofthisbookhasdeveloped fromlecturenotesforaonc-scmester graduate courseontransmission-linc theorywhichhMbeengivcnat Harvard Uniycrsity forthelast10yeaTS.Itisconcerned primarily withthehigh-frequency aspectsoftransmission linesandwiththeir steady-state operation. Itisdesigncd toserveasanecessary andfun damental introduction preccding seriousworkinwaveguidesandcavities aswellasantennas. Although thetreatment ispredominantly analyt ical,littlcmorethanasoundknowledge ofdifferential audintegral calculus andelementary differential equations ispresupposed, together withathorough background inalternating currents. Although the foundation oftransmission-line theoryonelectromagnetic principles is aspecialfcatureofthisbook,itispossibletopursueamoreconventional coursesimplybyomitting Chap.I,Sees.3to12,andChap.II,Sees.I to4.Itistobeexpected thatChaps.VandVIwouldnotbecoveredin suchaplan. Theauthorisindebted toseveralofhisstudents forcontributions and assistance. Theseinclude, inparticular, J.Eisenstein, D. D.King, J.Porter,L.S.Sheingold, J.E.Storer,C.T.Tai,andK.Tomiyasu. D.B.BrickandJ.E.Storerassistedwiththeproofs. Theentiremanu scriptwastypedbyPhyllisKennedy. Thefigureswereprepared by E.Risingandhisassistants. RONOI.DW.P.KINO CONTE 1'8 Prefau vii Nouon1MNumbering f)fEf{1J.otioru ondFiguruandon1MNiHatiQfl. xiii CIIAPTER I:TilEINFINITELY LONOLIN!:. 1 1.Methods ofAnaly~illg theTranllmiMion Line 1 2.TheConventional Derivation oftheDifferential EqualiODII oftheTrans- missionLine. 3 3.Potential Fum:t.iODS andE1ootromagnetic PTeliminariell 7 4.E1ectromagnetie Derivation QftheEquations andParameters forBal- ancedInfinitely LongTwo-wire Lince 13 6.TheBalanced Four-wire Line. 19 6.TheCoaxialLine 20 7.TheOoerlySpacedTwo-wire LinewithUnequal Conductors. 23 8.TheShielded LinelrithEcceotric InnerConductor. 31 9.TheShieided-lMir Line. :u 10.Three-wire Polyphue Line;Th~phue Cable. 39 11.TheCou:i.aJCage Tra.nem~ion !.ine 43 12.StripLinel! 4.S 13.GeneralSolution ofthoDifferential Equations foranInfiniteLine 48 14.Interpretation ofthoSolution forlheVoltagealonganInfiniteLine. PhAseandGroupV(!!ocitiell 50 Problema . 56 CHAPTEB II:Tu.ETEfUllNATED LI!;-I!:. 58 1.PDtential Functions foraTerminated Line 58 2.Generalized Differential Equationa 64 3.Terminal Zones;Couplinl5 andEndEffeetll 68 <t.Equivalent Uniform LinewithTerminal-wne Network 71 5.Evaluation ofConstantll inTermsofBoundary Conditions; Exponential Solution foraTerminated Line 73 6.Infinite-eerics FormoftheExponential Solution 77 7.lnddent- andReflected-wave FormoftheExponenti&1 Solution. 79 8.Hyperbolic FormaoftheSolution 83 9.Instantaneous ValuesoftheHyperbolic Solutions 86 10.ThePropagation Constant 91 II.TheCharacteristic Impedance 93 12.ThePhll.!!llandGI'QUPVeloeitiea oftheInfiniteLine 94 13.SpecialFormsoftheGeneralParameteMl oftheI.ine 95 14.Relation between Reflection Coefficient andTerminal Functions 10J 15.ThePhaaeandAt~nuation Functions oftheTerminations 102 16.Graphical Representation oftheTenninal Functiona intheNormalized Impedance orAdmittance PlanejCircleDiagram HU 17.Graphical Representation ofthe~ormali1ed Impedance orAdmittance in theRefteo:tion-eoeRieient Plane;SmithChart 108 ~ COXTENTS 18.SpecialFormlloftheTerminnl Funetionll andoftheRf!f1f!ction Cotfficient -Resistive Termination 112 19.SpecialFormaoftheTerminal Function9-the Predominantly lteactive Termination IIi 20.TheConducting \VireUridgeasaTermination; Resilltive Wire. 120 21.Conducting PiBtonslindDillklllUITerminations 127 22.Terminations withNegative Attenu:\tion Funclion orReflection Cocffi. eientGreaterthanUnity 128 Problema 130 CIIAPTER III:blPEDANCE ASD A[)~I,.r.o.SCE. 133 I.Normalized Input11ll1)C(lallce andAdmittance ofaTcrmillated Section ofLine 133 2.InputImpedallee nndAdmittnnce 147 3.Extreme ValuesoftheInputRC6illtnnce nndConductance 153 4.Extrcme VllhWIlortho1111111tll.cnclnncc lindSUllcpplancc 157 5.Summary ofCriticalValuCllofInputImpedancll andAdmittance forII. SectionofLow-10M Line 100 G.SectionofTransmission LineasanInsulator Hi-' 7.Impedance Transformation UsingaNetwork ofTransmission-line Scc7 tiona-General Formulation 172 8.TheSeriesTransformer 174 9.?o.fBtching SectionwithaSingleMovable Stub In 10.Matching SectionConsisting ofaDoublC-8tub Tuner 184 II.Matching withaShuntSection 190 12.Representation ofaSectionofTransmi!l8ion LinebyLumped Equiv- alenla;Impedance, Admitllln(..'C, andScattering Matrices 194 13.Unbalanced LoadTerminating aSymmetrically DrivenShielded·pair Line 203 14.SeriesStubsandUnbalanced Sections ofLine;FoldedDipolc; Balun; Shielded IAlop 209 15.TheHybridJunction forTrausmiuion Lines 225 Problems 241 CII...MEnIV:GE:<rERAL AMrLtTUOE REr,...·I'lo:<rsrORCURRE:<rT "'NoVOl,T"'UY. 24:\ LTheDi~tribUlion ofCurrentandVoltllgeandtheTran~fcr ofl'ower lllon~ aNonresonant Line 243 2.General F.xprell5ions forCurrent andVoltllge forlinArbitrarily Ter minated LineWhenDrivenbynSinglePilirofI~'l\lalandO"l'o~ite Point Generato!1l (orTheirl!:quivalent) Anywhere alongtheLine 244 3.General Expressions forCurrent andVoltll.ge foranArbitrarily Termi natedLineWhenDrivenbyTwoPairsofEqunlandOpposite l'ointGen- erators(orTheirEquivalent) Anywhere alongtheLine 216 4.General Expressions forCurrent andVoltage foranArbitrarily Termi natedLineWhenDrivenbyThreePairtlofGcnerators (orTheir ~uiv. alent)Anywhere alongtheLine 248 5.PolarFormoftheGeneral ll."'xpressiona forCurrent andVollllge . 249 6.TheTransfer ofPoweralongaTransmill8ion Line 251 7.Rc!lonance CurveaandtheCondition for!ll'..'lOnance 254 8.Distribution CurVell 257 9.ReilOnance-eurve andDislribution-eurve Ratios; theStanding-wave Ratio 25!'l CONT~:NTS xi 10.Distributions ofCurrent AndVoltage inIIResonant Line;Components of Current andVoltage 262 It.TheWidthsofRe!Jonance IlndI)igtribution Curves. 266 12.The"Q"ofaTransmil!llion Line 269 13.TheoryofTransmission-line 1\IenBuremenLS 272 Problems 286 CHAPTER V:DI5COSTINl;ITlEll AS"NOSUSIFOIU1ITrES ISTRASlWlBSIOS LINES. 288 I.Two-terminal-pair Networka inTrllnsmi8llion Lines 288 2.Equivalent TransIormer forTwo-terminal-pnir Network ThatIncludes Sections ofTransmission Line.Wei.floch Tangent Relation 294 3.Experimental Determination ofanEquivalcnt IdealTransformer fora Reactive Network 298 4.Deschamps's Graphical Method forDetermining theScatt.cring Matrix BndEquivalent CircuitofaJunction 304 5.Mell'!\lrement ofIrnpndllllee lindReflection Coefficient through 1.IJune- tioo. 314 6.TheoryofaDielcctric nlld:'.fagnetie SlnborIleadinaTrallSrni511ion Line 317 7.TheMaxiulIlm-:'.linimum-shift Melhod forDetermining Dicleetric Con stantsandPermCllhililiCll ofoolirlsBndLiquidsandEquivalent SectioDs ofTrnnsmillSion I.incforSymmetrical Two·terminal-pair Networks. 329 8.Determination ofLoesceinDielectric ::londl'.lagnctic Materials Usingthe 1'.1a.xim\lm-Bhift Method 341 9.TheDoubleDeOOa.nd thcSplicingofBelldllforl\oChangeinImpedance. 340 10.TheDouble-slug TransIormer . 351 11.LOllsyTerminations forNonresonant Shielded Lines 358 12.ClosedandOpcnEndsIl8Reactive Terminations inTwo-wire andColUia.l Lines 364 13.Junction ofTwoOpen-wire LilU~withConductors ofDifferent Hadii 368 1-1.ChangeofRadiusinIlCoaxialLine 377 15.Ikndin1.ITwo-wire Line 382 Hi.TJunction inaTwo-wire Une. 389 Ii..Junction Network>! forSericaBranches inTwo_wire Lines;Terminal-zone Networks forStub-supported lIudOmlcr-driven Antennas and ~'oldcd Dipolcs 397 18.Change inSpacingofnTwo-wirc Line 411 19.Right,..angle llcndinthePlaneoraTwo-wire !.inc 418 20.BendsnndTJunctions inBall\llced Shielded-pair Lines 426 21.BendinaCoaxialLine;TJunction 426 22.EndCorrection foraCoaxial LineWhenDrivinganAntenna overa GroundScreen 430 Problcms 437 CIlAI'TER VI:TRASllloll&8TON-LINC OllCILI.ATORll ANDCoUPLED SECTIONS OF TRANSYI&810S I.ISE 439 I.Frequency Characteristics ofSimpleTriodeOscillators withTrlllUlmil5- .'lionLinesIl8TankCircuits 439 2.Frequency Characteristics oraTmnsmiJlSion-line Oscillator withCoupled Secondary 447 3.Electric FieldofaConductor withSinusoidally Distributed Current. 454 4.TheElectric FicldofaDrivenScctionofTwo-wire Line 457 '"CO~ENTS 5.CurrentandVoltageinII.LineDrivenbyII.Coupled Sectionof'rrllonijrnis- ~ionLine;Dire<:tiOMI Coupler. 463 G.Coupled TraosmiSllion Lines 408 7.Admittance ofBridge-eoupled Sections of[,ow-lo~8 Trallsmisllion Linc; O.l1lplcd-eircuit Effcctl!Involving Minima /HIdDoublePeaks 470 8.Transmission-line )Ica.8urcmcnt.l:l with BMultiple-frequency Source; FilterSections 482 9.Radiation fromOpen-wire Liol's 487 Problems 492 8iMiographll . 494 lnduu SOl NOTEONTHENUMBERING OFEQUATIONS ANDFIGURES ANDONTHENOTATION Chapters arenumbered withromannumeralsj sectiolls arenumberod witharsbic numerals beginning withIineaehcha.pter; equations arenumbered eonsacutively (I),(2),...,ineachsactionwithnoreFerence tosactionnumber. Atthetopof eachlcft-haud JHlgeisthechapternumber; atthetopofeachright-hand pageisthe lJC1ltionnumber. Whenreference illmadetoanequation inthesamesection,onlythe equation numbcr ~giv<ln,e.g.,(5).Whcnrefercnce i.smfldetofillequation inanothcr tlOOtioninthcsamechapter, thesectionandequation numbt,rs arcgivenintheform Sec.6,];:q.(12).Whenreference ismadetolUIequation inauother chapter, the ehuplernumber, IICCtionnumber, Rndcqulltion numberaregiven,e.g.,Chap.1I,Sec.4, Eq.(36).Figuresarenumbered withbothsectionandfigurenumbeJ"llj thusFig.6.2 isthesecondfigurein800.6.Reference toa6gureinanotherehapter includes the chapter number, c.g.,Chap.II,Fig.7.5.Byreferring tochapterandsectionnum bel'llatthetopllofthepages,anyequation orfigureisquicklyfound. Superior numool1l refcrtotheBibliogrlljlhy fittheendofthebook.Thefollowing lIymbolillm isused:SpaceveelOl1I, whether realorcomple"" areinboldface roman type,A,%.Complex !lClllars(phllllOrs) areinboldface italic,Z,Orboldface Greek,y. RealllCalllrs lireinlightface italicorlillihtFace Greek,X,a.Matrice.'lanl repJ'Cllented byboldfaee roman,Y. CHAPTER I THEINFINITELY LONGLINE 1.Methods ofAnalyzing theTransmission Line.Thedistributions ofcurrentandpotential difference andthetransfer ofpoweralongopen andshielded transmission linesmaybedetermined byseveralmethods. Thechoiceofmethodmayappeartobeofnopractical concern, sinceit isassumed, quitenaturally, thatallmethods mustgivethesamecorrect answer. Actually themathematical analysis ofmanyphysical phenom ena.isnoteasilyreducedtorightandwrong. Aso-called solution isin almosteveryinstance anapproximation oridealization, andinconse quenceitscorrectness isamatterofdegree. Thisistrue,inapeculiar way,ofthecurrentinatransmission line.Butthisfactdoesnotin itselfdictateachoicebetween theseveraldifferent approaches toaprob lemiftheyleadtothesamefinalresult. Obviously, inthisevent,all methods arecorrecttothesamedegree,anditwouldseemthatonemust beasgoodasanother. Insofarastheultimate formula isconcerned, this istrue.Ontheotherhand,itisneversufficient tobeprovided merely withaformulathatistobeusedtocompute actualresultsinanengi neeringproblem without acomplete statement ofthecircumstances to whichitappliesandoftheconditions underwhichitwillyieldaccurate results. Suchastatement isanessential partofeverymathematical for mula,notwithstanding thefactthatitisoftennotprovided andthat correctanswers areoftenobtained without it.Itisthefunction ofa mathematical derivation ofaformula fromfundamental principles to supplyinformation regarding allrestrictions, approximations, andlimi tationsthatareimposed, aswellastoproduce theformula itself.But eventhisisnotenough.Itisnecessary alsotoexamine thegenerality andtheapplicability ofthefundamental principles thatareaccepted at theoutset. Themethods thatmaybepursued inanalyzing thetransmission line fallintotwogroups,thosewhicharebasedonelectric-circuit theoryand thosewhichproceedfromelectromagnetic theory. Inthefirstgroupare twowell-known methods. Theonedividesthetransmission lineinto smallelements eachofwhichisrepresented byan"equivalent" circuit ofsuitably arranged elements ofinductance, resistance,' andcapacitance towhichKirchhoff's lawsmaybeapplied. Byallowing thelengthof theelementtovanish,thedifference equations soobtained becomethe 1 2 TRANSMISSION-LINE THEORY [Chap.I familiar first-order differential equations ofthetransmission line.The secondmethodinthefirstgroupisrelatedcloselytotheonedescribed. Ittreatsthetransmission lineasalimitingformofanartificial linecon structed ofrecurrent sections ofunrestricted impedances andanalyzed ingeneraltermsbynetwork theory. Inthesecondgroup,whichdepends onelectromagnetic theory,the attempt maybemadetoanalyzethetransmission lineasaboundary valueproblem, oritmaybemerelyaquestion ofderiving thetransmis sion-line equations. Ineithercaseitispossible toproceed from the Maxwell equations defining theelectromagnetic fieldorfromthedefin ingrelations forthescalarandvectorpotentials. Adetailed, criticalevaluation oftheseseveralmethods cannotbemade atthispoint.Thefollowing comments, however, areofferedasan introduction tofurtherwork.Throughout thefirstgroupitisassumed withoutproofthatthemethods ofelectric-circuit theory,inparticular, theapplication ofKirchhoff's lawsandthedescription ofcircuitsentirely intermsofresistance, inductance, andcapacitance, aresufficiently gen eraltoserveasfirstprinciples. Actually thisistrueforthetransmission linesubjecttodefiniteconditions, which,moreover, cannotbedeter minedinanyderivation thatignorestheminitsinitialpostulates. None ofthemethods inthisgroupcanprovideformulas forthecircuitparam eters,whichmusttherefore alwaysbederivedseparately. Ontheother hand,theentirefirstgroup,andparticularly thefirstofthetwomethods mentioned, ischaracterized bytheanalytical simplicity ofnetwork theory. Thisisnomeanadvantage. Thesecondmethodalsobrings intoclearperspective theimportant relationships thatexistbetween transmission linesandartificial lines. Themethods ofthesecondgrouphavetheadvantage overallothers thattheydirectlydependuponthefirstandmostfundamental prin ciplesofmacroscopic electrodynamics. Theyhavethealmostequally greatdisadvantage ofsharinginthecomplexity ofelectromagnetic the ory.Thislatterisespecially trueofmethods thatattempttoanalyze thetransmission lineasaboundary-value problem, butalso,thoughto asmallerdegree,ofmethods thatspecialize theMaxwell equations or theequations forthepotential functions inordertoobtainthetransmis sion-line equations. Incarrying outsuchaspecialization allnecessary restrictions andapproximations maybemadeavailable, andformulas for allparameters ofthelinearederivedintheprocess. Asaconsequence, itmaybeshorteraswellasmorerigorousandcomplete thanthesimplest methodbasedoncircuittheory,iftheseparate determinations ofthe severalparameters areaddedtothis.Finallythemoregeneralanalysis usingelectromagnetic principles permitsastudyoftransmission-line end effectsandcoupling effectsinaregionnearthejunction ofthetransmis sionlinewithitsterminations orwithanotherlineofdifferent charac- Sec.2] THEINFINITELY LONGLINE 3 teristics. Theseplayasignificant roleindetermining theapparent impedance ofatermination asanactualload,asdistinctfromitsideal ortheoretical impedance asanisolatedentityindependent ofthelineto whichitisconnected. However, toadegreesatisfactory formostprac ticalpurposes, suchjunction effectscanberepresented byanappropri atecircuitoflumped reactances. Itfollowsthat,subjecttosuitable restrictions thatdefinethelimitsofconventional transmission-line anal ysis,theentireproblem canbesolvedinaformthatmakesuseofthe symbolism ofnetwork theory. Thatis,thevariables arecurrents, volt ages,andcharges; theparameters areresistances andreactances. Just asinnetwork theory,thespecificvaluesofresistance andreactance associated withaparticular configuration ofconductors musteitherbe determined theoretically fromelectromagnetic principles ormeasured. Itmaybeconcluded thatelectromagnetic investigations arerequired in orderto(1)specifythenatureofthecircuit,(2)evaluate theparameters involved, and(3)providetherestrictions onthegenerality ofthecircuit andtheformulas fortheparameters. However, electromagnetic theory isnotneededtodetermine theproperties ofagivennetwork ofresist ancesandreactances. Inordertosimplify theanalysis ofaproblemthatinitsfundamental senseishighlycomplicated, itseemsdesirable tobeginwithastudyof thetransmission lineasalimiting caseofarecurrent network oflumped resistances, inductances, andcapacitances. Thesoundness ofthis .methodisthenverifiedbyreanalyzing tbeinfinitelineusingthescalar andvectorpotential functions ofgeneralelectromagnetic theory. In thiswaytheformulas forthelineconstants areobtained withthedif ferential equations. Atalaterpointthissamemethodisappropriately generalized tothefinitelinesothataccount maybetakenofjunction andendeffects. 2.TheConventional Derivation oftheDifferential Equations ofthe Transmission Line.Shortsections oftwo-andfour-wire linesandof coaxialandshielded-pair linesareshownschematically inFig.2.1.For theopen-wire linestheconductors areidentical. Eachiscircularincross section;itsradiusisa,andtheseparation between centersisb.Inthe caseofthecoaxiallinethesmallerconductor hasanouterradiusaI,and thelargerconductor hasaninnerradiusa2andanouterradiusaa.In carrying outtheanalysisitisassumedthateachsectionoflengthL\zmay betreatedasifequivalent tothecircuitshowninFig.2.2,withfixed valuesofr,l,c,andginthelimitasLlzismadetoapproach zero.For theopen-wire linesrlandIIareassumed tobeequal,respectively, tor2 andl2,raandla,r4andl4.InthecaseofthecoaxiallinerlandIIare notequaltor2andl2.Clearly, toreplaceeachlengthL\zofatransmis sionlinebythecircuitofFig.2.2impliesthatallsuchlengthsareexactly alike,acondition trueonlyforalinethatisinfinitely long.Inductance 4 TRANSMISSION-LINE THEORY [Chap.I ,, i A2 B2 CoaxiallineIa3 I 4,21: a1III--;..!-----+- ...... rI1Z ir12z+6z A2 B2 Two·wifelineandcapacitance and,toasmallerdegree,resistance andleakage con ductance perunitlengthdifferneartheendsofalineoffinitelength, however terminated, fromtheirvaluesfarfromtheends.Moreover the loadmaybecoupledtotheconductors ofthelineinashortregionnear theircommon junctions. Iftheassumption ismade,nevertheless, that inductance andcapacitance aswellasresistance andleakageconductance Al B1 Al B1:"z I I I 1--- l----Az__ r-IIZ ~IIZHZ ~I2z :'-'12ztAZt'-----;.~---..;..-~ }2a ! ~ : I +r--~~-----!.L.-""7\ (a) (b) I , I I ,rI1ZI1Z+AZ;j) t:: ,-_..1-: .....:_~:) ;-I2z12z+Az: I I I • Cd)Shielded·pairline+Y.-B3fI1Z+6Z C=~~=:;:==~__--_:;;;;_-_-:.jo'o_o::::;-¥~. Z+6Z:-lI2zrlI2Z+Az t------A.z-...f A2 B2 Four-wire line (c) FIG.2.1.Sections ofinfinitetransmission lines.B4 ~tI2zHz /b/A 1,~_,'--~~--_---=t""'""~ }2a.'t I I I I I ~, I I l, perunitlengthareconstants independent ofthelocationoftheelement Az alongtheline,andcoupling between lineandloadisignored, theover-all errorsointroduced canbemadenegligible onlybymakingthesepara tionoftheconductors ofthelinesufficiently smallcompared withboth thelengthofthelineandthewavelength. Inpractice, theerrorinvolved inthisassumption eitherisdisregarded andconsequently included with theterminal impedance orisdesignated asanendeffect.Notethatthe assumed equivalence between thecircuitsinFigs.2.1and2.2isinno wayqualified byrestrictions limiting itsgenerality, thussuggesting that therearenorestrictions. Thisisaconsequence ofthefactthatthe Sec.2] THEINFINITELY LONGLINE 5 (2)(1)restrictions thatactually obtainarelimitations onnetwork theoryasa wholeandnotonthisparticular application alone.Noristheargument thatthecalculated resultsareverifiedexperimentally entirely satisfac toryexceptinalimitedway,sinceforanygivenlinetheagreement ceasestobeagoodonewhenthefrequency israisedtoasufficiently high value.Itisshownatalaterpointthatnetwork theoryisagoodapproxi mationintransmission linesonlyiftheconditions b«X aresatisfied, wherebisthespaeingofatwo-wire lineanda2istheinner radiusoftheouterconductor ofacoaxialline. ThecircuitofFig.2.2maybeanalyzed asfollows: First,sinceL1zis small,thecurrents andthepotential difference atthepointz+L1zmaybe +--------------IJ.% --------- expressed intermsofthecurrents andthepotential difference atthepoint zbymeansofMaclaurin's expansion: lz+~z=lz+(~~}L1z+(~:~)z(L1;)2+ . Vz+~z=Vz+(~~)zL1z+(~:~)z(L1;)2+ . Next,uponapplying Kirchhoff's emflawaroundtherectangle formed bytheinputandoutputterminals ofthesection,thefollowing resultis obtained: j(llz+11z+~z)(r1+jwl1)L1z+Vz+~z -j(12z+12z+Az)(r2+jWl2)L1z-Vz=0(3) With(1)and(2)thisgives i[211z+(~1)zL1z+...](r1+jWl1)L1z -i[212z+(~2)zL1z+..·1(r2+jwl2)L1z+(~~)zL1z+=0 (4a) 6 TRANSMISSION-LINE THEORY [Chap.I Collecting terms,dividing by~z,andthenallowing ~ztoapproach zero give (4b) Thecurrentinanopen-wire linecanberesolved intotwocomponents, tobedistinguished inthefollowing bysubscripts C(forcodirectional) and0(foropposite). Thesecomponents aredefinedtosatisfythefol lowingrelations: lIz=IClz+IOlz Ic2z=Iciz12z=IC2z+I02z I02z=-lolz(5a) (5b) Itispossible and,forpurposes oftransmission, desirable todriveand arrange open-wire linessymmetrically insuchamannerthat Icz=0I02z=-lolz=-lz (6) Open-wire linesthatsatisfy(6)arebalanced; if(6)isnottrue,thelineis unbalanced. Thecomponents ofcurrent lcz,iftheyexistonanopen-wire line,are codirectional antenna currents thatdonotdependupontheconstants r,l,C,andg;hencetheycannotbedetermined bytransmission-line the oryorordinary electric-circuit theory. Whether codirectional currents existornot,transmission-line theoryismeaningful onlyforthecompo nentloz.Iflczisnotzero,thetotalcurrentisgivenby(5a),withlcz obtained fromantenna theoryandlozfromlinetheory. Inthefollow ingonlylozisdetermined, andforsimplicity itisassumed that(6)is satisfied, sothatthesubscript 0maybeomitted. Forpractical purposes (6)isalwayssatisfied forthecurrents onthe innerconductor andontheinnersurfaceoftheouterconductor ofa coaxialline.Antenna currents, iftheyexistonacoaxialline,asthey oftendo,areontheoutersurfaceoftheouterconductor. Inashielded pairlineunbalanced currents treatthelinelikeacoaxialline,withthe twoinnerconductors inparallelasonelineandtheinnersurfaceofthe sheathastheother.Antenna currents ontheoutsideoftheshieldare alsopossible. Letthefollowing shorthand beintroduced andappliedtoabalanced open-wire lineoracoaxialline: z==r+jwl (7) Thequantities r,l,andzarethetotalresistance, inductance, andim pedance perloopunitlength.Intwo-andfour-wire linesandshielded pairlineswithidentical conductors, Sec.3] THEINFINITELY LONGLINE 7 Assuming (6)tobetrueandusing(7),(4b)becomeH zlz= _(dV) dzz(8) BeforeKirchhoff's currentlawisappliedatthepointP,itistobe notedthatthevoltageacrossPP'isi(Vz+VZ+4Z)'Then lIz=IIz+4z+i(Vz+VZ+4Z)(g+jwc)AZ (9) Ifuseismadeof(1)and(2)andtheexpression soobtained isdividedby AZbeforeallowing thistoapproach zero,thefollowing equation isthe result: yVz= -(dl) (10)dzZ In(10)thesymbolystandsforthetotalshuntadmittance perloopunit length: y==g+jwc (11) Thefirst-order differential equations (8)and(10)arethewell-known transmission-line, orlong-line, equations. Thevariables arereadilysep arated,andtheequations replaced bytwoofthesecondorder,asfollows: _z(dl)=(d2~)=yzVz (12)dzzdzz _y(dV)=(d2~)=yzlz (13)dzzdzz Itisconvenient todefineaquantity "(.,knownasthecomplexpropagation constant, asfollows: "(2==yz=(g+jwc)(r+jwl) (14) Therealandimaginary partsof"(areaand{3.Thus"(=a+j{3.The lawofconservation ofelectricchargeisexpressed bytheequation ofcon tinuity. Incomplex formitis (15) whereqzisthechargeperunitlengthononeconductor. With(10), qz=-(jyjw)V z. J.Potential Functions andElectromagnetic Preliminaries. Fromthe pointofviewofgeneralelectromagnetism thedetermination ofdistribu tionsofcurrentandchargeinaconfiguration ofmetallic conductors (such asatransmission line)embedded inapoorlyconducting ornonconduct ingdielectric medium isaboundary-value problem involving thefield equations ofMaxwelltsubjecttoappropriate boundary conditions. tMaxwell's equations areformulated instandard booksonelectromagnetic theory. See,forexample, Refs.9,21. 27, 28. 8 TRANSMISSION-LINE THEORY [Chap.I Maxwell's equations mayberegarded essentially asdefinitions ofthe fundamental electromagnetic-field vectorsEandBintermsofcurrent andcharge. Formanypurposes itisconvenient tointroduce thescalar potential cf>andthevectorpotential AintermsofEandB.Inahomo geneous isotropic medium characterized bythepermittivity (dielectric constant) E,thepermeability p.,andtheconductivity 0",thesearedefined asfollows: aA-gradcf>=E+at curlA=B acf>divA=-0"f.Lcf>-Ef.Lat(1) (2a) (2b) Thethreevectoroperators in(1)to(2b)havethefollowing formsin cartesian coordinates (unitvectorsinthedirections ofthecoordinate axesarex,y,2): gradcf>=xacf>+yacf>+tacf>axayaz divA=aAx+aAy+aAz axayazxy2 curlA=aaaaxayaz AxAyAz Withaperiodic timedependence(3) (4) (5) e!>inst=epeiwtAinst=Aeiwt (6) theformulas (1)and(2a,b)becomecomplex andindependent ofthetime. Theyare -grade!>=E+jwA curlA=B divA= -jWf.L~e!>=-j~e!>w wherethecomplex dielectric factor ~isdefinedby andthelosstangentis h=~..tII- WE(7) (8a) (8b) (9a) (9b) tIfthedielectric constant andtheconductivity arefunctions ofthetimeinthe sensethattimelagsoccurand ~andqbecomecomplex intheforme=e'-je"and Sec.3J andwhereTHEINFINITELY LONGLINE 9 (10) Therelative dielectric constant orpermittivity Erandpermeability Vr satisfytherelations Vr=.!. (lIa) P.O where EO=8.85X10-12farad/m P.o=411"X10-7henry/m (lIb) Inordertoemphasize symmetry between analogous electricandmag neticquantities, thereluctivity v(orreciprocal permeability) isuseful: 1 v=-yI Vr= yrI Po= - =7.95X106m/henry P.O(lIe) Inmostapplications therelativepermeability isreal.However, when timelagsinmagnetization areinvolved, itiscomplex. Thus ~==y==p.'-jp."=p.'(I-jhm)==p.(1-jhm) (12a)v where p.=p.'VI+h;'==p.' (U~b) Thelaststepin(12a)and(12b)assumes ht«I Subjectto(12c)and thefollowing approximate expressions areuseful:(12e) (13a) wherey~==p.E[l-j(he+hm)]=p.E(l-jhv) ~==~[I+j(he-hm)]=~(I+jhr) ~E E ~==~[I-j(he-hm)]=~(I-jhr)yp. p. hv=he+hmhr=he-hm(13b) (13c) (13d) (13e) Notethat,whenhm=0,hv=hr=he.Convenient combinations ofthese v=V'-jv",therealeffective quantities V" wV.=v'+wl' (9c) mustbeintroduced andsubstituted forfandVin(9a)and(9b).Inordertoavoid thedistinguishing subscripts, fandVareretained withtheunderstanding thatwhere required theymustbereplaced byf.andv.asdefinedin(9c). 10 TRANSMISSION-LINE THEORY [Chap.I •factorsincludethecharacteristic velocities: 1vo==_/-==3 X108m/sec vJ.l.oEo 1 vov==--=--~VJ.l.rEr V=_1_==v-Vy~VI-jh" thephaseconstants: fJo==!!!.=w~~vo w_/-_/- (J==-=wVy~==fJV1 -jh"v andthecharacteristic impedances: ro==I~==376.7ohms "EOr==I~=roI;'"E"Er r_Iy.r ":.=,,~=vI-jhr(14a) (14b) (14e) (15a) (15b) (15e) (16a) (16b) (16e) Theelimination oftheelectricandmagnetic vectorsfrom(1)and (2a,b)usingthefieldequations leadstotheequations V2«1»+(J2«1»=0 V2A+(J2A=0(17) (18) wherethelaplacian operator V2(nablasquared), whenappliedtoascalar, isdefinedby v2«1»==divgrad«I» Whenappliedtoavectoritis V2A==graddivA -curl curl A Incartesian coordinates(19) (20) V2A=iV2Az+yV2AII+.2V2A.. (21) and V~=(a~2+:;2+::2)'" (22) where'"standsforanyscalarsuchas«1»,Az,All,andAll' Solutions of(17).and(18)whichgivethescalarandvectorpotentials atallpointsinahomogeneous isotropic medium duetodistributions of Sec.3] THEINFINITELY LONGLINE 11 currentandchargeinarbitrary configurations ofconductors are c>=_1_((n'e-i~RdS' (23) 47r~JJR A=_1_[[[i'e-i~RdV' (24) 47r"JJJR wheren'isthechargedensityonthesurfaceelementdS'oftheconduc toratapointQ'(x',y',z') andi'isthevolumedensityofcurrentinthe interior element dV'atQ'(x',y',z'). Thepotentials arecalculated ata pointQ(x,y,z) outsidetheconductors inthemedium inwhichtheyare embedded. Thedistance between thepointQ'(x',y',z')locating theele mentofchargeorcurrentonorintheconductor andthepointQ(x,y,z) wherethepotential iscalculated is R=Vex-X')2+(y-y')2+(z-Z')2 (25) Theintegration in(23)isoverallcharged surfaces; thatin(24)isover theinteriorofallcurrent-carrying conductors. Notethatthesolutions (23)and(24)takeaccountoftheboundary conditions automatically. If thecurrentdensityisexpressed incartesian coordinates, i=ii:z;+jill+ziz (26) thethreecartesian components of(24)are A=iA:z;+jAlI+zAz (27) A:z;=4;"JJJi~e-;RdV' (28a) A"=4;"JJJi~e-;RdV' (28b) Az=4~"JJJi~e~~RdV' (28c) Ifallconductors areofsufficiently smallcrosssection,forexample, a circleofradiusawhichsatisfiesthecondition 1~la«1 (29) (30a)(21r q'=Jon'adO'itispropertodefinethetotalaxialcurrentandthetotalchargeperunit lengthineachconductor. Specifically thepotentials calculated fromcur rentsandchargesinacylindrical conductor ofsmallcrosssectionalonga directionzaregivenbyt 1fe-ifJR c>= -q'--dz' 47r~zR Az=~I.I~e-i~Rdz' I~=[a[21ri~r'dr'dO'(30b) 47r"zR JoJo tTheseformulas aregoodapproximations iftheaxialintegration extends over distances atleastasgreatas5a(Refs.9,10,61).Neartheendsofacylindrical 12 TRANSMISSION-LINE THEORY [Chap.I Sincenocurrents areexcitedaroundtheaxisoftheconductor, ie=0, andtherefore Ae=O.Asaconsequence ofthefactthattheradialcur rentdensityirmustbesmallcompared withizif(29)issatisfied, and thatthecontributions ofoppositp.ly directed elements ini~cos8'virtu allycancelincomputing (30c) itfollowsthatArissosmallcompared withAzthatitmaybeneglected. Thatis,At'==O. Notethat(30a)and(30b)mustsatisfy(8b),which,withA=tAz, reducesto dAz+j~c>=0 dzw(31) (32) (33b)(33a)Iftheconfiguration ofconductors consistsofseveralconductors indif ferentdirections, contributions tothevectorpotential bycurrents in eacharelike(30b),with2replaced byaunitvectorintheappropriate direction. Theresultant vectorpotential isthevectorsumofallthese contributions. Theresultant scalarpotential maybeobtained byinte grating(30a)overallsurfaces. Alternatively, asaconsequence of(31) andtheequation ofcontinuity: dlz+ . 0-Jwq=dz intermsofthetotalcurrentlzandchargeperunitlengthq,itispossible toassociate specificpartsofc?andAwithoneanother andwithspecific partsofthedistributions ofchargeandcurrent. Forexample,ifatrans missionlinewithitsterminations includes conductors lyingparallelto thexaxis,othersparalleltotheyaxis,andyetothersparalleltothez axis,thevectorpotential isgivenby(27)with 1fe-ifJR A:c=4rv 1~Rdx' 1fe-ifJR Ay=4rv 1~Rdy' 1fe-ifJR Az=4rv 1~Rdz' (33c) whereA:cisdetermined entirelybycurrents intheconductors parallelto thexaxis,Aybythecurrents intheconductors paralleltotheyaxis, conductor oratbends,smallerrorscorresponding toachangeinthelengthofthe zintegration byatmost±aareinvolved. Notethattheformulas are,ineffect, theaccurate potentials forachargeqoracurrentIconcentrated alongtheaxisofthe conductor. Sec.4] THEINFINITELY LONGLINE 13 etc.Theassociated partsofthescalarpotential are (34a) (34b) (34c) Thefollowing continuity relations mustbesatisfied: (35a) (35b) (35c) Notethatq(x)isthechargeassociated withthecurrentIxandthatC>(x), whichisderivedfromq(:t:),isthescalarpotential associated withthevec torpotential Ax,whichinturnisderivedfromthecurrentIx. 4.Electromagnetic Derivation oftheEquations andParameters for Balanced Infinitely LongTwo-wire Lines.9,lo,47-49tSincethetrans mission-line equations derivedbynetwork theoryapplystrictlyonlyto anunending lineinwhicheverysectionislikeeveryother,itisappro priateasafirstapplication ofelectromagnetic methods torederive these r-Load :plane I }~Q{ 1I~ Rbb:---..Infinite line / :: toload Z-Q;! Q2 Ic------w'---.~ 14----------- w---------------1 w=o FIG.4.1.Sectionofinfinitetwo-wire line.Infiniteline~ togenerator equations fortheinfinitely longline.Theequations soobtained apply approximately toallpartsofauniform lineoffinitelengthexceptnear terminations orotherdiscontinuities. Thespecificconditions arederived inChap.II. Consider auniform two-wire line(Fig.4.1)extending alongthezaxis ofarectangular systemofcoordinates. Thetwowireslieintheyzplane withthecenterofwirelaty=bj2andthecenterofwire2aty=-bj2. tSuperior numbers refertotheBibliography attheendofthisbook. 14 TRANSMISSION-LINE THEORY [Chap.I Theradiusofeachwireisa.Itsatisfiestheinequality 1~la«1 Letitbeassumed forthepresentthattheinequality b2»a2(1) (2) issatisfied, sothatdistributions ofcurrentandchargeineachconductor maybeassumed rotationally symmetrical. Letthesectionoflineto therightofaplanez=8orw=8 -Z=0bedesignated theload. Thedistance alongtheaxisofeachconductor totheleftofthisplaneto axialelements dw'atpoints Q~andQ;isw'.Thetotalaxialcurrentat thepoint Q~inconductor 1isIh(w');thechargeperunitlengthnearthis pointisql(W'). Thecurrentatpoint Q~inconductor 2is12z(w');the chargeperunitlengthnearpointQ;isq2(W'). Theconditions fora balanced lim'areassumed tobesatisfied. Theyare 12z(w)=-Ih(w)=-Iz(w) A2z(w)=-A1z(w)q2(W)=-ql(W) =-q(w) cl>2(W)=-cl>l(W)(3) (4) Thevectorandscalarpotentials in(4)aredefinedattheequipotential sur faceofthecrosssectionatwoftheconductor indicated bythesubscript. Thecurrentandchargeperunitlengthsatisfytheone-dimensional equation ofcontinuity: dlz(w) .()0~-Jwqw = (5a) Thecorresponding relationforthepotentials thataredefinedintermsof Iz(w)andq(w)isSec.3,Eq.(31),witha/aw=-ajaz,namely, aAz(w)_j~cl>(w)=0 aw w Notethat,withthezcomponent ofSec.3,Eq.(7),viz.,(5b) (6a) and(5b),thefollowing equations areobtained bydifferentiation and substitution: (6b) (6c) sothatforpointsonthesurfaceofaperfectconductor, whereEz=0, cl>andAzsatisfytheequations ~:~+~2cl>=0~~;+~2Az=0 (6d) Sec.4] THEINFINITELY LONGLINE 15 Thepotential differences between equipotential ringsaroundthesur facesofconductors 1and2atopposite pointsQlandQ2atequaldis tanceswfromtheplaneoftheloadatw=0aredefinedasfollows: Yew)==cl»l(W)-cl»2(W)=2c1»l(W) Wz(w)==Alz(w)-A2z(w)=2Az(w)(1) (8) (9b)(9a)Thelaststepineachequation followsfrom(4).Thedifferential equa tionssatisfied byYew)andWz(w)areobtained readilybycombining (6b,c)with(7)and(8).Theresultsare o2y(W) 2 _ 0~+~Yew)-OW(Elz-E2z) o2Wlw)+~2W( )= .~2(E_E )ow2 tfzwJwlz 2z whereElzandE2zaredefinedonthesurfaces oftheconductors. Theevaluation ofthepotential differences intermsofthecurrentis accomplished asfollows:UsingSec.3,Eqs.(30a,b),thegeneraldefinitions are where and(lOa) (lOb) (11) (12) Inordertoevaluate (9)and(10)thechargeperunitlengthandthe currentatthepointQ'maybeexpanded inTaylorseriesintermsofthe chargeperunitlengthandthecurrentatthepointQ.Thustheleading andfirstcorrection termsare q(w')=q(w)+(w'-w)oq(w)+... ow liw')=I;(w)+(w'-w)olae;)+ . Using(5a),Eqs.(13a,b)maybeexpressed asfollows: (').()+1(J21z(w)(') qw=qw-;--- w-w JWow2 lz(w')==lz(w)+jwq(w)(w'-w)(13a) (13b) (14a) (14b) Finallythesubstitution of(14a)and(14b)in(lOa)and(lOb)givesthe following generalexpressions forthepotential differences alonganinfinite 16 TRANSMISSION-LINE THEORY [Chap.I uniform line: whereYew)=2~~[q(w)ko(w)+j~a2:~~) kl~W)] (15a) Wz(w)=2~v[Iz(W)ko(w)+jwq(w) kl~W)] (15b) ko(w)==f-"'",PL(w,w') dw'=f-"'",(~a-~b)dw' +f-"'",[F(a}-F(b)]dw'(16) k1(w)f'"(')P(')d'f'"(')(1 1 ) d ' --=w-WLW,W W= W-W- - - w (J -'" -'" RaRb +f-"'<S)(w'-w)[F(a)-F(b)]dw'(17) e-i'JRG-1 e-i'JRb-1andwhere F(a)==R aF(b)==R b(18) Thefirstintegrals, intheexpanded forms(16)and(17),maybeevaluated directly. Theyyield f'"(11) b- - - dw'=2ln- (19)-'"RaRb a and f'"(w-w')(l--~)dw'=0 (20)-'" RaRb Itwillnowbeshownthat,subjecttothecondition l~bl2«1 (21) thesecondintegrals in(16)and(17)maybeneglected. Evidently, over thatpartoftheintegration forwhich(w'-W)2islargecompared with b2anda2,RaandRbdiffernegligibly fromIw'-wiandfromeachother. ItfollowsthatF(a)==F(b),sothattherearenocontributions tothe integrals. Theprincipal contributions totheintegrals occurwhen Iw'-wiissmall,sothatRaandRbareoforderofmagnitude b.But whenthisistrue,itfollowswith(21)that I~2R21«1e-i'JR==1 -j~R-~2R2+j~3R3 (22) whereRstandsforRaorRb•Hencetheintegrand ofthesecondintegral ontherightin(16)isoftheorderofmagnitude IF(a)-F(b)\=1~2(Rb-Ra)-j~3(Rl-R~)I (23) whereas theintegrand ofthefirstintegralis 1 1 Rb-Ra (24)Ra-Rb=RbRa Theratiooftheintegrand inthefirsttothatinthesecondintegral in (33b)(33a)Sec.4] THEINFINITELY LONGLINE 17 both(16)and(17)is 1:11PRaRb-j~3RaRb(Rb+Ra)1 (25) overarangewhere(22)isvalid,sothatwith(22)thesecondintegralis negligible. Itfollowsthat,fortheinfinitely longlinesubjectto(21), ko(w)~ko=2In~ (26)a k1(w) ~0 (27) Itisinteresting tonotethattheimaginary partof(25)determines theradi ation,which(21)makesnegligible. With(26)and(27)thepotential differences atapointQalongtheline are Yew)~q(w)~=dlz(w)! (28) Ydwy Wz(w)=Iz(w)le (29) Thefollowing symbolism isintroduced in(28)and(29)': +. . 21l"~ jW1l"~ (30)y==gJWC==JWk;=In(b/a) a g==In'(:/a) C==In~E/a) le=~In~ (30b) Thesearetheleakageconductance, thecapacitance, andtheexternal inductance perunitlengthoftheinfinitetwo-wire line.Ift'iscomplex, lealsoiscomplex 6Inthelaststepin(28)useismadeof(5a).Notethat ~2=W2t'~= -zey (31) whereze=jwleistheexternal impedance perunitlengthoftheparallel line. Itisnoteworthy thatthecorrection termsin(13a,b)and(14a,b)make nosignificant contribution tothepotential differences whenthelineis infinitely longand(21)issatisfied. Theassumption ofuniform current andchargeperunitlengthisadequate inevaluating thepotential differences andtheconstants foraninfiniteline. Oneofthedifferential equations forthetwo-wire lineiscontained in (28).Theotherisobtained from(6a)using(7)and(8).Thus dV(w) .(];U)-JWWzCw)=E1z(w)-E2z(w) (32) whereE1z(w)andE2z(w)aretheaxialtangential components oftheelec tricfieldatthesurfaces ofthetwoconductors. Thiselectricfieldis proportional tothetotalcurrent, sothat E()1().Iz(w)Zi lzW=lzwzi=-2- Iz(W)ZiE2z(w)=12z(W)Z~= --2- 18 TRANSMISSION-LINE THEORY [Chap.I Notethat12z(w)=-11z(w)forabalanced lineandz~=ziforidentical conductors. Hence (33c) (34) aviWJ.Le(Fe~10Thecomplex quantity Zi=ri+jxiistheinternal impedance perunit lengthofthetwo-wire line.Itsrealpartriistheinternal orohmicresist anceofaunitlengthofthetwoconductors. Theevaluation ofZifrom theratioEz(w)jlz(w)alongacylindrical conductor isintheliterature (Ref.9,Chap.V).Theformula forhighfrequencies subjectto(2)is ~=ri+'xi=1+j~ 1 1 J121l"a\j2;:; whereJ.Le and(Feapplytotheconductor. NotethatZi=2ziforatwo wireline.Using(33)and(29),(32)becomes aV(w)=(Zi+Jwle)l(w) (35a)aw With(30a)in(28),thisbecomes a~~)=(g+jwc)Yew) (35b) Thesearethefamiliar one-dimensional transmission-line equations. If desired,-ajazmaybesubstituted forajaw,sothat,with z=Zi+jwle=ri+jw(li+Ie)=r+jwl (36) thefinalequations are _aV(z)=zl(z) (37a) az _al(z)=yV(z) (37b) az Thesearetheequations derivedinSec.2byassuming thatthelineis equivalent toarecurrent network ofresistive andreactive networks. Thisassumption hasnowbeenjustified. Notethattheconstants ofthe linehavebeenderivedexceptforZi,whichpresents aspecialproblem. Intherestofthischaptertransmission-line theoryisformulated underthe assumption thatthepermeability ofallmediaisreal.If,asoutlined inSec. 3andimpliedin(30b),(35a),and(36),\'iscomplex andgivenbySec.3, Eq.(12a),i.e., \'=J.L'-JJ.L"=J.L(1-jhm) thecomplex external inductance Ie=le(l-Jhm) mustbeusedinsteadoftherealvalueleoThen ze=jwle=jwle(l-jhm)(38a) (38b) (39a) Sec.5] THEINFINITELY LONGLINE 19 andin(36)itfollowsthat z=Zi+jwle=ri+wlehm+jW(li+le)=r+jwl (39b) Clearlythegeneralized valueofris (40) 3 FIG.5.1.Four-wire line.wherehmistheratiooftheimaginary totherealpartofthecomplex permeability ta.Itfollowsthat,whenever amagnetic medium witha timelaginmagnetization response isinvolved, rasgivenin(40)mustbe used. Foraninfinitely longtwo-wire line(orapproximately forthesection ofalongfinitelinethatissufficiently farfromterminations ordiscon tinuities) (37a,b)arethecorrectequations forcurrentandvoltage. The following conditions havebeenassumed orimposed: (1)Thetwoconductors areparallelandidentical (2)I~al«1 (3)l~bI2«1 (41) (4)b2»a2 (5)Thelineisinfinitely long Itisshownlaterthat,byamodification intheformulas forthelinecon stants,condition (4)canberemoved andthat,byintroducing anappro priatelumped-constant networkat eachtermination ordiscontinuity, condition (5)maybeeliminated. Condition (1)ismodified toinclude conductors ofunequalsizeinSec.7. S.TheBalanced Four-wire Line.9Themethod ofanalysis usedinSec.4isreadilyappliedto thebalanced four-wire linecon structed ofidentical andparallel conductors (eachofradiusa)which aresodriventhatconductors 1and 3ononediagonal ofasquareofside- 4bareinparallel,asareconductors 2 and4ontheotherdiagonal. The twoparallelpairsformthetransmission line(Fig.5.1).Thefollowing conditions areassumed: I~al«1b2»a2(1) 14(w)=12(w)=-1l(w)=-13(w)=-j1(w) (2) q4(W)=Q2(W)=-ql(W)=-q3(W)=-jq(w) (3) 20 TRANSMISSION-LINE THEORY [Chap.I wherelew)isthetotalcurrentinthelineandq(w)isthetotalchargeper unitlength. (4) (5) (6) (7) (8) (9) (10) and Ra=v(w'-W)2+a2Rb=vI(w'-W)2+b2 Rc=yew'-W)2+2b2 Itisreadilyverifiedthat,subjecttothecondition II}bl4«1(11) (12) itfollowsthat (2)(1) (3) whereko(w)=1_00 00PL(w,W') dw'==2lna~=2(In~-0.3464) (13) 211"0"e 211"Ee J.lb Hence g=In(b/a0) C=In(b/a0) le=211"Ina0(14) Also Zi=z1+z~+z~+z~=4z1 (15) where,subjectto b2»a2(16) z1isgivenbySec.4,Eq.(34),athighfrequencies. Thesamedifferen tialequations asforthetwo-wire lineapplytothefour-wire line,but withthesenewvaluesoftheparameters. 6.TheCoaxialLine.Theanalysis oftheinfinitecoaxialline(con sistingofaconductor 1ofradiusalinaconducting sheath2ofinner radiusa2andouterradiusa3)maybecarriedoutinthemannerusedfor thetwo-wire line.Iftheconditions ofbalanceforcurrentandcharge inSec.4,Eq.(3),arepostulated, thepotentials atanarbitrary point Q(r,(),z)inthedielectric medium (al~r~a2)aregivenby 1J.2rf00 d8'c>(w)=47r~0 _ 00q(w')P r(w,w',8') dw'211" Az(w)=4~(2rf00Iz(w')Pt.(wtw',8')dw'd28' 11"11Jo-00 11" , , _[e-i()Rs1 e-itJR82]Pr(w,w ,())=-R- --R- d 82 21 THEINFINITELY LONGLINE Sec.6] and(Fig.6.1) R81=V(w-W')2+s;sf=r2+a;-2alrcos8'(4) R82=V(w-W')2+s~s;=r2+a~-2a2rcos8'(5) Thechargeperunitlengthandthetotalcurrentineachconductor are definedasfollowsintermsofthesurfacedensityofchargenandtheaxial component ofthevolumedensityof currentiz: (7)(6)ql(W)=21ralnl(w) q2(W)=21ra2n2(w) =-ql(W) llz(w)=21rfoalilz(w,r)r dr 12z(w)=21rfaai2z(w,r)rdrJas =-llz(w) Indefining 8in(5)itisassumed forsimplicity thattheentirecurrent 12z(w)isconcentrated inathinlayer ontheinnersurfaceofthesheathas foraperfectconductor. Foragood butimperfect conductor, thefieldin thesheathisincluded intheeval- uationoftheinternal impedance. FIG.6.1.Coaxialline. NotethatonlyR.inPr(w,w',8') involves 8'. Inevaluating (1)and(2)fortheinfiniteline,q(w')maybereplaced byq(w),andIz(w')byIz(w),asshownforthetwo-wire line.Thisleaves onlyPr(w,w',8') underthesignofintegration. Theintegration with respecttozmaybecarriedoutjustasforthetwo-wire line.Mterthe condition (8) isimposed toeliminate highermodes,thestepsintheanalysis parallel thoseinSec.4,Eqs.(16),(18),and(19).Theresultis f..Pr(w,w'8')dw'==f00(; -; )dw'=2ln~(9) _00 -0081 82 SI Theintegration withrespectto8'(whichdoesnotoccurwiththetwo wireline)maybecarriedoutusingPierceformula 523.Thus cI>(w)=q~:i{f:[In(r2+a;-2a2rcos8')]~~ -f:[In(r2+a~-2alrcos8')]~~} =q(w)Ina2 27T{r(10) 22 TRANSMISSION-LINE THEORY [Chap.I Similarly (11) Thepotential differences between pointsQl(al,(J,w) andQ2(a2,(J,w) on thesurfaces ofthetwoconductors are (12) (13) (16)(14) (15) (17)y=g+jwc with le=£.In~ 271'"al Asforthetwo-wire line,theinternal impedance perunitlengthis Zi=z1+z~Notethattheseresultsmaybeobtained eveniftheterm1/Raisomitted in(9)andYew)isevaluated directly. Thisshowsthatthepotential differences maybedetermined entirelyfromthechargesandcurrents in theinnerconductor. Thepotential differences in(12)and(13)maybe expressed intheformofSec.4,Eqs.(28)and(29),viz., Yew)=dlz(w)! dwy Wz(w)=Iz(w)le 271'"ug=~.,---~In(adaI) whereziandz~aretheinternal impedances perunitlengthoftheinner andouterconductors. Athighfrequencies (alviWUcJ.Lc~10) zi=1+jr;;; z~=1+jlJ.Lcw(18) 271'"al\j2;: 271'"a2\j2uc Theremaining stepsfollowthoseforthetwo-wire lineandleadtothe samedifferential equations, butwiththelineparameters (16)and(17) insteadofSec.4,Eqs.(30)and(33c).Thegeneralconditions onthe equations arelikethoseinSec.4,Eq.(41),forthetwo-wire lineexcept thatconditions (1)and(4)inSec.4,Eq.(41),arenotimposed. Theelectricandmagnetic fieldsinthedielectric medium inthecoaxial lineareobtained fromthepotential functions. Sincerotational sym metryobtainsandAzistheonlycomponent ofthevectorpotential, the vectorrelation[Sec.3,Eq.(2a)]reducesto Br=0BIJ= -aAzBz=0 (19)ar incylindrical coordinates. Hence,with(11), B=6BIJBIJ=-.!.L. (20)271'"vr Sec.7] THEINFINITELY LONGLINE 23 (21)TheelectricfieldisgivenbySec.3,Eq.(1).WithA=zAzandrota tionalsymmetry theonlytransverse component oftheelectricfieldis aepq Er= -ar=21l"~r Thelaststepfollowsfrom(10). Ifthespacebetweenthetwoconductors ofacoaxiallineisnotfilledby asinglehomogeneous isotropic dielectric, complications arisewhichcan notbesolvedwithouttheintroduction ofmoreadvanced mathematical methods. Twocasesareofinterest: (a)Iftheinnerconductor iscoated withagooddielectric ofuniform thickness andtherestofthespace between thetwoconductors isair,aso-called guidedfieldexistsinthe dielectric whichpropagates inamannerrelatedtothatofthesingle-wire dielectric-coated line.46Itcannotbeanalyzed byordinary transmission linetheory. (b)Ifthespacebetween theconductors isfilledwithtwo different dielectrics eachextending fromtheinnertotheouterconductor, buttheoneonlyoveranangle (Jandtheotherovertheremaining angle 21r-(J,thelinebehaves essentially inthetransverse electromagnetic (TEM)mannercharacteristic ofthecoaxiallinewithasingledielectric, provided a2issufficiently small.Thecapacitance perunitlengthofthe two-dielectric lineisaparallelcombination ofthecapacitances perunit lengthofthetwosectorswithdifferent dielectrics. 34 7.TheCloselySpacedTwo-wire LinewithUnequal Conductors.1.9•4o Inthegeneralstudyofthetwo-wire lineinSec.4theconductors are assumed tobeofequalradius,andthecondition b2»a2isimposed from theoutsetinordertokeepthetransverse partoftheanalysis simple whilethecomplications resulting fromtheoccurrence ofbothtransverse andaxialvariables areunresolved. Animportant resultofthisanalysis isthedemonstration that,subjecttotheconditions imposed, thetrans verseandaxialproblems areindependent toahighdegreeofapproximation. Indeedthefinaldifferential equations arethesameforallthelinesinves tigated. Theyinvolveonlytheaxialvariable z,whereas thesolutions oftheseveraltransverse problems appropriate tothecross-sectional boundaries arecontained intheformulas fortheparameters ofthepartic... ularline.Sincetheconditions ensuring thiseffective independence of axialandtransverse problems areprimarily thoserequiring thecross sectional dimensions tobesmall,thesearebettersatisfied formore closelyspacedtwo-wire lines.Solongastheconditions l~all«1and I~a21«1aresatisfied, thereisnothingintheanalysis whichrequiresthe tworadiitobeequal.Itfollowsthattheformulation oftheproblem ofabalanced two-wire lineconsisting oftwoparallelconductors ofradii alanda2separated adistance bbetween centerswhichsatisfies the conditions I~bl«1 (1) 24 TRANSMISSION-LINE THEORY [Chap.I wherethefirstinequality merelymeansthatthetwoconductors maynot actually makecontact,isreadilyachieved. Theseparation oftheaxialandtransverse partsofafunction, suchas thepotential difference or (2) (3)whichsatisfiesthescalarwaveequation iJ2Wz+iJ2Wz+iJ2Wz+~2JVz=0 Ox2oy2OZ2 isaccomplished bytheseparation ofvariables. Bysetting Wz=F(x,Y)f(z) andsubstituting thisin(2),thefollowing resultisobtained: ___1_[02F(X,y)+02F(X,y)]=_1o2j(z)+~2 (4) F(x,y) ox2oy2fez)OZ2 Inorderthatthemutually independent sidesofthisequation maybe equalforallvaluesofthevariables, theymustbothequalaconstant, which,however, maybemultivalued. Letthisconstant bek2•Then, with(3),thefollowing equations areobtained: (5) (6) (7)Foratwo-wire linewithidentical conductors, theaxialequation forWz isSec.4,Eq.(9b),withSec.4,Eqs.(33a,b,c), viz., 02Wz+rt2TJ'T _ •~2iJ OZ2 If".-:J;-zz whereziistheinternalimpedance perunitlengthofthetwo-wire line. Itisnowclearthattheapproximations involved intheanalysis ofSees. 4and5inwhich (8) areequivalent tosetting (9) in(5).Notethattheequation isthenvalidonlyonthesurfaces ofthe conductors, sincetherightsideisobtained fromtheelectricfieldson thesesurfaces. Withzi/wleextremely smallforgoodconductors, k2is smallinmagnitude compared with 1~12.Nevertheless k2mustbe retained in(5),.sincewhenthedielectric medium isperfectand ~2is Sec.7] THEINFINITELY LONGLINE 25 real,thepropagation constantyin "(2=k2_~2 (10) isapureimaginary unlesskdiffersfromzero.Anonzero valueofk impliesimperfect conductors andisnecessary tomaintain WzandIz finite.Ontheotherhand,thecontribution ofaverysmallvalueofk2 tothetransverse problem definedby(6)isinsignificant. Accordingly (6)isreplaced by a2wz+a2wz..:..0 (11)ax2ay2- Thesolution actually obtained inSec.4isforWzasdetermined from (7)and(11).Thusaccount istakenofthelargebutfiniteconductivity oftheconductors indetermining axialdistributions thatinvolvean unrestricted lengthofconductor, butnotindetermining thetransverse distributions thatinvolveonlytheverymuchrestricted crosssections. Theproblem istosolvethetwo-dimensional Laplace equation (11) subjecttothecondition thatAzhaveaconstant butdifferent valueon eachoftwocirclesofgivenradiialanda2.Thesearethecircularcross sections oftheconductors. Theappropriate solution is where rl=Vex-d)2+y2(12) (13) anddisaconstant lengthtobedetermined. Kin(12)isnotafunction ofxandy,butofzalone.Itisverifiedbysubstitution that(12)satisfies (11).Itremainstobeshownthattheequipotential surfaces arecircles. Forthispurposelet sothat andnIr2 t....p==n -=cons.rl ~=e2p rl Az=2Kp(14a) (14b) (15) Thuscontours ofconstant pdefineequipotential linesintheplaneor infinitecylinders inspace.Nextlet(14b)besquared and(13)substi tutedinit.Theresultis (16) Thiscanberearranged asfollows: d2 (x-dcoth2p)2+y2=d2(coth22p-c-1)-sinh22p(17) Thisequation definestwofamilies ofcircles-the onewithpositiveand theotherwithnegative valuesofpasparameter. Thusithasbeen 26 TRANSMISSION-LINE THEORY [Chap.I provedthatcontours ofconstant P,whichcoincide withcontours ofcon stantvectorpotentials, arecircles. Sinceanentirely similaranalysis canbemadeforthescalarpotential <p,itfollowsthatequipotential lines forbothscalarandvectorpotentials arecirclesofconstant p.Therangeof Pisfrom-00to+00.Thecentersofthecirclesareat Theirradiiarex=dcoth2py=o (18) d a=Isinh2pl (19) 2x/d o -1 -21----.--- 0- /.......,--------- -.....;0:::::':..........,~0.<5......."-:'i::J').~ 1',/~ "/"/"/ \ / /.--=-ifs-\/,,-/0.5--....",-4/.... \ / ......."\I/'\\\I/ \\ I\ ".~/- \\II/---.;(0 \",t //'\.;r--\\, IIrr(s\ \ p=_~'l-~/,II\\'P{p_ooI \I \\\ I '- I\/I\\/ I\, "/I '\ /1 "/I \"...... // \......., ...-/ .......-.._--_/ /\-----" /1 \ / '\ / '\./ ""- ,,-,,- .......,-....- .... 1--____----- ........................._------- -2oThetwofamilies ofcirclesareshowninFig.7.1.Theyaredividedby 2 FIG.7.1.Equipotential circlesfortwo-wire line. (20a) (20b)Y=YI=0 Y=Y2=0thestraight linex=0orp=O.Thepointsx=±d,Y=0arefor p=±·oo. Consider aparticular pairofcirclesdefinedbyp=PI=IPIIand P=P2=-lp21.Thecentersofthecirclescoincide withtheaxesofthe twoconductors at x=Xl=dcoth2PI X=X2=-dcoth21p2! Theradiiarethesameastheradiiofthetwoconductors, viz., Notethatd d al=sinh2PIa2=sinh2!P21 a2sinh21p2!=alsinh2PI(21a) (21b) Sec.7] THEINFINITELY LONGLINE 27 With(21a)III(20a,b)thelocations ofthecentersmaybeexpressed as follows: Xl=alcosh2Pl X2=-a2cosh2P2Yl=0 Y2=0(22a) (22b) Thedistance between centers, whichisthespacingbetween centersof thetwo-wire line,is (23) InordertoexpressPIandP2(andwiththemthepotentials onthetwo circles)intermsofbandalanda2,P2(orPI)maybeeliminated between (21b)and(23).Thus Similarlyb=alcosh2Pl+Va~+aisinh22Pl ThiscanbesolvedforPItoobtain b2+a2-a2 2Pl=cosh-12a:b2==cosh-11/;1 b2+a2-a2 2P2= -cosh-12 1==-cosh-I1/;22a2b(24) (25a) (25b) Thesymbols 1/;1and1/;2aredefinedin(25a)and(25b).Itfollowswith (15)thatthecomponents ofvectorpotential onthesurfaces ofthetwo conductors ofradiialanda2,withcentersseparated adistance b,are Alz=2Kpl=Kcosh-11/;1 A2z=2Kp2= -2Klp2!= -Kcosh-I1/;2 Thevectorpotential difference is Wz=Alz-A2z=K(cosh-11/;1+cosh-11/;2)(26a) (26b) (27) TheconstantKin(27)maybeevaluated bycomparing thesolution of Sec.4,Eq.(29),with(27)whenthisisspecialized towiresofequalradius andsufficiently greatseparation bysettingal=a2=aandutilizing the inequality b2»a2•Subjecttothesespecialconditions, b 1/;1=1/;2= 2a(28) Usingthestandard relation between arc-hyperbolic andlogarithmic functions, viz., cosh-1l!.-=In[l!.-+1(l!.-)2-1]2a 2a'\j2a itfollowsthatwithb2»a2 bcosh-11/;1=cosh-I1/;2 =Ina(29) (30) 28 TRANSMISSION-LINE THEORY [Chap.I Henceafterthesubstitution of(30)in(27)itfollowsfrom.acomparison withSec.4,Eqs.(29)and(30b),andl.'=J.1.=l/v,that K=~ 211'"V sothatthevectorpotential difference is Wz=2lz(cosh-11/11+cosh-11/12) 11'"V(31) (32) Thecorresponding solution forthescalarpotential difference hasadif ferentconstant. Itis (33) Itisnowconvenient todefinetheexternal inductance perunit lengthleandtheadmittance perunitlengthy=g+jweasfactorsof currentandchargeintherelati,ons Wz=lzleV=qjw (34)y asinSec.4,Eqs.(28)and(29).Theuseof(32)and(33)in(34)gives thefollowing formulas fortheparameters ofthecloselyspacedtwo-wire line: Thearguments arele=;11'"(cosh-11/11+cosh-11/12) g=21l"(T(cosh-11/11+cosh-11/12)-1 e=211'"E(cosh-11/11+cosh-11/12)-1(35a) (35b) (35e) (36) wherebisthedistance between centersofthetwoparallelconductors ofradiia1anda2. Forwidelyseparated conductors forwhichtheconditions b2»aiand b2»a~aresatisfied, (37) 11'"(1'g=------::==In(b/ya1a2)c= 1I'"E (38) In(b/ya1a2) Themostinteresting andimportant specialcaseiswhena1=a2=a. Inthiscase (39a) Sec.7] sothatTHEINFINITELY LONGLINE 29 ls=!:cosh-l~ 1r 2aU1r g=cosh-l(b/2a)E7r C=cosh- l(b/2a)(39b) Notethatwith(29)itispossibletoexpressthegeneralparameters for arbitrary spacingofidentical conductors inthesameformasforaspacing thatsatisfiestheinequality b2»a2merelybydefininganeffectivespacing bs,whichreplaces b.Thiseffective spacingisobtained from(29)tobe With(40a),(39b)maybeexpressed asfollows:(40a) ls=~In~ 7ra(1'7r g=In(bs/a)E7r C=;-In-(;-C;-bs-'/a~) (40b) Clearly, whenb2i&.sufficiently greatcompared witha2,theseformulas reducetothoseinSec.4,sincebs==bwhenthecondition b2»a2is satisfied. Asecondspecialcaseiswhenconductor 2hasaninfiniteradius,sothat itssurfacebecomes thexzplane,givenbyP=P2=O.Inthiscaseit followsfrom(22a)that XlPI=cosh-1- P2=0 (41)a where Xlisthedistance fromthecenterofconductor 1totheplane. Sincewith(26b)A2z=0,itfollowswith(27)and(31)that (42) Letthedistance Xlbeexpressed intermsofthefulllinespacingbbetween theconductor anditsimageintheconducting plane.Thatis,let b Xl=2" (43) Then,with(32)and(35), 11. b 27r(1' 27rEls=.!::-cosh-1-g c (44)27r2a =cosh-1(b/2a) =cosh-l(b/2a) Notethatleisone-half andgandcdoublethecorresponding valuefora two-wire lineofidentical conductors spacedadistance bbetween centers. ThisfollowsfromthefactthatWzandVmeasured between theconduc torandconducting plane,adistanceb/2,areone-half thevaluesmeasured between twoidentical conductors separated adistance bandcarrying equalandopposite currents andcharges. Thesameresultscanbe deriveddirectlyfromthetheoryofimages. 30 TRANSMISSION-LINE THEORY [Chap.I Theinternal impedance perunitlengthismodified whentwoparallel conductors areclosetogether bytheso-called proximity effect.The densityofaxialcurrentisincreased inadjacent partsofparallel con ductorswithoppositely directed currents andisdecreased atmoreremote parts.Thisincreases theeffective internal impedance, sincemorecur rbntisconfined toasmallervolume. Accurate formulas forz1forone cylindrical conductor inthepresence ofanotherwithdifferent radiusare notavailable. Ifthetwoconductors areidentical, anapproxiinate high frequency formula involves aneffective radius ae=aVI-(2a/b)2 (45) (46)inplaceofaintheformula forz1foracylindrical rotationally symmetri calconductor.tThusforeachconductor40 ..1+jI JJ.cW zi=21l"a'\j20'c[1-(2a/b)2] Theinternal impedance perunitlengthofatwo-wire lineisZi=2z1. Forthesinglewireovertheconducting plane,zi=zfiflossesinthe planeareneglected. Thetransverse electricandmagnetic fieldsinthemedium surrounding thetwoconductors arereadilyobtained. Sincebydefinition B=curlA, itfollowsfromSec.3,Eq.(5),that,whenA=zAz, B=aAzB 1I= _aAz (47) II:ay ax Theslopeofamagnetic lineis (48) sothatitscontourmustsatisfytheequation Bzdy-Bydx=aa~zdy+aa~zdx=dAz=0 (49) Integration yields Az=const. (50) Thusthemagnetic fieldisdirected alongcontours ofconstant vector potential, thatis,alongthecircles p=constant. Thedirection isspeci- tMoreaccurate formulas foruseatlowerfrequencies andwithtubularconductors ofouterradiusaandinnerradiuskaareS7 ri~r(1_k2)[BA_+B(2-B2)+B(9-lOB2+4B4)+...J(45a) 1 0 2y2 4 16y2A li~~[!InB+1+~_B(9-'-lOB2+4B4)+...J (45b) 11r2B-1 2y2 9y2Aa whereA==aV1J.cUcW,B==a/a.==[1-(2a/b)2J-l, andro=1/1rua2•Theseformulas aregoodapproximations provided thatrfro?;2.Stillmoreaccurate butalsomuch morecomplicated formulas aregivenintheliterature.as Sec.8] THEINFINITELY LONGLINE 31 fiedbytheright-hand screwrelationwithrespecttoA.SinceAreverses withP,thedirection ofBaroundthecirclesP=constant, withp>0, isopposite tothataroundthecirclesp=constant, withp<O. Theelectricfieldsatisfiestherelation E= -grad~-jwA Hence,withA=zAz,thetransverse components are(51) o~E=-- :r;oxo~E=-- 11oy(52) Sincethegradient ofascalarfunction isavectorinthedirection ofthe greatest rateofincrease ofthefunction, itmustbeperpendicular tothe equipotential linesgivenbyp=constant. Thustheelectriclinesmust becirclesperpendicular tothecirclesofconstant p.Assuchtheypass throughthepointsx=±d,y=O.Sincethevolumedensityofcurrent inthedielectric medium ifthisisimperfect isgivenby i=O"E (53) whereEisthefieldinthedielectric and0"isitsconductivity, itfollows thattheElinesarealsothelinesofflowofelectricchargesfromone conductor totheotherthroughthemedium. 8.TheShielded LinewithEccentric InnerConductor.I•33Iftheinner conductor (radiusal)ofacoaxiallineisdisplaced sothatitsaxisisata distanceDfromthecentralaxisofthe enclosing sheath(innerradiusa2),as showninFig.8.1,thesolutionofthe transverse problem maybeobtained fromtheresultsofSec.7.Itisshown inSec.7thatthecirclesp=constant areequipotential linesineachtrans verseplaneofatwo-wire line.The solution ofthetransverse problem of thetwo-wire linewasachieved by identifyingthecircularmetallic sur- facesofthetwoconductors which FIG.8.1.Shielded linewitheccentric haveradiialanda2andcentersatinnerconductor. X=Xl=lXII,y=0andX=X2=-IX21,y=0withthecircles p=PI=IPIIandP=P2=-lp21according totherelations (1) Thesameidentification withequipotential circlesmaybecarriedoutfor ashielded line(withcircularmetalsurfaces ofradiialanda2withcenters atX=Xl=lXII,y=0andX=X2=IX21,y=0,showninFig.8.1) 32 bysettingTRANSMISSION-LINE THEORY [Chap.I (6a)Xl=alcosh2PI X2=a2cosh2P2 (2) Notethatforthetwo-wire linePIispositiveandP2isnegative, whereas fortheshielded linePIandP2arebothpositive. Theaxialseparation of thetwoconductors oftheshielded lineis D=X2-Xl=a2cosh2P2-alcosh2PI (3) Since,asshowninSec.7,thefollowing relationissatisfied: alsinh2PI=a2sinh2P2 (4) itispossibletoeliminate firstPI,thenP2,from(3).Theresultsare a2-a2-D2 2PI=cosh-122a~D ==cosh-1tf;lc (5a) a2-a2+DZ2pz=cosh-l22a~D ==cosh-1tf;2c (5b) where tf;lcandtf;zcaredefinedby(5a)and(5b).Thusithasbeenshown thatashielded linewithinnerconductor ofradiusalandouterconductor ofinnerradiusa2>aI,withaxesseparated adistanceD,maybeidenti fiedwithtwocirclesofconstant Pdefinedby(5a,b).Thevectorand scalarpotentials areobtained fromthevaluesofPbymultiplying bythe appropriate constant. Thus Iz CAlz=-2.2Pl+A1rV «PI=2;~.2PI+C", «P2=2;~.2pz+C", (6b) Itisevidently possibletoaddthearbitrary constant potentials CAand C",andstillsatisfythetwo-dimensional Laplace equation. Thisisdone sothatthepotentials maybereferredtozeroattheshieldbysetting CA= -~.2P2C",= -..!L.2P2 (7) 21rV 21r~ sothatA2z=0and«P2=O.Thepotentials oftheinnerconductor are thenequaltothepotential differences asfollows: Wz=Alz-A2z=Alz=2Iz(cosh-1tf;lc-eu~h-ltf;2c) (8a) 1rV v=cj>l-«pz=«PI=2;~(eo~h-l tf;lc-co~h~ltf;2c) (8b) Withthegeneralformula cosh-1x-cosh-1Y=cosh-1[xy-V(x2-1)(y2-1)](9a) itfollowsthat a2+a2-D2cosh-1tf;lC-cosh-1tf;2c=cosh-1221..- (9b)ala2 Sec.8] THEINFINITELY LONGLINE 33 (15)1 2+2 -D2 WZh-la2a1 (10)Hence z=211"vcos 2ala2 a V=!Lcosh-l a~+a~-D2 (lOb) 211"~ 2ala2 sothattheexternal inductance perunitlengthofaninfinitelineis W 2+2 -D2le=...-!=E--cosh-la2al (1Ia)Iz211" 2ala2 Similarly t:!.=Y=_1_cosh-la~+ai-D2 (lIb) Yq 211"~ 2ala2 wherey=g+jwcistheadmittance perunitlength.Itfollowsthat 211"0" g=cosh-l[(a~+ai-D2)/2ala2] (12) 211"E c=cosh-l [(a~+ai_D2)/2ala2] (13) Notethat,whenthedistanceDbetween axesissufficiently smallsothat thefollowing inequality issatisfied: D2«ai+ai (14) itiscorrecttoset 2+2 _D2 2+2ah-la2a1 •h-la2a1 •I2cos =cos---=n-2ala2 2ala2 al sothatthelineparameters reducetothesimpleformofthecoaxialline, viz., ~~ 2~ ~ le=211"Inlhg=In(adal) C=In(adal) (16) Itissignificant thatthefunctions in(8a,b)maybeinterpreted as eitherofthefollowing: (1)The'potential difference between thesurface ",....---.......;-P=P2@P-P -:2al~ //' at~a2 -2~l: ~' a2\2-------bi-----t- .1 ) P=Pl p=-PI P=Pl''-......__....< Circlein (a) (b) space FIG.8.2.(a)Shielded linewitheccentric innerconductor. (b)Conductor 1with imageconductor 2,whichtogether maintain thesamepotential onthecircleofradius a2asexistsontheshieldofthesameradiusin(a). ofconductor 1(ofradiusalandwithcurrentIzandchargeperunit lengthq)andtheinnersurfaceofthesheath(ofinnerradiusa2and withcurrent-Izandchargeperunitlength-q),asshowninFig.8.2a. (2)Thepotential difference between thesurfaceofconductor 1(ofradius alandwithcurrentIzandchargeperunitlengthq)andacircleofradius 34 TRANSMISSION-LINE THEORY [Chap.I a2inspacewheninthepresence ofasecondconductor 2(ofradiusal andwithcurrent-Izandchargeperunitlength-q).Thissecond conductor (image)hasitscenteratadistance (17) fromthecenterofconductor 1,asshowninFig.8.2b.Thisdistance biin(17)isobtained from(5a)andSec.7,Eq.(23),withappropriate specialization andchangesinnotation simplybyrequiring thepotentials onconductor 1tobethesameinthetwocases,sothat(5a)maybe substituted inSec.7,Eq.(23),witha2=al.Iftheradiusalissuf ficiently smallsothat (18) (17)reducesto (19) Thisisthefundamental relationbetween thedistanceDofalinesource fromtheaxisofametalcylinder ofradiusa2andthedistance bi+D fromthesameaxistotheimageofthelinesource. 9.TheShielded-pair Line.Consider fourinfinitely longparallelcon ductorseachofradiusaarranged sidebyside,asshowninFig.9.1. ,...----- .../' ........., , , 1C----------b j-------i/..--toI ...-:'~------bj ------~ I I....-D·...-D-... \ I I 'I I , 'I0-------+----\---041 11O--~----+-------0 3 : \ 1,' 2II 4 I \ /,/: \ 42 , "'"'''!...... .,."".._---- FIG.9.1.Four-conductor linethatmaintains aconstant potential onthecircleof radiusa2. Fromlefttorighttheconductors arenumbered 3,1,2,4.Thecurrents andchargesinthefourconductors arerelatedasfollows: 12z=13z=-liz=-14z (la) q2=q3=-ql=-q4 (lb) Thedistance between conductors 1and2isb=2D;thatbetween 1and 3andthatbetween 2and4arebi,where,fromSec.8,Eq.(17), (2) Withthischoiceofdistances itfollowsfromSec.8thatthecurrents and Sec.9] THEINFINITELY LONGLINE 35 chargesinconductors 1and3makethecircleofradiusa2anequipotential surfacewithp=P2=constant, asgivenbySec.8,Eq.(5b).Bysym metrythecurrents andchargesinconductors 2and4(whichareoppo sitelydirected fromthosein1and3,respectively) alsomakethissame circleanequipotential surfacewithp=-lp21=constant. Itfollowsby superposition thatthecurrents andchargesinallfourconductors make p=P2-P2=0onthecircleofradiusa2,provided thetwopairsofcon ductorsdonotinteractsufficiently toaltersignificantly thedistribution ofcurrentintheconductors. Thisistrueapproximately ifthecondition (3a) Metal cylinder FIG.9.2.Shielded-pair line.issatisfied. Subjecttothiscondition, thecircleofradiusa2inspace maybereplaced byaconducting sheathof radiusa2enclosing conductors 1and2,and theimageconductors 3and4removed with outchanging anything electrically within thiscircle.Conductors 1and2inthe sheaththusformabalanced shielded-pair line,asshowninFig.9.2. Thepotential differences between con ductors 1and2inFigs.9.1and9.2are thesame.Whereas theycannotbereadily obtained whentheradiusalofthecon ductorsisunrestricted, theyareevaluated easilywhenalissufficiently smallsothattheconditions (3a)and a~-D2»ai (3b) aresatisfied. Sincethetwo-wire lineconsisting ofconductors 1and2 isbalanced, itfollows,justasfortheopentwo-wire lineinSec.4,that, inthenotation ofSec.4, 12z(w)=-llz(w) A2z(w)=-Alz(w)Q2(W)=-Ql(W) cP2(W)= -cPl(W)(4a) (4b) wherethepotentials aredetermined onthesurfacesoftheconductors. It followsthatthepotential differences are Wz(W)=A1z(w)-A2z(w)=2A1z(w) Yew)=cPl(W)-cP2(W)=2cPl(W)(5a) (5b) Henceitismerelynecessary todetermine thescalarandvectorpotentials onconductor 1asmaintained bythechargesandcurrents inallfourcon ductorsinFig.9.1.Thispotential isequaltothatmaintained bythe chargesandcurrents inthetwoconductors andinthesheathinFig.9.2. Usingthenotation ofSec.4,thetwo-conductor problem isreadily 36 TRANSMISSION-LINE THEORY [Chap.I extended tofour.Thusfortheinfiniteline whereWz(w)=Iz(w)koYew)=q(w)ko 2~p 2~~ foo(11 11),ko= - - - - - +-dw -00llll111211131114 andlln=v(w-W')2+ai1112=v(w-W')2+4D2 1114=v(w-W')2+(bi+2D)2R13=v(w-w')2+b;(6) (7a) (7b) Theintegration gives ko=2In2D_ 2Inbi+2D al bi(8) Thedistancebimaybeeliminated using(2).Thus,with(3b), _ 2D(a~-D2)_ b(a~-b2/4) ko-2Inal(a~+D2)-2Inal(a~+b2/4) (9) whereb=2Disthedistance between centersoftheshielded pair,each ofradiusaI,anda2istheradiusoftheshield.Itfollows,asinSec.4, thatthelineparameters forthebalanced shielded-pair lineare (lOb)(lOa)2~Ec=-ko2~u g=7C;;le=p,ko 2~ wherekoisasin(9). Notethat,whentheshieldissolargethattheinequality a~»b2/4is satisfied, (8)reducestothevaluesfortheopen-wire line. Sincetheconductors aresufficiently farapartandfarenoughfrom theshieldtosatisfy(3a,b),theinternal impedance perunitlength Zi=ri+Jxioftheshielded two-wire lineisobtained frqm.thesame formula[Sec.4,Eq.(34)]asfortheopentwo-wire line.Tothismustbe addedtheimpedance perunitlengthoftheshield,inwhichequaland opposite currents areinduced onopposite sides.Anapproximate for mula(Ref.4,page44)is Zi=4(1+J)jP,cw(b/2a2) 2 8 ~a2'\/2uc1 -(b/2a2)4 Thetotalinternalimpedance perunitlengthis2zi+z:.Iftheshieldis madeofmaterial different fromthatoftheinnerconductors, UcandP,cin (lOb)differfromthesequantities inSec.4,Eq.(34); Itispossibletodrivetheshielded-pair linesothatthetwoinnercon ductorsareinparallel, withequalandcodirectional currents andequal chargesofthesamesign,andthesheathisthereturnconductor, witha totalcurrentthatisequalinmagnitude tothesumofthecurrents in theinnerconductors, butopposite indirection, andatotalchargeper unitlengthwhichisequalinmagnitude tothesumofthechargesper Sec.9] THEINFINITELY LONGLINE 37 unitlengthinthetwoinnerconductors, butofopposite sign.Forthis methodofdriving,thelinehasproperties similartothoseofashielded linewitheccentric innerconductor. Iftheradiusaloftheidentical innerconductors issufficiently smalltosatisfy(3a,b),thereisnosignifi cantproximity effect.Foreachconductor thesheathisanequipotential surface,asanalyzed inSec.8.Itspotential withonlyoneoftheinner conductors presentisproportional toP2,where,fromSec.8,Eq.(5b), 2-h-Ia~-ai+D2 P2-cos 2a2D (11) (12b)(12a)Sincethesecondinnerconductor maintains thesamepotential onthis circle,thetotalpotential ofthesheathisproportional to4p2.Fora currentIz(w)/2andachargeperunitlengthq(U;)/2ineachinnercon ductorandacurrent-Iz(w)andachargeperunitlength-q(w)inthe sheath,thepotentials ofthesheathare A( )=Iz(w)/24+C_-Iz(w)h-Ia~-ai+D2+C 2zW2 P2 A2cos 2D .A1rP 1rP a2 "'"( ) =q(w)/24+C=q(w)h-Iai-ai+D2+C ,,*,,2W 21r~P2 '" 21r~cos 2a2D '" Thepotential oneachinnerconductor maybeobtained from A( ) -Iz(w)/2k+ClZW-~O A C>l(W)=q(w)/2ko+C'" 41r~(13a) (13b) (16)(15)(14a) (14b)wherekoisdetermined forthetwoconductors withtheirimagesshownin Fig.9.1,butwithcurrents andchargesthatsatisfytheconditions Ilz(w)=12z(w)=-Iaz(w)=-I4z(w) ql(W)=q2(W)=-qa(W)=-q4(W) Thuskoislike(7a)butwithdifferent signs.Specifically fex>(I I I I) Iko= - + - - - - - dw -ex>IlilIlI2IlIaIlI4 wheretheIl'sareasdefinedin(7b).Theintegration gives k=2 Ibi(bi+2D) o n 2Dal Itisassumed thattheconductors aresufficiently faraparttosatisfythe conditions wherebiisgivenbyai«4D2 (17) (18) 38 With(17)and(18) sothatTRANSMISSION-LINE THEORY [Chap.I (19) (20a) (20b) Iz(w)(a4-D4 a2+D2)Wz(w)=A1z(w)-A2z(w)=47rvIn~D3al-2cosh-122a2D =Iz(w)Inat-D4 (21) 41rv 2Dala~ _ _q(w) a~-D4Similarly Yew)-cPl(W)-cP2(W)--4~In2D2 (22)1r.. ala2 Fromtheirdefinitions, le=Wz(w)/Iz(w) andy=g+jwc=q(w)/V(w), itfollowsthattheparameters fortheshielded-pair lineusedwithitsinner conductors inparallelare jJ.ko 87re 81r(1 le=81r C=k;g=k;; (23a) where k=2Ina~-b4/16 (23b) o bala~ andb=2Disthedistance between theinnerconductors. Although theseformulas arerestricted by(17),sothatthedistance 2Dbetween theinnerconductors mustbelargecompared withtheirradiusal,the limiting caseinwhichthetwoinnerconductors coincide isreadily obtained bysettingD=b/2=0inthenumerator ofthelogarithm [sincethiscomesfrom(12a,b)]andsettingb=2D=alinthedenomi nator[sincethiscomesfrom(15),inwhichcoincidence isspecified by Rll=R12].Theresultistheformulaforthecoaxialline. Theinternal impedance perunitlengthoflineisobtained approxi matelybytreating theouterconductor astheshieldinthecoaxialline andeachinnerconductor asifrotationally symmetrical. Thus where(24a) (24b) Iftheshieldisofrectangular crosssection,asshowninFig.9.3,the following lineconstants applywhentheinnerconductors arebalanced, i.e.,haveequalandopposite currents andcharges :43 21r(1g=ko21rEc=-ko(25) Sec.10] whereTHEINFINITELY LONGLINE 39 [001+sinh2(rbj2h) ] ko=2In2htanh(rbj2h)_'"'In co~h2(mrwj2h) raL.t1 _smh2(rbj2h) m=l sinh2(mrwj2h)(26) 0241 0h I--b~Iw ofTheseformulas aregoodapproximations provided theradiusaofthe innerconductors issmallcompared withthedistance bbetween them andsmallcompared withthedistance fromthewiretoanysideofthe surrounding surface. Usuallyitisnotnecessary togobeyondm=I inthesumin(26)toobtainanadequate approximation. Asanumerical example witha=0.0625in.,h=0.4 in.,b=0.5in.,andw=0.9in.,(26) converges rapidlytogiveko=2.564. Theinternal impedance zi=ri+jxi perunitlengthistheinternal imped anceofthetwo-wire line,asgivenin Sec.4,Eq.(34),plusasmallcontribu-FIG.9.3.Two-wire lineinshieldoftionz;fromlossesintheshield.Since rectangular crosssection. thiscarriesonlysmallequalandoppo- sitecurrents onthetwosides--the totalaxialcurrentiszero-the valueof z;issmall.Intheabsenceofanaccurate formula, andsincetheshield isassumed farfromthewiresascompared withtheirradius,areason ableestimate isobtainedifthevalueofz~in(lOb)foracircularshield isused,ifitscircumference 2ra2ismadeequaltotheperimeter 2whof therectangle. Thatis,(lOb)isusedwitha2=whjr. Ifthelinewithrectangular shieldisdrivenwiththetwoinnercon ductorsinparallel,sothattheentirereturncurrentisintheshield,the lineconstants definedin(25)applywith [ ~oo 1+cosh2(rbj2h)] ko=2In2hcoth(rbj2h)+(_l)mInsinh2(mrwj2h) (27) ra 1 _cosh2(rbj2h) m-l cosh2(mrwj2h) Fora=0.0625in.,h=0.4in.,b=0.5in.,andw=0.9in.,(27)gives ko=0.677.Athighfrequencies theinternalimpedance perunitlength isgivenby(24a),withz1asin(24b)and (28) 10.Three-wire Polyphase Line;Three-phase Cable.I•59Athree wirepolyphase transmission line(Fig.10.1)consistsofthreeidentical paralleiwireseachofradiusallocatedattheverticesofanequilateral 40 TRANSMISSION-LINE THEORY [Chap.I triangleofsideb.Itisassumed thattheconditions (1) aresatisfied. oThelineisdrivensothatthecurrents inallthreewiresareequalinmagnitude andhaveaprogressive phasechangeof1200 fromonewiretothenext.Specifically 13z=pI2z=P2[lz (2) oda1 ,. b· .1 FIG.10'.1.Three-wire linewith conductors atthevertices ofan equilateral triangle.where p2+P+1=0(3) Thevectorpotential atapointQl(X,y,Z) onthesurfaceofconductor 1isthesuper positionofcontributions maintained byall threecurrents. Thus (4) (5) With(2)and(3)theexpression forthevectorpotential maybesimplified. Thus (7)(6) (8) whereA1z(w)=-41f""11z(W')P 1,(w,w')dW' 'Try-"" cPl(W)=4~~1_""""Ql(W')P L(w,w')dw' e-i'JRa e-i'JRb PL(w,W')=-----RaRb Thecorresponding expressions forthepotentials onconductors 2and3areSimilarly A2z(w)=pA1z(w) A3z(w)=p2A1z(w)cP2(W)=PcPl(W) cP3(W)=P2cPl(W)(9) (10) Thepotential differences between conductors 1and2are W12z(w)=A1z(w)-A2z(w)=14-Pf""11z(w')P L(w,w')dw'(lla) 'Try -"" V12(w)=cPl(W)-cP2(W)=14~~P1_""""Ql(W')P L(w,w')dw'(lIb) Thepotential differences between theouterpairsofconductors differ onlyinsubstituting for1 -PthefactorsP(l-P)orp2-1ifreferred tothecurrentIIinconductor 1.Theyarelike(lla,b)ifthesubscripts arecyclically permuted sothatW23(W)isreferredto12andW31(W)to13. Acomparison of(lla,b)withSec.4,Eqs.(lOa,b), showsthatthe Sec.10] THEINFINITELY LONGLINE 41 integrals arethesame.Itfollowsthattheirapproximate evaluation mustbethesame,subjecttotheconditions imposed in(1).Thus W12(w)=j(1-P)I12(w)le )1() ( )jw1(1)dl1z(w)V12(W="21 -Pqw-="2 -P-d-y yw(12) (13) (l4b)(14a) -8V~:(Z)=j(1-P)zllz(z) _811z(z)=_2_yV12(z) 8z1 -P where112isthecurrent inconductor 1 andV12(z)isthepotential difference be tweenconductors 1and2.Theequa tionsfortheothertwophasesareobtained from(14a,b)bycyclicalpermutation of thesubscripts. Thefactorj(1-P)isI unchanged. Thustheproblem ofthe three-phase lineisreducedtothatofthree b· two-wire lineswithcurrents relatedac-/' cordingto(2)and(3). Theanalysis isreadilyextended totheFIG.10.2.Shielded three-wire line withimageconductors equivalentn-phase n-wiretransmission lineandthetotheshield. single-phase multiwire transmission line.69 TheThree-phase Cable.Ifthethree-phase lineinFig.10.1isplaced symmetrically inacylindrical metalshieldofradiusa2,theconstants foreachphasemaybeobtained bythemethod usedinSec.9.This consists inimagining theconducting shieldremoved andthreeimage conductors soarr:wged thattheresultant pot.ential fromthesixcon ductorsvanishes onacircleofradiusa2corresponding tothecircumfer enceofthemetalshield. Thecurrents andchargesinthesixconductors showninFig.10.2arerelatedasfollows:whereleandyareasinSec.4,Eq.(30a,b). Subjecttob2»af,the internal impedance perunitlength z~ofeachconductor isthesameas inSec.4,Eq.(34).Since(12)and(13)differfromSec.4,Eqs.(28) and(29),onlyintheconstant factorj(1-p),thefinaldifferential equa tionscandifferfromSec.4,Eqs.(37a,b),onlyinthisfactor. Thusthe differential equations forthevoltageandcurrentinonepairofathree conductor three-phase lineare where13=pl2=p211 P==ei21r/3p2+P+1=0 16=-1316=-1214=-11(15a) (15b) (15c) Thedistance between eachconductor anditsimageisbi•Itisgiven 42 TRANSMISSION-LINE THEORY [Chap.I bytheequivalent ofSec.9,Eq.(2),inFig.10.2: a22-al2-D2 bi=D Subjecttotheinequalities a~-D2»afb2»af whichareimpliedinthesolution asexplained inSec.9, a2-D2bi==2D(16) (17a) (17b) (20)whereThedistancedfromtheimageofoneconductor tooneoftheothertwo conductors is d=V(D+b,)'+D'+D(D+bi)=~(~)'+D'+al(18) Thelaststepfollowswith(17b). Thepotential functions atWonthesurfaceofconductor 1withinthe shieldareequaltothepotentials calculated fromthecurrentsandcharges inthethreeactualconductors andinthethreeimageconductors without theshield. Thus,with(15c), A1z(w)=~v1-....\Ib(W') (e~RG_e;:R) +[I2z(w')+Iaz(w')](T-~)Idw'(19) Ra=Y(w-w')2+a~Rb=Y(w-w')2+b2 Ri=y(w-W')2+bjRd=y(w-W')2+d2 In(20)alistheradiusofeachconductor; bistheaxialdistance between pairsofconductors; biisthedistance between eachconductor andits image,asgivenin(17b);anddisthedistance fromtheimageofone conductor tooneoftheotherconductors. With(15a,b,c) thevectorpotential (19)maybeexpressed asfollows: A1z(w)=~vl_fIOfIOIz(w')Pd(w,w') dw' (21a) Similarly thescalarpotential atwonconductor 1is (22)(21b) where1f" cl»l(W)=41r~_..q(w')Pd(w,w') dw' ,(e-ifJRG e-ifJRI»(e-ifJRie-ifJRd)Pd(w,w)=-----------RaRb RiRd Thecorresponding expressions forthepotentials onconductors 2and3are A2z(w)=pAlz(w) cl»2(W)=Pcl»l(W) (23a) Aaz(w)=p2A1z(w) cl»a(w)=p2cl»1(W) (23b) Sec.11] THEINFINITELY LONGLINE 43 Thepotential differences between conductors 1and2atthecoordi nateware W12z(w)=Alz(w)-A2z(w)=14-Pf00Iz(w')Pd(w,w') dw'(24a) 1I"V -00 V12(w)=~l(W)-~2(W)=14-Pf00q(w')Pd(w,w') dw'(24b) 1I"V -00 Theseintegrals arelikethosein(lla,b). Theydifferonlyintheoccur renceofPd(w,w') inplaceofPL(w,w'). Hencethedifferential equations thataresatisfied byV12(z)andIl(z)mustbethesameas(14a,b)butwith different valuesofthelineconstants. Byimposing thecondition (25) andcarrying outtheanalysis asinSec.4,thefollowing resultsare obtained: ko=fooPd(w,w') dw'==foo[(~-~)-(~-~)]dw' -00 -00RaRb R~Rd bd=2Inalb i(26) wheredandbiareasin(17b)and(18).Thelineconstants are J.Lbd 1I"(T 1I"E le=:;Inalbig=In(bd/albl)C=In(bd/albl)(27) Theinternalimpedance perunitlengthofeachconductor zi=ri+jxiis thesameasinSec.4,Eq.(34),subjectto(17a). 11.TheCoaxialCageTransmission Line.Aconventional coaxial lineisshowninFig.11.1a.Ifitsoutercylinder isreplaced by2Ntcon ductorseachofradiusalsymmetrically arranged inacircleofradiusb //V ......, .,=~~~'......0/ (a) (b) FIG.11.1.(a)Coaxial line; Co=21rE/[ln(b/a)]and l~=[In(b/a)J/211"P. (b)Cage linewiththesamevaluesofCoand l~. aroundthecentralconductor, asinFig.11.1b,andthese2Nconductors areoperated inparallel, theproperties ofthecoaxiallineinFig.11.1a maybecloselyapproximated. Inordertodemonstrate this,letitbe tAnoddnumber2N+1conductors mayalsobeusedtoformthecage.For simplicity onlytheevennumbers areconsidered here. 44 TRANSMISSION-LINE THEORY [Chap.I assumed asusualthatthefollowing inequalities aresatisfied: b2»a2b2»ai (1) Theparameters ofthecagetransmission linemaybequicklydeter mined. Thusthescalarpotential atthecoordinate wonthesurfaceof thecentralconductor (number 0),whenthishasapositive chargeq(w') perunitlengthatw',whereas eachofthe2Nouterconductors (num beredfrom1to2N)hasacharge-q(w')/2N, isgivenby cPo(w)=_1_f"q(w')(e-iIJRG _e-iIJRb )dw' 411"~-.. RaRb ==q(w)f"(l--1-)dw' (2) 411"~-..RaRb where Ra=yew'-W)2+a2Rb=yew'-W)2+b2(3) Itisassumed thatthefollowing inequality issatisfied l~bl2«1 (4) Thepotential onthesurfaceofeachofthe2Nouterconductors atthe sameaxialcoordinate wisthesameasthepotential cPl(W)onconductor 1. Thisis N q(w)f"[1 ( 1 1L1)1], cPl(W)==- - - -+--+2 - - - dw 411"~_..2NRa1R1,N+I R1iRb i=2(5) where RI<=~(w'-w)'+ [2bsin"(;2;1)r2;;;;;;;N(6a) R1,N+I==yew'-W)2+4b2Ra1=yew'-wF+ai(6b) Thepotential difference is Yew)=cPo(w)-cPl(W)==q4(~)f"[R1+21N(R1+-R111"..-.. a al 1,N+l N +2~l-)-~]dw'(7)LtRliRb i=2 Thisexpression maybeintegrated termbytermandrearranged togive Yew)==q~W;~ko (8) where b1Ib ko=In-+-In- -(2N-1)In2a2Nal 1[.11"•211".371" - 2nsm2Nsm2Nsm2N. b1b=In-+-In--a2N2Nal.(N-1)11"]}. .sm2N (9) THEINFINITELY LONGLINE Sec.12] Itfollowsthat 21rE Co=kole=~=~oCo21rv45 (10) Sincetheformulas (10)applytothecoaxiallineofFig.11.1awith ko=In(b/a),itfollowsthatthecagelinewillhavethesamevaluesof Co,go,and19asthecoaxialline,provided theradiusoftheconductors ofthecagehasavaluesuchthat (11) Theinternal impedance perunitlengthofthecageisapproximately where(12a) (12b) whereasthatforthecoaxiallineis Zi=z~+zi where z~isasin(12b)and(13a) (13b) Sincetheprincipal ohmiclossisintheinnerconductor, whichisthesame inthecagelineasinthecoaxialline,Ziin(12a)usuallydoesnotdiffer sufficiently fromZiin(13a)tomakeitnecessary toadjust (TelSOthat zi=zt.Inthecaseoflow-loss linesitisadequate toimpose(11)in ordertomaketheproperties ofthecagelineequaltothoseofthecoaxial line. 12.StripLines.Thetwoconductors oftheopen-wire lineanalyzed inSec.7areofcircularbutnotnecessarily equalcrosssection. Inpar ticular,oneoftheconductors maybeofinfiniteradius,i.e.,mayconsist ofahighlyconducting imageplane.Although conductors ofcircular shapeareusuallymostconvenient inpractice, therearespecialappli cationswhereflatstripconductors areuseful. Thesemaytakeseveral forms,thesimplest ofwhichisshownincrosssectioninFig.12.1.A paralleltwo-conductor linemadeofflatstripsofsmallthickness isshown inFig.12.1a;thecorresponding single-conductor lineoveranimageplane isshowninFig.12.1b.Theproperties ofstriplinesofthesesimpletypes donotdiffersignificantly fromthoseoflineswithcircularcrosssection. Theanalyses inpreceding sectionsindicatethatthecapacitance perunit length Comaybedetermined byelectrostatic methods irrespective ofthe natureofthe('·rosssection, provided thewidthhofeachstripandthe 46 TRANSMISSION-LINE THEORY [Chap.I distance 2hbetween thetwostripsarebothsmallcompared withthe wavelength (orothermeansareprovided toensuretheexistence ofexclu sivelyaxialcurrents, which,forlosslessconductors, areintheTEMmode). Thecapacitance perunitlengthofaverythinstrip(conductivity O'e, permeability J.1.e,andthickness d)inahomogeneous infinitemedium ~b-toI Stripconductord; %ok ~&)//;;;;;;;;;;;} 922 (a)z.Conducting plane (b) A· Stripconductor O'It'Ir .rIeeerae ; 1,":'({f·::.·.·,....·.·.:.·.·.·.·.·.·.·.}h)))>>):;,)7/7)/;';) ) )/)7) ) ; ; 7>;,>>;,') Conducting plane (c) >~~;:~7u~T§:>:~:::!::~i~,~ti~!::·:! ::::::'.:::::::.'.:::'.'·lh;·.::···:··':·:'::··. ::1;;;;;);;;;; I;)»;;»)) ; )7>;;);// Conducting plane Cd) FIG.12.1.Crosssections ofstriplines:(a)two-conductor stripline;(b)striplineover conducting plane;(c)stripconductor ondielectric-coated conducting plane(micro strip);(d)stripconductor indielectric overconducting plane(sandwich line). (dielectric constant e,permeability J.1.,andsmallconductivity 0')overa highlyconducting infiniteplanesurface(conductivity O'eandpermea bility J.l.e)hasbeendetermined36byconformal transformation subject totheinequalities d«h«b.Itis c=~F~X) (1) where F(x)=1+x+In(1+x)7rbx=-2h(2) Itfollowsdirectlythat O'bF(x) g=hx (3) and le=~=J.l.h~ (4)cbF(x) Asusual,Ze=VF7C,using(1)and(4).Theapproximate internal impedance perunitlength Zi=Z~triP+Z~laneisobtained from i_!r;;;;:x[1+x+7r- 2In(0/2)] Z.trip -b'\}~ F(x) . 1 IWJ.l.ex(1+x) z~\ane=b'\}20'eF(x)(5a) (fib) Sec.12] THEINFINITELY LONGLINE 47 where (J'candP-capplytotheconductor and o=k2-1+kv!k2-1 (6) Theaboveformulas aregoodapproximations foraplaneoffinitewidth, provided itextendsadistance equaltothewidthbofthestriponeach side. Animportant practical application ofthestriplineistoprovideamore compact andmorereadilymanufactured substitute forwaveguidesand coaxiallines.Oneformofstriplineknownasmicrostrip83 ismadeof thinsheetsofalow-loss plasticdielectric material, withcontinuous films ofcopperlaminated tobothsides.Appropriate portions ofthefilmare removed ononesidetoleaveadesignhavingtheshapeofthedesired strip-line circuit. Thestriplineobtained inthismanner differsfrom thesimplelinealreadydescribed inhavingalayerofdielectric ofthick nesshcovering theentireconducting plane,asshowninFig.12.1c.Thus thestripconductor isnotcompletely immersed inasingleinfinitedielec tricasassumed inderiving (1).Actually thepresence ofthetwodielec trics,plasticforathickness habovetheconducting planeandairbeyond this,introduces complications thatresultfromthefactthatitisnolonger possibletomaintain currents onlyintheTEMmode.Anelectricfield between thestriplineandtheconducting planemayexcitemodesinthe thinlayerofdielectric whichpropagate outward inamannerquitedif ferentfromthatcharacteristic oftheTEMmodeandwithmagnitudes thatdecrease muchlessrapidlywithdistance.37Itfollowsthattwostrip linesonthesamedielectric-coated metalsurfacemaybecloselycoupled eventhoughsofarapartthattheirinteraction wouldbenegligibleifthere wereonlyasinglehomogeneous dielectric. Notethatthisinteraction is duenottoradiation intheTEMmodebuttoso-called guidedmodes. In anelementary sensethepropagation inthethinlayerofdielectric isa consequence oftotalinternalreflection attheair-dielectric boundary. An analysis oftheelectromagnetic fieldandtheconstants ofthelinewhena guidedmodeexistsisbeyondthescopeofthisbook.However, formany purposes satisfactory approximations areobtained byassuming thatonly aTEMmodeexists. Inordertoreducetheguidedmodes,thethickness ofthedielectric layermaybeincreased sothatthestripconductor iscompletely immersed init,asshowninFig.12.1d.Suchalineisknownasasandwich line. Ifthethickness tofdielectric isquitelargecompared withtheheighth ofthestriplineabovethemetalplate,conditions approximating thosein aninfinitedielectric areapproached, andtheformulas givenearlierin thissectionaregoodapproximations. Alternatively thedielectric may itselfconsistofastripnotmuchwiderthanthemetalstripconductor. Sincelossesinthedielectric mustbekeptaslowaspossible, itis 48 TRANSMISSION-LINE THEORY [Chap.I Conducting plane FIG.12.2.Shielded-pair stripline.advantageous tohaveairratherthanasolidmaterial between themetal stripandtheconducting plane. 44Asymmetrical arrangement resem blingaflattened shielded-pair lineoperated withthetwoinnerconductors inparallelisshowninFig.12.2.Inthisconstruction thetwoinnercon ductorsareseparated byadielectric thatservesasthesupport, asshown. Sincethetwoinnerconductors are atthesamepotential, thereisnofield inthedielectric, sothatnoguided modeisexcitedinit.Thusitis essentiallyonlytheTEMmode whichismaintained, andthereisno soliddielectric between theinner conductors andtheouterones,where thefieldisgreat.Moreover, iftheconducting shieldextendssufficiently faroutbeyondtheedgesofthestriplines,lossesbyradiation arereduced belowthosefortheunshielded striplineshowninFig.12.1c. 13.GeneralSolutionojtheDifferential Equations joranInfiniteLine. Thefirst-order differential equations thataresatisfied bythescalar potential difference andthecurrentinalltheseveraltypesoflineana lyzedinthepreceding sections havetheform where_aV=zl_alz=yvaz zaz y=g+jwc Z=Zi+jwle=ri+jwl(1) (2) Itiscustomary todefinethetotalinductance perunitlengthby l=le+lili=~ W(3) SinceXiisnotalinearfunction ofthefrequency, itfollowsthatliisnot independent offrequency. Theformulas forle,g,c,andZidifferfordif ferentcrosssections. Theequations forthepolyphase linesderivedin Sec.10arethesameas(1),butwiththeindividual currents multiplied bythefactor(1-P)/2,wherep=ei27r/3•ItfollowsthatlIzinconductor 1 isobtained fromthesolution of(1)forIbysetting 1-PIz=lIz-2- (4) Bydifferentiation withrespecttozandappropriate substitution, the first-order equations (1)maybetransformed intosecond-order equa tions.Sincetheequations inIzandVarealikeinform,itissufficient toexamine oneofthem. Thedifferential equation forthevoltageis (5) Sec.131 THEINFINITELY LONGLINE 49 where,asdefinedinSec.2,Eq.(14),thecomplex propagation constant is y==VZiJ=a+jj3 (6) Thegeneralsolution ofthewell-known Eq.(5)maybeexpressed indif ferentways,suchas VI:=BleTz+B2e-'Y1II=Clcoshyz+C2sinhyz=Dcosh(yz+8)(7) asmaybeverifiedbydirectsubstitution. Alternatively w=8 -Zmay besubstituted forzintheseveralformsof(7).TheB's,C's,D,and8 arecomplex constants ofintegration. Theexpression forthecurrentis mosteasilyobtained from zl= _(dV) III dz III Thus,fortheexponential formof(7), zIz=-y(Ble'YlII-B2e-'YIII)(8) (9) Itisconvenient tointroduce Zc,calledthecharacteristic impedance, for theratioz/y.Thus Z=R+'X=Ir+jwl c - c Jc -'\Ig+jwc Yc==~e==Ge+jRe Thecurrentisthengivenby II:=Yc(-Ble'YlII+B2e-rlll )(lOa) (lOb) (11) Similarexpressions fortheotherformsof(7)arereadilyderivedusing(8). Therelations (7)and(11)aregeneralsolutions forthecomplex currents andpotential differences. Although strictlycorrectonlyforaninfinitely longline,theyaregoodapproximations forfinitesections oflinewhich satisfytheconditions (8-Z)2»b2 (12) asisshownlater. Alternative exponential formsof(7)and(11)whichareconvenient in theanalysis ofjunctions (Chap.V)areobtained byredefining thearbi traryconstants. Theyare VI:=VZc(Ae-'Yz+Be'Yz) (13) 1111=vY:(Ae-'Yz-Be'Yz) (14) whereA=BlYYcandB=B2vY;:.Thecoefficients BlandB2in(7) and(11)aredimensionally voltages; thesquares ofthecoefficients in (13)and(14)aredimensionally powers. 50 TRANSMISSION-LINE THEORY [Chap.I 14.Interpretation ojtheSolutionjortheVoltagealonganInfinite Line.PhaseandGroupVelocities.17,28Beforeproceeding toevaluate B1andB2(Sec.13)intermsofgeneralterminal conditions, itisinstruc tivetoapplythesolutions obtained toaninfiniteline.Consider asec tionoflinebeginning atz=0andendingatz=8=00.Forphysical reasonsthevoltagemustvanishatinfinity, sothatB1=O.Itfollows directlyfromSec.13,Eq.(7),thatB2isthevoltage Voatz=O.Thus 8=00 (1) Uponmultiplying throughbyeiwtandselecting therealpartasthesolu tionthatisconsistent withanassumed timedependence oftheform Vo=Vocoswt=Re(Voeiwt) (2) whichrefersthephasetothemaximum valueoftheinstantaneous volt age,oneobtains Vz=Voe-azcos(wt-(3z) (3) Thissolution hasaninstructive physical interpretation. Notethat thevoltage Vzisafunction oftwoindependent variables, thetimetand thedistance zalongthewire.Atanyfixedpointz=Zlthevoltage variesperiodically. Thepotential ispositive ononewireandnegative ontheotherforonehalfperiod. Theamplitude increases fromzerotoa maximum ofVoe-az1anddecreases tozeroinasinusoidal fashion. Then thepolarity reverses, andthevoltagedecreases toanequalnegative extreme, thenagainisreduced tozero.Thephaselagofthevoltage atzbehindthevoltageatz=0is{3z.Thecyclerepeats. Thesame variation occursateveryotherpointz,buttheamplitude Voe-azisdif ferent,andthephaselagsthatatz=0by{3z.Theamplitude decreases exponentially, andthephaselagincreases linearly withdistance from z=O. If,insteadofconcentrating onafixedpointalongtheline,theampli tudeallalongthelineisexamined atagiveninstant,suchast=0,then Vz=Voe-azcos{3z Ataquarterperiodlatert=T/4,and Vz=Ve-azsin{3z Atahalfperiodlatert=T/2,and Vz= -Voe-azcos{3z(4a) (4b) (4c) Thethreedistributions areshowninFig.14.1.Itappearsthat,astime passes,anygivencurve,suchastheonefort=0,movesdowntheline withamplitude confined betweenthelimiting curves Voe-azand-Voe-az• Inordertoinvestigate thismotion,letattention befocusedspecifically onthephaseofthevoltage. Thisisgivenbytheargument ofthe Sec.14] THEINFINITELY LONGLINE 51 trigonometric function in(3),thatis,bywt-{3z.Pointsandtimesin thedistribution ofvoltagealongthesemi-infinite lineatwhichthevolt agesareallinthesamephaserelativetoacomplete cyclearedefinedby 1/1=wt-{3z=constant. Because thetrigonometric function ismulti valued,thecurrentatallpointsforwhichtheconstant differsby2n,,- t=I.4 Tt=- "---"~ 2'"-ttZ ,"~.-.--L voe.."--r-\./. \/\/ or-:o---¥.---+---f.--:--+---r--\---*--~:------:*--~--+-OQ /\2Tf :\ \ "/\:\i\L."f~--t,'........-LVoe-ucospz -Vo -Voe-ttZcospz FIG.14.1.Instantaneous distribution ofvoltagealongasemi-infinite lineatinstants differing byaquarterperiod. (wherenisanyinteger) isinthesamephaseasatz=0.Thecurrents atdifferent timesanddifferent pointsalongthelinewhichdifferinphase byintegral multiples of2,,-aredefinedby wt-{3z=1/In=1/10-2n,,-n=0,1,2,3, (5) (6) n=0,1,2,3,.where 1/10isaconstant. Thesignificance ofthisrelation maybedis closed,first,bydetermining thedistances zfromtheinputendatwhich thevoltages differinstantaneously inphasebyintegral multiples of2,, and,secondly, bydiscovering whathappens totheseparticular phases astimepasses. Ifanarbitrary instant t1isselected, thepointscharac terizedbyvoltages inthephases 1/10-2n,,-aregivenby 1 Zn=~(wt1-1/10+2n,,-) Thedistance between twopointsthatareadjacent anddifferinphase by2,,-is 2,, Zm+l-Zm=7imisanyinteger (7) Thisdistance isthesameforallchoicesofm.Itisafundamental con stantofthedistribution calledthewavelength ontheline.Itisassigned 52 TRANSMISSION-LINE THEORY [Chap.I thesymbol A.Thus,bydefinition, A==211'" f3(8) Atanygiveninstantoftime,voltages alongthesemi-infinite linewhich differinphaseby211'"areseparated bydistances A. Withthepointsinthedistribution whicharecharacterized byvoltages inaparticular phaseatagiveninstantdetermined, itremainstodiscover howthedistancezlocatinganyone suchpointvariesintime.Thisis determined bydifferentiating bothsidesof(5)withrespecttotime.In thisway or,defining Vp,dzw-(3-=Odt(9) (10)dzw Vp==dt=~ In(10)Vpisthevelocity withwhichagivenphasetravelsalongtheline. Ingeneral,ithasnothingtodowiththepropagation ofenergy,butonly withthearrangement ofphases;itappliesonlytoperiodic phenomena of infiniteduration. Thuseachparticular phaseofthevoltagetravelsalong theinfinitelineinthepositive zdirection withaconstant velocityw/(3. Thisphenomenon, inwhichpointsofconstant phaseareseparated by constant distances Aandalltravelwithaconstant velocity Vp,iscalled traveling orrunning wavesofconstant phase.Depending onwhether attention isdirected toaconstant phaseofvoltageorofcurrent, the traveling wavesarecalledvoltagewavesorcurrentwaves.Anyparticular phasereaching adistancezataselected instantmusthavestartedat z=0atanearliertimegivenbyt-z/vporbyt-f3z/w.Consequently avoltageinthisparticular phasealwayslagsthevoltageatz=0atany timetbyaphaseangle(3z.Similarly, ifthedistribution ofvoltageis viewedalongtheentirelineatanysingleinstant, asinFig.14.1,the phaselagatanydistancezfrom0(withrespecttothevoltageat.z=0 atthatinstant) isf3z.Thusf3measures thephaseanglecharacteristic of agivensemi-infinite lineperunitofitslength.Itisthephaseconstant (perunitlength)ofthe(infinite) line.Itismeasured inradiansper meterifzisinmeters. Theamplitude ofvoltageinaparticular phaseisreducedaccording to e-a••Thusameasures thenaturallogarithm oftheratioofamplitudes IVo/V.lperunitlength: 1IVol a=-log-zV. Itistheattenuation constant (perunitlength).ofthe(infinite) line. therelation (11)itismeasured inneperspermeterifzisinmeters.(11) In Sec.14] THEINFINITELY LONGLINE 53 Ifthephaseconstant {3isalinearfunction ofthefrequency, sothat {3=~ (12)v wherevisaconstant independent offrequency, thenthephasevelocity Vpisthesameforallfrequencies andequaltotheconstant vintroduced in(12).Underallotherconditions thephasevelocity isdifferent for eachfrequency, sothat,foranycomplex voltagethatisasuperposition ofcomponents ofseveralfrequencies, thesecomponents havedifferent phasevelocities. Inthiscasedispersion issaidtooccur. Thesignificance ofdispersion maybedetermined byinvestigating the propagation alongasemi-infinite transmission lineofavoltagethatis modulated inamplitude atanangularfrequency Owwhichissmallcom paredwithw.Inthiscasetheinputvoltageatz=0maybewrittenas follows: Vo=Vorl+mcos(owt)]coswt (13) wheremisthedegreeofmodulation (usually multiplied by100and expressed inpercent). Usingastandard trigonometric formula, this mayberewritten inthefollowing equivalent form: Vo=Vo[coswt+~cos(w+Dw)t+icos(w-ow)t] (14) Sincethedifferential equation islinear,thevoltageatanypointzalong theinfinitelineisthesuperposition ofthevoltages duetothethree components. Thus v.=VoIe-a.cos(wt-(3z)+ie-(a+8a).cos[(w+ow)t-({3+omz] +ie-(a-8a)zcos[(w-ow)t-({3-omz]l(15) Herea±oaand{3±o{3are,respectively, theattenuation constants and phaseconstants associated withtheangularfrequencies w±ow.Ifowis suffici~ntly small,itmaybeassumed thatthechangesinaand{3foran increase inwbyowarethesameinmagnitude asthechangeswhenwis decreased byow.Sinceaisverysmallalongahighlyconducting line, aswillbeshownlater,oaisasmallquantity ofhigherorderandof negligible importance indetermining thenatureofthepropagation, at leastovermoderate distances. Specifically e±8az==1±oaz.Thelast termisnegligibleifzisnotsogreatthatitisnotpossibletorequire oaz«1.Ifoazisneglected, theresultis Vz=Voe-az(cos(wt-(jz)+icos[(w+ow)t-({j+omz] +icos[(w-ow)t-({j-omz]l(16) 54 TRANSMISSION-LINE THEORY [Chap.I Thismaybetransformed trigonometrically, without furtherapproxi mation,intothefollowing expression: Va=Voe-aa[1+mcos(owt-0{3z)]cos(wt-(3z) (17) Thetransmission properties ofthemodulation-amplitude arecontained inthefunction insquarebrackets. Thusaparticular phaseinthe modulation amplitude isdefinedby owt-0{3z=const. (18) (19a)Differentiation withrespecttotyieldsthevelocityofpropagation (dz/dt) ofaparticular phaseofthemodulation amplitude alongtheinfiniteline. Itisthegroupvelocityandisdefinedby dzow Vg==dt==0{3 Inthelimitasowapproaches zero, .owdw Vg=a~~o0{3=d{3 Analternative formisobtained usingow=o({3vp)=vpo{3+{3oVp: dvp Vg=Vp+{3d{3(19b) (19c) Since(j=21r/A,(3(d/d{3) -X(d/dX), sothat dvp Vg=vp-XdX (19d) Ifthereisnodispersion, wislinearlyrelatedto{3bythesimplerelation {3=w/v,withvaconstant independent offrequency. Inthiscase(19) together with(12)gives Vg=v=Vp (20) Whenthereisnodispersion, amodulation envelope travelsalongthe transmission lineatthesamevelocity asanyparticular phaseofthe carrierfrequency. Ifthereisdispersion, thevelocity ofthemodulation envelope isdifferent fromthatofthecarrier.Ifthephasevelocity decreases withfrequency sothatdVp/d{3isnegative, aparticular phase travelsmoreslowlyatahigherfrequency thanatalowerone,thedis persionisnormal,andthegroupvelocity islessthanthephasevelocity. Ifthephasevelocity increases withfrequency sothatdvp/d{3ispositive, aparticular phasetravelsmorerapidlyathigherthanatlowerfre quencies, thedispersion isanomalous, andthegroupvelocity isgreater thanthephasevelocity. Itiseasilyshownthatthegroupvelocity isalsoapproximately the velocity ofpropagation ofapulsethatcanberepresented intermsofa Sec.14] THEINFINITELY LONGLINE 55 narrowfrequency bandbetween Wo+owandWo-ow,withowverysmall. Ifthisistrue, (21) (26)Ifavoltagepulsecomposed ofanarrowbandoffrequencies (notethat thisdoesnotmeananarrow,sharppulsethatiscomposed ofaverywide bandoffrequencies) isimpressed acrossaninfinitelineatz=0,the instantaneous complex valueoftheresulting voltagepulseontheline canberepresented intermsofacomplex Fourierintegral oftheform Va=f00V(mei(wt-fJz) d(3==(fJo+ofJV«(3)ei(wt-fJz)d(3 (22) -00 ho-. HereV«(3)isanamplitude function ofthefrequency, andhenceof(3, whichhasanyshapeintheinterval (30-0(3to(30+0(3butisvanishingly smalloutsidethisinterval. (Apulseofanyshapecanbeexpressed bya Fourierintegralwithlimitsextending from-00to+00.Asharppulse contains suchawiderangeoffrequencies, eachwithadifferent phase velocity, thattheshapeofthepulsechanges sorapidlythatagroup velocity cannotbedefined.) Because itisrequired thatowbesmall, theangularvelocity wintheintegrand canbeexpanded asafunction of (3inarapidlyconverging Taylorseriesaboutthevalueatflo,andhigher powertermsmaybeneglected: w«(3)=WfJ=fJo+(dw) «(3-(30)+ ... (23)d(3fJ=fJo CJJt-flz==[wo+(~;)o «(3-flo)]t-«(30+fl-flo)z =wot-(3oZ+«(3-(30)[(~;)ot-z] (24) Hence Thecomplex amplitude Vzvarieswithzonlyinthephasefactorinthe exponential. Accordingly- Vzisthesameatallpointsandtimeswhere (~;)ot-z=const. Differentiating withrespecttothetimegivesthevelocity dz (dw)dt=Vg=dfl0(27) ofthepulse.Itisthesameasthevelocity ofamodulation envelope (19). Theconceptofgroupvelocity ispreciseonlyinthelimitasowapproaches zero.Ifowissufficiently small,theshapeofthemodulation envelope 56 TRANSMISSION-LINE THEORY [Chap.I orofapulseremains approximately thesameoveralongdistance, so thatavelocity ofpropagation ismeaningful. Thisisthegroupvelocity. Thevelocity ofasignalwhosetransmission canbedescribed interms ofelectromagnetic wavesis,ingeneral, neitherthephasevelocity nor thegroupvelocity, butathirdvelocity calledthesignalvelocity. This isnoteasytodefineingeneralterms,butitcorresponds physically tothe arrivalofasufficiently largeamplitude toactivate areceiver. Inthe caseofnormalandsmalldispersion thesignalvelocity practically coin cideswiththegroupvelocity, andbotharesmallerthan3 X108m/sec. Whendispersion isnormalbutlarge,bothgroupandsignalvelocities aredifficulttodefineatall;whendispersion isanomalous, complicated conditions mayobtaininwhichthegroupvelocity maydiffergreatly fromthesignalvelocity. Innocasedoesthesignalvelocity exceed 3 X108m/sec;inallpractical casesitisless.Theoretically aninfinitely sensitive receiver shoulddetecttheextremely smallamplitude ofthe so-called firstprecursor ofasignal. Thisalwayshastheso-called wave frontvelocity, 3 X108m/secforallmedia. PROBLEMS 1.Derivethetransmission-line equations usingthegeneralmethod ofSec.2as appliedtoanequivalent IIsection. 2.Determine thelineconstants ofafour-wire lineinwhichadjacent pairs(instead ofdiagonal pairs)ofconductors areinparallel. Thefourconductors areatthe cornersofasquare. 3.Determine thelineconstants ofafour-wire linewithconductors arranged at thecornersofarectangle ofsidesbandc.Thediagonal pairsofconductors arein parallel. 4.Theinnerconductor ofahorizontal coaxialslottedlineissupported alongits entirelengthbyawedgeofpolystyrene (Er=2.6)whichoccupies a9°angle.Ifthe wavelength measured alongthelineis1.2m,whatwoulditbeifthelinewerecom pletelyair-filled? 5.Atroughlineconsists ofasinglewireplacedsymmetrically parallel tothe intersecting lineoftwohighlyconducting planes. Theplanesmeetatanangleof60°; theconductor liesonthebisector ofthisangleataperpendicular distance b/2from eachplane.Determine thelineconstants, indicating whatapproximations aremade. (HINT:Useimages.) 6.Ashielded cableconsists offourcopperconductors atthecornersofasquare inanironshieldofcircularcrosssection. Thedielectric ispolystyrene. (a)Determine thelineconstantste,c,and{3foreachofthepossible phase-sequence voltages, assuming simpleimagetheorytoapply. (b)Whataretheassociated phasevelocities? Obtainanestimate oftheirnumeri calmagnitudes byassuming thecopperconductors tobeNo.10wire,thesquareto haveasideof1em,andtheshieldtohaveaninnerdiameter of3em. 7.Atransmission lineterminated atz=8initscharacteristic impedance of 300ohmsisdrivenatz=0byagenerator withanemfof100voltsandanimpedance of8+j40ohms.Thefrequency is100Me/sec. Theattenuation constant ofthe lineis0.01neper/m.Determine theinstantaneous currentandvoltageatz=10m iftheinstantt=0ischosentooccurwhentheemfhasapositive maximum inits cycle. THEINFINITELY LONGLINE 57 8.Theamplitude ofthecurrentinalonglineterminated initscharacteristic impedance ismeasured attwopoints100mapart.Theratioofthetwovaluesis1.1. (a)Whatistheattenuation constant ofthelineinneperspermeter? (b)Whatistheratioofpotential differences between thetwoconductors ofthe lineattwopoints20mapart? 9.PlotcurvesshowingIlz/lolalonganinfiniteline(oralineterminated inZc)for whicha=10-3neper/moverarangefromz=0toz=2}"andoverasecondrange fromz=IOO}"toz=I02X.Sketchtheinstantaneous currentiz/loatt=0in bothrangeswith{3=3.14radians/m (io=10coswt). 10.Aflexibletwo-wire lineconsists oftwocopperwiresjoinedbyathinribbonof dielectric. Thecharacteristic impedance ofthelineisspecified bythemanufacturer. Thewiresizeandspacing canbedetermined bydirectmeasurement. Howcould thewavelength alongthislinebedetermined bycalculation foraspecified frequency? ,...CHAPTER II THETERMINATED LINE 1.Potential Functions foraTerminated Line.9,10,49Sincethediffer entialequations derivedinChap.Iarevalidstrictlyonlyforaninfinitely longline,itisnotcorrecttoassumethattheymaybeappHedtoalineof finitelengthwitharbitrary impedances astheloadatz=8andinseries withthegenerator atz=O.Inordertoinvestigate thisproblem of termination, lettheinfinitely longlinetotherightoftheline-load plane ·h ~~I------Z------il.+~ ....4w-iV I I dU~Tl /' R1T/·/· /' CI~,.,._-=~~./--- ~fzb1~--~ -.j/4 dw' \ \ \ FIG.1.1.Linewithtermination. atz=8bereplaced byaterminal impedance offinitelength. Since thereisadifferent specificsolution foreachtypeoftermination, itisnot possibletoderivegeneralresultsvalidforallterminations andtypesof lines.However, thegeneralmethod ofanalysis canbeformulated in termsoftheconfiguration ofconductors showninFig.1.1,consisting of asymmetrical coilterminating atwo-wire linewithidentical conductors 58 Sec.1] THETERMINATED LINE 59 ofradiusaandspacedadistancebthatsatisfiestheinequality b2»a2 (1) Thisrestriction mayberemoved asinChap.I,Sec.7. Asafirststepinthederivation ofageneralized setofdifferential equations, letthescalarandvectorpotential differences beevaluated. Letw=s-zbemeasured fromtheline-load planealongthetrans missionlinetotheringQL(W,X,y) onthesurfaceofeachconductor where thepotentials areevaluated. Similarly letw'bethedistance fromthe line-load planetotheelements dw'atQ~(w',x,y) atopposite pointson theaxesoftheconductors. Thecoordinate uismeasured fromtheline loadplanealongeachsideofthesymmetrical load.Thedistance from thisplanew=0,u=0totheelements du'atQ~ontheaxesofthe conductors formingtheloadisu'. Inorderthatthelinemaybebalanced withequalandopposite cur rentsandchargesonthetwoconductors, thatis, (2) (wherethesubscript Lstandsforline),itisnecessary thatthelineand theloadbesymmetrical, sothat (3) SinceC>2(W)andC>l(W)mustbecalculated fromallthechargesand A2z(w)andAlz(w)fromallthezcomponents ofcurrentinboththeline andtheload,thislattermustbesymmetrical initsgeometry andinits chargesandcurrents. Thusitisnecessary that (4) wherethesubscript Tstandsfortermination. Ifthehalvesoftheload aregeometrical imagesofeachotherintheplaney=0(Fig.1.1),but withsignsofchargesanddirections ofcurrents opposite tothoseof mirrorimages,allconditions (2)to(4)aresatisfied. However, there areconfigurations ofconductors inwhichthehalvesarenotgeometrical imagesintheplaney=0whichalsosatisfytheseconditions. With(2)thepotential differences between opposite pointsontheequi potential surfaces ofthetwoconductors are Yew)=C>l(W)-c>~(w)=2C>l(W) Wz(w)=A1z(w)-A2z(w)=2Alz(w)(5a) (5b) Notethatthesearethesumsofthepotential differences calculated from thechargesandcurrents intheline(subscript L)andinthetermination (subscript T): Yew)=VL(W)+Vr(w) Wz(w)=WzT,(w)+Wzr(W) (6) 60 TRANSMISSION-LINE THEORY [Chap.II Equations (5)aretrueifsubscripts LorTareaddedtoeachpotential. Theevaluation ofthepotential differences atwonthelinemaybe carriedoutasinChap.I,Sec.4,butwithfinitelimits.Thus WZL(w)=-21[8IzL(w')PL(w,w') dw' (7a) 7rVJo WzT(w)=-21[8'1'IzT(u')PT(w,u') du' (7b) 7rVJo VL(w)=2~~/,8qL(W')PL(w,w') dw' (8a) VT(w)=2~~.!o8TqT(U')PT(w,u') du' (8b) e-i'JRa e-i'JRb where PL(w,w')=----- (9a)RaRb e-i'JR1T e-i'JR2'l' PT(w,u')=R1T-R2T(9b) andwhere Ra=V(w-W')2+a2Rb=V(w-w')2+b2(ge) Thedistances R1TandR2Taremeasured fromthesymmetrically placed elements ofintegration du'inthetermination tothepointQL(W,X,Y) on oneoftheconductors ofthelinewherethepotentials arecalculated, as showninFig.1.1.Thehalfdistance aroundthecontour ofthetermi nationisST.Itisassumed thatthelength Softhelineissufficiently greatsothatthedirectcoupling between thegenerator andtheloadis negligible. Itfollowsthatitissufficient todetermine thepotential dif ferencesfarfromthegenerator endoftheline.Byinterchanging zandw theresultssoobtained applytothepartofthelinefarfromtheloadend. Inordertoevaluate thepotential differences in(7)and(8),thecharges andcurrentsatw'onthelineandatu'inthetermination areexpanded inTaylorseries,asinChap.I,Sec.4.Thedistributions ofcurrentand chargearecontinuous attheline-load junctions, sothat qL(w'~0)=qT(u'~0) (lOa) IZL(w'~0)=IUT(u'~0) (lOb) Withdls/ds+jwq=0,thefollowing expansions areobtained (asin Chap.I,Sec.4)forqL(W')andIzL(w')(onlythefirsttwotermsare retained) : Notethat(lla) (lIb) (He) (lId) (12) Sec.1] THETERMINATED LINE 61 whereIuT(u')isthetotalaxialcurrentatu'inthetermination and1/;(u')is theanglebetween thedirection ofthecurrentatu'andthezaxis. Thesubstitution of(11)and(12)in(7a,b)and(8a,b)andthesubse quentsubstitution oftheintegrals soobtained in(6)give Wz(w)=2~JI!IzL(W)[ko(w)+kOT(w)]+jWqL(w)[kl(~) +k1T(W)]j (13) {Ia2IzL(w)[k()k'()]} V()-1 -;-a2 1 W+1TW W- - ( () ,)]JW W 21l"~qI_w)[kow+kOT(w+ ~ (14) where,withw=8 -Z, ko(w)==fo8PL(w,w') dw'==fo8(~a-~)dw' =sinh-l~-sinh-I'!!!+sinh-I': -sinh-1~(15a)a b a b Forasufficiently longline(82)>b2)thisreducesto ko(w)==ko(w)==sinh-1~-sinh-I'!!!+In~a ba =21n~-Inw+yw2+b2(15b) aw+yw2+a2 {aT {8T(1 1 )kOT(w)==JoPT(w,u') cosl/t(u')du'==JoR1T-R2Tcos1/;(u')du' (15e) k~T(W)==fo8TPT(w,u')du'==fo8T(RlIT-R12T)du' (15d) k1(w)==~fo8(w'-w)PL(w,w') dw'==~fo8(w'-w)(~a-~b)dw' =~(yw2+b2-yw2+a2-yz2+b2+yz2+a2)(15e) k1T(W)==-~fo8T(u'+w)PT(w,u') cosl/t(u')du' foST(1 1 )==-~(u'+w)- - - cos1/;(u')du' (15!)o R1TR2T k~T(W)==-~foST(u'+w)PT(w,u') du' ==_~{ST(u'+w)(_1__1)du' (15g)Jo R1TR2T Notethattheintegrals (15a)and(15e)arethesameasthoseinChap.I, Sec.4,Eqs.(16)and(17),exceptthatthelimitsofintegration arefromoto8insteadoffrom-00to+00.JustasinChap.I,Sec.4,itisa 62 TRANSMISSION-LINE THEORY [Chap.II goodapproximation toreplacethefirstintegrals in(15a)and(15e)by thesecondintegrals, provided thefollowing restriction isimposed onthe separation bofthetwoconductors oftheline: l~bl2«1 (16) Itisnotclearwhether thisrestriction issufficient tomakethesecond integrals in(15c),(15d),(151),and(15g)goodapproximations ofthefirst integrals. Theapproximation actually madeis Ir(1..-1..)du'I»Ir(FiT-F'T)du'I(17a) wheree-i~R2T -1 F2T==R2T(17b) Inordertoevaluate (17a)itisnecessary tospecifythegeometry of thetermination. Asaconvenient andrathergeneralcase,letallsignifi cantcontributions tothepotential differences onthelinecomefromcur rentsandchargesinthestraight partsofthetermination inFig.1.1 whichmakeaconstant angle 1/1withtheline.Thismeansthatthese partsarerelatively longercompared withthelinespacingbthanin Fig.1.1.NotethatR1TandR2Tmaybeexpressed asfollows: RlT=V(w+u'cos1/1)2+(u'sin1/1)2+a2(18a) R2T=V(w+u'cos1/1)2+(u'sin1/1+b)2 (18b) Significant contributions tobothintegrals in(17a)areobtained onlyfrom valuesoftheintegrand forwhichRlTandR2Tareoftheorderofmagni tudeofsmallmultiples ofbandtherefore sufficiently smalltosatisfythe inequalities (19) ForlargervaluesofR1TandR2T,thesedistances approach eachother, andtheintegrands inbothintegrals in(17a)becomesmall.Overthe rangesspecified in(19)theexponentials in(17b)maybeexpanded asin Chap.I,Sec.4.Theresultis FIT-F2T=~2(R2T-R1T)-j~3(R~T-R~T). • • (20) Itfollowsthattheratioofthemagnitude oftheintegrand ontheleftin (17a)tothatontherightis (21) Sinceoverthisrange(19)issatisfied, theintegrand ontherightin(17a) issmallcompared withtheintegrand ontheleftovertheentiresignifi cantrangeoftheintegral. Therefore therepresentation ofthefirstinte gralsin(15c,d)and(15/,g)bythesecondintegrals maybeassumed tobe agoodapproximation, subjectto(16).Letthefollowing symbols be 63 (22f)(22e) (22y)(22b) (22c) (22d)(22a) NotethatSec.1] THETERMINATED LINE defined(notethatkin(15b,c)isessentially real): le(w)==l~(w)+l~(w)=ko(w):kOT(w) 7r'V .I()_ . [-I()+-I()]_ko(w)+k~T(W)JWllw=JWYo wYTw- 27r~ yew)==yew)+jwc(w) pew)=kl(w)+kIT(w) -ko(w)+kOT(w) P'()=kI(w)+k~T(W) W-ko(w)+k~T(W) _kl(w) Po(w)=ko(w) ~2==~5=-jwlg(w)y(w)[ko(w) +k~T(W)] 'V ko(w) If(22a)to(22f)aresubstituted in(13)and(14),thefinalexpressions forthevectorpotential difference andthescalarpotential difference are Wz(w)=le(w)[IzL(W)+jWqL(;)P(W)] (23a) Yew)=jw[L(W)+-!-iJ2IzL(w)P'(W)] (23b) yew)q JWiJw2~ Forsomepurposes theratiofunctions al(w)and~I(W)areuseful. Theyare (24c)(24b)(24a) (26a) (26b) (26c) (26d) (26e)()Wz(w). le(w).() alw==WzL(w)=19(w)=alw ~I(W)=VL(W)==yew) -Yew)yo(w) Notethat,whentheleakageconductance issmall,asisusual, ~I(W)==tI>1(W)==c(w)co(w) Itisnowreadilyverifiedfrom(15a)to(15y)that,subjecttothe inequalities w2»b2Z2»b2(25) thegeneralexpressions (23a)and(23b)forthepotential differences are wellapproximated bythesimpleformulas derivedinChap.I,Sec.4, fortheinfiniteline.With(25)itfollowsthat ko(w)==ko=2In~kl(w)==0a kOT(w) ==0 k~T(W) ==0kIT(w)==0 Po(w)==0pew)==0P'(w)==0 ~I(W)==1VL(W)==Yew) aI(w)==1WzL(W) ==W..(w) 64 TRANSMISSION-LINE THEORY [Chap.II SOthat wherejwYew)= -qL(W)Y y 211"~ jw=~(27) (28) Itfollowsthat,whereas thegeneralequations (23a)and(23b)forthe potential differences mustbeusedwithindistances ofthetermination at bothendswhichdonotsatisfy(25),thesimpleformulas fortheinfinite linearegoodapproximations atsufficient distances fromtheends. Although thediscussion inthissectionwascarriedoutspecifically for atwo-wire line,itapplieswithslightmodification indetailtotheother typesoflineanalyzed inChap.1.Inallcasesthereisaregionnear eachtermination wherethemoregeneralequations (23a,b)mustbeused, whereas theformulas fortheinfinitelineapplyatdistances fromthe termination whicharelargecompared withthecross-sectional dimensions oftheparticular typeofline.Foreachtypeoflinetheappropriate formulaforkomustbeusedinthegeneralexpressions (28)fortheparame tersleandy=g+jwc. Forexample, inthecaseofthecoaxialline, 1fo""fo00(11) ko(w)==- -- -dw'dO' 211"0 0RIR2 where,asinChap.I,Sec.6,withr=aI,(29) (32)RI=V(w-W')2+aiR2=V(w-W')2+Si2(30a) S12=Va~+ai-2ala2cose' (30b) Theintegration withrespecttow'maybecarriedoutdirectlytogive ko(w)=2ln~_(""Inw+vw2+a~+ai-2a2alcose'de'(31) alJo w+vw2+ai 211" Theintegralin(31)hasnotbeenevaluated, butasatisfactory approxi mationisreadilyobtained. Sincea2isalwaysgreater,andusuallymuch greater,thanatandsincetheexpression undertheradicalinthenumer- atorrangesbetweenvw2+(a2-at)2andvw2+(a2+al)2,itisclear thatareasonable meanvalueisobtained simplybyneglecting theterms inatinthenumerator. Theresultis k( ).2 Ia2Iw+Vw2+a~ow=n-- n------:;=====.- atw+vw2+ai Thisisseentobethesameinformas(15b)forthetwo-wire line,with a2occurring inplaceofbandalinplaceofa. 2.Generalized Differential Equations.9•10,49Thederivation ofthe differential equations forthescalarandvectorpotential differences which Sec.2] THETERMINATED LINE 65 (1)arevalidatallpointsalongaterminated lineparallels thederivation in Chap.I,Sec.4,fortheinfinitelinebutproceeds frommoregeneralforms ofthefundamental relations. Specifically, sincethevectorpotential at pointsontheconductors ofthelinenearitsterminations mayhave components perpendicular tothelineaswellasparalleltoit,thegeneral relation [Chap.I,Sec.3,Eq.(8b)]mustbeused.Thedesiredgeneral equation ofcontinuity forthevectorpotential atpointsontheconductors ofthelineis aAz+aAy+aAz+ .~.....=0axayazJwY whereAz,Ay,andAzarethecomponents ofthetotalvectorpotential due tothecurrentsIZLinthelineandsuchofthecomponents IzT,IyT,and IZTasmayexistintheterminations; ~isthetotalscalarpotential dueto chargesqLinthelineandchargesqTintheterminations. Aspointed outinChap.I,Sec.3,itispossible toreplacethesingleequation (1) byseveralequations involving relatedcomponents ofthepotentials such asthefollowing: Ay=AyTAz=AZL+AZT ~=~T+«>L(2a) (2b) (3b)(3a)AZLand~Larecomputed atpointsonthelinefromcurrents andcharges intheline,whereas AzT,AyT, AzT,and«>Tarecomputed atthesamepoints onthelinefromcurrents andchargesinthetermination. Thesecom ponentssatisfythefollowing equations: aAZT+aAyT+aAzT+ .r«>T=0axayazJw aAzL+ . ~2....._0 --J -yL-az w Ifthescalarandaxialvectorpotential differences areintroduced as definedinSec.1,Eqs.(5a,b),itfollowsfrom(3b)andwitha/aw=-a/az that (4) Similarly, proceeding fromthegeneralequation [Chap.I,Sec.4,Eq. (6a)],viz., (5) andmakinguseofthedefining relation [Chap.I,Sec.4,Eq.(33a)]for theinternalimpedance perunitlengthzi,namely, E1z(w)=11z(w)zi,the following equation isobtained directly: a~sw)=zilzL(w)+jwWz(w) (6) 66 TRANSMISSION-LINE THEORY [Chap.II whereYew)andWz(w)arethetotalpotential differences between points onthetwoconductors ofthelineatadistance wfromtheload-line junction, I1z(w)=IzL(w)isthetotalcurrentinconductor 1atthisdis tance,andZi=zf+z~'Thedesireddifferential equations maynowbe obtained from(4)and(5)withSec.1,Eqs.(23a)and(23b).Asafirst step,letqL(W)beeliminated fromSec.1,Eq.(23a),usingSec.1,Eq. (23b),togive Wz(w)=[e(w)[IzL(w)+V(w)y~w)P(w) _a2~~~w) P(W)~'(W)] (7) Sincethelasttermontherightin(7)isasmallcorrection term,itis satisfactory toassume inevaluating itsorderofmagnitude thatthe current satisfies theuncorrected equation. Moreover, sincetheterm includes thesmallfactorP(w)P'(w) ,itisnegligible beyondadistance lOb fromeachendoftheline.IntheshortlengthslObthesmallinternal impedance oftheconductors maybeignored, andj~substituted for"(. Undertheseconditions thedifferential equation forthecurrent as obtained inChap.I,Sec.13,is a2~~~w)+~2IzL(w)=0 (8) If(8)isusedin(7),thefirstandlasttermsontherightbecome IzL(w)[1+P(w)P'(w)] (9) However, sincebothpew)andP'(w)arecorrection terms,theirproduct is ofhigherorderandmaybeneglected. Hence,subjecttothecondition Ip(w)p'(w)I«1 (10) thezcomponent ofthevectorpotential difference in(7)maybeexpressed asfollows: Wz(w)==[e(w)[IzL(W)+V(W)Y~)P(w)] (11) Wz(w),asgivenin(11),maybesubstituted in(6)toobtain IzL(w)=Z(~)[a~;)-jWle(w)y~(w)P(w) V(W)] (12) wheretheimpedance perunitlengthhasbeendefinedasfollows: z(w)==Zi+jw[e(w) (13) Thecorrection factorontherightmaybeexpressed intermsoftheratios al(w)and.1(W)definedinSec.1,Eqs.(24a,b),ifuseismadeofSec.1, Eqs.(22).Theresult IzL(w)=Z(~)[a~~)+~al(w).l(W)P(W)V(w) ] (14) isthegeneralized first-order equation forthecurrent. Sec.21 THETERMINATED LINE 67 Thesecond-order equation forthevoltageisobtained bydifferentiating (6)withrespecttowandusing(4)together with ThusWz(w)=WzL(w)+WzT(w) a2:~~)+~2VL(W)=a~[ziJzL(w)+jwWzT(w)1 (15) Inthisrelation WzT(w)istheaxialcomponent ofthevectorpotential difference duetocurrents inthetermination only,andthetermwith Ziasafactortakesaccount oftheverysmallinternal impedance ofthe line.Thustheentiretermontherightin(15)isafirst-order correction inwhichthevectorpotential difference andthecurrentmayberepre sentedbytheirleadingterms,i.e.,bytheiruncorrected values. Thus withSec.1,Eq.(23a),theleadingpartofthetotalvectorpotential is Wz(w) ==IzL(w)le(w) =IZL(w)[l~(w)+l~(w)J (16) Evidently, sinceWz(w)=WzL(W)+WzT(w),itfollowsthat (17) (18)If(17)issubstituted inthebrackets in(15)andIzL(w)isreplaced byits leadingtermfrom(14),viz., I( )==_1_aV(w) zLWz(w)aw therightsideof(15)becomes ~[Z(w)-jwl~(w)av(w)] aw z(w) aw(19) Sincetheprincipal partofz(w)isjwl~(w),theleadingtermin(19)is z(w)-jwl~(w)a2V(w) z(w)~(20) (21)Thesubstitution of(20)in(15)andasubsequent rearrangement ofterms givethefollowing homogeneous equation: a2v(w)+ .z(w) ~2VL(W)=0 aw2)wl8(w) However, withSec.1,Eqs.(22)and(24b),itfollowsthat ~2 yew) jw18(w)= -«I»l(W) (22) Hence,sincewithSec.1,Eq.(24b),VL(W)/.l(W) =V(w)l.--the finalequa tionforthetotalvoltagealongthelineis (23) 68 TRANSMISSION-LINE THEORY [Chap.II Thegeneralized propagation constant y2(W)isdefinedby y2(W)==z(w)y(w) ==zo(w)YO(w)al(w)«Il 1(w) (24) Whenthereisnoinductive coupling between thetermination andthe line,al(w)=1;whenthereisnocapacitive coupling, «Il1(w)=1.At distances thatsatisfythecondition w2»b2, y2(W)=="(2=zy=(Zi+jwle)(g+jWC) (25) wherele,g,andcaretheparameters oftheinfiniteline.Thus,when w2islargecompared withb2,thegeneralized Eq.(23)reducestothe equation fortheinfinitelinegivenbyChap.I,Sec.13,Eq.(5). 3.Terminal Zones;Coupling andEndEffects.10•49Itwasshownin thepreceding sectionthatthescalarpotential difference between thetwo conductors ofatransmission lineoffinitelengthandterminated inarbi traryimpedances isgiveninfirstapproximation bytheequation where(1) (2) Thecurrentinoneoftheconductors ofthebalanced line,inwhich 12z(w)=-I1z(w)=-Iz(w),isobtained fromthescalarpotential differ encebydifferentiation: Iz(w)=Z(~)[a~~w)+~p(W)al(W)«Ill(W)V(W)] (3) Sincethevariable w,ingeneral, occursin"(2(W)inanintricate manner, Eq.(1)cannotbesolvedbyconventional methods thatapplytoequa tionswithconstant coefficients. Indeed,since"(w)isadifferent func tionofwforeachtypeoftermination andline,ageneralsolution of(1) isnotpossible. Fortunately, precise knowledge aboutthedistribution ofcurrentorvoltageinthepartsofalinenearitsendsatw=0and Z=8 -W=0,whichareexcluded bytheconditions Z2=(8-W)2»b2(4) isrelatively unimportant, provided thecurrents andvoltages areknown accurately everywhere else.Although atallpointsoutsidetheterminal zonesoflength d~lObd==O.L\ (5) currents andvoltages satisfythesimpleequations d2V(w)_"(2V(W)=0 "(2=zy dw2 1aV(w) Iz(w)="Zaw(6) (7) Sec.3] THETERMINATED LINE 69 forwhichgeneralsolutions aregiveninChap.I,Sec.13,thecurrents and voltages actually cannotbedetermined from(6)and(7)without speci fyingboundary conditions; andthesenecessarily involvetheterminal zones inwhich(1),(2),and(3)butnot(6)and(7)arevalid. Thedifferences between thegeneralEqs.(1)and(3)fortheterminated lineandthespecialEqs.(6)and(7)fortheinfinitelineandforpoints sufficiently farfromtheendsofafinitelinemaybesummarized under ko(w) 0.01 0.0010 4 6 101214161820 w b FIG.3.1.Thefunctions ko(w), kl(W)/~b, andPo(w)/{3b foratwo-wire lineinaperfect dielectric. theheadings ofcoupling between theloadandthelineandtransmission lineendeffects. 1.Coupling between theloadandthelinemaybeinductive owingto anonvanishing zcomponent ofcurrentintheload,sothatWZT(w)and l~(w)arenotzeroandQl(W)differsfromunity;itmaybecapacitive, so thatVT(w)andCT(W)inYT(W)arenotzeroand4»l(W)differsfromunity. Theabsence ofinductive coupling isdefinedbyQl(W)=1;theabsence ofcapacitive coupling isdefinedby4»l(W)=1.Itissignificant tonote thattheinfinitelineisnotcharacterized byanabsence ofeitherinductive orcapacitive coupling between thesections oflineoneachsideofan arbitrary line-load junction atw=O.Onthecontrary, intheinfinite linethefollowing relations aretrue: l~(w)+lo(w)=le YT1(w)+yr;l(w) =y-l=(g+jwC)-l(8a) (8b) 70 TRANSMISSION-LINE THEORY [Chap.II sothattheconstancy ofleandypresupposes inductive andcapacitive coupling. Notethat,atw=0, IT(O)=loCO)=ileYT(O)=Yo(O)=2y (9a) whereas, whenw2»b2orw~00, IT(00)=0lo(00)=le (9b) (11) d2V(w)_"(2V(W)=0(13) dw2 whichistheinfinite-line equation. Ontheotherhand, 1 Iz(w)=z(w) [a~~)+~Po(W)V(W)](14) where Po(w)=~:~:~ (15)sothat~l(W)=1 Itfollowsthat z(w)=Zo(W) yew)=Yo(w) "(2(W)=Zo(w)Yo(w) =zy="(2 (12)bii /~ 7~'rt-!-rr4V~ 7rr 11- I If40 152030200400 ISO500 300 100E90 ~80 ...:70 ~60 r<so1 2 wlb101-+---+--4--1 0.101+---+--4--1 0.02~::F1:=t'---==l{0.08 "0 ~0.061----+--4-~ i ~0.04~-+--4--I~ ~8 s::6JJ+--+---:,-+----1i ~4~-.-·~_10-1X13....--;-r----.,........., 12 o 10o 0 1 200 wlb wlb FIG.3.2.Thefunctions l~(w),co(w),and Rc(w).Itisclearthattheequations fortheinfinitelinedonotapplyeventoa sectionoflinewithanopenend,since12YI~ly(w)1 ~lyl. 2.Transmission-line endeffectsarisefromthefactthat,evenwhen thereisnocapacitive orinductive coupling between thelineandtheload orwhenthereisnoload,thegeneralEqs.(1)and(3)donotbothreduce tothesimpleforms(6)and(7). Thisisreadilyseenbysetting Notethatitisonlytheproductzo(w)Yo(w) whichisindependent ofw, notthefunctions zo(w)andYo(w)individually. Itisduetothefactthat zo(w)isproportional toko(w)andYo(w)tol/ko(w)thattheproduct zo(w)Yo(w) isconstant. Clearlytheratiozo(w)/Yo(w) isnotindependent ofw,anditisthisratiowhichdefinesthegeneralized characteristic impedance Zc(w). Thefunctions ko(w),k1(w),andPo(w)areshowninFig.3.1foranopen two-wire lineasafunction ofthenormalized distance w/bfromtheload Sec.4] THETERMINATED LINE 71 atw=o.Thelinespacingisb.Theinductance andcapacitance per unitlength,19(w)andco(w),areshowninFig.3.2together withtheratio Rc(w)=19(w)/co(w) ==zo(w)/yo(w). Aperfectdielectric isassumed. Thesectionoflinenearatermination (orotherdiscontinuity) inwhich theequations oftheinfinitelines,(6)and(7),arenotvalidiscalleda terminal zone,andtheconditions thatareresponsible forthedifferences between (1)and(3),ontheonehand,and(6)and(7),ontheother, arecalledterminal-zone effects. 4.Equivalent Uniform LinewithTerminal-zone Network. Sincethe principal purposeofananalytical solution ofthetransmission-line prob lemistopredetermine quantities actually measured onatransmission line,itisnecessary toformulate anapproximate solution ofthegeneral equations [Sec.3,Eqs.(1)and(3)]forpractical use.Transmission-line measurements usuallyinvolvethedistributions ofcurrentandvoltage onpartsofthelinewhichareoutsidetheterminal zones.Thedataso obtained aretheninterpreted usingconventional formulas derivedfrom thesolutions ofthespecialequations [Sec.3,Eqs.(6)and(7)]foran infiniteline.Although theseequations arevalidintheregionofmeasure ment,theirrangeofapplication doesnotextendtotheactualtermi nations. Hencethisprocedure iscorrectonlyifasufficiently longsection oflineisincluded asapartofthetermination, sothatz=smaynotbe theactualendofthesmoothline.Iftheconventional formulas are assumed (incorrectly) toapplytotheterminal zonesandz=sor w=s-z=0coincides withtheactualjunction ofthelinewithan impedance, theimpedance apparently terminating thelineincludes the effectoferrorsmadeinusingincorrect parameters andformulas inthe terminal zone.Thisapparent terminal impedance ZaGatz=s(orZOGat z=0)isnot,ingeneral,theratiooftheactualscalarpotential difference across,tothecurrententering, theterminating impedance. SinceZaG involves theproperties ofthetransmission line,thesameimpedance may havequitedifferent apparent impedances whenconnected asaloadto different transmission lines.Merelybyvaryingthespacingoftheline orbychanging therelativeorientation oflineandtermination, theappar entterminal impedance ofagivenloadmaybealtered. Forreasonssimilartothosewhichmakeitimpossible tohavetheuni formproperties ofalongtransmission linecontinue toitsjunction with anarbitrary impedance, itisalsoimpossible todefineforanarbitrary circuitelementanimpedance thatisindependent ofthecircuittowhich itisconnected. Thedegreeofcoupling ofsuchanelementtotheadja centpartsofthecircuit,e.g.,thetransmission line,varieswiththecon figuration ofconductors andtheseparation ofitsterminals; itmaybe largeoralmostzeroinspecially designed arrangements. Onlywhenthe separation oftheterminals ofacircuitelement isvanishingly small,as whenitisdrivenbyafictitious extensionless generator orbyanequally 72 TRANSMISSION-LINE THEORY [Chap.II ·1(8) ' 1(8) Vi8)=4>1(8)-.2(8) -12(8)=ld8) +2(8)fictitious transmission linewithzerospacing, isitpossible todefinea self-impedance Z.thatisanindependent characteristic ofthecircuitele ment,whichthenbecomes acomplete self-contained circuit. Itispossible toseparate formally thecircuitproperties of,andthe coupling between, twopartsofasinglecomplete circuitintotwoself impedances andamutualimpedance.9Exceptwhenthedistribution of currentisgreatlyaffectedbythemutualterm,theself-impedance ofthe loaddiffersnegligibly fromitsidealself-impedance whenisolated and drivenbyapotential difference maintained acrossitsterminals bya fictitious source. Thisistrueofthecoupling between atransmission lineanditsload.Accordingly thetransmission linemaybeanalyzed asifithadaphysically extensionless load,andtheloadmaybeanalyzed asifitweredrivenbyafictitious sourcethatmaintains therequired potential difference V(s)=4>l(S) 4>2(S)atitsterminals, asshowninFig. 4.1,providedseparate accountistaken oftheactualcoupling between them. Thismaybedoneapproximately by meansofasuitable equivalent net workthatrepresents thecoupling as iflumpedatthejunction insteadof distributed overshortdistances nearFIG.4.1.Typical termination fortwo- wireline. it.Byconcentrating coupling effects andtransmission-line endeffectsin suchanetwork oflumpedelements, theactualterminal zoneinwhichz(w) andyew)arefunctions ofpositionmaybereplaced byafictitious sectionof lineinwhichthevariable parameters z(w)andyew)arereplaced bythe constants zandyoftheinfiniteline.Thatis,thelengthoftheterminal zoneisreduced from,say,d==lObtozero,anditsdistributed circuit properties, insofarastheydepartfromthoseofasmoothline,arecon centrated asalumpednetworkattheline-load junction. Ifthisisdone, theimpedance terminating thehypothetical completely uniform linewith constant parameters everywhere istheapparent terminal impedance Z.a. Thisconsistsoftheimpedance oftheidealized isolatedloadZ.=V(s)/I(s) , asobtained fromFig.4.1,incombination withthelumpednetworkthat takesaccountofallterminal-zone effects. Thisisshownschematically in Fig.4.2,wherethelumpedelements oftheterminal-zone network consist ofaseriesimpedance ZT=jwLTandashuntadmittance YT=jwCT. Thelumpedelements ZTandYTaretocompensate forthedifference between theseriesimpedance andshuntadmittance oftheactualterminal zoneandtheseriesimpedance andshuntadmittance ofasectionofline whichisequaltotheterminal zoneinlengthbuthasthelineconstants Sec.5] THETERMINATED LINE 73 Conventional line Zjn=Z8a~tl Constant parameters Zo,Yoofaninfiniteline.Theseelements aredefinedasfollows: ZT=Iod[z(w)-z]dw==iwIodW(w)-le]dw=iwLT(1) YT=Iod[yew)-y]dw==iwIod[c(w)-c]dw=iwCT(2) whereyew)isasdefinedinSec.1,Eq.(22c),z(w)asinSec.2,Eq.(13), andzandyasinSec.2,Eq.(25).Alternatively, withSec.2,Eqs. (24a,c), LT=Iod[lg(w)al(w) -le]dw (3) CT=Iod [CO(W)l(W) -c]dw (4) WithZTinseriesandYTinparallelwiththeload(theorderisnot important), z(w)andyew)intheterminal zonemaybereplaced by ITerminal zone AI ~Load, 1-.(coupling toline) IVariableparameters B :z(wl,y(w);coupling toload ...·,-------7\./2 • Conventional lineiLTA Zj"~Z~-:J~ CTG(~I~:fing Constant parameters zo'Yo 1LB nocouplingtoload 2T FIG.4.2.Actualandequivalent transmission lines.Theconfiguration ofconductors between AandBisthesameinbothcases. zandy,sothatyew)becomes yandpew)=O.Itfollowsthatthe infinite-line equations [Sec.3,Eqs.(6)and(7)]applytotheentireline including terminal zones,provided anappropriate lumpednetwork iscon nectedbetween thelineandeachtermination, asshowninFig.4.2,so thattheapparent terminating impedances areZaaatz=8andZOaat z=O.Theconstants ofthisnetwork mustbeevaluated separately for eachimpedance andeachtypeofline.Specificapplication ofthisgeneral theorytoimpedances ofvarioustypesterminating different linesandto thejunction oftwodifferent linesismadeinlatersections. Forusein thenextsectionithasbeenshownthattheconstants ofintegration in thegeneralsolution oftheinfinite-line equations [Sec.3,Eqs.(6)and(7)] maybeappliedtofinitelines,provided theboundary conditions are expressed intermsofapparent terminal impedances Zaawhichincludean appropriate terminal-zone network. 5.Evaluation ofConstants inTermsofBoundary Conditions; Expo nentialSolution foraTerminated Line.17Ifthetransmission lineisof finitelengthextending fromz=0toz=8,asshowninFig.5.1a,the 74 TRANSMISSION-LINE THEORY [Chap.II endsoftheconductors maybeconnected byterminal impedances ofthe mostgeneralsort,provided asectionoftransmission linewhichislong compared withthelinespacingbisincluded asapartofeachtermination. Theimpedances aredefinedby vZ=y (1) whereVisthecomplex potential difference acrosstheterminals ofthe impedance and1isthecomplex currentineachterminal. Thecurrents II _lye+I:~orz,f +lVe-2I0 i I(a)I II_lVe+I I z:fItZ• I+lye_I I2.0 I I(b)I ! ! ~~~ ~~ -: I I I I I z=o (c) z=s FIG.5.1.Terminated transmission lines.(a)Terminations atbothendsinclude sections oftransmission line.(b)Lumped terminations. (c)Lumped terminations withsinglegenerator atcenterofZoo inthetwoterminals areequalandopposite. Attheinputendtheimped anceisZo=Ro+jXo;attheoutputenditisZs=Rs+jXs•The generator attheinputendisseparated intotwoidentical partseach maintaining anemfiV~.Thesuperscript eistodistinguish anexternally appliedpotential difference oremffromavoltagedrop.Thesubscript 0 locatesthegenerators atz=O. Forlinesinwhichbissosmallthatitmakesnosignificant difference inanylengthoflinewhetheritisincreased ordecreased byanamount b, terminal-zone effectsarenegligible, andthecircuitofFig.5.lborFig. 5.leisadequate withZsa==ZsandZOa==ZooIfthecurrentatallpoints inZoisthesameas10(thecurrentintooroutoftheline),thehalvesof thegenerator maybecombined intoasinglegenerator connected inseries withZoinanydesiredmanner.Ifthecurrentamplitude isnotconstant throughout Zo,thegenerator mustbeintwopartsinordertohavethe Sec.5] THETERMINATED LINE 75 currents equalandinreversed directions atopposite pointsalongthe twoconductors oratthecenterofasymmetrical structure. Ifthetermination doesnotincludeasectionoflineandthespacingis notsosmallthatterminal-zone effectsarenegligible, thesemaybe assumed tobelocalized inanetwork oflumpedelements atthejunction ofthelineandthetermination, asexplained inthepreceding section. Bycombining suchacorrective network withtheidealimpedance Zoor Zsoftheimpedance whenisolated, theapparent impedance ZOoorZsais obtained. Thisisthefictitious impedance thatwouldhavetoterminate thelineifuniform conditions prevailed totheendsandthesamecur rentsandvoltages existedeverywhere onthelineasontheactualline withtheactualtermination exceptintheterminal zones.Theapparent impedance isthatdetermined frommeasurements madeonthelineifsolu tionsoftheconventional oruniform-line equations areusedinthereduc tionofthedata. Inthefollowing itisassumed forsimplicity inthenotation that terminal-zone effectsarenegligible, sothatZoandZsarethetermi nations. Thesolution obtained maybeappliedtogeneralterminations merelybyaddingtheadditional subscript atoZoandZsandtoother functions introduced todescribe theterminations. Theboundary conditions forthecircuitofFig.5.1are Forz=0, Forz=8,Vo=V~-1oZo Vs=1sZs(2a) (2b) Iftheappropriate currents andvoltages asgivenbythefirstequation in Chap.I,Sec.13,Eq.(7),andbyChap.I,Sec.13,Eq.(11),namely, Vz=B1e"(z+B2e-TZand1z=i(-B1e"(z+B2e-"(z),aresubstituted in(2), twoequations areobtained forevaluating thearbitrary constants B1and B2intermsoftheimpedances ZoandZsandtheparameters oftheline. Theseequations are Vs=B1+B2+~(-B1+B2) !.!(-B1e"(s+B2e-"s)=B1e"s+B2e-Ts Zc Rearranging andcollecting termsleadto(3) (4) (5) (6) 76 TRANSMISSION-LINE THEORY [Chap.II Forconvenience letthefollowing shorthand beintroduced in(5)and(6): (7) Thecomplex factorsr0andrsarecalledcoefficients ofreflection ofvoltage. Theirproperties arestudiedinlatersections. Solving (5)and(6)for B1andB2,using(7),gives (8) (9) Uponsubstituting theseinChap.I,Sec.13,Eqs.(7)and(11),thefinal solutions inexponential formareobtained. Theyare v_V~~e-rz+rse-y(2s-z) z-Zc+Zo1 -rorse-2ys I=Vge-rz-rse-y(2s-z) zZc+Zo1 -rorse-2ys(10) (11) Thesolutions foraterminated linemust,ofcourse,reducetothesolu tionspreviously obtained inChap.I,Sec.14,Eq.(1),forasemi-infinite lineoflengthsthatisallowedtoincrease without limit.Ifs~00in (10)and(11),allexponential termsinvolving svanish,provided 'Yhasa positiverealpart,sothat vez TT=IZ= 0ce-rz=V:oe-rz"z zcZc+Zo ThisislikeChap.I,Sec.14,Eq.(1);notethat Vo=VgZc Zc+Zo(12) (13) Itissignificant thatsolutions like(12)and(13)areobtained foraline terminated sothat or (14) Theinputcurrentgivenin(12)islikethatinasimplecircuitcon sistingofZcinserieswithZooItfollowsthataninfinitely longtrans missionline,oralineofanylengthterminated inZs=Zc,behavesatits inputterminals likeanimpedance Zc.SinceZcisdefinedbyChap.I, Sec.13,Eq.(lOa),entirelyintermsofparameters characteristic ofthe lineitself,Zcisproperly calledthecharacteristic impedance oftheline. Notethatatransmission linebehaves likeanimpedance Zconlyifitis infinitely longorifitisterminated inZc.Itsbehavior underothercir cumstances isquitedifferent. Sec.6] THETERMINATED LINE 77 6.Infinite-series FormoftheExponential Solution.17Theexponen tialsolutions ofSec.5,Eqs.(10)and(11),maybemodified eitherin ordertomakethemmoreconvenient mathematically orinordertofacili tatetheirinterpretation intermsofaphysical pictureormodel.Inthis sectionSec.5,Eq.(10),isrearranged intoaformwithaphysicalinterpre tationthathelpstoexplainthesignificance oftheparameters appearing inthesolution. Fromtheanalytical pointofviewsuchaphysical inter pretation isnotrequired. Ontheotherhand,pictures ormodelsthat illuminate amathematical formulaintermsofareadilyvisualized physi calmechanism oftenserveavaluable purpose. Aphysically fundamental transformation ofSec.5,Eq.(10),isderived below. Theprocedure iscontrary tothatusuallyfollowed bythemathe matician, whoprefersaclosedformula toaphysical interpretation, in thatSec.5,Eq.(10),isexpanded intoaninfiniteseriesbydividing the numerator bythedenominator. Theresultis VeZVz=Zo-+Zc[e-OYz+r,-e-oy(2_z)+ror.e-oy(2.+z)+ror~e-oy(4_z) +r3r;e-oy(4.+z)+...](1) Inordertoobtaintheinstantaneous realvoltage, (1)mustbemultiplied byeiwtandtherealpartselected. Thisis NotethatVgIZo~ZcI{e-azcos(wt-(3z+<1» +I'.e-a(2_z)cos[wt-(3(2s-z)+t/I.+<1>]+I'oI'.e-a(2.+z)cos[wt-(3(2s+z)+t/lo+t/I.+<1>]+I'oI'~e-a(4_z) cos[wt-(3(4s-z)+t/lo+2t/1.+<1>]+I'~I'~e-a(4'+z) cos[wt-(3(48+z)+2t/1o+2t/1.+<1>]+...}Vz= (2) Zc_IZCIeiif! (3) Zo+Zc-Zo+.zc and ro=I'oei"'or.=I'.eN• (4) Thevelocity ofaconstant phaseassociated witheachtermin(2)is obtained bysettingthephaseequaltoaconstant anddifferentiating withrespecttothetime.Thus Forthefirstterm, ddi(wt-(3z+<1»=const. Forthesecondterm, ddt(wt-2{Js+(3z+t/I.+<1»=const. Forthethirdterm, ddt(wt-2{Js-(3z+t/lo+t/I.+cf»=eonst.dzw- = -=vdt{3 P dz wdi= -p=-vp dzwdi=73=Vp 78 TRANSMISSION-LINE THEORY [Chap.II Forthefourth,sixth,andeveryeven-numbered termdz/dt=-Vp;for thefifth,seventh, andeveryodd-numbered termdz/dt=Vp• Theseries(2)maybeinterpreted termbyterm.Byallowing the length 8ofthelinetoincrease without limit,thefirsttermin(2)isseen tobethecomplete solution fortheinstantaneous voltageatzona8emi infiniteline.ItislikeChap.I,Sec.14,Eq.(1),foragenerator imped anceZoinsteadofzero.Theinterpretation previously appliedtoSec.4, Eq.(1),maybeappliedtothefirsttermin(2).Thatis,thecontribu tiontotheinstantaneous voltageatzbythefirsttermmaybevisual izedasavoltagewavetraveling inthepositivezdirection withaconstant phasevelocity Vp,theamplitude ofthevoltagediminishing exponentially withz.Attheinstanttwhenthewavereachesthepointz,thewavewill havetraveled atotaldistance zfromz=0toz=z.Thevoltagemeas uredatzattheparticular instanttmayberegarded ashavingoriginated atthegenerator atanappropriate earliertimet1suchthatt1=t-z/vp• Thusthefirsttermmaybeassumed torepresent avoltagewavethathas traveled onlythedistancezfromthegenerator tothepointofobservation atzwithvelocity Vp•Itinvolves aphaselag{3zandadecrease inampli tudebythefactore-azcompared withthepointz=0atthesameinstant. Theinstantaneous voltageatzontheterminated lineattimet1differs fromthatwhichwouldbeobserved atthesamepointandtimeifthe linewereinfinitebytheaddition oftheseriesoftermsfollowing thefirst onein(2).Viewedinthesamelightasthefirstterm,thesecondterm in(2)represents avoltagewavemovinginthenegative zdirection which hastraveled thedistance 28-z,startingatthegenerator, proceeding totheendofthelineatz=s,andreturning tothepointz,whereit arrivessimultaneously withthefirstwave.Theentiredistance was traversed withtheconstant phasevelocity Vp•Thestarting timewas it-(28-z)/vp+I/;./w.Itinvolves aphaselag(3(28-z)anda decrease inamplitude bythefactore-a(2.-z).Inaddition, thereis anamplitude factorfs=I(Zs-Zc)/(Zs+Zc)Iandaphaseshift 1/;,=arg(Zs-Zc)/(Zs+Zc).Sincefsand1/;8depend onlyonthe terminal impedance Z8atz=8andontheparameter ofthelineZc, itisplausible toregardr8=f8ei~8asacoefficient ofreflection character izingtheimpedance Z8whenthisterminates thelineofcharacteristic impedance Zc.Theeffectofthecoefficient istochangetheamplitude ofanincident voltagewavebyafactorfsandthephaseby1/;8afterthe wavereaches Z8andbeforeitstartsbackasareflected wave. Thethirdtermin(2)maybeinterpreted inananalogous mannerasa wavewhichoriginated atthegenerator atatimeit-(28+z)+1/;8/W+ I/;o/wandwhichhastraveled totheendatz=8,backtothegenerator atz=0,andfinallybacktothepointofobservation atz,whereitarrives simultaneously withtheotherwaves. Whenitarrivesatz,itistraveling inthepositivezdirection alongwiththefirstwave.Intransitthewave Sec.7] THETERMINATED LINE 79 isattenuated bythefactore-a(2s+z)duetotheline,byafactorrsdueto reflection atZs,andbyafactorroduetoreflection atZooSimilarly thereisaphaselag{3(28+z)duetothedistance traversed ontheline andphaseshifts1/;sand1/;0duetoreflection atZsandZooAllsucceeding termsin(2)maybeinterpreted inamanneranalogous tothatusedto describe thefirstthreeterms.Eachisacontribution tothevoltage aJ; thepointzfromcomponents thatstartedatz=0sufficiently earlyto travelasaconstant phasebackandforthalongtheline.Insodoing theamplitude suffersacontinuous exponential attenuation, andthephase suffersalinearlyincreasing lagwithrespecttothevoltageatz=O.In successive reflections ateachofthetwoends,discontinuous changes in amplitude andphasesupplement theeffectoftheline.Thenumber of reflections ateachendisgivenbythepowerstowhichthefactorsroand rs,whichcharacterize asinglereflection, areraised. Thetotaldistance traveled byeachcomponent isgivenbythefactorofaintheexponents orof(3inthephases. Intermsofthisphysically attractive picturethe instantaneous potential difference atanypointalongaterminated trans missionlineistheresultant ofallthecontributions reaching thatpoint simultaneously frombothdirections afteraninfinityofsuccessive reflec tionsattheends.Theterminated lineisthusseentoplaytheroleof aninfinitelinefoldedbackandforthuponitself,withdiscontinuities at intervals equaltotheactuallengthandwiththepotential difference in thesefoldedpartsactually superimposed andcombined algebraically into asinglevalue. Thisinterpretation canbeobtained directlyfromthecomplex series (1)ifitisrecalledthatacomplex quantity involves arealamplitude andaphaseshift.Thatis,oncetherelationship between complex and realinstantaneous valuesisunderstood, theessential pointsmaybedeter mineddirectlyfromthecomplex formwithoutthereal solution. Asimi larexpansion andinterpretation maybeusedforthecurrent. 7.Incident- andReflected-wave FormoftheExponential Solution.ll Analternative physical pictureoftheexponential solution ofthetrans mission-line equations isoftengiveninaformofSec.5,Eqs.(10)and (11),inwhichthevoltageVsacross,andthecurrentIs=Vs!Zsin,the terminal impedance Zsareintroduced explicitly. Thevoltage Vsis obtained bysettingz=8inSec.5,Eq.(10).Itis V~Zce-ys(l+rs) Vs=Zc+Zo1 -rorse-2ys Similarly, fromSec.5,Eq.(11), v~e-ys(l-rs) Is=Zc+Zo1 -rorse-2ys(1) (2) 80 TRANSMISSION-LINE THEORY [Chap.II If(1)and(2)aresubstituted inSec.5,Eqs.(10)and(11),andthe notation w==s-z (3) isintroduced, thefollowing expressions areobtained: Vz=1:srs(cyw+rsc-Yw)=1~Z~s(eYw+rse-Yw) (4) lz=Vs(e'Yw-rse-Yw)=__ls_(eYw-rse-Yw)(5) Zc(1+rs) 1 -rs Itisclearthatthedistribution ofcurrentdepends onw=s -Z,noton zalone.Theinstantaneous realsolutions areobtained bymultiplying byeioJtandselecting therealparts.Thus,forexample, thefirstequation in(4)'leadsto Vz=ReVzeiwt=11:srsl[eawcos(wt+fJw+<1» +rse-awcos(wt-fJw+1/;s+<1»](6) where isgivenby (7) Thisinstantaneous voltageconsists oftwoterms.Thephasevelocity forthefirsttermisobtained byholdingthetotalphaseconstant and differentiating withrespecttotime.Thisgives dwdzw-di=dt=~=Vp Thephasevelocity ofthesecondtermis dwdz w -(jj=dt= - ~=-Vp(8) (9) Accordingly thefirsttermin(6)represents avoltagewavetraveling inthepositive zdirection withphasevelocity Vp,whereas thesecond termrepresents awavetraveling inthenegative zdirection withthe samevelocity. Thustheinstantaneous voltage VzatZmaybecon sideredtobemadeupofthesumofacomposite waveofamplitude IVs/(l+rs)leawtraveling towardZsandacomposite waveofamplitude IVs!(1+rs)lrse-awtraveling intheopposite direction. Thetwowaves differinphaseby1/;s-2fJw,corresponding toagreaterdistance oftravel forthesecondwavefromthepointztoZsandbacktoz,withaphase shift1/;sonreflection atZsoThewavetraveling towardZsistheincident wave;thattraveling awayfromZsisthereflectedwave.Notethatthese composite waveshaveamplitudes thatareintricate functions ofthe parameters ofthelineandofbothZoandZs,andthattheoriginofthe wavesisnotreadilydetermined bynotingthedistance traveled. (11)(10)Sec.7] THETERMINATED LINE 81 Therelativephasesandamplitudes ofthetwotermsin(4)and(5) arerepresented inthefollowing alternative formulas: V=~ eyw[1+re-2awei(-Jt-2/Jw)] z1+rs. s 1z=Vs)eyw[1-rse-2awei<-Jt-2/Jw)] Zc(1+rs Aplotofthebracketin(10)isgivenwithex=0inFig.7.1andwith ex~0inFig.7.2. Itisinstructive tocompare therepresentation oftheexponential solu tion[Sec.5,Eq.(10)],first,byaninfiniteseriesasinSec.6,Eq.(1)and, secondly, bytwotermsasin(6).Thefirstrepresentation expresses the instantaneous solution asaninfinitesumofindividually simpleterms, eachofwhichisthesolution ofaninfinitelinefoldedbackuponitself. FIG.7.1.Thefunction 1+re-i2{Jwwith r=rei-Jt;r=0.6,if;=30°. Thecontribution byeachtermismadeupofthefraction ofthegener atorvoltageimpressed acrosstheline,modified inamplitude andphase bytheeffectoftheover-alldistance traversed onthelineandbythe coefficient ofreflection ateachendappearing asafactorforeachreflection. Theinstantaneous voltageatagivenpointisthusmadeupofthesimul taneously arriving contributions ofaninfinitenumber ofsimplewaves thathavetraveled backandforth,withonetermforeachpossibledistance between generator andpointofdetermination. Thesecondrepresentation, usingonlytwocomposite terms,ineffect separates theinfiniteseriesintotwoparts,asdetermined bythedirection ofmotionatthepointzattheinstantt.Thusalltermsrepresenting wavestraveling inthepositivezdirection arecombined intoasingle composite wave,theincidentwave;similarly alltermsrepresenting waves traveling inthenegativezdirection arecombined intoacomposite reflectedwave.Eachofthetworesulting composite wavesisthesuper position ofaninfinitenumber ofsimplewavestraveling simultaneously inonedirection. Assuch,itsamplitude involves theeffectofallreflec tionsatbothendsanddoesnotrepresent eachbyanexplicitfactor. 82 TRANSMISSION-LINE THEORY [Chap.II (12) (13)Theregrouping ofaninfinitenumber ofsimplewavesintotwocomposite onestraveling inopposite directions isanalytically convenient butnot physically sotranspa,rent asthesuperposition ofaninfinitenumber of simplewaves.Inparticular, thephysical interpretation ofthecoef ficientsofreflectionroandraasmodifying amplitude andphaseateach reflection islost,andtheexpressions (4)and(5)mayaswellbemade moresymmetrical bysubstituting forrstheequivalent (Za-Zc)/ (Za+Zc).Theresulting expressions are Vz=;';a(Za+Zc)e'Yw+2~a(Za-Zc)e-Yw =jIa(Za+Zc)eYw+{Ia(Za-Zc)e-Yw Iz=2~Za(Za+Zc)eYw-2~Za(Za-Zc)e-Yw -la(Z+Z)yw-~(Z-Z)-yw-2Zcsee 2Zcace Somewritersintroduce thenotation W==8 -Z1+=Vt zZc V-I-=_--.!.. zZc Then whereVz==Vi+V;-Iz=Ii+I; Vi==;';a(Zs+Zc)eYw=jIa(Za+Zc)eYW V;==;';S(Za-Zc)e-Yw={Ia(Za-ZJe-Yw Vz=V}eYW+V:;e-Yw;Iz=I-;-eYw+I;e-Yw Vt=~(Za+Zc)=;';S(Za+Zc) V:;=~(Za-Zc)=2~a(Za-Zc) It=;zc(Za+Zc)=2~Za(Za+Zc) I;= -2~c(Za-Zc)= -2~Za(Za-Zc)(14) (15) (16) (17) (18) (19) (21)Clearly,from(10)and(11),withjl/t =2"(w=j2(jw,for"(=j(janda=0, Vz=Vte'rW(1+Ira\)=Vz,maxIat1/1-2{3w=0,271",471",...(20) Iz=Iieyw(1-Ira!)=Iz,min Similarly, atjl/t=2"(w+ilr, Vz=Vteyw(1 -Iral)=Vz,minI 3 I-1+w(1+\r\)-IatI/t-2(jw='Ir,'Ir,z- aeYa - z,max Thetermswiththesuperscript+areforwavestraveling inthepositive zdirection; thosewiththesuperscript -areforwavestraveling inthe negativezdirection. Sec.8] NotethatTHETERMINATED LINE 83 V;I;r,=Y+= -F-, ,-r; (22) sothatthereflection coefficientr,maybeinterpreted asmeasuring the ratioofthereflected composite voltagewaveatZ,dividedbythecom positeincident wave.Similarlyr'measures theratioofcurrentwaves. Nosuchinterpretation forroexists.Infact,rodoesnotappearinany oftheformulas fortwocomposite waves,beingcontained exclusively in theamplitudes V,andI,. 8.Hyperbolic FormsoftheSolution.708!Theformulas [Sec.5, Eqs.(10)and(11)]forthecomplex currentandvoltagemaybeexpressed intermsofhyperbolic functions ofcomplex argument byeliminating the reflection factorsroandr,usingSec.5,Eq.(7),andthedefinitions of thehyperbolic sineandcosine,viz., sinhu==j.(eu-e-ll) (1) (2) (3) (4)whereustandsfor"'(sinthedenominators and"'((s-z)inthenumerators ofSec.5,Eqs.(10)and(11).Withw=s-zthetwoexpressions are Yz=Vfe(Z,cosh"'(w+Zesinh"'(w) Ve . Iz=DO(Z,sinh"'(w+Zecosh"'(w) D=(Z~+ZoZ,)sinh"'(s+Ze(ZO+Z,)cosh"'(s where Theseexpressions explicitly involve "'(andZeandtheterminal imped ancesZoandZ,;coefficients ofreflection donotappear. Thesealterna tiveformsofSec.5,Eqs.(10)and(11),donotlendthemselves toa simplephysical interpretation intermsofsuccessive reflections. Theformulas (2)and(3)aretoocomplicated topermitadirectvisuali zationorsimplegraphical representation ofthedistributions ofcurrent andvoltage. Veryconsiderable simplification from.thispointofview maybeachieved byemploying thesamemethod previously usedin simplifying theexponential forms,viz.,introducing thevoltage and currentatz=8explicitly. Theseare V-VoZeZ,-IZ, -D-"I=VoZe, D (5) Ifthesevaluesaresubstituted in(2)and(3)toeliminate V~/D,thefollow ingformulas areeasilydeduced: Vz=V,cosh"'(w+IsZesinh"'(W Iz=i'sinh"'(W+Iscosh"'(w e(6) (7) 84 TRANSMISSION-LINE THEORY [Chap.II Itisclearfrom(6)and(7)thatthevoltageandcurrentatanypoint alongaterminated transmission linedependonthedistance w=s-z ofthepointfromtheoutputend.Furthermore theymaybeexpressed entirelyintermsofthevoltageacross,andthecurrentinto,theterminal impedance Za.Theformulas (2)and(3)areinthiswaydividedinto twosomewhat simplerparts.Thefirstpermitstheinvestigation ofthe currentandvoltageatanypointalongthelineintermsofthecurrent andvoltageattheoutputend.Thesecond,asgivenin(5),involves the currentandthevoltageattheoutputend. Itisevidently possible toexpressEqs.(6)and(7)inthefollowing generalform: Vz=AVa+Bla Iz=CVa+Dla(8a) (8b) where,inthepresentcaseofatransmission line,thefourcomplex coef ficientsA,B,C,andDhavethefollowing values: A=D=coshyw B=Z:C=Zcsinhyw(9a) (9b) (10)Since(8a)and(8b)are,infact,thegeneralequations ofafour-terminal network oflumpedelements, itisclearthatasectionoflinebetween the pointszandsmustbeequivalent tosuchafour-terminal network. This is considered inChap.III,Sec.12. Analternative methodofsimplifying (2)and(3)involves thedefinition offunctions toreplacethereflection coefficients roandra•Itwillbe recalledthatroandramaybeinterpreted asameasure ofthechangein amplitude andtheshiftinphaseproduced byeachofaninfinitenumber ofsuccessive reflections ofawavetraveling backandforthalongtheline. Thenewfunctions thataretoreplaceroandraaredefinedtobea measure ofthecomplete orover-allattenuation andphaseshiftbythetermi nation,thatis,torepresent asasingleeffectthecomposite effectofallthe reflections atagiventermination. Thecomplex terminal function 6anditsrealandimaginary parts,the terminal attenuation function pandtheterminal phasefunction <P,are definedasfollows: 6==p+j<P==coth-1i Asubscript 0or8isusedon6,p,<P,andZtodistinguish between the terminations atz=0andz=8.Analternative definition thatisat timesmoreconvenient is 6'==p+j<P'==tanh-1i (11) Itisreadilyverifiedthattheprimedfunctions differfromtheunprimed Sec.8] THETERMINATED LINE 85 bythefollowing verysimplerelation: 6'=6+i!!.- 2eP'=eP+~-2(12) Theproofreducestoshowing that coth(p+jeP)=tanh(p+jeP±~) (13) UsingDwightformulas 655.3and655.4,thisisaccomplished atonce. ItisshownlaterthatePismeasured fromzerocurrentintothetermi nationandeP'fromzerovoltageacrossthetermination. Ifthenumerator anddenominator of(2)aredividedbyZ:andZo/Ze andZs/Zearereplaced, respectively, bycoth60andcoth6s,according to (10)itfollowsthat VI:=Ve cothOscosh"(w+sinh"(w (14) o(1+coth60coth6s)sinh"(8+(coth60+cothOs)cosh"(8 Withsuitable rearrangements thisexpression becomes SimilarlyV z=Vesinh00cosh("(w+6s) osinh("(8+60+Os) I=Vosinh60sinh("(w+6s) zZesinh("(8+60+6s)(15) (16) (20)(19)Ifthedefinition (11)isusedinsteadof(10),thecorresponding formulas are Vz=VecoshO~sinh("(w+6~) (17) osinh("(8+O~+O~) I-Vgcosh O~cosh("(w+6~) (18) z -Zesinh("(8+6~+6~) Theserelations maybereferredtovoltageandcurrentatz=8.Thus, from(15)and(16), V s=Vesinh00cosh6s osinh("(8+00+Os) I=Vosinh60sinhOs sZesinh("(8+60+Os) Forconvenience letthemodified terminal voltageVsbedefinedasfollows: Vs==Ve sinh00 =~=IsZe ( ) osinh("(8+60+Os)cosh6ssinh6s 21 Using(21),(15)and(16)reduceto Vz=Vscosh("(w+Os)=Vscosh[Caw+Ps)+j({3w+ePs)](22) Iz=~ssinh("(w+6s)=ZVssinh[Caw+Ps)+j({3w+ePs)](23) e c 86 TRANSMISSION-LINE THEORY [Chap.II Theseformulas reducetoaparticularly simpleformforalosslessline withadissipationless load,forwhich,asisdiscussed laterindetail, a=0r=J/3ZC=Rc (24) P.=09.=ipB=J(<p~+;) (25) Vz=V.cos«(3w+<P.)= -V.sin(/3w+<p~) (26) IzRc=VBsin(/3w+<p.)=VBcos(/3w+<p~) (27) Forashort-circuited lineforwhichZB=0,<p.=1r/2,and <P~=0(as showninSec.15), Vz= -V.sin/3w (28) IzRc=V.cos/3w (29) Foranopen-circuited lineforwhichZ.=00,<p.=0,and <P~=-1("/2 (asshowninSec.15), Vz=V.cos/3w IzRc=V.sin(3w(30) (31) Thecompletely hyperbolic formsofthesolution, asexpressed in(22) and(23),represent thegeneralcasewitharbitrary terminations inaform analogous tothesimpleformofalosslessline.Thephasefunction <p. represents theover-all phaseshiftduetothetermination atz=s(as distinguished fromthephaseshiftperreflection givenbytheargument ofthereflection coefficient). Thefunction <Poplaysasimilarpartfor thetermination atz=O. Thepartplayedbytheattenuation functions PoandP.isbestseenin theamplitude factorV.in(21).Thusin V=Ve .sinh(po+J<po) (32) • - 0sinh[as+Po+P.+J({3s+<Po+<PB)] PoandP.contribute theover-allattenuation duetotheterminations in thesamemannerasascontributes theover-allattenuation duetotheline. 9.Instantaneous ValuesoftheHyperbolic Solutions. Theinstanta neousvoltageandcurrent areobtained bymultiplying therespective complex quantities byeiu>tandtakingtherealparts.Forthispurpose itisnecessary toexpressthecomplex quantities inpolarform,which) inturn,involves thepolarformsofcomplex hyperbolic functions. These are wheresinh(u+jv)=S=Seia cosh(u+jv)=C=Ceie S=y{(cosh 2u-cos2v)=ys-;i-nh~2;;--u-+"'--s-;-in-::2"-v C=y{(cosh 2u+cos2v)=ysinh2u+cos2v tanv (1'=tan-1---tanhu E=tan-1(tanvtanhu)(1) (2) (3) (4) (5) (6) Sec.9] THETERMINATED LINE Withthisnotation Sec.8,Eqs.(15)and(16),become V_VBSoCW=VBSoCWei(ero+e..-cr.> a- 0S. 0S. 1 -VgSoSw=VgSoSwei(cro+cr..-cr.-</Io> a-ZcS.ZcS.87 (7) (8) where So=Soeicro=sinh00=sinh(po+j4>o) (9) Sw=Sweicr..=sinh(yw+0.)=sinh[aw+P.+j({jw+4>.)] ==sinh(Aw+jFw) (10) Cw=Cweie..=cosh(yw+0.)=cosh[aw+p.+j({jw+4>.)] ==cosh(Aw+jFw) (11) S.=S.eier•=sinh(ys+00+0.)=sinh[as+po+P.+j({js+4>0+4>.)] ==sinh(A.+jF.) (12) Zc=ZceicP (13) Inparticular V.=Vg~=Vg~ei(ero-cr.>=V.ef(ero-tr.> (14) V-VeSo_VBcosh2po-cos24>0_Vesinh2Po+sin24>0(15a) •- 0S.-0cosh2A.-cos2F.-0sinh2A.+sin2F. -1tan4>0 -1tanF. 0'0-0'.=tant-h-tant hA (15b)anPoan. Theinstantaneous realvoltageisobtained bymultiplying bothsidesof (7)byeic.>tandtakingtherealparttocorrespond toadrivingvoltage vg=vgcoswi (16a) Thus Va=VoS~~wcos(wi+0'0-0'.+Ew) (16b) Forfixedterminations (16b)maybeinterpreted asasinglewaveofcom positeamplitude andphase. Thedistribution ofvoltagealongthelineatparticular instantsmaybe investigated conveniently bysettingwi'=wi+0'0-O'a,sothat Va=Vg~~Vsinh2Aw+cos2FwcosEw(coswi'-tanEwsinwi')(16c) With(6)and(11)thismaybeexpressed asfollows: So Va=VgS.sinh2Aw+cos2Fw[ , ( hAtF)· ']1+tanh2Awtan2Fw~oswi-,tanwanwsmwi (16d) Rearrangement gives Vz=Vo~:(coshAwcosFwcoswt'-sinhAwsinFwsinwi')(17a) Notethat,foramatched linewithp.=00,(17a)reducestotheexpres sionpreviously obtained inChap.I,Sec.14,usingtheexponential form 88 TRANSMISSION-LINE THEORY [Chap.II ofthesolution. Specifically, whenPs~00,Us~/38+CPo+CPs,and sinhAwcoshAellW+P•--- ~ w~ =e-POe-llZ wherez=8 -W(17b)Ss Ss ellS+PO+P. sothat Vz=VgSoe-poe-llZ cos(wt-/3z+Uo-<Po) (17c) (18a) (18b)Forwt'=0: 7rForwt'=2:ThisisthesameasChap.I,Sec.14,Eq.(3),whenPo=0andCPo=7r/2. Convenient instants forstudying thedistribution ofvoltageinthe generalcase(17a)arewt'=0andwt'=7r/2.Attheseinstants thevolt agedistributions are VeSoVz=8 scoshAwcos({1w+<ps) V8So•hA. ( )Vz=&sm Wsm{1w+CPs Notethatthesedistributions resemble (17c)inthattheyaresinusoidal, butthattheamplitude factorsbehavequitedifferently inthattheyhave different valuesatdifferent instants oftime.Thismeansthat,fora wavetraveling alongthelinewithafinite(butnotnecessarily constant) velocity, theamplitude varieswithlocation. Notethat,foraW«Ps, Awisessentially constant. Thephasevelocity ofthesinusoidal wavewithvariable amplitude may bedetermined intheusualmannerbyselecting anarbitrary phasey;and differentiating itwithrespecttotime.Let Then1/1=wt+Uo-Us+Ew=constant dy;=w+dEw=0 dt dt(19) (NotethatUoandUsareconstants thatdonotinvolvet,z,orw.)Intro ducingthevariable w=8 -z,(19)maybeexpressed asfollows: w+dEwdw=w_dEwdz=0 (20) dwdt dwdt whereHencethephasevelocity isgivenby dz dw w Vp=dt= -di=dEw/dw Ew=tan-1[tan({1w+<Ps)tanh(aw+Ps)](21) (22) Differentiation usingtheformula d 1dx-(tan-1x)-dw -1+x2dw(23) givesdEwd/dw[tan({1w+<ps)tanh(aw+Ps)] dw=1+tan2(/3w+<ps)tanh2(aw+Ps)(24) Sec.9] THETERMINATED LINE 89 Usingthestandard relations d dx--tanx=sec2x -dw dw d dxdwtanhx=sech2xdw(25a) (25b) (29) (30)in(24),theresultis dEw_/3tanh(aw+Ps)sec2(/3w+cPs)+atan(/3w+cP8)sech2(aw+P8) dw- 1+tan2(/3w+cPs)tanh2(aw+P8) (26) Accordingly, with/3=w/v,Fw==/3w+cPs,andAw==aw+P8, w 1+tan2Fwtanh2Aw vp=dEw/dw=vtanhAwsec2Fw+(a//3)tanFwsech2Aw(27) Notethatvisthephasevelocity previously obtained foraninfinitely long ormatched line.Equation (27)canberearranged bymultiplying numer atoranddenominator bycosh2Awcos2Fw.Thus cosh2Awcos2Fw+sinh2Awsin2Fw Vp=vsinhAwcoshAw+(a//3)sinFwcosFw (28) Thedivision of(28)bycoshAwsinhAwandtheintroduction ofdouble arguments give ()_cothAwcos2Fw+tanhAwsin2Fw Vpvoltage-V •2F1 _~sm w /3sinh2Aw Thecorresponding expression forthephasevelocity ofthecurrentis obtained inthesamemanner.Itis ()_cothAwsin2Fw+tanhAwcos2Fw Vpcurrent-V •2F1 _~sm w /3sinh2Aw Asimplespecialcaseisthatofatransmission linewithlowattenuation perunitlength(asmall)butsufficiently end-loaded sothat a= a«1 /3sinh2Aw/3sinh2(aw+Ps)(31) Inthiscasethephasevelocities forthevoltageandcurrentin(29)and (30)are (vp)voltage ==v(cothAwcos2Fw+tanhAwsin2Fw) (32) (vp)current ==v(cothAwsin2Fw+tanhAwcos2Fw) (33) Onalow-loss linea//3maybeoftheorderofmagnitude of10-3,sothat (31)issatisfied ifsinh2(aw+P8)~0.1.ThisistruewhenaW+PI'~ 90 TRANSMISSION-LINE THEORY [Chap.II 0.05.Ifthelineisnotloaded,sothatPB=0,itfollowsthat (34a) Thisisequivalent toa«1 (jsinh2aw or,witha/(j==10-3,sinh2aw==2aw~0.1. 2~W~10-2(jw~50w~8A (34b) Thatis,onalinewithlowattenuation andnoloadthephasevelocity givenby(32)or(33)isaccurate onlyateightormorewavelengths from theendz=s.Nearertotheendthanthis(29)or(30)mustbeused. Iftheattenuation onthelineissmallcompared withtheload, Aw===aW+PB==PBaw« PB (35) Theamplitude ofthevoltageandcurrentdistribution alongthelineas functions ofw=s-zisgivenby Vz1"./Cw=ysinh2Aw+cos2Fw Iz1"./Sw=ysinh2Aw+sin2Fw(36) (37) If(35)istrue,asisusualonaloadedline,Aw==PBisindependent ofw, sothatVzisgreatestwherecos2Fwisgreatestandsmallestwherecos2Fwis smallest. Similarly Izisgreatest wheresin2Fwisgreatest. Specifically ForFw=n1r, (Vz)max 1"./Cw=ysinh2Aw+1=coshAw (Vp)voltage =vcothAw=(Vp)max (Iz)min 1"./Sw=sinhAw(Vp)current=vtanhAw=(Vp)min ForFw=n1r+11"/2, (Vz)min 1"./Cw=sinhAw(Vp)voltage =vtanhAw=(Vp)min (Iz)max 1"./Sw=ysinh2Aw+1=coshAw (Vp)current =vcothAw=(Vp)max(38a) (38b) (39a) (39b) Notethat(Vp)max(Vp)min =v2•Thusthephasevelocity isgreatest (vcothAw)wheretheamplitude isgreatest (1"./coshAw);thephase velocity issmallest (vtanhAw)wheretheamplitude issmallest (1"./sinhAw).Fromthebehavior ofthehyperbolic tangentandcotan gent,theextreme valuesofthephasevelocity increase withdecreasing Awanddecrease withincreasing Aw•Inparticular, ifPB=00,sothat AB=00,tanhAw=cothAw=1,sothatVp=v=constant atall pointsalongtheline.Thisisthematchedorinfinitelinewithrunning ortraveling waves. Ontheotherhand,ifthelineisessentially lossless withaverysmallload,tanhAw==AwisverysmallandcothAwisvery large,sothatthephasevelocity isverylargewheretheamplitude is largeandextremely smallwheretheamplitude issmall. Sec.10] THETERMINATED LINE 91 Sincetheshapeofthevoltage(orcurrent) wavechangesasittravels, themotionofagivenphaseisnotreadilyvisualized. However, two pointsarealwayslocatedeasily. Thesearethezeropointsandthepoints wherethetrigonometric factorhasitsmaximum valueofunity.Since thevalue1ismultiplied byavaryingamplitude, itisnotalwaysatthe maximum ofthetraveling wave.However, itisalwaysatthepointof contactofthevarying traveling disturbance, withtheamplitude vari ationplottedasafixedfunction ofz.Thusthepictureisthatofawave ofvarying shapemovingbetween boundary linesdefinedbytheampli tude,insuchamannerthatthewaveisincontactwiththeselinesat onepointineachwavelength. Thispointofcontactandthezerovalue travelwithaphasevelocitythatincreases anddecreases alongtheline withthefixedamplitude distribution. Thus,whenthezeropointinthe instantaneous wavepassesthemaximum oftheamplitude, itismoving withhighspeedifthetransmission linehasonlyasmallload.Onthe otherhand,whenitpassestheminimum intheamplitude, ittravelsvery slowly. Thesameistrueforthepointofcontact. 10.ThePropagation Constant. Thecomplex propagation constant "(definedinChap.I,Sec.13,is "(=a+}(3=vCr+jwl)(g+}wc)=y(rg-w2lc)+jw(lg+cr)(1) Therealandimaginary partsoftheradicalontherightmaybesepa ratedintwoways leading toequivalent butdifferent formulas forthe attenuation constant aandthephaseconstant (3.Inthefirstmethod bothsidesof(1)aresquared, andtherealandimaginary partsequated separately. Bysolvingforaand/3andselecting positive rootstomake aand/3realandpositive, theresultsare wherea=yj(yz-w2lc+rg) /3=yi(yz+w2lc-rg) y=yg2+W2C2Z=y'r-=2:-+- w-:2=ZZ(2) (3) (4) Inthesecondmethodforobtaining explicitexpressions ofaand/3, (1)isarranged asfollows: where"(=a+j/3=jVw2lc-rgyl-jh-y =yrg-w2lcyl+jh-y 1+'=y£v'w(lg+rc) h=Iw(lg+rc)I 'Yw2lc-rgw2lc>rg rg>w2lc rg=w2lc(5) (6) (7) (8) Thesecondsquarerootin(5)and(6)isintheformusedindefining the tabulated functionsf(h) andg(h)(Ref.9,Appendix II).Thesefunctions 92 TRANSMISSION-LINE THEORY [Chap.II aretherealandimaginary partsofthesquareroot.Thus yl±jh=f(h)±jg(h) (9) Thefunctions f(h)andg(h)aredefinedasfollows: Forh2«1, Forh2»1,f(h)==Yj(yl+h2+1)=cosh(jsinh-1h) g(h)==Y·iCYI+h2-1)=sinh(jsinh-1h) f(h)==1g(h)==~2 f(h)==g(h)==~~(10) (11) (12) (13) (17) (18)(16)Usingthenotation of(9)in(5),theresultis a+jli=jYw2lc-rg[f(h-y)-jg(h-y)] w2lc>rg(14) a+jli=yrg-w2lc[f(h-y)+jg(h-y)] w2lc<rg(15) Forallpractical transmission linesw2lcisalwaysgreaterthanrg.How ever,insomeattenuators rgmayexceedw2lc.Itfollowsthat a=yw2lc-rgg(h-y)I Ii=Yw2lc-rgf(h-y) a=vrg-w2lcf(h-y)I (j=yrg-w2lcg(h-y) a=Ii=~lg+rc) Theseformulas aremoreconvenient forcomputing aandIithanare(2) and(3)iftablesoff(h)andg(h)functions areavailable. Thecondition w2lc>rgissatisfied forallpractical high-frequency transmission lines. Theimportant ratioaliiisgivenby (19a) (19b) Clearly aFor~<1, w2lc>rg a (20)For~>1, w2lc<rg Fora=Ii, w2lc=rg Itisreadilyverifiedthatforw2lc>rgthefollowing relations aretrue: h~«1~==hoy<1 (21) 4 Ii2 Sec.11] THETERMINATED LINE 93 or,specifically, foranerrornotexceeding 1percent, h~~0.2 ~~0.225 (22) Consequently agoodapproximation isobtained ifa2/{j2isneglected in comparison withunityifhydoesnotexceed0.45. Ontheotherhand,forrg>w2lc, h;»1 4{j•1-=1-a hy(23) (1) (4)(3)11.TheCharacteristic Impedance. Thegeneral definition ofthe characteristic impedance is _ . _ ~r+jwlZc=Rc+JXc=--+.gJWC Thisalsomaybeseparated intorealandimaginary partsintwoways corresponding tothoseusedforthepropagation constant. Inthefirst mannerZcisobtained inpolarform.Usingtan-1x=7r/2-tan-1(l/x), 4r2+w2l2 Zc= 2+2 2expI(j/2)[tan-1(g/WC)-tan-1(r/wl)]} (2)gwe Inordertomakeuseofthef(h)andg(h)functions, let w2lc+rg(1_ .were-19) Zc=W2C2+g2\jJw2lc+rg anddefine h=Iwere-19)I cw2lc+rg sothat(3)isequivalent to (5) Theuppersignistobeusedwhenrc<19;thelowersignistobeused when·rc>19,asisusualinalltransmission linesimmersed ingood dielectrics. Then w2lc+rgf(h) (6)w2e2+g2 c ,------;:::c---,--- X+w2lc+rg(h)=+Rg(hc} (7) c= - w2C2+g2gc -cf(hc) Heretheuppersignapplieswhenre<19;thelowersignapplieswhen rc>19,asisusual.Ifrc=19,Xc=0andf(hc)=1.Itisconvenient tointroduce thequantity cPc,calledthedistortion factor,bysetting (8) 94 TRANSMISSION-LINE THEORY [Chap.II Bythesamereasoning aswasusedtoestablish Sec.10,Eq.(20),it followsthatcPecannotexceed1.Thatis, cPe<1 Forhe<00: Inparticular, Forhe~0.45: cP~~0.05 or Intermsof(3)to(5)itfollowsthatcP~«1(9) (10) Ze=Re{l-jcPe) Ze=Re{l+jcPe) Ze=Rerc>19 rc<19 rc=19(11) Zemaybeexpressed inpolarformasfollows: Ze=ReVI+cP~e-itan-l</l. ==Reci<l>· forcP~«1andrc>19(12) Forrc<19thesignofcPeischanged; forrc=19,cPe=O. 12.ThePhaseandGroupVelocities oftheInfiniteLine.Theveloc ityofaparticular phaseofcurrentorvoltagetraveling alonganinfinite lineorreflected backandforthalongaterminated lineisdefinedin Chap.I,Sec.14,Eq.(10).Itis w 1 vp==~=-Vr:~l:=c=-===;=(r=g/7W=::=2)O=-j-:::(h-=-"(7) (1)W Vj[V(g2+W2C2)(r2+w2l2)+w2lc-rg] Numerical valuesmaybedetermined usingtablesofj(h).Atsuf ficiently highfrequencies ~,asdefinedbySec.10,Eq.(8),becomes small;j(h"(),asdefinedbySec.10,Eq.(10),approaches unity;andrg/w2 in(1)becomes negligible. HencetheupperlimitofVpasthefrequency is increased withoutlimitis 1vp~_J7:asw~00 Vlc(2) Thelowerlimitaswbecomes smallisobtained mosteasilyusingthesecond formof(1).Itis rg 2 2+212asw~0 reg(3) Since Vpisnotindependent ofthefrequency, theremustbedispersion. Thegroupvelocity asdefinedbyChap.I,Sec.14,Eq.(I9a),maybe calculated directlyfromSec.10,Eq.(3),orfromSec.10,Eqs.(16)to (18),usingChap.I,Sec.14,Eq.(19a)or(19b).Theexpression obtained isintricate, and,ingeneral,thegroupvelocity isnotequaltothephase velocity. Sec.13] THETERMINATED LINE 95 13.SpecialFormsoftheGeneral Parameters oftheLine.Thegen eralformulas forthecomplex parameters randZcofatransmission line maybesimplified byimposing restrictive conditions upontherelative magnitudes ofsomeoralloftheparameters r,l,c,andg.Although considerable simplification intheformofthegeneralsolutions forthe currentandvoltagealongthelinemaybeachieved inthismanner,itis obtained attheexpense ofgenerality. Wherever theparticular restric tionsimposed areconsistent toasatisfactory degreeofapproximation withtheexperimental circumstances forwhichthefinalformulas areto beused,suchsimplification isdesirable andvaluable. Otherwise itmust beusedwithgreatcaution, ifatall,andwiththerealization thatresults arenotdependable inanygeneralsense. Fromthepointofviewofhigh-frequency circuitsthemostimportant specialcasesarethoseassociated withlowvaluesofresistance andleak ageconductance. Thesimplifications leadingtotheso-called distortion lesslineortheoceancableareofminorimportance inthehigh-frequency field. TheLinewithLowAttenuation perUnitLength. Theconditions that defineatransmission linewithlowattenuation perunitlengthare (1) Subjecttotheseconditions, hy==~(1+~)he==~(1-~) (2) sothat h;«1 h~«1 (3) ItfollowsdirectlyfromSec.10,Eqs.(10)and(11),using(3),that,for h2«1, f(h)==1g(h)==; (4) Theserelations applyforbothsubscripts 'Yandc.Accordingly a==~(1+~) (5) fJ==wv1C (6) ~=2~[1+~]sothat;:«1 (7) Rc==~~ (8) <Pc==~c=2~(1-~)sothat<P;«1 (9) . 1 .vp=.ylC=Vg (10) 96 TRANSMISSION-LINE THEORY [Chap.II Inthisspecialcasethephasevelocity istoafirstapproximation, equal totheupperlimitgiveninSec.12,Eq.(2),whichisindependent ofthe frequency, sothatnodispersion occurs.Itistobenotedthatthisis onlyapproximately trueandthatevenasmalldifference inVpfordiffer entfrequencies mayleadtogreatover-alldispersion onalineofsufficient length. Asshowninthesmalltypebelow,moreaccurate formulas are {3=w~(1+0) (11) vp=1)6(1-0) (12) vg=V6(1-~) (13) ~(14) Rc=C(1+0) where 20r 1(15)=wlev=--8-vz;c FactorsofHigherOrderintheParameters ofaLinewithLowAttenuation atHigh Frequencies. Although theformulas givenin(6)to(10)areexcellent approximations formostpractical purposes usinggoodtransmission lines,itisofvalueinspecial instances todetermine thefirsttermsneglected inquantities suchasRc,{J,Vp,and Vgwhicharenotthemselves small,asareaand¢c.Thus,insteadof(4), f(h)=~I+¥ sothat,withSec.10,Eq.(16), Using(2)and(9)andneglecting higher-order terms, Accordingly Inordertodetermine Vgitisnecessary tointroduce thespecificformula forlbecause itinvolves thefrequency. Thuswith letl=2li+le l=l~+l~+le {~:i 20=£;+l~ leforatwo-wire line foracoaxialline foratwo-wire line foracoaxialline UsingChap.I,Sec.4,Eq.(34),andChap.I,Sec.6.Eq.(18),whicharevalidathigh Sec.13] THETERMINATED LINE 97 frequencies, 2wii21" l'20= - = --:-=-wiawlawi' w(i~+it) 1'~+rtl'20= =--=-wia wlawlaforthetwo-wire line forthecoaxialline Hence 0=_1'_and 02«12wla Clearly 4Je=2~G-nisneverlargerthan0,sothat 4J~isatermofhigherorder than4Jeor0,subjectto4J~«1and o~«1.Therefore Accordingly wherefj==wVlac(l+20)(1+4J~)==wVlac(l+20)==wv-,;c(1+0) w 1 1 - 0 Vp= -= ==-= =v,(1-0) fjVlac(l+0)Vl'c 1 Vo.3 X108m/sec Va==VFC=v;;;.=v;;;. andPorand fraretherelativepermeability anddielectric constant. Also 1 Va Vg=afj/iJw=1+0+wao/aw But,sinceathighfrequencies rvariesasV~, o=_1'_=~ -2wla~ wherePisindependent ofthefrequency. Hence ao P 0 aw-2w~= -2w sothatVa Va Vg=1+0-j-o=1+0/2 Rc=Itfollowsthatboth VpandVgaresmallerthan VabutthatVgisgreaterthanVp,sothat thedispersion isanomalous. Since0isverysmall,thedispersion isalsosmall. Higher-order termsinReareobtained using(6)withf(he)==VI+h~/4.Thus [a(I+20)(1+1'g/w2lc)(1+h2/4)~a~j--------__~ e_= _V(1+20)(1+h;)(l-g2/W2C2) c(I+g2/w2C2) C Theleadinghigher-order termistheonein0,sothat ~Ila _11 ~lfeRe==-\Jc(1+20)=~c==~c(1+0) Although 0isnosmallerthan4Je,theerrorisusuallyinsignificant if0isneglected, whereas inatleastoneinstance (tobedescribed later)anerrorof50percentis madeif4Jeisneglected. Thisdifference isduetothefactthat4JeReistheleading imaginary terminZe,whereas oReisanextremely smallfraction oftherealpartofRe. Itisalwaysagoodapproximation athighfrequencies toneglect0incomputing Re, sothat 98 TRANSMISSION-LINE THEORY [Chap.II TheLinewithLowAttenuation andNegligible LeakageConductance per UnitLength. Formosthigh-frequency linesthefollowing condition appliesinaddition to(1): JL«! wewl andfurthersimplification ispossible. Thus h.h.r -y=e=-;;;z a==;J=2~e A..•a.r 'l'e=~=2wl(16) (17) (18) (19) Ontheotherhand,{j,Re,andVpareasin(6),(8),and(10). TheLinewithoutAttenuation. Theextreme simplification afforded by neglecting theresistance ofthelineisattractive. Thephysically unrealiz ableconditions are r==0g==0 (20) sothat h-y==he==0 (21) a==0==cPe (22) Theexpressions for{j,v,andRearethesameasforlowattenuation. For thisreasonalimitednumber ofimportant resultsinvolving sections of lineofrestricted lengthmaybecalculated quiteaccurately apparently by assuming (20).Greatcaremustbeexercised intheuseofformulas that dependupon(20)and(21),sincecorrectresultsareobtained onlywhen thesameresultsareobtained with(20)aswith(1)and(16). TheLinewithLowDistortion. Alinewithlowdistortion isonethat needberestricted innoothermannerexceptthatcPeissufficiently small sothat cP~«1 Ithasalreadybeenshownthat(23)isequivalent to he~0.45 inwhichcase(23) (24) cP;~0.05 (25) TheDistortionless Line.Theconditions defining adistortionless line "'(=a+j{j=~~(r+jwl) Ze=Re(1-jcPe)=~~are Inthiscaseorrg[=c(26) (27) (28) Sec.13] HenceTHETERMINATED LINE fj=wVZC Rc=~~ ¢e=0 1 Vp=VlC=Vg99 (29) (30) (31) (32) If(26)issatisfied, theseformulas arenotapproximate, aswhen(1)is satisfied, butexact.Therewouldbenodispersion iflwerenotafunc tionoffrequency. Atlowfrequencies inlineswhererin¢eaffectsthe phasevelocity, considerable improvement canbemadebyusingloading coilstofulfill(26).Athighfrequencies thecondition (26)canbeful filledonlywithdifficulty, andsincetheeffectofrissmallerthanthatofli, littleisgained. Mosthigh-frequency transmission linesnormally fulfill theconditions (1),sothatlargechanges involving anincrease ineitherl orgorbotharerequired before(26)canbesatisfied. Athighfrequencies thishasnotbeenfoundpracticable. LineswithHighAttenuation: Attenuators.tTheextreme caseofhigh attenuation whichleadstoconsiderable simplification inthegeneral formulas involves thefollowing conditions: Hence sothat(5)2»1 (:C)2»1 h~==w(~+~) h;«1 f(h)==1rg ~<lc(33) (34) (35) (36) forboth h~andhe.Hence with Notethat'Y=a+jf3 a=yrg fj==rgh-y==wyrg(~+~)2 2 rg ~=~(~+~) (32«1 a2gra2 w2yrg V=-=--- p(319+rc(37) (38) (39) (40) (41) tAttenuators referred toherearelossyandnotthebeyond-cutoff typefamiliar in wave-guide theory. 100 Similarly withTRANSMISSION-LINE THEORY [Chap.II (42) (43) (44) LinewithLargeLeakage Conductance andNegligible Resistance. A so-called lossylinemaybeconstructed ofverygoodconductors anda poordielectric, sothatasatisfactory approximation is r==0g h'Y=he= -==hwC(45a) gVksothat ex.=-2cIntwospecialcasessimplification ispossible. moderately lowattenuation, asdefinedby g2«W2C2h2«1 {3=wViC Rc=~~(46a) X=gRe(46b) c2wc Thesecondspecialcaseassumes veryhighattenuation, asdefinedby W2C2«g2 h2»1 (47a) sothat ~{3~'wlg R~X~'wl (47b)ex.- -'\j2 c - e - '\j2g TheOceanCable.Anapproximation thatserveswellforspecialtypes ofcable,suchasthoseusedunderwater, is !-»1wlJL«1we(48) Thefirstcondition isduetolowfrequencies andverysmallinductance, nottohighresistance. InthiscaseSec.10,Eq.(1),andSec.11,Eq.(1), givedirectly y=ex.+j{3==~=(1+j)~w;c Zc=Rc+jXe=f!=(1-j)12r '\j;;:;c '\jwe Thephasevelocity is(49) (50) (51) Sec.14] THETERMINATED LINE 101 Since Vpincreases withfrequency, thegroupvelocity islargerthanthe phasevelocity andcanbecomputed usingr=P.v;,withPaconstant. Then Hence(52) (53) Evidently dispersion isanomalous andquitelarge;higherfrequencies travelmorerapidlythanlowerfrequencies. Ifthenumerator and denominator oftheexpression undertheradicalin(51)aremultiplied byl,theinductance perloopunitlengthofthecable,itbecomes v=j2wlII p'\lr'\lIe(54) Herethesecondsquarerootgivesthephasevelocity (10)foralinewith smallattenuation. Sincewlissmallcompared withr,according to(48), itisclearthatVpforthecableissmallcompared withthatalongacon ventional line.Linesforwhich(48)maybeusedarenotencountered athighfrequencies. 14.Relation betweenReflection Coefficient andTerminal Functions. Inthefollowing sections adetailed studyismadeofthephaseand attenuation functions andpforvariousterminal impedances. Since themagnitude andangleofthereflection coefficientraresimplyrelated topand<1>,theymaybeobtained directlyfrompand<1>.Therequired relationships arereadilydeduced using rI''.1._Z-Zc=e''1'=---Z+Zc o=p+j<l>=coth-1ZZcr'=I'eN'=Y-Yc=Zc-Z Y+YcZc+Z 0'=p+j<l>'=coth-1Y=tanh-1ZYc Zc(1) (2) Thedivision of(1)byZcandsubsequent substitution from(2)give r -I'fiji-coth0- 1 - -28 (3)- e-coth0+1 -e Since0=p+j<l>,theresultis r=I'eN=e-2(p+icl» (4) -2cothp-1 11 1+I'sothatr=eP=h ' orp=21n~=coth-1-- (5)cotp+1 I' 1 -r t/;=-2<1>or2(1l"-<1» (6a) Similarly t/;'=-2<1>'or2(1l"-<1>') (6b) Itisclearthat,whenp=0,I'=1;whenp=00,r=O.Using(5)and (6),randt/;areobtained readilyfrompand<1>,andrand1/1'frompand<1>'. 102 TRANSMISSION-LINE THEORY [Chap.II 15.ThePhaseandAttenuation Functions oftheTerminations.8lThe generaldefinition ofthecomplex terminal function 8(or8')oftheimped anceZwhenusedtoterminate alineofcharacteristic impedance Ze isgiveninSec.8.With Zl==Z/ZeandYl==Y/Ye=Ze/Z,8maybe expressed asfollows: Or,with8=p+jcf>=coth-lZl=tanh-lYl cf>'=cf>-~2 8'=p+jcf>'=coth-lYl=tanh-lZl(1) (2) (3) Explicitexpressions fortheterminal attenuation function pandthetermi nalphasefunction cf>areobtained readily. Thenormalized impedance Zl=rl+jXlmaybeexpanded asfollows: HenceZ R+jX_R-f/>eX+j(X+f/>eR) Zl=Ze=Re(l-jf/>e)- Re(l+f/>~) R-f/>eX .R X+f/>eR .X rl==Re(1+f/>;)=ReXl==Re(1+f/>~)=Rc Thecorresponding expansion ofthehyperbolic cotangent is(4) (5) .sinh2p-jsin2cf> •coth(p+Jcf»=h2 2cf>=rl+JXlcosP-cos(6) (7) (8)rl=cosh2p-cos2cf> -sin2cf>X---;---=--------::c-::- 1 -cosh2p-cos2cf>Byequating therealandimaginary partsof(6),explicitformulas forthe normalized resistance andreactance areobtained. Theyare sinh2p Withtheaidofsimpletrigonometric andhyperbolic transformations, theseformulas maybesolvedfortheterminal functions. Theresultsare 1h-l2rl p="2tanIzil+1 _ ,+ 11"_ 1t-I-2XI cf>-cf>'2-"2anlzil_1(9) (10) Forsomepurposes itisadvantageous toexpresstheterminal imped anceinadmittance form.Since withY==~=G+jB R G=R2+X2 B=-X(R2+X2)1 'bYl= -=gl+J1Zl(11) (12a) (12b) Sec.15] THETERMINATED LINE 103 itfollowsthat,withYe=1/Ze, sothat Since whereasyl=gl+jb1=~=(G+jB)R e(1-jc/>e) gl=Re(G+c/>eB)==ReG b1=RiB-c/>cG)==ReB Y1=~=coth(p+jip') %1=~=coth(p+jip)(13) (14a) (14b) (15a) (15b) itfollowsthatpandip'areexpressed intermsofglandb1inexactlythe sameformasarepandipintermsofr1andXl.Specifically 1th-12g1 (16)P=~anIgil+1 , 11"1t-1-2b1 (17) ip=ip-"2=~anIgil-1 Theformulas forp,ip,andip'applytoeithertermination. Thesub script0orsmaybeusedwhererequired todistinguish between theter minalfunctions associated, respectively, withZoatz=0andZ.atz=s. Inordertomakethedefinition ofthephasefunction ipin(10)unique, itisnecessary tospecifythequadrants inwhichipisfoundfordifferent typesoftermination. Thisinformation isobtained directlyifthefollow ingconventions areadoptedfortan2ip=a/b:2ipisinthefirstquadrant ifaandbarebothpositive,inthesecondquadrant ifaispositiveandbis negative, inthethirdquadrant ifaandbarebothnegative, andinthe fourthquadrant ifaisnegativeandbispositive. Obviously allthese statements locatingtheangle2ipapplytotheangleipifoctantiswritten throughout forquadrant. Sincetrigonometric functions of2ipareinno waychangedifthisisincreased by211",itfollowsthatfunctions ofipare notalteredifipisincreased by11".Accordingly whatever values Xland T1mayhaveinthefirst,second,third,orfourthoctants,respectively, are duplicated exactlyinthefifth,sixth,seventh, andeighthoctants. Thus, withXl=(X+c/>eR)/Re(1+c/>;),thevaluesindicated inTable15.1are determined. Thecorresponding valuesforip'areobtained byadding 11"/2toiporbyincreasing theoctantnumberby2. Thefollowing important characteristic ofpmaybenotedinthegeneral case:Whenever itfollowsthat%1=1 p=itanh-11=00(18) (19) Evidently, whenp=00,thetermination hasnoeffectontheincident trainoftraveling waves,sothatnofactorinvolving canoccur.The 104 TRANSMISSION-LINE THEORY [Chap.II mathematical limitforasXlapproaches zero,withrl=1,is1r/4or 31r/4. TABLE15.1 -2xISignsofa ReactanceIZII(or/lId)2<1>=tan-IjZI12_ 1andbinQuadrant forOctantforXI(orbl) ora2<1>(or2<1>') (or<1>')(-2bl)tan-I- 2<1>'=.tan-1_,-,-b 1f12-1 Capacitive: tan-I~ ± Large....- >1 I I,V Ilzlj'-11 +---- 12xd + ". ".5".- 1 tan-I- - -or'-0 0 2 44--- ----- Small....- <1 tan-I+2Ixl! +II II,VI-llzll'-11-- .-- 0".3". ". -or-0 <1.................. - 22- Inductive:-21.n1 -Small.... + <1 tan-I -IiZlj2-11- III III,VII--------- -12xll - 3". 3". ". + 1 tan-I--0 2"-or-- 0 4 4---- -21xll -IV,VIII Large.... + >1 tan-1 +\lzd2-11- IV + 0 0 >1.................. +0,2". 0,". 0±oo >1..................+0,2". 0,,,. 16.Graphical Representation oftheTerminal Functions intheNormal izedImpedance orAdmittance Plane;CircleDiagram.1l·47.81 Equations forcurvesofconstant pandofconstant arereadilyderivedinterms ofrlandXl,asgiveninSec.15,Eqs.(7)and(8).Thuscurvesofcon stantparedefinedby orri+xi-2rlcoth2p+1=0 (rl-coth2p)2+xi=coth22p-11 sinh22p(1) (2) Thisgivesafamilyofcircleswithoriginsat TI=coth2pXl=0 (3) andwithradiiequaltol/sinh2p.Thesameequation andfamilyof circlesareobtained intermsofglandbl. or sothatSec.16] THETERMINATED LINE Intercepts alongtherlaxisoccurwhenXl=0,thatis,when ri-2rIcoth2p+1=0 (rl-tanhp)(rl-cothp)=0 rl=cothportanhp105 (4) (5) (6) Notethatcothp~1,sothatXl=0,rl>1giveintercepts associated withcothp,andthattanhp~1,sothatXl=0,rl<1giveintercepts associated withtanhpwhenXl=0,rl=1,andp=00. Similarly curvesofconstant <J?aredefinedby orXi+ri+2XIcot2<J?-1=0 1 (X+cot2<J?)2+r2=1+coP2<J?=-.-1 1 m~~(7) (8) Thisequation definesafamilyofcircleswithoriginsat Xl= -cot2<J? (9) andwithradiiequaltoI/lsin2<J?1.Theintercepts alongtheXlaxisoccur atrl=O.Thus or sothattherootsarexi+2XIcot2<J?-1=0 (Xl+cot<J?)(Xl-tan<J?)=0 Xl= -cot<J?(ortan<J?)(10) (11) (12) Inordertobeconsistent withtheassumed convention thatXlisnegative for0~<J?~7r/2andpositivefor7r/2~<J?~7r,onlytherootXl= -cot<J? maybeusedtodefinetheintercepts. Thesignificance ofthisrestriction isbrought outbynotingthat,since cot(7r+2<J?)=cot2lJ> (13) sin2(7r+2lJ»=sin22lJ> (14) itfollowsthattheoriginsandradiiofcirclesforlJ>andlJ>-7r/2arethe same,sothatthefamilies ofcirclesfor<J?andlJ>-7r/2coincide. This meansthatinthefamilyofcirclesdefinedby(10)eachcirclerepresents twoelectrically different valuesoflJ>ineachrangefromzeroto7r,namely, lJ>andlJ>-7r/2.Thisambiguity isresolved bytheconditions imposed inSec.15tomakeeachpairofvaluesrlandXlhaveauniquevalueof<J? Since,bypostulate, Xlisnegative for0~<J?~~ Xlispositive for~~<J?~7r(15a) (15b) itfollowsthattoeachcircleofconstant <J?,withitstwopossible values of<J?,isassigned onlythevaluelessthan7r/2whenXlisnegative andonly 106 TRANSMISSION -LINETHEORY [Chap.II thevaluegreaterthan71"/2whenXlispositive. Thisimpliesthatthe valueof4>assigned toeachcirclejumpsby71"/2oncrossing therlaxis, where Xl=O.However, sincetheintersections ofthe4>circleswiththe TIaxissatisfyEq.(7)withXl=0,thatis,rl=1,itfollowsthatall4> curvespassthrough thepointXl=0,rl=1,wherep=00,andaredis continuous by71"/2atthispoint.Thus,ifep=300alongacircleofcon stant4>forXlnegative, itequals1200alongthesamecircleforXlpositive, thejumpof90°occurring atXl=0,rl=1.Thelimitingcircleofinfinite 21-----+-~--_+---_f_lL-..-~ <fJw)=180· -2l-------f-----:/---===+======---\--I-----1 o 2 3 FIG.16.1.Circlediagram-illustrative rectangular form. radiusthatistherlaxishasthevalue0or180°forrl~1andthevalue 90°forr~1. Sinceglandblsatisfyexactlythesameequations intermsofpand 4>'(=4>-11"/2),asdorlandXlintermsofpand4>,itfollowsthatthe samefamilies ofcirclesareobtained intermsofglandbl,pandtf>'. Moreover thevaluesofglandblwhichcorrespond toagivenpairof valuesrlandXlhavethesamevalueofpandavaluetf>'=4>-71"/2. Families ofcirclesplottedusingXl(orbl)andrl(orgl)asrectangular coordinates areshowninFig.16.1.Circlesofconstant angle4>(or4>') havetheircentersalongthepositive andnegative Xl(orbl)axis.They arescaledindegrees. Circlesofconstant phavetheircentersalongthe TI(orgl)axisextending fromrl=1.Theyarescaledinnepers. Only asmallnumber ofcirclesareshowninFig.16.1,whichisintended to Sec.16] THETERMINATED LINE 107 illustrate clearlytheconstruction ofthediagram. Amoredetailed dia gramthatmaybeusedfordetermining approximate valuesofpand or<1>'directlywithout computation isgiveninFig.16.2. Intermsofthereflection coefficient, circlesofconstant (or<1>')are alsocirclesofconstant 1('-1/;/2(or1('-1/;'/2).Thisfollowsfromthe 2 140° 1135° 130° 125° 120" 115° 110° 105° 100° 95°agoO 85°aoo .g~75" 1;j'~70° 1l~65° 8}60° 55° 50" -145° 40° -2Doublestub165°Alaspacing,. o· 0.5 0.4 Doublestub 3AAlspacing 150 o.... 20°2 -1 -2 FIG.16.2.Circlediagram-detailed rectangular form. relationt/I=2(1('-<1»or1/;'=2(1('-<1>').Similarly circlesofconstant p arealsocirclesofconstantr,sincer=e-2P• Thefollowing applications ofthecirclediagram maybelisted: 1.Determination ofpandcI>fromknownvaluesofrlandXlorofp andcI>'fromknownvaluesofglandbl. 2.Determination ofrlandXlfromknown(e.g.,experimentally deter mined)valuesofpandorofglandblfromknownvaluesofpand<1>'. 3.Determination ofZl=rl+jXlfromgiven Yl=gl+jbl.This merelyinvolves entering thecirclediagramatthepointgl,bl,notingthe valueof<1>',movingonacircleofconstant pto=<1>'+900 ,andreading 108 TRANSMISSION-LINE THEORY [Chap.II '1andXlappropriate tothispoint.Thedetermination ofY1withZ1 givenissimilar. Notethatthecirclediagram isactually beingusedto obtainthereciprocal ofacomplex number. 4.Determine P,givenitsintercept S=cothpontheaxisofreals; determine S=cothp,givenp.(Thequantity Sisthestanding-wave ratio,tobediscussed later.) 17.Graphical Representation oftheNormalized Impedance orAdmit tanceintheReflection-Coefficient Plane;SmithChart. 58Theconven tionalcirclediagram (Sec.16)consistsofcirclesofconstant attenuation pandconstant phaseshiftq,(orq,')inthecomplex Z1=r1+jX1(or Y1=g1+jb1)plane.Moregenerally itrepresents graphically thetrans formation fromthecomplex valuesofZ1=r1+jX1(orY1=g1+jb1)to thecomplex valuesof6=p+jq,(or6'=p+jq,'),according tothedefin ingrelations Z1=coth6Y1=coth6' (1) Circlesofconstant phavecentersonther1axisandenclosethepoint '1=1,Xl=0,whichisthecircleofzeroradiusforp=ex:>.Theaxisof imaginaries r1=0isthecircleofinfiniteradiusforp=O.Circlesfor allvaluesofpfromzerotoinfinityhaveintercepts withther1axisbetween '1=0and'1=1andagainbetween rl=0andr1=00.Itisevident thatanytwocirclesforspecified valuesofplierelatively veryclose together astheycrossther1axisbetween 0and1,whereas theyare relatively farapartastheyagaincrossther1axisbetween 1and00. Circlesofconstant q,havetheircentersontheXlaxis.Allcirclespass through thepointr1=1,Xl=o.Itfollowsthattwocirclesfordiffer entconstant valuesofq,lieclosetogether nearthepointrl=1,Xl=0 butarerelatively farapartatvaluesofrlandXlwhicharelargecom paredwithunity. Sincebothr1andXlvaryfromzerotoinfinity,itisevidentthata circlediagram ofthetypeshowninFigs.16.1and16.2cannotbepracti calsimultaneously intherangeofsmallvaluesandlargevaluesofrl andXl.Eitheranumber ofcirclediagrams drawntodifferent scales mustbeusedor,iftheconvenience ofasinglediagram isdesired, the scalemustbetransformed inamannertocompress therangeoflarge valuesofrlandXlandexpandtherangeofsmallvalues. Thismaybe accomplished bythewell-known bilinear transformation (Ref.1,page 106)zi=(azl+b)!(CZl+d),witha=2,b=0,andc=d=1.This distortsthecirclediagram inFig.16.1insuchamannerthatthecircles ofconstant pbecomeconcentric aboutthepointrl=ri=1,Xl=xi=0 withtheXlaxis(p=0),acircleofunitradius,andthecirclesofconstant q,becomeradiallines.Simultaneously thesimplestraight lines(circles ofinfiniteradius)rl=constant andXl=constant aredistorted into morecomplicated families ofcircleswithfiniteradii.Thesearenothing Sec.17J THETERMINATED LINE 109 elsethantherepresentation ofrlandXlinthecomplex planeofthereflec tioncoefficientr=reN'inpolarcoordinates. Thisisshownmoreexplic itlyafterthetransformation hasbeencarriedout. Thedesiredtransformation isachieved byrequiring thenewcirclesof constant ptohavecentersinthez~planeatri=1,311=0andtohave radiigivenbyjexp(-20)1=exp(-2p)=r.Theequations ofthese circlesare(r;-1)2+X;2=exp(-4p). Thecirclesofconstant ~in thez'planearetheradiallinesdefinedby2~= -tan-l[xV(r~-1)]. Theequation oftransformation is where z~=r;+jx;definestherectangular coordinates ofthenewcom plexplane.WithZI=coth0,itisreadilyverifiedthat(2)isabilin eartransformation fromthecomplex Zplaneintothecomplexz;plane, according to (3) (4b)(4a) (5)Thesimplerectangular netofTlandXlinthezplaneisdistorted into morecomplicated families ofcirclesinthez~plane.Theirequations maybederivedbysolving(3)successively forrlandXlintermsof r;andx~.Theresultsare r;(2-1"1)-X;2 Tl=(r;-2)2+X;2 _2xi Xl-(ri_2)2+X~2 Withconsiderable manipulation (4a)mayberearranged intothefollow ingform·: (, 1 r1)2+'2 1 r1- -rl+1 Xl=(rl+1)2 Thisistheequation ofafamilyofcirclesofconstant rlintheziplane. Thecircleshaveradiil/(rl+1)withcentersat1"1=1+rl/(r}+1), x~=O.Theentire Xaxisdefinedbyrl=0intheZlplanebecomes a circleofunitradiuswithcenterat1"1=1,Xl=0inthez'plane.Thus theentirehalfspacerl~0intheZlplaneismapped insidetheunit circleinthez'lplane.Thelinerl=00mapsintoacircleofzeroradius withcenteratri=2,xi=O.Thelinerl=1mapsintoacircleof radius0.5withcenteratrl=1.5,Xl=O.Thusthestrip0~rl~1 intheZIplanemapsintotheregionbetweenthecirclesrl=0andrl=l. Thecontours ofconstant Tl,whichwerestraight linesintheZIplane, mapintocirclesintheziplane,asshowninFig.17.1.Theintercepts Xl=0onther1axisintheZlplanetransform intothepointsobtained (6)[Chap.II X~=0TRANSMISSION-LINE THEORY 110 bysetting Xl=0in(4b).WithXl=0,Zl=rlin(2)and 2rl r~=-rl-1 Eq.(4b)mayberearranged sothatitbecomes (r~-2)2+(X~--x11)21 (7)-Xf Thisequation definesafamilyofcirclesofconstant Xlinthez~plane withradii1/lxl\andwithcentersatr~=2,x~=l/xl.Therlaxis, definedbyXl=0intheZlplane,be comesasectionofacircleofinfinite radiuswithcenteratr~=2andx~=00; thisisther~axis,definedbyx~=o. Theintercepts onther~axis, x~=0, 10occuratthesinglepoint r~=2.Hence XI=Ot---t---+--I---+-E;~_10 allcirclesofconstant Xlpassthroughthe point r~=2,x~=O.Thecirclesofcon stantrlandXlinthez'plane(orgland blinthey'plane)areshowninFig.17.1. Theycorrespond totherectangular..,1 FIG.17.1.Circlesofconstant rlandmeshofstraightlinesrl(orgl)=con Xlintransformed circlediagram. stantandXl(orbl)=constant intheZl plane(orYlplane). Theconcentric cir- clesofconstant pandradiallinesofconstant <1>(or<1>'),corresponding to thetwofamiliesofcirclesintheZlplane(orYlplane),areshowninFig.17.2. <J!.O 901--t---+-t---~c-+--t--+--t-----1<1>=O.180 90 54 144 FIG.17.2.Circlesofconstant p,<P,and<p'intransformed circlediagram. Thus,whereas theoriginalcirclediagram fortransforming fromZlto6(or fromYlto6')hasasimplerectangular meshforrlandXl(orglandbl) andtworathercomplicated familiesofcirclesforpand<1>(or<1>'),bothof whichextendtoinfinity,thetransformed circlediagram (Fig.17.3)(Smith chart)hascomplicated familiesofcirclesofconstant rlandXlandsimple Sec.17} THETERMINATED LINE 111 familiesofconcentric circlesandradiallinesforpand<1>.Significantly allvaluesofrlandXl,including infinitevalues,arecontained withinor areontheunitcirclerl=0orp=o. Sincethecirclesofconstant pinthez'planeare,ineffect,circlesof constantr=e-2p,where,intherangep=00top=0,rincreases from FIG.17.3.Smithchart. oto1,itisoftenconvenient tousealinearscaleofrastheparameter insteadoftheexponential scaleforp.Thusrissimplytheradialdis tancefromthecenterofthediagram onascalethathasthevalue1for thebounding circlerl=O.Fromsymmetry, thecirclesofconstant canbescaledintermsoftheangleofthecomplex reflection coefficient t/I=2(1r-<I»[ort/I'=2(1r-')}.Thus,whereas rangesfrom0to 1800clockwise aroundthecircle,t/Irangesfrom0to3600counterclock wisearoundthecircle.Therelationbetween thepandscalesandthe randt/Iscalesisindicated inFig.17.4. 112 TRANSMISSION-LINE THEORY [Chap.II Itisevidently possibletouseeithercirclediagram forconverting from theterminal function 0=p+J4?(or0'=p+J4?')tothenormalized impedance Zl=rl+JXI(oradmittance YI=gl+Jbl),orviceversa. Alternatively theconversion maybefromr=reN(orr'=reN')to Zl(orYI),orviceversa.Eachhasadvantages forcertainpurposes. Notethaty;=3600 -24>. 18.SpecialFormsoftheTerminal Functions and oftheReflection Coefficient-Resistive Termination.81Inordertoobtainaclearerpic tureofthedependence oftheterminal functions pand4?uponthetermi nalimpedance Z,consider alinewithanessentially resistive termination suchthatXl=(X+cPeR)/Re(l+cP;)=O.Thisisphysically possible overarangeofresistances extending fromverysmalltoenormous values. Notethatverylowvaluesofresistance areusuallyassociated withshort piecesofcopperwirewhichdonothavezeroreactance. Infact,their resistance isusuallynegligible compared withtheinductive reactance. Sec.18] THETERMINATED LINE 113 Zeroreactance withverysmall,although neverzero,resistance, canbe obtained withaseries-resonant circuit. Thetermination tobeinvestigated isdefinedby (1) Itfollowsthat without approximation. Similarly, with(1),(2) Also Notethatb=R(B-A.G)= _Rc(X+cPcR)=0 1 - c 'f'c R2+X2 =R(G+A.B)=Rc(R-cPcX)=Rc (/1-c 'f' R2+X2 R Rc1 (/1= - = -Rr1(3) (4) (5) Theterminal functions are: _ 1h-12r1_1h-12(/1 P-~tanri+1 -~tan(/i+1 1·1t1-2X1 =1m2"an- 2+21 X1-+0 r1X1- <1>'=r1t1· -2b1 b~~O"2an-(/i+bi-1 Letpbeinvestigated firstusingthetrigonometric formula 2tanhp tanh2p=1+tanh2p withwhich(6)becomes 2tanhp 2r1 1+tanh2p=ri+1 Thiscanbeexpressed intheform(6) (7a) (7b) (8) (9) withroots(tanhp-rl)(tanhp-~)=0 111r1 - tanhp=1or(/1 - (/1r1(10) (11) Sincethehyperbolic tangentcanneverexceedunity,itfollowsthat p=tanh-1rl p=tanh-1(/1(12) (13) 114 TRANSMISSION-LINE THEORY (rl=gl-l.0000 5 to4decimal placesforp>5) 4[Chap.II P3lowerrangep 0.10 Higher range 0.08 2 0.06 0.04 0.02 °o-=;,..--'-----'---...JL...---'---J......-JO 0.020.04 0.06 0.080.1lowerrangeo0.20.40.6 0.8 1.0Higherrange ReRg=R~1;r=-~1Re FIG.18.1.Theattenuation function p==tanh-lTI,withTI~1,andp==tanh-lgl, withgl~1,foraresistive termination, withTl==1/g1• ..------------~To co 900 ~1l'~I- " 2 0°0L.-- 'L-- ----','- ---II_ o 1 2 3 r-11-gl FIG.18.2.Thephasefunction foraresistive termination, withXl==O. Sec.18] THETERMINATED LINE 115 Thisincludes theentirerangeofvaluesofr1=1/g1fromzerotoinfinity. ItisplottedinFig.18.1. Forsmallvaluesofr1orsmallvaluesofgl=l/r1corresponding to verylargevaluesofr1,theinversehyperbolic tangentmaybereplaced byitsargument. Thus forri«3 forgi«3 Itisreadilyverifiedthattheconditions(14) (15) coincide with forwhichXl=0r1=1 Zl=1orZ=Ze=Re(1-jcPe) R=RcX= -cPcRc(16a) (16b) (17) Inthiscasep=00. Thephasefunctions cI>andcI>'asgivenby(7a,b)maybeexpressed as follows: 1Forr1= -<1,gl 1ForY1= -<1r1'ocI>=1tan-l-- 2"ri-1 ocI>'=1..tan-l--- 2gi-1 lcI>=1..tan-l~=~or31r 2 _2 2 <1>'=-~=0or1r2 l'1t-101r31r cI>=2"an-==2"or2 =<1>'+~=1ror02(18) (19) (20a) (20b) (21a) (21b) Forr1=1,and<1>'areindeterminate because discontinuous. Ifr10r glapproaches 1fromsmallvalues, cI>or<1>'is1r/2or31r/2;ifr1orgl approaches 1fromlargervalues, orcI>'is0or1r.Ontheotherhand, ifr1issetequaltounitybeforeXlismadetovanish,thefollowing resultis obtained: Sincetheargument becomes infinitewithsigns-/0,itmustbeat 31r 1r cI>= -or--4 4cI>'=-~=~2 4(22b) Thephasefunction ofaresistive termination isrepresented inFig.18.2. 116 TRANSMISSION-LINE THEORY [Chap.II Theangleofthecoefficient ofreflection isgivenby Hencey;= -2<1>or271"-2<1> y;=271"-71"=71"forr1<1orgl>1 y;=271"-271"=0forr1>1orgl<1 371" 71"y;=271"-2="2forr1=1=gl(23a) (23b) Summarizing, ifXl=0=b1,<1>=0andy;=0forgl<1orr1>1; <1>=71"/2andy;d:71"forr1<1orgl>1.Withr1=1,<1>and<1>'are 71"/4,andy;=71"/2. Ifitisrequiredthat x=0=B insteadofXl=0=b1,then(24) R-cPeX r1=Re(l+cP;) X+cPeR Xl=Re(l+cP;)\(25a) (25b) IfcP~isneglected compared withunity,itfollowsthatr1and,hencepare thesameforX=0asforXl=O.Ontheotherhand, .m. 1t-1-2cPerl'¥=~an---ri- 1(26) Withf1>1,2<1>isinthefourthquadrant(-/+);forr1<1,2<1>isin thethirdquadrant(-/-).Hence Also<1>= _ 1tan-12cPerl 71"~ iri-11 .m._71"+1t-12cP~rl '¥-"2~anIri-11 <1>- 1t-1-2cPe_371"-~an-0--4f1<1 (27) Alternatively .m.'_.m._~_~_!t-I2cPegi '¥-'¥2 - 2 2 anIgi-11 <1>'=<1>-;=~tan-IIg~cP:!\1 <1>'=~4gl<1 (28) Theprincipal rangeof<1>asafunction offlandglwithcPeasparameter isshowninFig.18.3. Summarizing thebehavior of<1>andpforaresistive termination with XI=0,thefollowing simplepicturemayhedescrihed: Asrlincreases Sec.19] THETERMINATED LINE 117 fromnearzeroto1,pincreases fromnearzerotoinfinity. tI>continues constant atw/2untilTlisexactlyequalto1.Itthenrisesabruptly to 311"/4.AsTlincreases furtherfrom1toverylargevalues, pdecreases frominfinitytonearzero. tI>risesabruptly from311"/4to11"asTlexceeds1. Thusanormalized resistance Tlthatissmallerthan1behaves likeazero resistance insofarasthephasefunction tI>isconcerned; similarly anormal izedresistance Tlthatisgreaterthan1acts,insofaras4>isconcerned, 0.2 1.0 -40· ~. -50· -60· -70· -80· -90·ISO·-0· ISO·-30·170·-10· 90·160·-20· 110" 100·120·140· 135· 130· 1.00.8.. %1-0,orx-o,+c-o/:I\ 10-X-O.+c-7.l8x 10-4. x-O.+c-l0-Z,....l ):_!r.11' 2 2 o3.0 2.0 1.82.6 -0.6-0.2 2.8 -0.4 -1.2 -1.42.4 7f311'-4"4 ~~iana 2.2 -1.0 0.20.40.6 rl=f;;~1 FIG.18.3.Thephasefunction tI>foraresistive termination, withX=0and <t>~«1. likeaninfiniteresistance. If,insteadofXl=0,thecondition isX=° onalineforwhichepcissmallbutnotzero,sothat ep~«1,thefunction pisjustasforXl=0,andthefunction 4>followsthesamegeneral behavior without quitesuchanabruptchangefrom4>=11"/2totI>=11" atTl=1.Thesmaller epcis,themorerapidisthechange. 19.SpecialFormsoftheTerminal Functions-the Predominantly Reactive Termination.8lItisnotpossibletoconstruct apurelyreactive termination. Ontheotherhand,aterminal impedance witharesistive component thatisnegligible compared withthereactive component is easilyobtained exceptforverysmallreactances andverylargepositive reactances. Negatively reactiveimpedances withextremely smallresist anceareeasilyconstructed forarangeofreactance whichextends frompractically negatively infinitetonearlyzero.Asthereactance 118 TRANSMISSION-LINE THEORY [Chap.II approaches zero,apointisreached wheretheresistance ceasestobe smallcompared withthereactance, andultimately asthereactance becomes stillsmaller,theresistance alwayspredominates. Inthecaseof positively reactiveimpedances exactlythesamesituation asfornegative reactances obtainsnearzerovalues. Althoughitisasimplematterto providepositively reactive impedances withresistive components small compared withthereactive overarangeofreactance extending from ~mallvaluestoextremely largeones,itisnotpossibletoapproach infinite values. Thisarisesfromthefactthatverylargevaluesofinductive reactance usuallymustbeobtained withparallelorantiresonant circuits. Astheresonant frequency isapproached fromthepositively reactive side,thislatterincreases toalargevaluebeforeantiresonance isreached, butthisisneverinfiniteandalwaysdropstozeroatantiresonance while theresistance increases toamaximum. Thusapredominantly reactive termination isphysically available intherangeofXfromnegative infinity tonearlyzeroandfromnearlyzeroonthepositive sidetoextremely largebutnotinfinitevalues.Nearandatzeroreactance theimpedance ispredominantly resistive. Letthepredominantly reactive termination bedefined bythe inequality r~« Ix~-11 Forlineswithlowdistortion (1)isequivalent to(1) cP:«1 (2) Subjectto(1)and(2),thegeneralformulas forpandbecome _ 1h-I2rl p-"2tanx2+1 ;F.._1t-1-2XI'i:"-"2an--- x~-1(3) (4) Since(1)includes r~« x~+1,itfollowsthattheargument ofthe inversehyperbolic tangentin(3)issmall.Hence .rl p=x~+1 Theexpressions formaybesimplified usingtheformula(5) (7)(6)2xtan-Ix=1tan-I-----"21 _x2 ;F.. 1t-1-2XI_1t-1 -2XI 'i:"="2anxi_ 1-"2an-(1-xi) Assuming Xlpositiveandlessthanunityforthemoment, theinverseThus Sec.19] THETERMINATED LINE 119 tangentin(7)isinthethirdquadrant. Hence ~=i(11"+tan-l1~IXi)=;+tan-lXl=11"+tan-lbl(8) Byreplacing 4>with ~'andXlwithblitfollowsthat 4>'=i(11"+tan-l1~lbi)=~+tan-lbl=11"+tan-lXl(9) Also ~'=~-~=tan-lXl (10)2 and Itisclearfrom(11) thatt-1 11"t-I1anX= - -an-2 X 1Xl=-bl(12) (13) (20)(19)(18)(17) orUsefulformulas forXlareobtained from(8)and(9).Thus Xl(==:)=tan(4)-~)= -cot~=tan~' (14) bl(==BRc)= -cot(4>-~)=tan4>= -cot~'(15) Thecondition (1)defining apredominantly reactive termination includes theentirerangeofXlfromzerotoinfinityexceptarangenear IXII=1.Insofaraspisconcerned, theinequality (1)couldbereplaced bythemuchlessrestrictive conditions ri«xi+1 (16) whichistobeinterpreted asarestriction onrlandnotaslimiting Xl. Accordingly formula (3)forpandothersderived fromitarevalidfor allvaluesofXlif(16)issatisfied. Thecondition (16)evidently isnotsufficient for~or4>',since ForIXII=1,~=itan-I-22=i(211"-tan-l..;)rl rl Using(12),itfollowsthat ~=11"-~(~-tan-l~) if>=311"+!tan-l!J==311"+:1 4 2 2 4 4 Thus4>asobtained from(4)isinerrorbyrU4intheextreme case IXII=1.If(16)issatisfied insteadof(1),itfollowsthat ri«4 120 TRANSMISSION-LINE THEORY [Chap.II SOthatri/4isnegligible compared with371/4.Accordingly (4)aswell as(3)maybeusedsubjectto(16)insteadofsubjectto(1).Thismeans that,if ri«1 (21) Xlmayhaveallvalues. Formostpurposes, therefore, thepredominantly reactive termination maybedefinedby ri«xi+1 (22) Asimilarsetofformulas maybeobtained bywritingblforXl,glforrl, and<p'for<P. 20.TheConducting WireBridgeasaTermination; Resistive Wire.81 Transmission-line measurements dependontheavailability ofastand ardterminal impedance. Theproperties ofsuchastandard necessarily Terminating Longline Asection: :b B ...-------8 t------~ fA c\ : Ietb d B FIG.20.1.Terminating sectionashalfofarectangle. includeterminal-zone effects. Whereas theseareanalyzed ingeneralin Chap.V,itisadvantageous toconsider simplestandards inthischapter. Forthetwo-wire openlineandtheshielded-pair lineastraightconducting wirebridgeisuseful.Itisanalyzed inthissection. Theconducting pistonordiskforuseincoaxialandothertypesoflineisconsidered in thenextsection. Theimpedance ofasectionoflength StofalonglineoflengthS»St, whenterminated inastraight conducting bridgeoflengthbequaltothe spacingoftheline,maybedetermined quiteaccurately ifthefollowing inequalities aresatisfied: SF»b2»a2 f35sF«1(1) (2) whereaistheradiusoftheconductors ofthelineandf30=271'/Ao.The methodconsistsintreating theterminating sectionasone-half ofalong andnarrowrectangle ofwireoflength2standwidthb,asshowninFig. 20.1.Theimpedance lookingtotherightfromABonthelonglineand ontherectangle isthesameifthedistance betweenthesidescandeof therectangle issufficiently greattomakeanycoupling between them insignificant. Thisisensuredby(1). Denoting theself-inductance ofatypicalsidejoftherectangle by Ljjandthemutualinductance between sidesiandjbyLij,thetotal Sec.20] THETERMINATED LINE 121 inductance oftherectangle ofwireisasfollows(notethatthemutual inductance ofmutually perpendicular sidesiszero): L=Lrr+Lee+Ldd+LJI+Ldi+Lid+Lee+Lee (3) Subject to(1),thecontribution toLbyLee+Leeisnegligible. By symmetry itfollowsthat (4) Theinductances in(4)areevaluated inRef.9,Chap.VI.Subjectto (1),thefinalexpression forLforwiresinairis L=~[b(sinh-1!!.-+~-/1+qi)+2stIn~+a-b](5)7rVo asb'\jb2 a where Vo=1/J.Lo=107/47rm/henry, aistheradiusofthewiresofthe longsides,andasistheradiusoftheshortsides.Theinductance per unitlengthofanidealuniform linehasbeenshowntobe 1b19= -In-7rVoa(6) subjecttothecondition b2»a2,whichisincluded in(1).Strictlythe squarerootoccurring in(5)maybereplaced byunityif(1)isimposed. However, byusingtheunrestricted formula 1 ble=-cosh-1-o7rVo 2a(7) fortheinductance perunitlengthofthelineandretaining thetermin a;/b2intheradicalin(5),thisformulaforLmaybegeneralized toapply approximately toallvaluesofb/a.With(6)or(7)theexternal induc tanceoftherectangle asgivenin(5)maybeexpressed asfollows: where andtLe=2Ls+2stlg+2LT Ls=_b_(sinh-1!!.-+~-~1+qi) 27rvo asb b2 b-aLT=--- 27rvo(8) (9) (10) Thetotalexternal inductance Leoftherectangle ismadeupoftwoparts: (1)thesumofthetwoinductances Lsoftheshortsidesoftherectangle, and(2)thetotalinductance ofthelongsidestreatedasatransmission lineandexpressed intheform2stIn(b/a)+2LT•Theterm2stIn(b/a) istheinductance ofauniform linewithconstant inductance perunit length,andLTisthecorrection fortheactualnonuniformity oftheinduc- tThesameformula forLTisderivedfromthegeneralintegral [Sec.4,Eq.(3)]in Chap.V,Sec.12. 122 TRANSMISSION-LINE THEORY [Chap.II tanceperunitlength. Theinternal inductance Liisnegligible, sothat L=Le+Li==Le. Sincetheshortendscandeofthelongrectangle inFig.20.1areby postulate sufficiently faraparttobeessentially uncoupled, asectionof length Btattheendofalongtransmission linehasaninductance given byone-half of(8),viz., L=Ls+Bel~+LT==Lsa+Btl~ where,bydefinition,(11) (12) istheapparent impedance oftheterminating wirebridgeifthelineis assumed tobeuniform andLsisthetheoretical, isolated impedance of thebridge. NotethatLsadiffersfromLsbyaterminal-zone inductance LTthatisnegative, indicating thattheinductance perunitlengthis greateralongauniform linethannearaterminating impedance atthe endoftheline.Notethattheinductance ofthebridgeapparently termi natingtheline,asdetermined bymeasurement andcalculation using(6) fortheentirelengthofline,including thesectionoflength Bt,isLsa• Sincetheresistance ofthewirebridgeis (13) where r~istheinternal resistance ofacylindrical conductor, itfollows thatthetheoretical impedance ofthebridgeis (14) whereas theapparent terminal impedance ofthebridgeattheendofan assumed uniform lineis (15) Theterminal functions 6s=Ps+jips,corresponding tothetheoretical impedance Zs,and6sadpsa+jipsa,corresponding totheapparent imped anceZsa,maybereadilydefinedfortheconducting wirebridge. HighlyConducting Bridge. Thenormalized resistance andreactance ofahighlyconducting wirebridgemaybeassumed tosatisfythefollow ingconditions: whereris«1 Rs-cPcXs TlB=Rc(1+cP:) Xs+cPcRs Xls=Rc(1+cP~)(16) (17a) (17b) Intheapproximate equalities in(17a,b)itisassumed thatcPcissuf ficiently smallsothat cP~«1. Sec.20] THETERMINATED LINE 123 With(16) Let_ 1th-12r1s P8-2"an 2+2+1==r18r1sXIs ..:F.._1t-1 -21Xlsi ..!...7r+t-1 ..!...7r+ 'J:'II-2"an-Iris+xiII-II -2"anXIII-2"XIs Xs=wLIIRs=br~(18a) (18b) (19) where r~istheresistance perunitlengthoftheconducting bridgeof radiusas. Foralow-loss lineinair,cPc==r/2wl=a/[3,whereristheresistance andlistheinductance perloopunitlength.Itfollowsthat(18a),with (17a)and(19),maybeexpressed intheform (20) Thefollowing newsymbols, whicharedimensionally lengthsinmeters, areintroduced conveniently fortheratiosin(20): Hence Similarly Sincemil==2~sks==~s Ps=a(ms-ks) 7rXs+Rs7rwlk a2 «1>11=2+RccPcRc=2+RcII+7im Rc=_1_=vt:!.==[3 lylC v(21) (22) (23) (24) itfollowsthat,with 0:2/[32«1, (25) Theapparent terminal functions fortheapparent terminal impedance Zsa=Rsa+jXsaareobtained inthesamemanner as(20)and(25). Since itisnecessary merelytodefine kLsak k LsLTsa==T= s+T=T+T inordertoobtain psa=a(ms-ksa)=Ps-akT 7r af>sa=2"+(3ksa=«I>s+[3kT(26) (27) (28) (29) 124 TRANSMISSION-LINE THEORY [Chap.II Itisconvenient torepresent theequivalent lengthskaandkaaasfollows: where,withk=fabk_faab a2 aa-2 b by==-Ya==-a aa sinh-1Ya+y;l-V::-1-+"--y-;""""2 fa= cosh-1(y/2) sinh-1Ya+2y;1- 1 -VI+y;2 faa= cosh-1(y/2)(30) (31) (32) (33) 6.907 4.605 2.3022 ~-t----:':5--"lL,-0-------::-:lOO:=-------:1-:OOO=-------' 0 y FIG.20.2.Thefunctions faandf.aofy=bfa. Thefunctions fsandfsaareshownasfunctions ofyinFig.20.2inthe specialcasewhereallwiresareofthesamesize,sothataa=aand Ya=y.Itisseenthat fa==0.95 (34) whereas faaisconsiderably smallerandmuchlessconstant wheny~5. Evidently thetheoretical, equivalent reactive lengthkaofastraight con ductingbridgemadeofwireofequalradiusasthelineis ka==0,475b ==£ (35) However, sincetheinductance perunitlengthofthelineisnotconstant neartheterminating bridge,theapparent equivalent reactive lengthksa, asdetermined undertheassumption thatlisconstant, isappreciably smallerthankR• Sec.20] THETERMINATED LINE 125 Ifthebridgeisnotonlyequalinradiustothewiresofthelinebut alsomadeofthesamematerial, itfollowsthat ms=2Rs=br(36) sothat,with(22), Ps=ab(1-~) (37) If(35)isagoodapproximation, Ps==iab (38) Bychoosing theradiusasofthebridgesomewhat smallerthanthe radiusaofthelinewires,fsorfsamaybemadeequaltounityforany particular valueofbja.Therequired valueofasisobtained bysetting (32)or(33)equaltounityandsolvingforYs=bjas.Forsimplicity letYsbequitelarge,sothattheradicalin(32)or(33)maybesetequal tounityandy-;1neglected compared with1.Specifically let (39) (40) Thenys»1 f~sinh-1Ys-1 _ 1f~sinh-1Ys-2 s-~sh-l(yj2)- sa-cosh- 1(yj2) Forlargearguments theinversehyperbolic functions approach natural logarithms oftwicetheargument, sothat(40)becomes In2ys==1+InyIn2ys==2+Iny (41) orI2ys~1 I2ys~2(42) n-- n--y y Theresultsare Ys==1.359y Ys==7.389y (43) Hence as==0.736a forfs=1(44)as==0.135a forfsa=1 Iftheradiusofthetermination ischosentohaveoneofthesevalues andbjaislarge,sothatbja~5, fs=1bks=2, b(45) or fsa=1ksa=2, Iftheterminating wirediffersfromthetwo-wire lineinbothdiameter andmaterial, thefollowing formula istrue,subjecttotheindicated condition: (46) 126 TRANSMISSION-LINE THEORY [Chap.II whereu.andJI.pertaintothematerial oftheterminating wire.Simi larlythecomparable condition andtheresistance perunitlengthofthe lineare Accordingly!#ur. 1~-;-a-~10-=rt= - -v- 2 271'"a2uv m._2R._!!:-.r;;; b-rb-a.\}~(47) (48) Thevalueofa.tomakef.=1hasbeenshowntobea.=O.736a; thevaluetomakef.a=1isa.=0.139a. Withthesevalues m.=1.36r;;; b \}~or (49) Thus,ifthematerials canbesochosenthat itfollowsthat and.!!.!..-=0.54u.v. P.=jabor or1.!!.!.-=0.183u.-v. P.a=jab(50) (51) (52) Itistobenotedthatitisnotpossible forphysical reasonstomake P.=jaborP.a=jabifthetwo-wire lineiscopper,sinceamaterial with 1.85or5.45timestheconductivity ofcopperwouldberequired forthe terminating bridge. Nosuch material isavailable. Bymakinguseofthetheorydeveloped inthissection,theapparent terminal functions ofaconducting wirebridgemaybedetermined. Note thattheaccuracy ofthesedeterminations involves errorsoftheorderof magnitude oftheradiusofthewireinthemeasurement oflengthsonthe two-wire line. Inactualuseaconducting bridgeoftenmustbemovable alongan extended sectionoftwo-wire line.AsshowninChap.VI,aconducting bridgemayserveasaninductance common tothesections oflinecon tinuingineachdirection whenever thebridgeisnotexactlyattheend. Inordertoavoidthecoupled-circuit effectsandpreserve theimpedance ofthebridgeasaconstant independent ofitslocationalongtheline,itis oftenadvantageous tousetwobridgesseparated adistance A/4-k.a andconnected soastomovetogether intandem. Thisarrangement is illustrated inFig.20.3.AsshowninChap.III,Sec.6,asectionofline ofthislengthwhichisterminated inaconducting bridgebehaves essen tiallylikeaninsulator ofseveralhundred thousand ohms.Evidently thepresence ofsuchahighimpedance thatmovesalongandisalwaysin parallelwiththeverylowimpedance bridgeterminating thelinehasno Sec.211 THETERMINATED LINE 127 significant effectontheproperties ofthisbridge,andtheseremainessen tiallyunchanged asitismoved. Resistive Bridge. Ifitisdesiredtoterminate atwo-wire lineinits characteristic impedance inordertohaveamatched linethatmaybe used,forexample, forphasecomparisons (Chap.IV,Sec.13),abridge consisting ofastraight conductor madeofresistance wireorastraight conductor withasmallcarbonresistoratitscentermaybeused.Such abridgehasaninductive reactance thatisessentially thesameasifit werehighlyconducting. Sincethisisverysmallcompared withthe resistance, thecondition ofmatchisapproximated closelywhenthe resistance ofthebridgeisRs=Re• Terminating Tandem bridge bridge Activepartofline -TogeneratorInactivepartofline FIG.20.3.Tandem-bridge sectionoflinetoformmovable insulating support con tinuously inparallelwithmovable bridge. 21.Conducting PistonsandDisksasTerminations. Aperfectly con ducting pistoninacoaxiallineorashielded-pair lineandaperfectly conducting diskofinfiniteextentonanopen-wire lineareterminations thatmaybeanalyzed together usingthetheorem ofimages. Ineach casetheelectromagnetic fieldandthedistributions ofcurrentandcharge onthetransmission lineareunchanged iftheconducting pistonordisk atapointPalongthelineisremoved andthelinecontinued asageo metricimage,including thegenerator withitspolarity reversed. Since thepotential difference acrossthelineatPduetotheimagegenerator andlinemustalwaysbeequalandopposite tothatmaintained bythe actualgenerator andline,itfollowsthatthepotential difference acrossthe lineatPisalwayszero.Hencetheterminal function oftheimageline andoftheequivalent conducting pistonorinfinitediskis6s=0+j7r/2. Accordingly ahighlyconducting pistonforacoaxiallineorshielded-pair lineandaninfinitediskforanopen-wire linemaybeassumed tobe represented byps==0and<l>s=7r/2.Theseareconvenient foruseas standard terminations. Inpractice, aconducting diskofinfiniteextentisunavailable, andthe question arises:Howlargemustadiskbeinordertoapproximate an infiniteoneasatermination? Asuitable criterion maybeobtained by investigating therateofdecrease ofthemagnetic fieldalong,orofthe surfacecurrentin,aninfinitediskduetoapairofconductors separated adistance bandwithequalandopposite currents. Forsimplicity, the conductors maybeassumed infinitely long,sincethecontributions from sectionsofthelinewhicharefarfromthediskcompared withthespacing ofthewiresarenegligible. Intheplanethatisequidistant fromthetwo 128 THANSMISSION-LINE THEORY [Chap.II (la)conductors, theresultant magnetic fieldisradialandgivenapproximately by B().1mb rr=21rJlor2+b2/4 Intheplanecontaining thetwoconductors thefieldisapproximately Bo(r)~1m(11)_1mb ( )-21rJlor-b/2-r+b/2-21rJlor2_b2/4 Ib 1misthemaximum ofthesinusoidally distributed current alongthe transmission lineandtherefore thecurrent entering andleavingthe planefromthewires. Asuitable criterion ofcomparison isthemagnetic fieldonthedisk midway between itsjunction withthetwoconductors. Thisisgivenby (la)withr=o.Itis (2) (3) (4)Theratioofthefieldattwolocations atradiusrtothatatr=0is B(r)<b2/4 B(r=0)=r2±b2/4 Letitbeassumed thatthemagnetic fieldandthesurface-current density atrarenegligible compared withthefieldandsurface-current density atr=0whentheratioin(3)isnogreaterthan0.01.Thus b2/4 r2±b2/4;£0.01 Thiscanbesatisfied onlywhenb2/4issmallcompared withr2,sothat (4)reducesto ~;£0.04rb--:::;;0.2r -(5) Itfollowsthatthemaximum density ofsurfacecurrentontheinfinite diskataradialdistance rfromthecenterofthelineislessthan1percent ofthedensityhalfway between thetwoconductors ofthelineif r~5b (6) Itmaybeassumed that,ifthecurrentdensityonaninfinitediskat r=5bisonly1percentofthatatitscenter,theeffectofthepart,of thediskbeyond r=5bisofnogreatsignificance andmaybeomitted. Thatis,(6)isasatisfactory criterion fordetermining theradiusofadisk toterminate anopentwo-wire linewithp==0and==1r/2. 22.Terminations withNegative Attenuation Function orReflection Coefficient GreaterthanUnity.Aninteresting specialtermination isthat Sec.22] THETERMINATED LINE 129 forwhichtheterminal attenuation function psbecomes negative. Since themagnitude ofthecoefficient ofreflection isfs=e-2p.,itfollowsthat negative valuesofPscorrespond tovaluesoff8greaterthanunity.t Thegeneralformula forP8is (1) (2)ForP8tobenegative, rlsmustbenegative. Thatis, R8-cPcXs0 rIs==Re(l+cP~)< Evidently thisispossible onlywhenXsispositiveandcPc~0,thatis, Rs<cPcX. (3) cPcXs,notthatR8=0.)Thesecond (NotethatrI.=0meansthatRs condition in(3)isequivalent to Rs<cPc=Xc Xs Rc(4) Forexample, onalow-loss linewithnegligible leakage conductance, cPc==a/{3=r/2wl,sothat,sinceXisinductive, orRsr wL.<2wl k8=L.>2R.(=R.•foratwo-wire line) lsrrt(5) (6) tThepossibility ofcoefficients ofreflection thatexceedunityevenslightly may appearparadoxical tothoseaccustomed totreatthecharacteristic impedance asa pureresistance. WhenZe=Re,sothatitisapureresistance, asforthedistortion lessorlosslessline,forwhich cPe=0,thecoefficient ofreflection canneverexceed unityinmagnitude, anditispossible toseparate notonlyincident andreflected volt agesandcurrents butalsoincident andreflected powers. WhenZeiscomplex, asin thelossyline,thecurrentandvoltagealonganinfinitely longorperfectly matched line arenotexactlyinphase. Although itisstillpossible toidentify traveling wavesof incident andreflected voltages andcurrents, aseparation intoincident andreflected powers isnolongerpossible owingtotheappearance ofcross-product terms. A termination withareflection coefficient ofunityreflectsincident wavesofcurrentand voltagewiththesamephasedifference. Ifthephasedifference ofthereflected cur rentandvoltage islC88thanfortheincident waves,thereflection coefficient must exceedunity.Thepowerfactorfordissipation inthelinehasbeenimproved. Inthe caseoftheusuallow-loss lines,cPeinZe=Re(l-jcPc)isverysmall,sothattheinci- dentvoltageandcurrentdifferbytheverysmallangletan-1.pe =cos-1(l/VI+cP;). Itisreadilyverified fromSec.5,Eq.(7),that,foratermination withR.=0and X.=-Xe/2=cPeRe/2,r.=VI+cP;,whichexceedsunityslightly. Notethat achangeinpowerfactorfromcostan-1cPetounitycorresponds toanincrease in magnitude byafactorVI+cP;.Thecontribution tothepowerbytermsinvolvin/!; .pcisreferredtoinChap.IV,Sec.6. 130 TRANSMISSION-LINE THEORY [Chap.II Foratermination onanopen-wire lineconsisting ofaconducting wire bridgeofthesameradiusasthewiresoftheline,Sec.20,Eq.(22),gives where(7) Notethatr;istheinternal resistance perunitlengthofthebridge,and riistheinternal resistance perunitlengthofeachoftheconductors of theline.Hencetherequirement fornegativep.,namely, reducesto(8) (9) sincef.isslightlylessthanunity.Thatis,theresistance perunitlength ofeachconductor ofthetwo-wire linerimustbeatleastdoublethe resistance perunitlengthoftheterminating bridge. Using thecondition becomes. 1r;; r'=21ra'\j2;;(10) or(lla) (lIb) (12) (13) sothatClearly, ifthelineisofcopperimmersed inair,nobridgewithnegative P.isavailable. However, withabrasslineandacoppertermination, m;=(~)2=~=g=021<023ri(f.11.5.65 . . m.==0.46b and,withf.==0.96, P.==ab(0.46 -0.48)==-0.02ab (14) Evidently thehighertheresistance ofthelinewirescompared withthe resistance oftheconducting bridge,themorenegative P.maybe.Note that,sincef.aissmallerthanf.,P.aisalwayslessnegative thanP.fora givenvalueofr;/ri•Inparticular, p.maybenegative, andP.a,positive. PROBLEMS 1.Asectionoftransmission lineistwowavelengths longandterminated inaload atz=8equaltoZc/2.Thegenerator atz=0hasanemfof10voltsandanimped anceequalto2Zc.Thelossesintherelatively shortlinearenegligible. Writeout thefirstfourtermsintheinfiniteseriesforthevoltageatadistance ofone-quarter wavelength fromthegenerator. Expressthereflection coefficients numerically. THETERMINATED LINE 131 2.Express thevoltageatthesamepointinthelinedescribed inthepreceding problem asthesumofanincident andareflected wave.Givenumerical values. 3.Express thevoltageonthelinedescribed inProb.1intermsofasinglewave traveling towardtheloadwithvariable phasevelocity. Plottheamplitude ofthe voltagewavealongtheline;alsoplotthephasevelocity. 4.Calculate thelineconstants T,l,andc(gisnegligible) foracoaxiallinecon sistingofasilverinnerconductor ofNo.20wireandatinouterconductor ofinner radius0.254cmatafrequency of3,000Mc/sec. Thedielectric ispolystyrene (Er=2.6;IJ.r=1).Usethefollowing conductivities: (J"(silver)=6.14X107mhos/m; (J"(tin)=0.87X107mhos/m. 5.Determine theattenuation constant a,thephaseconstant {j,thecharacteristic resistance Re,thedistortion factor cPe,andthewavelength forthelineinProb.4. 6.Calculate thelineconstants T,t,andc(gisnegligible) foratwo-conductor line madeofbrasswithaconductivity of1.22X107mhos/m. Thetwoconductors are identical andMradius0.04in.spaced1embetween centersinair.Thefrequency is750Me/sec. 7.Calculate theattenuation constant, thephaseconstant, thecharacteristic resist ance,andthedistortion factorforthelineinthepreceding problem. 8.Determine thephasevelocity andthegroupvelocity forthelinedescribed in Probe6,usingthemoreaccurate formulas giveninSec.13. 9.Calculate afewpointsandplotcurvesofRe==~forthefollowing trans mission linesfortheindicated rangesofb/a.Usesemilog paper. (Foropen-wire linesbisthedistance between centersofadjacent wires,andaistheradiusofeach wire;forcoaxiallinesbistheinnerradiusoftheouterconductor, andatheradiusof theinnerconductor.) (a)Coaxiallinewithairasdielectric. Range,1~b/a~600. (b)Coaxiallinewithpolystyrene (Er=2.6)asdielectric. Rangeasin(a). (c)Two-wire linewithairasdielectric. Rangeasin(a).Notethattheloga rithmicformula issatisfactory onlywhenb2»a2• (d)Four-wire linewithdiagonal conductors inparallelandwithairasdielectric. Range,5~b/a~600. 10.Deriveaformula fortheminimum valueoftheattenuation constant afora coaxiallinewithanouterconductor offixedradiusastheradiusoftheinnerconductor isvaried. Plotacurveofaasafunction oftheratiooftheradiusoftheouterto thatoftheinnerconductor neartheminimum todetermine whether thisissharp orflat. 11.Determine thecomplex reflection coefficient rainpolarformandtheterminal functions paand<l>aforthefollowing impedances Za(ohms)terminating alow-loss linewhhZe==Re=300ohms: (a)Za=3,000+jO;(b)Za=0+j3,000; (c) Za=3,000(1+j);(d)Za=0+j300;(e)Za=200+jlOO;(f)Z.=0;(g)Za=00. 12.Anapparent impedance Zaa=75+j200ohmsterminates alow-loss coaxial lineofcharacteristic impedance Ze==Re=50ohms.Whatarethecorresponding valuesoftheapparent reflection coefficient raa;theapparent terminal functions Paa, <l>aa,and;a;andtheapparent admittance Y.a? 13.Inanexperimental determination oftheimpedance ofanantenna, thefollowing typicaldatawereobserved onalineforwhichZe==Rc=50ohms: Electrical lengthofantenna, radians. ... .0.22 Measured P,nepers '"....... . .0.001 Measured <1>,radians................... 0.171.45 0.870 1.572.00 0.230 3.002.28 0.147 3.14 132 TRANSMISSION-LINE THEORY [Chap.II Calculate RandXfortheantennaateachlength,andcheckwiththecirclediagram. 14.Thecapacitance ofacapacitor is15p.p.f.Itisconnected astheterminal impedance ofalineofcharacteristic impedance Zc==Rc=400ohms.Whatare theterminal functions pand<Pat150Me/sec? Assumethecapacitor tobewithout loss. 15.Atwo-wire lineismadeofbrass!in.indiameter, with2embetween centers. Acopperrodofthesamediameter isusedasashort-circuiting barattheloadend. Determine TI,Xl,andforthisbarat300Me/sec. (Use (T=1.5X107mhos/m for brassand (T=5.65X107mhos/m forcopper.) A Z'in-; Z.~ BI<t---- W ~I FIG.1.2.Isolated sectionoftrans mission line.Theimpedance Z.nis notthesameasZ.inFig.1.1,owing totheendeffect.CHAPTER III IMPEDANCE ANDADMITTANCE 1.Normalized InputImpedance andAdmittance ofaTerminated SectionofLine.Atransmission lineextendsfromz=0toz=S+Se, wheresi»b2•Theimpedance terminating thelineatz=sisZs=I/Ys• Whatistheimpedance Z.oradmittance Y.terminating thelineatz<s'? Alternatively, whatistheimpedance Zin=Z.oradmittance Yin=Yz lookingintothesectionoflinetotheright(Fig.1.1)ofthepointz, A£---_----:z::;..:!:z'---.-::: :--- 2..::..8---_:-~~ z=O z,B z=s z=s+s, FIG.1.1.Terminated transmission line. expressed intermsofZsorYsandthelength 8-Z=wofthesection '? Notethatthisisnotthesameastheimpedance Zinoftheisolatedsec tionoflineoflengthwterminated inZsshowninFig.1.2.Theabsence ofthelinetotheleftofthepointsABinFig.1.2ascompared with Fig.1.1involves anendeffectandcou plingeffectinaterminal zonenearAB. Ifapotential difference ismaintained acrossABinFig.1.1,equalandoppo sitecurrents areinthetwoconductors atz.Letphasebereferredtothecur rentinconductor 1asheretofore. The inputimpedance oradmittance ofthe linetotherightofAB(whichisequaltotheloadimpedance oradmit tanceterminating thelinetotheleftofAB)isdefinedtobe (1) Notethat,sinceABisbydefinition farfrombothendscompared with thewavelength, theycomponent ofthevectorpotential iszeroorpracti callyzero(Ay==0)nearAB,sothatEy=-d4-/dy -jwAy==-d4-/dy. Hence v.==4-1z-4-2.=24-Jz==fobEvdy I~~(2) 134 TRANSMISSION-LINE THEORY [Chap.III Anumber offormulas maybeobtained forZinandYincorresponding totheseveraldifferent representations ofcurrentandvoltagealonga transmission line.Eachhascertainadvantages inspecialcases.Thus, forexample, theexponential solutions [Chap.II,Sec.5,Eqs.(10)and (11)]leadtothefollowing formulas forthenormalized inputimpedance andadmittance: Zin1+r8e-2oyw Zlin==Zc=1 -r8e-2oyw (3) Foralosslessline(a8=0)thesereduceto Zin1+r8ei(~.-2I3w) Zlin=Rc=1 - r8ei(~.-2I3w) Notethat .~Z18-1Y18-1 r3=r8e}=Z18+1= -Y18+1 where Zls==Z8/Zc,Y18==Y8/YC,and-r8eN'=-r~ (5) (6) Thecomplex hyperbolic formsoftheinputimpedance areobtained fromChap.II,Sec.8,Eqs.(2)and(3).Theyare AlternativelyZinZ8coshjW+ZcsinhjW Zlin==Zc=Z8sinhjW+ZccoshjW YinY8coshjW+YcsinhjW Ylin==Yc=Y8sinhjW+YccoshjW Z18cothjW+1Z18+tanhjW Zlin=Z18+cothjW=Z18tanhjW+1 Y18cothjW+1Y18+tanhjW Ylin=Y18+cothjW=Y18tanhjW+1 Forthelosslessline(a8=0)(7a) (7b) (8a) (8b) Zlin=Zin=1 -jZl."cot{3w=Z18~jtan{3w (9a) Rc[Z18-Jcot{3w1+JZ18tan{3w Yin1 -jY18cot{3w Y18+jtan(3w (9b)) Ylin=Gc=Y18-jcot{3w=1+jY18tan(3w Formanypurposes involving dissipative loadsonlow-loss lines,(9a,b) areadequate. Thecompletely hyperbolic formissimpleandespecially convenient for deriving explicitformulas forRinandXin,GinandBin.Thedesiredfor mulasareobtained fromChap.II,Sec.8,Eqs.(15)and(16).Theyare Zlin=coth(jW+68)=coth[Caw+P8)+j({3w+4>8)](lOa) Ylin=coth(jW+6;)=coth[Caw+P8)+j({3w+4>;)](lOb) Sec.1] IMPEDANCE ANDADMITTANCE 135 Forconvenience, letthefollowing notation beintroduced: Aw==aw+P. Fw==~w+4>. F~==~w+4>~ 8'8.7r'4> 7r•=•-J2"4>.=•-2" Notethat,with(l1a,b),(lOa)becomes %Un=coth(Aw+jFw)Ylin=coth(Aw+jF~)(l1a) (lIb) (12) (13) Theseformulas areliketheexpressions previously derived forthe normalized terminal impedance Zl.andadmittance YIs,namely, %Is=coth(P.+j4>.)Yls=coth(P.+j4>~) (14) Obviously (11)mustreduceto(12)whenever w=O. Since (15) itfollowsthatthenormalized inputresistance andreactance, conductance andsusceptance are sinh2Aw rlin=cosh2Aw-cos2Fw sinh2Aw glin=cosh2Aw-cos2F:U -sin2Fw Xlin=cosh2Aw-cos2Fw -sin2F'.bl'= w oncosh2Aw-cos2F:U Inverserelations aresinhAwcoshAw sinh2Aw+sin2Fw sinhAwcoshAw sinh2Aw+sin2F~ -sinFwcosFw sinh2Aw+sin2Fw -sinF~cosF:U sinh2Aw+sin2F:U(16a) (16b) (17a) (17b) A_Ith-l 2rlin w-"2an 2+2+1runXlin F-1-1 -2Xlin w-"2tan 2+2 _1runXlin=1tanh- l2glin "2 g~in+b~in+1 F'-1t-1-2blin • -"2an 2+b2_1glinlin(18) (19) Theseformulas havethesameformandobeythesamesignconventions asthosederived previously forZ.andY..Theydifferonlyinthe appearance ofAwinsteadofP.andFworF:Uinsteadof4>.or4>~.When w=0,Aw=P.,Fw=4>.,andF:U=4>~. Thenormalized inputresistance andreactance areexpressed numeri callyinTables1.1and1.2asfunctions ofAwandFw•Theinputphase function Fwisrepresented asaftinction oftheinputattenuation func tionAwinFig.1.3withrlinasparameter andinFig.1.4withXlinas parameter. Withthesubstitution ofthesymbol 4>.forFw,P.forAw, rIBforrlin,andXl.forXlin,thesetablesandcurvesapplytothenormalized terminal impedance %Is=rIs+jXl•. 136 TRANSMISSION-LINE THEORY [Chap.III Theanalogous significance ofA8andF8fortheinputimpedance and P8and4>8fortheterminal impedance revealsthesimpleandfundamental partsplayedbytheterminal functions. Theattenuation function P8 playsthesameroleforthetermination asdoesasfortheline.Indeed P8!aisanequivalent lengthoflineforthetermination fromthepointof viewofattenuation. Similarly thephasefunctions 4>8and 4>~playthe 90°-0N IId 80°c,,;.- 70° 60° i50° ~ ~ II ~~40° 0.20.4 0.6 0.8 1.0 1.2 1.4 Aw=otw+ps FIG.1.3.Contours ofconstant rlinwithFw=f3w+<1>.andAw=aW+P.asvariables. samepartindescribing theeffectofthetermination asdoes138forthe line.Theratio4>8!13or4>~!13isanequivalent lengthoflineforthetermi nationinsofarasphaseshiftisconcerned. Theequivalent lengthsfor attenuation andforphaseshiftarenotalikeunlessZ.orY8isitselfasec tionoftransmission lineoraforthesectionismodified sothatP8!a=4>.!13 orP8!a=4>~!13.InthisspecialcaseitispossibletoreplaceZ8orY.bya sectionoftransmission linewhichisitsequivalent bothinphaseshift andinattenuation. Itisreadilyverifiedthat,justasforP.and4>8or4>~intermsofrl.and Xl.orgl8andb18,thecontours ofconstant A8andthecontours ofconstant FworF~arefamilies oforthogonal circles. Theintercepts, centers,and radiiareasfollows; Sec.1] IMPEDANCE ANDADMITTANCE lOOD....---...--.....----r---.----,,----r--T- .... (Xlinnegativewhen0<F,<goo,positivewhen90°<F,<180°)137 OO~~~~~~~~=:I==--LJo 0.60.8 1.4 Aw=«w+ps FIG.1.4.Contours ofconstant XhnwithFw={lw+<1>.andAw=aW+P.asvariables. Forcirclesofconstant Aw: I hIXlin=0ntercepts areatrlin=tanhAw,cotAwblin=0 C h2AIXlin=0entersareatrlin=cot wb.-0 hn- Radiihavemagnitudes equaltocsch2Aw Forcirclesofconstant FwandF~: {Xlin= -cotFw(ortanFw)Intercepts areatrlin=0b.- _ t F'( tF')hn-co woranw {Xlin= -cot2FwCenters areatrlin=0b.- _ t 2F'hn-co w Radiihavemagnitudes equaltoIi~:~;~~: Thesamecirclediagrams previously constructed forP.and<1>.intermsof rl.andX18,P.and ~intermsofghandbIBmaybeusedforAw=aW+P. 138 TRANSMISSION-LINE THEORY TABLE1.1 sinh2Aw Tlin=cosh2Aw-cos2Fw[Chap.III ~0° 0.5° 1° 2° 3° 4° 5°Aw 180° 179.5° 179° 178° 177° 176° 175° 0.0............... 0.000 0.000 0.000 0.000 0.000 0.003...............9.524 2.441 1.091 0.6155 0.3945 0.004...............12.50 3.236 1.451 0.8195 0.5256 0.005...............15.15 4.016 1.808 1.022 0.6562 0.01100.0 57.14 24.69 7.576 3.521 2.014 1.300 0.02 50.01 42.12 28.38 12.35 6.371 3.800 2.502 0.03 33.36 30.79 24.91 14.16 8.247 5.207 3.534 0.04 25.03 23.91 21.02 14.20 9.227 6.194 4.355 0.05 20.03 19.45 17.86 13.46 9.558 6.800 4.961 0.06 16.68 16.34 15.38 12.47 9.479 7.101 5.370 0.07 14.30 14.09 13.47 11.46 9.180 7.185 5.616 0.08 12.52 12.38 11.96 10.52 8.776 7.122 5.734 0.09 11.14 11.04 10.74 9.688 8.332 6.968 5.758 0.10 10.03 9.957 9.736 8.944 7.880 6.756 5.710 0.11 9.127 9.071 8.903 8.294 7.447 6.517 5.616 0.12 8.373 8.330 8.200 7.722 7.040 6.266 5.491 0.13 7.736 7.702 7.599 7.218 6.662 6.014 5.346 0.14 7.189 7.162 7.079 6.770 6.312 5.767 5.191 0.15 6.716 6.694 6.627 6.373 5.992 5.530 5.031 0.16 6.303 6.285 6.229 6.019 5.698 5.304 4.870 0.17 5.939 5.924 5.877 5.700 5.429 5.090 4.712 0.18 5.616 5.603 5.564 5.414 5.182 4.889 4.559 0.19 5.327 5.316 5.282 5.154 4.955 4.701 4.410 0.20 5.067 5.057 5.029 4.919 4.746 4.524 4.267 0.22 4.619 4.612 4.590 4.5069 4.3750 4.2030 4.001 0.24 4.246 4.241 4.224 4.1599 4.0570 3.9214 3.7600 0.26 3.933 3.928 3.915 3.8643 3.7826 3.6740 3.5433 0.28 3.664 3.661 3.650 3.6096 3.5436 3.4554 3.3483 0.30 3.433 3.430 3.421 3.3881 3.3341 3.2615 3.1728 0.32 3.231 3.2286 3.222 3.1941 3.1494 3.0890 3.0148 0.34 3.054 3.0517 3.046 3.0229 2.9855 2.9348 2.8720 0.36 2.897 2.8952 2.890 2.8709 2.8393 2.7962 2.7428 0.38 2.757 2.7557 2.751 2.7350 2.7081 2.6713 2.6254 0.40 2.632 2.6308 2.627 2.6131 2.5899 2.5582 2.5186 0.45 2.3702 2.3694 2.3669 2.3569 2.3406 2.3181 2.2899 0.50 2.1640 2.1634 2.1615 2.1543 2.1423 2.1259 2.1051 0.55 1.9979 1.9975 1.9960 1.9907 1.98171.9693 1.9535 0.60 1.8620 1.8617 1.8606 1.8564 1.8495 1.8399 1.8278 0.65 1.7493 1.74901.74821.74491.7394 1.73191.7223 0.70 1.6546 1.6544 1.6537 1.65111.6468 1.6407 1.6331 0.75 1.5744 1.5743 1.5737 1.5716 1.5681 1.5632 1.5569 0.90 1.0 1.3130 1.3130 1.3127 1.3119 1.3104 1.3084 1.3059 1.2 1.5 1.1048 1.1048 1.1047 1.1045 1.1041 1.1036 1.1029 2.0 1.0373 1.0373 1.0373 1.0372 1.0371 1.0369 1.0367 Sec.1] IMPEDANCE ANDADMITTANCE TABLE1.1(Continued) sinh2Aw Tlin=cosh2Aw-cos2Fw139 ~6080100150200250300 Aw 1740172017001650160015501500 0.0 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.003 0.004 0.005 0.4566 0.2578 0.1657 0.07462 0.04273 0.02799 0.02000 0.01 0.9070 0.5136 0.3305 0.1491 0.08541 0.05596 0.03998 0.02 1.766 1.012 0.6547 0.2969 0.1704 0.1118 0.07989 0.03 2.539 1.481 0.9667 0.4422 0.2547 0.1672 0.1196 0.04 3.197 1.910 1.261 0.5839 0.3377 0.2222 0.1592 0.05 3.731 2.290 1.534 0.72080 0.41919 0.27655 0.19836 0.06 4.139 2.618 1.782 0.85203 0.49878 0.33009 0.23716 0.07 4.435 2.893 2.003 0.97684 0.57618 0.38269 0.27551 0.08 4.633 '3.116 2.197 1.0946 0.65108 0.43422 0.31332 0.09 4.751 3.292 2.364 1.2048 0.72330 0.48459 0.35055 0.10 4.803 3.424 2.505 1.3071 0.79258 0.53366 0.38714 0.11 4.806 3.518 2.621 1.40130.85875 0.58132 0.42300 0.12 4.771 3.580 2.715 1.4874 0.92168 0.62750 0.45810 0.13 4.709 3.615 2.788 1.5655 0.98130 0.67214 0.49241 0.14 4.627 3.627 2.843 1.6356 1.0375 0.71513 0.52583 0.15 4.532 3.622 2.8823 1.6983 1.0903 0.75648 0.55840 0.16 4.429 3.601 2.9075 1.7536 1.1397 0.79611 0.59004 0.17 4.321 3.569 2.9206 1.8021 1.1857 0.83401 0.62073 0.18 4.211 3.5287 2.9237 1.8440 1.2283 0.87017 0.65045 0.19 4.100 3.4810 2.9181 1.8799 1.2677 0.90455 0.67917 0.20 3.9910 3.4283 2.9053 1.91011.3038 0.93719 0.70689 0.22 3.7792 3.3137 2.8632 1.9555 1.3671 0.99728 0.75930 0.24 3.5802 3.1930 2.8055 1.9835 1.4191 1.0506 0.80762 0.26 3.3959 3.0719 2.7382 1.9973 1.4608 1.0974 0.85191 0.28 3.2263 2.9534 2.6654 1.9997 1.4934 1.1381 0.89226 0.30 3.0709 2.8395 2.5903 1.9930 1.5179 1.1732 0.92878 0.32 2.9289 2.7315 2.5150 1.9794 1.5355 1.2029 0.96165 0.34 2.7990 2.6296 2.4409 1.9604 1.5471 1.2279 0.99106 0.36 2.6803 2.5340 2.3688 1.93751.5535 1.24861.0172 0.38 2.5716 2.4445 2.2993 1.9117 1.5557 1.2653 1.0403 0.40 2.4719 2.3609 2.2329 1.8840 1.5543 1.2785 1.0605 0.45 2.2564 2.1756 2.0805 1.81021.5389 1.2989 1.1001 0.50 2.0803 2.0199 1.9477 1.7358 1.5124 1.3054 1.1267 0.55 1.93471.8885 1.8326 1.6644 1.48001.3021 1.1430 0.60 1.81311.7771 1.7331 1.5979 1.4450 1.2925 1.1517 0.65 1.7108 1.6821 1.64701.5372 1.4096 1.2788 1.1546 0.70 1.6238 1.6007 1.5722 1.4821 1.3751 1.2627 1.1535 0.75 1.5494 1.5306 1.5072 1.4325 1.3422 1.2455 1.1495 0.90 1.0 1.3027 1.2949 1.2850 1.2523 1.2105 1.1627 1.1118 1.2 1.5 1.1021 1.1001 1.0975 1.0887 1.0770 1.0629 1.0471 2.0 1.0365 1.0357 1.0349 1.0321 1.0282 1.0234 1.0180 140 TRANSMISSION-LINE THEORY TABLE1.1(Continued) sinh2Aw Tlin=cosh2Aw-cos2Fw[Chap.III ~F·I350400450500550600 Aw~, 145014001350130012501200 0.0 0.000 0.000 0.000 0.000 0.000 0.000 0.003 0.004 0.005 0.01520 0.01 0.03039 0.02420 0.02000 0.01704 0.01490 0.01333 0.02 0.06073 0.04837 0.03998 0.03407 0.02980 0.02666 0.03 0.09100 0.07250 0.05994 0.05108 0.04468 0.03998 0.04 0.12115 0.09655 0.07983 0.06805 0.05954 0.05328 0.05 0.15109 0.12049 0.099672 0.084987 0.074364 0.066558 0.06 0.18084 0.14431 0.11943 0.10187 0.089155 0.079810 0.07 0.21033 0.16798 0.13909 0.11868 0.10390 0.093031 0.08 0.23953 0.19147 0.15864 0.13543 0.11860 0.10621 0.09 0.26841 0.21478 0.17808 0.15209 0.13324 0.11935 0.10 0.29694 0.23787 0.19738 0.16867 0.14782 0.13245 0.11 0.32506 0.26072 0.21652 0.18513 0.16232 0.14550 0.12 0.35275 0.28331 0.23549 0.20149 0.17674 0.15848 0.13 0.37999 0.30562 0.25430 0.21773 0.19109 0.17141 0.14 0.40673 0.32764 0.27290 0.23384 0.20534 0.18427 0.15 0.43298 0.34934 0.29131 0.24981 0.21950 0.19706 0.16 0.45868 0.37072 0.30951 0.2656.4 0.23355 0.20977 0.17 0.48383 0.39176 0.32748 0.28132 0.24750 0.22241 0.18 0.50842 0.41243 0.34522 0.29684 0.26133 0.23496 0.19 0.53240 0.43273 0.36271 0.31219 0.27504 0.24742 0.20 0.55578 0.45266 0.37995 0.32736 0.28863 0.25979 0.22 0.60070 0.49133 0.41365 0.35718 0.31543 0.28425 0.24 0.64308 0.52835 0.44625 0.38623 0.34167 0.30830 0.26 0.68289 0.56369 0.47770 0.41447 0.36733 0.33190 0.28 0.72013 0.59732 0.50798 0.44188 0.39238 0.35506 0.30 0.75482 0.62921 0.53704 0.46843 0.41679 0.37773 0.32 0.78700 0.65938 0.56489 0.49410 0.44056 0.39990 0.34 0.81675 0.68782 0.59152 0.51887 0.46366 0.42157 0.36 0.84414 0.71457 0.61691 0.54274 0.48607 0.44270 0.38 0.86924 0.73965 0.64107 0.56568 0.50778 0.46329 0.40 0.89221 0.76313 0.66404 0.58773 0.52881 0.48334 0.45 0.94084 0.81506 0.71630 0.63888 0.57829 0.53103 0.50 0.97847 0.85817 0.76159 0.68456 0.62342 0.57521 0.55 1.0069 0.89349 0.80050 0.72504 0.66432 0.61593 0.60 1.0278 0.92208 0.83365 0.76070 0.70120 0.65326 0.65 1.0427 0.94498 0.86172 0.79195 0.73430 0.68735 0.70 1.0528 0.96311 0.88535 0.81921 0.76388 0.71836 0.75 1.0591 0.97729 0.90515 0.84293 0.79025 0.74648 0.90 ....... 1.0028 0.94681 0.89670 0.85293 0.81558 1.0 1.0604 1.0107 0.96403 0.92149 0.88369 0.85094 1.2 •••• 0••1.0154 0.98368 0.95387 0.92664 0.90247 1.5 1.0300 1.0125 0.99505 0.97818 0.96236 0.94797 2.0 1.0120 1.0057 0.99933 0.99302 0.98697 0.98136 Sec.1] IMPEDANCE ANDADMITTANCE TABLE1.1(Continued) sinh2Aw Tli"=cosh2Aw-cos2Fw141 ~65° 70° 75° 80° 85°90°Aw 115° 110° 105° 100° 95° 0.0 0.000 0.000 0.000 0.000 0.000 0.000 0.003 0.004 0.005 0.01 0.01217 0.01132 0.01072 0.01031 0.01008 0.009999 0.02 0.02434 0.02264 0.02143 0.02062 0.02015 0.02000 0.03 0.03651 0.03396 0.03214 0.03092 0.03022 0.02999 0.04 0.04866 0.04527 0.04285 0.04122 0.04029 0.03998 0.05 0.060791 0.056560 0.053537 0.051509 0.050341 0.049960 0.06 0.072903 0.067836 0.064215 0.061785 0.060386 0.059929 0.07 0.084993 0.079094 0.074878 0.072049 0.070419 0.069887 0.08 0.097051 0.090327 0.085520 0.082294 0.080435 0.079828 0.09 0.10908 0.10154 0.096145 0.092524 0.090438 0.089756 0.10 0.12108 0.11273 0.10675 0.10274 0.10042 0.099670 0.11 0.13303 0.12388 0.11732 0.11292 0.11039 0.10956 0.12 0.14495 0.13499 0.12787 0.12309 0.12033 0.11943 0.13 0.15681 0.14608 0.13839 0.13322 0.13025 0.12927 0.14 0.16863 0.15711 0.14887 0.14333 0.14013 0.13909 0.15 0.18039 0.16811 0.15932 0.15341 0.15000 0.14888 0.16 0.19209 0.17907 0.16973 0.16345 0.15983 0.15865 0.17 0.20374 0.18997 0.18010 0.17346 0.16963 0.16838 0.18 0.21532 0.20083 0.19043 0.18344 0.17940 0.17808 0.19 0.22683 0.21163 0.20072 0.19337 0.18913 0.18775 0.20 0.23827 0.22237 0.21095 0.20327 0.19883 0.19737 0.22 0.26094 0.24369 0.23129 0.22293 0.21810 0.21652 0.24 0.28329 0.26475 0.25140 0.24240 0.23720 0.23550 0.26 0.30530 0.28554 0.27129 0.26167 0.25611 0.25429 0.28 0.32695 0.30604 0.29094 0.28074 0.27484 0.27290 0.30 0.34823 0.32623 0.31033 0.29958 0.29335 0.29131 0.32 0.36911 0.34611 0.32946 0.31818 0.31164 0.30950 0.34 0.38960 0.36566 0.34831 0.33654 0.32971 0.32748 0.36 0.40966 0.38493 0.36686 0.35463 0.34754 0.34521 0.38 0.42929 0.40372 0.38511 0.372"46 0.36511 0.36271 0.40 0.44849 0.42221 0.40305 0.39001 0.38244 0.37995 0.45 0.49450 0.46678 0.44648 0.43262 0.42455 0.42190 0.50 0.53763 0.50894 0.48782 0.47334 0.46489 0.46212 0.55 0.57788 0.54862 0.52698 0.51209 0.50339 0.50052 0.60 0.61524 0.58581 0.56393 0.54882 0.53997 0.53705 0.65 0.64980 0.62054 0.59867 0.58352 0.57461 0.57167 0.70 0.68164 0.65284 0.63120 0.61616 0.60729 0.60437 0.75 0.71090 0.68280 0.66159 0.64678 0.63804 0.63515 0.90 0.78453 0.75956 0.74045 0.72697 0.71895 0.71630 1.0 0.82335 0.80094 0.78364 0.77136 0.76403 0.76159 1.2 0.88169 0.86450 0.85104 0.84140 0.83559 0.83366 1.5 0.93534 0.92469 0.91624 0.91011 0.90639 0.90515 2.0 O.976:~5 0.97206 0.96861 0.96609 0.96455 0.96403 142 TRANSMISSION-LINE THEORY TABLE1.2 -sin2Fw Xlin=cosh2Aw-cos2Fwt{Chap.III ~0 0.5° 1° 2° 3° Aw 180° 179.5° 179° 178° 177° 0.0 ............. .-57.21 -28.59 -19.07 0.003 ..0•••••••••• •-55.40 -28.38 -19.01 0.004 ............... -54.53 -28.22 -18.96 0.005 .............. -52.88 -28.02 -18.90 0.01 0 -49.86 -43.09 -26.42 -18.40 0.02 0 -18.37 -24.75 -21.53 -16.64 0.03 0 -8.95 -14.48 -16.45 -14.36 0.04 0 -5.21 -9.160 -12.37 -12.04 0.05 0 -3.39 -6.221 -9.376 -9.974 0.06 0 -2.371 -4.463 -7.229 -8.237 0.07 0 -1.750 -3.346 -5.690 -6.832 0.08 0 -1.344 -2.597 -4.568 -5.709 0.09 0 -1.065 -2.071 -3.734 -4.813 0.10 0 -0.8630 -1.688 -3.099 -4.091 0.11 0 -0.7137 -1.401 -2.609 -3.510 0.12 0 -0.5999 -1.181 -2.223 -3.037 0.13 0 -0.5111 -1.009 -1.915 -2.648 0.14 0 -0.4405 -0.8710 -1.665 -2.326 0.15 0 -0.3836 -0.7595 -1.460 -2.057 0.16 0 -0.3369 -0.6679 -1.290 -1.830 0.17 0 -0.2982 -0.5918 -1.147 -1.637 0.18 0 -0.2658 -0.5279 -1.027 -1.473 0.19 0 -0.2383 -0.4737 -0.9239 -1.331 0.20 0 -0.2148 -0.4273 -0.8353 -1.208 0.22 0 -0.1771 -0.3526 -0.69199 -1.0065 0.24 0 -0.1484 -0.2957 -0.58196 -0.85046 0.26 0 -0.1261 -0.2513 -0.49577 -0.72717 0.28 0 -0.1083 -0.2160 -0.42698 -0.62811 0.30 0 -0.09401 -0.1876 -0.37124 -0.54742 0.32 0 -0.08230 -0.1642 -0.32548 -0.48089 0.34 0 -0.07259 -0.1449 -0.28744 -0.42539 0.36 0 -0.06445 -0.1287 -0.25550 -0.37864 0.38 0 -0.05757 -0.1150 -0.22841 -0.33889 0.40 0 -0.05169 -0.1032 -0.20525 -0.30483 0.45 0 -0.04028 -0.08047 -0.16017 -0.23834 0.50 0 -0.03212 -0.06419 -0.12788 -0.19055 0.55 0 -0.02612 -0.05216 -0.10397 -0.15509 0.60 0 -0.02152 -0.04302 -0.085795 -0.12808 0.65 0 -0.01797 -0.03592 -0.071670 -0.10706 0.70 0 -0.01516 -0.03031 -0.060485 -0.090394 0.75 0 -0.01290 -0.02579 -0.051489 -0.076980 0.90 1.0 0 -0.006317 -0.01263 -0.025233 -0.037768 1.2 1.5 0 -0.001924 -0.003849 -0.0076912 -0.011521 2.0 0 -0.0006633 -0.001327 -0.0026514 -0.0039725 tXl\nisnegative for0°<F.<90°andpositive for180°>F.>90°. Sec.1] IMPEDANCE ANDADMITTANCE TABLE1.2(Continued) -sin2F1O XIin=cosh2A1O-cos2F10143 ~40 50 6080100 Aw 17601750174017201700 0.0 -14.30 -11.43 -9.515 -7.115 -5.671 0.003 -14.28 -11.42 0.004 -14.26 -11.41 0.005 -14.23 -11.39 -9.494 -7.106 -5.666 0.01 -14.02 -11.28 -9.429 -7.079 -5.652 0.02 -13.22 -10.86 -9.179 -6.971 -5.597 0.03 -12.07 -10.22 -8.791 -6.799 -5.507 0.04 -10.76 -9.443 -8.300 -6.572 -5.385 0.05 -9.448 -8.601 -7.143 -6.302 -5.237 0.06 -8.215 -7.752 -7.155 -5.999 -5.065 0.07 -7.119 -6.943 -6.565 -5.676 -4.877 0.08 -6.169 -6.197 -5.995 -5.345 -4.676 0.09 -5.359 -5.525 -5.458 -5.013 -4.468 0.10 -4.670 -4.925 -4.960 -4.687 -4.255 0.11 -4.090 -4.397 -4.505 -4.372 -4.042 0.12 -3.599 -3.935 -4.094 -4.073 -3.832 0.13 -3.183 -3.531 -3.723 -3.790 -3.627 0.14 -2.829 -3.177 -3.391 -3.525 -3.428 0.15 -2.527 -2.869 -3.094 -3.278 -3.2373 0.16 -2.268 -2.598 -2.829 -3.050 -3.0551 0.17 -2.044 -2.361 -2.592 -2.839 -2.8821 0.18 -1.850 -2.152 -2.380 -2.6443 -2.7185 0.19 -1.681 -1.967 -2.190 -2.4653 -2.5643 0.20 -1.533 -1.804 -2.0201 -2.3006 -2.4192 0.22 -1.2874 -1.5291 -1.7294 -2.0104 -2.1554 0.24 -1.0944 -1.3094 -1.4927 -1.7650 -1.9243 0.26 -0.94034 -1.1316 -1.2985 -1.5572 -1.7223 0.28 -0.81543 -0.98592 -1.1374 -1.3804 -1.5459 0.30 -0.71296 -0.86539 -1.0028 -1.2294 -1.3916 0.32 -0.62797 -0.76471 -0.88949 -1.0998 -1.2565 0.34 -0.55672 -0.67981 -0.79325 -0.98799 -1.1379 0.36 -0.49647 -0.60763 -0.71095 -0.89109 -1.0336 0.38 -0.44506 -0.54579 -0.64008 -0.80665 -0.94148 0.40 -0.40088 -0.49246 -0.57869 -0.73275 -0.85991 0.45 -0.31428 -0.38737 -0.45700 -0.58419 -0.69319 0.50 -0.25175 -0.31105 -0.36803 -0.47375 -0.56683 0.55 -0.20519 -0.25398 -0.30116 -0.38973 -0.46927 0.60 -0.16964 -0.21027 -0.24974 -0.32451 -0.39269 0.65 -0.14192 -0.17610 -0.20943 -0.27301 -0.33167 0.70 -0.11991 -0.14892 -0.17728 -0.23170 -0.28238 0.75 -0.10217 -0.12697 -0.15129 -0.19814 -0.24210 0.90 1.0 -0.050207 -0.062523 -0.074679 -0.098410 -0.12118 1.2 1.5 -0.015331 -0.019118 -0.022873 -0.030269 -0.037469 2.0 -0.0052880 -0.006597 -0.0078963 -0.010461 -0.012971 144 TRANSMISSION-LINE THEORY TABLE1.2(Continued) -sin2Fw Xlin=cosh2Aw-cos2Fw[Chap.III ~15° 20° 250300350 Aw 16501600155° 15001450 0.0 -3.7322 -2.7474 -2.1445 -1.7321 -1.4281 0.003 0.004 0.005 -3.7308 -2.7468 -2.1439 -1.7319 -1.4280 0.01 -3.7266 -2.7451 -2.1433 -1.7314 -1.4277 0.02 -3.7100 -2.7381 -2.1397 -1.7293 -1.4264 0.03 -3.6827 -2.7265 -2.1338 -1.7258 -1.4242 0.04 -3.6451 -2.7104 -2.1255 -1.7210 -1.4212 0.05 -3.5979 -2.6899 -2.1149 -1.7149 -1.4174 0.06 -3.5416 -2.6653 -2.1021 -1.7074 -1.4127 0.07 -3.4773 -2.6368 -2.0871 -1.6987 -1.4071 0.08 -3.4060 -2.6046 -2.0702 -1.6887 -1.4008 0.09 -3.3287 -2.5691 -2.0513 -1.6776 -1.3937 0.10 -3.2459 -2.5304 -2.0304 -1.6652 -1.3859 0.11 -3.1592 -2.4889 -2.0079 -1.6518 -1.3773 0.12 -3.0692 -2.4450 -1.9838 -1.6373 -1.3680 0.13 -2.9769 -2.3989 -1.9582 -1.6218 -1.3580 0.14 -2.8830 -2.3509 -1.9312 -1.6054 -1.3473 0.15 -2.7885 -2.3014 -1.9030 -1.5881 -1.3361 0.16 -2.6938 -2.2507 -1.8736 -1.5699 -1.3242 0.17 -2.5996 -2.1989 -1.8433 -1.5510 -1.3118 0.18 -2.5066 -2.1465 -1.8122 -1.5314 -1.2988 0.19 -2.4150 -2.0936 -1.7803 -1.5112 -1.2854 0.20 -2.3251 -2.0404 -1.7478 -1.4904 -1.2715 0.22 -2.1520 -1.9342 -1.6815 -1.4473 -1.2424 0.24 -1.9889 -1.8293 -1.6139 -1.4026 -1.2119 0.26 -1.8366 -1.7269 -1.5461 -1.3568 -1.1801 0.28 -1.6954 -1.6277 -1.4784 -1.3103 -1.1475 0.30 -1.5652 -1.5325 -1.4116 -1.2634 -1.1141 0.32 -1.4457 -1.4417 -1.3461 -1.2165 -1.0803 0.34 -1.3361 -1.3555 -1.2822 -1.1699 -1.0462 0.36 -1.2359 -1.2740 -1.2202 -1.1239 -1.0120 0.38 -1.1443 -1.1972 -1.1604 -1.0785 -0.97788 0.40 -1.0607 -1.1250 -1.1028 -1.0342 -0.94402 0.45 -0.88174 -0.96363 -0.96930 -0.92813 -0.86126 0.50 -0.73850 -0.82723 -0.85088 -0.83026 -0.78238 0.55 -0.62306 -0.71225 -0.74682 -0.74113 -0.70840 0.60 -0.52931 -0.61533 -0.65593 -0.66076 -0.63984 0.65 -0.45254 -0.53349 -0.57679 -0.58877 -0.57689 0.70 -0.38914 -0.46416 -0.50795 -0.52458 -0.51949 0.75 -0.33639 -0.40520 -0.44808 -0.46752 -0.46742 0.90 1.0 -0.17264 -0.21454 -0.24557 -0.26547 -0.27475 1.2 1.5 -0.054338 -0.069105 -0.081278 -0.090516 -0.096619 2.0 -0.018909 -0.024218 -0.028728 -0.032305 -0.034847 Sec.1] IMPEDANCE ANDADMITTANCE TABLE1.2(Continued) -sin2Fw Xli"=cosh2Aw-cos2Fw145 ~I40° 45° 50° 55° 60° 140° 135° 130° 125° 120° 0.0 -1.1918 -1.0000 -0.83910 -0.70021 -0.57735 0.003 0.004 0.005 0.01 -1.1915 -0.99980 -0.83896 -0.70010 -0.57728 0.02 -1.1906 -0.99920 -0.83853 -0.69979 -0.57705 0.03 -1.1892 -0.99820 -0.83782 -0.69927 -0.57666 0.04 -1.1872 -0.99681 -0.83682 -0.69854 -0.57612 0.05 -1.1846 -0.99502 -0.83554 -0.69761 -0.57544 0.06 -1.1815 -0.99284 -0.83398 -0.69646 -0.57459 0.07 -1.1778 -0.99028 -0.83214 -0.69512 -0.57360 0.08 -1.1735 -0.98733 -0.83003 -0.69357 -0.57246 0.09 -1.1688 -0.98402 -0.82765 -0.69183 -0.57117 0.10 -1.1635 -0.98032 -0.82499 -0.68989 -0.56973 0.11 -1.1577 -0.97628 -0.82208 -0.68775 -0.56815 0.12 -1.1514 -0.97187 -0.81891 -0.68542 -0.56643 0.13 -1.1447 -0.96713 -0.81548 -0.68291 -0.56456 0.14 -1.1374 -0.96204 -0.81181 -0.68021 -0.56255 0.15 -1.1298 -0.95663 -0.80789 -0.67732 -0.56041 0.16 -1.1217 -0.95090 -0.80374 -0.67426 -0.55814 0.17 -1.1131 -0.94486 -0.79935 -0.67103 -0.55573 0.18 -1.1042 -0.93853 -0.79475 -0.66762 -0.55320 0.19 -1.0949 -0.93191 -0.78992 -0.66405 -0.55053 0.20 -1.0853 -0.92501 -0.78488 -0.66032 -0.54775 0.22 -1.0650 -0.91044 -0.77421 -0.65239 -0.54182 0.24 -1.0435 -0.89491 -0.76278 -0.64387 -0.53544 0.26 -1.0209 -0.87853 -0.75066 -0.63480 -0.52862 0.28 -0.99749 -0.86137 -0.73791 -0.62523 -0.52141 0.30 -0.97331 -0.84355 -0.72459 -0.61519 -0.51382 0.32 -0.94854 -0.82516 -0.71078 -0.60473 -0.50589 0.34 -0.92332 -0.80629 -0.69652 -0.59389 -0.49765 0.36 -0.89778 -0.78704 -0.68189 -0.58271 -0.48912 0.38 -0.87204 -0.76748 -0.66694 -0.57124 -0.48034 0.40 -0.84622 -0.74770 -0.65173 -0.55952 -0.47133 0.45 -0.78194 -0.69779 -0.61292 -0.52937 -0.44800 0.50 -0.71914 -0.64805 -0.57:365 -0.49848 -0.42388 0.55 -0.65879 -0.59933 -0.53459 -0.46738 -0.39936 0.60 -0.60159 -0.55228 -0.49630 -0.43652 -0.37480 0.65 -0.54795 -0.50738 -0.45921 -0.40628 -0.35049 0.70 -0.49807 -0.46492 -0.42366 -0.37694 -0.32669 0.75 -0.45200 -0.42510 -0.38986 -0.34875 -0.30361 0.90 -0.33567 -0.32180 -0.30014 -0.27241 -0.24006 1.0 -0.27443 -0.26580 -0.25022 -0.22896 -0.20319 1.2 -0.18294 -0.17996 -0.17185 -0.15930 -0.14298 1.5 -0.099536 -0.099328 -0.096160 -0.090270 -0.081951 2.0 -0.036294 -0.036619 -0.035835 -0.033985 -0.031143 146 TRANSMISSION-LINE THEORY TABLE1.2(Continued) -sin2Fw Xlin=cosh2Aw-cos2Fw[Chap.III "- ~650700750800850 900 Aw 115° 1l0° 10501000950 0.0 -0.46631 -0.36397 -0.26795 -0.17633 -0.087489 0 0.003 0.004 0.005 0.01 -0.46625 -0.36393 -0.26792 -0.17631 -0.087481 0 0.02 -0.46608 -0.36381 -0.26783 -0.17625 -0.087454 0 0.03 -0.46579 -0.36360 -0.26769 -0.17616 -0.087410 0 0.04 -0.46540 -0.36331 -0.26749 -0.17604 -0.087349 0 0.05 -0.46489 -0.36294 -0.26723 -0.17587 -0.087270 0 0.06 -0.46427 -0.36249 -0.26692 -0.17567 -0.087173 0 0.07 -0.46353 -0.36196 -0.26655 -0.17544 -0.087059 0 0.08 -0.46269 -0.36135 -0.26612 -0.17517 -0.086928 0 0.09 -0.46174 -0.36066 -0.26564 -0.17486 -0.086779 0 0.10 -0.46068 -0.35988 -0.26510 -0.17452 -0.086614 0 0.11 -0.45951 -0.35903 -0.26450 -0.17415 -0.086431 0 0.12 -0.45823 -0.35810 -0.26386 -0.17374 -0.086232 0 0.13 -0.45685 -0.35710 -0.26316 -0.17329 -0.086016 0 0.14 -0.45537 -0.35602 -0.26240 -0.17281 -0.085784 0 0.15 -0.45378 -0.35486 -0.26159 -0.17230 -0.085536 0 0.16 -0.45209 -0.35363 -0.26073 -0.17175 -0.085270 0 0.17 -0.45031 -0.35233 -0.25982 -0.17118 -0.084990 0 0.18 -0.44843 -0.35096 -0.25886 -0.17057 -0.084695 0 0.19 -0.44645 -0.34951 -0.25785 -0.16993 -0.084383 0 0.20 -0.44437 -0.34800 -0.25679 -0.16925 -0.084056 0 0.22 -0.43996 -0.34477 -0.25453 -0.16782 -0.083358 0 0.24 -0.43520 -0.34128 -0.25208 -0.16626 -0.082602 0 0.26 -0.43010 -0.33754 -0.24946 -0.16459 -0.081792 0 0.28 -0.42470 -0.33357 -0.24667 -0.16282 -0.080927 0 0.30 -0.41900 -0.32938 -0.24372 -0.16094 -0.080013 0 0.32 -0.41303 -0.32498 -0.24063 -0.15896 -0.079050 0 0.34 -0.40681 -0.32039 -0.23739 -0.15689 -0.078043 0 0.36 -0.40036 -0.31561 -0.23401 -0.15474 -0.076993 0 0.38 -0.39370 -0.31068 -0.23052 -0.15251 -0.075903 0 0.40 -0.38685 -0.30559 -0.22692 -0.15020 -0.074777 0 0.45 -0.36902 -0.29229 -0.21747 -0.14414 -0.071819 0 0.50 -0.35045 -0.27837 -0.20755 -0.13776 -0.068694 0 0.55 -0.33143 -0.26403 -0.19727 -0.13113 -0.065446 0 0.60 -0.31223 -0.24946 -0.18680 -0.12436 -0.062118 0 0.65 -0.29309 -0.23486 -0.17625 -0.1l751 -0.058750 0 0.70 -0.27420 -0.22036 -0.16573 -0.1l066 -0.055378 0 0.75 -0.25576 -0.20612 -0.15535 -0.10389 -0.052034 0 0.90 -0.20426 -0.16594 -0.12583 -0.084508 -0.042433 0 1.0 -0.17390 -0.14195 -0.10803 -0.072741 -0.036581 0 1.2 -0.12356 -0.10166 -0.077846 -0.052646 -0.026545 0 1.5 -0.071522 -0.059332 -0.045730 -0.031072 -0.015711 0 2.0 -0.027407 -0.022896 -0.017747 -0.012108 -0.0061376 0 Sec.2] IMPEDANCE ANDADMITTANCE 147 (1) (2)andFw={jw+<1>BintermsofTlinandXlinorforAw=aW+PBand F~={jw+<1>~intermsofOlinandblin. InChap.II,Sec.16,severalapplications ofthecirclediagram are listedintermsofthefunctions P,<1>,Tl,XlandP,<1>',gl,bl.Evidently theseapplyequallyifAissubstituted forpandFfor<1>andifasub scriptinisaddedtoTl,Xl,gl,andbl.Additional applications include thefollowing: 1.FromknownvaluesofPBand<1>B(orcI>~)andofawand{jw,valuesof Aw=aW+PBandFw={jw+cI>B(orF~=(jw+cI>~)maybedetermined fromthecirclediagram, andfromtheseTlinandXlin(orOlinandblin) maybeobtained. 2.FromknownvaluesofTlinandXlin(orglinandblin),AwandFw (orF~)maybedetermined fromthecirclediagram, andfromtheseand knownvaluesofaWand{jw,PBandcI>s(or cI>~)maybeobtained. From these,inturn, TIsandXIs(orglsandbIs)maybefoundfromthecircle diagram. 2.InputImpedance andAdmittance. Theformulas relating input impedance andadmittance tonormalized inputimpedance andadmit tanceare Rin-cPeXin TUn=Re{1+cP;) Xin+cPoRin Xlin=Re(l+cP;) Theinverserelations are Rin=Re(Tlin+cPeXlin) Gin=Ge(Olin-cPeblin) (3) Xin=Re(Xlin -cPeTlin) Bin=Ge(blin+cPeglin) (4) 1Notethat ReGe=1+ct>: (5) Theformulas involving thephaseandattenuation functions are R.-Rsinh2Aw-cPesin2Fw (6a) ,n-ecosh2Aw-cos2Fw G.=Gsinh2Aw+cPesin2F~ (6b) ,n ecosh2Aw-cos2F~ Xin=-Rsin2Fw+cPesinh2Aw (7a) ecosh2Aw-cos2Fw Bin=-Gsin2F~-cPesinh2Aw (7b) ecosh2Aw-cos2F~ Notethat Aw=aw+PBFw={jw+cI>B F~=(3w+cI>~ (8) andalso cosh2Aw-cos2Fw=2(sinh2Aw+sin2Fw) (9) Theformulas (6)and(7)aregeneralandinvolvenorestrictions or 148 TRANSMISSION-LINE THEORY [Chap.III approximations otherthanthoseimplied inthederivation ofthediffer entialequations andtheirapplication toterminated sections ofline. Theimpedance andadmittance givenby(6)and(7)arestudiedcon veniently intwoforms. Thecriterion distinguishing themiswhether theinputreactance canbemadetovanishornotbyvaryingthephase functionFw=<1>8+{3woverarangefromzeroto7r.Thecondition of zeroinputreactance characterizes asectionoflinewithaninputimped ancethatistunedtoresonance orantiresonance. Anysectionofline forwhichtheinputreactance canbemadetovanishbyvaryingFwis potentially resonant. Asectioninwhichtheinputreactance cannotbe madezerobyvaryingFwisnonresonant. Anexamination of(7a,b)showsthattheinputreactance orsusceptance iszerowhen sin2Fw= -cPcsinh2Aw sin2F~=cPcsinh2Atv(lOa) (lOb) Sincethesinecannotexceedunityinmagnitude, itisessential thatthe following condition besatisfied: cPcsinh2Aw~1 (11) Thisisthecondition characterizing allpotentially resonant sections of line.Correspondingly thecondition fornonresonance is cPcsinh2Aw>1 (12) Nonresonant SectionofLine.Subject to(12),Xinisalwaysnegative. SincecPcisverysmallonalow-loss line,thecondition (12)impliesthat Awisquitelarge.Clearly, if(12)issatisfied together withcP;«1,it followsthat sinh2Aw»1 sothat,from(6)and(7), Rin=Rctanh2AwGin=Gctanh2Aw Xin= -Rc(tanh2Aw)(cPc+s~~~iA'w) (sin2F~)Bin=Gc(tanh2Aw)cPc-sinh2A w(13) (14) (15) wherethelargeparentheses arealwayspositive subjectto(12).More overforAw~2,sinh2Aw~27andI~tanh2Aw~0.9993. Hence [.( sin2Fw)]Zin==Rc1 -JcPc+sinh2Aw Y.G[1 . ( sin2F~)1in=c+JcPc-sinh2At"(16) (17) Sec.2] IMPEDANCE ANDADMITTANCE 149 Theimaginary partsof(16)and(17)areverysmall,sothatZinandYin areessentially realandthelineismatched forallpractical purposes. The perfectly nonresonant orexactlymatchedlineisdefinedby Aw=aw+Ps=00 Zin=Re(l-icJ>e)=Ze Yin=Ge(l+icJ>e)=YeZlin=I Ylin=1(18) (19a) (19b) Resonant SectionofLine.Thepotentially resonant sections ofline mayhavezeroinputreactance andsusceptance whenoneoftwopossible conditions issatisfied. Thetwopossibilities aredistinguished asinput resonance andinputantiresonance, asfollows: InputResonance Fw-n;=F~-(n-;1)1r=isin-1(cJ>esinh2Aw) =icos-1VI-cJ>~sinh22Awnodd (20a) Xin=0(Rin)res= Re(l+cJ>;)sinh2Aw (20b) cosh2Aw+VI-cJ>:sinh22Aw Bin=0(Gin)res= Gc(l+cJ>~)sinh2~w (20c) cosh2Aw-VI-cJ>;smh22Aw 1nputAntiresonance Fw-~=F~-(n-;1)1r=1r-isin-1(cJ>esinh2Aw) =1r-j-cos-1VI-cJ>;sinh22Awneven (21a) Xin=0(Rin)antires = Re(l+_~:)sinh2Aw(21b) cosh2Aw-vI-cJ>:sinh22Aw (G.).=_~-=-G-:-c---,---(1----:-+_cJ>"1=;)=s=in=h=::=2::::;A=w====::::::::::== (21c) Bin=0 onantores _/cosh2Aw+V1 - cJ>~sinh22Aw Foralow-loss lineandatermination thatsatisfythefollowing inequaiities: cJ>~«1cJ>:sinh22Aw«1 (22) (20)and(21)maybereadilysimplified. Since cJ>eisverysmallona goodline,(22)isnotasevererestriction onPsinAw•Mostterminations thatarenotadjusted tobematched satisfy(22). InputResonance (23a) (23b) (23c) 150 InputAntiresonanceTRANSMISSION-LINE THEORY [Chap.III F-n7r=F'_(n-1)7r==_ 0 2 4 6 w2 w 2"n="" Xin=0(Rin)antires ==RccothAw Bin=0(Gin)antires ==RctanhAw(24a) (24b) (24c) Foralow-loss lineandalow-loss termination thatsatisfythefollowing inequalities: A~«1 conditions ofresonance andantiresonance areasfollows: InputResonance F-n7r=F'_en- 1)7r=,I..A==0 1 3 5w2 w2 'f'cw n= , , , Xin=0(Rin)res ==RcAw Bin=0(Gin)res ==1: InputAntiresonance F_n7r=F'_(n-1)7r w2 w 2 =7r-cJ>cAw==7rn=0,2,4,6, Xin=0(Rin)antires ==1: Bin=0(Gin)antires ==GcAw Notethatin(20)to(27)(25) (26a) (26b) (26c) (27a) (27b) (27c) Aw=aW+ps F~={3w+If>~ (28) Itiswelltonotethat,subjectto(22),theapproximate formulas for inputresonance andantiresonance coincide withtheexactformulas for normalized inputresonance andantiresonance, definedasfollows: Normalized InputResonance F_n7r=F'_(n-1)7r=0 w2 w2 n=1,3,5, Xlin=0(rlin)res=(R~:res=tanhAw blin=0(glin)res=(G~~res=cothAw(29a) (29b) (29c) Sec.2] IMPEDANCE ANDADMITTANCE 151 Normalized InputAntiresonance F-n7r=F'_(n-1)7r=_ 0 2 4 6 w2 w2 IIn=""o ()(Rin)antir£s -thAXlin=0 rlinantires= R c-co w b()(Gin)antires -thAlin=0 glinantires=G e-an w(30a) (30b) (30c) Thegeneralformulas forinputresistance andreactance areconsidered conveniently intworanges. RangeofImpedance Including InputResonance. Thisrangeisdefined bythecondition sin2Fw»sinh2Aw (31) sothatFwisnotnearn7r/2,withneven.Itfollowsthatthenormalized inputresistance andreactance are sinhAwcoshAw_1 •h2A 2F rlin=sin2Fw-~sInwcscw Xlin= -cotFw(32a) (32b) (36a)IfAwissmallcompared withunity,thisistheprincipal range.Note thatisinh2Aw==AwifAwissmall.Theinputresistance andreactance are Rin=Re(rlin+cPeXlin)=RcC-isinh2Awcsc2Fs-cPecotFw)(33a) Xin=Re(Xlin -cPerlin)= -Re(cotFw-icPesinh2Awcsc2Fw)(33b) Thetermswith cPeasafactorareusuallynegligible exceptindeter miningRinforshortsections oflinewithterminations forwhich Psisvery small.Forexample, ifthefollowing conditions aresatisfied: A~=(aw+PsP«1Fw={jw+cPs({jW)2«1 cPe=~(34) itfollowsthat Rin=Recsc2Fw(aw+Ps-~sinFwcosFw) (35) IfcPs==0,sinFw==sin{jw=={jwandcosFw==1,sothat R.==RePs tn{32w2 IfthetermwithcPe=a/{jasacoefficient isneglected, theresultis (36b) Thisiscomparable with(36a)onlyifaWisnegligible compared withpso 152 TRANSMISSION-LINE THEORY [Chap.III Similarly, ifcI>sa=-rr/2+{3ksa,Ps=a(b-ksa),asforaconducting bridge, sinFw==1,andcosFw==-(3(w+ksa),sothat Rin==Rc[a(w+b-ksa)+~(3(w+ksa)]==Rca(2w+b)(37a) whereitisimpliedthat{32w2«1.IfC/>Cisneglected, Ps=ab,andthe termina/(3ismissing. Thatis, (37b) Clearlythecontribution fromtheconductors ofthelineis50percentin erroriftheterminC/>Cisomitted. RangeofImpedance Including InputAntiresonance. Thisrangeis definedby -isin2Fwcsch2Awsin2Fw«sinh2Aw sothatthenormalized inputresistance andreactance are rlin=cothAw sinFwcosFw Xlin= -sinh2Aw(38) (39a) (39b) IfAwislargecompared withunity,asforalinethatisalmostmatched, thisistheprincipal range.Evidently rlinisnearunity,andXlinisvery small.Ontheotherhand,ifAwissmallcompared withunity,thisrange includes onlynarrowbandsnearFw=n-rr/2,withneven. Theinputresistance andreactance are Rin=Rc(rlin+c/>cXlin)=Rc(cothAw-ic/>csin2Fwcsch2Aw)(40a) Xin=Rc(Xlin-c/>crlin)= -Rc(~sin2Fwcsch2Aw+C/>CcothAw)(40b) Thegeneralformulas fortheinputadmittance Yinmaybereferred in asimilarmannertotwoprincipal ranges. However, theserangesdonot correspond tothoseoftheinputimpedance. RangeofAdmittance Including InputAntiresonance. Thisrangeis definedbythecondition sin2F~»sinh2Aw sothatF~isnotnear(n-1)-rr/2,withnodd. conductance andsusceptance are glin=isinh2Awcsc2F~ b1in= -cotF~(41) Thenormalized input (42a) (42b) IfAwissmallcompared withunity,thisistheprincipal range.The inputconductance andsusceptance inYin=Gin+jBinare Gin=Gc(glin-c/>cb1in)=Gc(isinh2Awcsc2F~+C/>CcotF~)(43a) Bin=Gc(l>lin+c/>"glin)=-Gc(cot F:"+ic/>csinh2Au'csc2F~)(43b) Sec.3] IMPEDANCE ANDADMITTANCE 153 Thetermswith<Pcareusuallynegligible exceptindetermining theinput conductance ofshortsections oflinewithterminations forwhich Psis small.Thesituation parallels thatdiscussed inconjunction with(34) to(37). RangeofAdmittance Including InputResonance. Thecondition for thisrangeis sin2F~«sinh2Aw Thenormalized admittance isgivenby(44) blin=-isin2F~csch2Aw (45) IfAwislargecompared withunity,asforalinethatismatched ornearly matched, thisistheprincipal rangewithglinnearunityandblinsmall. IfAwissmallcompared withunity,thisrangeisverynarrowandnear F~=(n-l}n/2,withnodd. Theinputconductance andsusceptance are Gin=Gc(Ylin-<Pcblin)=Gc(cothAw+i<Pcsin2F~csch2Aw)(46a) Bin=Gc(blin+<Pcglin)=-Gc(isin2F~csch2Aw-<PccothAw)(46b) 3.Extreme ValuesoftheInputResistance andConductance. Sec tionsoftransmission linecanbesodesigned thattheinputresistance or theinputconductance isextremely smallorextremely great.Manyof themostusefulapplications ofsectionsoflinearisefromthesetwoproper ties.Theconditions underwhichRinorGinmayassumeextreme values bysuitably adjusting thelengthwofthesectionwithagiventerminal impedance mustbedetermined byequating thederivative ofRinorGin withrespecttowtozero.Differentiation ofSec.2,Eq.(6a),leadsto thefollowing equation: (cosh2Aw-cos2Fw)(acosh2Aw-cPcfjcos2Fw) -(sinh2Aw-cPcsin2Fw)(asinh2Aw+(3sin22Fw)=0(la) Similarly differentiation of(6b)givesthefollowing equation: (cosh2Aw-cos2F~)(acosh2Aw+<Pc(3cos2F~) -(sinh2Aw+cPcsin2F~)(asinh2Aw+fjsin22F~)=0(lb) Thissecondequation, (lb),differsfrom(la)onlyinhaving F~inplaceof Fwand-cPcinplaceofcPc.Bysuitable rearrangement theseequations maybeexpressed asfollows: sinh2Awsin2Fw=a+cP"fj=a(1+cPcf3/a)==215(2a) 1 -cosh2Awcos2Fw(3-<Pea(3(1-cPca/(3) sinh2Awsin2F~=a-<Pc(3=a(1-<pc(3/a)==215'(2b) 1 -cosh2Awcos2F~ (3+cPca(3(1+<Pca/(3) where8and8'areasdefinedin(2a)and(2b)fortemporary use.Fora 154 TRANSMISSION~LINE THEORY [Chap.III low-loss linewithnegligible leakageconductance, forwhich (3) itfollowsthat o=l/Jc=~ (30'=0 (4) Thusthebehavior ofresistance andconductance neartheirextreme values differsunless,asinadissipationless line,l/Jc=O. Since(2a)and(2b)areformally alike,theanalysis maybecontinued using(2a).Theparallelresultfor(2b)isobtained byaddingprimeson Fwando.Thefollowing rearrangement of(2a)isconvenient: sinh2Awsin2Fw+20cosh2Awcos2Fw=20 (5) Let D==vsinh22Aw+402cosh22Aw (6) Then(5)canbeexpressed asfollows: sinh2Aw•2F+20cosh2Aw2F_20Dsm w Dcos w-D (7) 20cosh2Aw •2.1,D=sm'YNowlet sinh2Aw 2.1,D=cos'Y sothat(7)becomes sin2(Fw+1/t)20 D20coth2Aw=tan21/t (8) (9) Sincetherightsidein(9)isessentially positive, theargument ofthe sinemustbeinthefirstorsecondquadrants. Thatis, F_n7r .1,+1 .-120 w-2-'Y2"smD n7r1.20Fw=2-1/t-2"sm-1Dn=0,2,4,6, n=1,3,5,...(10) (11) Theseformulas maybeputintomoreconvenient forms. with(6), .20 20sm-1-=tan-1---;=======-----:-:--:::--:--- DVI+482sinh2Aw Thesubstitution of(12)in(10)and(11),using(8),yields Fw=n7r_~(tan-1 28_tan-1 20 ) 2 2 tanh2AwVI+402sinh2Aw F=n7r_!(tan- 120+tan-1 28 ) w2 2 tanh2AwVI+482sinh2AwNotethat, (12) neven (13) nodd (14) Sec.3] IMPEDANCE ANDADMITTANCE 155 Theformulax+ytan-1x+tan-1y=tan-1----- 1=+=xy permitstheexpression of(13)and(14)inthefollowing forms: F=n1r_!tan-120VI+402sinh2Aw-tanh2Aw w2 2 402+VI+402sinh2Awtanh2Aw Fw=n1r_!tan-120VI+402sinh2Aw+tanh2Aw 2 2 VI+482sinh2Awtanh2Aw-402(15) neven (16) nodd (17) Thesearethegeneralformulas givingextremizing valuesofFw' Restriction toLow-loss Lines.Thelow-loss linewithlowover-all attenuation isdefinedby 402«14A~«1 (18) Subjectto(18),thehyperbolic functions maybeexpanded inseriesto obtain F=n1r_!t-120(2Aw)8(i+{)==n1r_A~o w2 2an 4(02+A~) 202+A~ Similarly (17)becomesneven(19) _n1r_~-12AwOFw-22tanA2_02 wnodd (20) Theargument in(19)issufficiently smallsothatitmayreplacethe inversetrigonometric function. Thisisnotnecessarily thecasein(20), sincethedifference inthedenominator maybesmall.Forsimplicity in interpreting (19)and(20),letthelineberequired tohavenegligible leakageconductance, sothat JL«!. WCwlepc=~=8 {3(21) With(21),(19)becomes F=n1r_(a/{3)(aw+Ps)8 n1ra2w;{3w p w2a2/{32+(aw+Ps)2=2-1+{32w;neven(22) where (23) SinceFw={3w+CPs={3wp-{3ps/a+<l>s,(22)maybesolvedfor{3wp• Thus {3(1+a2w;_)=n1r_+{3ps Wp1+{32w~ 2 saneven (24) 156 TRANSMISSION-LINE THEORY [Chap.III Sinceithasbeenassumed that A~=(aw+P8)2=(awp)2«1 itfollowsthat(24)reducesto(25) n1rFw={jw+<P8=2neven (26) (27a) noddThevalueof(jwdefinedby(26)locatesthemaximum valueofRin,asis shownlater. Subjectto(21),(20)becomes F=n1r_!tan- 12(a/{j)(aw+P8) w2 2 (aw+P8)2-a2/{j2 With(23),thisisequivalent to F-n1r1-12/{jW p w-2-2tan1 -(1/{jwp)2nodd (27b) Theuseofthetrigonometric identitytan-1x=jtan-1[2x/(1-x2)]in (27b)reducesthisto n1r 1(n-1)1rF= - -tan-1-= +tan-1{jwnodd(28)w2 {jwp2 p Hence tanFw=tan({jw+<P8)={j(w+;) (29) Thisrelationgivesthevaluesof{jwwhichlocatetheminimum valuesof inputresistance. Formulas corresponding to(26)and(28)fortheextreme valuesofthe inputconductance areobtained directlyfrom(19)and(20)byaddinga primeonFwandon0andnotingthat0'=O.Thus(19)gives F'=RW+<p'=n1rw1J 82neven (30) Similarly (20)gives F~={jw+<P~=~nodd (31) Itwillbeshownthat(30)locatesthemaxima ofGinand(31)locatesthe minimaofGin. Substitution of(26)inSec.2,Eq.(6a),using(21)and(25)gives n1rFw={jw+<P8=2n=0,2,4, (32) Sec.4] IMPEDANCE ANDADMITTANCE 157 Inordertosubstitute (27b)inSec.2,Eq.(6a),notethat,withnodd, .2F- . ( t-12{3wp) - •t-12{3wp_2{3wpsm w-smn1r-an{32w;_1 -sman{32w;_1 -{32w;+1 (33a) cos2Fw=cos(n1r-tan-1{32~W~1) _ -12{3wp__fJ2W:-1 (costan{32w;_1 - {32w;+1 33b) _2awp({32w;+1)-2awp_ Hence Rin-Rc{32w;+1+{32w;_1 -Rcaw p (34) sothatfinally tan({3w+<1>8)={3(w+~)(35) Substitution o((30)and(31)inSec.2,Eq.(6b),leadsdirectlyto {3w+<1>~=n; {3w+<1>~=n;neven(36) nodd(37) Notethat(Rin)lDAxand(Gin)minoccuratthesamevaluesoffJw,butthat (Rin)miDand(Gin)maxdonot. 4.Extreme ValuesoftheInputReactance andSusceptance. Extreme valuesofXinandBinareobtained bydifferentiating Sec.2,Eqs.(7a,b), withrespecttowandequating thederivatives tozero.Theresulting equation fordXin/dw=0is (cosh2Aw-cos2Fw)({3cos2Fw+a¢ccosh2Aw) -(sin2Fw+¢csinh2Aw)(asinh2Aw+(3sin2Fw)=0(1) whereCollecting termsandrearranging give cosh2Awcos2Fw-28sinh2Awsin2Fw=1 28=={3¢c+a fJ-aepc(2) (3) Asidefromconstant factorstheexpression [Sec.2,Eq.(7b)]forBindiffers fromSec.2,Eq.(7a),forXinonlyinhaving F~appearinplaceofFw and-¢cinplaceofcPc.Hencetheequation corresponding to(2)is wherecosh2Awcos2F~-28'sinh2Awsin2F~=1 20'==a-{3¢c {3+a¢c(4) (5) Notethat,forthelinewithnegligible leakageconductance, forwhich 158 TRANSMISSION-LINE THEORY [Chap.III ep~=a2/{32«1,itfollowsthat o=~ {30'=0 (6) Thesolution of(2)isreadilycarriedoutbydividing through by D'==vcosh22Aw+402sinh2Aw andsetting cosh2Aw_2.1/D'-cos'Y Theresultis20sinh2Aw_ •2.1,'D'-sm'Y(7) 20tanh2Aw=tan21/1' (8) F=n1r_.1/+cos-11:.. w2'Y-D'neven (9) neven(11)However, with(7), cos-1~,=tan-1VD'2-1=tan-1(VI+402sinh2Aw)(10) sothat,with(8)and(10),(9)becomes n1r1Fw=2-2tan-1(20tan2Aw) ±~tan-1(vI+402sinh2Aw) Thearctangents maybecombined into Fw=n1r_!tan-120tanh2Aw+=VI+402sinh2Aw 2 2 1 -20VI+402tanh2Awsinh2Aw Thisisthegeneralformula. Restriction toLow-loss Line.Subjecttotheconditionsneven(12) (12)reducesto(2Aw)2«1(20)2«1 (13) neven (14) With(6)andthenotation Wp=w+p/a,(14)becomes I3p. n1r_ {3wp-~+cp.="2+awpneven (15) wheretheterm20Awhasbeenneglected compared withunityasaresult of(13).Thesolution of(15)for{3wpgives (16) Sec.4] IMPEDANCE ANDADMITTANCE 159 Hence Finallypw=(~-~.+p;.)(1+~)-P;.neven pw=(n;-~.)(1+~)+P.neven(17) (18) ThisisthefinalformulaforthelengthwgivingextremevaluesofXin• Thecorresponding formulaforextreme valuesofBinisobtained from (14),withF~writtenforFwand0'=0foro.Sincethetermin0was neglected, thesameformula isobtained, viz., pw=(n;-~~)(1+~)+p.neven (19) Theextreme valuesofXinandBinareobtained bysubstituting the equivalent of(18)and(19)inSec.2,Eqs.(7a,b),using(6)and(13). Thatis, Fw=(n;+Aw) F~=(~+Aw)neven neven(20) nevenB.=-Gsin2F~-2Awa/p ,n c1+2A,;-cos2F~ (21) (22)sin(n7l"+2Aw)==+2Awneven cos(n7l"+2Aw)=cos2Aw==1 -2A~ itfollowsthataresubstituted in Xin=-Rsin2Fw+2Awa/p c1+2A~-cos2Fw Since (23a) (23b) (24a) (24b) (25a)pw=(~1r-~.)(1_~)_P. neven pw=(~71"-~.)(1+~)+P.wheretheuppersignsgotogether andthelowersignsgotogether in (23)and(24).Specifically (Xin)max=2~cw(1-~) neven(25b) 160 TRANSMISSION-LINE THEORY [Chap.III ~W=(n;-~~)(1-~)-P. neven(26a) ~w=(~1r-~~)(1+~)+P. neven(26b) 5.Summary ofCriticalValuesofInputImpedance andAdmittance for aSectionofLow-loss Line. Conditions Assumed et>;=(~)2«1 A;'=(aw+p.)2«I(Ia) (lb) InputAntiresonance, nEvenInputResonance, nOdd n1r ~w="2-~. ~=(n-1)1r_;r..' IJW 2'*'. Minimum InputResistance tan(~w+~.)=~(w+~) (~w=0For~.=0andP.=0,~w=1.57for(Rin)min for(Rin)re.10.90 10.997.72 7.854.49 4.71n1r ~w="2-~. ~w=(n-1)1r_~' IJ 2 • Extreme ValuesofInputReactance andSusceptance, nEven (Zin=Rin+JXinYin=Gin+JBin) ~W=(n;-~.)(1-~)-p. Xin=(Xin)msx=~c~w--:/~~j(Rin)max(1-~) ~w=(~-~.)(1+~)+p. Xin=(Xin)min= -~c~u;;1:~-j(Rin)max(1+~)(5a) (5b) Sec.5] IMPEDANCE ANDADMITTANCE 161 (5c) (5d) Relations betweenExtreme Values (Rin)~ax ==41(Xin)max(Xin)mini (Gin)~ax ==41(Bin)max(Bin)mini(Rin)max=(Xin)max -(Xin)min (6a) (Gin)max=(Bin)max-(Bin)min (6b) Itisinteresting tostudygraphically thegeneral behavior ofthe inputimpedance Zinandinputadmittance Yinofalow-loss sectionof 211'jJw I I3rr/2 I 3rr/2._0_._._.-. Xm1n Xm1n 1f rro I 2rrF.511'/2 w FIG.5.1.Schematic diagram ofRi,.andXi,.forsectionoflineofelectrical lengthpw. transmission lineoflengths,linespacing b,characteristic impedance Ze=Re(1-jcPe),andpropagation constant 'Y=a+j/jwhentermi natedinanarbitrary impedance Z.=R.+jX.withcomplex terminal function 8.=P.+j..Itisassumed thatthistermination consists of 162 TRANSMISSION-LINE THEORY [Chap.III asectionoflineoflength St(s;»b2)withanarbitrary impedance atits end,sothatendeffectsdonotexistatthelocation ofZs. Aschematic diagram oftheinputresistance andreactance ofaline withlowover-allattenuation isshowninFig.5.1.Actually thepeaks shouldbeverymanytimes higher andnarrower inordertorepresent correctly alow-lossline.Forthepresentthedistorted curvesinFig.5.1 areconvenient todescribe thesalientproperties oftheimpedance. ThecurvesshowninFig.5.1applytoasectionoflineofelectrical length{jwterminated inanimpedance Zsthatincludes aratherlow o fiw, , I CI>. 'IT 21tF.51t/2 '"FIG.5.2.Schematic diagram ofGinandBinforsectionoflineofelectrical length{Jw. resistance Rsandasomewhat greatercapacitive reactance Xs•The appropriate valuesareindicated at{jw=0orFw={jw+cf>s=cf>s.As thelength{jwisincreased fromzero,theinputreactance Xinrisesfrom XstozeroatFw='fr/2,whichdefinesinputresonance asgivenin(2a). Atthislengththeresonant inputresistance isquitesmall,namely, (Rin)res=Rc(aw+Ps).As{jwisincreased further,theinputresistance risesfirstslowly,thenveryrapidlytoamaximum valueof atFw='fr.Thisistheantiresonant value.Beyond {jw='fr-cf>sthe Sec.5] IMPEDANCE ANDADMITTANCE 163 FIG.5.3.Schematic diagram ofRin andXinforatransmission lineof electrical length (3wwithdifferent terminations. (a)Idealopenend, <1>.=0,P.=0,R.=0,X.= - 00. (b)Idealshortcircuit, <1>.=11"/2, P.=0,R.=0,X.=0.(c)Par allelresonant circuit, <1>.=0,P. small,R.large,X.=o.(e)(a) (b)<pw+cI>.> 7(/2resistance dropsfirstrapidly, thenslowlyuntilitreachesamInImUm (Rin)min=Rc{aw+Ps)atavalueof{jwdefinedby(4).Correspond inglytheinputreactance risestoamaximum (Xin)maxatavalueof{3w [specified accurately in(5a)]whichlies veryslightlytotheleftofFw=7r.ItBin dropsabruptly through antiresonance Xln withXin=0atFw=7randthende- creasestoanegative extreme value (Xin)min; itthenrisestopassthrough zeroagainatresonance. Theentire cycleisthenrepeated. Themaximum valuesofresistance areonthecurve Rc/{aw+Ps)atFw=n7r.Theex tremepositive valuesofthereactance areonthecurve;.2(awR.+Ps)(1-~); thenegative valuesareonthecurve Bin()~ aw-~cPs1+~.Notethatinmag- nitudeanegative extreme isalways greaterthantheassociated positiveex treme. Corresponding curvesforin putsusceptance andconductance are showninFig.5.2. Bin Iftheimpedance Zsofthetermina- Xln tionatw=0ischanged, thegeneral shapeofthecurvesisunaffected, but theyaremovedbodilytowardshorter orlongerlengthsdepending onthena- tureoftheimpedance. Letafewspe- cialcasesbeconsidered: 1.Rs=0,Xs= -00.Thesecon ditionscorrespond toanidealopenend withps=0,<I?s=o.Theimpedance curvehastheshapeillustrated inFig. 5.3a. 2.Rs=0,Xs=O.Thesecondi tionsdefineaperfectshortcircuitsuch asgivenbyaninfiniteperfectly con ductingdiskonanopen-wire lineora perfectly conducting pistoninacoaxial line.Thecorresponding terminal functions arePs=0,<I?s=7r/2.The behavior ofRinandXinforsuchatermination isshowninFig.5.3b. 3.Rsverylarge,Xs=O.Theseconditions applytoatunedparallel 164 TRANSMISSION-LINE THEORY [Chap.III resonant current. Thecorresponding terminal functions arePsvery small, CPs=O.Theimpedance curvesbehaveasshowninFig.5.3c. Complete setsofcurvesoftheinputimpedance ofaparticular section oftwo-wire lineterminated inanidealopenendwithRs=0,Xs= - 00 andinaperfectshortcircuitwithRs=0,Xs=0areshowninFig.5.4 asfunctions ofthelengthwofthesection. Similarcurvesforaparticu larcoaxiallinearegiveninFig.5.5.Notethatinbothsetsofcurves MetersMt2.ססOO 2.01.5000 1.51.ססoo 1.0 0.505000 0.49900.5010 0.999011.0010 1.49901.5010 1.99902.0010eers R' R" -r---1-Y'-X'I.'TIIl'I ,......~", '1'.+~"7~-X" R' ,.+X,l:~X', X"-r-f--f--"- II +.r\ I I 1\ f\ i\ I\ \ , "'.","I', /,'," , I ,II , \ I ! \ I I"X ./ ""- ./ f(J 0 0 0106- ~ 4 2 105 ~ 4 2 104- ~ 4 2 1038 l/I6 E4.co2 102 86 4 2 1086 4 2 1.08 6 4 2 10-1_o W FIG.5.4.Zinfortwo-wire line.Rc=439.8ohms, cPc=7.183X10-4,a=2.258X10-3 neper/m, fJ=3.144radians/m, a=5.118X10-4m,b=0.02m,Aair=2m,Aline= 1.992m.Z;"=R;"+jX;nforanidealshortcircuit.Z;:=R;:+jX;:forideal openend. antiresonances occurat{3w=n7r/2,wherenisoddfortheshort-circuited endandevenfortheidealopenend.Corresponding curvesoftheinput admittance areshowninFigs.5.6and5.7. Theinputimpedances andadmittances ofthesamesections ofline terminated inpureresistance ofRs=60and2,500ohmsareshownin Figs.5.8to5.11. 6.SectionofTransmission LineasanInsulator.70,81,99Oneofthe mostinteresting practical applications ofterminated sections oftrans missionlinemakesuseoftheveryhighvaluesofresistance whichcanbe obtained atantiresonance. Theformula formaximum inputresistance Sec.6] IMPEDANCE ANDADMITTANCE 165 MtS20000 2.0Meters15000 1.5 w10000 1.0 0.505000 0.99901.0010 1.49901.5010 1.99902.0010eer0.49900.5010 R' R' II) I};' .li' f--f--t-+X' X'r-+X" X" +X~X'+X"o--:~:...=~ "\. \ II\ \ \ ,'. " I' I Ii I ! \'\ II I\.I \ I \. / X / vto-L-- / 0 0 0'0106 86 4 2 105 86 4 2 104 86 4 2 103 ~ 4 III2 ~102 8o 6 4 2 1086 4 2 1.08 6 4 2 10-1 8 6 4 2 10-2o FIG.5.5.Z,nforcoaxialline.Rc=75.13ohms,cPc=1.384X10-4,a=3.577X10-4 neper/m, {3=3.142radians/m, a=0.01m,b=0.035m,Aair=2m,Aline=1.992m. Z;n=R;n+jX;lIforidealshortcircuit.Z;:=R;:+jX;lIforidealopenend. ofasectionoflow-loss lineforwhich is atr a=2Rc (Rin)max=~: n>..eps 8=2(3R2 =jr(s+~sa/a) n=0,1,2,..(1) (2) (3) Foratwo-wire lineterminated inabridgeoflengthbandmadeof thesamewireastheline,theapparent terminal functions are 7r epsa=2+(3ksa (4) wherekS4andmsareasdefinedandevaluated inChap.II,Sec.20.For 166 TRANSMISSION-LINE THEORY [Chap.III Mt 2.0Meters2.ססOO 1.51.5000 w1.01ססoo 0.505000 0.49900.5010 0.9990 UlOIO 1499015010 19990 2.0010eers G" I G' -I-+B"- I./.-cB"-G"G' '--~B' ,+B:13-~f=-B"- 17-,---B'=r=, I"\,\./\,1'\.-j,o<, ,_c 0,B' I, "+B'\ 1\ " : j\ \ -1-- 'X V ',,' v- I',,' .-,/I'..-,, ".- III !\ I I\. X )'" .......,,- - 0 0 ItO to108 6 4 2 1.08 .6 4 2 10-18 6 4 2 10-28 III 6o 4.s= :i: 2 10-38 6 4 2 10-48 6 4 2 10-5 8 6 4 2 10-6o FIG.5.6.Yinfortwo-wire line.Constants ofthelinearethesameasinFig.5.4. Y:..=G:..+jB:..foridealshortcircuit.Y::=G::+jS::foridealopenend. thecoaxialline Ps=PBa==0 (5) forthecoaxiallineTheappropriate conditions forantiresonance withnintegralare s=nzA-~;a=n;-~-ksaforthetwo-wire line nXXs=2"-4(6a) (6b) andthecorresponding antiresonant resistances are Rc ZR; a(s+b/2)r(s+b/2)forthetwo-wire line (7a) and (Rin)max=Rc2R;forthecoaxialline (7b)asrs Forconvenience let s'=s+PBa==bforthetwo-wire line (8) s+-a 2 8'=8 forthecoaxialline (9) Sec.6] IMPEDANCE ANDADMITTANCE 167 Meters2.ססOO 2.0Meters 1.51.5000 1.01ססoo 0.505000 0.49900.5010 0.99901.0010 1.4990I1.5010 1.99902.0010 G"G'G"G' f--f--I-+B'!...,-B"I I I ,'-I "+B'.";-B'+B"-:'i+B',--B-I-- ,1'1~-B" , , 11\ 1/'\ \ ,, I, I I " /,"/,I.'" [."1,. ~. /'. ."'-. , .- ,\ I \I V II f'.... ,,/ -I--'1--1-- 0 0 I'~102'8 6 4 2 lOs 6 4 2 1.086 4 2 10-1s 6 4 ell22.10-2_ :IEf 4 2 10-3-f 4 2 10-4 ~ 4 2 10-5 86 4 2 10-6_o W FIG.5.7.Yinforcoaxialline.Constants ofthelinearethesameasinFig.5.5. Y;n=G;n+jB;nforidealshortcircuit. Y;~=G;:+jB;:foridealopenend. Theshortest possiblelengthsandthegreatest resistance areobtained withn=1,sothat8'=X/4and (Rin)max=4a~c=8r~: (10) Foralinewithlowattenuation perunitlengthwhenoperated atsuf ficientlyhighfrequencies, thefollowing simpleformulas aregoodapproxi mations: Forthetwo-wire line Rc=rcosh-1..!!- 1r 2a Forthecoaxialliner=~Y;a(11) fbRc=-In21rar=m~(1+~)2b a(12) 168 TRANSMISSION-LINE THEORY [Chap.III Meters 2.0 1.5 w1.0 0.5I / \Rin R~' /1\RinR~' '\"n,n r---f-+Xin1r+X"/+XinI'\-+X"I "I\I\'-Xi; in,,1-.A,-Xi~ v/.\',-Xin ,in,I,A..-Xi~ ~. '..../I'....,,-"1./I " ,....,/ \I/ '\ I '/ '\"/x,'J. '>(IA. I"IX X'..I1\ "'\.'/ I"/\ I" ./""'"'"r-..i/'1\'":'V' -"/ \ t- V ,- I'"I If I! I; \: ,; :1001048 6 4 Rin R" ~in R"on in ti~Xin r-~X~'f~;1-X'7k~II\ +Xi,;- ....X!'+. Xinf-+Xi~/\.;Xi~_ on In In-....' I"~..,./\1'1' VI,\1'''- \.'I\I\',//: ......./V\",/"\'"/I""I \--',1\A \,".")',/ \,..... "IF",. '"'"/ '\.t X ~ X X /'........\//'.1 ............\1/ ......, V T7 ........-....1..J-/ r7 ......~i-J-V I,I, \; I, "I', 1/ " 0:",FIG.5.8.Zinfortwo-wire line.Constants ofthelineareasinFig.5.4.R;nand X;nforR.=60ohms,X.=0;R::andX::forR.=2500ohms,X.=o. 1038 6 4 1028 6 1IlE4.co 1.000.5 1.0 w1.5 2.0 Meters ,-UI~ I1\...o-U1n /....-Ui~ I1\-IUlnI--f---r--I-+Bi~++':'Bj,-+Bin 71\*,Bin-+Bi~'rJ-rT-BI~I-+Bin7h~~BiCf-- f---~ I'. T \, .' ,-'"-',".'" I, '//1\',/ ',//i1\x" "t'" \ --/ '\ I \.-/,'\ / \ X1\. ./\'x )(11, /\IA X X X X X. ~ .......:- ~ \ \i : iI ~I ,, III \1/ Ii "10-38 6 4FIG.5.9.Zinforcoaxialline.Constants ofthelineareasinFig.5.5.R;,.andX;n forR.=60ohms,X.=0;R;~andX;:forR.=2500ohms,X.=o. 10-1 8 6 4 0.5 1.0 1.5 2.0 Meters w FIG.5.10.Yinfortwo-wire lineundersameconditions asinFig.5.8. Sec.6] Intheabove UsingIMPEDANCE ANDADMITTANCE t-1 _to_120r =y;:;;;;-VErdPrd -VErdPrdohms169 G13) -~~-J§rd~3 X108~prd Vp- - - Vo- - - ~ E~ E~m/sec(14) thefollowing formulas areobtained: Forthetwo-wire line (15) 10-1 8 6 4 10-3 8 6 4..-tG;~ Gin1=f- .-+G;~ I~Gin I\.r-r---- f--f--+B~'-bkf7t;,-B;~-+BinW\it<-B;n-+Bi~l*~-B!'-+ Pinf7\;~-Bi~f- m"I'I·~,nc--- ~\. II\,'J'('/1/\"<.'7,XII,\'C::, 1\,</,\A)1\)(/ \ A/1\I.>, X",,~x. x .'\. -' A A "/ ....../ ..........1'..\i./ I'-\-I-J-V r--...\iV \"\:-I-t" ~t-1' ,\' \I \: \I i:, 0.5 1.0 1.5 2.0 Meters w FIG.5.11.Yinforcoaxiallineundersameconditions asinFig.5.9. Forthecoaxialline Let(16) (17) Thenforthetwo-wire line (R.) -8R;_bK_r2a(.h-1~)2 onmax-rX-VWbcos2a andforthecoaxialline 8R~ _r[In(b/a)J2 (Rin)max=--r>:=bKVW1+(b/a) Thenumerical valueofKis K==0.121Iqpr Prd'\jErdPrd(18) (19) (20) 170 TRANSMISSION-LINE THEORY [Chap.III 350 700800bRc=60loga 100 150 2000.10.2 200300400500 Rc=120cosh-1b/2a FIG.6.1.Functions ofseparation determining extreme valuesof(R,..)max.0.30.5 0.40.60.70.80.91.0Forcopperinair(J'=5.65X107,Pr=1,Era=1,andPro=1,sothat K=0.121XV5([5X103=0.91X103• Itistobenotedthat(Rin)max,max increases indefinitely withfre quencyandwithb.However, foragivenvalueofbthefunctions (2a/b)[cosh-1(b/2a)J2 and[In(b/a)J2/(l+-b/a)canbemaximized bysuitably adjusting a.Lety=b/2a.Themaximum valueof (l/y)(cosh-1y)2isobtained bydifferentiatir.g andequating tozero. l.l~---,r----r--r--'---r-'--r---r----Y Theresultisy=b/2a=3.95,sothatformaximum (Rin)max,withb fixed,thewireradiusshouldbe b a=7.90 (21) Thefunction(l/y)(cosh-1y)2isplott,edinFig.6.1.Itisseenthatits maximum valueis1.064.Similarly, withx=bfa,thefunction (InX)2/ (1+x)maybeshowntohavethemaximum x=b/a=9.2,sothat, forgivenb,theradiusoftheinnerconductor ofthecoaxiallineshouldbe ba'=- (22)9.2 Thefunction (Inx)2/(l+x)isplottedinFig.6.1asafunction ofx. Itsmaximum valueis0.481. Sec.6] IMPEDANCE ANDADMITTANCE 171 Thecharacteristic resistances corresponding totheseextremizing values ofaareasfollows: Forthetwo-wire line Forthecoaxialline(23) (24) Thecorresponding extreme valuesof(Rin)ma.xareasfollows: Forthetwo-wire line b- =7.9a (25) b- =9.2a=bKV~X1.064 Forthecoaxialline 133.1 (Rin)ma.x,ma.x =Vas VdrEdr =bKV~X0.481 ForcopperinairK=0.91X103,sothat Forthetwo-wire line(26) (Rin)ma.x,ma.x =0.97b~X103=2.43bV1X103(27) Forthecoaxialline (Rin)ma.x,ma.x =0.438bV~X103=1.098bV1X103(28) Inthecaseofthetwo-wire linedescribed inFig.5.4,forwhichb=2em anda=5.118X10-2ematf=1.5X108Me/sec, themaximum value ofRinis (Rin)ma.x=:::=2.258X4~g~~X0.50=397,000 ohms (29) Iftheradiusofthewiresischanged from5.118X10-2emtotheopti mumvalue,a=b/7.9=2/7.9=0.253em,sothatRc=246ohmsand a=0.828X10-3neper/m, I (Rin)ma.x,ma.x =0.02X2.43X1.225X104=595,000ohms (30) Forthecoaxiallinedescribed inFig.5.5,forwhichb=3.5emand a=1em,themaximum valueis(Rin)ma.x=Rjas=420,000ohms.By adjusting theradiustobea=b/9.2=3.5/9.2=0.38em,sothat Rc=133.1ohms, (Rin)ma.x,ma.x =1.098X0.035X1.225X104=471,000ohms(31) Sinceitisnotdifficulttoobtainresistances oftheorderofmagnitude ofhalfamegohm ormoreusingantiresonant sections oftransmission 172 TRANSMISSION-LINE THEORY [Chap.III (c)(b) ~ine Support ,StubUne-========== ~-k48Support Lin:~_--S_tub_~t-- __l (a)lineathighfrequencies, suchsections serveadmirably asinsulators for supporting transmission linesorcircuitelements whenever theseareto bedesigned forsingle-frequency operation. Because theshortest length ofasectiondesigned forthispurpose isaboutone-quarter ofawave length,suchinsulating stubsareusefulonlyatultrahigh andmicrowave frequencies. Sincethemaximum inputresist9,nce increases withfre quency(because thelengthdecreases morerapidlythantheresistance increases), theinsulating properties ofquarter-wave stubsimprove asthe frequency becomes higher. Atsufficiently highfrequencies theyare superior tomostinsulators (and oftenmorerugged). InFig.6.2 schematic diagrams areshownof parallel- andcoaxial-line stubsar rangedtosupportbothwiresinthe caseoftheparallellineandthein-L[mLnerconductor ofthecoaxialline.t:StubIII iTheproperlocationandspacingof Line~ h~ supporting insulators, whether con- structed ofstubsorofdielectric ma terial,arediscussed inasubsequent section. Thequestion ofendeffects isalsoconsidered later. 7.Impedance Transformation UsingaNetwork ofTransmission- lineSections-General Formula tion.Inordertoreducethepower lossesintransmission overlongdis tancestoaminimum, atransmis sionlinemustbeterminated inits characteristic impedance Zc.Such atermination isshowninalater sectiontoleadtominimum losses intheline.Schematic circuitdia gramsfortwo-wire andcoaxiallines withmatching networks areshowninFig.7.1.InFig.7.2averygeneral matching network isinserted between theendofthelonglineandthe loadimpedance Zs.Inordertodistinguish quantities associated withthe longlinefromthoseusedtodescribe thematching section,thelatterare designated withasubscript m.ThusZcisthecharacteristic impedance ofthelongline;Zcmisthatofthematching section. Thecondition thatmustbesatisfied is(d) ~-k8 FIG.6.2.High-impedance stubsusedas insulators. (a)Stubsupports fortwo wireline.(b)Closed-end supports for innerconductor ofcoaxial line.(c) Stubsupportatright-angle bendintwo wireline.(d)Quarter-wave high-imped ancestubintandemwithmovable bridge. Zin=ZcorRin=RcXin= -cPcRc (1) whereZinistheinputimpedance ofthematching sectionoflengths, Sec.7] IMPEDANCE ANDADMITTANCE 173 cw x(2) whereYA=1/ZAistheinputadmittance oftheentire matching networkatAAasseenfromthefeedingline andYAxandYAyare,respectively, theadmittance at AAofthepartofthematching network oflengthx withitstermination andofthelengthywithitster mination. Yc=1/Zcisthecharacteristic admit tanceofthefeedingline;Ycm=I/Zcmisthecharac teristicadmittance ofthematching sections.withtheloadZsasitstermination. InFig.7.2Zinistheimpedance at AA.Thegeneralrelations (1)fortheinputresistance andreactance ofaterminated sectionoftransmission linemaybeexpressed inadmit tanceformasfollows: y Zy FIG.7.2.General matching section.A B : ~Z8 A B Matching section A BTwo·wire line Longline Coaxialline Longline f- -}- 3*lJZ Matching 8 section FIG.7.1.Circuitformatching aloadtoalongline. Division of(2)byYcmgivesthenormalized valuesreferredtothematch ingline.Thus YlAx+YIAy=Ylc (3) Theseparation ofrealandimaginary partsgives Since(4) itfollowsthatforalow-loss line,withc/>;«1andc/>cc/>cm«1, Rem1-c/>emc/>e•Rem gic=J[; ~1+c/>~=R; b=Remc/>em-c/>e==0 IcRe1+c/>; Foralow-loss lineitiscorrecttoassume c/>~==(a/{3)2«1.(5b) (5c) Since {3is 174 TRANSMISSION-LINE THEORY [Chap.III thesameandaisverysmallonbothlines,thedifference 4>em-4>eis negligible compared withblAx+blAvin(4). Sincethematching sectionoflengthymaybeconstructed tohavean extremely smallinputconductance bymakingitessentially reactive, the following condition iseasilysatisfied: With(5c)and(6),(4)becomes simply(6) blAx+blAY=0 (7) Thesearethefundamental conditions formatching theloadwithits matching network tothelongline.Theintroduction oftheterminal functions definedinSec.1into(7)permits thesetobeexpressed as follows: sinh2Ax =Rem cosh2Ax-cos2F~Re sin2F~+sin2F~=0 cosh2Ax-cos2F~1 -cos2F~(8a) (8b) whereAx==ax+Px,F~=={1x+<P~,andF;=(1y+<P~.Notethatpz and <P~aretheterminal functions ofZxconsisting ofZsinparallelwith thestuboflengthx.Equation (8b)maybewrittenasfollows: sinF~cosF~ I .h2A+ .2F'+cotFy=0sIn xsIn x(8e) Ingeneral, twovariables arenecessary tosatisfyboth(8a)and(8e). Depending onthechoiceofthesevariables, thematching network may besimplified inanyone ofseveralways.Initscomplete formthe following arbitrarily adjustable quantities areavailable: 1.Thelengthsw,x,andy,withtherestriction thattheattenuation a(w+x+y)mustbesmallcompared withPs. 2.Theterminating impedances ZxandZy,withtherestriction that theirattenuation functions pzandpymustbenegligible compared with theattenuation function Psoftheload. 3.Thecharacteristic impedance ofthematching line withinsomewhat narrowpractical limits. Thecondition that4>embeas smallaspossiblemustbeobserved. 8.TheSeriesTrans!ormer.81,97,lOlAsimpleformofthegeneral matching network isshowninFig.7.1.Itisderivedbyremoving the sections oflineoflengthwandyandselecting xandReasthevariahles. Sec.8] IMPEDANCE ANDADMITTANCE 175 ThismeansthatF~=1r/2andcotF~=O.AlsoZs=Zzistheentire impedance terminating thesectionoflengthx.ThegeneralEqs.(8a) and(8e)reducetothefollowing: sinh2Az Rem cosh2Az-cos2F~Re sin2F~=0(1) (2) Therearetwoinfinitesetsofsolutions. Fromthesethesolution with thesmallest physically possible valueofn=0,1,2,shouldbe chosen. Thesetsare F~=(2n+1)~ F~=n1rhARemtan x=R e hARemcot x=JI:(3) (4) SincetanhAx~1andcothAz~1,itfollowsthat,withnaninteger, Re=RemcothAx Re=RemtanhAzF~=(2n+1)~ F~=n1r(5a) (5b) Foraproperly designed matching section,xissufficiently shortanda sufficiently smallsothatitiscorrecttosetax«Px.Hence Ax=aX+pz==pz (6) Itisshowninalatersectionthat,whenthecondition ax«pzissatis fied,thequantity cothPxisequaltothestanding-wave ratioSzonthe matching sectionoflineoflengthx.Thus Sz=cothpz Withthisnotation theconditions formatchare(7) (9)(8) Re<Rem F~=n1rF~=(2n+1)~ Rc=~c: wherenisaninteger. Although simpleinform,(8)and(9)areconvenient onlyifRemis givenandRcistobedetermined. Usuallythereverseistrue.Inthis caseRemisnotdirectlyavailable from(8)or(9),sinceitisinvolved inpz throughtherelation where(10) (11) 176 TRANSMISSION-LINE THEORY [Chap.III Anexplicitformula forRemisreadilyderivedusing forF;=n1r forF~=(2n+1)~together with2tanhPx tanh2px=1+tanh2Px 1Re RemtanhPx=Rem Re(12) (13) If(10)and(12)areequated usingeitherofthetwoformsof(13)insuc cession,thefollowing relationmaybederived: whererlx==RxlReandXIx==XxlReo IfRe>Rem, IfRe<Rem,F'=2n+11rx2 F~=n1r(15) Inthespecialcaseofapurelyresistive termination Xx=0,<1>;=0for Rem>Rx,and ~=1r/2forRem<Rx.Hence Rem=yReRx{3x=lF~-~I=l2n::r~=2n:l1r F--n1r- - x2 2 for(Re>Rem>Rxl(16) Re<Rem<Rx From(1)itfollowsthat,ifRem=Re,amatchispossible onlyif (17) ClearlyitisnotpossibletoachieveamatchforallvaluesofRxandXx, sincetheradicalin(14)mustremainreal.Thisistruesubjecttooneof thefollowing setsofconditions: ForRx>Re, ForRx<Re,R~+X~>ReRx R~+X;<ReRx(18) (19) Evidently (18)isalwaystrue,whereas in(19)thevaluesofRxandXx arelimited. Theregionsintherlx,XIxplaneinwhichamatchispossiblearebounded bythefollowing curves: rlx=1 (20) ri:l:+xix-rb=0or(rlx-1)2+xfx={-(21) Sec.9] IMPEDANCE ANDADMITTANCE 177 Evidently (20)istheequation ofastraight line;ontheotherhand, (21)istheequation ofacirclewithcenteratrlx=j,XIx=0andwith radiusj.ThiscircleisshowninFig.8.1.Amatchispossible onlyif rlxandXIxhavevaluesthatlietotheright ofthelinerIx=1orwithinthecircle. Matching isimpossible ifrIxandXIxlie bothtotheleftofthelinerIx=1and outsidethecircle. Thecomplete solution forRemandX canbeexpressed asfollows(normaliza- tioniswithrespecttoRe,andgeisby r1X definition theratioRem/ Re): Rem==geRe=ReIrix+xix~rIx(22)\IrIx- Forge<1, -1L.L-~"""""'-"'-~"""""'...L.&._"'--""" FIG.8.1.Lociofmatchforseries sections; match possible for rlx>1orwithin circlefor rlx<1. (24) 9.Matching SectionwithaSingleMovable Stub.8l,I09Awidelyused formofthegeneralnetwork described inSec.7dispenses withthesec tionoflengthwbutretainsthestuboflengthy(seeFigs.9.1and9.2). Thepointsofconnection AA'aremademovable, sothatbothXandy Ax FIG.9.1.Singlestubontwo-wire line. FIG.9.2.Movable adjustable stubon coaxialline. maybeadjusted, or,inanycase,{3xand({3y+CPy)aremadeavailable. Itisconvenient touselinesofthesamecharacteristic impedance through out.Thatis,Ze=Zem,sothatgle=Zem/Ze=1.Accordingly Sec.7, Eqs.(7),become glAy«1glAx=1 (1) NotethatZa=Zz.Mtertheterminal functions Ax==aX+pzand 178 TRANSMISSION-LINE THEORY [Chap.III F~=={3x+<p~areintroduced, thefollowing equations areobtained: glA==glAz=sinh2Az=1 cosh2Az-cos2F~ b b+bsin2F~ F'0 lA=lAz lAy=h2A 2F'+cot y=cos z-cos z(2a) (2b) Thefirstofthese,(2a),maybesolvedforF~.Thus,aftercombining terms, cos2F~=(coshAz-sinhAz)2 Thesecond,(2b),gives tF'- _sin2F~co 1/-sinh2Az(3) (4) Thesetwoequations, (3)and(4),mayberearranged asfollows,using cotiu=y(1+cosu)/(1-cosu); cotF'=/1+cos2F~=/1+cosh2Az-sinh2Az(5) z~1 -cos2F~~1 -cosh2Az+sinh2Az Butwith cosh2u+1=2cosh2ucosh2u-1=2sinh2u(6) andwithaX«pzitisreadilyshownthat cot({3x+~)==±ycoth pz==±vIS or,with<Pz=~-f7r/2,(7) (8) Similarly, sincecosu=cotu/yl+cot2uandsinu=l/yl+cot2u, cotF~= Butsince itfollowsthat2sinF~cosF~_ 2cotF~ sinh2Az- -(1+cot2F~)sinh2Az ±2ycothAz (1+cothAz)sinh2Az . 2cothusmh2u=th2 1cou -(9) (10) _ycothAzcoth2Az-1 cotF~=+1+cothAz·cothAx-cothAz-1(11) +ycothAz Finally,setting F~={3y+~andAz=aX+pz,thefollowing formulas areobtained: cot({3y+<p~)=+[vcoth (ax+pz)-ytanh(ax+pz)](12) Sec.9] IMPEDANCE ANDADMITTANCE 179 (15)Sinceaxissmallcompared withpz, cot({3y+~)==+(ycothpz-ytanh pz) =+(VB-Js)=+8;-sl (13) Analternative formisobtained using cI>~=<1>11-1r/2.Itis 8-1tan({3y+<1>1/)=±VB (14) Itispossibletocombine (7)and(12)usingtherelation cot(a+(3)=cotacot{3-1 cot{3+cota Theresulting formula is cot({3x+(3y+cI>~+~)=+[coth(ax+Pz)]f==+(cothPz)f=+:8f (16) or cot({3x+(3y+<1>z+<1>1/)=+8f (17) Forapurelyresistive termination Zs=Rs=Rz,thefollowing relations obtain: Forrlz<1,pz=tanh-1rlz Forglz<1,pz=tanh-1glz Alternatively, sincerlz=I/glz,cI>z=~,orcI>~=1ror0 cI>z=0or1r,or~=~(18) Forrlz<1, Forglz<1,pz=coth-1glz pz=coth-1rlzcI>~=0 cI>'=!z2(19) Thefollowing formulas followdirectly: Forglz>1orrlz<1: cot({3x+cI>~)=cot{3x=±yglz cot({3x+(3y+<1>;+cI>~)=cot[(3(x+y)+~]=+:glZ~(20) Forrlz>1: cot({3x+cI>~)= -tan{3x=±vr: (21) cot({3x+(3y+<1>~+<1>~)= -tan[(3(x+y)+~J=+rlzf Thefollowing specifications areusuallyconvenient: a.Chooseanopenstub (<1>~==1r12)forrlz<1sothat tan{3x=±yrlz with+signforshortest length (22) -cot[(3(X+y)+~]=tan[{3(x+y)J=glzf (23) 180 TRANSMISSION-LINE THEORY [Chap.III (24) (25)Thefinalformulas forthecircuitinFig.9.1are (3x=tan-I~:=tan-Irix (R)' (1)~(3(x+y)=tan-IR:=tan-Iri x b.Chooseaclosedstub (<p~=(3ky)forTix>1sothat tan{3x==+v'rh: with+signforshortest length tan(3(x+y+ky)=-rlx~ tan[11"-(3(x+y+ky)]=rlx~ 11"(3(x+y+kll)=11"-tan-Irlx~="2+tan-Igtxf sincetan-Ix=11"/2-tan-1(l/x).Thefinalformulas forthecircuitof Fig.9.2are {3x=tan-I~:=tan-Irlx (3(x+y+kll)=11'-tan-I(~:)f=~+tan-Irlxl wherekyistheequivalent lengthoftheterminating bridge.Ifthisisa pistoninacoaxialline,ky=0;ifitisaconducting bridgeonanopen wireline,kll=ksa,withksaasdefinedinChap.II,Sec.20. 1.4 1.2 ~ ~1.0r---~--.,.-_~o x+y cO.So ~0.6 u.. 0.4 0.2 O~_-'-...I-~"""""'..L..LJ_----JL.--.J-..JL.-J.,....l...L-I...l..L._--l----I--'--l...JU-I..L.l-_--I---L.....1..o..L.L..LLU ~ ID ~ T1X=Rx/Rc FIG.9.3.Stubmatching withpureresistance termination. Casesaandbforapurelyresistive termination areillustrated inFig. 9.3.Numerical andgraphical solutions aredescribed belowforapar ticularproblem. nlustrative Example 1forSingle-stub Matching Given. Aloadimpedance Z.=Z'"=1,600+j800ohmstobematched toaline withcharacteristic resistance Rc=400ohms. Problem. Todetermine thelengthsxandyasfractions ofawavelength forastub withanidealclosedend. Sec.9] IMPEDANCE ANDADMITTANCE 181 Analytical Solution 1.Determination ofnormalized resistance andreactance: Xl",=X",=2Rc 2.Determination ofterminal functions: _ 1 -12rt.. p",-"!"tanh 2 2 + 1=0.2rb+xb _ 1 -1-2xt.. «1>",-"!"tan 2+21=174.1°rbXb- 3.Determination ofthelength Xusing(8): tan(flx+«1>",)=±vcoth P",=±2.26 flx=180°±tan-l2.26-174.1°=72° X=0.20>.. 4.Determination ofthelengthyusing(14): tan(fly+«1>1/)=+(VcothP",-Vtanh p",)=+1.814 fly+«1>1/=180+61.2°=118.8°(or241.2°) Fortheshortest lengthychooseanidealclosedendwith«1>1/=90°.Then fly=28.8° y=0.080>.. x+y=0.280>.. Graphical Solution ontheCircleDiagram (Fig.9.4).Tochangetonormalized admittance, 1.Compute rh:=4;Xl'"=2. 2.Locate rl",=4andXl'"=2onthechart,andnotethatthisisatP",=0.2and «1>",=174°. 3.Moveoncircleofconstant P",through 90°tolocate «1>:=84°forusewithgl'" andbl",. Toobtainmatch, 4.Firstcondition formatch:gh:=1.Movealongcircleofconstant P",=0.2to flx+«1>:=156°atgt",=1.(Notethatb l",=+1.8.) Thenflx=72°andx=0.20lA. 5.Secondcondition formatch:bl",=-bll/'Locate bl1l=-bl..=-1.8 ontheaxisofb.Readofffly+«I>~=29°.Fortheshortest lengthychooseanideal shortcircuitwith «I>~=0(<I/=11'/2)sothatfly=29°andy=0.080>..; X+y=0.28>... nlustrative Example 2forSingle-stub Matching Given.Aloadimpedance Z.=Z'"=3,200+jl,600ohmstobematched toa two-wire linewithcharacteristic resistance Rc=400ohms. Problem. Todetermine thelengths Xandyasfractions ofawavelength forastub withanidealopenendorclosedend. 182 TRANSMISSION-LINE THEORY [Chap.III o 2 3 4 Tl"l FIG.9.4.Graphical solution ofsingle-stub matching (Example 1). Analytical Solution 1.Determination ofnormalized resistance andreactance: Tl'"=R",=8Rc 2.Determination ofterminal functions:X'"Xlz=R c=4 3.Determination ofthelengthxusing(8): tan({jx+<1>",)=±v'coth p",=±3.16 {jx=180°±tan-13.16-177.1°=75.4° x=O.21X 4.Determination ofthelengthyusing(14): tan({jy+<1>,,)==1=(vcoth p",-v'tanh p",)==1=2.844 {jy+<1>"=180° =1=70.6°=109.4°(or250.6°) Sec.9] IMPEDANCE ANDADMITTANCE 183 Fortheshortestlengthychooseanidealclosedendwith4>11=90°.Then (jy=19.4° y=0.054). x+Y=0.264), 40 00177.10 4»(CZ>') FIG.9.5.Graphical solution ofsingle-stub matching (Example 2). Graphical Solution ontheCircleDiagram (Fig.9.5).Tochangetonormalized admittance, 1.Asbefore,rl'"=8;Xl'"=4. 2.LocateTl'"=8andXl'"=4onthechart.Notethatthisoccursatp",=0.1and 4>",=177.1°. 3.Moveoncircleofconstant p",through 90°tolocate4>:=87.10forusewithgl'" andbl",. Toobtainmatch, 4.Firstconditionformatch:g 1",=1.Movefromp", =0.1,4>:=87.1°top", =0.1, fJx+4>:=1620atgl'"=1.(Notethatbl",=+2.85.) Then(jx=75°andX=0.21).. 184 TRANSMISSION-LINE THEORY [Chap.III 5.Secondcondition formatch:he=-blY'Hencelocate bIll=-he=-2.85 (g1:J:=0) ontheaxisofb.Readoff{3y+<1>:=19.5°.Choose ~=0(y=71"/2)sothat {3y=19.5°andy=0.054:\;x+Y=0.264:\. 10.Matching SectionConsisting ofaDouble-stub Tuner.ll,I09For thedouble-stub tunerthelengthswandyareadjustable, whereasx,w, andyarefixedbutarbitrary. Allsectionshavethesamecharacteristic impedance Zcasthelongline.Thelengthswandymaybesochosenthat glCw«1blew= -cot({3w+~) (1) glAy«1buy= -cot({3y+;) (2) wherethenotation isthatofSec.7.Circuits foropenandcoaxiallines areshowninFigs.10.1and10.2.Theconditions formatchare gu=gux=1 bu=bux+buy=0(3) (4) Because thetermination ofthesectionoflengthxconsists oftheload Zs=Rs+jXsorYs=Gs+jBsinparallelwiththesectionoflengthw, itisnotconvenient torepresent theircombined impedance byterminal u===== (6)FIG.10.1.Double-stub matching network FIG.10.2.Double-stub matching network fortwo-wire line. forcoaxialline. functions. IfZx=Rx+jXxappliestoZsinparallelwiththestubof lengthw,thehyperbolic formoftheadmittance YAxlookingintotheline oflengthxis sinhyx+Ylxcoshyx1+Ylxcothyx (5) YlAx=coshyx+Ylxsinhyx=.cothyx+YIx whereYux=YAx/YcandYlx=Yx/Yc.Sincethelengthxistobekept smallanditstermination includes theload,itisagoodapproximation tosetax«Pxand{3x~n7r.Thencothyx==-jcot{3x,sothat 1 -jylxcot{3xYlAx= .Ylx-Jcot{3x where Ylx=glx+jblx=glew+gl.•+j(blCw+bIs)==gls+j(blCw+bI.•)(7) and blew= -cot({3w+~) gICw«gls (8) Sec.10] IMPEDANCE ANDADMITTANCE 185 Itfollowsthat _+'b-1 -j(glx+jblx)cot{jx (9)ylAx-glAxJlAx-+'b.tRgIxJIx-Jco/-IX Theseparation ofrealandimaginary partsleadsto gIxCSC2{jX glAx=grx+(cot{jX_blx)2 (10) b-g~xcot{jx-(cot{jx-b1x)(1+bixcot(jx) (11) lAx- g~x+(cot{jx-blx)2 Application ofthecondition formatch(3)gives gIxcsc2j1x=grx+(cotj1x-bIx)2 (12) Thisequation mayberearranged asfollows: (gIx-jsec2j1X)2+(cotj1x-bIx)2=(jcsc2{1X)2 (13) Thisistheequation ofacirclewithcenterat andwithradius1bix=cot{1x (14) (16)(15) 371' 4 2R=1 2sin2j1x Thecircle(13)passesthrough thepointb1x=0,gIx=1.Asisshown later,especially usefulvaluesofj1xare1r/4,1r/2,31r/4.Forthesethe following tableapplies: ~xl 1 i~-----~------- --------- Centerat ,'"'jglx=1b1x=1-},0_1~ Radius " . , " . 1 -} 1 Theassociated circlesareshowninChap.II,Fig.16.2.Forafixedvalue ofXamatchispossible onlyifglxandbixlieontheappropriate circle. SincegIx=gI8isnotadjustable, whereas bix=bls+blewisadjustable inblew,amatchcannotalwaysbeachieved withagivenvalueofimped anceandagivenvalueofx. Thetwoquantities tobedetermined, F~=j1w+<t>~andF~=j1y+<t>;, arereadilyevaluated fromblewandblAY'Inordertodetermine blewitis possibletosolvetheequation glAx=1forbix=blCw+bls•Thus gIxcsc2j1x=gIx+(cotj1x-b1x)2 (17) cotj1x-bix=±glxI~2{1-1 (18)'\jgIxSInx blew=-bI8+cotj1x+gIxI,I2{j-1(19)'\jgIxsmx 186 Accordingly, withTRANSMISSION-LINE THEORY [Chap.III and blew= -cot({3w+<fl~) (20) itfollowsthat cot({3w+<1>~)=bls-cot{3x+glBI.12{3-1(21)"\jglssmx Inordertodetermine buy=-bux,(11)mayberearranged. Thus, using(12), buy=-bux=(cot{3x-bIz)(l+bIz;ot(3x)-gixcot{3x(22) gIzcsc{3x Theelimination ofbIzfrom(22),usingbb=blew+blsandgb=gIs, gives,afterconsiderable rearrangement, -buy= -cot{3x±I.12{3-1 (23)"\jglssmx Since,ingeneral, blew= -cot({3w+<1>~) (24) itfollowsthat(19)isequivalent to -blew=cot({3w+<fl~)=bls-cot{3x±gls/.12{3-1(25)'\Jglssmx Similarly, for(23), -buy=cot({3y+<1>~)= -cot{3x±I.12{3-1(26)"\jglssmx Intheseformulas theuppersignsgotogether, asdothelowersigns. Sincetheradicalmustbereal,therestriction onglsis glssin2{3x~1 (27) Thisisgreatest for{3x=7r/2andleastfor{3x=7r.However, noadjust mentinbuyispossible for{3x=7r,andtheadjustment isextremely deli catefor{3xnear7r;furthermore {3x=n7rwasexcluded whenattenuation wasneglected. Agoodcompromise, whichprovides areasonable range andaccurate adjustment, is{3x=7r/4or37r/4.Notethat{3x=7r/2 givesthesmallest range.Forpistonsincoaxialstubs <1>~=0=<1>~; forbridged two-wire lines<1>;={3kya;<fl~={3kwawherekyaorkwacanbe madesmall.Thesethreeimportant casesgive (a) {3x=!sin2{3x=icot{3x=14 cot({3w+<fl~)=-1+bls±gls/2- 1'\Jgls cot({3y+<1>~)= -1±I2 - 1'\jgls(28) (29) (30) Sec.10] IMPEDANCE ANDADMITTANCE 187 Sincetheradicalmustbereal,itfollowsthat gIs<2 (31) (b) {Jx=311"sin2{3x=i-cot{3x=-1 (32)4 cot({3w+4>~)1+bIs±gIs~2 - 1 (33)gb cot({3y+4>~)1±~2- 1 (34)gls Asbefore, gIs<2 (35) (c) {3x='!.sin2{3x=1cot{3x=0 (36)2 cot({3w+4>~)=bIs±gIs~1-1 (37)gls cot({3y+4>~)=±~1 - 1 (38)gb gIs<1 (39) Foridealpistons 4>'=O.Asusual,uppersignsgotogether, andlower signsgotogether. Instead ofcalculating thelengthswandyofthedouble-stub tuner usingtheappropriate formulas derivedabove,acirclediagram maybe used.Thisisespecially convenient ifthe'A/8and3'A/8matching circles areprovided asinChap.II,Fig.16.2.Thegeneralprocedure follows: 1.Entercirclediagram atgivenvaluesofglsandblsofload. 2.Adjustlengthwofstubinparallelwithloadsothatitssusceptance blewmakesthecombined admittance gIsandbb+blew=bIzfallonthe circleappropriate tothegivenvalueof{3x,say,atP.Thisdetermines w.ThepointPhascoordinates pz=Psand 4>~. 3.Add/3xto4>~togive{3x+4>~.Then{3x+4>~andpz=Psarethe terminal functions looking towardloadatjunction withsecondstub. Thesevaluesof/3x+4>~andpzoccuratguz=1andthesamevalue ofbuz. 4.Adjustlengthyofsecondstubsothatbuy=-buz;thatis, buy+buz=O.(Thesecondstubisusedtotunetheinputsuscep tancetozero.Thefirststubhasbeenadjusted togiveaninputconduct anceofunity.) Thisdetermines y. Numerical andgraphical solutions aredescribed belowforagiven problem. nlustrative Example forDouble-stub Matching Given.Aloadimpedance Z.=1,600+j800ohmstobematched toatwo-wire linewithcharacteristic resistance Rc=400ohms. Problem. Withlengthxfixedbutarbitrary, todetermine lengths yandzas fractions ofthewavelength forstubswithidealclosedends. 188 TRANSMISSION-LINE THEORY [Chap.III 90° 0° ct>(') FIG.10.3.Graphical solution ofdouble-stub matching usingSmithchart. Analytical Solution 1.Determination ofnormalized impedance andadmittance: rIa=0.2 gIa=r;8+X;8XIa=X8=2Rc 2.Determination ofthelengthwusing(19)and(20)withxfixed.(For)./8 double-stub spacing, (:Jx=11'"/4.)Firstcondition formatch: glA=guo:=1. orglO:=gIa=0.2 blew=-bIa+cot{3x+gb,yr----=.1- 2-{3----1 = {0 1.5 7 glo:smx . bl>:=bIa+bww=-0.1+{~:~= {~:: bww=-cot({3w+~) ~=0 fJw=116.6° w=0.324). fJw=149.5° w=0.415). 3.Determination ofthelengthyusing(23)and(26).Secondcondition formatch: bu=buo:+bUll=O. Sec.10) IMPEDANCE ANDADMITTANCE br.b=-bull= -cot(3x±.y.12(3-1gIasmx -b1Az=cot«(3y+~) ~=0 (3y=26.6° y=0.074>' or (3y=165° y=0.458>'{2.00 -4.00189 Graphical Solution ontheCircleDiagram (Figs.10.3and10.4).Tochangeto normalized admittance, 1.Compute ria=4;XIa=2. 2.Locateria=4andXla=2onthechart,andnotethatthisoccursatp,=0.2 and,=174°. 3.Moveoncircleofconstant p,through900tolocate ;=840forusewithYIa=0.2 andbla=-0.1. Toobtainmatch, 4.Firstconditionformatch:gu =guz=1.AddblcwtobIatomakeblCw +bl,=bIz 4 3 o bl--4 atih=1 FIG.10.4.Graphical solution ofdouble-stub matching usingcirclediagram.-1H---\\:---+-+t----t----+------l -2~~~-r~--+----1---~ 190 TRANSMISSION-LINE THEORY [Chap.III intersect the>../8circleatp=0.16(or0.06), ~tD=112°(or148°),giving bICtD=0.5 (or1.7),corresponding tofJw=116.6°(or149.5°);w=0.324>..(or0.415>..). 5.Secondcondition formatch: blA=blAz+bUll=O.Add45°to~tD=112° (148°)oncircleofconstant p=0.16(orp=0.06)toreachtheglA=1line,where buz=2.00(or-4.00). 6.Locate blAlI=-blAz=-2.00(or4.00); glA=O.ReadofffJy+~=26.5° (or165°).Fortheshortest lengthychooseanidealshortcircuitwith ~=0sothat fJy=26.5°forblAlI=-2.00(fJy=165°forbUll=4.00);y=0.074>..(ory=0.458>..). (Ifanidealopencircuitwith<1>'=90°ischosenwhenfJy+<f>~=165°,itfollowsthat fJy=75°andy=0.208>".) 11.Matching withaShuntSection.tAlesswidelyusedcircuitfor matching consists ofasectionoflineoflength 82whichisconnected in parallelwithalength 81,formingpartofalonglinethatisterminated inanimpedance Zs,asshowninFig. _____ A8111 11.1.Zsmaybetheinputimpedance Rc+~Z6 ofanadditional sectionoflinetermi-B~ natedinanarbitrary impedance. I 82 Anexactanalysis oftheshuntmatch- I ingnetwork involves thecoupling be- t-:------.~z tweenthetwoparallelsections inaddi- Z"O tiontotheusualterminal-zone and FIG.11.1.Shuntsections oftrans-junction effects. Inthepresentanaly mission lineforimpedance match-sisitisassumed thatthecoupling being.tweenthetwoshuntsectionsisnegligi- ble.Foropen-wire linesthismaybeapproximated bysufficiently small linespacingcompared withtheseparation ofthetwosections; forcoaxial orshielded-pair linesitissatisfied automatically. Thefirststepindetermining theconditions formatchinthecircuitof Fig.11.1istoderiveanexpression fortheimpedance lookingtotheright atAB.Thisisaccomplished byapplying Chap.II,Sec.8,Eqs.(6) and(7),toeachoftheshuntsectionsthatareassumed tohavethesame lineconstants. Letthevoltageandcurrentintheundivided lineat ABbeVoand10,andattheload,VsandIs.Itfollowsthat (1) Also (2) (3a) VBsinh'"(82+(1.-IB1)Zecosh'"(82 C3b)Vo=V.cosh'"(81+Is1Zesinh'"(81=Vscosh'"(82+Is2Zesinh'"(82 110Ze=VBsinh'"(81+IB1Zecosh'"(81 120Ze=VBsinh'"(82+IB2Zecosh'"(82= Theaddition of(3a)andC3b)andtheelimination oflsIusing(1)and tPartsofthissectionfollowtheworkofTaLlOl Sec.11] IMPEDANCE ANDADMITTANCE 191 (2)give loZ=2V8[cosh"(81+82)-1]~l8Ze[sinh "(81+82)] (4) e sinh"(81+smh"(82 Theelimination of1stfromtheleftsideof(2)gives V-V8sinh"(81+82)+l8Zesinh"(81sinh"(82 (5) o - sinh"(81+sinh"(82 Thenormalized impedance lookingtotherightatABinFig.11.1is ZoVo Z1Bsinh"(81+82)+sinh"(81sinh"(82 Z10==Ze=loZe=2z18[cosh"(81+82)-1]+sinh"(81+82)(6) Thenormalized terminating impedance ZlB==ZlB/Ze=V8/l8Zemaybe expressed as Z1B=coth9B (7) where98=PB+jifJ8isthecomplex terminal function. If(7)issubsti tutedin(6)andthisisrearranged, theexpression belowmaybeobtained. Sincethesections oflineareassumed tobehighlyconducting andquite short,itissatisfactory toneglect a(81+82)ascompared withP8ifZ8is adissipative load,asisassumed. Itfollowsthat"(=a+j{j==j{j. Thedesiredexpression is cosh298cos{381cos{382-cosh(98+j(381)cosh(9B+j(382) Z10=sinh298-sinh[29B+j{3(81+82)]+jcosh298sin(3(81+82)(8) Thecondition formatch Z10=1maynowbeimposed, andthefollow ingsolutions oftheresulting pairofequations obtained: cos2ifJB-e-2Pa 2sin2ifJ8 =[(5e-2pa+3cos2ifJ8)(e-2Pa-cos2ifJ8)]1 4(1-e-4Pa)(9) (10) Theseequations definethelengths 81and82ofthetwoshuntsections fordifferent valuesoftheterminal function 98=P8+jifJ8•Notethata physically meaningful solution requires realandpositive valuesofboth 81and82. Sincetherightsideof(9)isalwaysreal,thereisnorestriction on 81+82.Apositive realvalueof81+82canalwaysbefound. Onthe otherhand, 81-82isrealonlywhen P8andifJ8satisfythefollowing equation: (11) Itfollowsthatanormalized impedance Z10canbematched byapairof 192 TRANSMISSION-LINE THEORY [Chap.III shuntsections withtheirjunction attheterminal impedance onlyifPs and<JIscorresponding toZ1sin(7)satisfy(11).Thecorresponding equa tionsare T1s=1 4(T~s+x~s)-5T1s+1=0(12a) (12b) 271"81T=(381=tan-1A+\an-1B(13a) 271"82T=(382=tan-1A-tan-1B(13b) whereAisthequantity ontherightin(9) andBisthequantity ontherightin(10). Aplotof8dAand8dAisgiveninFig.11.3, with<JIsasvariableandPsasparameter. The rangeof<JIsislimitedto0~<JIs~90°,since itisclearfrom(9)and(10)that180°-<JIs ~90°givesexactlythesamevaluesofAandBforuseEvidently (12a)istheequation ofaverticallineintheTlsX1splane,and (12b)istheequation ofacircleofradius-lwithcenteratTls=i,X1s=o. Thesecurves(Fig.11.2)definetworegions. Amatchispossible only forvaluesofTlsandX1stotheleftoftheline T1s=1,butnotincluding thecircleofradius-l. Amatchisnotpossible ifT1sandX1sareto therightofthelineT1s=1orinthecircleto 'Xs theleftofthisline. Explicit expressions for(381and(382maybe obtained bysolving(9)and(10).Theyare -1L-.....I.---L._l.-....I.---L.""--Lu....u FIG.11.2.Lociofmatchfor shuntsections; matchpos sibleonlyoutside cross hatched area. for180°~<JIs in(13a,b). Rlustrative Example 1forMatching withShuntSections Given. Aloadimpedance Z.=200-j400ohmstobematched toatwo-wire linewithcharacteristic resistance Re=400ohms. Problem. Todetermine thelengths SIandS2asfractions ofawavelength using thesametypeoflinethroughout. Solution 1.Determination ofnormalized resistance andreactance: r18=~=0.5 2.Referring toFig.11.2,itisseenthatthispointlieswithintheregionofpossible match. Fromthecirclediagram orbycomputation, thevaluesofP.and<1>.corre sponding tothegivenvaluesofrl.andXl.arefoundtobe0.24neperand41.5°, respectively. 3.FromFig.11.3therequired valuesofsd"Aandsd"Aare0.572and0.350. Sec.11] IMPEDANCE ANDADMITTANCE 193 Rlustrative Example 2forMatching withShuntSections Ifamatchisnotpossible withshuntsections terminated directly intheload,an additional sectionoflineofappropriate length S3maybeinserted between thejunc tionoftheshuntsectionsandtheloadZa. Given. Za=800+jOohmstobematched toatwo-wire linewithRc=400. Problem. Todetermine thelengthsSdAandS2/Aasfractions ofawavelength and Sa/Aiftheadditional sectionisrequired. Solution 1.Thenormalized impedances arerh=2andXh=O.Thecorresponding termi nalfunctions arePa=0.55andcf>a=o. \ "- ..............----- -siiA --...........---sIA ........ 1020304050607080900.1I----f------+---+---+--+---+-----,I--~-J-~~0.8.....--..,-----..--.,.---.----r--"T"""""-.-r----,---, 0'-_.J-_---.l __-L-_-1.__.L-_-1..__L-_-.l-_-.l o0.7I----+---+--.f---I--t~ cf> FIG.11.3.Contours ofconstant SdAandSdAasfunctions ofpandcf>foruseinimped ancematching withshuntsections. 2.Referring toFig.11.2,itisseenthatthepointinquestion liesoutsidetherange ofpossible match. Letanadditional sectionoflength 83=A/8or/3sa=45°be inserted. 3.Thevaluesofpandcf>terminating theshuntsectionsattheirjunction ata distance A/8fromtheloadarep=Pa+a8a=0.55andcf>=cf>a+/3S3=45°.(Line lossesareneglected.) 4.Thenewvaluesofpandcf>arewithintherangeofmatch. FromFig.11.3, sdA=0.535andsdA=0.414. Itistobenotedthatjunction, coupling, andterminal-zone effectshavebeen neglected inthissection, sothatthequantitative accuracy oftheformulas inany particular application depends onthedegreetowhichtheseeffectsaresignificant.. Corrections forthemwhentheyarenotnegligible areconsidered inChap.V. 194 TRANSMISSION-LINE THEORY [Chap.III (a)12.Representation ofaSectionofTransmission LinebyLumped Equivalents; Impedance, Admittance, andScattering Matrices.16The "lumped equivalent" ofthesectionoftransmission linetotherightof thepoints11'inFig.12.Iaisanycombination oflumpedelements which, whenconnected across11'inplaceofthesectionoflineasinFig.12.Ib, leavesalldistributions ofcurrentandvoltageatallpointsalongtheline totheleftof11'unaltered. Actually nosuchnetwork canbeprovided inpractice inanygeneralsense,sinceevenapproximately lumped ele mentsdonotexistindependent offrequency. Evenintherestricted senseoftheresponse toasinglefrequency, whichisusuallyimpliedwhen an"equivalent lumped" network istobesubstituted forasectionof line,itisnotpossibletoprovidesuchanetwork owingtothechangein thecoupling between thelineandthenetwork-distributed orlumped totherightof11'.Endeffectsandcoupling effectsinthevicinity of 11'inthecircuitofFig.12.1adifferfromthoseinthecircuitofFig. 12.Ib.Itfollowsthatitisnotactu- ;1 allypossible tomaintain thesamec: tl~ distribution ofcurrentandcharge ',4Junctionzonesalongtheentirelinetotheleftof11' inthechange fromFig.I2.lato e~===========~! ::;ZeQUIv Fig.12.1b.Thebestthatcanbe :I'(b) achieved istokeepthedistribution FIG.12.1.Lumped equivalent oftwo-totheleftofajunction zoneof terminal sectionoftransmission line.lengthnear10timesthelinespac- ingfrom11'unchanged, whereas the distribution inthejunction zonemaybequitedifferent. Thismeans thattheapparent impedance terminating thelineat11'asseenfroma sufficient distancetotheleftof11'isthesameinthetwocases. Amoregeneralproblem isthe.substitution ofalumpedequivalent for apieceoflinebetweentwosetsofpoints,11'and22'inFig.12.2a,sothat excluding shortjunction regionsallcurrents andvoltages totheleftof 11'andtotherightof22'areunchanged. Itisassumed thattheequiva lenceistoapplyonlyatasinglefrequency. Thevoltageandcurrentinanycross-sectional planew=8 -zalong atransmission linemaybeexpressed intermsofthevoltageandcurrent atanother planes(wheres~z)inthefollowing form,asobtained from Chap.II,Sec.8,Eqs.(6)and(7): Yew)=YeO)coshyw+I(O)Zcsinhyw (Ia) lew)=V(O)Ycsinhyw+1(0)coshyw (lb) Alternatively Yew)=V(O)A+I(O)B (Ie) lew)=V(O)C+I(O)D (ld) where A=D=coshywB=Z~C=Zcsinhyw (Ie) Sec.12J IMPEDANCE ANDADMITTANCE 195 Zcisthecharacteristic impedance, andYc=1/Zcisthecharacteristic admittance oftheline;w=s-zisthedistance measured fromthe pointstowardz,asshowninFig.12.2. , w d 0 II Iw-I I I Iscal~ I I I I I Io z s-dss+d I I I I II-+zscale I I I II I I I I (a) (l? 2d2'i Junctionzones1ft:i\:~~------~lumped ~----'<l ~ -network -----!t-------::I~:l' 2'I (b) FIG.12.2.Lumped equivalent offour-terminal sectionoftransmission line. 2' 0' I'(b)(2a) (2b)Yew)=V(8)(W)+V(a)(W) lew)=[(8l(W)+[(a)(w) wherethesymmetrical combination consistsofthatpartofthevoltage whichisoddandthatpartofthe currentwhichisevenwithrespect totheplanew=0,andtheanti-(e) symmetrical combination consists FIG.12.3.Equivalent Tnetwork ofsec- tionoftransmission line.ofthecorresponding evenpartof thevoltageandtheoddpartofthecurrent. Thispairingofthecompo nentsisnecessary, sincelew)"-'aV(w)/a(w). Thefollowing explicit formulas arederived from (la,b):Consider theproblem ofrepresenting thelengthofline2dbetween the terminal pairs11'and22'byanelectrically equivalent four-terminal net- workoflumpedelements, asinFig. 1 1(0) 2 12.3.Theproblem isreadilyana- I(d)...'+....----+~---- ..+••.l(-d) lyzedusingsymmetry andtheodd V(d) V(O) V(-d) andevencomponents ofcurrent1'· ·2'andvoltagereferred tothecross- 1.--------- 2d-------~ sectional planez=sorw=0at(a) thecenterofthelength2d.Thus lettheactualcurrents andvoltages alongthelinebeseparated into symmetrical andantisymmetrical combinations suchthat 196 TRANSMISSION-LINE THEORY [Chap.III Symmetrical Combination Forevencurrent: I(s)(w)=j[I(w)+I(-w)]=1(0)cosh"(w (3a) Foroddvoltage: VCs)(w)=j[V(w)-V(-w)]=I(O)Zcsinh"(w (3b) Forsymmetrical admittance: I(B)(w) Yi~(w) ===VCB)(W)=Yocoth"(w (3c) Antisymmetrical Combination Foroddcurrent: ICa)(w)=j[I(w)-I(-w)J=V(O)Ycsinh"(w (4a) Forevenvoltage: VCa)(w)=j[V(w)+V(-w)]=YeO)cosh"(w (4b) Forantisymmetrical impedance: VCa)(w) Zi~)(w)=I(a)(w)=Zccoth"(W (4c) Notethat V(-w)=VCa)(w)-V(s)(w) (5a) I(-w)=-ICa)(w)+ICB)(w) (5b) Itisnowinteresting tonotethatthenormalized symmetrical admit tancelookingtotherightattheterminals 11'atw=dinFig.12.3a,viz., Yi~~(d)=coth"(d (6a) isprecisely thenormalized inputadmittance ofasectionoflineoflengthd whenitsendisanidealshortcircuit. ThisfollowsfromSec.1,Eq.(lOb), ' withw=dand 9~=O.Similarly thenormalized antisymmetrical impedance, viz., zi~~(d)=coth"(d (6b) isthenormalized inputimpedance ofasectionoflineoflengthdwhenits endisanidealopencircuit. ThisfollowsfromSec.1,Eq.(lOa),with w=dand9s=O.Accordingly thecurrentI(d)=ICB)(d)+ICa)(d)and thevoltageV(d)=V(s)(d)+V(a)(d)attheterminals 11'ofthelinesec tionarethesumsofthecurrents andvoltages thatwouldbeobtained if thelinesectionwereprovided successively withperfectshortcircuitsand perfectopencircuitsatw=0inFig.l2.3a.Thesymmetrical combi nationisthesolution oftheshort-circuited lineoflengthd;theanti symmetrical combination isthesolution oftheopen-circuited lineof lengthd.Ineachcasethesectionoflineisequivalent toasinglelumped admittance orimpedance, givenby(3c)or(4c). Sec.12J IMPEDANCE ANDADMITTANCE 197 Inordertoobtaintheequivalent circuitoftheoriginalsectionofline, letthesymmetrical (short-circuit) impedance andtheantisymmetrical (open-circuit) impedance berepresented asfollows: Zf~)(d)==Zll-Z12=Zctanhyd Z~:)(d)==Zll+Z12=Zccothyd(7a) (7b) wherethenewlyintroduced impedances ZllandZ12aredefinedby(7a) and(7b).Thus Zll==MZ~:;(d)+Z~:)(d)J=;c(tanh-yd+coth-yd) (7c) Z12==MZ~:)(d)-Z~~(d)]=;"(cothyd-tanhyd)=Zccsch2yd(7d) Itisnowreadilyverifiedthattheequivalent circuitofthetransmission linesectionoflength2dinFig.12.3aistheTnetwork inFig.12.3borits moresymmetrical equivalent inFig.12.3c.Sincethegeneralcaseisthe superposition ofthesymmetrical andantisymmetrical caseswhenthenet workisshort-circuited andopen-circuited atitscenter,itisnecessary merelytodemonstrate thatFig.12.3cyieldsthecorrectshort-circuit and open-circuit impedances. Thatthisistrueisseenbyinspection. It followsthatthefour-terminal network inFig.12.3borinFig.12.3cis theequivalent ofasectionoftransmission lineoflengthdiftheimped ancesareassigned thevaluesspecified in(7c,d). TheImpedance Matrix. Ifconventions regarding thesignsofthe voltages andthedirections ofthecurrents areadopted toconform with Fig.12.3c,itfollowsthat,with(2a,b)and(5a,b), VI=V(d)=V(O)cosh-yd+I(O)Zcsinh-yd (8a) V2=V(-d)=yeO)cosh-yd-1(0)Zcsinhyd (8b) II=I(d)=YeO)Ycsinh-yd+1(0)cosh-yd (9a) 12=-I(-d)=V(O)Ycsinhyd-1(0)cosh-yd (9b) Notethatthepositivedirection ofthecurrentatterminals 22'ischosen opposite inthelumpednetworktoconform withconvention. Bysolving (9a)and(9b)for1(0)andyeO)andsubstituting thesevaluesin(8a,b), thefollowing well-known equations areobtained fortheTnetwork: VI=I1Z11+12Z12 V2=I1Z21+12Z22(lOa) (lOb) whereZ21=Z12forallreciprocal elements andZ22=Zllinthisparticular case,sincethenetwork issymmetrical withidentical sides. ThetwoEqs.(IOa,b)maybeexpressed inthefollowing matrixtform: v=ZI tSeeRef.6forabriefdiscussion ofmatrices.(lla) 198 TRANSMISSION-LINE THEORY [Chap.III whereVandIarethecolumnmatrices givenby (Ub) CUe) (13b)(13a)Q' p'• •p' Q' 1+-- - - -2d-- - ----I +..---+-----+-----+-(a)andZisthesquareimpedance matrix Z==[ZllZ12] _Z21 Z22 Inthisrepresentation, usinganimpedance matrix,apointofviewisim- p Q pliedinwhichcurrents aretreated I(d-r·+-------- .....+.../(-d)asfundamental orgivenquantities V(d) V(-dl andthevoltages arecalculated from themusingthematrix. TheAdmittance Matrix.Ifde sired,thepointofviewmaybe changed, thevoltages maybe treatedasfundamental orgiven, andthecurrents maybecalculated fromthemusinganadmittance ma trix.ForthispurposetheIInet workinFig.12.4boritssymmetri callydrawnequivalent inFig.12.4c isconvenient. Itisevidentthat (8a,b)and(9a,b)applydirectly if thesignofV2in(8b)andof12in (9b)arereversed toconform with..---p.,-----Q.,-.... theconventions ofFig.12.4.Itis (e) readilyverifiedwith(6a,b)thatthe FIG.12.4.Equivalent IInetwork ofsec- tionoftransmission line. symmetrical orshort-circuit admit- tance Y~:;(d)=1/Z~:;(d)andthean- tisymmetrical oropen-circuit admittance Y~~)(d)=1/Z~~)(d)aregivenby Y~:;(d)=Yccoth"fd==Yll+Y12 (12a) Y~~)(d)=Yctanh"fd==Yll-Y12 (12b) whereYllandY12aredefinedin(12a,b). Theyare Yll=;c(tanh"fd+coth"fd) Y12=;c(coth"fd-tanh"fd)(b) (14a) (14b)11=V1Yll+V2Y12 12=VtY21+V2Y22Thecorresponding equations areobtained fromFig.12.4orfrom(9a,b), with1(0)andyeO)eliminated using(8a,b)(withsignsofV2and12 reversed) .Theyare Sec.12] IMPEDANCE ANDADMITTANCE 199 whereY21=Y12fromreciprocity andY22=Yllfromsymmetry. The direction ofthecurrentandthesignofthevoltageatterminals 2have beenreversed inFigs.12.4band12.4cascompared withthoseinFigs. 12.3band12.3c(thecurrentisnowinthesamedirection asinthetrans mission-line sectioninFig.12.4a,butthevoltageisreversed) inorder thatthefinalequations maybeintheform(14a,b),andY12(ratherthan - Y12)istheserieselementinthecircuitsofFigs.12.4band12.4c. Thecurrentequations inmatrixformare where1=YV I=[~:] v=[~:](15) andwheretheadmittance matrixis y=[YllY21 Ifthesignconventions usedfortheimpedance matrixareadopted, y= [Yll-Y21(16) (17) (18a) (18b)TheScattering Matrix. Instead ofrepresenting asectionoflineby equivalent lumpedimpedances oradmittances, itisadvantageous for someapplications tointroduce itsreflecting andtransmitting properties. Thisisaccomplished byexpressing currentandvoltageintheexponential formgiveninChap.I,Sec.13.Withslightchangesinthenotation to suitpresentrequirements, thesolution isasgiveninChap.I,Sec.13, Eqs.(13)and(14),namely: V(z)=VZc(Ae-Yz+BeYz) l(z)=YYc(Ae-Yz-BeYz) whereAandBarearbitrary constants tobeevaluated fromtheboundary conditions, "(isthecomplex propagation constant, Zcisthecharacteristic impedance, andYcisthecharacteristic admittance. Asdiscussed in Chap.I,thefirsttermin(18a)and(18b)represents awavetraveling in thepositivezdirection, andthesecondtermrepresents awavetraveling inthenegativezdirection. Fromthepointofviewofthetraveling-wave description givenin Chap.II,Secs.6and7,thesectionoflinebetween terminals 11'and 22'inFig.12.2mayberegarded asreceiving atraveling waveofdiffer entamplitude andphaseapproaching fromeachside.Apartofeach incident wavemaybereflected andaparttransmitted intothesection. Letthecomplex amplitudes oftheincident wavereaching terminals 11'fromtheleftbeAi,andletthoseofthewavereaching terminals 22' 200 TRANSMISSION-LINE THEORY [Chap.III fromtherightbeA2•Letthewaveleaving11'andmovingtowardthe lefthaveanamplitude Bl,andletthatleaving22'andmovingtoward therighthaveanamplitude B2•Aportionofthewaveleaving11', namely, SUAl,isthereflected partoftheincident waveAl;therestis thewavethathastraveled from22'to11',whereitemerges withampli tudeS12A2.Owingtothelinearity oftheequations thesetwoparts combine linearly. Thus Bl=SUAl+S12A2 Similarly, ontheotherside, B2=S2lAl+S22A2(19a) (19b) Inthesetwoequations theS'sarecomplex coefficients thatcharacterize thenetwork between terminals 11'and22'. Inthesimplecaseunderdiscussion thesectionbetween 11'and22'is liketherestoftheline.Therefore itispossibletowritedownformulas fortheS'sdirectly. Owingtothefactthatat11'thelinecontinues smoothly, thereisnoreflection oftheincident Al,sothattheentireout goingwaveistheemerging wavethathadtheamplitude A2at22'and mustbeequalto (20a) at11'.Similarly Itfollowsthat,forthesmoothlinebetween 11'and22',(20b) Su=0 (21) Thegeneralcaseof(19a,b)maybeexpressed inmatrixform.Thus thescattering relationis whereB=SA B==[~~] A==[~J(22) (23) andwherethescattering matrixisdefinedby (24) (Notethatthismatrixhasbeendefinedintermsofthesignanddirection convention forvoltageandcurrentwhichagreeswiththatusedindefining theimpedance matrix.) Inthesimplecaseathand S=[0e-2Yd ]e-2yd0(25) Sec.12] IMPEDANCE ANDADMITTANCE 201 Thematrixelements inthegeneralmatrix(24)maybeinterpreted as follows: 811isthecomplex amplitude ofthewavethatisreflected (orscattered) attheterminals 11'whenawaveofunitamplitude isincident onthese terminals. 812isthecomplex amplitude ofthewaveemerging at11'whenawave ofunitamplitude isincident ontheterminals 22'. 822isthecomplex amplitude ofthewavethatisreflected (orscattered) attheterminals 22'whenawaveofunitamplitude isincident onthese sameterminals. 821isthecomplex amplitude ofthewaveemerging at22'whenawave ofunitamplitude isincident ontheterminals 11'. Forreciprocal elements 821=812•Fornonreciprocal elements such asgyrators, 821~812•Theidealgyratorischaracterized by821= -812, 8p=822=O.Thenotation T:::::812iscommon, andthisquantity is calledthetransmission coefficient. Evidently 811and822arereflection coefficients. Notethatinthesimplespecialcasedescribed by(25)the reflection coefficients arezero. Although inthesimplecaseathandthecharacteristic impedances of thelinestotherightandleftofthejunction arethesame,thedefinition ofthescattering matrix(24)isvalidwhentheyaredifferent. This moregeneralcaseisconsidered inChap.V,Sec.4. Relations between Impedance, Admittance, andScattering Matrices. Theterminals 11'arelocatedatz=s-d,andtheterminals 22'at z=s+d.Itfollowsfrom(18a,b)that VI=V(s-d)=VZc[Ae-y(s-d)+Bey(s-d)]=VZc(AI+Bl)(26a) V2=V(s+d)=VZc[Ae-y(s+d)+Bey(s+dl]=vz:(A2+B2)(26b) 11=I(s-d)=vY:[Ae-y(s-d) -Bey(s-d)]=vY:(AI-Bl)(27a) 12=-/(s+d)= -vY:[Ae-y(s+d) -Bey(s+d)]=vY:(A2-B2)(27b) whereAl=Ae-y(s-dl, Bl=Bey(s-d),A2=Bey(s+d), B2=Ae-y(s+d). With(19a,b)theB'smaybeeliminated, andthefollowing equations obtained: VI='\IZc[(1+811)Al+S12A2] V2=VZc[S2lAl+(1+S22)A2] 11='\lYe[(1-811)Al-S12A2] 12=vY:[-S21Al+(1-S22)A 2](28a) (28b) (29a) (29b) Bysolving(29a,b)forAlandA2andsubstituting thesevaluesin (28a,b),thefollowing equations areobtained: VI=IlZ11+12Z12 V2=IlZ2l+12Z22(30a) (30b) 202 TRANSMISSION-LINE THEORY [Chap.III where Z11=~c[(1+Sll)(l-S22)+S12S21] Z12=2Z;12 (31a) Z22=~[(1-Sll)(l+S22)+S12S21] Z21=2Z;21 (31b) where D=(1-S11)(1-S22)-S12S21 (31c) Ifthesectionissymmetrical, S22=S11andS21=S12,sothat Z-Z-Z1 -S~1+Si2 22-11- c(1-S11)2-S~2 Z Z Z .2S12 21=12=c(1-S11)2-S:2 Theserieselements oftheequivalent symmetrical Tsectionare Z1+S11-S12 11-Z12=Zc1(SS)-11-12(32a) (32b) (33) (34a) (34b)Theshuntelement isZ12in(32b). Bysolving(28a,b)forAlandA2andsubstituting thesevaluesin (29a,b),thecurrents areexpressed asfunctions ofthevoltages and admittances, withthelatterexpressed intermsoftheelements ofthe scattering matrix. Theimpedance andadmittance matrices maybeformulated directly intermsofthescattering matrix. Asafirststep,thematrixequivalents of(28a,b)and(29a,b)are V=VZc(U+8)A I=VYc(U-8)A wheretheunitmatrixis (35) andwhereV,I,andAarecolumnmatrices: (36) Bypremultiplying bothsidesof(34b)byVh(U-8)-1andsubsti tutingtheexpression soobtained forAin(34a),itfollowsfromthe defining equation V=ZIfortheimpedance matrixZthat Similarlyz=Zc(U+8)(U-8)-1 Y=Yc(U-8)(U+8)-1(37a) (37b) Therelations between theimpedance, admittance, andscattering matrices havethusbeenestablished. Evidently aknowledge of8permitsthe directevaluation ofZorY.Conversely aknowledge ofZorYpermits Sec.13] IMPEDANCE ANDADMITTANCE 203 theevaluation ofSusingthefollowing easilyverifiedrelations: S=(Zl-U)(Zl+U)-l S=(U-Y1)(U+Y1)-1(38a) (38b) whereZl=Z/Zcand Y1=Y/Yc• 13.Unbalanced LoadTerminating aSymmetrically DrivenShielded pairLine.72,l06Theloadterminating ashielded-pair lineoflength 8 (Fig.13.1)consists oftwoimpedances, ZalandZ82,inseries.Their junction isconnected totheshieldthroughanimpedance Zp.IfZ81and Z82areunequal, theloadZ8=Z81+Z82isunbalanced. Inthiscaseit !~ 1 z z.~+~:_o 3 81 'r~r X~2 ~Rzp l~ ~2 FIG.13.1.Shielded-pair linewithunbalanced loadwhenZ.l~Z.2. sothatisconvenient tointroduce thedifference impedance Zd,definedby Zd=Z81-jZ8=jZs-Z82=j(Z81-Zs2) Z81=jZ8+ZdZ82=jZ8-Zd(la) (lb) Letthecurrents inthetwoinnerconductors 1and2ofthelinebe separated intosymmetrical (codirectional) andantisymmetrical (equal andopposite) partsasfollows:t Ii=I;=j(/l+12)=j/8 11=-/~=j(/l-12)=la(2a) (2b) Notethateachoftheinnerco.nductors carriesonlyone-half ofthetotal symmetrical current18,whichisequalandopposite tothecurrent-18 intheshield(conductor 3).Thefactorjontherightin(2a)isintro ducedforthisreason. Thetotalcurrents ineachconductor are 13=-(/~+12)=-18(3) Thevoltagedropsacrossthetwoparts,Z81andZ82,oftheloadare (4) wherethesignconventions ofFig.13.2areassumed. Ifuseismadeof(1)and(3),thevoltagedropsmaybeexpressed as follows: V1(s)=jla(s)Z8+{Is(S)Z8+la(s)Zd+j/8(s)Zd (5a) V2(s)=jla(s)Z8 -{]8(S)Z8 -la(s)Zd+418(s)Zd (5b) tNotethatthesymmetrical andantisymmetrical currents inthissectionarenot thesameasthoseinSec.12. 204 TRANSMISSION-LINE THEORY [Chap.III NowletthevoltagedropsacrossZdberepresented byequivalent gener atorswithappropriately definedemfs.Specifically let sothatV;==_[a(S)Zd V~==-[·(S)Zd VIeS)=j[a(S)Z.+j[·(S)Z.-V:-jV: V2(S)=j[a(S)Z.-j[·(S)Z.+V:-jV~(6) (7a) (7b) Theequivalent generators withemfsV:andjV~areshowninFig.13.2. Thecurrents II,12,and13atanypointalongtheshielded-pair line maybeformulated intermsofthecircuitofFig.13.2ifuseismadeof theprinciple ofsuperposition to determine separately thecurrents maintained byeachofthethree pairsofemfsjV8, jV~,andV:. Sincetheimpedances ofthenet workarebalanced, thetwopairs ofemfsjvgandjV~whenoperat ingalonecanmaintain onlyanti symmetrical (equalandopposite) FIG.13.2.Equivalent circuitforFig.13.1.currents; thetwoemfsV:whenop- eratingalonemaintain onlysym- metrical (codirectional) currents intheinnerconductors oftheline. Theantisymmetrical (equalandopposite) currents maintained atany pointatadistance ZfromZo(orw=s-zfromZ.)intheinnercon ductorsbytheemfsV8andV~aregivenby (9b)(8) (9a) whereli(z)=-I~(z)=[a(z)=VgF(w,s)+V~G(z,s) F()=ysinh90sinh("(aw+9.) w,s-casinh("(as+90+9.) G()=ysinh9.sinh("(aZ+90) z,s-casinh("(as+90+9.) Asusual,90=coth-I(Zo/Zca)and9.=coth-I(Z./Zca);Zca=I/Yrais thecharacteristic impedance, and"(aisthepropagation constantt ofthe shielded-pair linewhendrivenantisymmetrically withequalandopposite currents initsinnerconductors andnocurrentintheshield. Thesymmetrical (codirectional) currents maintained bythegenerators V:aredividedequallybetween conductors 1and2,whichareinparallel. Theentireequalandopposite currentisintheshield. Thesumofthe codirectional currents intheinnerconductors isgivenby [fez)+12(z)=-13(z)=I·(z)=V:H(z,s) (10) tNotethatexceptforthesmalleffectoflithephaseconstants (30and(3.areequal ifthelineisfilledwithahomogeneous dielectric. Ontheotherhand,(3.and(30may differgreatlyifthedielectric between thetwoinnerconductors differsfromthat between themandtheshield. Sec.13] where whereIMPEDANCE ANDADMITTANCE H()=ysinh9psinh('"(sz+9q)z,s-cs.h ( )sm'"(ss+9p+9q 9=th-1Zp+Zs/4 9=th-1Zq+Zo/4pco Zcs qco Zcs205 (11) Zoistheimpedance inserieswiththegenerators. Itiscenter-tapped to theshieldthrough animpedance Zq.Zcs=I/Yc8isthecharacteristic impedance, and'"(8isthepropagation constant oftheshielded-pair line whendrivensymmetrically withtheinnerconductors inparallel. With(6)in(8)and(10),currentsatz=sorw=°are la(s)=V8F(0,s) -Is(s)ZaG(s,s) 18(s)=-la(s)ZaH(s,s)(12) (13) whereF(O,s)isgivenby(9a)withw=0,G(s,s)isgivenby(9b)with z=s,andH(s,s)isgivenby(11)withz=s.Thesubstitution of(13) in(12)permitsthedetermination ofla(s).Thus la(s)_VgF(O,s) - 1 -ZjG(s,s)H(s,s) Similarly, with(14)in(13),(14) (15) (16) (17)V8ZdF(0,s)H(s,s) 1 -ZjG(s,s)H(s,s) Itisnowpossibletosubstitute (14)and(15)in(8)and(10)using(6) andinthismannertoobtainexpressions fortheantisymmetrical and symmetrical currents atanypointalongthelineintermsofV~.The resultsare la(z)=V8[F(w,s)+NZaH(s,s)G(z,s)J 18(z)= -V8NH(z,s) wherew=s -zandthedimensionless factorNisdefinedby N=ZdF(O,S) (18) - 1 -ZjG(s,s)H(s,s) Thetotalcurrents ineachconductor aredefinedin(3).With(16) and(17)theyare 11(z)=V81F(w,s)+N[ZdH(s,s)G(z,s) -j-H(z,s)]} (19) 12(z)= -V8IF(w,s)+N[ZdH(S,S)G(z,s) +j-H(z,s)]} (20) 13(z)= -V8NH(z,s) (21) Theratioofthetotalunbalanced currenttotheprincipal partofthe balanced currentisNH(z,s)/F(w,s). Itisseenthat,whenZd=0,sothattheloadisbalanced, N=0,and theentirelineisbalanced with (22) 206 TRANSMISSION-LINE THEORY [Chap.III (24) (26)Thesameresultisobtained whentheimpedance Zpjoiningtheloadto theshieldisremoved ormadeinfinite. WithZp=r:£J,Op=0and H(z,s)=R(s,s)=O. Thepowerdissipated bythesymmetrical currentistherealpartof thecomplex power ps=-}V:[s*(s) =--}Zd[a(S)[s*(s) (23) Thelaststepin(23)ismadeusing(6).With(14)and(15)itisseen that(23)isequivalent to Ps_1IVgF(O,S)Zd 12H*(n) -"2"1 _ZJG(s,s)H(s,s) o,s Therealpartofpsin(24)canbereducedbymakingtherealpartof R*(s,s)assmallaspossible. Thisisgivenby(11)withz=s.Thus H(ss)=YcssinhOpsinh(rss+Oq)= Ycs ( ) , sinh(rss+Op+Oq)coth(rss+Oq)+cothGp25 Thelaststepfollowsafterexpanding thedenominator anddividing through bythenumerator. SincecothOp=(Zp+Zs/4)/Zcs and coth(rss+Oq)=Zins/Zcs, whereZinsisthesymmetrical orcoaxial modeinputimpedance ofthelinelookingfromtheloadtowardthe generator, itfollowsthat H*(s,s)=Z~*+Z~+Z*/2.ns p s Rins+Rp+Rs/4-j(Xins+Xp+Xs/4) =(Rins+Rp+Rs/4)2+(Xins+Xp+Xs/4)2 Evidently H*(s,s)(andwithitthepowerdissipated bythesymmetrical currents) vanishes whenZpismadeinfinite, i.e.,whenthelumpedload isnotconnected totheshield. SincetherearecaseswhenZpissmall orevenzero,thismethodofeliminating unbalanced currents maynotbe available. However, evenwhenZp=0,therealpartofR(s,s)canbe madeverysmallbymaking Rinssufficiently great. Theinputresistance RinsisgivenbySec.2,Eq.(6a),viz., Rs-R sinh2(ass+pq) (27) ins-cscosh2(ass+pq)-cos2({3ss+et>q) wherethetermwithcPcasafactorhasbeenomitted asnegligible. Its maximum valueoccurswhen{3ss+et>q=11",forwhich Rins=Rcscoth(ass+pq) (28) Since(28)involves theattenuation asSoftheentirelengthoflineand, inaddition, theattenuation function pqoftheimpedances Zq+Zo/2, averygreatvalueofRinsisunavailable, ingeneral. Fortunately this difficulty canberemoved byasimpleexpedient. Sec.13] IMPEDANCE ANDADMITTANCE 207 Coaxial-mode Suppressor; Unbalance Squelcher. Inordertoobtaina highinputresistance Rinsforthecoaxialmodewhenlookingtowardthe generator fromtheload,itispossibletoconnectataquarterwavelength fromtheloadadouble-stub reactive networkthathaslittle ornoeffect ontheantisymmetrical currents butisessentially equivalent toashort circuitforthesymmetrical currents. Suchanetwork, originally intro ducedbyTomiyasu106andcalledbyhiman"unbalance squelcher," is illustrated inFig.13.3. Insofarastheantisymmetrical (equalandopposite) currents arecon cerned,thecircuitinFig.13.3consistsoftwoinsulating stubsconnected acrossthelineatAA'andBB'.ThelengthsACandBDfromthe twinlinetothebridgesCC'andDD'areA,a/4-k,wherekistheequiva lentlengthofthebridgeandA,aisthewavelength fortheantisymmetrical A B-------,----,-------4~--~~---~~---- Z,2• I IlA'lIB' \-----+,-+ ').,,----'I III Balanced :~-kI III Unbal~nced linel1II I sectIonsit!eND~ 2I,II :I I I I : I I I !Balanced~ input Short·circuiting pistons FIG.13.3.Tomiyasu's unbalance squelcher; >'aisthewavelength forthebalanced currents, and>..isthewavelength fortheunbalanced currents. mode.Therefore theimpedance lookingintoeachstubatAA'andBB' isveryhigh-several hundred thousand ohmsiftheadjustment iscare fullymade.Itfollowsthatthesestubshaveanegligible effectonthe balanced currents onthelineifthisisterminated inadissipative load. Theunbalanced (codirectional) currents aregenerated attheasym metrical loadbythegenerators V:inFig.13.1.Thesearelocatedon therighttowardtheoutputinFig.13.3.Forcodirectional currents the lineandthetwostubsbehavelikecoaxiallineswiththetwoinnercon ductorsinparallel. ThebridgesCC'andDD'contribute nothing, since theyjoinequipotential points. Therefore, ifthestubatAA'isA,s/2in lengthandisterminated inashort-circuiting piston,itsinputimpedance isextremely low-afewtenthsofanohmatmost-so thattheshielded pairlineiseffectively short-circuited intheplanecontaining AA'forall symmetrical-mode currents ontheshielded-pair line.Forsuchcurrents 208 TRANSMISSION-LINE THEORY [Chap.III (29)theimpedance lookingtowardAA'fromthecrosssectionatBB'isthe veryhighvalueofaquarter-wavelength lineterminated inashortcircuit. Ontheotherhand,theparallelimpedance atBB'lookingtowardDD'in thestubistheextremely lowvalueforahalf-wave closed-end stub.It followsthatthelineiseffectively terminated inaverylowresistance at BB'insofarascoaxial-mode currents generated ontheoutputsideare concerned. Theinputresistance asseenfromtheloadaquarterwavelength from BB'istheverylargevalue Rs-Rth3aX..:...4Rcsins-csCOT-3aX IfZp=0and(Rins+Rs/2)2isverymuchgreaterthan(Xs/2)2,itfollows from(26)that ReH(s,s) ==R~~R/2tnss(30) Accordingly, ifRinsissufficiently great,thepowerdissipated bythesym metrical currents issmall,andthesecurrents areconfined essentially to thesectionoflineandthestubstotherightofAA'inFig.13.3. Ithasbeenshownthatanasymmetrical loadisequivalent toasym metrical loadinserieswithgenerators thatmaintain bothbalanced and unbalanced currents. Inparticular, theemfV~ofthegenerator ofbal anced(antisymmetrical) currents isproportional tothecodirectional (symmetrical) currents intheload.Thisfollowsfrom(6).Accord ingly,iftheamplitude ofthesymmetrical currents intheunbalanced loadislarge,thesewillgenerate acorrespondingly largebalanced volt age.If,asinsometypesofprecision measurements onshielded-pair lines,itisundesirable tohaveaneffective generator ofbalanced currents intheload,itisnotsufficient merelytolocalizethesymmetrical cur rentsbymeansofanunbalance squelcher. Itisnecessary alsotoreduce theamplitude ofthesymmetrical currents. Thismaybeaccomplished bymodifying theunbalance squelcher, asshowninFig.13.4,whereone ofthestubsisterminated inZcs,thecharacteristic impedance oftheline forthesymmetrical mode.Thesymmetrical currents arethusdissipated without reflection, andtheiramplitude iskeptsmall.Notethatthe terminated stubistheoneadjacent totheunbalanced line.Thesym metrical impedance lookingintothisstubfromthelineisZcs,anethisis stillverysmallcompared withtheverygreatimpedance lookingintothe linetowardtheotherstub. Ifanunbalanced currentisexcitedinashielded-pair linebyanasym metrical orunbalanced generator, thecoaxialmodemaybeconfined toa sectionoflinenearthegenerator byinserting thecircuitofFig.13.3or Fig.13.4closetothegenerator. Sec.14] IMPEDANCE ANDADMITTANCE 209 Oncethesymmetrical currents havebeensuppressed fromtheprinci palpartofashielded-pair line,thismaybereplaced byanopentwo-wire lineifdesired. Instead ofeliminating unbalanced currents fromthemainlineby locating anunbalance squelcher nearanunbalanced loadorgenerator, itisadequate forsometypesofmeasurements merelytoconstruct a detector thatresponds onlytothebalanced currents. Insuchcasesit isdesirable toterminate thelineinitscharacteristic impedance insofar asthesymmetrical currents areconcerned. Adoublebridgethatper mitstheseparation ofbalanced andunbalanced currents onashielded pairlinehasbeenconstructed byMatthews. 86Byconnecting thetwo innerconductors andbringing acentertapoutthroughtheshieldasthe innerconductor ofacoaxialline,onlytheunbalanced modeisobtained. Byplacingatubularbridgewithagapatitscenteraquarterwavelength nearerthegenerator, thebalanced voltagemaintained acrossthegap Balanced current Short·circuiting pistons FIG.13.4.Coaxial-mode suppressor forshielded-pair line.Unbalanced current maybeusedtodriveacoaxiallineplacedinsideonesideofthebridge, provided itsinnerconductor crossesthegapandisconnected tothe othersideofthebridge. Thecoaxiallineforthebalanced modemaybe contained insideoneoftheinnerconductors oftheshielded-pair lineand broughtouttoadetector. 14.SeriesStubsandUnbalanced Sections ofLine;FoldedDipole; Balun;Shielded Loop.Sections oftransmission lineservemanypurposes whenconnected inparallelwiththeline.Someusefulproperties maybe realizedbyconnecting sections oftransmission lineinseries. Consider firstanopentwo-wire line.Ifthisiscutatanypointalong oneofitsconductors andthetwoterminals socreatedareconnected to anauxiliary two-wire lineasinFig.14.1a,theauxiliary lineisinseries withoneoftheconductors ofthemainline,andthis,quiteobviously, is unbalanced. Thecodirectional currents inanunbalanced open-wire line donotdifferfromtheradiating currents inanantenna. Sinceanycir- 210 TRANSMISSION-LINE THEORY [Chap.III cuitthatradiatessignificantly isnotusefulasatransmission line,further studyofthecircuitinFig.14.1aisofnointeresttotransmission-line theoryandcannotbeanalyzed bytransmission-line methods. Themaintransmission lineinFig.14.1amaybebalanced bythe expedient ofconnecting another auxiliary two-wire lineidentical with thefirstoneinserieswiththesecondconductor ofthemainline,as showninFig.14.1b.However, thefactthattheentirecircuitisnow geometrically symmetrical withrespecttoaplaneperpendicular toand bisecting thedistance betweenthelinesdoesnotensurethattheauxiliary transmission linesarebalanced. Actually theymaybeunbalanced so completely astoconstitute oneofthemostusefultypesofantenna, the TI I I t4 I I I I 1 (c)(a) (b)v:rr-----------'Zo ~z. Iv.e2D 1 1000....--....1-------.::--.J ---'1-----....2 Z2. ----82------I I I I I I1. I, FIG.14.1.(a)Two-wire linewithunbalanced seriessection. (b)Balanced two-wire linewithbalanced seriessections. (c)Folded-dipole antenna. so-called foldeddipole. Thisisachieved whentheauxiliary sectionsare eachaboutaquarterwavelength longwhenterminated inawirebridge andwhentheimpedance Z2lookingintothelength 82ofthemainlineis aslowaspossible. Intheusualarrangement (Fig.14.1c) 82=0,and Z2sistheimpedance ofashortstraight conductor.tAlternatively the seriessections oflinebehaveliketransmission lineswithvirtually bal ancedcurrents whenZ2isverygreat,forexample, whenZ2=00.In general,seriessections ofopen-wire lineareusefulprimarily asantennas. Aninteresting modification ofFig.14.1bisshowninFig.14.2a,where tThefoldeddipoleisanalyzed inRefs.10and11. Sec.14] IMPEDANCE ANDADMITTANCE 211 alllinesareshielded pairs.Thisdoesnotalterthefactthatthemain lineisbalanced andtheauxiliary linesingeneralareunbalanced. How ever,themetallically enclosed unbalanced linesdonotradiateandcan beanalyzed bytransmission-line methods asinthepreceding section. InFig.14.2athesymmetrical generator withemfVoinserieswithan internalimpedance Zoisshowncenter-tapped andconnected totheshield. Thesameistrueoftheterminating impedance Z2softhemainline.Since themainline(including loadandgenerator) isbalanced, thereisnonet flowofchargealongtheshieldandnocurrentinthecentertapstothe loadandgenerator. Evidently thesemayberemoved ifdesiredinsofar ;)'=0(',r *t ,F1y(Y)by 13,.(y)12y(YLz ly,e'20 - +~1Y(O)-. -12.)1(0) ~10Ilz(z) 'lz(8) '2,:(0) '2,:(z)Z2S~ -.llz(Z) -'2z(z) + - ll'oe.fTubular shield lj,.(y)'~11(Y)~ ~2iY Ita~ I I1/IO r.o I21= Zl=Sl Z2= z2=s2 FIG.14.2a.Crosssectionofbalanced shielded-pair linewithseriessections. asbalanced currents areconcerned. Theimpedance Zlaterminating each oftheauxiliary seriessections oflineoflengthlissymmetrical andcon nectedtotheshieldatitscenterthrough anarbitrary impedance Zr,so thattheeffective terminating impedance forthesymmetrical modeis Zzs=jZza+Zr.Zrmaybealumped impedance, asinFig.14.2a, including ashortcircuitZzs=0andanopencircuitZzs=00,orthe inputimpedance ofasectionofcoaxiallineofarbitrary length,asin Fig.14.2b. Theanalysis ofthecircuitinFig.14.2ainvolves thedetermination ofthecurrents inbothbranches ofthemainlineandintheseriessec tions.Thecurrent /lz(Z)intheleft-hand partofthemainlineisbal ancedandreadilydetermined fromconventional transmission-line for- 212 TRANSMISSION-LINE THEORY [Chap.III (la)mulasassoonastheterminating impedance Zl=Vl(S)=Vl(S) 8Ilz(S) 111/(0) isknown. Correspondingly thecurrent12z(z)intheright-hand partof themainlineisalsobalanced andiseasilyevaluated ifthevoltage (lb) acrossitsinputterminals isknown. Z2istheinputimpedance ofthe right-hand sectionofthemainlineoflength S2.Thecurrents Ill/(Y)and 1211(y)inthetwoconductors ofthetwinlineintheseriessections arenot Tobalanced.. generator,t......J.,J... ZII=Zr+ZltJ/2tZin=Zr "'r---Z14 z,J-{ Z.l_ ....Z2 f-r-7'"r:lY=1 y=O FIG.14.2b.Crosssectionofbalanced shielded-pair linewithseriessections endingin coaxiallines. necessarily equalandopposite. Iftheyareunbalanced, theremustbe acurrent1311(y)intheshield,asindicated inFig.14.2a. Forsimplicity itisassumed inthepresentanalysisthatthelinespacings aresufficiently smallcompared withthewavelength sothatjunction and coupling effectsmaybeignored.Ifthisisnotthecase,accountmaybe takenofthemusingmethods described inChap.V. Inordertodetermine thecurrents Ill/(Y)and121/(y)inthetwocon ductorsofeachoftheserieslines,itisconvenient toseparate theminto antisymmetrical andsymmetrical components. Theformerare,bydefi nition,theequalandopposite currents ofthebalanced twinlinewith zerocurrentintheshield,namely, Thelatteraretheequalcodirectional currents 1~I/(Y)=Ifll(Y)'Thetotal Sec.14] IMPEDANCE ANDADMITTANCE 213 symmetrical currentintheconductors is 1;(y)==1~,,(y)+1211(y)=21~,,(y)=-13,,(y) where1311(y)isthecurrentintheshieldand1;(y)isthesumoftheequal symmetrical currents inthetwoinnerconductors. Thetotalcurrents ineachofthetwoinnerconductors are 111l(y)=lill(Y)+Ify(Y)=j;1;(y)+1~(y) 1211(y)=liuCy)-Ify(Y)=jl;(y)-1:(y)(2a) (2b) Theseexpressions maybesolvedforthesymmetrical andantisymmetri calcurrents asfollows: (2c) Theequalcodirectional orsymmetrical currents mayberepresented asif maintained byequalin-phase emfsV";theequalandopposite oranti symmetrical currents mayberepresented asifmaintained byequaland opposite emfsVaand-Va.Thustheeffective emfsinthetwocon ductorsare sothatVI(S)=V"+Va V"=MVI(s)+V2(0)]V2(0)=Vs-Va Va=MVI(s)-V2(0)](3a) (3b) Sincethereisnoactualgenerator atthecenterofconductor 2,butan impedance Z2isconnected inserieswithitaty=0,theCompensation Theorem statesthat andthereforeV2(0)=-1211(0)Z2=12z(0)Z2 Vs=MVI(s)-1211(0)Z2] Va=MVI(s)+1211(0)Z2](4) (5a) (5b) Letthecomplex propagation constant andcharacteristic impedance ofallpartsoftheshielded twinlinebe"(aandZca,respectively, when thelineisoperated antisymmetrically withcurrents andchargesinthe twoinnerconductors equalandopposite andnocurrentorchargeonthe shield. Theseconstants maybeevaluated inanyparticular caseusing Chap.I,Sec.9,Eqs.(10).Similarly letthepropagation constant and characteristic impedance oftheshielded twinlinebe"(sandZcswhenthe lineisoperated symmetrically withequalandcodirectional currents in thetwoinnerconductors andwiththeshieldcarrying currentequal in magnitude tothesumofthecurrents intheinnerconductors butopposite indirection. Theseconstants maybeevaluated usingChap.I,Sec.9, Eqs.(23a,b). Theantisymmetrical ortwin-mode currents intheidentical seriessec tionsoflinemaybedetermined withtheaidofFig.14.3,inwhichtwo equalandopposite generators eachwithemfVaareconnected inseries withtwoidentical sections oflineeachoflengthlandterminated inan 214 TRANSMISSION-LINE THEORY [Chap.III arbitrary impedance ZCa.Theantisymmetrical currents intheupper sectionaregivenbyChap.II,Sec.8,Eq.(16),withl-ysubstituted forW,Olasubstituted for08,and00setequaltoj7r/2.(Thislastsubsti tutiondepends ontheequivalence ofthelowerseriessectionandone- 2Zz. FIG.14.4.Symmetrical problem inthe analysis oftheseriessections.1;1=!1; I·I,.12=2'7 I~ 1~==-11'1;3=-I; YlZ~v· ~YlZ I~ 1~==11' Zla FIG.14.3.Antisymmetrical problem in theanalysis oftheseriessections. halfofeachgenerator toaperfectly conducting infiniteimageplane.) 'Thusthenormalized currentis I~(y)=1:1(y)=ysinh["(a(l-y)+Ola] Va Va cacosh("(al+Ola) Theantisymmetrical inputadmittance isdefinedby(6a) liY(O)=I~(O)=ya=Ytanh(Yl+Ol)1Va Va ca Ia a=Za(6b) With(6b)theantisymmetrical currentis la()=]a(o)sinh["(a(l-y)+Ola] IIy IIsinh("(al+Ola)1;(0)=yaVa (6c) Theantisymmetrical currents inthelowerseriessectioninFig.14.3are equaltoandcodirectional withthoseintheuppersection. Thatis, 1;(-y)=I;(y). Thesymmetrical orcoaxial-mode currents intheupperseriessection arederived usingFig.14.4.Sincethetwoinnerconductors arein parallelwithequalandcodirectional currents, itfollowsthat (7) Sec.14] IMPEDANCE ANDADMITTANCE 215 (8a)wherel;(y)isthesumofthecurrents intheinnerconductors andthe negative ofthecurrent 13Y(Y)intheshield. [Notethat,since l~y(Y)=0, 13Y(Y)=13y(Y).] Withappropriate changesinthenotation, thenormal izedsymmetrical currentl;(y)=liy(Y)+12Y(Y)= -131/(Y)isgivenby l~(y)_1Ysinhh's(l-y)+als]VB-"2"cscosh(Ysl+als) Thefactorioccursasaconsequence ofthefactthattheparallelgener atorsaredrivingidentical sections oflineinseries.Thesymmetrical inputadmittance seenbyeachemfVSisdefinedby li~?)=IJ~)==ys={-Ycstanh(ysl+als)=:s (8b) SinceZSistheimpedance oftwoidentical sectionsoflineinseriesdefined intermsofone-half ofthetotalcurrent, theinputimpedance Zincofthe upperorlowersection,whendriveninthesymmetrical modeandreferred tothetotalsymmetrical current, is Vs 21y(0)=Zinc={-Zs=Zcscoth(ysl+als) (8e) IS(y)=P(O)sinh["(sU-y)+als] II/S(O)=2Y"V" (8d) 1/ 1/sinh("(sl+als) Thesymmetrical currents inthelowerseriessectionarethesameasthose intheuppersection,sincely(-y)=ly(Y). Thetotalcurrentineachconductor isgivenby(2c).Itfollowsthat 1;(0)=11Y(0)+12y(0)=Ills)-12z(0) (9a) I~(O)=j[/1y(0)-12y(0)]=j-[/1z(s)+12z(0)] (9b) Hencetheamplitude factorsin(6c)and(8e)areexpressed intermsof thecurrents 11y(0)=11z(S)and12y(0)=-/2z(0),whichentertheseries sections fromthemainlinesontheleftandright.Itfollowsthatthe totalcurrents intheconductors oftheseriessectionsmaybedetermined assoonas11z(s)and12z(0)areavailable. Thecurrents 11y(0)=11z(s)and12y(0)=-12z(0)maybeobtained using(3a)inthefollowing equivalent form: V1(s)=liY(O)Zs+l~y(o)Za =}l;(O)Zs+l;(O)Za (lOa) V2(0)=Iill(O)Zs -Iill(o)Za=iI;(O)Zs -I;(O)Za (lOb) Thesymmetrical andantisymmetrical currents maybeeliminated from (lOa,b)with(9a,b).Thus VI(s)=11y(0)i(Zs+Za)+12y(0)i(Zs-Za) (lla) V2(0)=11y(0)}(Zs -Za)+12y(0)}(Z"+Za) (llb) 216 TRANSMISSION-LINE THEORY [Chap.III NowletV2(0)bereplaced bythevoltagedropacrosstheinputimped anceZ2oftheright-hand sectionofline,asin(4).Alsoletthefollowing definitions bemade: Zll==i(Zs+Za)Z22==i(Zs+za)+Z2 (12a) Z21==Z12==i(Zs-Za) (12b) Itfollowsthat(lla,b)maybeexpressed intheform V1(s)=11y(0)Zll+12y(0)Z12=11z(s)Zll-12z(0)Z12 (13a) o=11y(0)Z21+12y(0)Z22=11z(s)Z21-12z(0)Z22 (13b) Theseequations aresolvedreadilyforthetotalcurrent11y(0)=lu(s) entering theseriessections fromthemainlineontheleft.Thus where(14a) (14b) istheinputimpedance oftheseriessectionasaloadonthemainline. Similarly, from(13b), Z21VI(S)-/211(0)=12z(0)=I1z(s)-Z==-Z (15) 22 e where Ze=ZllZ22-Z12Z22=2ZsZa+Z2(Zs+Za) (16) Z12 Zs-Za isthetransfer impedance. ThevoltageV1(z)andthecurrent11z(z)atanypointintheleft-hand partofthemainlinemaybeevaluated usingChap.II,Sec.8,Eqs.(17) and(18),ortheirequivalents, sincetheterminal impedance Zlsisgiven by(14)using(6b)and(8b).ThevoltageV2(z)andthecurrent12z(z) atanypointintheright-hand partofthemainlinearealsogivenby Chap.II,Sec.8,Eqs.(17)and(18),with V~replaced by V2(0)=12z(0)Z2=V1(S)Z2 Ze whereVIeS)isthevoltageacrosstheloadoftheleft-hand partofthe mainline,Z2istheinputimpedance oftheright-hand partofthemain line,andZeisobtained from(16)with(6b)and(8b). Theantisymmetrical andsymmetrical partsofthecurrentintheseries sections aregivenby(6c)and(8d),with1;(0)and I~(O)obtained from (9a)and(9b)usingthevaluesof11z(s)and12z(0)asdetermined inthepre cedingparagraph. Thetotalcurrents ineachconductor, IIII(y)and1211(y), aregivenby(2a)and(2b).Thecurrentintheshieldisla(Y)=-I;(y). Thusthecurrents inallconductors ofthenetwork inFig.14.2ahavebeen determined. Sec.14] IMPEDANCE ANDADMITTANCE 217 Threespecialcasesaresignificant: a.SeriesSectionsExcitedinTwinModeAlone.Ifthesymmetrical impedance zsismadesufficiently greatsothattheinequality \Zs\»\Za+2Z2\ issatisfied, theinputimpedance Z18in(14b)becomes Z=2Za+Z2(1+Za/Zs) ==2Za+Z 181+(Za+2Z2)/Zs 2 Thetransfer impedance Ztin(16)becomes Z=2Za+Z2(1+Za/Zs) ==2Za+Z t 1 _Za/Zs 2(17) (18) (19) Sincethetransfer impedance ispractically equaltotheinputimped ance,itfollowsthat12z(O)=l1z(s),or (20) Thismeansthattheseriessections behavelikeordinary sections of shielded-pair linewithbalanced currents. Thisisindicated bytheform oftheinputimpedance (18),whichisthesumofthreeimpedances. Generator....------- .... FIG.14.5.Seriessections excitedintwinmodealone.T ~+k4 -l- I I I=A_k2 I I -i.. Thecondition (17)maybesatisfied byselecting thelengthloccurring in(8b)tobenearanintegral, evenmultiple ofaquarterwavelength, withZlaverysmallandZlsverygreat,asforaX/4closed-end stub. Thisisaccomplished inFig.14.5byhavingtheseriesstub,including its termination Zls,a3X/4closed-end stub.Inthiswayzsistheveryhigh inputimpedance ofthe3X/4closed-end stub,whereas Zaistheverylow inputimpedance ofaX/2 closed-end stub.Thelength S2andtheload 218 TRANSMISSION-LINE THEORY [Chap.III Z2smustalsobesochosenthatZ2isnottoolarge.Apossiblecircuitis showninFig.14.5. b.SeriesSections ExcitedinCoaxial ModeAlone.Iftheantisym metrical impedance Zaismadesufficiently greatsothattheinequality (21) (23)(22)issatisfied, theterminating impedance Z18in(14b)andthetransfer impedance Ztin(16)become Z=2Zs+Z2(1+Zs/Za) ==2ZS+Z 181+(zs+2Z2)/Za 2 Z,=2Zs+Z2(1+zs/za) ==-(2Zs+Z) t Zs/Za_1 2 Sincethetransferimpedance isthenegative oftheinputimpedance, itfollowsthat12z(0)=-llz(s),or (24) Therefore theseriessections behavelikesections ofcoaxiallinewitha doubleinnerconductor. Thismeansthat,insteadofatermination ZlB T L.L,L.L.L..I.-L.L.L.L..l'..U,t l..~-k .L. OJ FIG.14.6.Seriessections withcodirectional currents converted tocoaxiallines. equivalent toaseriescombination ofthreesections ofbalanced shielded pairline,asincasea,thetermination isequivalent toaseriescombi nationoftwoidentical sections ofline,withcurrents inthecoaxialmode andcombined impedance 2Zs,andasinglesectionofbalanced shielded pairlinewithimpedance Z2.WhenZ2=0,thisstructure istheshielded- T I I I A '2 I 1.219 (25) Asaconsequence Zla=2+~a/Z2=={Za=2Zinc(26a) [J!J Zit=Z2 (26b) FIG.14.7.Balanced twinlinefeeding seriessections inwhichthetwinand whereZincistheinputimpedance ofcoaxialmodesareequal. eachofthetwoidentical seriessec- tionsdriveninthecoaxialmode.Thecurrents entering thetwoinner conductors oftheseriessections areSec.14] IMPEDANCE ANDADMITTANCE pairtransmission-line analogue ofthefolded-dipole antenna fortheopen two-wire line. Acircuitarrangement inwhichtheseriessections conduct onlythe coaxialmodeisshowninFig.14.6.Itisassumed thattheinputimped anceZ'aofeachcoaxiallineatPP'has avaluesuchthattheinputimpedance ofeachseriessectionatitsjunction withthemainlineisverymuchsmaller forthesymmetrical (coaxial) mode thanfortheantisymmetrical (twin line)mode,asrequired by(21).Thisis accomplished, forexample, ifZ'a==0, l==),,/4-k,andZ'adoesnotexceed thecharacteristic impedance Zca. c.SeriesSectionsExcitedwithEqual Twin-mode andCoaxial-mode Currents. Letthefollowing conditions beim posedon(14b)and(16): Za=021Z21»IZal l1y(0)=l1z(s) VIeS)-12y(0)=12z(0)=--z;-(27a) (27b) Acircuitthatsatisfies (25)isshowninFig.14.7. Bymaking Z2sufficiently greatcompared withZa,thecurrentinthe secondconductor maybemadeassmallasdesiredcompared withthe currentinthefirstconductor. Thismeansthatthecodirectional cur rentsarepractically equaltotheoppositely directed currents, sothat theyaddtotwicethevalueofoneinthefirstconductor andcancelin thesecondconductor. Undertheseconditions thesecondconductor may bedispensed with,andthecircuitarranged asinFig.14.8aorasinFig. 14.8b,whereZ2isinfinite. Theseriessections arenowcoaxiallines insteadoftwinlinesinwhichoneconductor carriesallthecurrent. 220 TRANSMISSION-LINE THEORY [Chap.III Evidently, from(26a), (27c) where Zincistheinputimpedance ofeachofthecoaxiallines.Thusthe twosections ofcoaxiallinearesimplyinseries,andZincin(27c)isthe inputimpedance ofeachcoaxial-line section. Baluns.19Itisoftennecessary toconnectabalanced line,suchasa shielded-pair lineoratwo-wire line,toacoaxiallinethatisinherently unbalanced. Balanced-to-unbalanced converters, orbaluns,maybe derivedfromsomeofthespecialcircuitsinvolving seriessections ofline. I--~'-'4 (a) (b) FIG.14.8.Balanced shielded-pair linesfeedingidentical matched coaxiallinesusing twodifferent circuits. Consider firstcasebwithitsdefining condition (21).Evidently this lattermaybesatisfied equally wellwithZ2=O.Suppose thatthe terminations fortheseriessectionsinthecircuitofFig.14.2bareZla=0 andZla=Zr=Zc,whereZcisthecharacteristic impedance ofthecoaxial line.Theresultant circuitisshowninFig.14.9.Forit (28) Thiscircuitconverts fromabalanced shielded-pair linetotwocoaxial lines. Ingeneral,interestisinconverting toasinglecoaxialline.Sincethe twolinesareinseries,itmightbesupposed thatitismerelynecessary to replaceoneofthembyashort,unloaded section,asshowninFig.14.10, provided therequirement (21),thatis,IZal»IZs!,ismaintained. This Sec.14] IMPEDANCE ANDADMITTANCE 221 couldbeaccomplished bymakingthesectionaclosed-end half-wave stub forthecoaxialmodeandaclosed-end quarter-wave section for thetwin mode.However, thisprovides acompletely unbalanced loadwhich,as described inSec.13,generates anunbalanced codirectional currenton themainshielded-pair line.Notethat,eventhoughnoconnection is showninFig.14.10fromthepointLtotheshield,thisdoesnotmean thatZpinFig.14.1isinfinite. Evidently averyappreciable fraction ofthepowerwouldbedissipated bythecurrents inthesymmetrical mode inthemainshielded-pair lineunlessanunbalance squelcher, suchas f ~-k4 ~ taIe '\ FIG.14.9.Balanced twinlinefeeding twomatched coaxiallines.OJIII II I AT--k ~ ~--kt ~+k ~ FIG.14.10.Shielded-pair linefeedingone matched coaxialline. showninFig.14.3,wereinserted intheshielded-pair linenearitsjunc tionwiththecoaxialline.Ifsuchasymmetrical-mode suppressor were usedinthecircuitofFig.14.10,thiswouldconstitute aneffective, albeit constructionally somewhat complicated, balun. Without anunbalance squelcher abalanced shielded-pair linecanbe maintained onlywhentheloadisequallydividedbetween thetwoseries sections, asinFig.14.9.Thissuggests thepossibility ofcontinuing the balanced shielded-pair lineastwocoaxiallines,eachterminated inZc. Thisisillustrated inFig.14.11.Asaconsequence ofsymmetry itis immaterial whether thetwoseriessections aredividedbyametalwall atSS'ornot. Although thecircuitofFig,14.11provides acompletely balanced con versionfromashielded-pair linetoaloadinacoaxialline,ithasthe undesirable featureofrequiring twocoaxiallinestotheload.Fortu natelythismaybeavoidedbyarranging theoutputsofthetwocoaxial 222 TRANSMISSION-LINE THEORY [Chap.III linesinparallelandthenconnecting theload,asisshowninFig.14.12. Sincethetwolinescarrycurrents exactly1800outofphaseatpoints thatareequidistant fromthejunction withtheshielded-pair line,itis necessary toinsertanextrahalfwavelength oflineinoneofthelinesin / Balanced -J inputtA-k4lIe -S' Ie FIG.14.11.Shielded-pair linewithbalanced loadconsisting oftwoidentical, matched coaxiallinesfoldedsothattheirloadsareinseries. \ '------4-----1 } /'",/---81-- FIG.14.12.Balunforconnecting fromshielded-pair linetocoaxialline Fig.14.9).(basedon ordertohavetheoutputsfromthetwolinesinphase.Withthishalf wavelength sectionaneffective balunisachieved whichdoesnotrequire theuseofanunbalance squelcher. ThecircuitinFig.14.8bwhichconverts fromabalanced shielded-pair linetotwoidentical coaxiallinesmaybeusedasabalunineitherofthe Sec.14] IMPEDANCE ANDADMITTANCE 223 twoconnections showninFigs.14.11and14.12.Inthiscasethesection ofshielded-pair lineextending forthefirstquarter wavelength ofthe serieslinesisabsent. Asaconsequence thelength 81inthecircuitcorre sponding toFig.14.12maybereduced toassmallalengthasisphysi callypracticable, including zero.Theresultisthesimplecircuitshown inFig.14.13.Thisprovides aneffective balun. Asaconsequence ofthereciprocal theorem, thebalunsdescribed in thepreceding paragraphs maybeusedtoconvertfromacoaxiallineto abalanced shielded-pair line. Sincemanyofthecircuitsdescribed dependonaninequality that requires acertainimpedance tobeverylargeorverysmallcompared Unbalanced line ...------"",.",.-, ......." -" "\ Balanced A 2line_-+-~~;;;~~;;;~,yj/ /...../------- FIG.14.14.Balunforconnecting fromFIG.14.15.Balanced shielded loopdriven two-wire linetocoaxialline. fromshielded-pair line. withanother impedance, theoperation ofthecircuits isfrequency sensitive. Inpractice, thesignificant impedances canbemadeadjust abletopermittheirusewithanyone ofarangeoffrequencies, butthis doesnotimplybroadband operation. Itistobenotedthattheentirediscussion inthischapterhasassumed thatthecross-sectional dimensions ofallshielded-pair andcoaxiallines 224 TRANSMISSION-LINE THEORY [Chap.III FIG.14.16.Squareshielded loopwithoutersurfacere placedschematically bya coilofequalimpedance. RLisequaltotheradiation resistance.I-- I I I IL__•,__.J II II II IIaresosmallthatjunction effectsarenegligible. Ingeneral, thisisnot necessarily thecase.Forexample, inFig.14.13thefactthatseriesline2 branches offatrightangles,whereas seriesline1doesnot,involves a different terminal-zone correction, sothattheshielded-pair linemaybe slightlyunbalanced. Application toTwo-wire OpenLine.Whenever theunbalanced cur rentonthemainshielded-pair lineisvanishingly small,thislinemaybe replaced byatwo-wire linewithappropriate changes inconstants with outmodifying theformulas. Inparticular, thebalunsdescribed inthe preceding paragraphs maybeusedforconverting fromanopentwo-wire linetoacoaxialline.Thisisillustrated in Fig.14.14.Iftheunbalanced currents onthe shielded-pair linearenotinsignificant, thecor responding currents onatwo-wire linecannot bedetermined fromtransmission-line theory. ShieldedLoopAntenna.llAnimportant type ofantenna (especially fordirection findingand exploring electromagnetic fieldsintheformof aprobe)istheshielded loop.Thisconsistsofa sectionofcoaxiallinewhichisbentintoaclosed loopanddrivenfromashielded-pair line,as showninFig.14.15.Asectionoftheshield isremoved opposite thejunction ofthetwo typesofline,sothattheoutersurfaceoftheco axiallineconstitutes aloopantenna. Forpres- entpurposes itisequivalent toalumpedload ZL=RL+jXLconnected inserieswiththe shieldofthecoaxialline,asshowninFig.14.16 forasquareinsteadofacircularloop.t ThecircuitofFig.14.15or14.16satisfiesthe symmetry condition presupposed inconjunction with(27c),sothattheloadterminating the shielded-pair lineistwoidentical coaxiallineseachterminated inZL/2. Itfollowsthattheloadterminating theshielded-pair lineis where(29a) (29b) and8isthelengthofeachline.Referring toFigs.14.15and14.16, sisone-half thelengthoftheinnerconductor intheloop.Thetwo sections ofcoaxiallinewithcharacteristic impedance Zcandpropagation constant "(areseentobeinseries. tFormulas forR1.andX1.forcircularandsquareloopsaregiveninRef.9,Chap.VI. line4D+Iv."104-204Sec.15] IMPEDANCE ANDADMITTANCE 225 15.TheHybridJunction forTransmission Lines.8,13,16,90Auseful circuitelement forcoaxial,shielded-pair, andtwo-wire linesisthehybrid junction. Thisconsists ofacombination ofshuntandseriessections connected atthesamecrosssectioninatransmission line.Itisadvan tageousfirsttoanalyzethecircuitthatissimplest informandthento consider otherpossibilities. InFig.15.1isshownahybrid-junction circuitforusewithtwo-wire or shielded-pair lines.Itconsistsoffourtransmission lines,eachextending fromapairofgenerators withemfs iV~andimpedances Zotowarda common junction region. Lines1and2extendcontinuously fromA r''::;: 84 Iv.'1 1",2~l+line1 line2 _202 AEo~~~----;:.::~...rfJP-+~..----~.·.z;t +B + _-----i:!:::~4-7-~~--- ~_ jV01 u.s. iv~ ,-i~~""'.~J::~bol,"ce '1 / sq.,',h" C:v:;~383 i~3', FIG.15.1.Hybridjunction forshielded-pair ortwo-wire line. toB.Attheirjunction line3isconnected inparallelandextendsat rightanglestoC.Lines1and2arecoupled byidealtransformers to line4,whichextendstoD.Sinceline4inthesimplecircuitofFig.15.1 iscoupledtoonlyoneoftheconductors oflines1and2,itconstitutes anasymmetrical loadthatgenerates unbalanced currents, andthesein turncanexciteunbalanced currents inlines3and4.Therefore an "unbalance squelcher" (preferably oftheterminated typeshownin Fig.13.4)isconnected ineachlinenearthejunction region,asdescribed inthepreceding section. Notethatthefourunbalance squelchers have littleornoeffectonthebalanced currents. Owingtothepresence of unbalanced currents onallfourlinesbetween theunbalance squelchers 226 TRANSMISSION-LINE THEORY [Chap.III andthecommon junction, thesesections aswellasthesquelchers them selvesmustbeconstructed ofshielded-pair line.Between thesquelcher andthegenerator onlybalanced currents exist,sothatinthisrangeeach linemaybeconstructed eitherofshielded-pair cableoroftwoopenwires. Sincecodirectional currents arerelatively smallandareexcluded from thefourtransmission lines,theymaybeignoredwithout seriouserror intheanalysis oftheequalandopposite currents. Toincludethem wouldaddgreatcomplications without significantly alteringtheresults.t Line3 C FIG.15.2.Hybridjunction forcoaxialline. ThecircuitofFig.15.2isacoaxial-line equivalent ofFig.15.1.Actu allyitconsists ofonlythreecoaxiallines-the collinear lines1and2 andtheshuntline3.Theseriesline4isashielded-pair line.Since eachofthethreecoaxiallinescansupport onlyasinglemode,anunbal ancesquelcher isrequired onlyinline4. Theessential characteristics ofthehybridjunction maybedetermined byinspection. ItisclearfromFig.·15.1or15.2thatcurrents main tainedbyV83divideequally between lines1and2atthejunction and inducenovoltageinline4.Similarly currents maintained byV~4are tAnanalysis ofbothcodirectional andequalandopposite currents isgivenin Ref.86. Sec.15] IMPEDANCE ANDADMITTANCE 227 FIG.15.3.Equivalent circuitfor hybridjunction.+ (la) (lb)nd1+n2/2+n4/4=0 ~=~=e4 n1n2 n4limitedtolines4,1,and2.Evidently thereisnocoupling between lines 3and4. Inordertofacilitate theanalysis ofthehybridjunction, letthe transmission-line problems beeliminated byintroducing lumpedcircuits thataretheirequivalents insofarasthejunction isconcerned. Thisis accomplished byapplying Thevenin's theorematthecrosssections where thelinesjoin.Lettheopen-circuit voltages acrosstheseterminals be V1,V2,V3,andV4•Alsolettheimpedances lookingbackintothelines withthedrivinggenerators short-circuited beZl,Z2, Z3,andZ4.Then eachlineatthejunction isequivalent toitsopen-circuit voltageinseries withitsinputimpedance andthejunction, asshowninFig.15.3.The fourcurrents 11==11(Sl),12==12(S2),13==13(s3),and14==14(s4)inFig. 15.3arethesameasinFigs.15.1and15.2.Intheformerandinline4 ofthelattertheindicated currents areonlytheantisymmetrical partsof thetotalcurrentinthejunction region. Letitbeassumed thatthenumber of turnsonthetransformerwindings in lines1and2isthesameandgivenby n2=n1.Letthenumber ofturnson thewinding inline4ben4.Thegen eralcurrentandvoltage equations for anidealtransformer withthreewindings are wheree1=e2=eisthevoltageacrossthewindings inlines1and2 ande4isthevoltageacrossthewinding inline4.11,12,and14arethe currents through thethreewindings. Forthecaseathand, sothatn4=rn1=rn2 11+12+rl4=0(2a) (2b) (20) Thefollowing meshequations areobtained directly usingFig.15.3and (2b,c): VZ4 4=re+14Z4=re- -(/1+12)r V1+V3=e+I1Z1+laZa=e+11(ZI+Za)-12Za V2-Va=e+12Z2-laZa=e+12(Z2+Za)-I1Z3 In(3b,c)useismadeoftheequation 1a=II-12(3a) (3b) (3c) (4) 228 TRANSMISSION-LINE THEORY [Chap.III Bysolving(3a)foreandsubstituting thisin(3b,c),thefollowing equa tionsareobtained: I1A+12B=Va (5a) lIB+12C=Vb (5b) whereThesolutions for11and12areobtained directly. Theyare 11=V1YU+V2Y12+V3Y13+V4Y14 12=V1Y21+V2Y22+V3Y23+V4Y24 Y_Z2+Z3+Z4/r2Y_Z3-Z4/r2 11- D 12-D Y-Z2+2Z4/r2Y14= -(Z2+2Z3)(rD)-113-D Y_ZI+Z2+Z4/r2 Y21=Y12 22- D(7a) (7b) (7c) (7d) (7e) where With11and12determined, theothertwocurrents arereadilyfoundfrom (4)and(2b).Theexpressions areasfollows: 13=11-12=V1Y31+V2Y32+V3Y33+V4Y34 (9a) 14= -11+12=V1Y41+V2Y42+V3Y43+V4Y44 (9b)r where Y31=Y13 Y32=Y23 (9c) Y33=ZI+Z~+4Z4/r2 Y34=(ZI-Z2)(rD)-1 (9d) Y41=Y14 Y42=Y24 (ge) Y43=Y34 Y44=(ZI+Z2+4Z3)(r2D)-1(9f) The-fourequations (7a,b)and(9a,b)maybeexpressed inmatrixform asfollows: where(10) (11a) Sec.15] IMPEDANCE ANDADMITTANCE 229 (Ub)andtheadmittance matrixis [YllY12Y13Y14] Y==Y21Y22Y23Y24 Ya1Ya2YaaYa4 Y41Y42Y43Y44 Important applications ofthehybridjunction involvetheuseofonly oneofthefourgenerators. SpecialCaseA SpecialCaseB SpecialCase0 SpecialCaseDVI=V2=Va=0V4~0 11= _V4(Z2+2Za) rD 1 2= _V4(ZI+2Za) rD 1a=V4(ZI-Z2) rD I_V4(ZI+Z2+4Za) 4 - r2D VI=V2=V4=0Va~0 11=Va(Z2+2Z4/r2) D 1 2= _ Va(~1+2Z4/r2 ) D 1a=Va(ZI+Z2+4Z4/r2) D 1 4=Va(ZI-Z2) rD VI=Va=V4=0V2~0 11=V2(Za-Z4/r2) D 1 2=V2(Zl+Za+Z4/r2) D I _ -VZ(ZI+2Z4/r2)3----J)-- 1 4= -V2(ZI+2Za) rD V2=Va=V4=0VI~0 II=V1(Z2+Za+Z4/r2) D 1 2=V1(Za-Z4/r2) D(12a) (12b) (12c) (12d) (12e) (l3a) (l3b) (13c) (l3d) (l3e) (14a) (14b) (14c) (14d) (l4e) (15a) (15b) (15c) 230 TRANSMISSION-LINE THEORY [Chap.III (l5d) (15e) TheHybridJunction asaBridge. Therelations (12d)and(13e)lead tothefollowing important conclusions: V4~0;13=0 V3~0;14=0whenZl=Z2 whenZl=Z2(16a) (16b) Thecondition Zl=Z2requiresthattheimpedances lookingbackinto lines1and2fromthejunction bethesame.Ifthetwosections ofline areidentical incrosssectionandlength,thiscondition canbesatisfied onlyif ZOI=Z02 (17) (18a) (18b)Clearlythesufficient evidence that(17)issatisfied isthevanishing of 13ifvg4(Figs.15.1and15.2)istheonlyemforthevanishing of14if Vgaistheonlyemf.Accordingly itispossibletocompare twoimpedances ZOIandZ02byanullmethod. IfZOIisavariable standard impedance andZ02isanunknown imped ancethatistobemeasured, theseimpedances canbeconnected astermi nationsoftheidentical linesections 1and2.Byinserting agenerator inline4andadetector inline3(orviceversa),thecurrentinthedetector vanishes whenthevariable standard impedance isadjusted toequalthe unknown impedance. Thus,when13=0(or14=0),Z02=ZOI. TheHybridJunction asaLineStretcher. Alinearshiftinphasecan beintroduced inamatched transmission linebychanging itslengthwith theaidofatelescoping sectionknownasalinestretcher. Thehybrid junction makespossible theaccomplishment ofthesameresultwithout changing thephysical lengthoftheline.Thisisachieved byselecting lines1and2asthecontinuous lineinwhichashiftinphaseistobe produced. LetVOlbetheonlyactivegenerator emf,andletbothlines 1and2beterminated intheircharacteristic impedance Zc==Re.Thatis, VIy;=Zl+Rc With(7)and(8),theserequirements areequivalent tothefollowing: VI1 Dy;=Yll=Re+Z3+Z4/r2=Zl+Re Itisreadilyverifiedusing(8)and(18a)that(18b)issatisfied if 4Z3Z4=R2r2 c(19) Sec.15] IMPEDANCE ANDADMITTANCE 231 Thisissatisfied whenlines3and4areterminated inmovable highly conducting pistons, sothatZoa=Z04==0and90a==904==j1r/2.Fur thermore letline4bemaintained exactlyaquarterwavelength longer thanline3.Inpractice, thismayberealizedbyhavingthetwopistons gangedtogether sothat (20) (Itisassumed thatthephaseconstants ofthelinesareallthesame.) Sincethelinesections canbekeptshort,theirlossesmaybeneglected. Asseenfromthehybridjunction, theimpedances are Za=jReatan{38a Z4=-jRe4cot{38a (21) Thecondition (19)requiresthat r2R2ReaRe4=T (22) Theratioofthecurrent12entering line2tothecurrentIIleaving line1isobtained from(7a,b).Thus /2Y21 Za-Z4/r2_j[Reatan{38a+(Re4/r2)cot{38a1 h=Yll=Re+Za+Z4/r2-Re+j[Reatan{38a-(Re4/r2)cot{38a1 (23) Thisexpression isreducedtoverysimpleformifthedimensions oflines 3and4canbesochosenthattheircharacteristic resistances havethe following values: (24a) (24b) '"=1r-2{38awhereReisthecharacteristic resistance oflines1and2.Withthis choiceitfollowsdirectlythat 12 ",I.- =e1rII Byvarying {38a(with{384=(38a+1r/2)between1r/2and1r,thephase'" ofthecurrententering line2maybeshiftedlinearlyfrom0to1r,whereas bothlines1and2remainterminated intheircharacteristic impedance Re• Asdescribed laterinthissection,important typesofhybridjunctions haver=2,sothat(22)requires Rca=Re4=Re•Inthiscase(23) becomes 12 1+icot2{383 =1 -(j/2)cot{38a=eN( )h=1+jcot{38a-tcot2{38a1+(j/2)cot{38a 25a where '"=-2tan-1(jcot(38a)=2tan-1(jtan(384) (25b) 232 TRANSMISSION-LINE THEORY [Chap.III Thisformula alsoprovides arangeofI/;from0to1ras{3saisvariedfrom 1r/2to1r,butthevariation ofI/;isnotlinearin(3saaswith(24b). TheMeasurement ofPhasewiththeHybridJunction. Thephasedif ference Vtbetween twovoltages maybedetermined withthehybridjunc tionbyapplying thesevoltagestolines1and2andobserving thecurrents 13and14•WithVIr=0,V2r=0,Va=0,andV4=0,thegeneralexpres sions(9a,b)reduceto 13=VIY31+V2Ya2 14=VIY41+V2Y42(26a) (26b) Intheseequations VIandV2aretheopen-circuit voltagesattheendsof thelineswhenthesearedisconnected atterminals 1and2.Inaphase comparison thesignificant voltages arethosemaintained acrossterminals 1and2whenthelinesareconnected. Theseare (27) whereZIandZ2aretheimpedances lookingbackintolines1and2. IfthevaluesofVIandV2in(27)aresubstituted in(3b,c),itisseenthat thetermsinvolving ZIandZ2cancel. Evidently thegeneralexpressions forlaand14maybeexpressed intermsofV~andV;insteadofVIandV2 simplybysettingZIandZ2equaltozero.Hence la=V~Y~I+V;Y:2 14=V~Y~I+V2Y~2 whereY;jisobtained fromYijbysettingZIandZ2equaltozero. (7c-f)and(9c,d)itisfoundthat Y'Y'1al=a2=2Za Y'Y' r41=42= -2Z4(28a) (28b) With (29a) (29b) (30b)(30a)If(28a)and(28b)aresolvedforthecurrents using(29a,b),theresultsare 13=V~-V;=V~(1_i"')2Za2Zave 14= _V~+V;= _V~r(1+veN) 2Z4/r 2Z4 withthecomplex ratiofactorvdefinedasfollows: V'v=veN==V~ (31) Theformulas (30a,b)mayberearranged intwowaysthatleadtodiffer entmethods ofmeasuring thephase1/;.Theyareconsidered inturn. RatioMethodforMeasuring PhaseofVoltagesofEqualAmplitude. If anattenuator isavailable sothatthemagnitudes ofthetwovoltages Sec.15] IMPEDANCE ANDADMITTANCE 233 (32b)(32a)V~andV~maintained acrossterminals 1and2ofthehybridjunction canbekeptequal,itfollowsthatv=1,sothatthemagnitudes ofthe currents in(30a)and(30b)aregivenby 113\=I;]31Y2(1-cos1/1)=I;]31sin* 1141=I:£Iy2(l+cos1/1)=I:£Icos* Thesecurrents maybenormalized bynotingthefollowing conditions: For1/1=0, For1/1=11",=12VZ~r41I14maxl 113maxl=I;)31(33a) (33b) Byintroducing normalized currents thefollowing relations areobtained: . 113I . 1/1 h==--=sIn-13max 2 i4==I~I=cost14max 2(34a) (34b) Fromtheseequations anexplicitexpression forthephasedifference is obtained: (34c) Ifoneofthevoltages Vg1orVg2appliedattheinputterminals oflines 1and2isvariable inphase,thiscanbevarieduntil13=0isobserved andI14maxIisdetermined. Byagainvarying thephaseuntil14=0is observed andI13maxIisdetermined, thecircuitisstandardized forphase comparison. Byapplying astandard reference signaltoline1andthe signalofunknown phasetoline2,thephasedifference isobtained directly fromtheratioofthenormalized currents inlines3and4using(34c). Notethatthismethodrequiresthetwovoltages V~andV~tobeequalin amplitude. Balanced-detector MethodforComparing PhaseofUnequal Voltages.If thetwovoltages arenotequalinmagnitude, aconvenient alternative methodofphasecomparison isavailable. Although itcanbedirectly basedon(30a,b),itisadvantageous toobtainthecorresponding expres sionswhenthevoltages areappliedtoarms3and4insteadofarms1 and2,since,ingeneral, lines1and2canbemadealikemorereadily than3and4. Letthevoltages appliedacrosslines3and4beV~and V~,where thesearerelatedtotheopen-circuit voltages V3andV4bytheformulas Va=V~+1aZaandV4=V~+14Z4•Iftheseareusedin(3a,b,c)to 234 TRANSMISSION-LINE THEORY [Chap.III eliminate VaandV4,thetermsinZaandZ4cancel,sothat(7a,b)maybe expressed asfollows: Bynowrequiring thattheimpedances lookingintolines equal,i.e.,(35a) (35b) (35c) (35d) 1and2be (36)V~Y~a+V~Y~4 V~Y~a+V~Y~4 Y;a= --lZ2 Y~4= --.!..rZ2II= 12= Y~a=-lZl Y~4=__1_rZlwhere andbysetting ".f. V~ V=ve1 '1'==-V'ra Eqs.(35)become II=V~(1-veN)Zl(37) Theseequations correspond to(30a,b)ifZaismadeequaltoZ4/r.Since itismoreconvenient tosatisfyZl=Z2thanZa=Z4/r,Eqs.(37)are preferred to(30a)and(30b). Themagnitudes ofthecurrents entering thetwolinesare 1111=I~:I(1+v2-2vcost/I)! 1121=Ii:I(1+v2+2vcost/I)!(38a) (38b) Itisnowclearthatanyquantity thatinvolves thedifference between anarbitrary powernofthetwocurrents, thatis, IV'In II21n-II11n=Z:[(1+v2+2vcost/I)n/2-(1+v2-2vcost/I)n/2] (39) vanishes whent/I=7r/2.Evidently, ifacircuitcanbeprovided which measures aquantity proportional tothedifference currentin(39),anull readingindicates thatthetwovoltages are90°outofphase.Ifarefer encevoltagewithknownvariable phaseisavailable, thephaseofan unknown voltageisreadilydetermined byvaryingthephaseoftherefer encesignaluntilitis90°outofphasewiththeunknown. InFig.15.4anarrangement isshownfordetermining therelative phasedistribution oftheelectromagnetic fieldnearanantenna system. Twobalanced detectors areusedtorectifyindividually thecurrents 11 and12•Therectified outputofeachofthetwoidentical detectors is Sec.15] IMPEDANCE ANDADMITTANCE 235 proportional tosomepoweroftheradio-frequency current. Thediffer enceoftheseoutputsisappliedtothereceiver. TheHybridJunction asaDirectional Coupler. Thefunction ofa directional coupleristopermittheseparate andindependent determi nationofthecurrents maintained bygenerators (ortheirequivalents) atopposite endsofamatched transmission line.Inparticular, ifin Fig.15.1lines1and3together constitute acontinuous transmission line withV81andV8aasactiveemfswhileV82=Vg4=0,thehybridjunction Reference signal (4) Coaxial (1")hybrid (2) junction '----- ...... (3)Radio frequency transmitter Antennasystem undertest Toarm3 "Signal whosephase istobedetermined FIG.15.4.Balanced-detector methodofrelative phasemeasurement withahybrid junction. actsasadirectional couplerif14isameasure ofthatpartofthecurrent inlines1and3whichismaintained byvg1,and12isameasure ofthe partofthecurrentmaintained byV8a.Evidently thismeansthat12 mustvanishwhenV81istheonlyactiveemfandthat14mustvanish whenVgaistheonlyactiveemf.Itfollowsfrom(l5c)and(l3e)that thenecessary conditions are (40) If(40)issatisfied, itfollowsfromspecialcasesDandB,using(8), thatthecurrents are CaseD: (41a) (41b) (41c) (41d) [Chap.III (42a) (42b)TRANSMISSION-LINE THEORY CaseB:236 VI=V2=V4=0Va~0 11=-12=-~ZI+2Za 2Va1a=ZI+2Za(42c) 14=0 (42d) Nowletitberequired thatlines1and3bematched atthejunction, sothat VI11=ZI+Zcl=ZI+2Za Va ZIr;=Za+Zca=Za+2 Theseequations requirethefollowing: ZclZIZa=Zca=2=2(43a) (43b) (44) Evidently eachlinemustbeterminated initscharacteristic impedance attheendremotefromthejunction. With(40)and(44)theserela tionsmustbesatisfied: ZOI=ZI=Zcl Z02=Z2=Zc2=Zcl (45a) Zoa=Za=Zca=~1Z04=Z4=Zc4=r2Zca=r2;c1(45b) Thecurrents maintained byVg1aregivenby VI -VI11=1a=- 12=014=-- (46)2Zc1 2rZcl Thecurrents maintained byV~aare Va VaII= -12= - 1a= - 14=0 (47)2ZcI Zcl IfVo~istheonlyemfandline3isterminated initscharacteristic impedance, 12iszero.Ifline3isnotterminated initscharacteristic impedance butinZoa=Zca+Z~a,itispossibletoreplace Z~abyanemf Voa=-loaZ~a' Thecurrent12isthenameasure ofthisvoltageand therefore ofthereflected wavefromthetermination. Thus14measures thedirectwave,and12measures thereflected wave. Varioustypesofdirectional couplers areincommon use.Inprinciple theiroperation corresponds tothatofthehybridjunction asdescribed above,butdifferent constructions areinvolved. Sometypesconsistof twolinescoupledbyholesatoneortwopoints. Thetheoryoftrans mission-line directional couplers isformulated inChap.VI,Sec.5. HybridJunctions withoutTransformers. InthecircuitsofFigs.15.1 and15.2transformers areusedtocoupleline4tolines1and2andto providethecentertapleadingtoline3.Inactualpracticeathighfre quencies itispossibletoeliminate thecoilsinFig.15.2bylettingthe Sec.15] IMPEDANCE ANDADMITTANCE 237 innerconductors oflines1and2continue smoothly toajunction and substituting astraight conductor forthecoilattheendofline4.This innowayalterstheanalysis. Ifacoaxialoutputisdesiredforline4, thecoupling loopmaybetheequiv- I alentofashielded loop,asdescribed I(4) inconjunction withFigs.14.14and ffi}illl 14.15.Asimplecircuitofthistypeis ~=== illustrated inFig.15.5;usefulmodifi-(1)1~ ~IIIr'(21cationsareshowninFigs.15.6and ! ! 15.7·t Insteadofcoupling line4tolines1 and2,asinFigs.15.1and15.2,itmay I bejoineddirectly, provided line3be- (3) ginswithahigh-impedance stub,as~~~hs~~~~de~~~:i;:in~~~:i:. junction showninFigs.15.8and15.9.Inthis casetheantisymmetrical currents atthejunctions satisfythefollowing equations: 11+14=i-1s12+14=-41s (48) sothat 11+12+214=011-12=Is (49) Ifthetwoequations in(49)arecompared with(2b)and(4),itisseen r~ltl~~ 10lZ,f I ~~-+--~--l--~-l424 FIG.15.6.Modified coaxialhybridjunction. (3)Zc FIG.15.7.Modified coaxialhybridjunction. tTheseareduetoMoritaandSheingold.90 238 TRANSMISSION-LINE THEORY [Chap.III FIG.15.8.Hybridjunction forcoaxiallineusingshielded-pair section. lv.e-n+lv. ..~04+_204 . ! line4 ~•..i.ine3 i~~+ Z03 +ty~ FIG.15.9.Hybridjunction forshielded-pair ortwo-wire linewithhigh-impedance stub. Sec.15] IMPEDANCE ANDADMITTANCE 239 thattheycorrespond exactlyifthetransformer ratiorissetequalto2. Itfollowsthatalltheresultsderivedforthetransformer-coupled lines applyifrisreplaced by2. Thehybridjunctions forshielded-pair andtwo-wire linesinFigs.15.1 and15.9areunbalanced inallfourlinesinsteadofonlyinline4(inwhich codirectional currents mustbesuppressed withanunbalance squelcher). Thisisaconsequence oftheinsertion ofaseriessectionoflineinonly ly,e-nZ04+ly,e 204+ _204 .. iY~ ~ L_in_e_2__:::z~f +- .nr-~., ......Unbalance -21y~squelcher1.,.. 2"01-+ K=:~·._L_in_e_l __",-~ +1.,e2"01 ~~.~~ "V• i'Ol+. Zoo Iv.e203+ FIG.15.10.Hybridjunction forshielded-pair ortwo-wire lineincompletely symmetri calformwithhigh-impedance stubs. oneofthetwoconductors. Evidently theunbalance onlines1,2,and3 canbeeliminated byusingtwocomplete andidentical lines4,asinFig. 15.10.Alternatively theloadandgenerators inthelowerline4maybe omitted, andthelinefixedinlengthaty+]../4.Theopenendsofthe lowerline4arethenconnected totheupperline4atadistance yfrom thejunction. Thelengthymaybekeptasshortaspractical conven iencedictates. Itisreadilyverifiedthattheanalysis ofthebalanced circuitofFig. 15.10differsinnoessential mannerfromthatofFig.15.9. RingCircuit. Analternative methodofconstructing ahybridjunc tionforusewithcoaxiallinesistheringcircuitshowninFig.15.11. 240 TRANSMISSION-LINE THEORY [Chap.III Itisreadilyverifiedbyinspection thatithastheessential properties of thehybridjunction ifthelinesaretreatedaslossless. Thus,ifavoltage isappliedtoline4,theresulting currentdividesequally between lines 1and2,andnovoltageismaintained acrossline3.Similarly, ifavolt ageismaintained acrossline3,equalvoltages areestablished acrosslines 1---'" 2t---4 FIG.15.11.Hybridjunction informofringcircuit,or"ratrace." FIG.15.12.Modified ringcircuitwithreversed connections. 1and2,andthevoltageacrossline4iszero.Clearlylines3and4are notcoupled. Unlikethetransformer-coupled circuitofFig.15.2,the ring,or"ratrace,"ishighlyfrequency-sensitive, sinceitsoperation depends onthespacing ofthefourlinesaroundtheringspecified in Fig.15.11. Sec.15] IMPEDANCE ANDADMITTANCE 241 Aringcircuitofdifferent construction isshowninFig.15.12.Itcon sistsofaringthatisonewavelength ratherthanoneandone-half wave lengthsincircumference. Thefourtransmission-line connections are uniformly spacedaroundthering.However, oneofthefourquarter wavelengths between connections isspiraled through 180°,sothatthe connections atoneendarereversed.Itisreadilyverifiedthatthisring alsohastheessential properties ofthehybridjunction andismuchless frequency-sensitive thanthemoreconventional ringinFig.15.11.On theotherhand,itisdifficulttoadapttheringinFig.15.12forusewith coaxiallines. PROBLEMS 1.Determine theinputimpedance ofasectionoflineoflength20.2mforwhich Zc=400(1-jex./(3)anda=2 X10-3neper/m atafrequency of300Mc/sec. The lineisterminated inanapparent impedance Z.a=100-j800ohms. Z.Whatwouldbetheinputimpedance ofthelinedescribed inProb.1ifitwere lossless? J.Asectionofcoaxiallineistobedesigned toprovideamaximum possibleinput impedance. Itisspecified thattheinnerradiusoftheshieldmustbe1.5cm.Deter mineotherspecifications usingthebestphysicalIy available materials. Whatisthe maximum inputimpedance? 4.Designatransmission systemusingaseriestransformer tomatchanimpedance Z.a=40+j20toa50-ohm coaxial cablesothatthelineisterminated initscharac teristicimpedance. 5.Designasingle-stub matching network foraloadZ.a=20-j500ohmssothat thelineisterminated-in itscharacteristic impedance Zc==Rc=440ohms.Dothis analytically andalsographically usingbothtypesofcirclediagram. Explain the graphical solutions indetailwiththeaidofconstruction lines. 6.Designadouble-stub matching network foraloadZ.a=800+j600ohmsona lineforwhichZc==Rc=72ohms.Theinputimpedance ofthenetwork andload istobe72ohms.Useanalytical andgraphical methods. 7.Investigate thebroadband properties oftheseriestransformer bydetermining thestanding-wave ratioasafunction ofthefrequency ifthelineismatched ata givenfixedfrequency /0.Useappropriate constants. Thestanding-wave ratiois givenbyS=cothP•• 8.Repeatthepreceding problem forthesingle-stub matching network. 9.Investigate theimpedance-matching properties ofatriple-stub tunerconsisting ofthreeshuntstubsappropriately spacedatfixeddistances alongatransmission line. 10.Animpedance Z.=20-j500ohmsterminates alinewithRc=440ohms, ex.=2.26X10-3neper/m, and¢c=ex./f3.Thefrequency is150Mc/sec. (a)Determine P.and<1>.bycalculation andbycirclediagram. (b)Determine theshortest distance fromtheloadalongthelineatwhichthe impedance lookingtowardtheloadisapureresistance. Whatisthisresistance? Whataretheassociated valuesofP.and.?Whatisthestanding-wave ratioas definedbyS=cothP.? (c)Repeatpart(b)forthenextshortest distance forwhichtheimpedance isa pureresistance. 11.Acoaxiallineisconstructed ofanaluminum tubewithinnerdiameter of1in. andwallthickness ofiin.Theinnerconductor issteeldrillrodiin.indiameter thatissilver-plated foradistance ofexactlyone-half wavelength fromtheterminating piston. Determine thevaluesoftheterminal functions andthereflection coefficients 242 TRANSMISSION-LINE THEORY [Chap.III terminating thelineatthebeginning ofthesilveredsection. Assumethepistonto beperfectly conducting. Thefrequency is300Me/sec. 12.Anapparent impedance ZaG=616-j2,096ohmstermina.tes atwo-wire line withcharacteristic impedance Zc=400ohmsandanattenuation constant a=10-3 neper/matafrequency forwhichthephaseconstant isf3=0.30radian/m. Deter mine(a)theapparent terminal functions; (b)theapparent reflection coefficient; (c) theinputimpedance ofa2Q-mlengthoflinewhenterminated inZaG;(d)thelength oflinerequired inadditiontothe20min(c)sothatthelineischaracterized byinput resonance. Whatistheinputimpedance atinputresonance? CHAPTER IV GENERAL AMPLITUDE RELATIONS FORCURRENT ANDVOLTAGE 1.TheDistribution ofCurrentandVoltageandtheTransfer ofPower alongaNonresonant Line.Thedistribution ofvoltagealonganinfi nitelylonglineisdescribed inChap.I,Sec.14,andillustrated inChap.I, Fig.14.1.Thedescription involves traveling wavesofconstant phase. Thedistributions ofvoltageandcurrentalongalineoflengthsthatis terminated inZearethesameasthosealongthefirstsmetersofan infinitely longline.FromChap.II,Sec.5,Eq.(12),theyaregivenby (1) wherezisthedistance fromz=0alongalineoflengths,Zoisthe impedance ofthegenerator, andVoisitselectromotive force.Since Ze=Re(1-jcPe),itisevidentthatthecurrentandvoltageatthecross sectionzsatisfytherelation (2) Thusthevoltageleadsthecurrentinphasebytheusuallyverysmall angletan-1cPc.Foralinewithlowdistortion definedbycP~«1,(2) becomes simply (3) Thelossesonmanylinesaresufficiently lowsothatcPcisverysmall (10-3orless),andtheexponential in(3)maybereplaced byitsleading termofunity,asontherightin(3). Thepowersupplied tothenonresonant lineatitsinputterminals at z=0istherealpartofiVo/ri.Thatis, (4) where10isthepeakvalue.Thepowerdissipated inthematched loadat theendofalineoflengthsis Ps=~-IIsl2Rs=il/sl2Re=il/5lRce-2us Thisfollowssince,bydefinition ofanonresonant line,Zs=Ze. 243(5) 244 TRANSMISSION-LINE THEORY [Chap.IV (7)Theefficiency oftransmission is W=Ps=I/sl2=e-2as (6) Po1/012 Thetransmission lossisusuallyexpressed bythepowerratioPo/Ps• Thusthelossindecibels isdefinedby 1Po PoL(db)=10og-=4.3429In-Ps Ps Using(6) L(db)=8.6858as (8) whereaismeasured inneperspermeterandsisinmeters. Thepowerdissipated inheatingthelineis P,=Po-Ps=Re(1/51-I/s\2)=Rel/ol2(1-e-2as)(9) Itisinteresting tonoteatthispointthattheforms[Chap.I,Sec.13, Eqs.(13)and(14)]reduceto Vz=VZcAe-Yz (10) Iz=vY:Ae-Yz (11) fortheinfiniteline,sothattheinputpoweratz=0is Po=Re-}Vilz=-}AA*=-}A2 (12) Z.General Expressions forCurrent andVoltageforanArbitrarily Terminated LineWhenDrivenbyaSinglePairofEqualandOpposite PointGenerators (orTheirEquivalent) Anywhere alongtheLine.80,81 Generalformulas forthecurrentandvoltageatanycrosssectionz'ofa transmission lineextending fromz'=0toz'=s'aregiveninChap.II, -lx'i z'=Q z' s' I-j------=;e!!::t Pi~e----il----~, Vo~Vr"2""'2 lz Zx.=Zo~'----z.---1o' ~~----.rrt~-· ----.tZs.=;:Zs Vo~VI I2=2 : I I I , z=Q x z S FIG.2.1.Transmission linedrivenbyonepairofequalandopposite pointgenerators atanarbitrary distance xfromoneendoftheline. Sec.8,Eqs.(15)and(16).Norestrictions areimpliedontheimpedance ofeithertheloadatz'=s'orthegenerator atz'=0'.Itfollowsthat, ifthecircuitofChap.II,Fig.5.1a,inthemodification ofFig.2.1(upper scale)isused,inwhichtheemfisequivalent toapairofequaland opposite pointgeneratorst eachofmagnitudeiV8.,thegenerator imped- tApointgenerator isanimpedanceless, extensionless emf.Itsphysical realiza tionisdiscussed inChap.VI. (1) (4)Sec.2]AMPLITUDE RELATIONS FORCURRENT ANDVOLTAGE 245 anceZo'istheinputimpedance ofasectionoflineofarbitrary lengthx' terminated inanequallyarbitrary impedance Zx"UsingChap.II,Sec. 8,Eq.(16),withallcoordinates primed,thecurrentatz'is 1 V g,sinh60,sinh("(w'+6.,) Is'=Zcsinh("(8'+60,+68,) wherew'==8'-z',60,isthecomplex terminal function oftheimped anceZo'lookingtotheleftfromthegenerators, and68,isthecomplex terminal function oftheloadZ8"Theimpedance Zo'maybeexpressed asfollows~ Zo'=Zccoth("(x'+6x') (2) where6x'isthecomplex terminal function ofZx"Bydefinition Zo' , ( )60,==coth-1Zc="(x+6x' 3 If(3)issubstituted in(1),theresultis 1=Vg,sinh("(x'+6x')sinh("(w'+68,) z'Zcsinh("(8'+"(x'+6x'+68,) Nowlettheoriginbetransferred from0'atthegenerators tothe actualleftendofthelinewheretheterminating impedance Zx'islocated. Letzbemeasured fromthisend;letthetotallengthoflinebe8=8'+x'. Thedistance fromthenewandunprimed origintotheprimedorigin locating thegenerators isx=x';also6x'=60,68,=68,w'=8'-z', w=8 -z,andvg,=V~.Withthisnotation (4)maybeexpressed as follows: 1=V~sinh("(x+60)sinh("(w+6.) aZcsinh("(8+60+6.) Similarly V-Vesinh("(x+60)cosh("(w+6.) a-xsinh("(8+60+68)(5) (6) Theserelations givecurrentandvoltageatanycrosssectionzalonga linewhichisterminated inZoatz=0andinZ.atz=8andwhichis drivenbyapairofequalandopposite pointgenerators, eachofemfiV~, atz=x.ThecircuitisshowninFig.2.1,usingthelowerscale. InordertoexpressI.andVaatpointsbetweenz=0andthegenerator atz=x,itisnecessary merelytointerchange ends,Le.,substitute -I. forI.and-V;forV;.Thisisequivalent tomeasuring distances from Z8insteadoffromZooItalsoinvolves achangeinsubscripts from0to8 andviceversain(5)and(6)andthesubstitution ofy==8 -xforx andofzforw==8 -z.(Notethatthegenerators areinserieswiththe 246 TRANSMISSION-LINE THEORY [Chap.IV conductors oftheline.)Theresulting formulas are l~=V~sinh(yy+98)sinh(yz+90) (7) ~ O~z~xZcsinh(Y8+90+98) Vz= -Vesinh(yy+9s)cosh(yz+90)0~z~x(8) zsinh(y8+90+9s) Evidently (7)and(8)areformally like(5)and(6).Together with (5)and(6)theydefinethecomplex currentandvoltageatanarbitrary crosssectionalongalineterminated atbothendsinunrestricted imped ances.Thelineisdrivenatanycrosssectionxbyapairofequaland opposite pointgenerators ortheirequivalent, asdescribed inChap.VI, Sec.3. Inusing(5)and(6)or(7)and(8),notethatlzandVzdependon threeindependent variables. In(5)and(6)thesearethedistance xfrom Zotothegenerators, thedistance w==8 -zfromZstothepointwhere lzandVzareevaluated, andtheover-alllength 8oftheline.In(7) and(8),ontheotherhand,thethreevariables arethedistance y==8 -x fromZstothegenerators,' thedistancezfromZotothepointwherelz andVzarecalculated, andthelength 8oftheline. 3.General Expressions forCurrent andVoltage foranArbitrarily Terminated LineWhenDrivenbyTwoPairsofEqualandOpposite PointGenerators (orTheirEquivalent) Anywhere alongtheLine.8o,81 Inordertotreatvariousmethods ofdrivingatransmission linebydriving unitscoupled anywhere alongtheline,itisnecessary toobtainexpres sionsforlzandVzforalinethathastwoequalandopposite pointgener atorsineachline,withthetwoseparated asmalldistance. Thecircuit veve22e--e Iz Zof'--------1~f------f·I_t _---'fZs I ,I • I I z~o ~ei~ek ~ :~Ix-glx:x+g FIG.3.1.Transmission linedrivenbytwopairsofequalandopposite pointgenerators symmetrically locatedwithrespecttoanarbitrary pointatadistance xfromoneend oftheline. arrangement isshowninFig.3.1.Letthedistance fromZotoapoint oneachconductor halfway between eachpairofgenerators bex.Let thedistance fromthispointtoeachgenerator beg,sothatthecoordi natesofthegenerators arez=x-gandz=x+g.Eachgenerator is impedanceless andhasanemf}Ve,asinFig.3.1. Thecurrentatanypointzduetothepairofgenerators atx+gis givenbySec.2,Eqo(5),withx+gsubstituted forx;similarly thecur- Sec.3]AMPLITUDE RELATIONS FORCURRENT ANDVOLTAGE 247 rentduetothepairofgenerators atx-gisgivenbySec.2,Eq.(5), withx-gsubstituted forx.Thecurrentduetobothgenerators oper atingsimultaneously isthealgebraic sumofthecurrents obtained for eachoneseparately. UsingSec.2,Eq.(5),andthepolarities forthe generators showninFig.3.1,thefollowing currentisobtained: 1=V:sinh(yw+98 ){sinh[y(x+g)+90]-sinh[y(x-g)+90ll 21Zcsinh(y8+90+98) (1) Afterthehyperbolic sinesinthebracesareexpanded usingthecombi nationsyx+90andyg,thefollowing formula isobtained: 1=W~cosh(yx+90)sinh(yw+98) 21Zc sinh(y8+90+98) where W:==2Vesinhyg(2) (3) Thecorresponding formula forthevoltagefromSec.2,Eq.(6),is Y z=Wecosh(yx+90)cosh(yw+98) zsinh(y8+90+98)(4) Ifthesamecombination iscarriedoutwithSec.2,Eqs.(7)and(8), thefollowing resultsarereadilyderived(notethatx+gcorresponds to y-gandthatx-gcorresponds toy+g,sincebydefinition y==8 -x): W;cosh(yy+98)sinh(yz+90) 121= -Zcsinh(y8+90+98) Y z=Wecosh(yy+98 )cosh(yz+90) zsinh(y8+90+98)o~z~x-g(5) o~z~x-g(6) Actually norestrictions havebeenimposed onthedistance 2gbetween thepointgenerators ineachconductor. However, thecurrents and voltages aredefinedonlyatpointsoutsidethedistance 2gin(2)and (4)andin(5)and(6).Notethat,ifgissufficiently smalltosatisfythe inequality lygl2«1,(3)reducesto (7) Thelaststepin(7)impliestheinequality a«{lIf2gisallowedto become infinitesimally smallwhileVeismadecorrespondingly great, W:remains finite. Thegeneralrelations (2),(4)and(5),(6)givethecurrentandvoltage atanypointzalongaterminated linethatisdrivenbytwopairsofequal andopposite pointgenerators (ortheirequivalents) symmetrically placed oneachsideofthepointxalongtheline. Thenatureofthediscontinuity atthepointxinthelimitas2g approaches zero,whileVebecomes infiniteandW;remainsfinite,isinter- 248 TRANSMISSION-LINE THEORY [Chap.IV esting. Aszapproaches xfromabove,(2)and(4)reduceto I=W~cosh(yx+60)sinh(yy+6s) zZc sinh(ys+60+6s) V-Wecosh(yx+60)cosh(yy+6s) z- xsinh(ys+60+6s) Aszapproaches xfrombelow,(5)and(6)reduceto W;cosh(yy+6s)sinh(yx+60) -Zc sinh(ys+60+6s) Wecosh(yy+6s)cosh(yx+60) xsinh(ys+60+6s)(8) (9) (10) (11) ItisseenthatVziscontinuous whileIzjumpsfromIzto-Izatz=x. Thediscontinuity isreadilyevaluated tobe WeIz(z~xfromabove)-Iz(z~xfrombelow) =Zcx(12) Thusadiscontinuity inVzbyVeatx+gandby-Veatx-gwith continuous Izisequivalent, inthelimitasgapproaches zero,toacon tinuous Vzandadiscontinuity inIzbyW;/Zcatz=x.Physically thesemathematical resultsmaybeinterpreted asfollows:Thetwopairs ofequalandopposite pointgenerators tendtosetupequalandopposite currents intheconductors oflength2gbetween them.Depending on theimpedances inthetwodirections, oneortheotherpairmayproduce alargercurrent. Inthiscasetheequalandopposite generators atthe endsoftheinfinitesimal distance 2gareequivalent toacurrentgenerator atthecenteroralinedrivenbyashuntgenerator atz=x. 4.General Expressions jorCurrent andVoltagejoranArbitrarily Terminated LineWhenDrivenbyThreePairsofGenerators (orTheir Equivalent) Anywhere alongtheLine.8o,81Inordertorepresent analyt icallyasymmetrical drivingunitscoupledtoatransmission line,itis necessary toconsider alinedrivenbythreepairsofequalandopposite generators symmetrically oriented withrespecttothepointx,asshown inFig.4.1.'Thetotalcurrentandvoltagemaintained bythethreepairs vevive 2""22"oee lz ZO~L.--_~f--~" tf_Z----!fZs : ~Yivell : 2 2 2 jIo;~I z sx-glx:x+g FIG.4.1.Transmission linedrivenbythreepairsofequalandopposite pointgenerators symmetrically locatedwithrespecttoanarbitrary pointatadistance xfromoneend oftheline. Sec.5]AMPLITUDE RELATIONS FORCURRENT ANDVOLTAGE 249 ofgenerators areobtained bysuperimposing thesolutions obtained in Sees.2and3.Theresulting expressions arecompactly writtenusing thefollowing shorthand notation: Sm==sinh("(m+60)Sn==sinh("(n+68) em==cosh("(m+60)Cn==cosh("(n+68) Sa==sinh("(8+60+6a) Ca==cosh("(8+60+6a)(1) (2) (3) (4) In(1)and(2)mmaystandforxorz,andnstandsforyorw.With thisnotation theexpressions forthecurrentandvoltageatanypoint alongatransmission linethatisdrivenbythreepairsofpointgenerators arranged asinFig.4.1are 1=1-(V~Sx+W~Cx)Sw zZc Sa V z=(V~Sx+W~Cx)Cw Sa I z=1-(V~Sy-W~Cy)Sz Zc Sa V_(-V~Sy+W~Cy)Cz z - SaO~z~x-g O~z~x-g(5) (6) (7) (8) Asbefore,zisthedistance fromZotothepointatwhichcurrentand voltagearemeasured; w==8 -zisthedistance fromZaattheotherend ofthelinetothesamepoint;xisthedistance fromZotothemid-point ofthethreegenerators; andy==8 -xisthedistance fromZatothe samepoint.Thepairofgenerators maintaining V~isatx,andthetwo pairsmaintaining W~arelocatedatx±g. 5.PolarFormoftheGeneral Expressions forCurrent andVoltage. Thegeneralexpressions forthecomplex currentorvoltageatanypoint alongatransmission linewhendrivenbyone,two,orthreepairsofpoint generators maybeexpressed conveniently inpolarformintroduced in Chap.II,Sec.9.With"(=a+j(3and6=p+j4.>,thehyperbolic functions areeasilyseparated intorealandimaginary parts.Ifall phasesarereferredtoV~(whichisthusassumed tobereal),theformulas forthecurrentandvoltageduetoonepairofgenerators maybeexpressed asfollows: (1) (2) (3) (4) 250 TRANSMISSION-LINE THEORY [Chap.IV where (7)(5c) (6b)(5b)(5a) (5d) (6a)S:z;=[sinh2(ax+po)+sin2({jx+<l>o)]i iT:z;=tan-1[tan({jx+<1>0)coth(ax+PO)] SlI=[sinh2(ay+P.)+sin2({jy+<1>.)]1 iTlI=tan-1[tan({jy+<1>.)coth(ay+P.)] SW=[sinh2(aw+P.)+sin2({jw+<1>.)]1 iTw=tan-1[tan({jw+cI'.)coth(aw+P.)] S.=[sinh2(as+Po+P.)+sin2({js+<1>0+~.)]i iT.=tan-1[tan({js+~o+~.)coth(as+Po+P.)] CW=[sinh2(aw+P.)+cos2({jw+~.)]1 Ew=tan-1[tan({jw+cI'.)tanh(aw+P.)] CZ=[sinh2(az+po)+cos2({jz+~0)]1 Ez=tan-1[tan({jz+~o)tanh(az+po)] Notethatthesubscript onSorCalwaysreferstothevariable. The characteristic impedance is Ze=Re(l-jcPe)==Ree-i</>· sinceitisassumed that cP~«1. Thecorresponding expressions fortwopairsofpointgenerators are O~z~x-g O~z~x-g(8) (9) (10) (11) where,inaddition to(5a,b,c,d) and(6a,b),thefollowing shorthand sym bolsareused: (13)(12) (14)C:z;=[sinh2(ax+po)+cos2({jx+~o)]i E:z;=tan-1[tan({jx+~o)tanh(ax+po)] ClI=[sinh2(ay+P.)+cos2({jy+~.)]1 Ell=tan-1[tan({jy+~.)tanh(ay+P.)] SZ=[sinh2(az+po)+sin2({jz+~o)]i iTz=tan-1[tan({jz+~o)coth(az+po)] Thepolarformulas forcurrentandvoltageinalinedrivenbythree pairsofpointgenerators arecomplicated. Theyareobtained byadding (1)and(8),(2)and(9),(3)and(10),and(4)and(11)withappropriate phaserelations between W~andV~andreducing topolarform.Note thatanyoftheformulas (5), (6), (12),(13),or(14)maybeexpressed in termsofdoublearguments usingChap.II,Sec.9,Eq.(3)or(4),viz., S=vj(cosh 2u-cos2v)=vsinh2u+sin2v (15) C=Vj(cosh 2u+cos2v)=Vsinh2u+cos2v(16) Sec.6]AMPLITUDE RELATIONS FORCURRENT ANDVOLTAGE 251 If,intheexpressions forIVzland11zl,theover-alllengthsofatrans missionlineisvariedinsuchamannerthat8.istheonlyvariable factor, theresulting variations inIVzland11z1arecalledresonance curves. Ifthe singlevariable iswOf'zlocating thepointwhereVzand1zaremeasured, theresulting variations inIVzland11z1arecalledvoltageorcurrentdis tribution curves. Ifthelocation ofthedrivingpointxoryisvariedwith allelseconstant, theresulting variation inIVzlor11z1atanarbitrary fixedpointziscalledadriving-point distribution curve.Thesethreetypes ofcurves,corresponding tovariations insalone, worzalone,andxory alone,areconsidered inSec.7.Notethatw==s-zandy==s-x. 6.TheTransfer ofPoweralongaTransmission Line.81Oneofthe principal functions ofatransmission lineistotransfer powerfroma generator toaload.Letitbeassumed thatthesourceofpowerisequiva lenttoonepairofpointgenerators atapointz=xonatransmission linethatisterminated inaloadimpedancet Z.atz=s.Thetime average powertransferred tothesectionoflineoflengths-zandits termination Z.is (1) (3)(2)whereVzisthecomplex (peak)voltageand1:isthecomplex conjugate ofthe(peak)currentatthepointzontheline.Thesequantities are givenbySec.2,Eqs.(5)and(6),orbySec.5,Eqs.(1)and(2).The desiredformula forpowerisobtained from(1)usingSec.2,Eqs.(5)and (6),byexpressing thehyperbolic functions involving thevariables xand sinpolarform,asinSec.5.Thedesiredformsare 1z=~~:z;[sinh(yw+9.)]ei(lTz-IT.) Vz=V~~:[cosh(yw+9.)]ei(lTz-IT.) Thesubstitution of(2)and(3)in(1)gives 1V1*1(V~)28;.h(A'F) h"(A+of)2z z=2Z*82sm w-Jwcos wJw e •(4) whereAw==aw+p.andFw==fJw+<p..SinceZe=ReO-jc/Jc)and sinh(Aw-jFw)cosh(Aw+jFw)=i-(sinh2Aw-jsin2Fw)(5) thetime-average powertransferred tothelineatadistance wfromthe loadis Pz=(:i2 ~i[sinh2(aw+P.)-c/Jesin2(fJw+cp.)] (6) tTerminal-zone effectsareassumed negligible forthesakeoff'implicity. Ifthey aresi~nificant, theapparent terminal impedance Z.GmustreplaceZ•. Thepowerintheloadatz=sisgivenby(6)withw=o. P(Y~)2s;(.h 2 . 2<1», =4RcS~smp,-cJ>csm,252 TRANSMISSION-LINE THEORY [Chap.IV Itist (7) (11) (12)Thepowerinthelineandloadisgivenby(6)withz=xorw=s-x. Itis (ye)2S2 Pz=4R cS~[sinh2(as-aX+p,)-cJ>csin2({3s-(3x+cI>,)](8) Theratioofthepowerintheloadtothepowerinthelineandloadisthe efficiency. Itis W=P,= sinh2p,-cJ>csin2<1>, (9) Pzsinh2(as-ax+p,)-cJ>csin2({3s-(3x+<1>,) Whenthegenerator isattheendoftheline(x=0),asisusual,the powerratiois W=P,= sinh2p,-cJ>csin2<1>, (10) Posinh2(as+p,)-cJ>csin2({3s+<1>,) Onlow-loss lines cJ>cisoftheorderofmagnitude of10-3or10-4•If sinh2p,islargecompared withthisvalue,asisusualifZ,isadissipative impedance, theratioreducestothesimpleform W=P,==sinh2p, Posinh2(as+p,) TheratioofthepowerPI,dissipated inthelinetothetotalpowertrans ferredtothelineis PL=1 _P,==1 _sinh2p, Po Po sinh2(as+p,) Theinsertion lossinthelineisdefinedintermsoftheratioofthepower tNotethatinordertodissipate nopowerinthe10ad,P,=0,itisnecessary that sinh2p,=q,csin2<1>..Suppose thetermination consists ofasmallinductive imped anceR.+jwL.suchasawirebridgewith<1>.=(3k,+7r12andP.=a(m.-k.), wherek.=L.llandm.=R.lri=0fornodissipation. SinceP.issmall,p.=0in (7)reducestoP.=-ak.;alsor.=e-2P•=e2ak••NotethatP.isnegative andthat r.isgreaterthan1.Evenifm.isnotzerobutsufficiently smallsothatitislessthan k.,P.isstillnegative andr.greaterthan1,asdiscussed inChap.II,Sec.22.Alterna tively,ifP.=0andr,=1,k,-=m.andthepowertotheloadis p_(V~)2S;2aR, , -4RcS; ri Thusareflection coefficient ofunityisstrictlynotpossible withanidealdissipation lesstermination ifthisisasmallinductive reactance andthelineitselfisdissipative. Onmostlow-loss linesq,cissufficiently smallsothatthereflection coefficient can exceedunitybyonlyanextremely smallamount. Sec.6]AMPLITUDE RELATIONS FORCURRENT ANDVOLTAGE 253 totheloadwithouttheline(s=0)tothepowertotheloadwiththeline. L(db)=1010Po==1010sinh~(as+P.) (13)gP. gsmh2p. Theconvenience andobvioussignificance oftheterminal function P.are evidentin(9)to(13). Optimum Termination. Theoptimum termination minimizes thelosses onthelineandmaximizes theefficiency. Thecondition formaximum efficiency is oW=0 (14)op. Ifthisdifferentiation iscarriedoutusing(11),theresulting condition is tanh2(as+P.)=tanh2p. (15) Foralinethatisnotlossless, as>0,sothat(15)canbesatisfied only when p.=00Z.=Zc (16) Thisisthecondition formaximum efficiency. With(16),(11)becomes W( ) 1·sinh2p. 1·e2p • 2(7)P.=00=1m•h2(+ )=1m--=e--1pc-> 00SInasP. pc-> 00e2(aB+p.) Thecorresponding minimum insertion lossis L(db)=10loge2a•=8.686as (18) whereaistheattenuation constant inneperspermeter. Notethat(16) isthecondition foramatched line,sothat(17)and(18)coincide with Sec.1,Eqs.(6)and(8). Iftheimpedance Z.terminating thelineispredominantly resistive and, inaddition, differsconsiderably fromthecharacteristic resistance Rcof theline,itisshowninChap.II,Sec.18,that ForR.<Rc, p.==~:P:«3 (19a) ·ForR.>Rc, P.==~:P:«3 (19b) Iftheover-allattenuation isnogreaterthanp.in(19a,b),thefollowing condition issatisfied: (as+p.)2«3 (20) sothatthehyperbolic sinesin(11)maybereplaced bytheirarguments. With(20)and(19a,b),(11)becomes w=P.==_P_._= Poas+P.R.<Rc (21) 254 TRANSMISSION-LINE THEORY [Chap.IV SOthatthepower(insertion) lossindecibelssubjectto(19a,b)and(20)is L(db)==10log(1+~:as) L(db)==10log(1+~:as)(22a) (22b) Agraphical representation isgiveninFig.6.1ofthepowerlossL(db) asafunction oflinelengthinatypicaltransmission linewhenmatched andwhenterminated inapureresistance thatisconsiderably smaller 5r------~-----...,...--------. 100020a=2.26x 1O-3nepers/m p=3.144 radians/m +c=7.l8x 10-4 Rc=440ohms ~--Rs=60ohms;Xs=O;Ps=O.137 ~15t---_-_-_z-"'s_=_z-"-c_t- -+ -I--.M ~"C.sm10!"------i------i----/-_+_----I.s 10 100 Lengthofline,8(meters) Fro.6.1.Powerlossinamatched lineandinalinethatisnotmatched.5t------+--------I7,£:---,;-':.----I thanthecharacteristic resistance. Forthecaserepresented, itisevident that,ifthelineisshort(5morless),thelinelossisnegligible, sothat littleisgainedbyusingamatching network. Ontheotherhand,fora linethatislongenoughsothatlinelossissignificant, itisessential to matchtheloadwithanappropriate networkifgoodefficiency istobe maintained. 7.Resonance CurvesandtheCondition forResonance.8,78,81The magnitudes ofthevoltageandcurrentatadistance wfromanimped anceZsatz=sorw=0duetogenerators locatedatadistance xfrom Zoatz=0aregivenbythefollowing expressions: V=VeSxCw ZxSs(1) Ifthelengthsofthelineistheonlyvariable, whilethedistance wofthe detector fromZsandthedistance xofthegenerators fromZoarekept Sec.7]AMPLITUDE RELATIONS FORCURRENT ANDVOLTAGE 255 constant,t Thusthefunction 8;1characterizes thedependence ofbothV.andI. onthelengthoftheline.Thisfunction, whenplottedagainst {js,is calledaresonance curve.Atransmission linewithitstwoterminations ZoandZ.issaidtoberesonant when8;1hasitsmaximum value. Withthenotation A,==as+Po+p,F.=={js+<1>0+<1>, (3) theconditions determining theextreme valuesof8;1withrespectto changes insareobtained bydifferentiating (2)withrespecttosand equating theresulttozero.Thefollowing equation isobtained: (cosh2A.-cos2F,)-i(a sinh2A,+(jsin2F.)=0(4a) Sincecosh2A.alwaysexceedsunity,theonlypossiblerootsaredefinedby -sin2F.=~sinh2A. Thisleadstothefollowing extremizing valuesofF,:(4b) F2n+1+1 • 1(a.h2A)• =--2- 1f'VSln-7JSIn , forminimum 8;1andn=0,1,2, (5a) F.=n1f'-i-sin-1(~sinh2A.) formaximum 8;1andn=0,1,2, (5b) ThesecannotbesolvedreadilyforsunlessA,issmall.However, ifA, issmall,sothat itfollowsthatA:«1sinh2A,==2A, (6) isin-1(~sinh2A.)==~A.«1 (7) With(7),(5a)becomes 2n+1aF,=={js+<1>0+<1>.=-2- 1f'+7J(as+po+P.) forminimum 8;1(8a) Asimilarexpression isobtained for(5b).Sinceitisassumed thatthe tNotethatchanging 8bymovingZ.(orZo)requires thatthedetector (orthe generator) bemovedintandem. 256 TRANSMISSION-LINE THEORY [Chap.IV condition a2/{32«1issatisfied, itfollowsthat(5a)and(5b)become 2n+1aFa=={3s+<1>0+<1>s=-2- 7r+~(PO+Pa) Fa=={3s+<1>0+<1>s=n7r-~(PO+Pa) Whenever thecondition ~(po+Ps)«1 issatisfied, thefollowing expressions arevalid:forminimum S-;l(8b) formaximum S-;l(8c) (9) .2n+1Fs=={3s+<1>0+<1>s=-2- 7r forn=0,1,2,...andS-;laminimum (lOa) Fs==(3s+<1>0+<1>s==mr forn=0,1,2,...andS-;lamaximum (lOb) Thecondition (lOb)maximizing S;liscalledthecondition forresonance. Notethattheangle eTsinSs=Sseiu,hasthefollowing values: u.='tan-1(tanF,oothA.)~{~ Theextreme valuesofS-;1areforminimum S-;l formaximum S-;l(11) (S-l)_1 _ 1 smax----;---hA - .h(+ + )SInsSInasPoPs(S-;l)min =-hIA-=h ( 1+)cos acosas+PoPs2n+1 {3Smin+<1>0+<1>s=-2- 7r n=0,1,2,..,(12a) {3smax+<1>0+<1>s=n7r n=0,1,2, (12b) Amongthespecialcasesthefollowing areimportant: (as+Po+Ps)2«1 Po+ps»as(S-l)...!... 1 smin-h (+ )COSPoPs(S-l)...!... 1smax-+ +asmaxPoPs (S-l)...!... 1·smax-·h(+)SInPoPs(13) (14) Notethatin(14)theextreme valuesareindependent ofs. Thegeneralshapeoftheresonance curvesforlowover-allattenuation (A;«1)isobtained readilyfrom(2)ifitisnotedthat,exceptnear resonance whereFs=={3s+<1>0+<1>s=n7r, (15) Itfollowsthattheresonance curveisacosecant curvelimitedbythe bounding curvescschAandsechA,asshowninFig.7.1foralinewith Sec.8]AMPLITUDE RELATIONS FORCURRENT ANDVOLTAGE 257 2TT TT47Tps 311' 41T Fs=fls+~o+~s FIG.7.1.Resonance curves(qualitative) foralinewithnegligible lossesinthelineand inthegenerator; as+Po«Pa. \.csch(Po+Ps), "-"-"..... sech(cts+Po+Ps) 311j3s cJls+cJlo 1T 21T 31TFs=ps+cf>o+cf>.'1 FIG.7.2.Resonance curves(qualitative) foramoderately damped line. ........._liAs--- 11:==~~:::::I===~=±==~"':::=::Lo I . cJlo+cJls 1T 27T 37TFs=(j1s+<1>0+<l>s> FIG.7.3.Resonance curves(qualitative) foralinewithlowover-allattenuation. negligible lossesinthelineandgenerator, inFig.7.2foralinewith moderate over-all attenuation, andinFig.7.3foralineinwhichthe over-allattenuation issmall. 8.Distribution Curves. 8,78,81Thevariation ofthevoltageandcur rentalongafixedtransmission linemaybeexpressed asafunction of thedistance w==s-zfromtheloadimpedance Zs.Theformulas for 258 TRANSMISSION-LINE THEORY [Chap.IV themagnitudes ofthevoltageandcurrentdistribution functions are Vz(W) t"'..ICw=[sinh2(aw+Ps)+cos2({3w+<Ps)]l (1) Iz(w) t"'..ISw=[sinh2(aw+Ps)+sin2({3w+<Ps)]l (2) Analternative formula forVz(w)whichislikethatforIz(w)isreadily obtained iftheterminal function <P:=<Ps-7r/2isintroduced in(1). Theresultis Vz(w) r-.JS~=[sinh2(aw+Ps)+sin2({3w+<p~)]l (3) Evidently thevoltagevariesinjustthesamemannerasthecurrent,but thedistribution isshiftedalongthelineanelectrical distance 7r/2with respecttothecurrent. Distributions ofthevoltageandcurrent as definedby(1)and(2)or(3)and(2)maybeobserved inpractice by movingalooselycoupledvoltageorcurrentdetector alongthelinewhile allotherquantities arekeptconstant. Thevariations ofthemagnitudes ofthevoltageandcurrentatagiven fixedpointzalongatransmission line,asthelocation xofonepairof equalandopposite pointgenerators (ortheirequivalent) ischanged, are expressed asfollows,usingSec.5,Eqs.(1)and(2): Notethat(4)impliesx~,z~sandthatxistheonlyvariable, with allotherquantities constant. Thismaybeaccomplished inpractice by movingalooselycoupled oscillator orcoupling unitparalleltotheline whilethevoltageorcurrentisreadfromastationary detector. Iftwopairsofequalandopposite generators symmetrically located with respe~ttothepointxaremoved,thevoltageandcurrentatthe fixedpointz(withx~z~s)varyasfollows, usingSec.5,Eqs.(8) and(9): Vz(x) t"'..IIz(x) t"'..ICx=[sinh2(ax+Po)+cos2({3x+<Po)]l (5) Alternatively, intermsof<P~=<Po-7r/2, Vz(x) t"'..IIz(x) t"'..IS~=[sinh2(ax+po)+sin2({3x+<p~)]1 (6) Plotsofthefunctions SwandCw=S~against {3worofSxandCz=S~ against {3xarecalleddistribution curves. Withappropriate changes invariables andparameters, thefunctions 8w,S~,Sx,andS~areessentially thereciprocals ofS;I.Therefore the extreme valuesareobtained directly fromtheanalysis ofS;1inSec.7. Forcurrentandvoltagetheyare [Iz(w)]max r-.J(Sw)max=cosh(aw+Ps) [Iz(w)]min t"'..I(Sw)mio=sinh(aw+Ps)2n+1when{3w+<Ps=-2-7r (7a) when(3w+<Ps=n7r (7b) Sec.9]AMPLITUDE RELATIONS FORCURRENT ANDVOLTAGE 259 [Vz(W)]max "'-'(S~)max=cosh(aw+Ps) [Vz(W)]min "'-'(S~)min=sinh(aw+Ps) wheren=0,1,2,. . . . Foronepairofpointgenerators atx,2n+1when(3w+~=-2- 7r(8a) when(3w+~=n7r (8b) [Iz(X)]max"'-' [Vz(X)]max "'-'(Sx)max=cosh(ax+po) 2n+Iwhen(3x+<1>0=--2- 7r(9a) [Iz(x)]min"'-' [Vz(X)]min "'-'(Sx)min=sinh(ax+po) when(3x+<1>0=n7r(9b) Fortwopairsofpointgenerators symmetrically spacedaboutx, [Iz(X)]max"'-' [Vz(X)]max "'-'(S~)max=cosh(ax+po) 2n+1when{3x+~=--2- 7r(lOa) [Iz(x)]min"'-' [Vz(X)]min "'-'(S~)min=sinh(ax+po) when(3x+~=n7r(lOb) Thegeneralshapesofthedistribution curvesforlowover-allattenu ationaregivenby Cw=S~==Isin({3w+~)I Sw==Isin({3w+<l>s)I Cx=S~==Isin({3x+~)I SX==Isin({3x+<1>0)Isinh2(aw+Ps)«sin2({3w+~) sinh2(aw+Ps)«sin2({3w+<l>s) sinh2(ax+po)«sin2({3x+~) sinh2(ax+PO)«sin2({3x+<1>0)(lla) (lIb) (12a) (12b) Thenatureofthedistribution curvesSwandCw=S~forthecurrent andvoltagemaybeseeninFig.8.1.Notethat{3w=(3(s-z)isscaled toincrease fromlefttoright. o sinh(aw+ps) ~spw+<t>s <t>~j3w+<t>~ FIG.8.1.Current andvoltagedistributions alongatransmission line. 9.Resonance-curve andDistribution-curve Ratios;theStanding waveRatio.8•78,81Quantities thatareusefulinvarioustransmission-line measurements aretheratiosofmaximum valuesofresonance ordistribu tioncurvestoadjacent minima. Forlineswithlowover-allattenuation 260 TRANSMISSION-LINE THEORY [Chap.IV (1) (2b)(2a) n=0,1,2,... wherethemaxima ofresonance curvesandtheminimaofdistribution curves areverysharp,whereastheminimaofresonance curvesandthemaxima ofdistribution curvesarebroad. Thisbehavior isreadilyunderstood if itisrecalledthattheexpression fordistribution curvesisformally the reciprocal oftheexpression forresonance curvesintermsoftheappro priatevariables. Forlineswithadissipative loadbothresonance-curve maximaanddistribution-curve minimaarebroad. Theratioofthemaximum ofaresonance curvetoanadjacent mini mumisobtained directlyfromSec.7,Eqs.(12a,b). Thus R=(S~?max=c?sh(aSmin+Po+P.) (S.)minsmh(asmax+Po+P.) n7r-«1>0-«1>. Smax= {J X Smin=Smax±4" Iftheattenuation constant aofthelineissufficiently smallsothat aX4«asmax+Po+P. (3) (1)reducesto R(S;l)max. th(+ + )=(S-l).=coasmaxPoP8 amin(4) IfaSmaxissmallcompared withP8,asisusual,itfollowsthat sothataSmax«po+P. (S;l)max. th(+ ) (S-l).=coPop. •miD(5) (6) Finally,ifitiscorrecttoset PO«p. itfollowsthat(7) (8) (S;l)max ==cothP.==S=SWR(S;l)min Thequantity S==cothP8isknownasthestanding-wave ratio.tItis customarily definedintermsofthedistribution curvesofcurrentand voltageratherthanintermsofresonance curves. Thisiscarriedoutin thefollowing. Theratioofanygivencurrentmaximum alongatransmission lineto anadjacent minimum isobtained directlyfromSec.8,Eqs.(7a,b).Itis [Iz(W)]max (Sw)maxcosh(awmax+P8) [Iz(W)]min={Sw)min=sinh(aWmin+P8)(9) tTheletterSwithoutsubscript orwithsubscript VorIisusedforthestanding waveratio.Swithsubscript 8,w,x,etc.,isanamplitude asinSec.8,Eq.(2). Sec.9)AMPLITUDE RELATIONS FORCURRENT ANDVOLTAGE 261 where[(2n+1)/2]11"-4t. Wmax= {3 X Wmin=Wmax±4n=0,1,... (lOa) (lOb) Subjecttothecondition aX4«awlllax+p. itfollowsthat(9)maybeexpressed intheform [111(W)]max_(Sw)max..!. th(+) [I()J--(S)-coawmaxP. IIWmin wmin(lla) (lIb) Theattenuation constant ofmostlow-loss linesissufficiently smallso that,whenthelineisloaded,thefollowing conditions aresatisfied: P.»awmaxP.»aWmin (12) Inthiscasetheratioin(lIb)reducestoaconstant. Thus [111(W)]max..!. th-S [1.(w)]min -coP.=1(13) (14)wherethequantity SIisthecurrentstanding-wave ratioforthesectionof linebetween thegenerator andZ..Thevoltagestanding-wave ratiois obtained inthesamemanner. Subjectto(13)itis [Vll(W)]max =(S~)max=cothP.==Sv [Vll(W)]min (Sw)min Notethatsubjectto(12)thestanding-wave ratioofthecurrentand voltagedistribution curvesdepends onlyontheterminal functionP., sothat (15) (16)Theratioofthemaximum currentatztotheminimum currentatthe samepoint,asobtained bymovingthegenerators, hasthefollowing value foronepairofequalandopposite pointgenerators atx: [Ill(X)]max (Sz)maxcosh(aXmax+po) [Ill(X)]min=(Sz)min=sinh(axmin+po) Subjecttothecondition po»aXmax Po»axmin (17) whichrequiresthattheattenuation ofthesectionoflineoflengthxmax orXminbenegligibly smallcompared withtheattenuation ofthegener ator?(16)reducesto [Vll(X)]max =[Iz(x)]max=(Sw)max ==cothPo==SI (18) [Vz(X)]min [Iz(x)]min (Sw)min 262 TRANSMISSION-LINE THEORY [Chap.IV Fortwopairsofequalandopposite generators, thecorresponding ratiois (19) (20a) (20b)forr18~1 forg18~1ismosteasilymeasured fromthe current orvoltage distribution curvesusing(13)or(14).But theothermethods alsoservetheir purposes inspecialcircumstances. Acurveofthestanding-wave ratio asafunction oftheterminal at tenuation P8isshowninFig.9.1. Thestanding-wave ratiohasa particularly simpleformifthe terminal impedance Z8ispre dominantly resistive, sothat,as defined inChap.II,Sec.18, P8=tanh-1r18 P8=tanh-1gl8S=cothPs 101-----'1<----+------1S100,...-----,-----,---------,Evidently (18)and(19)arethestanding-wave ratiosforthesectionof linebetween thegenerators symmetrically placedwithrespecttox andZoo Thestanding-wave ratiousually X18=0and1'-- ..1.- ....L..--===_-..I 0.01 0.1 10Ps FIG.9.1.Standing-wave ratioasafunction oftheterminal function P•• Similarlywheregl8=l/Tl8=ReiR8•Since P8=coth-1S,itfollowsthat r18=tanhP8=tanh(coth-1S) 1 g18=-=SforT1~1ris g18=tanhP8=tanh(coth-1S) 1 r18= - = Sforr1~1 g18(21a) (21b) (22a) (22b) Forlineswithlowattenuation, sothattermswiththedistortion factor C/>Cascoefficient arenegligible, . R 8•Sr18= - =Re . Re•Sg18=R8=(23a) (23b) Thesearesimpleandusefulformulas. 10.Distributions ofCurrentandVoltageinaResonant Line;Compo nentsofCurrent andVoltage. Thedistribution curvesdiscussed in Sec.8andpictured inFig.8.1represent themagnitude ofthecurrentand voltage. Alsoofinterestarethecomponents thatareinphasewiththe Sec.10]AMPLITUDE RELATIONS FORCURRENT ANDVOLTAGE 263 emfofthegenerator andinphasequadrature withit.Forsimplicity letthelinebedrivenbyasinglepairofpointgenerators (ortheirequiva lent)atz=0,sothatx=0inthegeneralformulas ofSec.2.The expressions forcurrentandvoltageare TT_VIIsinh80cosh("(w+98)=V-h (+4)".- 0sinh("(s+90+98)- •cos"(wu. I-V~sinh90sinh("(w+9.)=V..h (+4)• -Z.h ( ) -ZsIn"(Wu.esm "(S+90+98 C(1) (2) (5) (6)whereV.isasdefinedin(1)and(2).Forconvenience letitbeassumed thattheimpedance ofthegenerator isapureresistance Zo=Ro,which ismuchsmallerthanRe•Then Ro«Repo==Ro«14>0=! (3)Rc 2 and sinh90=sinh(po+j4>o)=jcoshpo (4) Letthecircuitbetunedtoresonance sothat {3s+4>0+4>.=n1f'nodd Then sinh("(s+90+68)= -sinh(as+po+Pit) Hence,expanding cosh("(w+9.)andsinh("(w+9.), II-jcoshpo ) ( )V.=Vo.h (++)[cosh(aw+P.cos{3w+<1>.SInas.Pop. +jsinh(aw+P8)sin({3w+«1>.)](7) V~ -jcoshPo •1.=R(1.).h (+ + ) [smh(aw+P8)cos({3w+4>.) c -:JcPesmasPoP. +jcosh(aw+P.)sin({3w+4>.)](8) Letthefollowing notation beintroduced: Aw==aW+P.Fw=={3w+«1>. (9) From(1)and(2)andwith(4)and(6)itfollowsthat V=VII sinh90=VII-jcoshPo (10) •- 0sinh("(s+90+9.)0sinh(as+Po+P.) Then,sinceV.in(10)isequalto-jV.,itfollowsthat VOl=V.(sinhAwsinFw-jcoshAwcosFw) (11) 1.=~:(1+jcPc)(cosh AwsinFw-jsinhAwcosFw)(12) ItisreadilyverifiedthatthetermswithcPcascoefficient contribute nothing ofsignificance inpractically important cases.Thus I.=~:[coshAwsinFw+cPcsinhAwcosFw -j(sinhAwcosFw-cPecoshAwsinFw))(13) 264 TRANSMISSION-LINE THEORY [Chap.IV Usingtheformulas Asinx+Bcosx=VA2+B2cos(x-tan-l~) (14a) =VA2+B2sin(x+tan-1~) (14b) andneglecting termsmultiplied by<P~, Iz=~:{coshAwsin[FlO+tan-1(<PctanhAw)] - jsinhAwcos[FlO+tan-1(<PccothAw)]}(15) SincetanhAwcanneverexceedunity,theangle tan-1(<PctanhAw)==<PctanhAw (16) (17)isofmagnitude <Pc,whichisoforder10-3onalow-loss line.Inpractical application anglesarenotusuallymeasured tothenearestthousandth ofaradian,sothatnoobservable errorisintroduced byassuming <PctanhAw==O.Theangletan-1(<PccothAw)in(15)isnegligibly small solongascothAwdoesnotbecomelarge.Itbecomes largeonlywhen Awhecomes small,thatis,whencothAw==1/Aw•Inthiscase tan-1(<PccothAw)==tan-1~Aw 1==tan-1---:---~.....,.. (3(w+ps/a) Clearly, ifPsislargecompared witha,thisangleisverysmall.Onthe otherhand,ifps==0,theangledefinedin(17)issmallonlywhen{3wis sufficiently large;as{3w~0,theangleapproaches 1r/2.Thustheangle isimportant onlywhenps==0and(3wbecomes small.However, when wbecomes small,sinhAw==Aw=aWlikewise becomes extremely small. Thus,whenthephaseofthecosinein(15)issignificantly affected by thetermincJ>c,theamplitude oftheentiretermbecomes vanishingly small.Hencenosignificant errorismadeifthetermsin<Pcareomitted, sothat Vz=V;'+jV;==VAsinhAwsinFlO-jcoshAwcosFlO)(18) Iz=I~'+j/~==;~(coshA10sinFlO-jsinhA10cosF10)(19) Thevoltageandcurrentfortheloadareobtained withw=O.Theyare Vs=V~'+jV~==Vs(sinh pscos<1>s-jcoshPssin<1>s)(20) Is=I~'+j/~==~:(coshP.sin<1>s-jsinhpscos<1>s)(21) ThefunctionsV;',V~,and\Vzlareshowngraphically inFig.10.1,and I;',I;,andIIzlinFig.10.2.Themagnitudes \Vz\andIIzlare,ofcourse, identical withdistribution curvesforthevoltageandcurrentinSec.8. f3s+4>s+rr/2=3Tt; 4>o=Tt;Po=O ,, \ \......IVzl \ \ \ \ \ \ \" 2Tt" 3Tt/2-l'.cosh (aWma.+ps) 11'/2 ov."• 2Tt 3rr/2 11' rr/2 4>. Fw-j3w+ cl>s FIG.10.1.Distribution ofvoltageinaresonant line. I;;.....-__ o/", \ I \ I \I~I I \ I \ I \ I \ I \ I \t. Ksinhps-.....r· -------"h.li -Kcoshps -10-Kcosh(eI!w+Psl I(3w2rr 3rr/2 TT TT/2 0 2rr 3rr/2 TT 11'/2 4>. Fw=f3w+cl>s Flo.10.2.Distribution ofcurrentinaresonant line. 265 266 TRANSMISSION-LINE THEORY [Chap.IV 11.TheWidthsofResonance andDistribution Curves.ll,7S,SIThe greatertheover-allattenuation ofatransmission line,asdefinedbythe function a8+po+ps,wherea8istheattenuation oftheline,Pothatof thegenerator, andpsthatoftheload,thelowerandbroader arethe maxima oftheresonance curvesdescribed inSec.7.Similarly, the greatertheattenuation oftheterminated sectionoflinebetween that pointzwherethecurrentandvoltagearemeasured andtheloadat z=8,ascharacterized bythefunction aW+Ps,wherew=8 -z,the higherandbroaderaretheminimaofthedistribution curvesforcurrent andvoltagedescribed inSec.~. Thesameistrueofthedistribution curvesobtained bymovingthepointofcoupling xoftheemfintermsof thefunction ax+po.Itmaybeconcluded thattheratioofmaximum tominimum andthewidthataspecified fraction ofthemaximum or minimum oftheresonance ordistribution curvesareusefulintrans mission-line measurements ofattenuation. WidthofResonance Curve.Thesquareoftheamplitude ofaresonance curveisshowninSec.8tobeproportional tothefunction 8;2=(sinh2As+sin2Fs)-l wheretheshorthand notation(1) As::::::a8+Po+Ps (2) isused.Asusual,aistheattenuation constant, {3isthephaseconstant, and8isthelengthofthelinebetween theimpedances Zoatz=0and Zsatz=8.Thephasefunctions oftheseimpedances are«1>0and«I>s; thecorresponding attenuation functions arePoandps. Themaximum amplitude squaredisdefinedby(1)whenthecondition ofresonance Fs={38+«1>0+«I>s=n7f' issatisfied.Itisn=0,1,2,. . .,8~0(3) (8;2)max=csch2As (4) Thesquareoftheamplitude (1)isreducedtoanarbitrary fraction 1/p2 ofthemaximum whenFsischanged from(3)tooneofthetwovalues Fs1=Fs-oFI (5) Notethattherequired changeinFsmaybemadebyvarying(a)the lengthoftheline8,(b)thefrequency, sothat{3=w/vchanges, or(c) ZoorZs,sothat«1>0or«I>sischanged. Inanyone ofthesecasesAsis alsoaffectedsothat,paralleling (5), Thus(6) (7) Sec.11]AMPLITUDE RELATIONS FORCURRENT ANDVOLTAGE 267 wheretheuppersignisforFs1,withsubscript 1onoAandof,andthe lowersignisforFs2,withsubscript 2onoAandof.Notethat,ifthe amplitude isreducedentirelybychanging thelengthofthelinefrom8to itfollowsthator (8) oF1={1081 oA1=a081oF2={1082 oA2=a082(9a) (9b) Thesubstitution of(4)inthemiddletermin(7),usingthecondition forresonance (3),gives p2sinh2As=sinh2(As+oA)+sin2(n1l"+of) (10) wherethesubscript 1isusedonoAandofwiththeuppersign,andthe subscript 2withthelowersign.Byexpanding thehyperbolic andcircu larsinesthefollowing transformed expression maybederived: (p2-cosh2oA)sinh2As+2sinhAscoshAssinhoAcoshoA -sinh2oAcosh2As=sin2of(11) SolongasthechangeinFsismadebyvaryingeitherthelength 8orthe frequency, theassociated changeinAsisverysmallif,asisassumed, alow-loss lineisinvolved. Thatis, (12) (13) (15)(14)coshoA==1 (OA)2«1sinhoA==oA With(12),(11)maybeapproximated by (-/-2--1 .hA+oAcoshAs)2- .2~F vp-sm s__~ -Sillu Vp2-1 Thetwoequations from(13)are -/-2--1· hA+oA1coshAs '0rFIvp-SIn s•~ =sInu1Vp2-1 -/-2--1 .hAoA2coshAsI'rFIvp-SIn s - _~ =SInu2Vp2-1 Let(14)beaddedto(15)togive IsinoF11+IsinoF21=2Vp2-1sinhAs(1+VOAl-oA2 .)2p2-1tanhAs (16) SinceoAlandoA2arebothsmallandnearlyequal,thecondition loAl-oA2!«2Vp2-1tanhAs (17) usuallycanbesatisfied easily. Subjectto(17),(16)canbesolvedfor Asasfollows: (18) 268 TRANSMISSION-LINE THEORY [Chap.IV Thisisthefundamental equation relatingtheover-alldamping factor A.=as+po+P.tothechanges ~F1and~F2required toreducethe resonance curvetolipofitsmaximum valueinbothdirections. Notethattheresonance curveissymmetrical if~F1=~F2and 8Al=~A2.Forasymmetrical curve(17)issatisfied automatically, so that(18)isagoodapproximation. Itisevidentfrom(14)and(15) that ~Fland~F2approach equality asthetermsin~Abecomesmaller. Hencesatisfactory conditions foranapproximately symmetrical reso nancecurveare ~A1«(p2-1)tanhA.8A2«(p2-1)tanhAll(19) With(19), ~Fl==~F2==~F (20) sothat(18)reducesto A. .h-1Isin~FI (21).=Sln _~ Vp2 -1 Iftheover-alldamping issufficiently smallsothat sin2~Fl==sin2~F2==(p2-1)sinh2A.«1 (22) itfollowsthatsin~Fl==~Flandsin~F2==~F2,sothat(18)reducesto A.==~Fl+~F2=6.F. (23) 2Vp2-1 2V1J2=1 where 6.F.==~Fl+~F2 (24) istheangular width oftheresonance curveatalevellipofmaximum. Bysetting p2=2andthusdefining theso-called half-power level,the following verysimplerelationisobtained: A.==~. (25) (26b)(26a) Withp2=2,Thatis,theover-allattenuation factorisone-halfthewidthofthereso nancecurveatalevellip=0.707. Iftheonlyvariable isthelengthsoftheline,sothat~F={j~sand 6.F.=6.8=(j(~Sl+~82),(23)reducestothefollowing: A+ +.{j6.s.=a8 poP.=_~2Vp2-1 A+ +.(j6.8• =asPoP.=2 Notethat6.8isthewidthoftheresonance curveatpowerlevellip whenthelineisresonantatthelengthsdefinedby(3).Theconditions impliedin(26a,b)are a88==a:8«1(88)2«(:~)2 (27) Sec.12]AMPLITUDE RELATIONS FORCURRENT ANDVOLTAGE 269 WidthofDistribution Curve.Letthedistribution curveobtained by movingacurrentorvoltageprobealongthelinebeexamined. (The distribution curveobtained bymovingthepointofcoupling ofthegener atormaybeobtained byasimplechangeinnotation.) Thesquareof theminimum amplitude is s~=sinh2Aw (28) whereA",=aw+P..Theminimum current amplitude occurswhen F",=(jw+4>8=n1rw~0 (29) Thesquareoftheamplitude increases top2timestheminimum attwo valuesofFw,onelocatedoneachsideoftheminimum asdefinedin(28). Thesevaluesare (30a) Ifwistheonlyvariable, thelocations ofthetwovaluesdefinedin(30a) are (30b) ThevaluesofAcorresponding to(30a)are (30e) ThevaluesofS~atF"'1andF",2mustsatisfytheequations p2sinh2Aw=sinh2(A",=+=~A)+sin2(n1r=+=~F) (31) wheretheuppersignisfor~Fland~Alandthelowersignisfor~F2 and~A2.Since(31)islike(10),withasubscript winsteadofs,itfol lowsthatthesamesolutions areobtained subjecttocorresponding con ditions. Thefundamental relationis(18),withthesubscript sreplaced byw;~F1and~F2arethechangesinFwwhichincrease theamplitude to thelevelp.Therelationcorresponding to(26a)is A+.{j~w",=awP.=_/2Vp2-1 Withp2=2,thisreducesto A+.fJ~w",=awP.=-2-(32a) (32b) In(32a,b) ~wisthedistance between half-power pointsoneachsideof thedipinthedistribution curvewhich occurs atw.Conditions corre sponding to(27),withwsubstituted fors,mustbesatisfied if(32a,b) areused.Theconditions forasymmetrical distribution-curve dipare like(19),withsubscript wsubstituted fors. 12.TheIIQ"ofaTransmission Line.AqualityfactorQmaybe definedforacomplete transmission-line circuit.consisting ofalengthof lines,aloadimpedance Z.,andagenerator impedance Zoorforatermi- 270 TRANSMISSION-LINE THEORY [Chap.IV (1)natedsectionconsisting ofalengthoflinewandaloadimpedance Zs. Inbothcasesthedefinition forthereciprocal oftheQis 1 _of'+of"ow'+ow"o{3'+o{3" Q=f w (3 wheref+of'andf-of"arethe"half-power frequencies" atwhichthe squareofthecurrentorvoltageatanarbitrary pointalongthelineis reducedtoone-half themaximum valueatthefundamental frequency j.t Experimentally Qmaybeobtained fromaresonance curvebyvarying thefrequency inordertodetermine f,f+of',andf-of".Inthefre quencyrangef-of'tof+of'letthefollowing condition beagood approximation: oAs«As==as+po+Ps (2) Also oFs=o({3s+<1>0+<l>s) (3) Itisunderstood in(2)and(3)thatoAsstandsforeither oA~oroA;'and oFsforoF~orof;',whereAs+oA~andFs+oF~correspond tow+ow' andAs-oA;'andFs-of;'correspond tow-ow",wherewisthe fundamental resonant frequency. Ifinterest isinaterminated sectionoflineoflengthwwithloadZs, (2)and(3)arereplaced bythefollowing: oAw«AwAw=aw+ps (4) ofw=o({3w+<l>s) (5) Sincewith(3)thecondition impliedinSec.11,Eq.(23),isfulfilled, itfollowsthat,withp=2, As=j(oF~+oF~')==~(o{3'+o{3")+j(o<l>~+o<l>~')+j(o<l>~+o<l>~') (6) Similarly Aw=j(oF~+of;:)==~(o{3'+0{3")+j(o<l>~+0<1>;') (7) If(6)issubstituted in(1),anexpressiop isobtained fortheloadedtotal Qoftheentireresonant transmission-line circuitincluding theline,the load,andthegenerator. Itis Q Q {3s (8)=t=2[as+Po+Ps-(o<l>~+o<l>~'+o<l>~+o<l>~')] wheresisthelengthoflineatthelowestresonant frequency. Itis givenby {3s+<1>0+<l>s=n7l" wherenisthesmallest integerforwhich{3sispositive.(9) tAQfactorforagivencircuitateachofanunlimited number ofharmonic fre quencies canbedefined ifdesired. InthissectiononlytheQatthefundamental frequency isintroduced. Sec.12]AMPLITUDE RELATIONS FORCURRENT ANDVOLTAGE 271 Alternatively, if(7)issubstituted in(1),anexpression isobtained for theloadedQatinputresonance ofasectionoflineoflengthwtermi natedintheloadZ,.Itis {3w Q=QL=2[aw+p,_(c5~+c5~')] (10) Theresonant lengthwisdefinedby {3w+,=n7r wherenisthesmallest integerforwhich{3wispositive. Iftheendsofthelineareeitheropen-orshort-circuited,(11) po==p,==0 (12) (15)and(8)and(10)reducetothesimpleformcharacteristic ofthelinealone. TheunloadedQis Q=Qo=fa (13) NotethatQoisindependent ofthelengthoftheline.Since ~=2~(1+~) (14) itfollowsthat 1rg Qo=wl+wC Foralow-loss linethecondition (a/{3)2«1issatisfied, sothat Q~»4 (16) Theexternal Q'softheresonant circuitoflengthsandtheresonant sectionoflineoflengthw,respectively, maybedefinedby {3s QE=2[po+p,-(c5~+c5~'+c5~+c5~')] (17a) (3w QE=2(p,_«5~_«5~') (17b) 1 1 1 sothat Qt=Qo+QE (18a) ~=1..+~ (18b)QLQoQE Itissignificant that,ifc5F,=«5({38)and«5=0,theeffectofvarying thefrequency withsfixed(sothat«5F,=8c5(3)isessentially equivalent tovaryingthelengthwiththefrequency fixed(sothatc5F,=(3«58).In thislattercaseQtisgivenby 1158'+c5s" Qt= S(19) 272 TRANSMISSION-LINE THEORY [Chap.IV where 8+08'and8 -08"arethelengthsoflineatafixedfrequency for whichthesquareofthecurrentorvoltageatanarbitrary pointalong thelineisreduced toone-half ofthemaximum valueattheresonant length 8.Similarly, withoFwo({3w)ando<l>s=0, 1 QLow'+ow" w(20) wherewisthelengthofthesectionforinputresonance andw+ow'and w-ow"arethehalf-power lengths. Itmaybeconcluded that,when 0<1>=0,theQofthetransmission linecircuitmaybedetermined fromresonance curvesusingeitherthe frequency orthelengthoflineasthevariable. 13.TheoryofTransmission-line Measurements. 8,14,62-64,66-68,71-73, 76-79,81,88,89,91,98,103 Although insomerespects thetransmission lineis notconvenient formakingelectrical measurements, itisbothversatile andvaluable atfrequencies thataresufficiently hightomakebridge circuitswithlumpedelements unavailable. Ifthedistance between the conductors ofthetransmission lineissufficiently smallcompared with thewavelength tomakehigherpropagating modesimpossible andto keepradiation fromopen-wire linesnegligible, transmission-line theory provides ahighlyaccurate analogue ofexperimentally observable and measurable conditions. Thisistrueexceptneartheterminations ordis continuities alongthelinewhereappropriate corrections usuallymustbe madetotakeaccount ofendeffectsinthelineandcoupling effects between thelineandthetermination, asdiscussed inChap.II,Secs.3 and4.Letitbeassumed inthefollowing thataccount hasbeentaken ofsucheffectsandthattheactualtransmission linewithnonuniform properties nearitsendshasbeenreplaced analytically byanequivalent uniform lineterminated inmeasurable apparent impedances. Theactual determination ofanapparent impedance ZOaorZsafromthetheoretical idealimpedance ZoorZsisconsidered inChap.V. Ageneraltransmission-line systemconsists ofasectionoflineextend ingfromz=0toz=sandterminated atz=0inZoandatz=8inZs. Owingtoterminal-zone effectsthissystemisequivalent toanidealuni formlineoflengthsterminated intheapparent impedances ZOaandZsa atz=0andz~8,respectively. Thelineisdrivenbytheequivalent ofoneortwopairsofequalandopposite pointgenerators symmetrically placedwithrespecttoanarbitrary andmovable pointx.Experi mentally available equivalents ofsuchgenerators aredescribed in Chap.VI. Anumber ofquantities thatcanbemeasured onsuchatransmission linearedescribed inthefollowing. Thepurpose istooutlinethetheo retical foundations, nottodiscusstheexperimental technique.sNote Sec.13]AMPLITUDE RELATIONS FORCURRENT ANDVOLTAGE 273 thatbridgeandphasemeasurements usingthehybridjunction aredis cussedinChap.III,Sec.15. TheMeasurement ofPhaseConstant andWavelength. Thephasecon stant{3andthewavelength A=27r/{3maybedetermined usingeither resonance ordistribution curves. a.Resonance-curve Method. Theparticular lengthsoflineSnforwhich themaximum ofaresonance curve (obtained bychanging theover-all lengthofthelinewitheverything elseconstant) maybeobserved are definedbythecondition n=0,1,2,. . . , Sn>0(la) Iftwosuchlengths SnandSn-laredetermined experimentally byvary ingthelengthsoftheline,itfollowsfrom(la)that {3=7r Sn-Sn-l(lb) Inordertoobtainsharpresonance curves,bothterminations ZOaand Zsashouldbereactive andpreferably shouldbepistonsinshielded lines (asinFig.13.1)andmetaldisksor conducting bridgesonopen-wire lines. Ge;~::;f-f:f---------.:at;- ~et Highprecision indetermining {3and ~I'---- s --+1.1 Amaybeachieved byplotting the z=o z-s vicinity ofeachresonance peakandFIG.13.1.Coaxiallinewithpistonsand drawing mid-point linestolocatethecoupling loopsformeasuring wave- lengthandphaseconstant bythe peakaccurately. resonance-curve method. Sinceithasbeenassumed thatsis theonlyvariable, neitherthedistance w=S-zfromthecurrentorvolt ageprobetoZsanorthedistance xfromZOatothecenteroftheconfigura tionofpointgenerators mustbechanged asSisvaried.Apossiblepro cedureistocouplethegenerator orthedetector tothelinebymeansof aloopwhichisattached directlytoamovable piston,asinFig.13.1,or toamovable diskorwirebridge. Ifboththepointofcoupling ofthegenerators andthelocation ofthe currentorvoltageprobearefixedatdistances xandz,respectively, from theimpedance Zo,whereasthelengthsofthelineisvariedbymovingZs (usually intheformofapistonwithZs==0),itisclearthatw=s -z varieslinearlywiths.Inthiscasethemagnitude ofthecurrentatzis proportional to sinh2(aw+Ps)+sin2({3w+cps) IIzlr-./sinh2(as+po+Ps)+sin2({3s+CPo+cps) (2a) Ifthelocation zofthecurrent-detector probeisfixedsothat 2n+1(3(s-w)+lJIo=-- 7r2 274 TRANSMISSION-LINE THEORY [Chap.IV withnintegral, thisexpression becomes (2b) Forsmallover-allattenuation 11z1variesasIcot({js+~o+~B)I[instead ofas\sec({js+~o+~B)Iwhenthedetector ismovedwithZB,sothat Wisconstant]. Thelocations andamplitudesoftheextreme valuesare givenby 2n+1 {3s+~o+~B=--2- 7r {3w+~B=m7r 111. -sinh(aw+PB) zmm-cosh(as+po+Ps) {38+~o+~B=n7r Q+~_2m+1fJWB-2 7r III-cosh(aw+PB) zmax-sinh(as+Po+PB)(2c) (2d) Gen-<J E."L-------JJ----- -p Det!--w---t FIG.13.2.Slottedcoaxiallinewithmov abledetector probeformeasuring wave lengthandphaseconstant usingthe voltage-distribution curve.Ifaloopis substituted forastraight probe,the currentdistribution curvereplaces the voltagedistribution curve.Sincethedistances between maxima arethesameaswhen 8aloneis varied,withWfixed,andtheresonance maxima arealmostassharp, {jandAmaybedetermined equallywellifonlyaterminating pistonis moved,withdetector andgenerator fixed,provided thedetector isplaced exactlywhereitgivesmaximum deflection. Notethattheabovefor mulasapplytoVzifeJ>aisreplaced byeJ>~• ItisshowninChap.VIthatthe condition forthedrivingsourcetobe equivalent toaconfiguration ofpoint generators alongthelinerequiresthat thisbecoupledlooselytotheline. b.Distribution-curve Methods CurrentandVoltage.Sincedistribu tioncurvesasfunctions ofworx arereciprocals ofresonance curves asfunctions ofs,thesameprocedure inmeasuring Amaybefollowed usingminimaofdistribution curvesasusingmaximaofresonance curves. Ifacurrentorvoltageprobeismovedalongtheline,asinthecircuit ofFig.13.2,toobtainacurrentorvoltagedistribution curveand,in particular, tolocateitsminimaatWn,theequations thatthesesatisfy areasfollows: [Iz(W)]min occursat{jwn+~Ba=n7r [Vz(W)]min occursat{jwn+~:a=n7r(3a) (3b) Sec.13]AMPLITUDE RELATIONS FORCURRENT ANDVOLTAGE 275 wheren=0,1,2,. . .andWn>O.Evidently {3= 7r' Wn-Wn-l(4) (5b)(5a)FIG.13.3.Slotted coaxial linewith movable generator loopformeasuring wavelength andphaseconstant using thedistribution curveofthepointof excitation.p,L-t---------::F;~~ k-X--ll!kGen.Sn --..IThedistribution-curve minimaaresharponlyifZsaisessentially reactive. Ifrequired, aplotofthecurvesmaybe made,andmid-point linesdrawntolo cateeachminimum moreaccurately. Sincethecurrentorvoltageprobe isnotincluded intheanalysis, itmust becoupled sufficiently looselytopro videnosignificant reaction onthe generator. c.Distribution-curve Methods-Point ofExcitation. Ifadistribution curve isobtained asafunction ofxbymovingthegenerators (asbymoving alooselycoupled oscillator orcoupling unit,asinFig.13.3)relativeto theline,minimainthecurveoccurwhen Foronepairofpointgenerators: {3xn+<Poa=n7l' Fortwopairsofpointgenerators: {3xn+<P~a=n7r' wheren=0,1,2,...andx>O.Evidently {3= 7r' Xn-Xn-l(6) TheMeasurement ofFrequency orPhaseVelocity. Thetransmission linepermitsthemeasurement offrequency onlyindirectly. Since f=~={3v (7)A27r' itisclearthat,ifthevelocity visknown,fcanbeevaluated from(7)if {3orAismeasured. Alternatively, ifthefrequency isknown,thephase velocity vmaybedetermined using(7).Theoretical valuesofvare giveninChap.II,Sec.12. TheMeasurement oftheAttenuation Constant. Theattenuation con stantaofalow-loss lineisusuallysosmallastomakeitsexperimental determination difficult. Although methods basedontheuseofSec.9, formulas (1), (9), or(16),aretheoretically possible, theratiosofmaxi mumtominimum amplitude inaresonance curveoradistribution curve areverygreatiftheonlyattenuation isthatofasectionofline.The measurement ofsuchlargeratiosinvolves seriousdifficulties. These includetheproblem ofadetector accurately calibrated overarangefrom verylowtoveryhighlevelsandsufficiently sensitive topermitveryloose 276 TRANSMISSION- LINETHEORY [Chap.IV coupling. Notethatthelossesintroduced bythedetector mustbe negligible compared withthesmalllossesinthelineitself. Otherformulas uponwhichthemeasurement ofamaybebasedare Sec.11,Eqs.(26b)and(32b).Theseinvolvethehalf-power widths ~sn ofresonance curveswithmaximaatSnorthehalf-power widths ~Wnof distribution curveswithminimaatwn•Theappropriate formulas are {j~sn aSn+POa+P8a=-2- +{j~Wn aWn P8a=-2-(8) (9) Inordertomakeuseof(8),thelengthofthelineisvariedbymoving oneoftheterminations, asinFig.13.1,andthelocations SnandSn+mof tworesonance curvesandtheirrespective widths ~snand~sn+mat0.707 ofmaximum amplitude aremeasured. Thequantities sodetermined are relatedbytheformula (10) Inordertouse(9),thecurrentorvoltageprobeismovedtolocate twodistribution-curve minimausingFig.13.2atdistances WnandWn+m fromtheterminating pistonorbridgeatz=s.Thehalf-power width ofeachisthenmeasured bydetermining W+ow'andW-ow"and setting ~W=ow'+ow".Theformula foraobtained from(9)when writtenforWnandWn+mis (11) Sincetheseveraldistances andwidthsin(10)and(11)mustbemeasured withgreataccuracy, itisoftennecessary toplotresonance orextended sections ofdistribution curves. TheMeasurement oftheApparent PhaseFunction ofaTermination. Theapparent phasefunction <P8aforanunknown impedance maybe determined usingthesamemethods described forthemeasurement of{j. Notethatresonance-curve methods maybeusedtodetermine thephase function ofthegenerator impedance aswellasthatoftheload;distribu tion-curve methods usingvoltageandcurrentprobesareusefulonlyfor theload. a.Resonance-curve Method. Themeasurement of<Pa(either <P8aor<POa) depends ontheavailability ofastandard termination suchasashort circuiting pistoninashielded lineorasufficiently largeconducting disk inanopenline.Forbothofthese <Pa=<P=7r/2.Analternative for anopenlineisaconducting bridgeforwhich <Pa=7r/2+{jk8a(seeChap. II,Sec.20). Ifthephasefunction ofthestandard impedance is<Psa=7r/2,the Sec.13]AMPLITUDE RELATIONS FORCURRENT ANDVOLTAGE 277 shortest resonant length Snloflineisgivenby 1r {3Snl+cI>oa+2=1r Alternatively, ifthestandard iscI>oa=1r/2,(12a) (12b)1r {3Snl+2+cI>8a=1r Theshortest res,mant lengthwiththeunknown impedance [Z.aforuse with(12a)andZOaforusewith(12b)]isSn2,givenby Subtraction of(13)from(12a)leadsto 1r cI>8a=2+(3(Snl -Sn2) Subtraction of(13)from(12b)gives cI>oa=~+(3(Snl -Sn2)(13) (14a) (14b) Thetworesonant lengths SnlfortheshortcircuitandSn2fortheunknown impedance maybedetermined experimentally bychanging thelengthof thelineasinthedetermination of{3andsubjecttothesameconditions. Twocasesareillustrated inFigs.13.4and13.5.Thelattershowsan Movable loosely standard Tandem coupled bridge bridge generator I<D,--------Z'='S-l+---;- IMovablen-\wI~ z=O probet z~sC~-ksa L~:~4 Movable unknown impedance Movablen-,w2=w1 probe I~ 'To detector 1+-----s2----~ FIG.13.4.Circuitsforthetwostepsinthe measurement oftheapparent phasefunc tionandattenuation function ofanun knownloadimpedance Z.2usingthe resonance-curve methodandatwo-wire line.~----Sl----""tl c~;~I--- I--------"*"- Movablepiston looselycoupled withdetector generator probe probe Receiving antenna ~infieldof distant ~--S2---~~~ transmitter--------'1'1!3::::= - Movablepiston Ground withdetector plane probe FIG.13.5.Circuits forthetwostepsinthe measurement oftheapparent phasefunc tionandattenuation function ofare ceiving antenna usingtheresonance curvemethodandacoaxialline. 278 TRANSMISSION-LINE THEORY [Chap.IV (15a) (15b)cP8a=7r-{3wn cP;a=7r-{3wnimportant application ofthismethodtothemeasurement oftheimped anceofareceiving antenna whileusedforreception. Inthiscasethe antenna isthegenerator fortheattached transmission line,sothat dis~ tribution-curve methods arenotapplicable. b.Distribution-curve Method. Thefunction cP8amaybedetermined usingamovable probewithalineoffixedlength. Bylocating the minimum ofthecurrentorvoltagedistribution curvefortheunknown impedance atadistance Wnfromthe endoftheline,thefollowing equa tionsareobtained, respectively, from (3a)and(3b):-Togener.:.;at::::or ----::-- (.....~ current-l- p~obeat wn--l minimum FIG.13.6.Measurement oftheapparent phasefunction .ausingamovable cur rentprobe. IShort·____ -::- circuitingj-diskCurrent p~o~eat Wnl~ minimumt DetectorTheapparent phasefunction isthus determined usingamovable currentprobe(asinFig.13.6)with(15a)and amovable voltageprobewith(15b).NotethatcP:a=cP8a-7r/2. Analternative procedure thatpermitsthemeasurement ofadifference inlengthsinsteadoftheactuallength Wnandmakesuseofastandard termination suchasashort-circuiting pistonordiskistoapply(3a)or (3b)tothestandard termination andthentotheunknown. Ifthemini mumusingthestandard termination isatWnlandtheminimum withthe unknown isatWn2,thefollowing equa tionappliesforacurrentprobeasde rivedfrom(3a),assuming thestandard termination tobeaperfectshortcir cuit(thisisillustrated inFig.13.7): cP8a=~+(3(Wnl-Wn2)(16a) Similarly for(3b)andavoltageprobe______ --::- ..;)lZs Current1- p~o~eatwnZ--l minimumt Detector FIG.13.7.Twostepsinthemeasure mentoftheapparent phasefunction .ausingamovable currentprobe. Thisisthedistribution-curve analogue oftheresonance-curve formula (14).Ithasanadvantage over(15)inthatitinvolves onlythediffer encebetween thescalereadings forthetwominimawiththetwotermi nationsinsteadoftheactualdistance ofoneminimum fromtheendof theline. Adistribution-curve methodofmeasuring cPousingamovable source maybebasedonaformula like(15)or(16)withxsubstituted forW andcPoforcP•• Sec.13]AMPLITUDE RELATIONS FORCURRENT ANDVOLTAGE 279 TheMeasurement oftheApparent Attenuation Function ofaTermi nation. Thefunction Psaofanunknown impedance maybedetermined usingoneofseveralmethods. SincePsaforthetermination isusually quitelargecompared withtheattenuation oftheline,ratiomethods as wellashalf-power-width methods maybeused. a.Resonance- andDistribution-curve-ratio Methods. Byvarying the over-alllengthofaterminated linetheratioofmaximum tominimum alongaresonance curvemaybeobtained. Thegeneralformula forthis ratioisSec.9,Eq.(1).Sincetheattenuation ofaquarterwavelength ofthelineisusuallynegligible compared withtheattenuation oftheload, Sec.9,Eq.(4),issatisfactory. IftheratioR2ofthemaximum tothe minimum ofaresonance curveisdetermined forZsaandagainasR1for astandard termination suchasaperfectshortcircuit (Psa=0),thefollow ingtwoequations areobtained: aS2max+POa+Psa=coth-1R1 aSlmax+POa=coth-1R2(17a) (17b) ThecircuitsofFig.13.4maybeused.Itfollowsthat Psa=coth-1R2-coth-1R1 (18) sincewith !S2max -slmaxl~X/4theterm a(S2max -Slmax)isnegligible. Notethattheattenuation ofadetector maybeincluded inPOa,sothat theonlyerrorinvolved in(18)istheneglectofaquantity oftheorderof magnitude ofaX/4. Ifamovable probeisusedwiththecircuitofFig.13.6todetermine theratioofthemaximum totheminimum ofadistribution curvefor currentorvoltage-the so-called currentorvoltagestanding-wave ratio- thegeneralequation [Sec.9,Eq.(lIb)]isapplicable. Sincetheattenu ationofthelineisusuallynegligible, Sec.9,Eqs.(13)and(14),arevalid. Thatis, Psa=coth-1S (19) whereSisthecurrentorvoltagestanding-wave ratio.Themostcon venientmethodofdetermining Psaisusuallyfromthecurrentorvoltage standing-wave ratio,asgivenin(19).Itiswelltonote,however, that thereactive andresistive loadingoftheprobeisneglected. Theapparent attenuation function POaoftheapparent impedance ZOa attheotherendofthelinemaybedetermined inasimilarmannerfrom theratioofthemaximum totheminimum ofthedistribution curve obtained bymovingthegenerators. b.Resonance- andDistribution-curve-width Methods. Byvarying the lengthsofaterminated lineonbothsidesofaresonant length Sn2foran unknown Zsainordertodetermine thehalf-power width ~Sn2andrepeat ingthisprocedure todetermine thehalf-power width ~Snlofaresonance curvewithmaximum atSnlforastandard termination forwhich PSG=0, 280 TRANSMISSION-LINE THEORY [Chap.IV thefollowing equations areobtained using(8): (3.1sn2 aSn2+pOa+Psa=-2- (3.1sn1 aSnl+POa=-2- Bysubtraction thefollowing formula forPsaisobtained:(20a) (20b) (21) IfSn2andSnlarechosenasclosetogether aspossible, thelasttermin (21)withaasafactorisusuallynegligible, sothat (22) Notethatthecontribution totheattenuation byadetector maybe included inPOaandthatitseffectissubtracted ifitremains constant. Bymovingacurrentorvoltageprobealongthelineonbothsidesofa distribution-curve minimum atWninordertodetermine thehalf-power width.1wn,therightsideof(9)isdetermined if(3isknown.Ifaisalso known, Psamaybeobtained directly. Alternatively, iftheattenuation duetothelineisnegligible, sothataWn«Psa,asatisfactory approxi mationis .(3.1wn Psa=-2- (23) Determination ofReflection Coefficient. Theapparent reflection coef ficientrsa=I'saeN.aisdetermined directly fromvaluesofPsaandepsa usingtherelations (24) (25)Theterminal functions psaandepsamaybeobtained usinganyofthe severalmethods. Ifthestanding-wave-ratio method isused,acon venientformula relates I'sadirectlytothestanding-wave ratio.Thus, since e-2psa=cothPsa-1 cothPsa+1 whenPsa»awand8=cothpsa,itfollowsthat 8-1 I'sa=S+1 Determination ofApparent Impedance. Oncetheapparent terminal functions Psaand<Psahavebeendetermined foragiventermination, the apparent normalized impedance zlsa=flsa+jXlsamaybecalculated (26a)Sec.13]AMPLITUDE RELATIONS FORCURRENT ANDVOLTAGE 281 directlyusingChap.II,Sec.15,Eqs.(7)and(8).Iftheactualimped anceZsaisrequired, thecharacteristic impedance Zcofthelinemustbe known. Ifresistance andreactance curvesforagiventermination aretobe determined asfunctions ofthefrequency orsomevariable characteristic ofthetermination, itisusuallymoreaccurate toplotsmoothcurves through theexperimentally determined valuesofPsaandcPsaandthen compute rlsaandXisafromthecurvesratherthanfirsttocompute rlsa andXisadirectly. ThisisduetothefactthatPsaandcPsaaresmoother, moreslowlyvaryingquantities thanrlsaandXisa' Three-probe MethodofMeasuring Apparent Impedance. Impedance maybemeasured usingthreefixedvoltageorcurrentprobesinsteadof asinglemovable oneasinthestanding-wave-ratio method. Themeas uredquantities required aretherelativevoltages (orcurrents) atthree fixedpointsalongthelineneartheunknown terminating impedance. 6S,64,71 Suppose thattheapparent impedance Zsa=Rsa+jXsaisatz=8or w=O.Thevoltageatadistance Wfromthetermination isgivenby Sec.2,Eq.(6),orSec.5,Eq.(2).Itsmagnitude is V-V~8xC z-8 8W where c~=sinh2(aw+Psa)+cos2((3w+cPsa) =Mcosh2(aw+Psa)+cos2((3w+cPsa)](26b) (27c)(27a) (27b)Letthethreevoltageprobesbelocatedat(3WI,(3W2=(3WI+11'"/4,and (3ws=(3WI+11'"/2.Therelativevoltages aregivenby C~l=([cosh2(awl+Psa)+cos2((3WI+cPsa)] C~2=i[cosh2(aw2+Psa)+cos2((3W2+cPsa)] =i[cosh2(aw2+Psa)-sin2((3WI+cPsa)] C~s=i[cosh2(aws+Psa)+cos2((3ws+cPsa)] =i[cosh2(aw2+Psa)-cos2((3WI+cPsa)] Itisnoweasilyverifiedthat,subjecttothecondition cosha(wl-ws)==1or(a4A)2«1 thefollowing expressions arevalid: Vi+V~-V~sinh22(aw2+Psa)+cos22((3WI+cPsa) V~=[cosh2(aw2+Psa)-sin2((3WI+cPsa)]2 Vi-V~sinh2(aw2+Psa)sinha(wl-ws)+cos2((3WI+cPsa) 2V~ cosh2(aw2+Psa)-sin2((3WI+cPsa) Ifthefirstofthethreeprobesislocatedat 11'" 11'" 311'" (3WI=4sothat(3W2="2and(3ws="4(28) (29a) (29b) (30) 282 TRANSMISSION-LINE THEORY [Chap.IV (31b)(31a)theaboveformulas reduceto V~+Vi-V~sinh22(aA/4+P8a)+sin228a V~ [cosh2(aA/4+P8a)-cos28a)2 V~-Visinh2(aA/4+P8a)sinhaA/4-sin28a 2Vi cosh2(aA/4+P8a)-cos28a Subjecttothefollowing conditions (whicharereadilysatisfied forawide rangeofimpedances terminating alow-lossline): aA"4«P8a (32a) Isinh2(~A+P8a)sinh~AI«Isin28al (32b) thesemaybereducedfurther. Theresultsart V~+Vi-V~==sinh22P8a+sin228a=r2+x2(33a) V~ (cosh2P8a-cos28a)2 lBalBa V~-Vi. - sin28a (33b)= =Xlsa2V~ cosh2P8a-cos28a whererlsaandX18aarethenormalized apparent terminal resistance and reactance, asdefinedinChap.III,Sec.2.Onlyifterminal-zone effects arenegligible maythesubscripts abeomitted. Itisseenthat,bymeasuring thevoltageamplitude atthreefixedpoints locatedatexactlyA/8,A/4,and3A/8fromtheload,theimpedance of theloadmaybedetermined. Notethatthethreeprobesareassumed tobesotunedandsolooselycoupledastohavenosignificant effecton thelineandononeanother. Ifdesired,theprobesmaybelocatedat otherdistances thanthosespecified in(30).withcorresponding changes intheformulas. Determination oftheCharacteristic Impedance oftheLine.Thecharac teristicimpedance ofatransmission lineisascalefactorrelatingcurrent andvoltage.Itisthenormalizing factorintheimpedance. Itcanbe computed directlyfromthedimensions ofthelineusingthetheoretical formula, butingeneralitcannotbemeasured onthelineitself,since thereisnoabsolute standard forimpedance. Thecharacteristic impedance ofonetransmission lineisreadilymeas uredusinganotherlinewithaknowncharacteristic impedance. Thisis accomplished bymeasuring theapparent impedance ofasectionofthe linewhenusedasatermination forthemeasuring line.Theusualpro cedureistoselectasectionoflineoflengthsandmeasure itsinput impedance Zicwhenitisshort-circuited withapistonoradiskandZio whenitisopen-circuited. InthefirstcaseP8=0and8=7r/2;inthe secondcaseP8=0and.=O.Thetwoinputimpedances are Zio=Zccoth"(8 (34) Sec.13]AMPLITUDE RELATIONS FORCURRENT ANDVOLTAGE 283 Theproduct oftheseexpressions is (35) (36a) (36b)IfZioandZicaremeasured, Zcmaybecomputed from(35). Notethat,wheretwotransmission lineswithdifferent crosssections areconnected, adiscontinuity existsofwhichnoaccountistakeninthe aboveprocedure. Moreover anidealopencircuitsuchasispresumed in (34)doesnotexistowingtothecapacitive endeffectthatcharacterizes everypractically available transmission line.Ananalysis ofjunction andendeffectsisgiveninChap.V.Butformanypurposes, especially withlinesofsufficiently smallcross-sectional dimensions, theyarerela tivelyunimportant, and(35)with(34)isadequate. Inthissection(35) isconsidered without correction. . Byseparating therealandimaginary partsof(35)using Zio=Rio+jXio,Zic=Ric+jXic,andZc=Rc(1-jcPc) thefollowing twoequations areobtained: R;(1-cP;)=RioRic-XioXic -2cPcR~=XioRic+XicRio Itfollowsfromtheseequations orfromthetypicalcurvesoftheinput reactance ofasectionoftransmission linethateitherXioorXicisnega tiveandtheotherispositive. Sinceonalllow-loss linestheinequality cP~«1isagoodapproximation, (36a)maybeusedtoevaluate Rcand (36b)toevaluate cPc.Thus,withcP;«1, Rc==VRioRic-XioXic ,I..~1XioRic+XicRio 't'c- -2RioRic-XioXic(37a) (37b) Ifthelengths 8ofthesectionoflineofunknown characteristic imped anceZcmaybeassigned freely,greatsimplification isachieved by selecting alengthforwhichtheinequalities R'fo«!Xio!2R'fc«IXic\2 (38) aresatisfied. Thisistruewhentheelectrical lengthisnearoneofthe following: {38=1r/4,31r/4,. ...Withsuchachoiceof{38(37a,b) reduceto (39a) (39b) InthiscaseRccanbedetermined fromameasurement ofreactances only. Typical numerical valuesillustrating theorderofmagnitude ofXio andXiccompared withRioandRicwhen{38=1r/4and31r/4arelistedin Table13.1. 284 TRANSMISSION-LINE THEORY TABLE13.1[Chap.IV Typeofline {3sRic Rio-Xio=Xic(Ric/Xic)2(Rio/Xio)2----- Two-wire......!0.90.2 440 4.2X10-62.1X10-7 4 Two-wire......311"1.32.0 440 8.3X1O~ 2.1X10-54 Coaxial.......~0.020.005 75 7.1X10-84.5X10-9 4 Coaxial.......311"0.30.5 75 1.6X10-54.5X10-54 Inmostcasesthecalculation ofRcfrom(39a)isconvenient and reasonably accurate. Ontheotherhand,theverysmallquantity cJ>cis usuallymoreeasilydetermined fromtheapproximate relation cJ>c==al(3 andthemeasured valuesofaand(3. Comparison ofPhases.Ifatransmission lineisterminated inits characteristic impedance, thecurrentandvoltagevaryprogressively anduniformly inphasealongtheline.Useofthisfactmaybemade forcomparing relativephasesatdifferent pointsinacircuitorinanother , Slottedconductor Togenerator ~ FIG.13.8.Circuitforcomparing phases(attenuators, linestretchers, etc.,arenot shown). transmission lineorfordetermining thecomplete relativedistribution of phasealongalineoranantenna. Theprocedure isillustrated inFig.13.8fordetermining thephase alonganantenna.Itconsistssimplyincoupling afraction ofthepower supplied tothecircuitinwhichmeasurements ofphasearecontemplated intoanauxiliary transmission linethatisterminated initscharacteristic impedance Zc.Twomovable probes,eachconnected toatransmission lineandbothjoinedtoamixeranddetector, areprovided. Oneofthe Sec.13]AMPLITUDE RELATIONS FORCURRENT ANDVOLTAGE 285 probesservestoexplorethecircuitundertest(theantenna inFig.13.8), whereastheothermusttravelalongtheauxiliary matched line.Acon venientbutarbitrary reference pointforphaseisselected inthecircuit undertest,andthefirstprobeismovedtothispointandtherecoupled tothecircuit. Thesecondprobeisnowmovedalongthematched line untilaminimum (or,iftheoutputs arecorrectly adjusted withattenu ators,anull)isobserved inthedetector. Thelocation oftheprobe alongthelineismarked; itisthereference pointintheline.Inthis adjustment thesignalsreaching themixerfromthecircuitandfromthe matched lineareinphaseopposition. Thefirstprobeisnowmovedtoapositionatwhichphaseistobe determined inthecircuitundertest,oritismovedprogressively from pointtopoint.Ateachlocationthetraveling probeismovedalongthe matched lineuntilaminimum (ornull)isobserved inthedetector. Fromthedistance inthematched linebetween thereference pointand thelocation oftheprobe,therelative phaseattheparticular pointor succession ofpointsmaybedetermined. Measurement ofDielectric Constants, Permeabilities, andConductivities ofPoorlyConducting Media:Drude's Method. Atheoretically simple methodofmeasuring thedielectric constant Eandtheconductivity CTof solidsandliquidsusesthematerial inquestion asthedielectric ina transmission lineofcoaxialorshielded-pair type.Bydetermining the phaseconstant {3andtheattenuation constant ausingmethods described earlierinthischapter, formulas ofChap.II,Sec.10,maybeusedto compute Eand CT.Ifgoodconductors areusedforthetransmission line, thecondition w2lc»rgiseasilysatisfied bythelineconstants l,c,T, andg.ItfollowsfromChap.II,Sec.10,Eq.(16),that (3=wv'lcf(h-y) a=wy'kg(h-y) (40) where,fromChap.II,Sec.10,Eq.(8), h-y==JL+!-=!!.-+~ (41)wCwl WEwl Foralow-loss line(asishereassumed) thefollowing inequality mustbe satisfied: h~«1 (42) sothat f(h-y)==1g(h-y)==~ (43) Afterthesubstitution of(43)in(40)andtheuseof(42),thefollowing expressions areobtained forthephaseandattenuation constants ofthe linewithalow-loss dielectric medium withrealeffective dielectric con stant Ee=EOEer,conductivity CTe,andpermeability J.L=J.LoJ.Lr: (44) 286 TRANSMISSION-LINE THEORY [Chap.IV where J.Lrlg==leistheinductance perunitlengthoftheline.Ifthesame lineisevacuated sothat(Fe=0,J.Lr=1,andEer=1,theconstants are (3=w~~ra=-2-[8(45) Thecombination of(44)and(45)gives _ 2~EoEer ( ~Eer) (Fe---ae-a- J.LoJ.Lr J.Lr Ifthemedium isadielectric withP-r=1,(46)reducesto(46) (47) If(3eand(3aremeasured, fermaybedetermined; if,inaddition, aeanda aremeasured, (Femaybeevaluated. Ifthemedium isamagnetic material with Eer==1, (48) SOthatJ.Lrand(Femaybedetermined if(3e,(3,ae,andaareknown. Although theoretically simple,themethodreferredtoaboveformeas uringdielectric constants, permeabilities, andconductivities isnotalways convenient fromtheexperimental pointofview.Thisisprimarily due tothelargesampleofdielectric required tofilltheentirelineandthe highover-allattenuation withanevenslightlyconducting medium.It maybeaddedthattransmission-line measurements inadielectric-filled lineareoftenawkward. Othermethods without thesedifficulties are described inChap.V. PROBLEMS 1.Alow-loss transmission line(Ro=300ohms, epo=a/[3=10-3, 8=4.2m) connects agenerator (ZOa=10+jOohms,V8=100volts,f=300Me/sec) toa load(Z.a=4,000+jOohms). Determine thefollowing: (a)Thecurrents inthegenerator andtheload. (b)Thecurrentandthevoltageonthelinehalfway between thegenerator and theload. (c)Thelocationandmagnitude ofeachmaximum andminimum ofcurrentalong theline. (d)Thestanding-wave ratio. (e)Theefficiency ofpowertransmission totheload. 2.Atwo-wire lineofcharacteristic impedance Zo==Ro=300ohmsisloadedat oneendbyanantenna withimpedance Z.=90+j45ohms.Atadistance of one-half wavelength fromtheloadasecond,identical antenna isconnected across theline.Determine thestanding-wave ratioandthedistribution ofcurrentonthe mainlineandonthesectionbetween thetwoantennas. Neglect lossesintheline andassumeterminal-zone effectstobenegligible. Chap.IV]AMPLITUDE RELATIONS FORCURRENT ANDVOLTAGE 287 3.Animpedance of800-j50ohmsismeasured onatransmission lineforwhich Zc==Rc=400ohmsanda=10-3neper/m. Thefrequency is100Me/sec. (a)Whereisthevoltageminimum nearesttheload?\Vhatisthestanding-wave ratio? (b)Whatisthewidthoftheresonance curveobtained byvaryingthelengthofthe lineaboutthepeakoccurring atthenexttotheshortest lengthofline? 4.Anapparent impedance Z.a=100-j40ohmsistobemeasured bythedistribu tion-curve-dip methodonalow-loss lineforwhichZc==Rc=50ohms.Neglecting linelosses,predictthedistance fromtheloadandthehalf-power widthofthecurrent distribution-curve dip. 5.Thefrequency ofagenerator isdetermined accurately tosixsignificant figures atavaluenear300MeIsec.Describe amethodfordetermining experimentally the phasevelocity ofpropagation alongacoaxialline.Selectpractical valuesforthe lineconstants anddetermine theaccuracy withwhichsuchameasurement might bemade. 6.Acoaxiallineisterminated initscharacteristic impedance Zc==Rc=50ohms byaseriestransformer connected between theendofthelineandaresistive load of800ohms.Neglecting lossesinthelineandjunction effects,determine andplot thedistribution ofthemagnitude ofthenormalized currentalongthelineandalong thetransformer. 7.Atwo-wire lineisterminated initscharacteristic impedance Zc==Rc=400ohms byasingle-stub matching network inserted between thelineandaresistive loadof 50ohms.Determine thenormalized currents onthelineandinthetwopartsofthe matching circuit. Neglectlossesinthelineandinthematching section,andassume junction effectstobenegligible. Useaclosedoropenstub,whichever isshorter. 8.Acoppertwo-wire lineisplacedsymmetrically inabrasspipetoformashielded pairline.Thepipeistobefilledwithwaterinordertomeasure thedielectric con stantbymovingacopper-wire bridgealongthetwo-wire linetodetermine thelocation ofsuccessive resonances. ThewiresareNo.9separated 2embetween centers;the pipeis8emindiameter. Investigate theaccuracy oftheresultstobeexpected by determining thesharpness oftwosuccessive resonance curves. Thismaybedone bycomparing thehalf-power widthswhenthepipeisfilledsuccessively withairand withwater. UseU'=2 X10-4mho/m (Er=81)fordistilled water, U'=5.65X10'1 mhos/m forcopper,andU'=1.5X107mhos/m forbrass. 9.RepeatProbe8forlakewaterwith U'=10-2mho1mandEr=81. CHAPTER V DISCONTINUITIES ANDNONUNIFORMIJ'IES INTRANSMISSION LINES 1.Two-terminal-pair Networks inTransmission Lines.15Thecon tinuityofauniform transmission linemaybeinterrupted innumerous ways.Inearlierchapters shuntandseriessections oflineandcombi nations oftheseareconsidered undertheidealconditions inwhich lvo~ 11(81)1 212(0) -I:OIz..~~~;)D~~~)~Z2 }z"l;e Line!~ Line2 2"01'\'Z"CI Two·terminal pair"'i2,Z"C2 (a) (c) FIG.1.1.(a)Two-terminal pairjoiningtwolines.(b)Tnetwork ascommon load fortwolines.(c)IInetwork ascommon loadfortwolines. terminal-zone andjunction-zone effectsarenegligible, sothattheformu lationofuniform-line theorymaybeappliedtoeachsectionasifisolated. Ingeneral, theuniformity ofatransmission linemaybeinterrupted by anarbitrary network thathastwopairsofterminals. Suchanetwork isaspecialcaseofageneralfour-terminal network; itmaybecalleda two-terminal pairoratransducer. Threeindependent parameters are required todescribe itselectrical properties provided nonreciprocal cir- 288 Sec.1] DISCONTINUITIES ANDNONUNIFORMITIES 289 Line1 (a)cuitelements suchasgyrators areexcluded. Initsmostgeneralform itconnects twodifferent butindividually uniform lines,asillustrated in Fig.I.Ia.Theproperties ofsuchanetwork mayberepresented bythe inputandoutputcurrents andvoltages associated withthetwopairsof terminals, asinFig.l.Ia;byanequivalent Tsection, asinFig.1.Ib; orbyanequivalent ITsection,asinFig.I.Ic.Ineachcasethesignand direction conventions forvoltages andcurrents aredifferent. Thechoice inFig.1.Iaisthatcharacteristic ofacontinuing transmission line.In Fig.1.Ibthecurrents andvoltages arethosecharacteristic oftwolines forwhichthetwopairsofterminals oftheTsectionconstitute theload. Notethatthischoicemakesthecurrents maintained bythetwogener atorscodirectional inthemutualelement Z12.InFig.1.Icthecurrents andvoltages atterminals 2arereversed ascompared withFig.l.Ib,so thatthecurrents maintained bythetwogenerators arecodirectional in themutualelement Y12.Notethat 12(0)= -12=I~andV2(0)=V2= -V~ Balanced TandIInetworks forusewithbalanced linesareshownin11"2(Zll-ZI2) 2(Z22-~12) II\I12 Line_l_~; ..~_l~_=..[~;__Line2 1~2(Zll-ZI2) 2(Z22-Z12) FIG.1.2a.Balanced Tnetwork ascom-FIG.1.2b.Balanced ITnetwork ascom- monloadfortwobalanced lines. monloadfortwobalanced lines. Fig.1.2aandb.Thesameequations applytothesefiguresastoFig. 1.Ibandc. Input-Output Current-Voltage Equations. Theinputandoutputcur rentsandvoltages inFig.1.Iaandbarerelatedasfollows: VI(s)=VI=V2(0)A+12(0)B=V2A-12B= -V~A+I~B(Ia) 11(s)=11=V2(0)C+12(0)D=V2C-12D= -V~C+I~D(Ib) Theseequations maybeinverted andsolvedforV2and12•Withthe relation AD-BC=1 (2) (whichappliestoallreciprocal circuitsandisderivedveryreadilyby application ofthereciprocal theorem tothecircuitofFig.1.Ia)theresults are V2(0)=V2= - V~=DVI-Bl1 (3a) 12(0)=-/2=I~=-CV 1+All (3b) Theseexpressions maybewritteninmatrixformasfollows[onlyfor mulasforV2and12foruseinFig.1.Ibaregiven,butthesubstitutions V2(0)=V2and12(0)=-/2orV~= -V2and12=-/~maybemade 290 TRANSMISSION-LINE THEORY [Chap.V toadapttheformulation toFig.l.laorc]: (4) (5) Asgivenin(2),itisnecessary thatthedeterminant I~~\=I~~I=I(6) ifthereciprocal theorem istobesatisfied. Itisclearfrom(4)and(5)thatasymmetrical four-terminal network is definedby D=A (7) Anidealtransformer ofNturnsconnected tostepupthevoltagebya factorNandstepdownthecurrentbyafactorliNfromterminals Ito terminals 2isgivenby [~~]~[t:] (Sa) V2sothat VI=NII=NI2 (8b) Thisisillustrated inFig.1.3. TheTnetwork ofFig.l.lbisconveniently represented bythevoltage equations Thematrixequivalent is11(81)l:N12(0) Line1~~(81) _~'2(:0')~ Line2 FIG.1.3.Idealstep-uptransformer.VI=I1Zu+12Z12 (9a) V2=I1Z21+12Z22 (9b) wheretherelationcorresponding to(6)anddefiningareciprocal circuitis Z12=Z21 (10) If(9a,b)or(11)issolvedforthecurrents, theresultsare II=V1Yu-V2Y12 12= -V1Y21+V2Y22 where Y- Z22 U -ZUZ22-Z12Z21 Y_ Z12 12-ZUZ22-Z12Z21 Y22=Zu ZUZ22-Z12Z21(12a) (12b) (13a) (13b) (13c) Sec.1] DISCONTINUITIES ANDNONUNIFORMITIES 291 Notethat(12a,b)maybewrittenwithallplussigns,ifthesignofY12in (13b)isreversed. Alternatively, ifthesignconvention ischanged from thatofFig.1.lbtothatofFig.1.lebysubstituting V~= -V2and 1~=-12in(12a,b),onlyplussignsappearwithY12,asdefinedin(13b). InthiscasetheZ'saredefinedfortheconventions ofFig.1.lb,andthe Y'sfortheconventions ofFig.1.le. TheIInetwork ofFig.1.Iemayberepresented bythefollowing current equations: where11=V1Yn+V;YI2 1~=V1Y21+~Y22 where Y21=Y12 Thematrixequivalent is [1:]=[Yn Y12][V:]12Y21Y22V2 If(14a,b)aresolvedforVIandV;,theresultsare VI=I1Zn-I~Z12 V~=-I1Z21+I~Z21 Y22Zn===--==------==-~YllY22-Y12Y21 Y12Z12=-=-=--==------==-=-YnY22-Y12Y21 YnZ22===--==------==-=-YnY22-Y12Y21(14a) (l4b) (15) (16) (17a) (17b) (18a) (18b) (18e) Notethat(17a,b)maybemadeformally like(9a,b)ifZ12ischanged to thenegative ofthevaluedefinedin(l8b).Alternatively thesigncon ventionmaybechanged fromthatinFig.1.1etothatinFig.1.1bby introducing V2= - V~and12=-1;,whichchanges (17a,b)into(9a,b) andleaves(18b)unchanged. Inthiscase,however, theY'sarealldefined usingtheconventions ofFig.1.1e,theZ'susingtheconvention ofFig. 1.1b. Therelations between theelements oftheimpedance andadmittance matrices andtheABCDcoefficients areobtained bysubstituting (9a,b) and(14a,b)intheappropriate formsof(la,b).Theresultsare (19a) (19b) (19c) (I9d) 292 TRANSMISSION-LINE THEORY [Chap.V Notethattheequations ontheleftimplythe"impedance" signcon ventions ofFig.1.1b,andthoseontheright,the"admittance" signcon ventions ofFig.1.1c.Ifthe"impedance" conventions aretobeused throughout and(12a,b)aretobewrittenwiththeplussigns,negative signsmustbeattached toY12in(19b,c). Similarly, ifthe"admittance" convention istobeusedthroughout and(17a,b)aretobewrittenwith plussigns,negative signsmustbeattached toZ12in(19b,c). Asaconsequence ofthereciprocal relations (2),(10),and(15),only threeinsteadoffourparameters arerequired inordertospecifythe properties ofthetwo-terminal-pair network. Thesemaybeanythree oftheABCDcoefficients-usually A,D,andBorC;Zu,Z12,andZ22;or yu,y12,andY22. Theimpedance oradmittance lookingintothenetworkatterminals 11 whenterminals 22areconnected toanarbitrary loadZL=I/YL(orvice versa)maybeexpressed intermsoftheABCDcoefficients, theimped ancecoefficients, ortheadmittance coefficients. Proceeding from(la,b)usingthe"impedance" or"admittance" con ventions ofFig.l.lborl.Icandsetting or (20) theinputimpedance andadmittance atterminals 11are (21) TheABCDcoefficients maybeexpressed intermsoftheimpedance or admittance coefficients using(19a,b,c,d), or(9a,b)and(14a,b)maybe solveddirectlytoobtainthefollowing equivalent formulas: (22) (23) ZlinandYlinare,ofcourse,independent oftheconvention adopted for thedirections ofthecurrents andvoltages. Ifthetwo-terminal-pair network isdrivenfromterminals 22andthe loadZLif:connected acrossterminals 11,theinputimpedance andadmit tanceare (24) (25) (26) Sec.1] DISCONTINUITIES ANDNONUNIFORMITIES 293 Inordertodetermine threecoefficients using(21)and(24),(22)and (25),or(23)and(26),convenient valuesofthearbitrary loadZLmaybe selected. Ingeneral,thesearetheshort-andopen-circuit valuesZL=0 andZL=00,although anyothervaluesmaybeselected. Consider the following values: ForZ/,=00andYL=0: AZlin==ZloC=C=Zll DZ2in==Z20c=C=Z22Y1in=Y10e= Y2in=Y20e=YllY22-Y~2 Y22 YllY22-Y~2 Yll(27a) (27b) ForZL=0andYL=00: Z.=Z-!-ZllZ22-Zi2 hn-he-D-Z22 Z.=Z=!!..=ZUZ22-Z~2 2m-28eA Zll With(2)itfollowsthat(28a) (28b) Also(29) (30) Itisclearthat,ifanythreeofthefourquantities Zloe,Z20c,ZlBe,and Z28Caredetermined, theABCDcoefficients, theimpedance coefficients, andtheadmittance coefficients canbeevaluated, andwiththesetheinput impedance andadmittance ofreciprocal networks. Ifthenetwork ispurelyreactive, itfollowsfrom(19a,b,c,d) thatAand Dmustbereal,BandCimaginary. Symmetrical Networks. Therepresentation oftwo-terminal-pair net worksisreadilyspecialized totheimportant caseofsymmetrical net worksbysetting,inaddition toZ21=Z12andY21=Y12, A=D (31) Analternative representation ofasymmetrical network makesuseof theresolution ofthetotalcurrents andvoltages intosymmetrical and antisymmetrical combinations referredtotheplanethroughthecenterof thenetwork.IthasalreadybeenshowninChap.III,Sec.12,thatthe elements oftheimpedance matrixofasymmetrical Tsectionmaybe expressed asfollows: Zll= MZI~)+ZI~)] Zl2= -MZI~-ZI~)](32a) (32b) ZI~)istheinputimpedance ateitherpairofterminals when(a)equal voltages inphaseareappliedsimultaneously acrossbothpairsoftermi- 294 TRANSMISSION-LINE THEORY [Chap.V nals,asinFig.l.4a,or(b)theTsectionisshort-circuited acrossits center,asinFig.l.4b. Z~~)istheinputimpedance ateitherpairof terminals when(a)equalvoltages inphaseopposition areappliedsimul taneously acrossbothpairsofterminals, asinFig.lAc,or(b)theTsec tionisopen-circuited acrossitscenter,asinFig.l.4d. InChap.III,Sec.12,theelements oftheTsectionwerecalculated to represent alength2dofsmoothline.Obviously thesameprocedure and Zll-Z12 Zll-Z12 I(s)+~>!V'wI'--1~"""""IVV""'" }'(S)2Z122Z12 (a) (b) (c) (d) FIG.1.4.(a)Circuitfordetermining Z(a).(b)Equivalent circuitfordetermining Z(a). (c)Circuitfordetermining zeal.(d)Equivalent circuitfordetermining Zeal. setofformulas applytoanysymmetrical network forwhichanequiva lentTsectionistobedetermined. Thecorresponding formulas fortheelements ofasymmetrical IIsection are Yu= {[Y~~)+Y~~)] Y12= -MYf~)-Y~~)](33a) (33b) where Yt~)=l/Z~~)and Yt~)=I/Zt~). 2.Equivalent Transformer forTwo-terminal-pair Network that Includes Sections ofTransmission Line.Weissjioch Tangent Rela tion.16,33,137Asimpleandusefultransformation ofthegeneralfor mula[Sec.1,Eq.(21)]fortheinputimpedance ofatwo-terminal-pair network maybecarriedoutifthisnetwork includes sections oftrans mission lineofadequate length. Consider thecircuitinFig.2.1,in whichanarbitrary two-terminal-pair network isconnected between two transmission linesthatmayhavedifferent characteristics. Lettheinput terminals 11belocatedinline1atanadequate butarbitrary distance fromthenetwork; lettheoutputterminals beat22inline2,alsoatan adequate distance fromthenetwork. Theinputimpedance isgivenby Sec.2] DISCONTINUITIES ANDNONUNIFORMITIES 295 Sec.1,Eq.(21),intheform AZL-B(A/C)ZL -B/C Zlin=CZL-DZL-D/C (1) wheretheoutputimpedance ZListheimpedance lookingintoalineof length8terminated inanarbitrary impedance ZT.Thecharacteristic r-so-r- 8-80~ 2J24I ZL-- line2ZZC2,12T t3 I ~4I t--8---l ~ S----+I FIG.2.1.Arbitrary two-terminal-pair network connecting twotransmission lines. (3)(2) Dc=-CZc2impedances andpropagation constants ofthetwolinesareZCl,Zc2and "(1,"(2.Itfollowsthat ZL=Zc2coth("(28+OT)=Zc2tanh("(28+O~) where O~=tanh-1(ZT/Zc2). Forconvenience let A Ba=--b=--CZc1 CZc1Zc2 Alsolet Zlin=Zclcoth("(18+PI+j~)=Zcltanh("(18+PI)(4) where 8isthedistance fromtheinputterminals 11tothepointbetween theseterminals andthenetworkatwhichthecurrenthasitsmaximum. Theimpedance lookingtotherightatthispointischaracterized by 01=PI+j7r/2. If(2), (3),and(4)aresubstituted in(1),thisbecomes atanh("(28+O~)-b tanh("(18+PI)=tanh("(28+O~)_c (5) Nowletthereference planeinline1beshiftedfrom11to33,adistance 80nearerthecurrentmaximum. Theinputimpedance lookingtoward theloadat33isgivenby tanh("(18+PI)-tanh"(180Z3in=Zc1tanh["(1(8-80)+PI]=Zcl1 t h (+ )t h (6)-an"(18PIan"(180 Thesubstitution of(5)in(6)leadstotheequation (a-fo)tanh("(28+O~)-(b-cfo) tanh["(1(8-80)+PI]=bfo_c_(afo-1)tanh("(~+O~)(7) where fo==tanh"(180 (8) Thisexpression mayberearranged asfollows: a-fotanh("(28+O~)-tanh"(280 tanh["(1(8-80)+PI]=bfo_c1 -tanh"(280tanh("(28+O~)(9) 296 TRANSMISSION-LINE THEORY [Chap.V wherethesuhstitutions b-C/na/o-1 _tanh"(2S0=---=--- =poa-/0b/o-C(10) havebeenmadeasadefinition ofthecomplex distance So.Thecom plexquantity Poisdefinedin(10).Itisassumed thatSoandSocanbe sodetermined that(10)maybesatisfied. Forconvenience let k==a-/0 (11)b/o-c With(10)and(11),(9)reducestothesimpleform tanh["(l(S-so)+PI]=ktanh["(2(8-So)+0;] (12) Ifthethreenewcoefficientsio, Po,andkaredetermined experimentally, thethreeparameters a,b,andcintheoriginalEq.(5)maybeqbtained bysolving thefollowing threeequations simultaneously [theseare obtained from(10)and(11)]: aio-bioPo+cpo=1 apo-b+c/o=/oPo (13) a-b/ok+ck=io Thesolutions are a=/oPo-k b=/0-Pokc=/oPok-1 (14) po-/ok Po-iok po-iok Alternatively, ifa,b,andcaregiven,thethreeparameters /0,po,andk maybeevaluated. Theparameter /0iseasilyobtained fromthemiddle equation in(10).Thisreducesto sothat12_ /1+a2 -c2 -b2+1=0 JO 0a-bc(15a) =1+a2 -b2 -c2+/[1+a2 -b2 -C2J2_III/0 2(a_bc)- 2(a_bc) (15b) Withioobtained from(15b),Poandkcanbeevaluated directly from (IO)and(11).Notethat,ingeneral,/0,po,andkarecomplex and whereas Soisarealdistance, Soiscomplex andnotaphysically meaning fuldistance. The±signin(I5b)indicates twopossible valuesofioand henceofkandpo. Thecoefficients a,b,andccanbeexpressed directly intermsofthe impedance elements ofa Tsectionortheadmittance elements ofaIIsec tionusingSec.1,Eqs.(19a,b,c,d), together with(3).Thus,fortheT section, Zna=Zcl fo=tanh"(ISOb=ZllZ22-Zr2 ZclZc2 Po=tanh"(2S0Z22c=Zc2 a-/0k=--bio-c(16) Sec.2] DISCONTINUITIES ANDNONUNIFORMITIES 297 Thecorresponding relations forthe1rsectionareeasilyderived. They are Yua=Yc1 fo=coth"(1SoY22c= Ye2 a-fok=-bfo-c(17) Reactive Network. Themostimportant specialcaseisthatofapurely reactive two-terminal-pair network connected between twoessentially losslesstransmission lines.Thefollowing simplifications apply: Zn=jX nZ12=jX 12Z22=jX 22 Zc1==Re1Ze2==Re2 "(1=j{31 "(2=j{32P1=0 6~=jCf>~(18) With(18),thefundamental Eq.(12)reducestotheWeissfloch tangent relation :33,137 (19) Intheoriginal Weissfloch form Cf>~=0,.sinceaperfectshortcircuitis assumed tobethetermination ofline2.Forthetangent relation (19), whichinvolves onlyrealquantities, thegeneralformulas (16)reduceto (20). .X22C=JC=JRe2 k=k=a-fo bio-C Therealquantities a,b,andcmaybeexpressed intermsofthereal parameters fo,po,andkasfollows:..Xua=Ja=JRc1 fo=jfo=jtan(31S0 fopo+ka='------=----,-po-fokb=fo-Pok po-fok1+fopokC=----'----":-_::_po-fok(21) (22a)Therelation corresponding to(ISb)is 1-a2 -b2+c21[1-a2 -b2+C2J2)t- f0=2(a-bc)± 2(a-bc)+1 Withfodetermined froma,b,andcin(22a),kmaybedetermined from (20),andpofromt aio+1b+cfopo=---=---bfo-cfo-a(22b) Notethatthe±signindicates twopossible valuesoffoandhenceofk andpo.Sincepo=jpoisapureimaginary, So=Soisreal. Thesignificance ofthetangent relation (19)isthatitsleftsideisthe normalized inputimpedance atterminals 3lookingtowardthenetwork tNotethatbandcarethenegative ofthecorresponding parameters intheoriginal paperofWeissfloch. 298 TRANSMISSION-LINE THEORY [Chap.V ontheright(Fig.2.1),whereas therightsideisktimesthenormalized impedance lookingtowardtherightatterminals 4.Thatis, Zin3=jXin3=jRc1tan(jl(S-so) ZL4=jX1A=jRc2tan[(j2(S-So)+~](23a) (23b) Itfollowsthattheinputimpedance atterminals 33isaconstant n2times theoutputimpedance atterminals 4: (24) Accordingly theentirenetwork between terminals 33inline1andtermi nals44inline2isequivalent toanidealtransformer withratioofturnsn. Thisisillustrated inFig.2.2. OwingtothefactthatSoin(12)iscomplex andnotaphysically measurable distance, adissipative 3 4 network cannotberepresented by ....Li.-ne--:-l---:--,~ ~]l Line2asimpletransformer, andnosimpleZC1.11 n zC2.12-':::':--:'_--",,*....) '-* .JIinterpretation of(12)isavailable. 3 4SincethereareothermoreconvenFIG.2.2.Idealtransformer equivalent to network between terminals 33and44ofientprocedures fordetermining the Fig.2.1. equivalent Tornofadissipative two-terminal-pair network thanus ing(12),onlythesimplerelation(19)forreactive networks isconsidered. Thetheoretical resultsobtained forthereactive network maybesum marized asfollows:Givenanarbitrary reactive network withtwopairs ofterminals towhichareconnected uniform transmission lines,itis alwayspossible tolocateinputandoutputterminals soastoinclude withtheoriginalnetwork sections ofthetransmission linesofappropri atelengths, sothatthethusaugmented network canbereplaced inits entiretybyanequivalent idealtransformer. Moreover foreachnetwork therearetwopossible combinations ofaddedsections oflineandtrans formerratios. ThisisWeissfloch's transformer theorem. Thepractical importance ofthetheorem depends ontheexperimental ortheoretical determination ofthethreeessential parameters, namely, thelocations of theinputandoutputterminals alongthefeedingandloadinglines(which permitsthereplacement oftheincluded network byatransformer) and thefactorkthatdetermines thetransformer ratio. 3.Experimental Determination ofanEquivalent IdealTransformer foraReactive Networky),33.137 Inordertocombine suitable sections oftransmission linewithanarbitrary reactivenetwork sothattheproper tiesofthecombination arethoseofanidealtransformer, itisnecessary to locatethenewinputandoutputterminals (33and44inFigs.2.1and2.2) anddetermine therationoftheequivalent transformer. Theexperi mentaldetermination oftheseunknowns depends oncertainsimple Sec.3] DISCONTINUITIES ANDNONUNIFORMITIES 299 properties ofthefunctional relationship between thevariables 8and8. Asillustrated inFig.2.1,8isthedistance between thearbitrary output terminals 22inline2andthereactive termination ZT=jXT,whichhas thephaseconstant cI>~;8isthedistance between thearbitrary inputter minals11andthelocation ofavoltageminimum orcurrentmaximum between theseterminals andthereactive network undertest. f31S=j31Sp/ k=0<;>-..... r-,6-=--~'""'""'=7''--' 25 5 2 1 -1T1f "2 0.. I0.:::. ~ -T(-TT/2 0TT/2 j32(S-SO)+ 4>7- FIG.3.1.Plotof(3(s-so)asafunction of(3(S-So)-if>~. Iftheelectrical length(31(8-80)isplottedasafunction of(32(8- 80)+cI>~according tothefundamental relation[Sec.2,Eq.(19)],viz., (1) where(3180,(3280,and cI>~areconstants, thecurvesinFig.3.1areobtained forarbitrarily chosenvaluesoftheconstant parameter krangingfrom 1toinfinity. Fork=1thecurveisthestraight 45°line;fork=00 itisaseriesofverticalandhorizontal lineszigzagging acrossthe45°line. Forallintermediate valuesofkthecurveoscillates symmetrically along the45°line. Itisnoweasilyprovedthattheoriginofcoordinates inFig.3.1,viz., thepointswhere(328=(3280-cI>~and(318=(3180,occursatthepointof maximum slopeandthatthismaximum slopeisprecisely k.Thisis accomplished bywriting(1)inthemorecompact form tan(y-Yo)=ktan(x-xo) (2) wherey={31S,Yo=(3180,X={32S,andxo=(3280-cI>~,anddifferenti- 300 TRANSMISSION-LINE THEORY [Chap.V atingwithrespecttoxinordertoobtaintheslopeofthecurve.The resultis dy=kcos2(y-Yo)=k1+tan2(x-xo) dxcos2(x-xo)1+k2tan2(x-xo) =k1+k-1tan2(y-Yo) 1+tan2(y-Yo)(3) Theseveralformsin(3)areobtained byelimination ofy-Yoorx-xo using(2).Theextreme valuesofthisslopeareobtained byequating d2y/dx2tozero.Theresultsare (dY)_kdxmax- (:;)miD=~whenx=Xoandy=Yo 7r 7rwhenx=Xo+2"andy=Yo+2"(4a) (4b) whereitisassumed thatk~1.(Ifk<1,themaximum andminimum valuesareinterchanged.) Theminimum valueisobtained usingthe secondandthirdformsof(3). Thefollowing significant conclusions maybedrawn:IfSisvaried experimentally withrespecttoarbitrarily selectedoutputterminals (22in Fig.2.1)bymovingareactive termination ofknownandconstant q,~ (preferably ashortcircuitforwhich q,~=0)andifthecorresponding distances 8fromthearbitrary inputterminals (11inFig.2.1)toavoltage minimum aredetermined, 8maybeplottedasafunction ofS.The resulting curvemustfallbetween thelimiting curvesk=1andk=00 inFig.3.1.Bylocating thepointofmaximum slope,valuesof{3180and {32S0-q,~aredetermined. Since {3Iand{32aswellasq,~areknown, 80andSoaredetermined. Bymeasuring themaximum slopeofthecurve throughthepoint{3I8={3I80,{32S={32S0-q,~,kisdetermined. Itfol lowsthat,bylocating theinputterminals 33at8=80alongline1and theoutputterminals 44atS=Soalongline2,thetwosections ofline andthereactive network lyingbetween thesepairsofterminals maybe replaced byanidealtransformer withn2=kReI/Re2.Thecharacteristic resistances ReIandRe2ofthetwolinesareassumed tobeknown. Sincethedetermination ofthemaximum slopekdirectly fromthe oscillating curvecannotalwaysbeachieved withsufficient accuracy, especially ifkislarge,analternative andmoreaccurate procedure is desirable. Thisdepends onthedetermination oftheamplitude and location ofthemaximum excursion oftheoscillating curvefromthe45° line.Referring toFig.3.2(whichshowsanenlarged andsimplified sec tionofFig.3.1),thepointofmaximum excursion islocatedbythecoordi nates(inradians) {3I8mand{32Sm-q,~alongthetwoaxes.Thedistance (inradians) between linesthroughthemaximum andminimum excursions (5)Sec.3] DISCONTINUITIES ANDNONUNIFORMITIES 301 fromtheparalleltothe45°lineisD.Itfollowsbyplanegeometry that 7rD 07r-DV2 (32(8m-80)=4-2'cos45=4 j31sm------ / / / / / /(6) /1 (7) It---L.--;-' -----r-+--r-------' (#280-<1>;'-fl (j328o-¢'r) (j328m-¢'r) FIG.3.2.SectionofFig.3.1.k=cot2(xm-xo) =coV(i-D1~dy_k1+tan2(Xm-Xo)=1 dx-1+k2tan2(Xm-Xo) whereXm=(328m-cI>~andXo= (3280-cI>~.Thisequation maybe solvedforktogiveSincetheslopeoftheoscillating curveatthepointofmaximum excursion is1,itfollowsfrom(3)that Thus,bydetermining thedistance Dfromthemeasured curve,itis possible tocalculate kusing(7). Usually Dismoreeasilydetermined accurately thanisthemaximum slopekdirectly. Ifthereference pointisshiftedfromthepointofmaximum slope (31S0,(3280-cI>~tothepointofminimum slope(31Sp==(31S0+7r12, (328p-cI>~=(3280-cI>~+7r/2,thefundamental Eq.(1)becomes (8)1tan(31(S-sp)=Ietan[(32(8-8p)+cI>~] whichislike(1)exceptthattheminimum slope11kreplacesthemaxi mumslopek.Itfollowsthattheinputimpedance attheterminals definedby(31Sp(atadistance >"014from33)isn2timestheoutputimped anceattheterminals definedby(328p(atadistance >"0/4from44),where now (9)2Rei n=kRe2 Thus,iftheinputandoutputterminals areatthepoints(31S0and(3280 locating amaximum slopeinthe(31Svs.(328curve,theequivalent trans formerfortHelinesandnetwork between theseterminals musthave n2=Rc1klRe2,wherekisthemaximum slope.Alternatively, ifthe inputandoutputterminals areatthepoints(31Spand(328plocating amini mumslope,theequivalent transformer musthaven2=Rc1lkRe2,where 11kistheminimum slope. 302 TRANSMISSION-LINE THEORY [Chap.V Alternative Procedure forMovable Reactive Network. Ifthereactive network undertestismovable alongauniform lineandtheterminating reactance XTandthearbitrary inputterminals 11(Fig.2.1)arefixedin position, aslightlydifferent formula isconvenient. Inaddition tothe obvious simplifications re~ulting fromthefactthatbothsections ofline arealike,sothat{32={31={3andZe2=Zcl=Ze,thedependent vari ableintheinputlineisnots,aspreviously defined,but l=s-S (10) Evidently, astheentirereactive network ismovedtowardtheinput terminals 11,theoutputterminals 22movealongwiththenetwork, so thatSisincreased. Insofarastheoutputlineisconcerned, itmakesno difference whether Sisvariedbymovingtheterminating reactance XT withthereactive network fixedorbymovingthereactive network with XTfixed.However, ifSisincreased bymovingthereactive network towardtheinputterminals 11insteadofbymovingXTawayfromthe reactive network, avoltage maximum occursatadistance l=s-S insteadofsfromterminals 11.Alsothepointcorresponding to{3s={3so when{3S={3So-~occursat{3l={3lo=(3(so-So)+~. Therelationship betweenlandSisreadilyestablished usingthewell knownformula forthetangent ofthedifference oftwoquantities-in thiscase(3(s-so)and(3(S-So)+~.Thus tan(3(l-lo)tan(3(s-so)-tan[(3(S-So)+~] 1 -tan(3(s-so)tan[(3(S-So)+~] (l-k)tan[(3(S-So)+~] 1 -ktan2[(3(S-So)+~](11) Thesecondstepin(11)followswhentan(3(s-so)iseliminated using(1) with{32={31={3andZe2=Zcl=Ze. If(3(l-lo)isplottedasafunction of(3(S-So)+~,asdefinedin (11),thecurveinFig.3.3isobtained. Itisseenthatthisisacurvethat oscillates aboutthehorizontal (3(l-lo)=0axisinsteadofaboutthe 45°line.Sincethevaluesof(3(S-So)+~arethesameasinFig.3.1, thelocation ofthepointofmaximum slopeisstillat(3(S-So)+~=O. Thisoccurswhen(3(l-lo)=O.Itisreadilyverifiedbydifferentiation thattheslopeofthecurvedefinedby(11)isgivenby dy_ 2 1 - ktan2x dx-(1-k)(secx)(1+ktan2X)2(12) wherey==(3(l-lo)andx==(3(S-So)+~.Sinceitisknownthat themaximum slopeisatl=0,x=0andtheminimum slopeatl=0, x=±'n/2,formulas fortheseextreme valuesareobtained directlyfrom Sec.3] DISCONTINUITIES ANDNONUNIFORMITIES 303 (12).Theyare mi=(dY)=1 -katy=0,x=0 (13a)dxmax m2=(dY) k -k1aty=0,x=!2 (13b)dxmin Thesevaluesareillustrated inFig.3.3. Theequivalent idealtransformer ofareactive network whichcanbe movedalongauniform linemaybedetermined asfollows:First,input terminals 11arefixedatanarbitrary point,andareactive termination Slopeml=OO-o..t-24 I~I,,-4'i../ ,I I Itanf3(I-1 1=[Cl-k) tan[13(8-8 01+<1>;')] o[l-ktan2[P(8-80)+;.J ml=l-k m2-1-1/k 1T "2 Slopem2= 1 0.96 0.8 _ Q5 _0 ~Ot----#---::------:~OL----___,¥_--__I ~ o-r!L-----L..---'---...L..:---...1-_---.:ll..- __L-I -1T -rr/2 f3(8-80)+4?'-r FIG.3.3.PlotoffJ(l-lo)=fJ(s-so)-fJ(S-So)-<1>;againstfJ(S-So)+<1>;. XTislocatedataconvenient fixedpoint.Outputterminals 22arespeci fiedatanarbitrary distance fromthereactivenetwork andareimagined tomovewiththenetwork. Thedistance fromthemovable outputter minals22tothetermination XTisS.Thedistance fromthefixedinput terminals 11toavoltageminimum orcurrentmaximum isl.Sisnext variedinstepsbymovingthereactive network, andthedistancelfrom theinputterminals toavoltageminimum isdetermined. Acurveof {3lasafunction of{3Sisplotted. Thepointofmaximum slopelocates {3l={3loand{3S={3So-~.Since ~isassumed known, (3Soisdeter mined. Theterminals oftheidealtransformer areat80andSo.The transformer ratioisn2=k=1 -ml,wheremiisthemaximum slope determined fromthecurve. Thepointofminimum slopealsomaybeusedtodetermine analtefna tiveidealtransformer. 304 TRANSMISSION-LINE THEORY [Chap.V Themethod ofmovingthereactive network isconvenient, especially whenthisconsists ofapieceofdielectric forwhichanequivalent ideal transformer isrequired. 4.Deschamps's Graphical Method jorDetermining theScattering MatrixandEquivalent CircuitofaJunction. Agraphical method for determining theelectrical behavior ofatwo-terminal-pair network has beendescribed byDeschamps.ll6,135 Consider thearbitrary passivenet workconnected between twopairsofterminals, asshowninFig.1.1a or4.1.Theinputterminals 11aredrivenfromtransmission line1with 11 12 1 Junction +20 t t1nputyTZ. or Zy.upu terminals 11_C1network C2_ 22terminals FIG.4.1.Junction ornetwork connecting twotransmission lines. characteristic impedance Zcl;theoutputterminals 22areconnected to theloadedtransmission line2withcharacteristic impedance Zc2.As discussed inSec.1,thecomplete specification oftheelectrical proper tiesofthenetwork atagivenfrequency involves aknowledge ofthe relations between theinputcurrentandvoltage (11andVI)andtheout putcurrentandvoltage(12andV2).Ifthenetwork isrepresented by anequivalent Tsection,theserelations arecontained inthesimultaneous equations VI=I1Z11+I2Z12 V2,=I1Z21+12Z22 orintheequivalent matrixequation(la) (lb) (3a) (3b)(2b)(2a) I=l~:]v=IZ Z=[ZllZ12] Z21 Z22 TsectionareZll-Z12andZ22-Z12;thewhere Theserieselements ofthe shuntelement isZ21=Z12. ItisshowninChap.III,Sec.12,thatasectionoftransmission line mayberepresented byanequivalent Tsectionusinganimpedance repre sentation, byanequivalent IIsectionusinganadmittance representation, orintermsofreflection andtransmission coefficients inatraveling-wave representation. Thisisalsotrueofanarbitrary two-terminal-pair net work.InChap.III,Sec.12,atraveling waveofcomplex amplitude Al reaching terminals 11fromtheleftandasimilarwaveofamplitude A2 reaching terminals 22fromtherightwererelatedtotheoutgoing travel ingwavesofcomplex amplitudes B1leavingterminals 11towardtheleft andB2leavingterminals 22towardtheright.Theappropria.te equa tionsareChap.III,Sec.12,Eqs.(19a,b),or B1=SllAl+S12A2 B2=S2lAI+S22A2 Sec.4] DISCONTINUITIES ANDNONUNIFORMITIES 305 (13)(10)Thematrixequivalent is B=SA (4a) where B=[:~]s=[~~:~~:] A=[~~] (4b) Thematrixelements maybeinterpreted asinChap.III,Sec.12. 811(822)isthecomplex amplitude ofthereflected waveatterminals 11 (22)duetoawaveofunitamplitude incident onterminals 11(22).The equation 822=811istrueonlyifthenetwork issymmetrical. 812(821) isthecomplex amplitude ofthetransmitted waveatterminals 11(22) duetoawaveofunitamplitude incident onterminals 22(11).Therela tion821=812istrueforallreciprocal networks. 812isoftendenotedby Tandcalledthetransmission coefficient ofthejunction ornetwork. Therelations between thecomplex amplitudes AlandA2ofthetravel ingwaveandtheinputandoutputvoltages andcurrents arelikethose giveninChap.III,Sec.12,exceptthatthecharacteristic impedances of thetwolinesarenownotnecessarily equal.Theappropriately general izedformulas fortwodifferent transmission linesare VI=~[(1+811)A1+812A2] (5a) V2=VZc2[821A1+(1+822)A2] (5b) 11=VYcl[(1-811)A1-812A2] (6a) 12=VYc2[-821A1+(1-822)A2] (6b) If(6a)and(6b)aresolvedforAlandA2andthevaluessoobtained are substituted in(5a,b),thevoltageEqs.(Ia)and(Ib)areobtained with Zll=~1[(1+811)(1- 822)+812821]Z12=2V~ 812(7) Z22=~2[(1- 811)(1+822)+812821]Z21=2V~CIZC2821(8) where D=(1- 811)(1- 822) -812821 (9) Thematrixformoftheseequations isreadilyderived. Firstletthe following diagonal matrices bedefined: Zi==[Z~10]yi==[Y:l0 ]=Z-ic0 Z~2 c0 Y~2 c With(10)thematrixformsof(5a,b)and(6a,b)are V=Z~(U+S)A (11) I=Y~(U-S)A (12) whereUistheunitmatrix;V,I,andAarecolumnmatrices, asdefined inChap.III,Sec.12,Eq.(36);andSisthescattering matrix S=[811812J821822 Bypremultiplying bothsidesof(12),firstwith Z~andthenwith 306 TRANSMISSION-LINE THEORY [Chap.V (U-8)-1,andsubstituting theexpression soobtained forAin(11), thematrixequation corresponding to(la,b)isobtained. Itrelatesthe impedance matrix Z=[ZllZ12J Z21Z22 inV=ZItothescattering matrix: Z=Z~(U+8)(U-8)-1Z~ Theinverserelationis(14) (15a) 8=(Y:ZY~-U)(Y:ZY~+U)-1 (15b) Analternative formulation ofthegeneralrelations (5a,b)and(6a,b) isobtained byintroducing thefollowing transformed voltageandcurrent attheoutputterminals 22: V;=~ I;=12N (16) where N==/Ze2==/Re2(17) ~Zel~ReI If(5b)isdividedbyNand(6b)ismultiplied byN,theybecome V~=YZc1[S21A1+(1+S22)A2] (18) I;=YYel[-S21Al+(1-S22)A2] (19) Thesearetheequations forthenetwork connected between twoidentical lineswithcharacteristic impedance Zel.Ontheotherhand,therela tions(16)coincide withSec.1,Eq.(8b),foranidealtransformer for whichNisreal.SinceepeinZe=Re(1-jepJisanextremely small quantity inlow-loss lines,Nmaybetakentobereal.Itfollowsthat thecircuitofFig.4.1maybereplaced bythatofFig.4.2,wherethe Junction or network11 Input 1+ terminals VIZCI1 -ZC2 12282+ 2 Vo' ;Output _2 ~2terminals 2l:N N=.JZC2/ZCl FIG.4.2.Junction ornetwork together withanidealtransformer connecting two transmission lines. sameunknown junction ornetwork isnowconnected between linesof equalcharacteristic impedance. ThemethodofDeschamps isbasedonthemeasurement ofthecom plexreflection coefficient rs=rseN•attheterminals 11inFig.4.1when terminals 22areconnected toalineofvariable length1terminated ina shortcircuit. Theeffective loadimpedance across22isZL,andthisis alsotheinputimpedance oftheshort-circuited line.(Ifaperfectshort circuitisnotavailable, anyreactive termination ofknownreactance may Sec.4] DISCONTINUITIES ANDNONUNIFORMITIES 307 (21) (25)(24)beused,sinceitisequivalent toashortcircuitataknowndistance fromit.) Referring toFig.4.3andEqs.(3a,b),consider awaveofcomplex amplitude Alincident ontheterminals 11fromtheleft.Thecomplex 1 2 Y~~ShortS~circuit 1 2j3l Go(locusofrsfor losslessjunction asplisvaried) G(locusofrsfor dissipative junction asj3lisvaried) FIG.4.3.Lociofthecomplex reflection coefficientr.asafunction offll. amplitude oftheoutgoing wavetraveling towardtherightatterminals 22isgivenby(3b),viz., B2=S2IA1+S22A2 (20) whereA2isthecomplex amplitude oftheWavereaching theterminals 22 fromtheright,i.e.,reflected fromtheinputimpedance ZLofline2.By definition ofrL, If(21)issubstituted in(20)andthisissolvedforB2,theresultis B2=S21 (22) Al1 -S22r L Thecomplex amplitude ofthewavetraveling totheleftfromterminals 11isgivenby(3a).Itis BI=S11AI+S12A2 (23) With(21)and(22),(23)maybeexpressed asfollows: BIS B2 S12S2IrLr8==-A=11+S2IrL-A=811+1SrI I -DL Forareciprocal junction thismayberearranged into r 8=(Si2-S11S22)rL+S11 -S22rL+1 whichisthewell-known bilinearformusuallyexpressed asfollows: Az+Bw=Cz+D (26) WhenRL=0,themagnitude rLofthereflection coefficient rLisunity, 308 TRANSMISSION-LINE THEORY [Chap.V sothat,asitsangle1/ILisvaried,rLdescribes acircleofunitradiusin thecomplex plane.Itisafundamental property ofthebilineartrans formation thatcirclesinthezplane(rLplane)aremapped intocircles inthewplane(raplane). Therefore thelocusofraasrLisvaried(by changing l)isalsoacirclesuchasGo(Fig.4.3)ifthejunction islossless andIral=1,oradisplaced andsmallercirclesuchasGifthejunction isdissipative. Usingtheorems ofnon-Euclidean geometry andthewell-known proper tiesofthebilineartransformation, Deschamps hasderived 116graphical procedures fordetermining themagnitude andangleofthethreematrix elements S11,S22,andS12in(25).Simplified proofsbasedonplane geometry havealsobeenderived,l36 together withmodified procedures. Sincethedetailsoftheproofandinterpretation ofthegeometrical con structions arelongandnotdirectly relatedtotransmission-line theory, theyareomitted. Thestepsintheapplication ofDeschamps's method areasfollows: a.Determination ofReflection Coefficients. Theoutputterminals 22 (Fig.4.1)ofthenetwork orjunction undertestareconnected toashort circuited line2ofelectrical length{3l.Thecomplex coefficient ofreflec tionraoftheterminal impedance Zapresented tothedrivingline1by thenetworkatitsinputterminals 11ismeasured foreachoffourelectri callengths {3lioftheline.Theseelectrical lengthsarechoseninpairs tohavethefollowing values: {3l1,{3l2,{3l3={3l1+1('/2,{3l4={3l2+1('/2. Although thevaluesof{3l1and{3l2arearbitrary, itisadvisable toselect {3l2sothat{3l2={3l1+1('/4.Inthismannerthefourlengthsareequally spacedalongahalfwavelength ofline2.Thereflection coefficients are measured usingamethoddescribed inChap.IV,Sec.13,e.g.,bydeter miningthestanding-wave ratioandthelocations ofcurrentorvoltage minimarelativetotheterminals 11. b.Construction ofCircle.Thefourcomplex reflection coefficients rai, withi=1,2,3,4,areplottedatPiinthecomplex plane,asshownin Fig.4.3.ThecenterCofthecircleGuponwhichthefourpointsmustlie isobtained bydrawingthelinePIP3andP2P4anderecting theirperpen dicularbisectors. Thedesiredcenteristhepointofintersection ofthese bisectors. WithClocated, thecircleGwithradiusrmaybedrawn. Ifthejunction islossless,thefourpointsmustlieonGo,andCisatO. c.Graphical Determination ofSu.Thepointofintersection S~lofthe linesPIP3andP2P4iscalledthecrossover point.Inordertodetermine S11,thefollowing construction ismade:ThelineS~lCisdrawn(Fig.4.4). Onopposite sidesofthisline,beginning at8~1andC,perpendiculars are erectedwhichintersect thecircleGatAandB.ThelineABisdrawn. Itsintersection with S~lCat811isknownastheiconocenter. Thecom plexnumbercorresponding tothispointinthecomplex planeisthecoef ficient S11ofthescattering matrix. Sec.41 DISCONTINUITIES ANDNONUNIFORMITIES 309 d.Graphical Determination of512and522•Thedetermination of512 and522isfacilitated ifoneofthepointsPicorresponds toanelectrical length {3l,whichlocatestheshortcircuitexactlyaquarterwavelength fromtheoutputterminals 22(thisisequivalent toanopencircuitat P4 B FIG.4.4.Constructions inthedetermination oftheelements ofthescattering matrix. theseterminals). If{3l1ischosentobezero,sothatpointPIcorresponds toashortcircuitdirectlyacrosstheterminals 22,Paat{3la=(3l1+1r/2 isthedesiredparticular point.Referring toFig.4.4,letthelinePaSl1 bedrawnandextended tointersect thecircleGatK.FromKthe diameter KCP~maybedrawn. Nexttheperpendicular tothelineS~lC atSl1isdrawn;itintersects thecircleGatEnearPa•Usingtheknown radiusrofthemeasured circleG,thefollowing valuesaretrue.Forthe sakeofcompleteness thevalueof811isalsoincluded. Let8=ISIei9 withappropriate subscripts. Then,referring toFig.4.4,theelements ofthescattering matrixare 511: 18111=OSl1 011=L(OP,OSl1) 812: 1812\=S;!f. 012=jL(OP,CP~) (27) 822.' 1822\=Sl1C 0L(SCCP') 22=11,ar Eachoftheanglesdefinedin(27)isthatthrough whichthefirstline segment intheparentheses mustbeturnedinordertomakeitcoincide withthesecondlinesegment intheparentheses. Thesignispositive iftherotation iscounterclockwise andnegative iftherotation isclock wise.Theangle012of812isindeterminate by1r. 310 TRANSMISSION-LINE THEORY [Chap.V Withthescattering matrix8known,theimpedance oradmittance matrixandtheelements ofa TorIIsectionthatiselectrically equivalent totheunknown junction ornetwork maybedetermined. Ifthemovable impedance ZLterminating line2isnotaperfectshort circuit,ashasbeenassumed, andtheconstruction todetermine thematrix elements iscarriedoutasforZL=0,thevaluesdetermined are8~~,8~~, and S;~.Thesearerelatedtothecorrectvaluesasfollows: 8~~=811 (28) whererListhecomplex reflection coefficient ofZL.Evidently aknowl edgeofrLisrequired if812and822aretobedetermined from 8~~and8;,'2" Ontheotherhand,theratio (29) isavailable withoutaknowledge ofrL,andthismakesthedetermination of812and822possiblewithoutrLintwospecialcases:(a)Symmetrical junction:Ifthejunction ornetwork issymmetrical, 822=811,sothat (29)canbesolvedfor812•(b)Reversible junction:Ifthejunction or network canbereversed sothatterminals 22aretheinputandtermi nals11aretheoutput,both811and822canbedetermined successively together withtheratio(29).Inthiscasetwoindependent valuesof812 maybeobtained, theoneservingasacheckontheother. Ifajunction ornetwork isnotreciprocal, sothat821¢812,theentire Deschamps technique is stillvalidexceptthaty821812isdetermined insteadof821=812•Inordertodetermine 812and821itisnecessary toperformatleastoneadditional measurement, andthiscannotbecar riedoutonjustonesideofthejunction sinceanymeasurement that involves thetransmission through ajunction inbothdirections succes sivelyisincapable ofresolving theeffectofitsnonreciprocal property. Possible measurements includethedetermination ofthemagnitude and phaseofthevoltageorthecurrentonbothsidesofthejunction when drivenfromoneside.Analternative procedure makes useofsymmetri calandidentical generators oneachsideofthejunction. Avoltage measurement (magnitude andphase)ismadeononesideorpreferably onbothsidesofthejunction. Inthiscaseitfollowsfrom(3a,b),with A2=Alequalincident voltages oneachside,that Bymeasuring thesetworeflection coefficients ofthejunction whendriven simultaneously andequallyfrombothsides,812and821maybeevalu- Sec.4] DISCONTINUITIES ANDNONUNIFORMITIES 311 ated,sinceS11andS22areassumed known. Evidently oneofthemeas urements issufficient ifyS12S21 isalsoknown. However, sinceS12and S21maybequitedifferent inorderofmagnitude--for somedevicesinvolv ing ferrites S21maybeextremely smallwhereas S12isnearunity-inde pendent determinations ofS21andS12maybepreferred. Avaluable featureoftheDeschamps method istheopportunity it provides forestimating errorsintheexperimental data.Thisisillus tratedinthefollowing example: nlustrative Example.tTheexperimental datainthisexample arederivedfrom standing-wave measurements obtained usingadissipative junction inthecircuitof Fig.4.5.Intheexample tobedescribed thejunction issymmetrical, butthisdoes notneedtobethecaseinorderthatthemethodmaybeapplied. Thereflection coefficients r.attheinputterminals weredetermined experimentally, withtheshort-circuiting pistonatthefollowing valuesof~l:0,1r/S,1r/4,31r/S,1r/2, ~---~Short ~ ...Jcircuit 2-(31- FIG.4.5.Circuitformeasuring thereflection coefficient ofajunction. 57r/8,31r/4,7"8/8.Thevalue ~l=0corresponds toashortcircuit,andthevalue fJl=7r/2corresponds toanopencircuitacross22.Eightpointsratherthanfour wereusedinordertoestimate theexperimental errorsinvolved. Theeightexperi mentally determined reflection coefficients areshownplottedinFig.4.6.(IfaSmith chartwithsuperimposed reflection-coefficient circlesisavailable, theimpedance Z. maybedetermined experimentally andplottedassuchusingtheRXcoordinates of thechart.Theseplottedpointsarethedesiredreflection coefficients usingthe r,t/Icoordinates.) Sincetheaccuracy oftheelements ofthescattering matrixwhich aretobedetermined depends ontheaccuracy ofthegraphical constructions, itis advantageous tousealarge-scale chartofthereflection coefficient. Sinceonlycon centriccirclesandradiallinesareinvolved, anappropriate chartisreadilyconstructed. Following theprocedure outlined ingeneralearlierinthissection,thepairsof pointsforwhichfJl=0,7r/2;1r/S,51r/S;1r/4,31r/4;and37r/8,h/Sarejoinedby straightlines.Theoretically thesechordsshouldintersect atthecrossover point S~l' Owingtopossible experimental errortheymaynot,andthedegreetowhichtheir intersections defineapointisameasure oftheconsistency oftheexperimental data. Fortheexperimental dataunderstudytheenlarged viewofthecrossover regionin Fig.4.7indicates thatthechorddefinedbythepoints7r/8,51r/8isnotconsistent with theotherthree,whichsatisfactorily locatethepoint S~latthecenterofaverysmall triangle. ThecenterCofthecircleGthatshouldbethelocusoftheeightreflection coefficients isreadilydetermined asthepointofintersection oftheperpendicular bisectors ofthethreechordsthatvirtually intersect atS".Itisfoundthatacircle canbedrawnthrough sevenoftheeightpointsbyusingasradiustheaverageofthe distances fromCtotheplottedpoints.Itmaybeconcluded thatthepointfor fJl=1rIS,whichaloneliesoffthecircleG,isinerrorandisresponsible1'or thefailure ofthefourthchordtointersect theotherthreeatS~linFig.4.7.Sincetwoextra pairsofpointshavebeendetermined, thispointmaybedisregarded. tThisexample isadapted fromRef.135. 312 TRANSMISSION-LINE THEORY goofil=3TT/4 180°t--+--+--+---I-L...f---1--+-.t-t--:::JJIE--I--+--+--+--+---+--!l--+--+~00 1.0[Chap.V 2700 FIG.4.6.Locusofexperimentally determined reflection coefficients r.=r.eil/l•• FIG.4.7.Enlarged sectionofthecrossover region. Sec.4] DISCONTINUITIES ANDNONUNIFORMITIES 313 FIG.4.8.Graphical determination oftheamplitudes oftheelements ofthescattering matrix. Opencircuit atoutput terminals /31=1f/2 180·I-t-+-+--+--+--+-+-+-+--3IiE--l-=-li;---=~t-P+--++--+--+=1 270· FIG.4.9.Graphical determination ofthephaseanglesoftheelements ofthescattering matrix. 314 TRANSMISSION-LINE THEORY [Chap.V With8;1andGlocatedandthecircleGdrawn,thegraphical constructions previ ouslydescribed maybeperformed todetermine themagnitudes andanglesofthe scattering-matrix elements. Forthree-place accuracy areflection-coefficient chart 1mindiameter wasused.Theconstructions areshowninFigs.4.8and4.9.The valuesobtained forthematrixelements 8=8ei8are S11=OS11=0.331 S11E S12=vr=0.808 S22=S11G=0.328r811=L(OP,OS11) =1350 812=iL(OP,GP~) =70.6°or109.40(30a) (30b) (30e) Theangle812of812isindeterminate by180°,sothattherearetwopossible values. Sincethejunction isknowntobesymmetrical, 811and822shouldbeequal.The resultsobtained areingoodagreement inviewofthefactthatthedistances required aremeasurable onlywithin0.001unit. Inmostcasesonlythemagnitudes S11,S12,and822arenecessary. Theanglesare notrequired inordertocalculate thedivision ofpowerduetothejunction. Since thepowerreflected attheinputterminals 11isproportional toS~1andthetrans mittedpowerisproportional to8~2'thefollowing resultsareobtained: Powerreflected: S~1X100%=11.0% (31a) Powertransmitted tomatched load:si2X100%=65.3% (31b) Powerdissipated inthejunction: (1-Sil-S;I)X100%=23.7% (31c) Ifitisdesiredtodesignamatching network toeliminate thereflected power,the angle811of811mustbeknown. Allthreeanglesarerequired inordertocalculate theequivalent impedance matrixandtheelements ofanequivalent Tsection(Chap. III,Fig.12.3).Formulas forthelatteraregiveninChap.III,Sec.12.Forthe symmetrical casethenormalized seriesandshuntimpedances areobtained from Chap.III,Sec.12,Eqs.(32b)and(33).Theyare Z11-Z12 %1=Zic Z12 %2=ZIC=(11+811-812 1 - 8 11+812 2812(32a) (32b) Sincethereisanindeterminacy of180°in812,t~oequallyvalidTsectionsareobtained. Itisusuallyadvantageous toselecttherepresentation inwhichtheresistive elements oitheimpedances arepositive. Thetwopossible setsofvaluesare _ {0.190-j0.414 %1-0+j1.0197_ {-0.095+jO.729 %2-0.095-jO.729(33) wheretheuppervaluesbelongtogether, asdothelowerones. 5.Measurement ofImpedance andReflection Coefficient througha Junction.tIfanunknown impedance isconnected toameasuring line through anadapting sectionorotherjunction, theproblem arisesto determine thisimpedance oritsreflection coefficient fromobservations madeonthemeasuring line.Thatis,measurements aretobemade tThissectionisbasedonRef.135. Sec.5] DISCONTINUITIES ANDNONUNIFORMITIES 315 ___z._s_~..iJu_nc_tio_n---l~9 FIG.5.1.Junction terminated inanun knownload. Qthrough ajunction. Thisispossible byanextension ofthegraphical methodofDeschamps described inthepreceding section. Consider thecircuitrepresented schematically inFig.5.1.The adapting sectionorjunctIon isbetween terminals 11and22.The measuring line1isattached attheleftof11,sothattheimpedance Za looking intothejunction at11is theloadterminating themeasuring line.Itscoefficient ofreflection is ra=rae""'.Ontherightanunknown impedance isconnected totheoutput terminals 22;itpresents animped anceZLwithreflection coefficient rL=rLe""L.Theproblem istodeter mineZLorrLfrommeasurements online1. Thefirstpartofthemeasurement consistsinreplacing ZLbyalossless lineterminated inamovable shortcircuitandproceeding asiftodeter minethescattering matrixofthejunction, justasdescribed inthepre cedingsection. Thestepsincludethemeasurement ofthefourreflection coefficients andfromthemthedeter mination ofthecrossover point S~1Jthe centerCofthecircleG,andtheicono centerS11,justasinFigs.4.6to4.8. Thesepointsandthecircleareshown GinFig.5.2. Thenextstepistoreconnect theun knownimpedance ZLacrossterminals 22andmeasure thereflection coefficient ra=rae"'"nowterminating line1at terminals 11.Thisisplottedonthe complex reflection-coefficient plane,as showninFig.5.2.Referring tothis figure,thefollowing constructions areFIG.5.2.Construction fordetermin-nowcarriedout:Draw S~lAperpeningr•.diculartoS~lC,DrawCDperpendicu- lartoS~lC,DrawDAextended tomeetCS11(extended) atQ.Draw Qraandsura.ItthenfollowsthatrL=rLei~Lisgivenby (1) whereS22=S22ei822istheelement ofthescattering matrixdetermined in Sec.4andtheangle0isasshowninFig.5.2. Itistobenotedthatthismethodrequires theprevious determination ofS22.Notealsothat,asS~lapproaches C,thepointQmovesfarther andfartheraway.Ifforacertainjunction andacertainloadthepoint 316 TRANSMISSION-LINE THEORY [Chap.V (2) (3) ~4)(Qr.)S22=UY ~==L(Qr.,Sllr.) =L(UY,Sllr.) sothatA / / / / //// //y/ QQistoofarawayforpractical use,analternative construction isavail able.Thisconstruction maybeusedinallcases. Referring toFig.5.3,thelinesDr.andCr.aredrawn. Fromthe pointUwhereQCintersects thecircleG,alineisdrawnparalleltoAD, intersecting CDatV.Through Van otherlineisdrawnparalleltoDr.and intersecting Cr.atY. Itcanbeshownthat Rlustrative Example. Letthejunction be thatinvestigated inthenumerical example FIG.5.3.Alternative construction inSec.4.Mteritsproperties havebeende fordetermining r.. termined usingavariable lengthofshort-cir- cuitedline,letanunknown impedance ZLbe attached toitsoutputterminals 22.Thereflection coefficientr.=O.70ei60•ismeas uredattheinputterminals andplottedinFig.5.4.Mtertheconstruction outlined inthesecondmethod described aboveandindicated inFig.5.4isperformed, the 2700 FIG.5.4.Graphical methodforobtaining thereflection coefficient ofaloadterminatin~ asymmetrical junction. Sec.6] DISCONTINUITIES ANDNONUNIFORMITIES 317 rellection coefficientrL=ry:i'ltisfoundtobe rL=Sur.=0.787 UY1/IL=a-822=-115.8° (5) wherea=L(UY,Sllr.). Thenormalized impedance isgivenby ZL1+rL . ZIL=Zc=1 _rL=0.165-}0.615 (6) 6.TheoryofaDielectric andMagnetic SlaborBeadinaTransmission Line.tInmostapplications transmission linesarenotuniform along theirentirelength. Usuallyjunctions, connectors, orsupports occurat intervals. Itfollowsthattheanalysis ofalineinwhichalengthdis characterized byparameters thatdifferfromthosealongtherestofthe lineisofpractical importance. Themodified sectionmaycontinue from agivenpointallthewaytotheload,oritmaybeashortpieceinserted between thegen~rator andtheload.AsshowninFig.6.1,suchasec tionmaydifferfromtherestofthelineinvariousways:(a)Themetallic conductors inthesectionareidentical withthoseoutsidebutare immersed inadifferent dielectric medium. (b)Thedielectric medium inthesectionisthesameasfortherestoftheline,butthecross-sectional dimensions andspacing oftheconductors aredifferent. (c)Boththe dielectric andtheconductors inthesectionoflengthddifferfromthose elsewhere alongtheline.(d)Thesectionconsists ofasymmetrical recurrent network ofapproximately lumpedreactiveelements distributed uniformly alongthelengthd. Theprincipal andcommon characteristic ofthesefourtypesofcircuit isthattheparameters ofthelineinthelengthd(region1)arenotthe sameaselsewhere alongtheline,sothatthepropagation constant 1'1=al+j{handthecharacteristic impedance Zel=Rel(l-j¢el)differ fromthevaluesl'=a+j~andZe=Re(l-j¢c)intherestoftheline. Owingtoendandcoupling effectsitistobeexpected thatthebehavior ofasmoothlinethatisinterrupted byasectionoflengthdwithphysical properties suchasthosedescribed in(a)to(d)maynotberepresented completely andaccurately simplybyassuming uniformbutdifferent line constants onthemainlineandinthesectionoflengthd.Ingeneral, junction effectsresulting fromthenonuniformity oflineparameters and fromtransverse conducting surfaces atthejunctions actually obtain exceptincasea.However, sincesucheffectscanberepresented by equivalent lumpedelements atthejunctions, account maybetakenof themaftertheeffectofchanges inl'andZchasbeendetermined. It tItisimplicit intheanalysis inthissectionthattheTEMmodeistheonly propagating mode.Forthecoaxiallinethismeansthattheinnercircumference of theouterconductor islessthantheTEMwavelength intheline.Notethat >'TEH=27f/fJ=27f/W~. 318 TRANSMISSION-LINE THEORY [Chap.V followsthat,although thecomplete analysis ofFig.6.1aisthespecific purpose ofthissection,theresultsofsuchananalysis maybegeneralized toapplytoFig.6.1b,c,anddbysubstituting appropriate valuesof1"1 andZclandsupplementing theselaterwiththeanalyses ofthejunction problems. B A I 1 E,p,(j~1 4-To}2a}2b-To generator ~fJ:!~1,zcload (a)YT (b) (c)I}2a}2b11,Zcl,2al 11,zc E,p,(jE~}2al~}2al l.zc~;l~ (d)E.p I2~-}2al 1,Zc~-1-1'-Z.-cl-----· FIG.6.1.Sectionofacoaxiallineinvolving changes intheparameters. (a)Dielectric slabinauniform line.(b)Sectionwithinnerconductor ofreduced size.(c)Under cutsectionwithdielectric. (d)Sectionwithaddedcapacitance andchangeinsizeof innerconductor. Theleft-hand boundary Bisatz'=0,theright-hand boundary Aatz'=d. Consider alongtransmission linethathasair(anyothergooddielectric maybeusedwithsmallchanges innotation) asitsdielectric everywhere exceptinasection(region1)between AandB(Fig.6.1)whichcontains ahomogeneous medium ofthickness dwhichmaybecharacterized bya complex dielectric factor: ~1==El-j(jl=El(I-jhe) (Ia)w andacomplex reluctivity VI(orpermeability til=I/Vl): "1==Pl(1+jhm)til==JLl(I-jhm) (Ib) Sec.61 DISCONTINUITIES ANDNONUNIFORMITIES 319 Itisassumed thatthemedium inregionIinvolves onlysmalllosses, sothatthedielectric factorandthereluctivity arepredominantly real. Since Eland0"1in(Ia)maystandfortherealeffective dielectric constant andconductivity, ohmiclossesarisingfromtimelagsinpolarization response aswellasfromactualconduction areincluded. Theimaginary partof(lb)represents ohmiclossesarisingfromtimelagsinmagneti zation. Theassumed restrictions h~«Ih;«I (Ie) although notnecessary intheanalysis, leadtoconsiderable algebraic simplification. Thesidesoftheslabofmaterial ofthickness dareplane,parallel, and perpendicular totheconductors ofthetransmission line.Inthecaseof Fig.6.Iatheconductors piercetheslabofmaterial without changein cross-sectional sizeorshape. Letitbeassumed thattheyaresohighly conducting thattheelectromagnetic fieldinthesurrounding medium differsnegligibly fromthatwithperfectconductors. Thisimpliesthat theaxialcomponent oftheelectricfieldisnegligible, thatis,Ez==0, inthedetermination ofthetransverse field.Itcanbeshownthatunder theseassumptions theboundary conditions fortheelectromagnetic field aresatisfied whentheinductance, capacitance, andleakageconductance ofeachsectionoflineareassumed tobethesameasiftheentireline werelikethatsection. Forexample, inthecaseofacoaxialline,the constants oftheair-filled partsaregiveninChap.I,Sec.6.Toavoid confusion withregionsIand2,theradiioftheconductors ofthecoaxial lineareaandbinsteadofalanda2: le=J.l.oIn~ 211"a211"Eo c=:;-----~~In(b/a)g=0 (2) Forthedielectric-filled sectioninwhich !II=!lrJ.l.OandEl==ErEO, 211"0"1 gl=In(b/a) WithChap.I,Sec.4,Eqs.(38a)to(40),itfollowsthat(3a) (3b) Sincetheconductors themselves areuniform, thesmallinternal imped anceperunitlength Zi=ri+jwliisthesameinallsections oftheline. Theproofthattheboundary conditions fortheelectricandmagnetic fieldsaresatisfied isreadilygiven. Consider thecoaxiallineasatypical example. Thefieldsinahomogeneous dielectric-filled linearegivenin Chap.I,Sec.3.Fordetermining thetransverse fieldtheymaybe 320 TRANSMISSION-LINE THEORY [Chap.V assumed toconsistofthefollowing twocomponents: Eqlrl=-2t" 7I'"~lr Foranair-filled linethefieldsare Er=_-.!l. 271'"Eor11Bel=- 271'"\llr IBe=- 271'"vor(4) (5) Theboundary conditions ontheelectromagnetic fieldbetween theair andtheimperfect dielectric (region1),ontheonehand,andbetween eachoftheseandtheconductors (assumed perfectforthispurpose), on theotherhand,areasfollows:Thesurfacedensityofchargeisdescribed byn;thesurfacedensityofcurrentonaperfectconductor isdenoted byl;thechargeperunitlengthisq;thetotalaxialcurrentisI.Between thetwodielectric mediathetangential electricfieldiscontinuous; the tangential magnetic B-fieldisdiscontinuous: z'=0,d a ~r~bEr=ErlvoBe=\lIBel (6a) Between theairandtheconductors thenormalelectricfieldisdiscon tinuous; thetangential magnetic fieldisdiscontinuous: q Iz'~0r=a EoEr=na=-2 voBIJ=la= - (6b)7I'"a 271'"a -q -Iz'~0r=bEoEr=-nb=271'"b voBIJ=-h=271'"b(6c) Between thedielectric medium andtheconductors thenormalelectric fieldisdiscontinuous, asisthetangential magnetic field: ql 11o~z'~dr=a ~lEr1=nal=-2 vlBo=lal= -7I'"a 271'"a (6d) -ql -11o~z'~d r=b ~lErl=-nbl=271'"b vlBe=-lbl=271'"b (6e) Thesubstitution of(4)and(5)in(6a)leadstothefollowing relations between thechargeperunitlengthqlandcurrent 11intheconductors wheninthedielectric medium andthecorresponding valuesqandI whentheconductors areinair: q~1ql=-11=I (7) EO Ifthesevaluesaresubstituted in(6b)to(6e),allequations arefoundtobe consistent with(6a).Since,fromChap.I,Sec.10,Eqs.(10)and(11), et»=2q t"In~Az=-21In~ (8) 7I'"~r 71'"\1r Sec.6] DISCONTINUITIES ANDNONUNIFORM ~"TIES itfollowsthat,with(7),321 sothat+1=+"lA1z=voAz VI=(+a-+bh=V=+a-+b "1Wlz=\'l(Aza-Azbh=VoWz=vo(Aza-Azb)(9a) (9b) Significantly bothVand1arecontinuous acrosstheboundaries atz'=0 andz'=d,whereasbothqandWzarediscontinuous. Sincetheboundary conditions aresatisfied ifthelineconstants le,C, andg=0areusedoutsidetheslabandli,Cl,andgl,asgivenin(3), areusedinsidetheslab,itfollowsthattheconventional lineequations maybeusedthroughout, withVand1continuous atz'=0andd,with "'(=a+j{3andZe=Re{1-jq,e)asthepropagation constant and characteristic impedance outsidetheslab,andwith"'(1=al+j{3land Ze1=Re1(1-jq,el)asthecorresponding quantities insidetheslab. Referring toFig.6.1,theadmittance lookingtotherightintotheline atitsjunction withregion1atAisYA=Yecoth 6~,where 6~=PA+jcI>~="'(8+6~=a8+PT+j({38+cI>~) Theadmittance terminating thelineatadistance 8fromthenearest pointoftheslabisYT=GT+JBT;itsterminal function is 6'+·holth-lYT T=PTJ'*'T=coY e(10) (lla)Thecharacteristic admittance andpropagation constant oftheair-filled sections areYe=Ge{1+jq,c)and"'(=a+j{3. Atthesamejunctionbutwithreference tothecharacteristic admittance ofthedielectric-filled line,theadmittance terminating thedielectric-filled lineis YA=Yclcoth 6~1 where 6~1=PAl+jcI>~lisdefinedby(lla)intheform 6~1=coth-l~:1=coth-1(~:~e)=coth-1(Tccoth6~)(lIb) wheretheratioofcharacteristic admittances is Inmostcasesthedielectric materials aresufficiently poorconductors to permittheneglectofq,eandq,cl.Thatis,q,c«1,q,c1«1,andq,eq,e1«1. 322 TRANSMISSION-LINE THEORY [Chap.V Theimportant ratio Tehasthefollowing significance forthecoaxialline illustrated inFig.6.1c: (12a) ForFig.6.la,al=a,sothatTe=VElr/1J.lr.ForFig.6.1b,Elr=1J.lr=1, sothatTeissimplytheratioofthelogarithms. Itissignificant tonote that b b To=1forv;;;.In-=~In- (12b)al a or al=b(~)k k=~= fElr (12c)b ~1J.~ If,asinFig.6.1a,al=a,thenTe=1when1J.lr=Elr;alsoTe>1when Elr>1J.lr,andTe<1whenElr<1J.lr.If1J.lr=1andthedielectric isan undercut slaborbead,asinFig.6.1c,Te=1whenal=b(a/b)kand k=~. Junction effectsowingtothechangeinsizeoftheinner conductor areneglected. Theadmittance lookingtotherightintothedielectric slabofthick nessdatitsinputjunction withthelineatBinFig.6.1is YB=Yelcoth(yld+a~l)=TeYecoth(yld+a~l) (l3a) Thesameadmittance viewedasthetermination oftheair-filled lineto theleftofthedielectric slabandreferredtothecharacteristic admittance ofthelineinairis YB=Yecoth a~ Byequating (l3a)to(13b)andusing(Ub),itfollowsthat a~=coth-1ITecoth[jId+coth-1(r;1coth a~)]} Useisnowmadeoftheidentity coth(x+ )=1+cothxcothy ycothx+cothy inordertotransform (14)into Of=coth- 1cotha~+retanhrid B 1+r;;-Icoth a~tanhrId Inlogarithmic formusingtheidentity x+lcoth-Ix=j-In--1x- (16)is(l3b) (14) (15) (16) (17) a'=1Incotha~+1+(r;1cotha~+Te)tanhrid (18) B2"coth 6~-1 -(r;1coth 6~-Te)tanhrid Byseparating therealandimaginary partsoftheexpression ontheright in(18),formulas forpnand ~maybeobtained. Sincethesearelongin thegeneralcase,theyarenotwrittenout. Sec.6] DISCONTINUITIES ANDNONUNIFORMITIES 323 (20) (21)Matched LinewithDielectric andMagnetic 8lab.Considerable simpli fication resultsifthelineismatched atitsfinaltermination andthe distortion factors cPclofthedielectric andcPcoftheair-filled linearesuf ficiently small.Lettheseconditions beexpressed asfollows: YT=Ycsothat 9~=00,9~=00,andcoth 9~=1(19a) cPr=cPcl-cPc«1sothatTc==rc (19b) Subjecttotheseconditions, (18)maybereducedto 6'=+'cp'=lIn2+(rc+r;;-1)tanh'nd BPBJB2(rc_r~l)tanh'rId Therealandimaginary partsofthelogarithm maybeseparated using theidentity tanh(x+ .)=tan~x+jtany JY1+Jtanhxtany Thefinalresultsforthematched lineare PB=-21InA A=1(2+klT)2+(kl+2T)2t2(22a)'J k~(T2+t2) ,T,1=n1f'+!t-1(kl+2Tt)+!t-1T. (22b) ~B4 2an2+kiT 2ant wheren=1forrc<1andn=3forTc>1.Thefollowing abbrevi ationshavebeenused: kl==rc+r;;-lk2==Tc-r;;-l T==tanhaIdt=tan{3ld Thecorresponding formulas forthereflection coefficient (YB-Yc) r~=rBeN'B=-'----"...:..(YB+Yc)(22c) (22d) areobtained directly. Theyare 1rB=e-2PB=A (22e) Thevoltageorcurrentstanding-wave ratiois 8=cothPB=coth(jInA)A+1A-I(22f) Thepowerstanding-wave ratiois82• Matched LineUnaffected byDielectric 8lab.Animportant specialcase of(22a,b)isobtained whenrc=1.Thisoccursunderconditions speci fiedin(12a,b,c). Ofthese,themostpractical istheundercut dielectric bead.Inthiscasekl=2andk2=0,sothatPB=00,fB=e-2PB=0, and8=1.Sincethereisnoreflection, thephasefunction ep~and thephaseangle y;~= -2cp~ofthecomplex reflection coefficient are meaningless. (23) (24c)(24a)324 TRANSMISSION-LINE THEORY [Chap.V Matched Low-loss LinewithDielectric andMagnetic Slab.Ifthe attenuation constantalofthedielectric medium issufficiently small sothatthefollowing inequalities aregoodapproximations: Tthd· d//Tc+T~lkl==anal=al.............. 2==2 (22a)and(22b)reducetosimplerforms.Theseare • 1I[4+(Tc+~1)2tan2(jld]i PB=~nI(Tc_~1)tan{jldl n-.f...:..'If'+1t-1[1(+1)tIJd](n=1,Tc<1(24b) ":t'B-:42an2Tc~an1Jl n=3,Tc>1 rB=I(Tc-~1)tan{jldl [4+(Tc+~1)2tan2131d]i Thepowerstanding-wave ratioS2=coth2PB,withPBobtained from (24a),isplottedinFig.6.2asafunction of{jld/2'1f'foranonmagnetic 50 5 2 1 1 o0.10.2030.40.50.60.7 J31d{€,.d2Ti=)c" FIG.6.2.Powerstanding-wave ratioproduced byadielectric beadoflengthdina coaxialline. dielectric bead.With JLr=1itfollowsthatTc=v;,:and{jl=y';;'I3, sothat{jld/2'1f'=v;,:d/A.Theparameter inFig.6.2istherelative dielectric constant Er•Itisseenthatthemaximum effectofthebead occurswhenitselectrical lengthisanintegraloddmultiple ofaquarter Sec.6] DISCONTINUITIES ANDNONUNIFORMITIES 325 (25)wavelength, andthattheminimum effect(actually zeroeffectforaloss lessdielectric) occurswhenitselectrical lengthisanintegralmultiple of ahalfwavelength. a.ThinSlabinMatched Line.Inthecaseofanelectrically shortlow lossbeaddefinedby (26a)forJl.r=1andre=V;': 1forEr=1andre=_~ VJl.r forre=1 00thefollowing expressions aregoodapproximations: 11 2 PB=ynIre-r;-ll{jld 11 2 yn(Er-l){jd iIn2 (Jl.r-1){jd (27b)1(27a)forEr=1andre=_~ VJl.r forre=1for/J.r=1andre=v;,: forre>1 forrc<100fJl~==~7r+i(re+r;-l){jld __~3;+i(Er+1){jdforJl.r=1andre=v;,: (26b) ~+!(Jl.r+1){jd forEr=1andre=_1_ 4 4 ~ (wheren=1forr~<1andn=3forre>1) ~i(Er-1){jd forJl.r=1andre=v;,: r'"ilr,-r,'lllId=~i<",-l)fjdfor"=1andr,=..); (26c) o forre=1 b.Quarter-wave SlabinMatched Line.Inthespecialcaseofalow lossbeadorslabhavinganelectrical lengthofexactlyaquarterwave length,thatis,{jld=!7r/2andt=tan(jld=00,thefollowing formulas maybeobtained fromthegeneralrelations (22a,b)[itisassumed that (23)issatisfied]: B=1InIre+r;-1IPyr-r-1e c iInEr+11==coth-1Er Er- iInJl.r+1=coth-1/J.r Jl.r-1 326 TRANSMISSION-LINE THEORY [Chap.V Er-1 Er+1 r'=Irc-r~lI=!J.r-1 Brc+r;;-l-- !J.r+1 o -l~8=cothp.=r.=~for!J.r=1andrc=V~ forEr=1andrc=_~(27c) V!J.r forrc=1 forJ.(r=1andrc=V~ 1forEr=1andrc=_r(27d) vJ.(r forrc=1 c.Half-wave SlabinMatched Line.Theformulas foralow-loss slab thathasanelectrical lengthofexactlyahalfwavelength, thatis,f3ld=11", sothatt=tanf3ld=0,areobtained fromthegeneralformulas (22a,b). Theyare PB=tInI211d-00foraIdsmallrc-r;;-al {11"forrc>1 ~= 11"forrc<12 r'=ilrc-r;;-l/ald==0foraIdsmall(28a) (28b) (28c) (29)SincePB==00andr==0,itfollowsthatthedielectric slabhasnosig nificanteffectandthelineremains matched. IdealOpenCircuitatEndofDielectric andMagnetic Slab.Ifthesec tionoflinetotherightoftheslabinFig.6.1isadjusted tomakethe impedance lookingintoitatAextremely great,sothatitisapproxi matelyequivalent toanidealopencircuitforwhich 6~=0+j1l"/2and coth 6~==0,thegeneralexpression (18)fortheterminal function 6~ reducestothefollowing muchsimplerform: 6'=1.In(1+Tctanh'rIdei7r) BO 21 -Tctanh'rid Usingstandard identities, itfollowsthat t hd_tanhaIdsec2f3ld+jtanf3ldsech2aId an'rl- 1+tanh2aIdtan2f3ld(30) Ifthelossesinthedielectric slabareatleastmoderately low,sothatthe following inequality isagoodapproximation: (a1d)2«1 therelation (30)reducesapproximately to tanh'rId==aIdsec2f3ld+.ftanf3ld(31) (32a) Sec.6] DISCONTINUITIES ANDNONUNIFORMITIES 327 provided {3Idisnotnearanoddmultiple of7r/2.When{3Id=7r/2,(30) reducesto hd1 . 1tan'YI=t hd=-danalal(32b) With(32a)and(Uc),together withtheshorthand notation H0==aIdsec2I3Id-ePrtanI3Id 6~o=PBO+jcf>~oin(29)becomes 6'=1In~I+Ho+jtanI3Id+ '~ BO "2"~I_Ho-jtanf3IdJ2 _ 1I 1+00+'cf>' -4"n1 _00JBO(33) (34) Since00asdefinedin(34)issmallcompared withunity,thelogarithm maybeexpanded inseriestogive •00 reHo (3) PBO=2"=1+r:tan2f3Id 5a J..'=~(t-1tanf3Id+t-1tanf3Id.)+!~2~+H ~-H 2~ 0 ~ 0 7r=tan-I(~Itanf3Id)+2 (35b) Interesting specialcasesaresummarized asfollows: cf>'7r BO=2(36a) (36b) (37) (38) IdealShortCircuitatEndofDielectric. Ifthesectionoflinetothe rightoftheslabinFig.6.1isadjusted tomaketheimpedance looking intoitatAextremely small,sothatitapproximates anidealshortcir cuit(inacoaxiallinethedielectric slabmaybeplacedincontactwith theshort-circuiting piston),itfollowsthat 6~=0andcoth 6~=00,and (18)reducesto 6'=1In1+r;Itanh'Yid BS '2"'1 -r;Itanh'Yid(39) Sincere=re(l+jePr)andr;I==~I(1-jePr),itfollowsthat,asidefrom theaddedtermj7r/2,(39)islike(29)exceptthat~Ioccursinplaceof 328 TRANSMISSION-LINE THEORY [Chap.V reand-cPrinplaceofcPr.Withthesechanges (35a)and(35b)applyto (39).Theresultsare ~s=0 ~s=~.reHs PBS=r~+tan2(3ld i£..1 1(t-1tan(3ld -1tan(3ld). -1( '¥BS=2'anr;l+Hs+tan171+Hs=tanretan(3ld) where HS==aIdsec2(3ld+cPrtan(3ld Forthespecialcasesthefollowing areobtained: For«(3ld)2«1:PBS==re(3ld(;~+<PI') For(3ld=1l": PBS=aldre 1l"For(3ld=2: PBS=aldre(40a) (40b) (40c) (41) (42) (43) LumpedCapacitance inMatched Line.Theparticular valuesofPB and<1>;whenalumpedcapacitance Cisconnected acrossthetransmission lineatBmaybederivedfromtheformulas forthedielectric slabby takingthelimitasthethickness dapproaches zero.Ifcisthecapaci tanceperunitlengthoftheair-filled lineandCl=ErCisthatofthe dielectric-filled line,thecapacitance perunitlengthaddedbythedielec tricis C=(el-c)d=Cd(Er-1) (44) Anyderivedlumpedconstant mayberepresented inthisequation asthe assigned constant valueofCinthelimitasd~0andEr~00. Consider thevalueof wCRe=WCd(Er-l)Re=Wd(Er-1)vZC=(3d(Er-1)(45) Inthelimitasd~0andEr~00thisbecomes wCRe~(3dEr=(3ldy;,: (46) ThusthevaluesofPBand<l>Bmaybeobtained fromthegeneralonesfor adielectric slabbysubstituting wCRefor(3ldy;,:andthenletting ~~0 andEr~00.Thisprocedure isillustrated below,beginning with(22a) specialized foralosslessline: where(47) (48a) (48b) (48c) Sec.7] NotethatDISCONTINUITIES ANDNONUNIFORMITIES 329 (49) (50a) (50b) Ifthecondition (wCR c)2«1issatisfied, thefollowing simplerformulas givethephaseandattenuation functions introduced intoamatched line byasimplelumpedcapacitance: • 1I 2 PB=2"nwCRecp'=311'"+wCRe B4 4(51) LineLoadedwithUniformly Distributed Lumped Capacitances or Inductances. IfNapproximately lumped shuntcapacitances C1or seriesinductances L1areconnected inalineinasectionoflengthd (Fig.6.1d),thelinemaybeanalyzed byassuming thattheeffective capacitance andinductance perunitlengthoflineareaugmented by theamounts NCtfdandNLtfd,sothattheeffective capacitance and inductance perunitlengthbetween AandBinFig.6.1dare (52) wheret;and l~arethevalueswiththelumpedelements absent. The characteristic impedance Zclandpropagation constant "{Iforthelength daredefinedasusual.ByproperchoiceofC1orL1orboth,aloaded linewithvarious properties canbeobtained. Forexample, atsuf ficiently lowfrequencies lmaybeincreased sufficiently bylumpedseries inductances L1sothatr/l=g/candthedistortion factor cPevanishes. Athighfrequencies variable reactive tunersmaybeconstructed foruse onopen-wire orshielded-pair linesbyproviding smallcapacitor plates thatmayberotatedfromcloselymeshedpositions towidelyseparated ones.l2oAlternatively thereactance ofordinary variable capacitors may bedetermined athighfrequencies byanalyzing themassections of capacitively loadedtransmission lineinwhichthestackandrotorrods aretheparallelconductors. 120 7.TheMaximum-Minimum-shift MethodforDetermining Dielectric Constants andPermeabilities ofSolidsandLiquidsandEquivalent Sec tionsofTransmission LineforSymmetrical Two-terminal-pair Net works.l23.129.130 Asimpledirectprocedure fordetermining boththe dielectric constant andthepermeability ofaslabofmaterial ofcon- 330 TRANSMISSION-LINE THEORY [Chap.V venientthickness isavailable inthemaximum-minimum-shift method.t Initsoriginalformitwasdescribed onlyformeasuring relativedielectric constants. However, itisextended without difficulty tothedetermi nationoftherelativepermeability. Thefundamental principle ofthemethod issimple. Ineffect,it involves thesuccessive measurement oftheimpedance ofasectionof transmission linewhenimmersed inthematerial undertestwhentermi natedinanopenandashortcircuit. Sincethesampletobeusedmay bechosentobesymmetrical, itisconvenient tomakeuseofthesym metrical andantisymmetrical combinations involving, respectively, an opencircuitandashortcircuitintheplanethroughthecenteroftheslab (seeChap.III,Sec.12). Alocation inwhichavoltagemaximum andacurrentnullareatthe centeroftheslabissymmetrical withrespecttothevoltageandanti symmetrical withrespecttothecurrent.Itisequivalent toanopen circuitatthecenter. Alocation inwhichavoltagenullandacurrent maximum areatthecenteroftheslabisantisymmetrical withrespectto thevoltageandsymmetrical withrespecttothecurrent.Itisequiva lenttoashortcircuitatthecenter. Actually, completely symmetrical distributions ofcurrentandvoltage(inwhichcurrentorvoltagenulls ratherthanminimaoccuratthecenteroftheslab)areachieved onlyif theslabitselfisexactlyatthecenterofaresonant symmetrical section oflinethatisdrivenbyidentical generators looselycoupledatbothends. Ifthegenerators areinphase,thereisavoltagenullatthecenterof theslab;iftheyare1800outofphase,thereisacurrentnullatthecenter oftheslab.Inpractice, theslabmaybeplacedwithitscenterata voltageorcurrentmaximum, withsections oflow-loss lineoneachside. Onlyoneofthesesections needbedrivenbyalooselycoupledgenerator ifthematerial intheslabisnothighlydissipative, asindicated bythe sharpness oftheresonance curves. Preferably thedetector shouldbe coupledtothesamesectionasthegenerator. Mathematical Formulation. Thefirststepinderiving thetangent relation onwhichthemaximum-minimum-shift method depends isto compare theinputadmittance oftwosections oftransmission line.Of these,thefirst(Fig.7.1a)hasonlyair(vacuum) asthedielectric from thearbitrarily locatedinputterminals atz'=0tothereactive termi nationwithadmittance YT=JBTatz'=s'.Thesecondsectionofline (Fig.7.1b)isimmersed inamedium withcomplex dielectric factor ~l andcomplex permeability tilfromz'=0(planeBinFig.6.1a)toz'=d tItisimplicitintheanalysis giveninthissectionthattheTEMmodeistheonly propagating mode.Forthecoaxiallinethismeansthattheinnercircumference of theouterconductor islessthantheTEMwavelength intheline.Notethat >'TEM=27r/(j=27r/wV;;'.Iff.l=f.lof.lrorE=EOErdiffersgreatlyfromf.loorEO,the cross-sectional sizeofthelinemayhavetobeverysmall. 331 ! s+d (C)(b)z'=Od z'=Od[Region 1(E1,Pl'0"1) _Y_in_\~t~,'4;-ii ~tyT=jBT s+d FIG.7.1.Sections oftransmission line. (a)Uniform section. (b)Slabofdielec tricoflengthdalongotherwise uniform line.(c)Slaboffluiddielectric between thinwallsrepresented bylumpedcapaci tancesalongotherwise uniform line.Yw=jBw=-jwC w atz'=0andz'=d(Fig.7.1c). Inthefollowing themoregeneral problem involving aliquidenclosed inthin,solidretaining wallsisformu lated,sincetheresultsarereadilyspecialized tothesimplerandmore important casesinvolving liquidswithPolyfoam wallsorasoliddielectric withnoadditional wallsbysettingBw=O. Theinputadmittance Y'forthesectionoflineinFig.7.IaisSec.7] DISCONTINUITIES ANDNONUNIFORMITIES (planeAinFig.6.1a)andinairfromz'=dtoz'=d+8,whereitis terminated inYT==jBT•Thematerial parameters havetheproperties discussed inconjunction withSec.6,Eqs.(la,b,c). Itisassumed that thetwosections oflineareidentical exceptfortheaddedpresence ofthe dielectric medium inthesecondsection. (Inpracticethemeasurements aremadesuccessively onthesame sectionoflinewithandwithoutthe Yi~'L~ J::; *;YTslab.)Bothlinesarehighlycon- -~t------------~ ducting, andtheconductivity ofthe z'=o (a) z'~s' dielectric ormagnetic sampleissuffi- cientlysmallsothatSec.6,condi tion(lc),issatisfied. Iftheme diumisaliquid,retaining wallsare required. Thesemaybeignoredif madeofamaterial likePolyfoam whichhasarelativedielectric con stantandarelative permeability differing negligibly from1.Ifthey aremadeofasoliddielectric, they maybesufficiently thintopermit theiranalytical representation as smalllumpedadmittances Y'=G'+jB'=Yccoth(ys'+6~)==-jGccot({38'+cI>~)(1) Thecharacteristic admittance ofthelineisYc==Gc=1/Rc;thepropa gationconstant isy=a+j{3.Theterminal function ofYTis foranidealshort-circuit 6~=O.Thevaluesfollowing theapproxi matelyequalsignin(1)andtherelateddefinitions applyonlyifthe attenuation ofthelineisneglected andthetermination isapure reactance. Theinputadmittance Y2ofthesectionoflineoflength 8inFig.7.1c inparallel withthelumped admittance Y",==jBwoftheright-hand 332 retaining wallisTRANSMISSION-LINE THEORY [Chap.V Y2=G2+jB2=Yw+Yecoth("(s+9~)==j[Bw-Gecot({3s+cI>~)](2) Theinputadmittance YoftheentirelineinFig.7.1eis Y=G+jB=Yw+YelY2coth"(Id+Yel (3) Y2+Yelcoth"(Id whereYc1=Gel(l+j</Jel)isthecharacteristic admittance and "(I=al+j{31 isthepropagation constant ofthelinewhenimmersed intheslabof material medium betweenz=0andz=d. Thefundamental stepinthederivation ofthedesiredequation isto requirethelengthss'andstobesorelatedthattheinputsusceptances BandB'areequal.Thus B=B'or1mY=1mY' (4) Theproperties ofthedielectric material arerepresented by"(IandZel; itsthickness disarbitrary. Itisreadilyverified, using(1)and(3)with(2),that(4)maybe transformed intothefollowing generalequation: withwhereRe{C1[coth ("(s'+9~)-coth("(s+9~)] +coth("(s'+9~)coth("(s+9~)-C2} CI=Clr+jCli==Tecoth"(Id+YwZe C2=C2r+jC2i==T~+2TeYwZccoth"(Id+Y~~ Ze Te==-zel=0(5) (6a) (6b) (7) Withthesedefinitions ofCIandC2,itistherealpartof(5)whichis derivedfromthesusceptance. Itisreadilyverifiedthat(5)withYw=0 isobtained directlybyequating Sec.6,Eq.(13b),withSec.6,Eq.(16), to(3)withYw=O.Equation (5)expresses therelationship between all valuesofs'andsforwhichtheinputsusceptances ofthetwosections (theoneoflengths'inair,theotheroflengthdinthedielectric ormag neticmediumandlengthsinair)areequal.Inthecomplete absenceof thematerial medium (d=0,Bw=0)theinputadmittances areequal whens'=Sl.Ifadielectric ormagnetic medium ispresent, d+sis takentobelessthans'. Thegreatest effectontheinputsusceptance Binisproduced bythe material medium whenitissosituatedthatthevaluesofs'andswhich satisfy(5)aresuchthats'-sisamaximum. Withthiscombination ofs'andsthecircuitreachesitsgreatestsensitivity tothereactiveeffect ofadielectric ormagnetic sample, sothatitrepresents theoptimum condition fortheprecisemeasurement ofErorp.r. Sec.7] DISCONTINUITIES ANDNONUNIFORMITIES 333 Theparticular formsofEq.(5)forwhich8'- 8hasitsextremevalues areobtained bysettingthederivative of8'-swithrespecttos'equal tozeroor,whatisequivalent, bysetting ds=d(ys)=1 ds'd(ys')(8) Usingfamiliar formulas, let(5)betransformed intothefollowing equivalent expression: I. 1+C2 )Re-C1smhyes'-s)+-2-coshyes'-s 1 -C2I +-2-cosh[yes'+s)+28T]=0(9) If(9)isdifferentiated withrespecttos'and(8)isimposed, thefollowing condition isobtained: (C2-1)sinh[yes'+s)+28~J=0 (10) SinceC2isnotequaltounity,ingeneral, (10)isequivalent tothe following: sinh[a(s'+s)+2PT]cos[pes'+s)+24>~] +jcosh[a(s'+s)+2pT]sin[P(s'+s)+24>~)=0(11) whereonlytheimaginary partisrelevant forthecondition (4).This partof(11)issatisfied when fJ(s'+8)+2~~=krk=0,1,2,. . . (12) Using(12)in(9),thefollowing equation isobtained forthemaximum andminimum valuesofs'-s(indicated bythesubscript m): ClisinP(s'-S)m+j(I+C2r)cosP(S'-S)m±j(l-C2r)=0(I3a) wheretheuppersignisforkevenandthelowersignisforkoddin(12), andwhereCliandC2raretheimaginary partofC1andtherealpartof C2,respectively. Inderiving (I3a)itisassumed thatthefollowing inequalities aregoodapproximations: IClil»IC2ia(S'-s)1[a(s'+s)+PT)2«1 (I3b) Equation (I3a)isreadilytransformed intothefollowing twoequations: cot2A+2ClicotA-C2r=0kevenin(12) (14a) tan2A-2ClitanA-C2r=0koddin(12) (14b) wherethenotation A==j{3(s'-S)m=ip(d+8m) (15) isintroduced. In(15)8misthemaximum orminimum shiftinthe position ofthetermination whenadjusted forresonance successively 334 TRANSMISSION-LINE THEORY [Chap.V withoutandwiththematerial medium. TheshiftSisshowninFig.7.1. Thesolutions of(14a)and(14b)are cot~=-Cli±vCri+C2r tan~=Cli±vCri+C2r(16) (17) where,for2~=(3(s'-s)positiveandlessthan 7r",onlytheuppersignsare relevant. Thecomplex constants Cl=Clr+jCliandC2=C2r+jC2i aredefinedin(6a,b). Although therealandimaginary partsofCland C2areseparable, ingeneral, without restricting theproperties ofthe material intheslabunderinvestigation, resonance curvesaresharponly formaterials thatarenotverygoodconductors. Accordingly itiscon venienttoobtainthesimplerformulas thatapplytosamples ofmoder atelyloweffective conductivity. Thisisinagreement withSec.6,con ditions(Ie).Therefore letthefollowing restrictions beimposed onthe propagation constant "(1=al+j{3landthecharacteristic impedance Zel=:Rel(l-jt/Jel): (ald)2«1 Subjecttotheseconditions,(18) coth"(ld==-jcot{3ld+aIdcsc2{3ld provided theadditional requirement(19) (20) issatisfied. With(7)and(18)itfollows(asinSec.6,Eqs.(Uc,d)]that (21) where Erand f.Lraretherelative dielectric constant andpermeability of thematerial medium. Notethat (3l=n{3n==VErf.Lr (22) wherenistheindexofrefraction. Inanonmagnetic dielectric material f.Lr=1andrc=n=~;inanondielectric magnetic material Er=1 andre=1/vJ;;=l/n. With(18)to(22)itfollowsfrom(6a,b)that Cli=BwRc-recotn{3d (23) C2r=r~+2rcBwRecotn{3d-B;R~ (24) sothat -Cli+vCri+C2r=-BwR c+rccoti;n{3d (25) Cli+vCri+C2r=BwRc+rctanjn{3d (26) Sec.7] DISCONTINUITIES ANDNONUNIFORMITIES 335 If(25)and(26)aresubstituted in(16)and(17),thesemaybeexpressed asfollows: cot.6.+BwRc=rccot{nfjd tan.6v-BwRc=rctanjnfjdkevenin(12) koddin(12)(27a) (27b) Itisreadilyverifiedthatcondition (12),fj(s'+s)+2<1>~="k1r,with k=0,1,2,...,ensures thattheextreme values.6.=j,8(s-S')m.and .6v={,8(s-S')mvoccurwhenthecurrentandvoltagedistribution pat ternsaresymmetrical withrespecttothecenteroftheslab.With(12) thepartofthesusceptance B2in(2)duetothelineisgivenby B2in==B2-Bw=-Gccot[k".-(fjs'+~)]=Gccot(,8s'+~)(28a) Sincethesusceptance B'in(1)isequaltothesusceptance Bin(3)and sinceB2'nin(28a)isthenegative ofB'in(1),itfollowsthat,when locatedforextreme shift,B2in=-B.However, sincetheentirecircuit isadjusted forresonance, thesusceptance Blookingintotheslabmust bethenegative ofthesusceptance atthesamepointsbutlookingaway fromtheslabbackintotheline: (28b) Thatis,thesusceptances lookingintothelineinbothdirections from theedgesofthedielectric slabarethesame.Thisispossibleonlywhen thecurrentandvoltagedistributions aresymmetrical withrespecttothe centeroftheslab.Inparticular, theextreme value.1.definedin(27a) alwaysoccurswhenthelargestnumber ofcurrentmaxima consistent withtheelectrical thickness n,8dofthesamplearecontained withinit. Whennfjdislessthan".,thismeansacurrentmaximum atthecenterof theslab.(Whennfjdisbetween". and2".,itmeansvoltagemaximum atthecenter,withtwosymmetrically placedcurrentmaxima withinthe slab.)Alternatively theextreme value.6"definedin(27b)occurswhen thelargestnumber ofvoltageorchargemaxima arecontained within thesample. Forn,8dlessthan".,thismeansavoltagemaximum atthe centeroftheslab.Thequestion astowhichofthetwoextreme values .6.and.6 visamaximum andwhichisaminimum depends ontherelative magnitudes ofErandJJ.r.IfJJ.r=1andEr>1,.6visthemaximum and .6.istheminimum. IfEr=1andJJ.r>1,.1.isthemaximum andLlvis theminimum. IfEr=JJ.r,thereisonlyonevalueofs'-sforallpositions oftheslab,sothat.1"and.1.areequal. Equations (27a)and(27b)maybesolvedforrc=v'Er/JJ.rand n=viJJ.rEr'Thesquarerootoftheproductof(27a)and(27b)is ~=rc=[cot.6.tan.6v+BwRc(tan .6.-cotLlv)-B~R~]i (29a)'\}; 336 TRANSMISSION-LINE THEORY [Chap.V Theratioof(27b)to(27a)yields _j-_ _2-1(tan!:l"-BwRc)'vJ.l.rEr-n-{jdtancot!:li+BwRc(29b) Ifthesampleincludes nosolidretaining walls,Bw=0,andthefollowing simplerexpressions areobtained: I~=rc=(cot!:litan!:l,,)1'\j/-Lr _j- 2vEr/-Lr=n={jdtan-1(tanl:1itan!:l,,)'(30a) (30b) Itfollowsthat 2 Er=rcn={jd(cot!:l,tan1:1,,)'tan-1(tan!:litan!:l,,)' (31a) J.l.r=~=2tan-1(tanl:1itan1:1,,)' (31b) rc{jdcotl:1itanI:1v Ifthetwoextreme values(s'-S)iand(s'-s)"in1:1.and!:l"aredeter minedexperimentally, therelativedielectric constant Erandtherelative permeability J.l.rofthesamplemaybedetermined from(29a,b)orfrom (31a,b). Theonlyotherquantities required arethethickness dofthe sampleandthewavelength Xin(j=27r/Xforthelineinair.Thus,since onlyfourlengthmeasurements areinvolved, anabsolutemethodforthe determination ofErandJ.l.risavailable. Ifaliquidmaterial iscontained between solidretaining wallsforwhich Bwisnotzero,BwRcmaybedetermined experimentally using(27a)or (27b)withthecellempty. Inthiscaserc=n=1,sothat BwRc=tanl:1"e-tanj{jd=cotj{jd-cotl:1ie (32a) where !:lV6=jf3(s'-S)m"and!:lie=j{j(s'-S)miforthecellempty. Alternatively thesusceptance Bwofthewallsmaybeeliminated by subtracting theequations fortheemptycellfromthoseforthefullcell. For(27b),forexample, theresultis tan1:1"-tan!:l"e+tanj{jd=rctanjn{jd (32b) (33b)(33a) keven koddf3(S'+s)+2q,~=k7r (j(S'+s)+2q,~=k7rImportant andverysimplespecialformsof(31a)and(31b)areobtained forasufficiently thinsample. Subjecttothecondition (n{jd)2«1and withBw=0,(27a)and(27b)reduceto n (s'-S)mir:;=J.l.r=d (s'-S)m"nrc=Er=d Itfollowsthatforanelectrically thinsample J.l.rcanbedetermined directly fromtheoneextreme shift,andErfromtheother.Theextreme shiftfor Sec.7] DISCONTINUITIES ANDNONUNIFORMITIES 337 determining J.Lrfrom(33a)occurswhenthecenterofthesampleisata currentmaximum; thatfordetermining Erfrom(33b)occurswhenthe centerofthesampleisatavoltagemaximum. IfJ.Lr=Er,thegeneralEq.(5)reducesto - s'-sn=VJ.LrEr=J.Lr=Er=-d- (34) xz=oInthiscasetheshiftS=s'-s-disconstant foralllocations ofthe cell. Thelocusoftheextreme values(s'-S)masafunction ofs'maybe determined usingtheextremizing condition (12).Theresultis {3Cs'-S)m {3s'+~-krr/2=2 (35) Accordingly, ifvaluesof{3(s'-s)mforarangeofpossiblevaluesofn,rr., anddareplottedasafunction of{3s'+~-krr/2,allpointsmustbe onalineofslope2.Thisappliestoboth{3(s'-S)m1land{3(s'-S)mi. Experimental Procedure. Inordertomeasure dielectric constants and permeabilities bytheextreme-shift method, asampleofthematerial of T To dt°t tgenerator s' b'•~II y,_r-I-----"'------,~-o ~ F,M/?vable--z-,=--.,..o-------------'s'"':-, -piston z=1 (a) 1+s' z=o xEr,Ur s bS=s'-s-d =E=II===y=j~\~=*6==:D~ie::;:le=c:;=:"-.:=lc:s-l:=a-b:= -:=-=-\::;:~~=~=iS=to=~=b' z'=Od d+s s' 11+d 1+d+s 1+s' L-'----......,yr--------" s' (b) FIG.7.2.(a)Location ofpistonatb'forresonance withnodielectric slab.(b) Location ofpistonatbforresonance withdielectric slabbetweenz=1andz=1+d. thickness dmustbemovedalongatransmission linethathasaloosely coupledgenerator andalooselycoupleddetector fixednearoneendand amovable reactive termination (e.g.,apiston)YT=JBTattheother, asshowninFig.7.2,withYT=o.tThefirstoperation istolocatethe position b'(Fig.7.2a)ofthereactive termination atwhichthecircuit withoutdielectric istunedtoresonance, asindicated byamaximum deflection ofthedetector. tIfpreferred, thedielectric maybefixed,andthedetector, generator, andpiston movedrelativetoit.Forsimplicity theprocedure isdescribed onlyforamovable dielectric slab. 338 TRANSMISSION-LINE THEORY [Chap.V Thesecondoperation istomovetheslabofmaterial (orthecellcon tainingtheliquid)fromthepointb'towardthedetector stepbystep, thusincreasing thedistance s'between b'andtheleft-hand surfaceof thedielectric. Foreachposition ofthedielectric thereactive termi nationismovedtowardthedielectric tob,wherethecircuitisagain tunedtoresonance, asindicated byamaximum deflection ofthedetector. Thedistance between thetermination attheresonant position bandthe 80 Voltage minimum withno 60dielectric ~ I8-8 I , I 40Locusof(8-8>,hi (minimum) : I I I I I 20 : I I I I IVoltage maximum withnoj/dielectric I I 120 140 160 180 200 220cm Positionofdielectric (center> FIG.7.3.Shiftcurves:a-distilled waterincellofthickness 2.08cm;a'-thesame cellempty;c-distilled waterincellofthickness 0.52cm;c'-thesamecellempty. Thelargecircleslocatethemaxima oftheshiftcurvesofwatersolutions ofethyl alcohol; bisoneofthesecompletely plotted; A=188.8cm. rightsideofthedielectric iss.Ass'andsareincreased stepbystep but,ingeneral,atdifferent rates,a"shiftcurve"maybeplottedofthe difference s'- sasafunction ofthelocation alongthelineofthecenter oftheslab.Theoriginofthelinearscalealongthelineisarbitrary. Typical"shiftcurves" forwatersolutions ofethylalcoholforwhich Er>1,withJLr=1,areshowninFig.7.3. Ass'isincreased bymovingthesampletowardthedetector, apointis reachedwherethereactive termination mustbemovedawayfromrather thantowardthedielectric inordertotunethecircuittoresonance. At thispointafurtherincreaseins'resultsinadecrease ins'-s-evidently themaximum value(s'-S)mvofs'-shasbeenreached. Foracertain rangebeyondthismaximum, s'-sdecreases ass'isincreased. Then s'- sreachesaminimum (s'-S)miandagainstartsincreasing with Sec.7] DISCONTINUITIES ANDNONUNIFORMITIES 339 continually increasing s'.Asindicated inFig.7.3,thecenterofthe dielectric slabisatavoltageminimum whens'-sisaminimum. If thereactive termination isaperfectshortcircuit(e.g.,apiston)andthe slabiselectrically thin,thecenterofthedielectric is'A/2fromb'when s'-sisaminimum. Inordertodetermine thedielectric constant ofamaterial withILr=1, itissufficient tomeasure themaximum value(s'-S)mvofs'-s;itis notnecessary toplotacomplete shiftcurvelikethoseinFig.7.3.Several experimentally determined maximum values(without therestofthe associated shiftcurves)arealsoshowninFig.7.3.Notealsothatthey r-...~ """'"~ ~~ '~ """-~ ............60Er 80 40 20020406080100 %ethylalcoholbyvolume FIG.7.4.Dielectric constant ofwatersolutions ofethylalcoholreferredtowaterat 15.5°C. Thecirclesareexperimental pointsobtained usingthecellofthickness 2.08em.SoliddotsarefromdatagivenbyWyman[J.Am.Chem.Soc.,53:3297 (1931)]. Roomtemperatures. alllieonthestraight lineofslope2,asrequired by(35)(inFig.7.3 s'increases fromlefttoright). With(s'-S2)mvmeasured and{3=27r/Xknown (ormeasured), n=rc=VZmaybeevaluated from(27b)withBw=0orfrom(34)if therearesolidretaining walls.Thevaluesof€robtained fromthe measurements represented inFig.7.3aregiveninFig.7.4. TheSizeoftheSample. Themathematical theoryassumes thatthe sampleundertestconsists ofaflatslabofthickness dwithitsparallel sidesperpendicular totheaxesoftheconductors andcompletely filling thespacebetween andaroundthem.Foruseinacoaxialorshielded pairlineitconsists ofadiskthatfitsintotheouterconductor orshield andhasaholeorholesfortheinnerconductor orconductors. Foruseon anopen-wire linethedielectric mustideallyextendtoinfinity, although a relatively smallproperly shapedsamplemaybeusedifitsrelativedielec tricconstant orpermeability isnottoonear1andacorrection ismade forthefraction ofthefieldoutsidethesample. 123Ingeneral, measure mentsaremostconvenient withacoaxialline. 340 TRANSMISSION-LINE THEORY [Chap.V Inordertodetermine themostusefulvalueforthethickness dofthe sample,itisnecessary toconsider boththemagnitude ofthedielectric constant andpermeability andthefrequency atwhichitistobemeas ured.InFig.7.5theoretical curvesareshownoftheindexofrefraction n=~asafunction oftheargument ~v=j{1(s'-S)mvforarangeof n 8 4 2 0.2 0.4 0.6 0.81.0 1.2 1.4 1.8 1/3('S-8)2 m FIG.7.5.Theoretical curvesoftheindexofrefraction nasafunction of!13(s'-S)m, with!l3dasparameter. valuesofj{1dasdetermined fromthefundamental Eq.(27b),withJ.Lr=1 andBw=0: tan~.=ntanjn{1d n=V;,koddin(12) (36) Withtheaidofthesecurvesitispossibletoestimate thethickness dof thesamplerequired toproduce anadequate maximum valueofs'-s, iftheorderofmagnitude oftheunknown dielectric constant isknown aswellasthefrequency. Ifthedielectric constant ofaliquidistobemeasured, aclosedmovable cellisrequired. ItsparallelsidesmaybeofPolyfoam orverythinsolid dielectric; itsinnerandoutercircularwallsshouldbemetalsleevesthat slideovertheinnerandintotheouterconductor ofthecoaxialline.By meansofmetaltubesofthesamesizesasthesleeves,theentiresection oflinefromthefrontofthedielectric sampletothereactive termination (piston)atbmaybemadetohaveconstant innerandouterradii.The factthatthesedifferfromthevaluesbetween thedetector andthefront Sec.8] DISCONTINUITIES ANDNONUNIFORMITIES 341 ofthecellisimmaterial, sinceonlythedistances s',s,anddoccurinthe finalformula. Thedetermination ofJJ.rformaterials withEr==1parallels thedetermi nationofErformaterials withJJ.r=1.WithBw=0andEr=1in(27a), thisbecomes tanA..=ntan!npdn=v'""ir,kevenin(12) (37) Sincethisisthesameas(36)exceptforadifferently definednanda differentkin(12),thecurvesofFig.7.5maybeused. Sinceav=jf3(s'-S)mvprimarily depends onEranda.=jf3(s'-S)mi depends onJJ.r,thecurvesofFig.7.5aresatisfactory forestimating the thickness deveninthegeneralcasewhen JJ.randErbothdifferfromunity. Ingeneral, (s'-s)"isthemaximum and(s'-s).theminimum shift when ErisgreaterthanJJ.r;(s'-s).isthemaximum and(s'-8)"the minimum shiftw4en JJ.risgreaterthanEr•AsJJ.randErapproach each other,themaximum andminimum flattenuntiltheshiftcurveisa straight linewhen JJ.r=Er• Measurement ofSmallSusceptances. Themaximum-shift methodisa highlysensitive procedure formeasuring smalllumped susceptances. Theappropriate formula isobtained directlyfrom(27b)bysettingd=0 andcombining thetwolumpedsusceptances Bwintothesinglelumped susceptance tobemeasured. Thus,withB=2Bwandd=0,(27b) becomes B=2Gctanjf3(s'-S)max (38) whereGc=1/Rcisthecharacteristic conductance oftheline.Forsuf ficiently smallsusceptances B==Gcf3(s'-S)max (39) Theextreme-shift methodpermitstheaccurate experimental determi nationofdielectric constants, permeabilities, andlumped susceptances from,measurements oflength,namely, (s'-S)max,(s'-S)min,andd. Theaccuracy isenhanced bythefactthatinmeasuring thedielectric constant thesampleislocatedatavoltagemaximum, whereitseffectis greatest; similarly, inmeasuring permeability, thesampleislocatedata current maximum, whereitseffectisagaingreatest. Incidentally themethod mayalsobeusedtodetermine thereactive properties of loadedsections oftransmission lineandofvariable capacitive tuners. IllS Theadaptation ofthemethodtomeasure lossesisdescribed inthe nextsection. 8.Determination ofLossesinDielectric andMagnetic Materials UsingtheMaximum-shift Method. Inthepreceding sectionamethod isdescribed fordetermining therealeffective dielectric constant Ee=EOEer andtherealpermeability JJ.=JJ.OJJ.rofasampleofmaterial. Section7, 342 TRANSMISSION-LINE THEORY [Chap.V conditions (18),requirethatthissample(region1)havesmall(butnot necessarily zero)attenuation constant alanddistortion factorcPcl.These quantities aredefinedasfollowsforamoderately low-loss line: (lb)(la) cP~l«1;:==2~(~+~) cPcl==1..-(~-~)2wIIel Intheirusualapplication rlinvolves onlyohmiclossesresulting from imperfect conductors, andgiinvolves theohmiclossesofanimperfect dielectric. Asoutlined inChap.I,Sec.4,lossesinthelinemayresult fromtimelagsinthepolarization response ofadielectric medium witha contribution totheeffective conductivity andhencetogl,orfromtime lagsinthemagnetization response ofamagnetic medium withacon tribution totheeffective resistance ri.Timelagsintheconduction response ofamedium involvecontributions totheeffective dielectric constant aswellastotheeffective conductivity. Allthesepossible effectsareincluded inthefollowing generalformulas formoderately low-loss lines,asobtained fromChap.I,Secs.3and4: r riJ.L"ri -l=-l+-' ==-l+h mW W J.LW g•(je (j'+WE"-=-= ==heweWEeWE'-(j"(2a) (2b) Notethat,iftheconductors areperfect, sothattheohmicresistance ri=0,andthelossesinthedielectric medium arenotfromconduction ((j'=(j"=0)butexclusively fromtimelagsinpolarization andmag netization, (2a)and(2b)reducetothefollowing symmetrical forms: rJ.L" gE" wl=J.L'=hmwe=7"=he (3) Itisassumed inthefollowing thattheimaginary partsofthecomplex permeability, complex dielectric constant, andcomplex conductivity are smallcompared withtherealparts,sothattheseformulas aregood approximations: t'=J.L'~jJ.L"==J.L-jJ.L" t=f.'-jE"==E-jE" d=(j'-j(j"==(j-j(j"J.L=VJ.L'2+J.L"2==J.L' E=VE'2+E"2==E' (j=V(j'2+(j"2==(j'(4a) (4b) (4e) Itfollowsthat (5) +" he==~==(jWE WEe WE-(j"""hm==!!:,==!!:- J.L J.L Byaddingsubscripts 1andsubstituting appropriate quantities in (la,b),theattenuation constant anddistortion factorofthedielectric Sec.8] DISCONTINUITIES ANDNONUNIFORMITIES 343 andmagnetic medium are ~==ri+h+h ~lwl m e ri c/>c1==-;;;z+hm-he(6a) (6b) Forthesamelineinair(vacuum) thecorresponding quantities are (7) Withthesepreliminary definitions summarized, attention canbe directed totheevaluation oftheeffective terminal function pofthe Dielectric Movable slab piston genJ~tor--!rM ~wA Coaxialline II I+d.11 S "I FIG.8.1.Dielectric slabincoaxiallineterminated inamovable piston. sectionoflinetotherightofB(Fig.8.1),including thedielectric and magnetic sampleandthereactive sectionoflengths. Theadmittance lookingtotherightatAinFig.8.1isY2,asdefinedin Sec.7,Eq.(2).Theadmittance lookingtotherightatBisY,asgiven inSec.7,Eq.(3).Sincetheeffectofsolidretaining walls(ifthematerial understudyisaliquid)isassumed tobepurelyreactive,thereisnocon tribution tothedissipation, anditisadequate totreatonlythesimpler casewithout walls.Thisisobtained withYw=0inSec.7,Eqs.(2) and(3).Theresulting expressions are Y2=G2+jB2=Yecoth('Y's+8~)==Ye(ascsc2~s-jcot~s)(8) Y=G+jB=Y IY2coth'Y'ld+Yel (9) eY2+Yelcoth'Y'ld Thecharacteristic admittance ofthelineinthedielectric ormagnetic medium isYel,andthatwiththelineinairisYe•Thetwoquantities aregivenby (10) (11) where c/>elandc/>eareasgivenin(6b)and(7)andEer=EelEOandJ.1.r=J.1./J.1.o aretherelativevaluesoftherealeffective dielectric constant andthe realpermeability. Whenthedielectric ormagnetic sampleisinaposition ofextremeshifl, thesusceptance B2isthenegative ofB,asshowninSec.7,Eq.(28b). 344 TRANSMISSION-LINE THEORY [Chap.V Thatis, sothat(9)becomes G'B-Y(G2+jB2)cothyld+Yel - J 2 -elG'BYhd2+J2+elcotyl(12) (13) Lettheadmittances benormalized bydividing byYe,thecharacteristic admittance oftheair-filled line.AsinSec.7,Eq.(21),let where_ Yc1_Re(1-jcPc)-=-(1.) Te=-Y-R(1_'cP) -re+JcPreelJcl re=Rc=GGelcPr=cPel-cPe=hm-he cP~«1ReIe(14a) (14b) whereWith(14a,b), (13)becomes 'b (02+jb2)cothyld+ Teg-J2=Te-"g'-2-+----"'J:-::-·b--'-2 -+-Te-c-'o'-t-=-"h-y-I--=-d 'bG-jB2+'bG2+jB2g-J2= g2J2=Ye Ye(15) (16) Therealandimaginary partsmaybeseparated using(14a)andSec.7, Eq.(19).Forconvenience let Clr+jCli==Tccothyld==re(a.ldcsc2/3ld+cPrcot/3ld)-jrecot/3ld (17a) whereusehasbeenmadeofSec.7,Eq.(19),andahigher-order term withcoefficient a.lcPrdhasbeenneglected. Alsolet C2r+jC2i==T~==r~(1+j2cPr) (17b) Withthisshorthand notation introduced in(15),thefollowing funda mentalequations areobtained: b~+2b2Cli-C2r-Clr(g2-g)+Og2=0 (18a) (g-02)(b2+Cli)-2b2C1r-C2i=0 (18b) Section7,conditions (18),implythefollowing inequality: C2r»IClr(o-02)+Og21 (19) since,when(19)issatisfied andwithSec,7,Eq.(15),and (k7f')Icot.6.kevenb2= -cot(/3s+~~)= -cot"2-.6.= _tan.6. kodd(20) (18a)reducesexactlytothefundamental equations [Sec.7,Eqs.(14a,b)] fortheconditions ofextreme shift.Bycombining Sec,7,Eqs.(14a,b), with(20)thefollowing alternative expressions areobtained forb2: b_{rccoti/3ld keven 2 --rctani/3ld kodd (21) Sec.8] DISCONTINUITIES ANDNONUNIFORMITIES 345 Theremaining equation, (18b),istobeusedtodetermine gand,from it,p.Sincethesectionoflinetotherightofthedielectric slab(Fig.8.1) isessentially reactive, itmaybeassumed thatg2isnegligible compared withg.Hence . .2b2Clr+C2i g-g2=g=b2+CIi With(17a,b)and(21),(22)maybeexpressed asfollows: g==2re[(aldcsc2{1ld+<Prcot(1ld)cotj{1ld+<Pr]sin(1ld Usehasbeenmadeoftheidentities(22) keven (23a) kodd (23b) tanjx=cscx-cotx cotjx=cscx+cotx(24a) (24b) (25)Theterminal attenuation function Pofamoderately low-loss linemay bedetermined from _ 1th-l2g •gP-~an1+b2+g2=1+b2 (26b)(26a) koddkevenThesubstitution of(23a)or(23b)in(25),together withtheappropriate formula from(20),leadsto 2re[(ald/2)sec2j{1ld+cf>rtanj{11dJ Pi= r~+tan2j{11d 2re[(ald/2)sec2j{1ld-cf>rtanj{1ldJ Ptl= 1+r~tan2j{1ld (Thesubscripts iandvindicate Pwithcurrentandvoltagemaximum, respectively, atthecenteroftheslab.)Inderiving (26a,b)usehasbeen madeof(24a)toexpressallarguments asj{1ld. Since,withkeven,thedielectric samplehasacurrentmaximum atits center({11d<1r)and,withkodd,avoltagemaximum, thevaluesofpin (26a)and(26b)shouldbetwicethevaluesobtained byplacingaslabof dielectric ofthickness d/2atanidealshort-circuited endandanideal openend,respectively. Thatis,pin(26a)shouldbetwicethevalueof PBSinSec.6,Eq.(40a),andpin(26b)shouldbetwice PBOinSec.6, Eq.(35a).Itisreadilyverifiedthatthisistrue. Sincethecircuitisalwaysadjusted toresonance indetermining the extreme shift,itisconvenient todetermine PiandPtlusingtheresonance curvemethod. Oncethesetwoquantities areknown,alandcf>rmaybe evaluated from(26a)and(26b),andfromthesehmandheusing(6a,b) with(14b).Itisassumed thattheconstants ofthelineinairareknown, aswellasReIand/31,whichinvolveErandILr. 346 TRANSMISSION-LINE THEORY [Chap.V 9.TheDoubleBeadandtheSpacingojBeads JOTNoChangein Impedance. Dielectric beadsorslabsoftenmustbeplacedatintervals alongatransmission lineinordertosupportoneormoreconductors and maintain thedesiredspacing. Sinceevenasingledielectric beaddis turbsacondition ofmatch,alargenumberofbeadsdistributed alonga linemighthaveaseriouseffectonthetransmission properties ifthey happened tobesolocatedthattheireffectswerecumulative. Inorder todetermine howtheeffectofasinglebeadmaybemagnified orreduced bythepresence ofotherbeads,lettwoidentical slabsofdielectric, each ofthickness dandseparated anarbitrary distancevbetween adjacent parallelsides,beinvestigated. Theconfiguration isshowninFig.9.1, wheretheslabnearertheloadhasitsnearestsideatadistance wfrom theload. Fortheair-filled sectionsoflinethepropagation constant isr=a+j{j, andthecharacteristic impedance isZe=Re(l-jq,e).Although the ~~ ~...triobood'~:::::~]}z, =m ~----._..Js I+d~ V .14d-tk-w--------t FIG.9.1.Coaxiallinewithtwodielectric beads. moregeneralcasecanbeanalyzed, letitbeassumed forsimplicity that lossesinthedielectric arenegligible, sothatEisrealandgtfwelisnegli giblecompared withrtfWll.Thepropagation constant andcharacteristic impedance oftheair-filled sections oflineare r=a+j{j (la) Theparameters ofthedielectric-filled sectionsare Sincethedielectric islossless,itfollowsthat(~~)2_q,:l«1(lb) {jl=n{jRel=Re nn=v;,. (2) InorderthatthesectionoflinebetweenAandE(inFig.9.1)which includes thetwobeadsmayhavenoeffectontheimpedance oftheline, theimpedance lookingtowardtheloadatEmustbeequaltotheimped ancelookingtowardtheloadatAforanarbitrary load.Thatis, ZE=ZA (3) Sec.9] DISCONTINUITIES ANDNONUNIFORMITIES 347 Complex terminal functions maybedefinedforeachimpedance interms ofthecharacteristic impedance ofeithermedium. Thus ZA=ZccothOA=ZclcothOAI=~cothOAI (4a)n ZE=ZccothOE=ZclcothOEI=~cothOEI (4b)n Evidently (3)isequivalent to OE=OAor (5) Thequestion, therefore, iswhether (3)and(5)arepossibleand,ifso,under whatconditions. Thefirststepistoobtainexpressions forthecomplex electrical length 'rv(visthedistance between theadjacent surfaces ofthetwopiecesof dielectric) intermsofthefunction OEI=OAIandthethickness dofeach beadorslab.Thisisaccomplished asfollows: Firstnotethat,since On=yv+OB,itfollowsdirectlythat 'rV=On-OB (6) Thenextstepsinvolvetheevaluation ofOnandOB.Since OAI=OEI='rId+Onl (7) and Zn=ZccothOn=ZccothOD1 (8a)n itfollowsthat On=coth-l(~cothOD1)=tanh-l(ntanhOD1) (8b) Bysolving(7)forOD1andsubstituting in(8b),thedesiredformula for Onisobtained. Itis On=tanh-l[ntanh(OEI-'rId)] (9) Similarly, since OBI='rId+OAI (10) and ZB=ZccothOB=ZccothOBI (lla)n sothat OB=coth-l(~cothOBI)=tanh-l(ntanhOBI) (Ub) itfollowsthat OB=tanh-l[ntanh(OAI+'rId)] (12) Thesubstitution of(9)and(12)in(6)givesthedesiredformula for'rV: 'rV=tanh-l[ntanh(OEI-'rId)]-tanh-I[ntanh(OAI+'rId)](13) Usingtheidentity tanh-lx±tanh-ly=tanh-Ix+y (14)1±xy 348 TRANSMISSION-LINE THEORY [Chap.V (15) (16)(13)maybetransformed into thIn[tanh (OEI-''(ld)-tanh(OAI+"ild)]"iv=an-1 -n2tanh(OEI-"ild)tanh(O.n+"ild) Eachofthehyperbolic tangents in(15)maynowbeexpanded usingthe identity tanh(x+ )=tanhx±tanhy - y1±tanhxtanhy Aftersomealgebraic manipulation theresultis t h 2ntanh''(ld(tanh2OAI-1) an"iV=1 _tanh2OAItanh2"(ld-n2(tanh2OAI-tanh2"ild)(17) Thisisacomplex equation for"iV=(a+j(3)v.However, since(3)can notactually besatisfied unlessthelossesinthesectionoflinebetween EandA(Fig.9.1)arenegligible, itisconvenient toreduce(17)toa singleequation in{3vbysettingtheattenuation equaltozero.Thus "i==j{3"il==j{31=jn{3 (18) With(18),(17)reducesto t. 2ntan{3ld(tanh20AI-1)an{3v= (19)1+tanh2OAItan2{3ld-n2(tanh2OAl+tan2(3ld) Sincetheleftsideof(19)isreal,therightsidemustalsobereal.Thisis possible onlyiftanhOAIiseitherrealorpurelyimaginary. Inthelatter caseOAI=PAl+jCPAImustbeimaginary, sothatPAlmustvanish. Since thispossibility limitstheloadtoapurereactance, itmustberejected. Thealternative is tanhPA+jtanCPA•tanhOA=1+ .t h t cPISreal (20)JanPAanA Forthisequation therearetwopossible solutions. Theyare tanhOA=tanhPA tanhOA=cothPA(21a) (21b) Thecondition CPA=CP,+{3w=7rmeansthattheedgeAofthedielectric (Fig.9.1)isatavoltagemaximum; thecondition CPA=CP,+{3w=7r/2 meansthatAisatacurrent maximum. Ingeneral, PA=P,+aw. Forasufficiently smallattenuation onthelineitispossibletoset P,»aw With(22)andthenotationP,==PA (22) S==cothP, (23) whereSisthestanding-wave ratioonthelinebetween theedgeAand Sec.9] DISCONTINUITIES ANDNONUNIFORMITIES 349 tanh6A=8theload,theconditions (21a,b)become 1 1 1tanh6A=Stanh6Al=ntanh6A=n8 (24a) 1 8tanh6A1= -tanh6A= -n n~A=~.+{jw=7r 7r ~A=~.+{jw="2 (24b) Ifthesevaluesaresubstituted in(19)andtheresultsarerearranged, the following expressions areobtained fortheelectrical distance {jv(notethat n=y;,:).: For4>A=11': t1n(n282-1)sin2n{jd{jv-an-- (n2S2-I)(n2+1)sin2n{jd-n'l(S2-1) (25a) 1n(n2-82)sin2n{jd{jv-tan-- (n2-S2)(n2+1)sin2n{jd+n2(S2-1) (25b) 2520~,8=1, 5'1\ \ \ \/--\I520 10 15 i\em FIG.9.2.Square ofstanding-wave ratioSinamatched linewithtwo dielectric beads, eachofthickness d=0.0635cmanddielectric constant Er=2.6,separated adistance Vc=2.03 cmbetween centers.(28b)~d Vc=4-"2(n2-1)and{3vc='(j(v+d)=!-!(3d(n2-1)2 2 (28a)Matched LinewithTwoBeads.Ifthelineismatched, PA= and~Ahasnosignificance. Theelec- 50 tricaldistance {jvbetween thetwo beadsis {jv=tan-1(n22~1cotn{jd)(26) Ifn{3dissufficiently smallsothat 10 cotn{jd==I/n{jd,(26)reducesto 82 {jv==!-!(jd(n2+1)(27)2 2 foramatched line.Inthiscasethe electrical distance {jvcbetween the centersofthebeadsis 2 Itisseenthattwosufficiently thin beadsmustbespacedatadistance slightlylessthanaquarterwavelength alonganonresonant line.The squareofthestanding-wave ratiointroduced inanoriginally matched line bytwobeadsisshowninFig.9.2asafunction ofwavelength. Itisseen thatthelineismatched attwowavelengths thatsatisfy(26). 350'i":ANSMISSION-LINE THEORY [Chap.V Byarranging beadsordielectric supports inpairs,witheachpair spacedadistance v,asgivenin(26)or(27),amatched lineremains nonresonant alongallsectionsbetween thepairsofbeads. Resonant LinewithTwoBeads.IfPAissmallandthestanding-wave ratioSisverygreat,sothattheinequalities S2»1 (29) For4>A=1r:areallsatisfied, thegeneralexpressions (25a)and(25b)reduceto _ _ -12ntann{jd {jv-1rtan1 2t2{jd- nann r:1 t-12ntann{jdIJV=1r-ann2-tan2n{jd(30a) (30b) Forsufficiently thinbeadsorslabsthatsatisfytheinequality (n{jd)2«1 thefollowing resultsareobtained: For4>A=1r:(31) (jV=1r-2n2{jdv=~-2n2dA(32a)2Ve=2-d(2n2-1) 1rFor4>A=2: (jv=1r-2{jdA A(32b) v= - -2d Ve=2-d2 Itisseenthatforaresonant linethepairofbeadsmustbeplacedalmost ahalfwavelength apart.NotethatVeisthedistance between centers. LinewithTwoLumped ShuntCapacitances. Asconsidered inSec.6, asinglelumped capacitance Cmayberepresented analytically bya dielectric beadinthelimitasitsthickness vanishes anditsdielectric constant becomes infinite. Therelations are C=limcd(n2-1)=limcdn2 (33a) d-+O d-+O n-+00 n-+00 wCRe=lim(jd(n2-1)=lim(jdn2 (33b) d-+O d-+O n-+00 n-+00 limntann{jd=limntanwCRe=wCRe (33c) d-+O n-+00 n n-+00 (34b)(34a)Withtheserelations (25a)and(25b)maybereducedtothefollowing: _-1 2wCR c {jv-tanW2C2R;_ 1+1/S2 r:1_-12wCRe IJV-tanW2C2R;_ 1+S2For4>A=0: Sec.10] DISCONTINUITIES ANDNONUNIFORMITIES 351 Forthematched linewithS=1thesereduceto with1r wCRefJv=2-tan-l-2- AC(WCRc)2«1v="4-2c(35a) (35b) wherecisthecapacitance perunitlengthoftheair-filled line. Fortheresonant line, For4>..=0: fJv=1('- 2tan-l(wCR~) (36a) A2Cwith (WCRc)2«1v=2-C (36b) 1r AForcf>..=2:fJv=0,1(',21(',. .v=2'A,. (37) Ifthefirstcapacitance isplacedatavoltage maximum, thesecond capacitor mustbesomewhat lessthanA/2fromit.Ifthefirstcapaci tanceisatacurrentmaximum, thesecondcapacitor mustbeexactlya halfwavelength fromit. 10.TheDouble-slug Transjormer.16Acoaxialtransmission linecon tainstwoidentical slugs,eachoflengthd,consisting eitherofdielectric material withrelativedielectric constant Er,asshowninFig.10.la,orof metalsleeves,asshowninFig.10.lb.Thecharacteristic impedance of EDBA genJ,~tor-~_-J~~~0L0"",~,-,/j~ __ ----,,,I2!,-,/j-'~,/;.J.0:.L0.L.~",,--- __--"_Toload WP& F000/d t+-d-+J-v~d --l (a) E B A /+-d--t+- V~d---+l (b) FIG.10.1.Double-slug transformers inacoaxialline.(a)Dielectric-slug transformer ({jld=7r/2).(b)Metal-sleeve transformer ({jd=7r/2). themainlineandofthesectionoflengthvbetween theslugsisZc,and thepropagation constant is"'(.=a+j{3.Forthepartsofthelinecon stituting theslugs(regions 1),theparameters areZcland"'(1=al+j{3l. Sincethetransformer istoserveasareactivedevice,onlygooddielectrics andlow':'loss conductors areinvolved. Forthesethereactive properties areessentially thesameasforperfectdielectrics andconductors, andthe lossesarenegligible compared withthedissipation intheload.There- 352 TRANSMISSION-LINE THEORY [Chap.V foreitisadequate toneglectattenuation intheslugsandtheregion between bysetting "(=j{1 "(l=j{11 Iftheslugsaredielectric, {11={1vz.(Ia) (Ib) (2a) (2b)Iftheslugconsistsofmetalsleevesofradiusalgreaterthantheradiusa oftheinnerconductor, In(bla) Re Rei=ReIn(blal)=ym Thesymbolmhasbeenintroduced in(2a)and(2b)tostandforeither [In(blal)]2 ErorIn(bla). Theproperties ofthetransformer maybestudiedbyderiving aformula forthecomplex terminal function OEofthearbitrarily loadedlinetothe rightofEinFig.10.1.Stepsinthederivation areoutlined below. Theimpedance lookingtowardtheloadatAis sothatZA=ZecothOA=.JmcothOAl 1tanhOAl=vmtanhOA Theimpedance lookingtotherightatBis(3a) (3b) sothat However,ZB=ZecothOB=.JmcothOBi OB=coth-l(JmcothOBi) OBi=j{11d+OAl(4a) (4b) (5) Letthelengthoftheslugsbesochosenthat (11d=~ If(6)isusedin(5),itfollowswith(3b)that (fir) 1cothOBi=coth2+OAl=tanhOAl=vmtanhOA Bysubstituting (7)in(4b)thefollowing resultisobtained: 6B=coth-l(~tanhOA)(6) (7) (8) Sec.10] DISCONTINUITIES ANDNONUNIFORMITIES 353 Byasimilarprocedure itisreadilyshownthat OE=coth-1[~tanh(j{3v+OB)] (9) sothat,with(8)in(7), OE=coth-1(~tanh[j{3V+coth-1(~tanh6A)]) (10) Thisisthedesiredexpression. Byexpanding thehyperbolic tangent of thesum,(10)maybetransformed into (11)6-+'<1>-th-11+(jIm)tanhOAtan{3v E-PEJE-co tanhOA+imtan(3v Ifthenormalized admittance lookingtotherightatAisintroduced by setting (12) (13a) (13b)tanh6A=coth 6~=YA=gA+jbA sinh2(aw+Ps) gA=cosh2(aw+Ps)-cos2({3w+<1>~) b A= - sin2({3w+<1>~) cosh2(aw+Ps)-cos2({3w+<1>~)where andwhere,asusual, O~=PA+j<1>~if..1 7f' '±"A=<1>A-2 <1>sandpsaretheterminal functions oftheimpedance Zsterminating a lengthwoflineextending fromtheendoftheslugatAtotheload. If(12)issubstituted in(11),therealandimaginary partsmaybe conveniently separated byintroducing thefollowing identity: x+lcoth-1x=jIn--1x-(14) Theresultsare wherePE=jInA (15) (16c)(16b)(16a) ([1+gA-(bAlm)tan{3v]2+[bA+(m+gAlm)tan f3ve)! [1-gA-(bAlm)tan{3v]2+[bA+(m-gAlm)tan{3v]2 .f,_t-1bA+(m+gAlm)tan{3v'YN-an- 1+gA-(bAlm)tanf3v .f,_t-1bA+(m-gAlm)tan(3v 'Yj)-an (bI )1 -gA- Amtan{3v Ifthelineismatched sothatZA=Zs=Zc,gA=1,andbA=0,greatA= 354 TRANSMISSION-LINE THEORY [Chap.V simplification isachieved. Thus PE=j-In4+(m+limptan2{3v (m-1/m)2tan2{3v iI'.=311"+!t-1(m+11mtR)'¥E4 2an 2anfJV(17a) (17b) BO°60 201020 40 flu FIG.10.2.Standing-wave ratiointro ducedinamatched linebyadouble slugtransformer whenitscenterisfixed andtheelectrical distanceflvbetween theslugsisvaried.Thestanding-wave ratioS=cothPE,withPEasin(17a),isshownin Fig.10.2asafunction of{3v,with masparameter. Itisseenthatby varying {3vthedouble-slug trans formercanintroduce avalueofSin thematched linerangingfrom1when {3v=0andPE=00toamaximum S=m2when{3v=11"/2andPE= j-In[(m2+1)/(m2-1)]=cothm2. Thedouble-slug transformer may beadjusted intwoways:(a)The pointmidway between theslugsis fixed,andbothslugsaremovedsimul taneously towardorawayfromthis point,thusvaryingv. (b)Bothslugs aremovedintandemwithvfixed. a.CenterofTransformer Fixed; Spacing ofSlugsVaried. Letthe centerofthetransformer befixed atadistance ufromaconvenient reference pointbetween thetrans formerandthegenerator. Thephasefunction atthereference pointis <1>"=(3(u-d-jv)+<l>E (18) Whentheslugsareincontact, v=0,and (19) Thephasedifference between (18)and(19)is .:l<l>"=(,,)v=o-<1>"=(E)v=O-<l>E+j{3v (20) Thevalueof(E)v=Oisobtained fromthegeneralformula (15)with (16b,c)..Itis (E)v=O =j(tan-11~AgA+tan-11~AgA) _ 1 -1-2bA (21) -2"tan g~+b~_1=A Sec.10] DISCONTINUITIES ANDNONUNIFORMITIES 355 (23)With(15)and(21)in(20),thephasedifference isasfollows: ~epu=epA-i(Y;N+Y;v)+i{3v (22) Forthematched lineepA=31r/4,sothat dIP.~}[I3V-tan-1(m+21/mtanI3V)] 7Thisfunction isrepresented inFig.10.3asafunction of{3v,withmas parameter. Theelectrical separation {3vofthe slugsforthemaximum valueof~epu foragivenmisobtained byequating thederivative of~epuin(23)with respectto{3vtozero.Theresultis 6 40° fJv FIG.10.3.Changeinthephasefunction <fl,.introduced inamatched linebya double-slug transformer whenitscenter isfixedandtheelectrical distance {3v between theslugsisvaried.{3vmax=tan-1~m+21/m(24) Ifthisvalueof{3vissubstituted in (23),themaximum phaseshiftturns outtobe (~epu)max=i(tan-1~m+21/m _tan-1~m+21/m) =tan-1~-:::+~-i(25) Thelocusofmaximum valuesof~epu isindicated inFig.10.3. Notethatthechangeinthephase function produced bythedouble slugtransformer, whenitscenteris fixedinposition inamatched line andthetwoslugsaremovedtogether, isrelatively smallincomparison with theratherlargechangeinattenuation function andstanding-wave ratio. Itisforthisreasonthattheadjust mentofthetransformer byvarying v,withthelocation ofitscenterfixed,issignificant primarily asameans forvaryingtheattenuation function PEratherthanepE. b.Spacing ofSlugsFixed;EntireTransformer Moved. Inorderto investigate theeffectofmovingtheentiretransformer alongtheline, 356 TRANSMISSION-LINE THEORY [Chap.V consider anarbitrary reference pointPuatadistance ufromtheleft sideE(Fig.10.1)ofthetransformer. Letthecomplex terminal func tionofthelinetotherightofPu(Fig.lOA)beOu=pu+jq,u.The lengthoflinefromtherightsideAofthetransformer totheloadZ.isw, Double-slug transformer zu""\11 ,--A--, gen~~tor----+:t:::;""""'-------r7::~"'~'7'i------:Z~Z8 _________ -.JU._<.<L- ......_""'_~ ___..(load) ~l.--- u--4II*T"f ~*'f""*""'f**--w--.l dvd FIG.lOA.Double-slug transformer withthedistance vbetween slugsfixedandthe entiretransformer movedbetween afixedloadZ.andafixedreference pointPu. sothatthedistance fromthereference pointPutotheloadis 8t=U+2d+v+w (26) Thisdistance iskeptconstant asuandwarevariedequallyandoppo sitely.Evidently u+wisalsoconstant foranygivenvalueofv. Thephasefunction q,uisgivenby (27) whereq,Eisgivenby(15)with(16b,c). Inordertoexamine q,uasu andwarevaried,itisnecessary toexpressthenormalized admittance YA=gA,+jbAintermsofthedistance w.Theappropriate formulas are (13a)and(13b). Since,whenthelineismatched (Z.=ZC,P.=00),theimpedance terminating thetransformer isalwaysZcforallvaluesofw,itfollows thatthereisnoeffectwhatever onOuasthetransformer ismoved,with vfixed,whenthelineismatched. Whenthelineissufficiently mis matched sothatP.isquitesmall,thefollowing condition maybeimposed: (aw+p.)2«1 (28) sothat(13a,b)reducetothefollowing: gA==(aw+P.)csc2({jw+q,~)=(aw+P.)sec2({jw+q,.)(29) bA==-cot({jw+q,~)=tan({jw+q,.) (30) Exceptoverasmallrangenear{jw+4>~=n7r,gAissmallandmaybe entirelyneglected indetermining thegeneralnatureofthevariation in q,u.If(30)isusedin(16b,c)andgAissetequaltozero,thefollowing expression isobtained forq,u,asdefinedin(27): 4>={ju+tan-1 mtan{jv-cot({jw+4>~) (31) u 1+(11m)tan{jvcot({jw+4>~) When{jv=0,4>u=(j(u+w)+4>8'whichisconstant forallvaluesof uandwwhichhaveaconstant sum.Forallothervaluesof{jv,4>uvaries Sec.10] DISCONTINUITIES ANDNONUNIFORMITIES 357 significantly withuandwwhenu+w=constant. Forexample, when {jv=7r/2, (32) 40° GO° j3w+~~ FIG.10.5.Phasechangeintroduced by moving adouble-slug transformer withslugsseparated aquarterwave lengthbetween inneredges.20°(33)cflu={ju+tan-1[m2tan({jw+<1>;)] Thisformula isstudiedconveniently 70°r--"""'--.~-r--...,.--.,-----r--r-..., iftheconstant quantity 40"thechangeinphasefunction maybe definedasfollows:(juo=cflu({jw+cfl~=0)isintroduced. Notethat,with{juo=(j(u+w)+~ Thisfunction isplottedinFig.10.510°with{jw+cfl~asthevariable andmas parameter. Clearly,iftheloadisfar frommatched, thephasefunction cfl"0°00 maybevariedoverawiderangeby movingadouble-slug transformer for whichmneednotevenbegreat. Itfollowsdirectly from(15)with (16a)thatPEissmallifgAissmall. Thisisconveniently shownbyexpressing (16a)asfollows:.1cfl==cflu-<l>u({jw+cfl;=0) =tan-1[m2tan({jw+cfl~)] -({jw+cfl~) (34) where P==(1+gl+b1)cos2{jv +2bA(m-~)sin{jvcos{jv+(~+~~+m2)sin2{jv(35) WhengAissmall,thefollowing inequality isvalid: p2»g1 (36) sothat(34)reducestoA==1+2gA/Pand - 1In(1+2gA)~gA (37)PE-~p-P ItfollowsthatPEissmallandthattherefore Pu=au+PEalsoissmall, sincetheattenuation aUofthesectionoflineisassumed tobeinsignifi- 358 TRANSMISSION-LINE THEORY [Chap.V cant.Byinserting (30)in(35)andneglecting gi,itfollowswith(32) that p==(cos2{jv+~2sin2(jV)cot2({jw+~) -(m-~)sin2{jvcot({jw+~)+cos2{jv+m2sin2{jv(38) Thevariation ofPEwith{jw+~isconveniently studiedwith{jv=7r/2. Inthiscase (39) sothat,with(29), . aW+ps ( ) PE=m2sin2({jw+~)+(l/m2)cos2({jw+~) 40 Short-circuiting plug Resistivedisk BAThisformulashowsthatpu=PE+au==PErangesbetween m2(aw+Ps) and(aw+ps)/m2asthetransformer ismovedanelectrical distance 7r/2. Notethat,formoderate valuesofm,Puissmallovertheentirerange. 11.LossyTerminations forNonresonant Shielded Lines.Twotypes oftermination whichresultinmatched lineswithstanding-wave ratios S=1areuseful. Theyarearelatively thindiskorslabofmaterial withlargeleakageconductance placedataquarterwavelength froma highlyconducting piston,andaratherlongsectionoflinewithadielectric medium characterized byamoder atelysmallleakageconductance. Resistive Disk.Theproperties of theresistive-disk termination maybe deriveddirectlyfromSec.6,inwhich thegeneralformulas foradielectric slabatanarbitrary location alonga FIG.11.1.Resistive diskwithhigh-transmission linearegiven.The impedance stubterminating acoaxialline. complex terminal function 6B=PB +j~ofasectionoflineoflengthd, propagation constant "(1,andcharacteristic impedance Zclbetween the planesBandA(Fig.11.1)alonganarbitrarily terminated lineisgivenin Sec.6,Eq.(18),intheform 6'_1Incoth6~+1+(1;1coth6~+1c)tanh"(Id (1) B-"2"coth 6~- 1 -(r;Icoth 6~-rc)tanh"(ld where,inthegeneralcasehereconsidered, rc=Zc/Zclandcoth 6~isthe admittance lookingtowardtheloadatsurfaceAnormalized tothechar acteristic impedance Zcofthemainline.Ifthesectionoflinetothe rightofthediskinFig.11.1isessentially reactiveandadjusted formaxi- Sec.11] DISCONTINUITIES ANDNONUNIFORMITIES 359 muminput impedance,8~ =PA+jif>~==j1r/2,sothat coth 8~==0 Itfollowsthat(2) 0'=IInTetanh'rId+1 B~Tetanh'rId-1(3) (5a) (5b)Sincethethickness doftheresistive diskmaybekeptverysmall,the conditions (ald)2«1(Pld)2«1 (4) areappropriate. Withthemthehyperbolic tangent in(3)maybe replaced byitsargument. Sincetheleakageconductance oftheslab mustbequitehigh,thecondition g2»ClJ2C2isappropriate. Theparam etersforthiscasearegiveninChap.II,Sec.13,Eqs.(47b).Theyare al=PI=~wl~gl Iwllal ReI=XcI='\j2gl=fi;. Itfollowsthat (7)(6a) (6b)tanh'rId=='rId=ald(1+j) Ze. Re Reglr=-=- .eZelRel(1+j)al(1+j) With(6a)and(6b),(3)becomes /).f_+'if>'_1IRegld+1 VB-PBJB-~nRegld_1 Threecasesariseasfollows(notethatSisthestanding-wave ratio): ForRegld>1: ~,0 IIRegld+1th-lR d'¥B= PB=~nRegld_1=co egl ForRegld<1: if>'=!B2II 1+Regldth-lR d PB=~n1 _Regld=an eglS=R~ld(8b) ForReg1d=1: if>'=!B4PB=00 8=1 (8c) Acomparison oftheseresultswiththoseforapredominantly resistive termination inChap.IIshowsthatthethindiskisequivalent toalumped resistance (9) 360 TRANSMISSION-LINE THEORY [Chap.V Short-circuiti~g plug ______ L_OS_S-=:Yt~I:k_.:::_ A ~ffi~~~_-~:I-d o._.---I FIG.11.2.Lossylineterminating a coaxialline. (15)LossyLine.Ifthelossesarenotprimarily confined toathindiskof relatively highleakageconductance buttoaratherlongsectionofline oflengthd(Fig.11.2)withanimperfect dielectric, formula (1)isalso applicable. Letitbeassumed thatthelossysectionisterminated ina conducting plug,sothat6~=PA+jCP~==O.Inthiscasecoth 6~==«l, sothat(1)reducesto 6'=1In1+171tanh'rId(10) B2"1 -171tanh'rId Sincethelossysectionistobequite long,itispossibletohavetheattenu ationperunitlengthmoderately low, sothatChap.II,Sec.13,formulas (1) to(9),apply.IfT/wlisnegligible compared withg/wc,Chap.II,Sec.13,formulas (46b),areapplicable. In eithercaseSec.6,Eqs.(11c,d),applyintheform Te=Te(1+jlPr) Te=RelPr=lPel-lPe==-1.-(~-r!.)(11)ReI 2wClC Itisassumed thatlP;«1. Inordertoseparate realandimaginary partsin(10),notethat 1 -e-2alde-i2{hd 1 -0 tanh'rId=1+e-2alde-i2{jld =1+0 (12) where 0=5r+j5i==e-2alde-i2{jld =e-2a1d(cos2f31d-jsin2f31d)(13) Forsufficient attenuation itisnecessary thatdbelongenoughsothat 5;==e-4a1d«1aId~0.6 (14) If(11)and(12)aresubstituted in(10)andonlytheleadingandfirst ordersmalltermsareretained, thefollowing resultsareobtained: • 1ITe+1+5r{Te -1) PB=2"n (Te -1+5rTc+1) CP~==jtan-1 5i(Tc-1)+lPr_jtan-15i(Tc+1)+lPr)(16) Tc+1+5r(Tc-1) Tc-1+5r(Tc+1 Neglected termshavethecoefficients OrlPr,OilPr,lP;,and 5~. Inordertohavethemainlineexactlymatched tothelossysectionatB, itisnecessary thatPBbecome infinite. Thisrequires thefollowing condition: 1 -T5r=e-2a1dcos2f31d=__ e (17) 1+Te Bysolvingthisexpression fore2a1dandthentakingthelogarithm ofboth sides,thefollowing equation isobtained: Tc=tan-1Te+Incos2f31d (18) Sec.11] DISCONTINUITIES ANDNONUNIFORMITIES 361 Alternatively, from(17), Re1 - ~r1 -e-2a\dcos2{3Id Te=ReI=1+~r=1+e-2a\dcos2{3Id(19) Evidently ReImustbesomewhat greaterthanRedepending onthelength doftheshort-circuited lossysectionoflineanditsattenuation constant alandphaseconstant {31. CoaxialResistor. Sincetheresistive-disk termination mustbeX/4in lengthandthelossylineconsiderably longer,botharephysically cumber someexceptatveryshortwavelengths. Thisisnottrueofthecoaxial Z~:zce:____ --..;':;--__z._c_r__ j%:! I I I -.lSpf4- (b) FIG.11.3.Coaxial resistorterminating acoaxialline. resistor,1l6,128 whichmaybequiteshort(lessthanO.D.inlength)and yetmaintain astanding-wave ratioaslowas1.01overawidefrequency band. Initssimplest formthecoaxialresistorconsists ofashortsectionof coaxialtransmission lineoflength Srterminated inashortcircuit,as showninFig.11.3a.Theinnerconductor ofthissectionconsists ofa cylinder coatedwithalayerofresistive material. Thiscoatingcanbe madesothin(e.g.,oftheorderof10-6mm)thatskineffectisnegligible anditsresistance perunitlengthisvirtually independent offrequency upto10,000Me/sec.Itfollowsthatthed-cresistance Roofthiscylin derisalsoitshigh-frequency resistance inthesensethattheresistance perunitlengthoftheinnerconductor isrr=Ro/srindependent ofthe frequency. Theouterconductor maybeassumed tobelossless,sinceits contribution totheresistance perunitlengthisnegligible. Thecharac teristicimpedance andthepropagation constants ofthecoaxiallineform- 362 TRANSMISSION-LINE THEORY [Chap.V ingtheresistor(subscript r)andofthemainlineare Zcr=II;/1-JrlrZc=Rc=I~ (20)~~~ Wr ~c "(r=ar+J{jr=JwYl,.cr~1 -J:lr"(=J{j=JwVi£(21) wherethelineconstants aredefinedintheusualmanner. Forsimplicity themainlineisassumed tobelossless. Notethatintheabsenceofmag neticanddielectric materials lc=lrcr=l/v~,sothat {j=wYl,.cr=wVlC Letitberequiredthattheresistance Roofthecoaxialcylinder bemade equaltothecharacteristic resistance ofthemainline.Thatis, Ro=Rc Forconvenience, letthefollowing ratiofactorbeintroduced: rc=rte:.=Rc(rr=0)=Ro(rr=0)~Cf,.Rer Rer With(23),(20)and(21)maybeexpressed asfollows: Zer=~r;_J~=1~{jsr-J Rcre~.L {jSryre{jSr rc Jvr:fj8,.~~."(= --J8rre(22) (23) (24) (25) Theinputimpedance oftheshort-circuited coaxialresistoroflengthSris Zr=Zcrtanh'"(rSr.Thenormalized impedance terminating themainline referredtoZcis ZrZerh 1 ~Sr .h(._;;::-r.;::~~.)Zir=Z.=-z.tan"(rsr=_j- - -JtanJVrc{jSr - -J c c vre{jSrrc rc (26) Sincethecoaxialresistoristobekeptshortcompared withthewave length(Sr~O.IX),itfollowsthat{jSrislessthan1,sothat(26)maybe expanded inpowersof{j8r.Thisleadstotheformula116 Zir=1+m+In (27) where _t)22(222)+t)44(2682+624)+m - fJ8r3"-15rc fJ8r5-315re2,835rc _{j8r[112+t)22(122+174)n-r:-3"refJ8r3"-5rc315rc +44(2342+6241,3826)+ {jSr15-105rc567rc-155,992rc(28a) •••](28b) Sec.11] DISCONTINUITIES ANDNONUNIFORMITIES 363 (29)Inpractice, r~liesintherange4~r~~4.8,inwhich(27),with(28a) and(28b),isinerrorbylessthan0.005ifsr/X~0.1.Ifmandnare small,thestanding-wave ratioonthemainlinewhenterminated inZiris s=1+r==1+Vm2+n21 -r wherer=I(ZIr-l)/(zir+1)1.Ifm,n,andSareplottedasfunc tionsof{3srorsr/Xwithr5asaparameter, itisfoundthat,intherange o~Sr~0.1,mrisesmostslowlywhenrc=0andthattheinitial slopeofnGanbemadetovanishwhenrc=y"3'.Theloweststanding waveratiooccursverynearlywhenrc=0,andthis,then,isthe optimum valueforthesimplecoaxialresistorshowninFig.11.3a.How ever,evenwhenrc=V3,thestanding-wave ratioincreases withsri>" fromunityatSr=0to1.11whensr/X=0.1. Thecondition rc=0isthatforwhichthereactive partnofthe normalized terminal impedance Zirhasazeroinitialslope.Clearly, if thereactance couldbecompensated byothermeansandrcwereadjusted nearthevalue0,forwhichmin(27)isvirtually zerointherange o~sri>"~0.1,thecondition ofmatchcouldbeimproved greatly,and thestanding-wave ratiokeptmuchnearer1asXisvariedinthisrange. Onemethodofaccomplishing thisistoundercut theinnerconductor of themainlineforalengthSe,asshowninFig.11.3b.Byapplying the methodoutlined inSec.6,itcanbeshownthatanormalized impedance Zir=1+m+jnterminating themainlineontherightoftheundercut section(characteristic impedance Zce)givesanormalized impedance ZIB=ZBIZCterminating themainlineontheleftoftheundercut section. rrhisisgivenby ZIB==(1+m)[1+~~tan{3se+(1-h)tan2{3se] J[(1+m)2-n2 ]I +jtn+re- retan{3se(30) wherere;:RcelRc>1.Theformula isagoodapproximation when re>1.5,m<0.1,n<0.5,andtan{3se<0.3. Although boththeresistive andreactivepartsoftheimpedance termi natingthemainlinearealteredbytheaddedundercut section,thechange inthereactive partismuchgreater. Thus,byadjusting theunder cuttingtomakethereactance vanishandselecting rctomakethenormal izedresistance asnear1aspossible, averylowstanding-wave ratiomay beachieved. Ingeneral,thismeansavalueofrcslightlygreaterthanvS-. Thereactance vanishes when Se1t-1 -n- = - anX211" re-[(1+m)2-n2J/re(31) 364 TRANSMISSION-LINE THEORY [Chap.V andthenormalized impedance oftheresistorbecomes ZIB=rB=(l+m)(I-r;_(1;2m)2+n2 - 2[Te2_(1~2m)2+n2rl(32) Although Temaybechosenfreelyintheoryandshouldpreferably belarge, practical limitations usuallyrestrictittotherange1.5~Te~2.Inthis mannerastanding-wave ratioof1.11forasimplecoaxialresistor(Fig. 11.3a)maybereducedto1.01whensri>..=0.1. Theanalysis oftheundercut linewhichculminates in(32)ignoresthe junction effectsresulting fromnonuniformity intheparameters ofthe lineovershortdistances oneachsideoftheundercut section.Itis showninSec.14thatachangeinradiusoftheinnerconductor ofa coaxiallinemaybeexpressed intermsofuniform-line theoryoneach sideofthejunction, provided asmalllumpedcapacitance ofappropriate magnitude isconnected inparallelwiththelineateachdiscontinuity. Sincethesecapacitances havebeenignoredinderiving (32),afurther correction isrequired. Thisinvolves smallchanges inTcandthedefi nitionofmodified valuesofmandn.Thedesignofacoaxialresistor thattakesaccountoftheseeffectsisgivenintheliterature,115 aswellas descriptions ofothermethods thanundercutting forcompensating the reactance. 12.ClosedandOpenEndsasReactive Terminations inTwo-wire and CoaxialLines.49,104Anidealclosedendforatransmission linehaszero impedance, orterminal functions PB=0andepB=1r/2.Terminations ofthiskindarediscussed inChap.II,Sec.21. Byananalysis ofaclosedrectangle theapparent (measurable) induct anceLBOofabridgeisdetermined inChap.II,Sec.20,asthesum LBO=LB+LTofanidealinductance Laofthebridgewhenterminating afictitious linethatisuniform evenintheterminal zoneandaninduct anceLTthatcorrects forthedifference between theactualnonuniform inductance intheterminal zoneandtheconstant valueassumed incon ventionalline theory. Theidealinductance Laofthebridgeisgivenby Chap.II,Sec.20,Eq.(9).Thelumpedcorrective inductance LTofa straightbridgemayalsobeevaluated bythegeneralmethoddescribed in Chap.II,Sec.4,whereLTisdefinedbyChap.II,Sec.4,Eq.(3),with al(w)=0(sincethereisnoinductive coupling). Thatis, whereLT=Ld[lg(w)-19]dw le(w)=ko(w) o 21rJ'(1) (2) Sec.12] DISCONTINUITIES ANDNONUNIFORMITIES 365 ko(w)=sinh-1~-sinh-1'!!!b+In~a a 1ble=le(w~(0)= -In-o0 rJla(3) (4) Theintegration of(1)iseasilycarriedoutusing(2),(3),and(4),with d2»b2•Theresultis b-aLT=--2rJl(5) 0.3oL.-__.l..-__ ~__ .... o1.0r------r-----r--.., 0.1 0.2 Radiusasofwirebridge inem FIG.12.1.Apparent terminal inductance ofawirebridge(Tomiyasu).inagreement withChap.II,Sec.20,Eq.(10).Theapproximate general methodbasedonChap.II,Sec.4,Eq.(3),isthusverifiedinonespecial case. Anexperimental determination oftheapparent terminal inductance L.a=L.+LThasbeenmadeforthreevaluesoftheradiusa.ofthe terminating wirebridge,namely, all=0.0794, 0.1588, and0.2382em. Theratio(L.+LT)/lgasafunction ofa.isgiveninFig.12.1.The experimental dataarethoseof5 Tomiyasu; thetheoretical values ~o havebeendetermined usingChap.J0.5~---I----:~~----f II,Sec.20,Eqs.(9)and(10).The+ CI)agreement isgoodforthesmaller 104 radii.Forthelargerradiitheas sumption ofrotational symmetry foreachconductor, withthecur rentconcentrated alongtheaxisin determining thevectorpotential, is notagoodapproximation nearthe corner. Themeancurrentfollowsasomewhat shorterpaththatreduces theeffective inductance L.a.Itisthisdecrease, notincluded inthetheo reticalcurveinFig.12.1,whichaccounts forthesmallermeasured value of(L.+LT)/l8. Anidealopenendforatransmission lineisaninfiniteimpedance, with P.=0andcf>1I=O.Inpractice, anopenendisobtained byproviding noconnections between theconductors. Butsuchanopenendisnot idealinthesensethattheapparent (measurable) impedance terminating thelinedoesnothaveXlla=00orcf>.a=0,eventhoughthetheoretical reactance ofthetermination isXB=00.Owingtotheriseincapaci tanceperunitlengthastheopenendisapproached (Chap.II,Fig.3.2), theuseofconventional transmission-line formulas withXB=00doesnot leadtothecorrectapparent impedance. Thelumpedcapacitance CT required tocorrecttheerrorintroduced bytheuseofCoinsteadofco(w) isgivenbyChap.II,Sec.4,Eq.(4)[withcl>l(W)=1,sincethereisno 366 TRANSMISSION-LINE THEORY [Chap.V coupling toaload],viz., CT=fod[co(w)-co]dw where,forthetwo-wire line,(6) 21rE co(w)=ko(w) withko(w)givenby(3). ForthecoaxiallineCo=Co(w=00)In(b/a)(7) 41rE 21rE co(w)=ko(w) Co=Co(w=00)=In(adal) (8) Asshowninconjunction withChap.II,Sec.1,Eq.(32),ko(w)asgiven in(3)forthetwo-wire lineisalsoagoodapproximation forthecoaxial lineifbisreplaced bya2,theinnerradiusoftheshield,andabyaI, theradiusoftheinnerconductor. Itfollowsthatko(w)inChapII, Fig.3.1,maybeusedforthecoaxialline. CTin(6)hasnotbeenevaluated inclosedform,butspecificcasesmay becomputed numerically. Asanexample, consider atwo-wire linewith a=0.1588cmandb=2cmoracoaxiallinewithal=0.1588cmand a2=2cm.Forthetwo-wire lineCo=10.96p.p.f/cm,andforthecoaxial lineCo=21.92p.p.f/cm. ForbothlinesCT=0.416co.Itfollowsthat, insofarasmeasurements onthelineatsomedistance fromtheendare concerned, anopenendisnotequivalent toaninfinitereactance but rathertothereactance ofasmallcapacitance CT. Anapproximate simpleformula forCTmaybeobtained intheform ofaseriesininversepowersofthequantity 2In(b/a)forthetwo-wire line[or2In(adal)forthecoaxialline].Theapproximate formula is derivedusingko(w),asdefinedin(3),inthefollowing equivalent form: ko(w)=(1-15)(2ln~) (9) where 15=In(w+Vw2+b2)-In(w+vw2+a2) (10) 2In(b/a) With(7)and(9)itfollowsthat co(w)-Co=_15_=5+52+ (11) Co 1 -15 CT=Cofod15dw (12) Withd=lOb(sothatb2andA2maybeneglected compared withd2)Since 15isalwayslessthan1,thedenominator inthemiddletermin(11) maybeexpanded inseriesasindicated ontheright.Iftheleadingterm isretained asafairapproximation if2In(b/a)issufficiently great,it followsfrom(6)that Sec.12] DISCONTINUITIES ANDNONUNIFORMITIES 367 theintegration of(12)with(10)leadstothefollowing simpleformula forthetwo-wire line: .co(b-a)1rE(b-a) CT=2ln(b/a)=2[ln(b/a)]2 (13) (Thesameformula appliestothecoaxiallineifa2issubstituted forb andalfora.)Fora=0.1588cmandb=2cm,theapproximate for mula(13)givesCT=0.365co,whereasthemoreaccurate valueobtained bynumerical integration isCT=0.416co. Inthiscaseb/a=12.6.The accuracy of(13)isimproved asb/aisincreased. CTwasmeasured onatwo-wire line(a=0.1588cm,b=2cm)with openendandbyTomiyasu usingthreetypesofends:(1)flatclosedends, (2)hemispherical ends,and(3)opentubing. Themeasured capaci tancesintheformCT/coarelistedinTable12.1;thesmallest valueis forhemispherical ends.Thisistobeexpected sinceboththeflatend surfaceandtheinteriorofthetubingarecharged neartheends.The theoretical valueisCT/CO=0.42cm.Ifthelengthoflineismeasured alongthesurfaceinsteadofalongtheaxis,CT/co=0.58cm,andthis corrected valueisinverygoodagreement withexperiment, asshownin Table12.1.Inanyevent,small errors oftheorderofmagnitude ofthe radiusaoftheconductors aretobeexpected inthequasi-one-dimensional analysis underlying theseresults. TABLE12.1.CTFORTWO-WIRE LINE Typeofend CT/eo,em Flatclosed.. . . . .. .. . . . . .. . . .. . ..0.60 Hemispherical. . .. . . .. . . . . .. .. . ..0.58 Opentubing 0.65 Theoretical. .... .. . . . . .. .. . . ... ..0.42 Theoretical (corrected). ...... .. . ..0.58 Itisinteresting toexamine thedistributions ofscalarandvector potential difference, chargeperunitlength,andcurrentnearanopen end.Aqualitative pictureisreadilyobtained iftheconductors are assumed tobeperfect.ItfollowsfromChap.I,Sec.4,Eqs.(9a,b), thatbothYew)andW.(w)satisfytheone-dimensional waveequation, sothatboththesefunctions aresinusoidally distributed evenintheter minalzone.Itfollowsthattheknowndistribution ofYew)outsidethe terminal zone,asobtained withatermination CT,maybeextended into theterminal zone.ThusinFig.12.2bthefunction Yew)isshownasa cosinecurvewithitsmaximum atadistance CT/cobeyondtheactual endoftheline.Foradissipationless lineitfollowsfromChap.I,Sec.4, Eqs.(6a),(7),and(8),that W.(w)=;:.aV(w) (14) JWaw sothatthecurveforW.(w)isacosinecurvelikethatforV(w),but 368 TRANSMISSION-LINE THEORY [Chap.V (15) (16)Iz(w)==Wz(w)+~oPo(w)Yew) 19(w) )wlg(w) Alternatively itisgivenbytheequa tionofcontinuity:Z<;QO(WI Vo(w) Io(w) (a) Wzo(w)shiftedaquarterwavelength alongtheline(andaquarterperiodintime). Suchacurvewithanarbitrary amplitude isgiveninFig.12.2b. Theleadingterminthedistribution ofchargeperunitlengthis obtained fromtherelationq(w)-V(w)Co(w) ,whereco(w)risesrapidly fortheterminal zonefromaconstant Coto2coatw=0,asshownin Chap.II,Fig.3.2.Anestimated curveforq(w)issketched inFig.12.2, withamplitude scaledtoequalthat ofYew)forconvenient comparison. Thedistribution ofcurrentmay bederived fromChap.II,Sec.3, Eq.(14),which,with(14),maybe expressed asfollows: Actually itisreadilysketched simply bynotingthatoutsidetheterminal zoneIz(w)=Wz(w)/le andthatat w=0itmustvanish. Anestimated curveofI(w)isshowninFig.12.2,withamplitude arbitrarily setequal tothatofW(w). Itistobenotedthatthebehavior ofthepotentials, current, and chargeperunitlengthinFig.12.2bdiffersgreatlyfromthatatanideal openend,asshowninFig.12.2a. 13.Junction ofTwoOpen-wire LineswithConductors ofDifferent Radii.Anopentwo-wire transmission lineconsists ofaleft-hand part extending fromthegenerator atz=0toz=Szandaright-hand part extending fromz=Sztoaloadatz=Sz+ST=s.Thedistance between thecentersofthewiresinbothpartsisb;theradiusoftheconductors in theleft-hand partisaz,andthatintheright-hand partisaT'Without restricting thegenerality, letaz~aT'ThelineisshowninFigs.13.1 and13.2. Consistent withtheconditions imposed ontheuniform two-wire line inChap.I,Sec.4,letthefollowing inequalities beassumed satisfied:I I:q(w) I ~V(w) ;~(W) w4 CT (b) CO FIG.12.2.Distributions ofscalarand vector potentials, charge perunit length,andcurrentneartheendofa two-wire line.(a)Idealdistributions derivedfromuniform-line theory. (b) Actualdistributions thattakeaccount ofendeffect. LBbl2«1 b2«Sfb2«S~(1) (2) Sec.13] DISCONTINUITIES ANDNONUNIFORMITIES 369 Forthepresent, lettherestriction (3) beimposed. Themoregeneralcaseinwhichbisnotsorestricted is introduced laterinthissection. Atdistances totheleftandrightofthejunction JJeachpartofthe transmission lineisuniform, withpropagation constants 'Yland'Y,.and <D/4o--w'---+I t-W --..l Ral);F,..----u'--+ fb!ar du' ,< 2a,...r/'"---ff2ar *dw'I'"'J l <I/'"I /Rbl..........Jlbr b /......... '",jJ ......... ~i .1~ dw' du' F, w.I~u z-s, FIG.13.1.Junction oftwotwo~wire lineswithconductors ofdifferent radii. z=o tZ~81 14-A/2+f4-A/2-.j 1J2~ IJZcr }Zs Zcl 1J2 !(Zll-Z12), !(Zll-Z12) 1 2 1 2 1 12(Zll-Z12) 2(ZU-Z12) FIG.13.2.Junction oftwotwo-wire linesandequivalent Tnetwork ofthejunction. characteristic impedances ZelandZeroAsthejunction isapproached. fromeachside,theinductance andcapacitance perunitlengthbecome functions ofthedistances wandufromthejunction owingtochangesin cross-sectional dimensions andtheexistence ofthenewchargeable and current-carrying annular surfaces. Inaregionextending adistance d,......,lOboneachsideofthejunction, variable parameters 'Y(w)andZe(W) mustbesubstituted fortheconstant parameters 'YandZe. Thepurpose oftheanalysis istoreplacetheactualcircuit,involving regionsinwhichuniform-line theoryisnotapplicable, withanidealized circuitthatiselectrically equivalent insofarasmeasurements onthelines outsideajunction zoneoflength2dareconcerned. Thisisequivalent to replacing thesectionoflineoflengthAextending from11to22inFig. 13.2byanequivalent Tnetwork, asshown.Ifthelinewereuniform 370 TRANSMISSION-LINE THEORY [Chap~V between 11and22,theseriesandshuntelements wouldbegivenby Chap.III,Sec.12,Eqs.(7c,d).Neglecting lossesinthislength,theele mentsarei(Zll-Z12)=0andZ12=00.Thatis,thesectionofloss lessuniform lineoflengthAhasnoeffect;itmaybeincluded oromitted asdesired. Evidently, bydetermining theinductance andcapacitance perunitlengthforthelinesections between 11and22,subtracting from thesetherespective constant valuesfortheuniformlines,andintegrating thedifferences overthejunction region,theseriesandshuntelements of theequivalent Tsectionmaybeobtained intheformZl1-Z12=jwLT andZ12=-j/wC T,whereLTandCTareevaluated usingthescalarand vectorpotential differences. Following thegeneralmethodofChap.I,Sec.4,andChap.II,Sec.1, thescalarpotential difference between pointsFzonthetwoconductors (Fig.13.1)atadistance Wtotheleftofthejunction maybeevaluated asthesumofthefollowing threeparts: Vz(w)=Vll(w)+VZr(W)+VZj(W) (4) whereVll(w)isthecontribution fromthechargesintheleft-hand section withachargeperunitlengthql(W'),Vlr(w)isthecontribution fromthe chargesintheright-hand sectionwithachargeperunitlengthqr(u'), andV1j(w)isthecontribution fromthechargesonthetwoannularsur facesatthejunction onwhichthechargedensityisftj(r',O'). Theaxialcomponent ofthevectorpotential difference ismadeupof onlytwoparts,asfollows: Wzl(w)=Wzll(w)+Wz1r(w) (5) wherethefirsttermontherightisduetothecurrentIzl(w')totheleft ofthejunction andthesecondtermisduetoIzr(u')totherightofthe junction. Theprimarily radialcurrents [rj(r',0')ontheannularsurfaces contribute nothingtotheaxialcomponent ofthevectorpotential differ ence,sincetheyareperpendicular toit. Theseveralcomponents are Vll(w)=2~~foBlqz(w')Pll(w,w') dw' (6a) Vlr(w)=2;~foBrqr(U')PZr(w,u') du' (6b) Wzll(w) =-21(BIIzl(w')Pll(w,w') dw' (7a) 7r'JIJo Wz1r(w) =2-.!--(BrIzr(u')P1r(w,u') du' (7b) 7r'JIJo e-i(}Ral e-i(}Rbl 1 1 where Pu(ww')=-----==~- - (8a), R azRbZRalRbi e-i(}Rar e-i(}Rbr 1 1 Plr(w,u') =-----==- - - (8b)RaT RbrRaTRbr Sec.13] DISCONTINUITIES ANDNONUNIFORMITIES 371 AsshowninChap.II,Sec.1,theapproximations ontherightin(8a,b) arejustified if(1)issatisfied. Thedistances are Raz=v(w-W')2+at Rar=V(w+U')2+alRbZ=V(w-W')2+b2 Rbr=V(w+U')2+b2(ga) (9b) Thecontribution toVz(w)bythechargesontheannularsurfaces is where1/.211"fal VZj(W)=27r~0}arnj(r',fJ')PZj(w,r',fJ')r' dr'dfJ' PZj(w,w')=_1 1_ RaZiRbli(10) (11) Raliisthedistance fromthepointFltoanelement ofchargenj(r',fJ')r' dr'dfJ'ontheannular surfacebelonging tothesameconductor asthe pointFz;Rbliisthedistance fromFltothecorresponding elementofcharge -nj(r',fJ')r' dr'dfJ'ontheotherannularsurface. Inordertoevaluate thepotential differences, itisnecessary toknow thecurrents andchargesonbothsections oftransmission lineandonthe annular surfacesatthejunction. Sincethelinesareassumed tohave lowlosses,theregionnearthejunction maybetreatedasiflosslessin determining LTandCT,sothatcurrents aswellaschargesareconfined tosurfacelayers. Aconvenient procedure forevaluating theexternal inductance le(w)andcapacitance c(w)perunitlengthistoresolvethe problem intosymmetrical (evencurrents andvectorpotentials, odd chargesandscalarpotentials) andantisymmetrical (oddcurrents and vectorpotentials, evenchargesandscalarpotentials) parts,asinChap. III,Sec.12.Theevenandoddproperty iswithrespecttotheplane throughthejunction perpendicular totheline.Thesymmetrical prob lemisequivalent toproviding ashortcircuitat22.Theantisymmetri calproblem isequivalent tohavinganidealopencircuitat22. Consider firsttheevencurrents andoddchargeswiththeidealshort circuitastermination. Inthiscasethecurrents mustbecontinuous alongtheconductors atthejunction, sothat Izl(w'---+O) =-Iri(r'---+al) (12a) Izr(u'---+0)= -Iri(r'---+ar) (12b) whereIrlr')istheoutward radialcurrentontheannular surface.By integrating the()component ofthemagnetic fieldaroundoneofthecon ductorsatthejunction, itisreadilyshownbysuccessive application of theMaxwell-Ampere theorem9thatthetotalaxialcurrententering the junction fromtheleftmustequalthecurrentleavingontheright.It followsthat(12a)and(12b)become Izz(w'---+0)= -Irj(r')=Izr(u'---+0) (13) 372 TRANSMISSION-LINE THEORY [Chap.V Important conclusions followdirectly from(7b)with(13).Since Plr(w,u') isthesameasitwouldbewithar=aZ,itfollowsthat,with (13),(7b)isexactlywhatitwouldbeiftherewerenochangeinthe radiusoftheconductors andnoannularsurfaces. Therefore, insofaras theinductance perunitlengthisconcerned, ithastheconstant value l1fromz=0toz=8landthelikewise constant value l~fromz=8zto z=8l+8r=8.Notethat e1bibII=-In- l~= -In- (14) 'lrPal 'lrPar Thusthereisnoinductive junction effect,andnolumpedinductance LTisrequired oneithersideofthejunction; Z11-Z12=JwLT=O. Thismeansthattherequired lumpednetwork canconsistonlyofashunt element. Theproblem ofoddcurrents andevenchargeswhentheloadsideof thejunction isterminated inanopencircuitinvolves continuity ofcharge intheform ql(w'~0)=[2'1rr'n(r',O')]r'-+az (15a) qr(u'~0)=[2'1rr'nCr',0')]r'-+ar (15b) However, sincetheradialcomponents oftheelectricfieldnearthejunc tiondependonthechargesontheannular surfaces aswellasonthe cylindrical conductors, theapplication ofGauss'stheorem doesnotlead totheconclusion thatql(w'~0)isequaltoqr(u'~0).Ontheother hand,sincethetransverse dimensions oftheconductors onbothsidesof thejunction aresmall,asrequired by(2)and(4),thescalarpotential difference between thetwoconductors ofthetwo-wire linemustbethe sameonbothsidesofthejunction. Therefore Vl(w~0)=Vr(u~0) (16) (17b)(17a)whereVl(w)isgivenby(4),with(6a),(6b),and(10),andVr(u)isgiven bycorresponding expressions, withuandwandsubscripts randlinter changed. (Notethatinthevicinity ofthejunction theamplitude of theperiodically varying electricfieldisdistributed essentially asifthe twoconductors weremaintained ataconstant, Le.,electrostatic, potential difference. ) Outsidethejunction zonewhereuniform-line theoryisvalid,theratio ofchargetovoltageoneachlineis ql(W) 'irE Vl(w)=COl=In(blal) qr(W) 'irE Vr(U)=COr=In(blar) Notethat,withal>ar,itfollowsthatCOl>COr.Itissatisfactory to usethesevaluesasapproximations inthejunction regioninorderto Sec.13] DISCONTINUITIES ANDNONUNIFORMITIES 373 (19) (20) (21)determine CTasacorrection forconventional linetheory. Inother words,theuncorrected approximate distribution ofchargeisusedin ordertodetermine thecorrection. Thismeansthat,inevaluating CT fromVz(w),(16)isappliedin(17a,b)inordertodetermine qr(U')for usein(6b).Thevalueis qr(U')==ql(W')COr=ql(W')In(bla,) (18) Co In(blar) Alsoletitbeassumed thatthechargedensityontheannularsurfaces is approximately rotationally symmetrical foreachconductor, sothat nCr',8')==n(r').Thisfunction satisfies theconditions ofcontinuity of charge(15a,b)ifitisapproximated asfollows: .(,8')==.(')==q,(w'=0)In(bla,) nJr,nJr 21rr'In(blr') Incalculating thepotential atanypointF,(Fig.13.1)justoutsidethe surfaceofthecylindrical conductor duetothechargesallalongthecon ductor,theformulas (6a,b)basedonthefundamental integral[Chap.I, Sec.3,Eq.(30a)]areused.Asindicated inthefootnote discussing Chap.I,Sec.3,Eq.(30a),thisfundamental integral forthepotential atthesurfaceofacylindrical conductor actually impliesthattheentire chargeisconcentrated inathinlinealongtheaxisofeachconductor ratherthanonthesurface.Itisagoodapproximation oftheactual distribution alongthesurfaceifpossibleerrorsinthelengthofthetrans missionlineoftheorderofmagnitude ±aareacceptable. Asimilar approximation involving errorsinlengthofthesameorderofmagnitude maybemadeinevaluating thepotential duetothecharges onthe annularsurfaces. Thatis,thetotalchargeoneachannularsurfacemay betreatedasifconcentrated atthecenterindetermining thepotential atFl.Thistotalchargeisobtained byintegrating (19)overtheannulus: f.al , , , blaldr'lbQj=21rrnj(r)dr =ql(w=0)bln--1('Ib) ar alarnr Byachangeofvariabletoy= -In(r'Ib),theintegralin(20)becomes a tabulated exponential integral.tTheresultmaybeexpressed asfollows: lrEKbQj=--ql(w=0) COL where COLisasdefinedin(17a)andwhere (22) t"Tables ofSine,Cosine,andExponential Integrals," vol.I,W.P.A., National BureauofStandards, 1940. 374 TRANSMISSION-LINE THEORY [Chap.V Theexponential integral is Ei(x)=f-Xe-U.du 00u(23) (24)Since,incalculating Vlj(w)from(10),theentirechargeistobetreated asifconcentrated ontheaxisatW=u=0,(10)reducestothefollow ingsimpleintegrated form,whichisagoodapproximation exceptwhen Wisassmallasal: V-(w)-Oi(1 _ 1) l)-21rEyw2+alyw2+b~ Thepotential difference between pointsFl(Fig.13.1)isgivenby(4) intermsof(6a),(6b)with(18),and(24).AsshowninChap.I,itisa goodapproximation tosetq(w)==q(w')inthejunction zone.Bythe sametokenandinthesamerange,ql(w=0)==ql(W).Theresultsare V( ).ql(W)fo81P(')d,.()[11]IIW=-2- IIw,wW=qlW- --(-) 1rE0 COlCllW VZr(W).ql(W)fo8T p(')d,.()[COr]=-2- lrW,UU=qlW--(-)n0 C~IlW VZj(w)==ql(W)Kpl(W) COl(25a) (25b) (25c) wherethevariable capacitance perunitlengthCH(W)isdefinedby 21rECll(W)== . (26a)In[(w+yw2+b2)/(w+vw2+af)] andwherethedimensionless function Pl(W)is b(11)Pl(W)==2yw2+al-yw2+b2 (26b) Theaddition of(25a),(25b),and(25c)toobtainVl(w)andthedefinition ofCz(w)astheratioofchargeperunitlengthtovoltagegive _1_==View)=~[1_COl-COr+KPl(W)] (27) Cl(W) ql(W) COl Cll(W) Clearly, wheneitheroftheconditions w2»b2orar=azissatisfied, Cl(W)==COl. Thecorresponding expression forcr(u)ontherightofthejunction is obtained byinterchanging subscripts landrwheretheyoccurexplicitly in(27)andsubstituting uforw.[Notethatthesubscripts occurring in Kasdefinedin(22)shouldnotbeinterchanged.] Thedesiredformulais _1_==Vr(u)==![1+COl-COr+KPr(U)] (28) cr(u) qr(U) COr Clr(U) (29a)Sec.13] DISCONTINUITIES ANDNONUNIFORMITIES 375 whereClr(U)andPr(U).aregivenby(26a)and(26b),withtheindicated changes andsubstitutions. NotethatClr(U)andPr(U)differverylittle fromCu(w)andPI(W)e~ceptwhenbisassmallasa,. Thetotalerrorincapacitance madealongtheleft-hand sectionofline ifCOlisusedinplaceofthetruevalueCI(W)isgivenby CTI=fad[CI(W)-COl]dw Thecorresponding errorontheright-hand sectionoflineis CTr=fod[cr(U)-COr]du (29b) In(29a,b)dhastheorderofmagnitude lOb.Thetotalshuntcapaci tancerequired tocompensate fortheuseofuniform-line theoryisthe sumCTl+CTr.Tothismustbeaddedthecapacitance between the twoannularringsCTj.Thismaybeobtained approximately asfollows: Bydefinition,CT'=Qj ==7rEKbq,(w=0) JV,(w=0) COlV,(w=0)(30a) Intheevaluation ofthesmallcapacitanceeTjitisadequate toassume thatq,(W)andV,ew)arerelatedaccording touniform-line theory,namely, q,(w)/V,(w)=CO,sothat (30b) whereKisasin(22). Thetotallumpedcapacitance required intheequivalent network in Fig.13.3is (31) wherethethreetermsontherightareasdefinedin(29a),(29b),and (30b).Notethat,whena,isgreaterthanar,enandeTjarepositive, whereasCTrisnegative. Theequiv- alentcircuitshowninFig.13.3 involves onlyuniform sections of1/,zc/ feT1r,zcr lineandalumped capacitance to correctfortheover-all errormade FIG.13.3.Uniform lineswithCTtocor- rectforthejunction-zone nonuniinassuming constant capacitances formities. perunitlengthatallpointsinboth lines,including theregionnearthejunction whereactually theyare variable. Sincearislessthanai,thelinetotheleftofthejunction mustbehave inamannerintermediate between itsbehavior forar=0andthatfor ar=a,.IthasbeenshowninSec.12thatnearanopenend,whichis equivalent tvar=0,thecapacitance perunitlengthandwithitthe chargeperunitlengthincrease sharply. Therefore itistobeexpected thatasimilarbutsmallerincreasemustoccurasthejunction withaline 376 TRANSMISSION-LINE THEORY [Chap.V (32)ofsmallersizeisapproached. Account ofthisincreaseinchargemaybe takenbyincluding alumped positivecapacitance CTlattheendofauni•formsectionwithuniform capacitance perunitlength COL. Sincetheuniform chargeperunitlengthontheleft-hand sectionof largerdiameter isgreaterthanontheright-hand sectionofsmallerdiam eter,andsinceitisfurtherincreased nearthejunction, asjustexplained, itisevidentthatthecontribution tothepotential difference ontheright bythechargesontheleftofthejunction isrelatively greaterthanwhen ar=al.Accordingly asmallerchargeperunitlengthisrequired onthe right.Thisdecrease inchargecorresponds toadecrease incapacitance perunitlengthbelowtheuniform value COr.Iftheuniform valueisto beused,acorrection intheformofanegativelumpedcapacitance CTris required. Theactualevaluation ofCTlandCTrhasnotbeenachieved butis unnecessary, sinceitisshowninSec.14thatCTforthetwo-wire line maybedetermined fromCTforthecoaxialline,forwhichgraphsand tablesareavailable. Abetterapproximation oftheratioofchargeperunitlengthonthe twolinesnearthejunction isobtained byequating Vl(w==0)in(27) toVr(u==0)in(28).Theresultis • COr+Kb(l_!) qr(u=0)_COrCOL2alb ql(w==0)-CoCOL+Kb(!_!) COr2arb I I!4---b,.--~ ..!Since COL> COrandal>ar,itisclearthatthecorrection factor[thefinal factorin(32)]islessthan1.Since qr(u==0)islessthanthevaluefora uniform lineandql(w==0)isgreater thanthevalueforauniform line,this isasitshouldbe. Iftheright-hand partofthecondi tions(3)isnotsatisfied inthesense thatalisnotnecessarily verysmall compared withb,theresultsobtained abovemaybemodified bymakinguseFIG.13.4.Crosssectionoftwotwo- wirelinesatjunction plane. oftheeffective radiusbe,definedin Chap.I,Sec.7,Eq.(40a).Sincethe surfaces ofallconductors mustbeequipotentials, theireffective separa tionbeinFig.13.4ratherthanthedistances between centersmustbethe same.Thatis, (33) Sec.14] DISCONTINUITIES ANDNONUNIFORMITIES 377 whereblandbrareasshowninFig.13.4.Ifthetworadiiandblare known,brmaybedetermined bysolving(33)togive Ifbllalissubstituted forblal'andbriarforblarintheformulas derived subjectto(3),theybecomeapproximations forusewhenbisnotsuf ficiently greattosatisfy(3),provided (33)issatisfied. 14.ChangeofRadiusinaCoaxialLine.Iftheradiusoftheinner conductor ofacoaxiallinechanges froma,toaratz=81,whereas the outerconductor hastheuniform innerradiusb,asshowninFig.14.1, R---TI>';;-=:::'-:' _ R.t../ T'--_-_-_T-o-;::..-_::-::------2a,-&-------- -------.T---- t2ar ~ IJdU' %=8, FIG.14.1.Junction oftwocoaxiallineswithinnerconductors ofdifferent radii. aproblem arisesresembling thatinthepreceding section. Byconsider ingthegeneralcaseinitsseparate symmetrical andantisymmetrical parts,asinSec.13,itfollowsthatthereisnoinductive correction near thejunction andthatthevaluesoftheexternal inductance perunit length: e1bll=-ln- 271'"Vale1bl=-ln- r271'"Var(1) characteristic ofuniformlinesarevalidtotheleftandright,respectively, ofthejunction.Itmaybeconcluded, asinSec.13,thattheentire corrective networkatthejunction consistsofashuntcapacitance CT. Inordertodetermine CTitisnecessary toevaluate thepotential dif ferenceVl(w)atanarbitrary distance wtotheleftofthejunction ofthe twolinesatz=8,.Thisisaccomplished asfollows: Thepotential <1>1atapointFlbetween theinnerandouterconductors (Fig.14.1)ataradiusrandatadistance wfromthejunction maybe evaluated inthreeparts,viz.,thepotential <l>uduetochargesonthe left-hand sectionoflineofradiusal,thepotential <l>lrduetochargeson theright-hand sectionofradiusar,andthepotential ,jduetocharges ontheannularringatthejunction. Thus (2) 378 TRANSMISSION-LINE THEORY [Chap.V Thefirsttwocomponents aredefinedasfollows: where where"'"( ) •ql(W)/.211"/.00p( I8')dId8''I'llW=47rE 0 0 IIW,W,W211" "'"( ) •qr(U)/.211"/.oop(I8')dId8''I'lrW=-4- lrW,U,u-2 1I"E0 0 11" P('8').1 1IIw,W,= ---RrlR.z P(I8').1 1lrW,U,=-R--Rrr Br Rrl=V(w'-W)2+r2RBl=y'~(W-,o---W----:-:)2-+-S-::2 Rrr=V(U'+W)2+r2RBr=y(U'+W)2+S2 S2=b2+r2-2brcos8'(3a) (3b) (4a) (4b) (5a) (5b) (5c) Theintegrations withrespecttow'andu'in(3a)and(3b)maybecarried outasfollows: /.00p( I8') d I=2I~_ IW+Yw2+S2IIW,W, W n n _/o r W+vw2+r2 /.00Plr(w,u',8')du'=InW+yw2+S2 o W+yw2+r2(Ba) (Bb) Iftheseexpressions aresubstituted in(2)using(3a)and(3b),theinte grationwithrespectto8'whichinvolves thetermIn(sir)maybecarried outjustasinChap.I,Sec.B.Theresultis C>l(W)=ql(W)[21n~_/.211"(1_Cor)InW+w2+S2d81 ]+C>dw)(7) 411"E r 0 COlW+w2+r2211" 1 Informulating (7)ithasbeenassumed thattheentireinnerconductor nearthejunction isessentially anequipotential surface,sothat (8) Itisalsoassumed thatthechargeperunitlengthcharacteristic ofthe uniform linemaybeusedtodetermine thecorrective capacitance CT. Thatis, 211"EC-~-=-.,------,-Ol-In(blal)whereqr(U)•COr ql(W)=COl 211"E COr=In(blar)(9) (10) arethecapacitances perunitlengthcharacteristic ofuniform lineswith radiiarandal,respectively. Sec.14] DISCONTINUITIES ANDNONUNIFORMITIES 379 Since0'occursonlyin82,thepotential difference is Vz(w)==qz(w)[2In!_(1_cor)InW+yw2+b2 41rE az Cozw+yw2+al +(1-cor)/.211"Inw+yw2+86dO']+Vj(W)(11) Coz0W+Yw2+8~21r where 8~=b2+al-2azbcos0' (12a) sg=2b2(1-cos0') (12b) Thelogarithm intheintegral in(11)hasitslargestvaluewhen W=0 andrapidlyapproaches unityasWincreases. However, whenw=0,it isreadilyverifiedbydirectintegration (usingPierce523)thattheinte gralvanishes. Itmaybeconcluded, therefore, thatthelasttermwithin thebrackets in(11)contributes negligibly toVz(w),sothat Vz(w)==qz(w)[21n!!.._(1_cor)Inw+yw2+b2 ]+V/w)(13) 41rE az Cozw+yw2+al Thepotential difference Vj(w)=[<Pi(W)]r=al- [4>/w)]r=bduetothecharges ontheannularringinthejunction planemaybeevaluated asinSec.13, wheretherearetworingsinsteadofonesuchring.Theresultislike Sec.13,Eq.(24),dividedby2,thatis, where(14) (15) whereCozisgivenby(10)andKisasinSec.13,Eq.(22).Bydefining (16b)(16a)41rECH(W)== - In[(w+yw2+b2)/(w+yw2+af)] b(11)Pl(W)=="2yw2+al-yw2+b2 thefinalformof(13)is Vl(w)==_1_=~[1_COl-COr+KPl(W)] (17) ql(W) Cz(W) COl CH(W) Thecorresponding expression forcr(u)ontherightofthejunction is Vr(U)=_1_=~[1+COl-COr+Kp(U)] (18) qr(U)-Cr(U) COr elr(U) r 380 TRANSMISSION-LINE THEORY [Chap.V (19)whereclr(u) andpr(U)areasgivenby(16a)and(16b),withthefollowing substitutions: Uforwandsubscript rforl. Formulas (17)and(18)arethesameinformasSec.13,Eqs.(27)and (28),fortheopen-wire line.However, allcapacitances in(17)and(18) refertothecoaxiallinewithouterradiusbandinnerradiia,totheleft andartotherightofthejunction. Sincetheformulas forthecapaci tancesofacoaxiallinedifferfromthosefortheopen-wire lineonlybya factor2,itfollowsthatCl(W)/COl andcr(w)/CO rarenumerically thesame forthecoaxialandopen-wire linesiftheratiosb/a,andb/ararethesame. Accordingly CTl/COlandCTr/CO r,asdefinedinSec.13,Eqs.(29a)and (29b),arealsonumerically thesameforthecoaxialandtwo-wire lines, withthesameratiosofb/a,andblareAsinSec.13,thetotallumped shuntcapacitance required atthejunction tocompensate fortheuseof uniform-line theoryonbothsidesofthejunction isCTl+CTr•Tothis mustbeaddedthecapacitance CTjbetween theannular ringatthe junction andthecoaxialshield. Thisisgivenapproximately by C T'=Qj=21rEKbq,(w=0) JV,(w=0) COLV,(w=0) whereusehasbeenmadeof(15).Fortheevaluation ofthissmall capacitance itisadequate toassumethatV,(W=0)andq,(w=0) arerelated,aspredicted byuniform-line theory. Thatis, Accordinglyq,(w=0) V,(w=0)=COl CTj==21rEKb (20) whereKisasgivenbySec.13,Eq.(22).Notethat(20)differsfrom thecorresponding formula[Sec.13,Eq.(30b)]byafactor2. Thetotallumpedcapacitance CTrequiredatthejunction ifuniform linetheoryisusedoneachsideis (21) Forthesameratiosb/a,andb/ar,thevalueofCTobtained from(21) forthecoaxiallineisdoublethevalueobtained fromSec.13,Eq.(31), fortheopen-wire line.Sincethelumpedsusceptance BT=wCTforthe coaxiallinehasbeendetermined byavariational analysis beyondthe scopeofthisbook,andsincetablesandgraphs13ofnumerical valuesare available forarangeofvaluesofb,a"andar,theanalyses ofthisandthe preceding sectionsservetojustifytheuseofthesetablesandgraphsfor thenumerical evaluation ofCTforthetwo-wire line. Incidentally datafordetermining BTforacoaxiallinewithuniform innerconductor andanouterconductor thatchangesfromb,tobratZ=S, arealsoavailable (Ref.13,p.311). Sec.14] DISCONTINUITIES ANDNONUNIFORMITIES 381 Bifurcation ofCoaxialLine.Abifurcated coaxiallineisshownin Fig.14.2a.Ifjunction effectsareneglected, coaxiallines1and2with characteristic impedances ZclandZc2areconnected inseriesacross coaxialline0withcharacteristic impedance Zc.Neglecting losses,the characteristic impedances are z..RrIaacl=cl=-2n 7ra2z..RrIa2c2=c2=-2n 7ral (22) wherer==1207rohms. Anapproximate analysis ofthejunction effectsinthebifurcated line couldbecarriedoutwiththemethodusedinthefirstpartofthissection. ;;;;;;;;;;;; ;;;;;;;;;; ;;;;;;;> (a) r-----------. I .line.....;.......::.:.....--+_~I Xo:Line0;Zco Line~~"-_I--~~~~l _, L .J Lumpednetwork (b) FIG.14.2.(a)Bifurcated coaxialline.(b)Thesamelinewithanequivalent network totakeaccount ofjunction effects. However, sincetheresultsofamorerigorous analysis areavailable (Ref.13,pp.369-370), itissufficient torepeatthese.Theequivalent lumpednetwork consistsofashuntreactance acrosseachlineatthejunc tionplane,asshowninFig.14.2b.Formulas forthethreereactances are Xo=-Rccot/3d whereaa-a2Xl=Rc---cot/3daa-ala2-alX2=Rc---cot/3das-al (23) (24) NotethatXoisnegative, alsothatXo= -(Xl+X2),sothatthereis nojunction correction lookingfromline0towardlines1and2.Curves of7rd/(aa-al)asafunction of(aa-a2)/(aS-al),aswellasformulas forhigher-order termsford,areavailable intheliterature. Theapproxi mation(24)isinerrorbylessthan2percent,provided 2(aa-al)<0.3~. 382 TRANSMISSION-LINE THEORY [Chap.V FIG.15.1.Bendinatwo-wire line.Z=S,;U=O z=O15.BendinaTwo-wire Line.104,131Whenatwo-wire lineintheyz planeisbentthrough anangleefromthisplane,asshowninFig.15.1, thetwowiresremainidentical, sothatnounbalanced currents aregener ated.However, theparameters ofthelinearenotconstant inthe vicinityofthebend.Sinceresistance andleakageconductance perunit lengtharesmall,changes inthem maybeignored. Butthisisnottrue oftheexternal inductance perunit lengthandthecapacitance perunit length. Inordertoevaluate these quantities nearthebend,lettheusual coordinate system beintroduced. Onepartofthelineextendsfromthe generator atz=0tothebendat z=8z;forconvenience thevariable w=8z-z,measured fromthebend towardthegenerator, isintroduced. Thesecondpartofthelineextends fromthebendatu=0totheloadatu=8rinadirection differing from theoriginalonealongthezaxisbyanarbitrary anglee.Asusual,the radiusofthewiresisa;theirseparation betweencentersisb.Itisassumed thatthecondition a2«b2issatisfied. Thedetermination ofle(w)andc(w)ononesideofthejunction andthe equalquantities le(u)andc(u)ontheothersideparallels theprocedure in Chap.II,Sec.1,wherethetwowiresofthelineflaredoutward. Chapter II,Sec.1,formulas (22a)and(22b),apply.Thepertinent partsare le(w)=ko(w)+kOT(W) 21rJl 21rE c(w)=ko(w)+k~T(W)(1) (2) where,asinChap.II,Sec.1,Eqs.(15b)and(26a), ko(w)==(00(~ _~)dw'==ko-FI(w) (3)JoRaRb ko=2ln~FI(w)==Inw+Vw2+b2 (4) a w+Vw2+a2 Ra=V(w'-W)2+a2Rb=V(w'-w)2+b2(5) Also kOT(w) ==f,00(RIIT-RI2T)cosedu'=F2(w)cose (6) k~T(W) ==f,00(R~T-RI 2T)du'=F2(w) (7) wherenow F2(w)==Inwcose+Vw2+b2(8) wcose+Vw2+a2 Sec.15] and.DISCONTINUITIES ANDNONUNIFORMITIES R1T=v'W2+U'2+2wu'cos(J+a2 R2T=v'w2+U'2+2wu'cos(J+b2383 (9) Thedistances RaandRbfromthepointQonthesurfaceofconductor 1 atadistance wfromthebendtotheelements ofintegration dw'along theaxesofthetwoconductors ata j4----w---+1 distance w'towardthegenerator Ra...---(rw'-----~' ...u' fromthebendareshowninFig.15.2, ......'" asarethedistances RITandR2Tto thecorresponding elements du'on theothersideofthebend.In(3), dWi (6),and(7)theupperlimitsinthe integration (Slor8r)arereplacedby infinity,sinceitisassumedthatboth FIG.15.2.Enlarged sectionofbendina two-wire line.sections oflinearesufficiently long sothatcontributions fromthepartsthatarefarfromthebendarenegli gible.Theintegrations arecarriedoutusingPierceformula 160. Itfollowsthat (lOa) (lOb) Sincetheinductance andcapacitance perunitlengthoftheuniform (infinitely long)lineare le=~o21rJl21rE Co=ko(11) itfollowsthatthelumpedinductance LTandcapacitance OTrequired to yieldthesametotalinductance andcapacitance when(11)isusedinthe sectionoflinebetween thebendandthegenerator, insteadof(9)and (10),aregivenby 1.00 11.00 LT=W(w)-19]dw=2- [F2(w)cos(J-FI(w)]dw(12)o 1rJl0 CI.00[ ( ) 21rEI.00FI(w)-F2(w) T=0C W-co]dw=~0ko_FI(w)+F2(w)dw(13) Exactlyequalvaluesareobtained forthesectionoflinebetween thebend andtheloadbyusing(lOa),(lOb),(12),and(13),withusubstituted for W,sothatthetotallumpedseriesinductance atthebendis2LTandthe totallumpedshuntcapacitance is2CT,asshownintheequivalent circuit ofFig.15.3. Theintegrands in(12)and(13),asevaluated byTomiyasu,l°4 are giveninFig.15.4forthreevaluesof(Jandb=2cmanda=0.1588cm. Itisseenthattheprincipal contributions occurintherangew~lOb. 384 TRANSMISSION-LINE THEORY [Chap.V Thesecondintegral in(12)maybeevaluated bynotingthat 10coIn(wcos0+YW2+b2)dw =locoIn(w+yw2+b2)dw+bfof$)In[1-K(x)]dx(14a) sothat 10coF2(w)dw=10coFI(w)dw+(b-a)fof$)In[1-K(x)Jdx(14b) Notethattheshorthand K(x)==(1-cosfJ)x(Yx2-1 -x) (14c) andthechangeofvariable x=w/bhavebeenintroduced in(14a).Itis noweasilyverifiedthatK(x)rangesbetween thevalue0whenx=0 !LT!LT Generalo<- UniformIin:::X Uniformline~Load 1 I1 .'2LT1'2LT, lumpednetwork atbend FIG.15.3.Equivalent circuitforbentline. toj(1-cosfJ)whenx=00.Since,forfJ~90°,K(x)~0.5,thelast logarithm in(14a)maybeexpanded intheseries In[1-K(x)]= - [K(x)+K2~X)+Kix)+...](15) andintegrated termbyterm.Thefirsttwotermsgiveasatisfactory approximation, viz., 10coIn[1-K(x)]dx==j(1-cosfJ) +15(1-cosfJ)2+15(1-cosfJ)3+ (16) Thefirstintegral ontherightin(14a)isreadilyintegrated usingDwight formula 625.Thus locoFI(w)dw=b-a (17) With(16)and(17)usedin(12),thefinalformula forLTis b-aLT= - 27rVG(O) (18a) where,withonlytwotermsretained intheseries(15)and(16), G(fJ)==(1-cosfJ)[1-jcosfJ+15cos0(1-cosfJ)J =(1-cos0)(1--A-cosfJ-15cos20) (18b) Thefunction G(O),ascalculated from(18b),isshowninFig.15.5asa function offJandislistedinTable15.1. Sec.15] DISCONTINUITIES ANDNONUNIFORMITIES 385 w/b 3 4 5 6 7 Q)-0.6 2 567 ~-0.8 I-0.01 ~-1.0 0-0.02 u ~-1.2 --0.03 ~-1.4~ ~~-0.04 -1.6 ~N+ -1.81...~-0.05 ~-:.. -k;-0.06 -2.0~-I ~-0.07 -0.08IH--+---I -0.09l----t---t -0.10 L-I-.....I- ......... FIG.15.4.Graphsofintegrands inEqs.(12)and(13). 1.0r-----r---r--,----,--..,--""r"'---r---,.----" 2.0 M(6) 0.40.8121.6 0.4G(6)=(1-cos 6)(1-~cos 6--fscos26) 0.8M(6)=(1-cos 6)+~(I-cos6)2+/s(1-COS 6)3 0.6 0.2G(6) o~-=::::=~--:~-..L--..l-_-l-_...L-_-L-_-L_---J 0 ~ ~WW~~~ 8 FIG.15.5.Graphsofthefunctions G(fJ)inLT==-(b-a)G(fJ)/21r/l andM(fJ)in CT=-21rE(b -a)M(fJ)/3k~ forabendinatwo-wire line. 386 TRANSMISSION-LINE THEORY TABLE15.1.THEFUNCTION G(fJ) 8,deg G(fJ) o 0.00 10 0.01 20 0.04 30 0.10 40 0.18 50 0.29 60 0.43 70 0.60 80 0.80 90 1.00[Chap.V Thedetermination ofCrusing(13)maybecarriedoutbytreating thedifference (19) (23)[whichoccursinboththenumerator andthedenominator of(13)]ina mannersimilartothatfollowed intheevaluation ofF2(w)cos(J-FI(w) in(12).Thus H(w)=Inwcos (J+vW2+1J2_Inwcos()+vw2+a2(20a) w+Vw2+b2W+Vw2+a2 Thismaybeexpressed inthefollowing form: Hew)=In[ 1 -K(r)J-In[ 1 -K(~)J (20b) where,asin(14c), K(x)=(1-cos()x(~ -x) (21) withx=wjborwja. Asbefore,thelogarithms in(20b)maybeexpanded intheseries(15), withtheresult Hew)=K(~)+jK2(~)+!K3(~)+... -[K(~)+jK2(~)+!K3(r)+...J(22) SinceK(x)~0.5for8~900 ,itisreadilyverifiedthat,withtheassumed condition b2»a2,ko=2In(bja)issufficiently largesothatthequantity H(w)jkosatisfiesthecondition H(w)<1 ko Subjectto(23),thesecondintegrand in(13)maybeexpanded inpowers Sec.15] DISCONTINUITIES ANDNONUNIFORMITIES 387 ofH(w)jko,asfollows: C T= _27rErooH(w)jkodw (24a) koJo1 -H(w)jko = _27rEroo[H(W)+H2(W)+H3(W)+...]dw(24b) koJoko kij k~ Theintegralsintheleadingtermcanbeevaluated asin(14b).The resultis rooH(w)dw=(b-a)(1-cos8)[1+i(l-cos0) Joko 3ko+-Js(l-cos8)2+...](25a) Theexactintegration ofthehigher-order termsiscomplicated bythe occurrence ofcross-product termsoftheformKn(wja)Km(wjb). Since thecontribution bythetermsinK(wja)isinanycasesmall,thesemay beneglected inthehigher-order terms,sothat H2(W)==[K(~)+jK2(~)+··r=K2(~)+K3(r)+ H3(W) ==[K(~)+jK2(~)+..·r==K3(r)(25b) (25c) (25e)Theintegrations cannowbecarriedouttogive rooH2(W) bJokrdw==ij[-ft(1-cos0)2+-Js(1-cos0)3+...](25d) rooH3(W) bJokrdw==kg;5(1-cos0)3+... With(25a,d,e) itfollowsthat(24b)becomes CT==-27rEO~kg a)[1-cos8+i(1+~)(1-cos0)2 +~(1+~+ij)(1-cos8)3](26) Sincethetermsinahavebeenneglected inevaluating (25d)and(25e), thefactorboccurs. Since,withb2»a2, b-adifferslittlefromb,the factorb-amaybesubstituted forbin(25d)and(25e)withlittleerror. Formostpurposes theleadingtermsin(26)areadequate; theypermit thedefinition ofafunction of0whichisindependent ofko.Theleading termformula is CT==-27rE(:k-; a)M(O) (27a) o where M(O)=1 -cos0+i(1-cos0)2+~(1-cos0)8(27b) Thefunction M(O)isplottedinFig.15.5andtabulated inTable15.2. 388 TRANSMISSION-LINE THEORY TABLE15.2.THEFUNCTION M(8) fJ,deg M(fJ) o 0 10 0.015 20 0.061 30 0.14 40 0.26 50 0.42 60 0.62 70 0.88 80 1.20 90 1.57[Chap.V Numerical valuesofLT/loandCT/COforabendinatwo-wire line,with b=2cmanda=0.1588cm,aregiveninTable15.3for(J=30,60,and 90°.Threecolumns aregivenforCT/CO:oneiscomputed usingthemore complete formula (26);thesecondcolumnmakesuseoftheleading-term formula(27a);thethirdlistsvaluesreported byTomiyasu 104whichare determined bynumerical methods. Allthreeareinreasonably good agreement. Twocolumns aregivenforLT/lg:oneiscomputed from (l8a);theotherisevaluated bynumerical methods byTomiyasu.104 Theagreement isgood. Theexperimental determination oftherelatively smallinductance 2LT andcapacitance 2CT(Fig.15.3)maybecarriedoutindependently by adjusting ashort-circuited sectionoflineinlengthtohaveacurrent maximum atthebendwhen2LTismeasured andachargeorvoltage maximum when2CTismeasured. Usingatwo-wire lineforwhich b=2cmanda=0.1588cm,Tomiyas1,1104 determined theexperimental curvesshowninFig.15.6.Thetheoretical valuesof2CT/coobtained from(26)andtabulated inTable15.3andthoseof2LT/lgobtained from (18a)andtabulated inTable15.3arealsoshowninFig.15.6.Itisseen thatthetheoretical curvefor2CT/coisinquitegoodagreement withthe experimental one.Ontheotherhand,whereas theoretical valuesof 2LT/loareallnegative inFig.15.6,theexperimental curveshowsasmall TABLE15.3.THEORETICAL VALUES FORCdco ANDLT/Ig FORBEND INTWO-WIRE LINE, WITHb=2CMANDa=0.1588 CM CT/Co,em LT/l~,cm fJ,degNumerical NumericalEq.(26) Eq.(27a)integrationtEq.(18a)integration 30 -0.017 -0.017 -0.016 -0.035 -0.033 60 -0.081 -0.077 -0.071 -0.152 -0.141 90 -0.224 -0.189 -0.189 -0.353 -0.353 tReported byTomiyasu.104 Sec.16] DISCONTINUITIES ANDNONUNIFORMITIES 389 0.2r----r----r---. -0.8~_...I--......&._~ 0°30°60°90·-0.61---+---+--T1-0.2I----I---"'rr~r-t-0.41----1---1--~-0.2I----I----"""d'''c---t 9 FIG.15.6.Comparison oftheo reticalcurves(solidlines)with experimental curves(broken lines)for2Cr/co,2LT/l~,and (2LT+LN)/l~(Tomiyasu).(28) for0=30°.Byaddingthisvalueto2Lr/lg toobtainatotalinductive correction (2Lr+ LN)/lo,apointat0=30°isobtained which isinexcellent agreement withexperiment. Owingtoincreasing complications asthebend ismadesharper,Tomiyasu evaluated nopoints for0=60and90°butmerelyassumed that thesamevalueofLNgivenin(28)applies for30°~0~90°.Acurveof(2Lr+LN)/loisshowninFig.15.6.It isseentoagreewellwithexperiment. 16.TJunction inaTwo-wire Line.Animportant typeofjunction consistsofthreesectionsofsimilartwo-wire linemeeting inaT.Com monexamples include(1)amatching ortuningstubconnected atright anglestoatransmission line,asshowninFig.16.1a;(2)adrivenline thatdividesintotwoloadedsections inparallel, asinFig.16.1b;(3)a linethatmakesaright-angle bendbutissupported byaninsulating stub, asinFig.16.1c. Intheelementary analysis ofthetransmission linewithashuntstub orinsulating support inChap.III,itisassumed thatconventional transmission-line formulas arevalidatallpointsalongallthreesectionspositive valuewhen0islessthan60°.Acorrection forthetheoretical curvehasbeenobtained byTomiyasu,104 whohasshownthatthedis crepancy between thetheoretical curvefor2LT/loandtheexperimental resultsisaconsequence ofthefactthattheelementary analysisforLras carriedoutinthissectionassumes rotational symmetry forthecurrentin,andthevector potential on,thesurfaceofthetwowires throughout thebendandalsoanabrupt changeinthedirection ofthecurrentattheem bend.Actually, inthecaseofconductors withfiniteradiusa,neitherthecurrentnor thevectorpotential isrotationally symmet- ricalaroundthebendevenwhenthecondi- tionb2»a2issatisfied. Thecurrentismore concentrated inthebend,sothatthecon ductorbehaves asifithadaradiussmaller thana.However, Tomiyasu hasshownthat asmallpositiveinductance LNmustbeadded to2Lr.Theanalysis involves ellipticinte grals,andforalinewithb=2cmandema=0.1588em,Tomiyasu obtained thevalue LN16=0.22cm o 390 TRANSMISSION-LINE THEORY [Chap.V 1 J 2 ~---: __-.,;;..W"'~----t- ..,...-.~ ~ Generator l'IJ' 2'load t 3VM~\~~ng (a) ~3' J 2 l'--+---:-~---~J'2'load1x"-_ load2 (b)3 3' 1IGenerator IJ 2 ""-~-~_.~Supporting l'J' 2' Lo:;d stub 3 :3' (C)Ii ~Generator FIG.16.1.ThreetypesofTjunctions inatwo-wire line.(a)Linewithmatching stub. (b)Singlelinedrivingtwoloadedlines.(c)Right-angle bendwithsupporting stub. FIG.16.2.Equivalent junction network forashuntstubonatwo-wire line. ofline,including theregionsclosetothejunctionJJ'(Fig.16.1).Actu allytheexternal inductance andthecapacitance perunitlengtharenot constant atthevalues[0andCointhejunction zone.Asaconsequence, itisnecessary tocompensate fortheinaccurate useoftheconstant values loandCobyintroducing afictitious lumpednetwork thatconsists ofa seriesinductance LTandashuntcapacitance CTconnected toeachline atthecommon junction, as showninFig.16.2.ThevaluesLTandCT Sec.16] DISCONTINUITIES ANDNONUNIFORMITIES 391 foreachlinearetobesoevaluated thattheimpedances lookingtoward thejunctionJJ'from11',22',and33'arethemeasurable apparent imped anceswhencomputed usingconventional transmission-line formulasin conjunction withthelumpednetwork. Sincethenetwork inthevicinityofthejunction issymmetrical with respecttoaplanethroughJJ',J3J~,and33'(Fig.16.2),theproblem of determining theelements ofthelumpednetwork issimplified. Itisclear that LT2=LTlCT2=CT1 (1) Theanalysis maybecarriedoutusingevenandoddcurrents andvolt ages.Lettheoriginofasystemofcoordinates belocatedmidway between theterminals JJ',asinFig.16.1a.Thecoordinate wincreases fromJJ'toward11'inline1,sothatcurrents andvoltagestotheleftof JJ'maybedesignated byII=I(w)andVI=V(w),andthosetothe rightofJJ'inline2,by12=I(-w)andV2=V(-w).Thecoordi natevincreases toward33'inline3,andthecurrents andvoltages inthe shuntorstubsectionarela=I(v)andVa=V(v). Letthecurrents andvoltages inlines1and2beseparated intoeven andoddpartstoformsymmetrical andantisymmetrical combinations, asfollows: . I(w)=I(B)(W)+I(a)(w) V(w)=V(B)(W)+V(a)(w) I(-w)=I(B)(W)-I(a)(w) V(-w)=V(B)(W)-V(a)(w)(2a) (2b) (3a) (3b) Bydefinition, letthesymmetrical currents beevenandthesymmetrical voltages odd,sothat (6) (7)V(B)(W)=j[V(w)-V(-w)] V(a)(w) =j[V(w)+V(-w)]I(B)(-w)=I(B)(W) V(B)(-w)= -VB(W) (4) I(a)(-w)=_Ila)(w) V(a)(-w)=V(a)(w) (5) Itisreadilyverifiedthatdefinitions (4)and(5)areconsistent with(2a,b) and(3a,b)andwith I(B)(W)=j[1(w)+I(-w)] I(a)(w)=j[1(w)-I(-w)] Oncethecurrents andvoltages onlines1and2havebeenresolved intosymmetrical andantisymmetrical components using(6)and(7), eachsetofcurrents andvoltages maybedetermined independently, and aseparate setofLT'sandCT'Sdefinedforeach.Notethat,ingeneral, junction-zone networks arenotthesameforsymmetrical currents and voltages asforantisymmetrical ones.Letthejunction zonebethe regioninallthreelineswithinadistance dofthejunction whichisshort compared withthewavelength andbeyondwhichthelineparameters maybeassumed constant. Usuallydisoftheorderofmagnitude of lOborless. 392 TRANSMISSION-LINE THEORY [Chap.V TheSymmetrical Problem. Asymmetrically drivenTjunction isshown inFig.16.3.Thetwosymmetrical branches areexcitedbygenerators thatmaintain currents andvoltages thatsatisfy(4).Itfollowsthat V(8)(w=0)=0 (8) sothattheremaybenogenerator inline3.Moreover, sincethegener atorsinlines1and2maintain zerovoltageacrossthejunction points + ,'(w) J ,s(-w)=,'(w) ~-'--- ---~---~w~ --__----r:-_-,y(')",(,)~~ ...:;w__+-0't-:" ~_-_--5 J' + l(v)=O FIG.16.3.Symmetrically drivenTjunction. l(a)(w) J l(a)(-w)=_~a)(W) +~----~·--n·----~...(CI)..,(a)~_ " __ ..-5 r J' - lev) +1I ~'eJ FIG.16.4.Antisymmetrically drivenTjunction. JJ',allcurrents, chargesperunitlength,andvoltages inthestubline3 arezero.Therefore lines1and2behavelikeasmoothlinewithoutthe stubsection,and C~i=C~~=0 L~i=L~~=0 (9) Sincetherearenocurrents orchargesinline3,itisunnecessary to introduce C~~andL<;~. TheAntisymmetrical Problem. Anantisymmetrically drivenTjunc tionisshowninFig.16.4.Ingeneral, theremaybegenerators inall threelines,butthoseinlines1and2mustbesoadjusted that(5)is satisfied. Therearenorestrictions onthegenerator inline3. Kirchhoff's currentlawappliedatJJ'inFig.16.4gives I(a)(w=0)=I(a)(-w=0)+I(v=0) (lOa) With(5)itfollowsthat I(v=0)=2/(a)(w=0) (lOb) Sincethejunction zoneissmallcompared withthewavelength, thelead ingtermsinMaclaurin expansions ofthecurrents areadequate. Thatis, I(a)(w)=I(a)(-w)==jI(v)Iwl~d,v~d (11) Sec.16] DISCONTINUITIES ANDNONUNIFORMITIES 393 Similarly thevoltages acrossthethreeconductors inthejunction zone areessentially equal: V(a)(w)=V(a)(-w)==V(v)Iwl~d,v~d (12) Notethattheequations ontheleftin(11)and(12)areexactandthose ontherightareapproximate fornonzero valuesofbfA. Capacitance perUnitLength. Thescalarpotential differences across thethreelinesinthejunction zonemaybeexpressed asfollows: V(a)(w)=Vu(w)+V12(w)+VJ3(w)=V(a)(-w) (13) V(v)=V31(v)+V32(V)+V33(V) (14) whereVl1(w)isthecontribution toV(a)(w)bythechargeperunitlength q(a)(w)online1,V12(w)isthecontribution toV(a)(w)byq(a)(-w)on line2,andV13(w)isthecontribution toV(a)(w)byq(v)online3.Simi larlyV31(V)isthecontribution toV(v)byq(a)(w)online1,V32(V)isthe contribution toV(v)byq(a)(-w)online2,andV33(V)isthecontribution toV(v)byq(v)online3.AsinChap.II,Sec.1,theexpressions for theseseveralvoltages areasfollows: where1f"" Vl1(w)+V12(w)==2- q(a)(w')PL(w,w') dw'7rE_"" 11,"" V13(w)==2-q(v')PT(w,v') dv' 7rE0 1f"" V31(V)+V32(V)==2- q(a)(w')P8(v,w')dw' 7rE_"" 1I,""V33(V)==-2q(v')PL(v,v') dv' 7rE0 Ra=V(w'-w)2+a2 Rb=V(w'-W)2+b'l(15a) (15b) (16a) (16b) (17a) (17c)(17b) (17d)P8(v,w')==_1__~ R8aR8bPT(w,v') ==_11_ RTaRTbRTa=yv'2+w2+a2 RTb=yv'2+w2+b2 R8a=YW'2+v2+a2 R8b=VW'2+v2+b2 Ra=V(v'-V)2+a2 Rb=V(v'-V)2+b2 Incarrying outtheintegration in(15a,b)thechargeperunitlengthis expanded inaTaylorseriesaboutthechargeperunitlengthatw.Simi larlyin(16a,b)thechargeperunitlengthisexpanded aboutthecharge perunitlengthatv.AsinChap.II,Sec.1,onlytheleadingtermsneed 394 TRANSMISSION-LINE THEORY [Chap.V beretained. UsingChap.II,Sec.1,Eqs.(l5b)and(26a),thefollowing formulas maybeobtained: Yll(W)+Y12(w)=q(2a )(w)ko (18a) 1I"E Y33(V)=q(v)(ko_Inv+v~) (18b) 211"E v+vv2+a2 where ko=21n~ (19)a Thefollowing additional resultsareobtained ifuseismadeofSec.15, Eq.(8),with(J=11"/2: Similarlyqa(w)w2+b2 Y13(w)=~In~+ 2':t7rE Wa q(v)v2+b2 V31(v)+Y32(V)=-2In~+ 21I"EVa(20) (21) Thevariable capacitances perunitlengthoflinemaynowbedefined asfollows. Forlines1and2 _q(a)(-w)_ _ q(a)(W) c(-W)=ya(-W)-C(W)=y(a)(W) (22a) 211"E where c(w)=ko+L(w) (22b) w2+b2 with L(w)==iIn2+2 (22c)wa Similarly, forline3, _q(v)_ 211"E c(v)=Y(v)-ko+2L(v)_M(v) (23a) v2+b2 where L(v)==iIn-y-+ 2 (23b)va M(v)==Inv+vV2=Fb2 (23c) v+vv2+a2 Notethat,whenIwlbecomes sufficiently great,c(w---+00)=Co=211"E/ko. Similarly c(v---+(0)=co.Ontheotherhand,atthejunction ) )2coc(w=0=c(v=0=3 Inductance perUnitLength. Sincethestubline3isperpendicular to boththeotherlines,thereisnoinductive coupling, andthejunction effectinline3reducestoatransmission-line endcorrection. Thus,with Sec.12,Eqs.(2)and(3),itfollowsthat le(v)=~o(v)=_1_(sinh-I!!. -sinh-I!!.+In~) (24) ~v ~v a ba Sinceforlines1and2theantisymmetrical currents areequalandoppo- Sec.16] DISCONTINUITIES ANDNONUNIFORMITIES 395 (31)(26) (27) (28b) (29a)(28a)siteatequaldistancesIwlfromthejunction, itisreadilyshownthat l'(w)=:.(sinh-1~-sinh-1~) (25) Lumped Elements. Thelumped inductive elements required ineach lineatthecommon junction aredefinedasinChap.II,Sec.4.Useis alsomadeofSec.12inthischapter. LT3=(dW(v)_loJdv==_b2- aJo ~p LT2=LTl=(d[le(w)_lo]dw==_b-a=2LT3Jo ~p Notethatdislargecompared withb. Thelumpedcapacitive elements aredefinedasfollows: CT3=Ld[c(v)-co]dv With(23a)thisintegral becomes Co(d 2L(v)-M(v) CT3= -koJo1+(ljko)[2L(v)-M(v)]dv CTl=fod[c(w)-co]dw With(22b)thislastintegral becomes Co(d2L(w) CT2=CT1= -2koJo1+L(w)jkodw (29b) Anapproximate integration of(28b)and(29b)maybecarriedout following aprocedure introduced inSec.15.Sincekoismoderately large compared withunityformostlinesandsince2L(v)-M(v)andL(w) havequitesmallvaluesovermostoftherangeofintegration, thedenomi natorsin(28b)and(29b)maybeexpanded inseries,sothat CT3==-~(d(2L(V)-M(v)-~[2L(v)-M(v)]2-..·1dv(30)koJo ko Cofod[1 ] CT2=CT1= - - L(w)--L2(W) -...dwko0 ko Ineachintegral theleadingorfirst-order termsarereadilyevaluated, sincetheyhavethesimpleformsgivenbelow: (CT3h= -Co{d[In(v2+b2)-In(v2+a2)-In(v+~)koJo +In(v+yv2+a2)]dv(32) (CT2h=(CTlh==-;~old[In(w2+b2)-In(w2+a2)]dw (33) 396 TRANSMISSION-LINE THEORY [Chap.V Thesemaybeintegrated usingDwight623and625.Theresultsare (C) - _ Co(1/'+l)(b-a)__1/'(1/'+l)(b-a)E (34) T31 - 2In(b/a) -2[ln(b/a)]2 c01/'(b-a) 1/'2(b-a)E (CT2h=(CTlh= -4In(b/a)= -[2In(b/a)]2 (35) Theseexpressions aregoodapproximations forsufficiently largevaluesof b/a.Ifb/aissmall,additional termsmustbeusedin(30)and(31), or(28b)and(29b)mustbeevaluated numerically. Notethatallthelumpedelements arenegative, indicating thatthe actualvariable inductance andcapacitance perunitlengtharesmaller than19andCointhejunction region. Therefore, if19andCoaretobe J Une2 tLr Lr=-5(b-a) 27Tv Cr=_1T(27T+1)(6-a)E 2(In%)2 FIG.16.6.Equivalent circuitforTjunc tion.line1 J,'2J{ J'3 FIG.16.5.Equivalent circuitforantisym metrically drivenTjunction. used,negative lumpedinductances andcapacitances thatdecrease the totalinductance andcapacitance arerequired. Itisimportant tobearinmindthatthevaluesofLTandCTgivenby (26),(27), (34), and(35)applyonlytothatpartofthecurrentwhichis oddwithrespecttothejunction andtothatpartofthevoltagewhichis even.Thelumpednetwork fortheantisymmetrical currents andvolt agesisshowninFig.16.5.Notethat C-C+C+C~_co(21/'+l)(b-a)- T-Tl T2 T3- 2In(b/a)-1/'(21/'+l)(b-a)E 2[ln(b/a)]2 (36) Sincetheentireantisymmetrical (odd)currents mustenterline3from lines1and2,itisobviously immaterial whetherthelumpedinductances LTlandLT2areconnected inlines1and2atthejunction, asinFig.16.5, orwhethertheyareconcentrated inline3,asinFig.16.6.Inthelatter casethetotalseriesinductance inline3is Lr=LTl+LT2+LT3=5LT3= -5(b2-a) (37) 1/'V Theadvantage ofFig.16.6overFig.16.5isthatitisalsothecorrect equivalent circuitforthesymmetrical problem andtherefore forthe Sec.17] DISCONTINUITIES ANDNONUNIFORMITIES 397 Tjunction ingeneral. Sincethesymmetrical voltageiszeroatthe junction, thepresence ofCThasnoeffect,andsincenosymmetrical cur rentsenterthestubline3,thelumpedinductances inthislinealsohave noeffectonthesymmetrical currents. Therefore theequivalent circuit ofFig.16.6maybeusedforthetotalcurrentandtotalvoltage,anditis notnecessary toseparate theevenandoddparts. 17.Junction Networks forSeriesBranches inTwo-wire Lines;Ter minal-zone Networks forStub-supported andCenter-driven Antennas andFoldedDipoles.10,124Abalanced seriesjunction inatwo-wire line isshowninChap.III,Fig.14.1b,andinFig.17.1.Itconsists of twocoplanar linesmeetingatrightangles. Thetwoconductors ofthe yt4a .t.c=:======~ yt..=::======:::J) ir2a b w.! 4z ~c::========~ ~bCl-ef•2=8 FIG.17.1.Balanced seriesjunction intwo-wire line. mainlineareparalleltothezaxis;theyareseparated adistanceband haveradiia.Theconductors ofthesecond(orauxiliary) lineareparallel totheyaxis;theyareseparated adistance baandalsohaveradiia. Inordertoevaluate theconstants oflumpednetworks foreachlineat thejunction, letthecurrents, voltages, andcharges beresolved into symmetrical andantisymmetrical parts,asinChap.III,Sec.14.tItis assumed thatthemainlineisbalanced, sothat12(z)=-11(z). Letthesymmetrical caseillustrated inFig.17.2bedefinedasfollows: (1) wherew===s-zismeasured towardtheleftonthemainlinefromthe tThesymmetrical andantisymmetrical designations differinthissectionfromthose inSec.16inorderthatcodirectional "antenna" currents ontheauxiliary linemay bedesignated symmetrical, asinantenna theory.l0 398 TRANSMISSION-LINE THEORY [Chap.V I~(z)r'-------::...-__--J b1 V'(Z) I .I.r-----~---- 1;(z)+ V'(z) 1;(z)=-lj(z) I :II Iw=o w=b2/2w--b2/2 FIG.17.2.Symmetrical currents andvoltages inbalanced seriesjunction. vacy) I~(y)t+ -tI;(y)=-If(y) l~(z) If(z)-. ~ + y Vt·)(z) w4 Vt·)(z) + ~ ~ ·2(Z) '2(Z)-=-'i(z,T'-------.,,;------1 I I I I I I Iw=o: I w=b2/2w=-b2!2 FIG.17.3.Antisymmetrical currents andvoltages inbalanced seriesjunction. Sec.17] DISCONTINUITIES ANDNONUNIFORMITIES 399 middleofthejunction. With(1)thefollowing relations alsoaretrue: y,,(-w)=y,,(W) ya(_y)=ya(y)=0q"(-W)=qa(W) qa(_y)=_qa(y)(2a) (2b). Lettheantisymmetrical caseillustrated inFig.17.3bedefinedby la(-w)=la(w) Accordingly Va(-w)= -ya(w) ya(_y)= _Va(y)I~(y)=-I!(y) qa(-w)=-qa(w) qa(-y)=qa(y)(3a) (3b) (3c) Ingeneral, thecurrents andvoltages aresuperpositions ofthesym metrical andantisymmetrical components. Thus I(w)=la(w)+la(w) I(y)=la(y)+la(y)Yew)=ya(w)+ya(w) Y(y)=ya(y)+ya(y)=ya(y)(4a) (4b) Thecomponents ofcurrentonthemainlinearedefinedasfollowsin termsofthetotalquantities: la(w)=j[1(w)+I(-w)] la(w)=j[1(w)-I(-w)]ya(w)=ilY(w)-V(-w») ya(w)=![Y(w)+Y(-w»)(5a) (5b) Thesymmetrical partsofthecurrents andvoltages aredefinedsothat themainlineisbalanced withequalandopposite currentsinitstwocon ductors, whereas theauxiliary lineiscompletely unbalanced withequal andcodirectional currents initstwoconductors. Thesymmetrically drivenseriessectionsdonotcarryequalandopposite transmission-line currentsbutequalandcodirectional antenna currents. Thesecannotbe evaluated fromtransmission-line formulas. However, theappropriate junction-zone network canbedetermined usingthemethods outlined in thischapter. Suchanetwork isuseful,forexample, inconjunction with linesusedtocenter-drive balanced antennas, e.g.,atunedfolded-dipole antenna. 10Byapplication ofthetheoryofimages,antennas overcon ducting planesmaybeanalyzed whendrivenbyasingleconductor paralleltotheimageplane. Theantisymmetrical partsofthecurrents andvoltages aretruetrans mission-line currents onboththemainlineandtheseriessections. Inordertocorrectforthenonuniformity oftheinductance andcapaci tanceperunitlengthinthemainlinenearthejunction, lumpedseries inductances LTandshuntcapacitances CTmustbeconnected oneach sideofthejunction. Lumped SeriesInductance LTforMain-line Network.Itfollowsbya simplemodification oftheformulation inChap.II,Sec.1,thatthe inductance perunitlengthinthemainlineontheleftofthejunction is 400 TRANSMISSION-LINE THEORY [Chap.V givenby iNherelo(w)ko(w) =211"11 j.ClO(11)f-ba /2(1 1)ko(w)==- - - dw'+= - - - dw' ba/2RaRb -ClORaRb(6a) (6b) Infinitelimitshavebeensubstituted forthefinitedistances fromthe junction tothetwoendsofline1,withtheunderstanding thatthe actualdistances areverygreatcompared withthelinespacing b.The Mainline Mainline FIG.17.4.Junction network oflumpedelements forusewithuniform lines. uppersignin(6b)isforthesymmetrical case,andthelowersignis fortheantisymmetrical case.NotethatRa=yew'-W)2+a2and Rb=yew'-w)2+b2•Theintegrations arereadilycarriedouttogive where{b2ln- ko(w)=B+=C+0a B.h1W-ba/2.h1W-bal2==sln---a---sm- b C==sinh-1w+aba/2_sinh-1w+bba/2(7a) (7b) (7c) Thelumpedinductance required inthecircuitofFig.17.4tocorrect fortheerrormadeinusingl8inplaceofle(w)=lO(w)al(w) isgivenby Chap.II,Sec.4,Eq.(3),viz., LT=(d+ba/2[le(w)-l8]dw=(d+ba/2[l8(w)-lo]dw (8) }ba/2 Jba/2 Sec.17] DISCONTINUITIES ANDNONUNIFORMITIES 401 where19(w)isgivenby(6a)with(7)andwhere2rllig=2Inbfa.Note thatal(w)=1,sincethetwolinesaremutually perpendicular [theratio factoral(w)isdefinedinChap.II,Sec.1,Eq.(24a)].Theintegration in(8)maybereadilyperformed using(7).Theresultsareasfollows: L;.j=-1[b_a+(Vb2+b2_Vb2+a2_bInbtl+Vb:+b2)] L~2rll - tI tI tIbtl+Vb:+a2 (9) whereL;.appliestothesymmetrical caseandL~totheantisymmetrical case.Inthespecialcasewherebtl=banda2«b2,thefollowing simpler formulaisobtained: L;.jb[a(_/0 1+~] =--1--±v2-1-lnL: 2rll b 2(10) whereLumped ShuntCapacitance CTforMain-line Network. Thedetermi nationoftheshuntcapacitance intheseveralnetworks forwhichLThas beenevaluated isbasedonChap.II,Sec.4,formula (4),viz., kd+b../2 kd+ba/2CT= [c(w)-co]dw= [CO(W)l(W) -co]dw(11) b../2 ba/2 2rE rE co(w)=ko(w) Co=In(bfa) (12) whereko(w)isgivenby(5b)and(7).Theratiofactorl(W)isdefined inChap.II,Sec.1,Eq.(24c).Itis (13) whereVL(w)isthepotential difference maintained exclusively bythe chargesonthesamelinewhereVL(w)ismeasured, andwhere isthepotential difference maintained byallchargesthatcontribute sig nificantly, including thoseontheauxiliary serieslines.Sinceco(w)is definedby itfollowsthat sothat SinceqL(W) CO(W)l(W) =Yew) /.d+ba/2[qr.,(w) ]CT=---Codw b../2YeW) 2rE Co(w)=ko(w)(14) (15) (16) (17) 402 TRANSMISSION-LINE THEORY [Chap.V whereko(w)isasdefinedin(5b),itfollowsthat YL(w)=qL(w)ko(w) 211"E(18) Using(7),thesymmetrical andantisymmetrical voltages maintained by chargesonthemainlineare Vt(w)=q~(w)(B-C+2In~) 211"E a ya(w)=q!(w)(B+C) L 211"E(19) (20) (21) (22b)(22a)whereBandCareasdefinedin(7b)and(7c). Thescalarpotential difference maintained acrossthemainlineata distance wfromthemiddleofthejunction isgivenby YT(w)==~1."qa(Y')[_1-~±(~-~)]dy' 211"Eb/2 RITR2TRibR'}b where R'T=~(Y'-£)'+(w-~)'+a' R'T=~(y,+£)'+(w-~)'+a' R,.=~(Y'-~)'+(w+~)'+a' R..=~(Y'+£)'+(w+~)'+a' andwhereqa(Y')isthechargeperunitlengthontheauxiliary lineatthe locationy'oftheelement ofintegration dy'.Asshowningeneralin Chap.II,Sec.1,onlytheleadingtermneedberetained intheexpansion aboutthepointwofthechargeperunitlengthqa(y')ontheauxiliary line,sothat qa(y')==qL(W) (23) With(23)theintegration in(21)isreadilycarriedout.Theresultis VT(w)==qL(W)(AI±A2) (24) 211"E where Al==sinh-l by'(ii)=-b a/2)2+a2 Theuppersignin(24)defines V~(w),andthelowersigndefines V~(w). Thenextstepintheevaluation ofCTistoexpresstheintegrand in (16)intermsofVL(w),asgivenin(19)and(20),andVT(w),asgivenin Sec.17] DISCONTINUITIES ANDNONUNIFORMITIES (24).Since403 (26) 7r€ where Co=In(b/a) (27) itfollowsthattheintegrand in(11)maybeexpressed asfollowsinthe symmetrical andantisymmetrical cases: A]+A2+B-C H ([c(w)-co]s=-co2ln(b/a)+Al+A 2+B_C==-co sw)(28) [c(w)-cola=-coAl-A2+B+C-2In(b/a)==-coHa(w) (29) Al-A2+B+C whereAlandA2areasin(25)andwhereBandCareasin(7b)and(7c). Thedetermination ofC~andC~depends onthesubstitution of(28) and(29)in(11)andtheevaluation oftheintegrals. Sincetheintegra tionhasnotbeencarriedoutinclosedform,graphical ornumerical methods mustbeusedifquantitative resultsaredesired. Asanalter native,theintegrands (28)and(29)maybeexpanded inseries,and approximate reasonably simpleformulas forCTobtained. Theirderi vationfollows. Anapproximate evaluation oftheintegrals in(11)maybecarriedout provided b/aissufficiently great.Asafirststep,itcanbeshownwith outapproximation that bB=In- -InD1a bC=In- -InD2a(30a) (30b) D1_=W-ba/2+vi(w-ba/2)2+b2 where (31)w-ba/2+vi(w-ba/2)2+a2 D2==w+ba/2+vi(w+ba/2)2+b2(32) w+ba/2+vi(w+ba/2)2+a2 B-C=InDD2B+C=2In~-InDID2 (33) 1 a If(33)isusedin(28)and(29)andthesearethendividedby2In(b/a) innumerator anddenominator, thefollowing expressions areobtained: 01=Al+A2+In(DdD 1) - 2 In(b/a) 02==Al-A2-InDID2 2In(b/a)(34a) (34b) Iftheratiob/aissufficiently great,both01and02arelessthan1,sothat H(w)==0-02+03- • • • (35) withappropriate superscript onHandsubscript ono.Itfollowsthat, 404 TRANSMISSION-LINE THEORY [Chap.V (36)ifonlytheleadingtermisretained, J.d+bG/2 J.d+bG/2 CT=-co H(w)dw==-co 0dw bG/2 bG/2 Co(J1±J2±J3 -J4) 2ln(b/a) wheretheJ'sareasdefinedbelow. Thefollowing approximate results applywhenthecondition d2»b2isvalid(ingeneral,thechoiced=lOb yieldsasatisfactory approximation): J.d+bG/2 J.d+bG/2b(d) J1== Aldw== sinh-1b/2dw==b1+In~ bG/2 bG/2 W-0 b (37a) J.d+bG/2 J.d+bG/2 b J2== A2dw== sinh-1+b/2dw b../2 bG/2 W 0 =(d+ba)Sillh-1d~b o-basinh-1~ +bInd+ba+V(d+baP+b2(37b) ba+Vb:+b2 J8==J.~/:bG/2InD2dw==bo[~1+~-1 -Inj(1+~1+~)] (37c) (37d)J.d+b../2 J4== InD1dw==b-a b../2 With(37a)to(37d),CTmaybedetermined directlyfrom(36)inallcases forwhichb/aisnottoosmall.Ingeneral,b/a>10leadstoafair approximation, asshownlaterforcertainspecialcases. Lumped Networks inGeneral. WithLTandCTdetermined forthe mainlineforboththesymmetrical andtheantisymmetrical cases,an appropriate network maybeconstructed foruseatthejunction withan auxiliary seriesline.Sinceinmostpractical applications theseriessec tionsinvolvepredominantly eitherthesymmetrical ortheantisymmetri calcasealone,thecomplication involved inaseparation intosymmetrical andantisymmetrical partsisavoided, andeitherL;.andC~orL~and C~mustbedetermined ratherthanbothpairs. Thedetermination ofsymmetrical andantisymmetrical valuesofLTG andCToforconnection intheauxiliary lines,asshowninFig.1704,is straightforward. Inthesymmetrical casethetwoconductors ofthe auxiliary lineareatthesamepotential, andtransmission-line theory hasnoapplication. Theimpedance oftheauxiliary conductors when drivenwithcodirectional currents mustbedetermined bythemethods ofantenna theory. Note, however, thataccount hasbeentakenof coupling between theauxiliary conductors andthemainlineinthe evaluation ofC~and1.J;'. Sec.17] DISCONTINUITIES ANDNONUNIFORMITIES 405 Whenthemainlineisdrivensothattheantisymmetrical caseobtains, theauxiliary linesdifferinnowayfromthemainline,sothatthefor mulasforL~aandC~aarethesameasthoseforL~andC~ifbandbaare interchanged. VAn"''' +[-------l __--L.---3+ ....Non·uniformlineW+ Non·uniformline'" ----(a)'f · ~~~-------olIf----~...---I (b) FIG.17.5.Antenna symmetrically drivenfromtwogenerators in(a)andwithstub supportin(b). Antenna r·· _'L._~niform line (a)••_~+ Uniform Iin~••...:i_ 1rAn...,..~:[~~~~iform line 12CT zLT (b) FIG.17.6.Lumped junction networks forthecircuitsofFig.17.5. Antenna DrivenfromTwoLines;Antenna withStubSupport.Ifthe separation baofthetwoconductors oftheseriessectionisreducedto zero,thesemaybereplaced byasingleconductor, asinFig.17.5or 17.6,insofarasthesymmetrical caseisconcerned. Thisconductor is acenter-driven antenna whoseimpedance cannotbedetermined from transmission-line theory. However, thetransmission-line junction effect andthecoupling between thefeedlineandtheantenna maybeobtained byspecializing thegeneral formulas derived inthissection. Inthis 406 TRANSMISSION-LINE THEORY [Chap.V mannertheconstants ofthejunction-zone networks showninFig.17.6 maybedetermined forusewithuniform-line theoryandtheimpedance oftheisolatedcenter-driven antenna, whichishereassumed tobeknown. Sincetheantenna inFig.17.5aisatrightanglestothelineandcarries thesumoftheequalandopposite currents ofthetwofeedinglines,the inductive junction effectisobtained directlyfromL;in(9)bysetting baequaltozero.Theformulais L}=b-a(38) anditistobeusedinthecircuitofFig.17.6a,inwhicheachfeedingline involves atotallumpedinductance L}. 2ln(b/a)+sinh-1(b/Jw2+a2) r-;,::;--l IO.lI-4\ll~~~-+--I___-+--+--+----4-_+____l ::l!<:J00.051--~~~~,..p..-.;:-=:I"-::::-+--+--4----4-_+____l ::l0.03I----t~~~~""':;::-P-=+:::::::-.~--=:;-c=:l-.,----+-___l-0~ I0.31k--+--+---+----jI-----+--+--+--+---:..-+---1 0.01~---+--l---l--=~~~~-""";;;J::::::::-+""""",,;:+:::::::::f--lO 2~40 0.005 ~--J..-J...-'-""--..L...-.i-.",;:..,.,...J,-.i........J.-...J-....L-.l....-L--l.--I..--'--l-..J 200100o2 4 6 8 101214 16 1820 w/b FIG.17.7.Thefunction -[CO(W)<l>l(W) -l]/coforantenna withstubsupport. Alternatively, ifinthecircuitofFig.17.5bthestubisadjusted in lengthtopresentasufficiently highinputimpedance atitsterminals compared withtheimpedance oftheantenna, thecurrententering the stubisnegligible compared withthatentering theantenna orleaving themainline.Itfollowsthatthepresence ofthestubmaybeignored exceptinsofarasitcontributes capacitively. Aseriesinductance L~need beusedonlyinthemainline,asshowninFig.17.6b. Thelumpedcapacitance CTrequired totransform thecircuitsinFig. 17.5aandbwithnonuniform linesintoFig.17.6aandbisobtained byomit tingtheterminA2in(24)andsettingbaequaltozero.Theresultis Al=sinh-1(b/vw2+a2)andB=C,sothat,withA2absent,(28) reducesto sinh-1(b/yw2+a2)+2ln(b/a)(39) Aplotofthisquantity, proportional totheintegrand in(11)forthe specialcaseunderconsideration, isgiveninFig.17.7usingw/basvari- Sec.17] DISCONTINUITIES ANDNONUNIFORMITIES 407 (40)ableandb/aasparameter. Theresultofanumerical evaluation of(11) using(39)isshowninFig.17.8insolidline.Forcomparison, acurve ofthesamequantity asevaluated fromtheapproximate formula(36) whichisvalidforsufficiently largevaluesofbfa-isalsoshown. For thecaseathand,J2in(37b)isomitted, andbaissetequaltozeroin J1,J3,andJ4.Withd=lObtheresultis •~_ 1+In(2d/b) ~_ 2 CT-bco2ln(b/a)-bcoIn(b/a) Theagreement oftheapproximate curve(shownindashedlineinFig. 17.8)withthemoreaccurate solidcurveisincreasingly goodasb/ais 1.2 0.8 Cr-beo 0.6 0.4 0.2 O'-'-........................... .L.-.........&..-...L--L...I ............L-'--...L-...L.- ..........L....I 4 6 810204060100200400 b/a FIG.17.8.Capacitance CTforthejunction network ofFig.17.5;computed for b=O.OIA. madelarger. Formostpractical purposes (40)maybeusedwhenever b/aexceeds20. Notethatinthestub-supported arrangement inFig.17.5bthecapaci tivecoupling between thestubandtheantenna maynotbeignored, since,asafirstapproximation, thechargeperunitlengthoneachwire ofthestubisthesameasonthemainlineandontheantenna. The equivalent network involves CTinparallel withtheantenna foreach sectionofline,oratotalof2CT,asshowninFig.17.6b. AntennaasEndLoadonTwo-wire Line.Ifthedistance babetween thetwoconductors oftheauxiliary lineismadeinfinite,thesimplecir cuitofFig.17.9aremains. Thisconsists ofatwo-wire lineend-loaded 408 TRANSMISSION-LINE THEORY [Chap.V byasymmetrical antenna. Inthiscasetheseriesinductance LTforuse inthecircuitofFig.17.9bisobtained from(9)bysettingbaequalto infinity. Theresultis b-aL8=La=--- T T 211'"11(41) Theshuntcapacitance CTisobtained byallowing batoapproach infinityin(28)or(29),whileWt==W-ba/2remains finite.Notethat I!LT ~L-- ~£---___~niformline CT IiLT Ca) (b) FIG.17.9.Antenna center-driven fromtwo-wire line.(a)Actualcircuitwithnon uniform linenearjunction. (b)Equivalent circuitforusewithuniform-line theory. Wtisthedistance 8 -zfromtheendofthelineandWisthedistance fromthepointmidway between thetwoconductors oftheauxiliary line. Forthespecialcaseathand At=sinh-t(vb) w~+a2 A2=0,B=sinh-t(wtfa)-sinh-t(wtfb),andC=In(b/a),sothat c(Wt)-Co Co sinh-t(b/Vw~+a2)+sinh-t(wtfa)-sinh-1(Wt/b)-In(b/a) sinh-t(b/vw~+a2)+sinh-1(wtfa)-sinh-t(wtfb)+In(b/a) (42) Thisquantity isplottedinFig.17.10asafunction ofw/b,withb/aas parameter. Bysubstituting (42)in(11)andevaluating theintegral numerically, thesolidcurveinFig.17.11isobtained. Ontheother hand,theapproximate formula(36)using(37a)to(37d)givesthefollow ingsimpleresults,withba:=00andd=lOb: In(2d/b) CT==-bco21n(b/a)-3 -bco21n(b/a)(43) Theapproximate valueof-CT/bco determined from(43)isshownin Fig.17.11inthedashed-line curve.Itisseentobeingoodagreement withthemoreaccurate numerically determined solid-line curveforvalues Sec.17] DISCONTINUITIES ANDNONUNIFORMITIES 409 ofb/agreaterthan10.Since(43)isactually thefirstterminaseries ininversepowersof2In(b/a),thisagreement forlargevaluesofb/ais verysatisfactory. o~ I 0.01t---+-+-- 0.OO51----+--+---j--t--+---'~-_+_-~::::=t_.......:;:3___i o2 4 6 8 10121416 18 20 w/b FIG.17.10.Thefunction -[CO(W)cI>l(W) -ll/coforantenna asendloadonatwo-wire line. 1.0 0.8 0.6 CT-bco 0.4 0.2lOb _ CT=.!.~[Sinh-l~-Sinh-l -F+sinh-l~-ln .£-jdw beobsinh-l~-sinh-l~+sinh-l_b_+ln.!!.. \ 0 a 0 .{iii+C a \ \\.,-_CT=_3_ \.beo2In(b/a) ".,....,...., "~"~......... I····~ _ CT=~(fOlded ~i·~I:;::-:-·::-:·. beoIn(b/a) o.......--'-..........--'-....L.--..L.--L..o--l.--'-...L.-l---L....I-.....L--'----l.....L...I 4 6 810204060100200400 b/a FIG.17.11.Capacitance CTforjunction network ofantenna asendload(Fig.17.7) andfoldeddipole(Fig.17.12);computed forb=O.OD.. Theanalysis ofthecircuitofFig.17.9amaybecarriedoutbysubsti tutingtheequivalent circuitinFig.17.9b,inwhichuniform-line theory maybeusedwithLTasin(41)andwithCTobtained fromFig.17.11. Theimpedance oftheantennatobeusedisitsisolatedvalue,sinceCT includes capacitive coupling tothelineandthereisnoinductive coupling. Itisevidentfromthetwospecialcasesrepresented inFigs.17.8and 410 TRANSMISSION-LINE THEORY [Chap.V 17.11thatreasonably accurate valuesofCTmaybeevaluated fromthe approximate formulas (36)and(37a)to(37d),provided b/aissufficiently great.Moreaccurate resultsdependontheevaluation ofthegeneral integrals obtained fromthesubstitution of(28)and(29)in(11). Junction Network forFoldedDipole. Thefoldeddipolediffersfrom thebalanced seriesjunction inFig.17.1inthatthereisnocontinuation Uniform'line:,: bNon·uniformline.t FIG.17.12.Foldeddipoleandequivalent circuit. ofthemainlinetotherightofthejunction. Atypicalfoldeddipoleis showninFig.17.12a. Inthiscase 19(w)=ko(w) 211"voko(w)=/.CO(l--~)dw' ba/2RaRb(44) whereRaandRbareasdefinedin(6).Theintegration gives ko(w)=B+In~a(45) whereBisasdefinedin(7b).Itfollowsdirectlythattheseriesinduc tanceforuseinFig.17.12bisgivenby b-aLT=--211"vo(46) Theevaluation ofCTparallels theanalysisfollowing (11).Thus VL(w)=q~S:)(B+In~) (47) VT(w)=-.!-[/.eoqa(Y')(_~-_1)dy' 211"E b/2 R1TR2T +~coqa(Y')(i1b-i?b)dy'](48) whereR1T,R2T'RIb,andR?bareasdefinedin(22a)and(22b).With Sec.18] DISCONTINUITIES ANDNONUNIFORMITIES 411 thesameapproximations asbefore,theintegrations leadto VT(W)==qL(W)A' 21rE 1(49) where A'=sinh- 1 b/2+sinh-t b/2 (50) 1 v(w-ba/2)2+a2 v(w-ba/2)2+b2 NotethatA~differsfromAtin(25)onlyintheoccurrence ofb/2in placeofbinthenumerators. WithVL(w)andVT(W)asdefinedabove, itfollowsthat A~+B-In(b/a) A~-InD1 c(w)-Co=-coA~+B+In(b/a)=-coA~-InD1+2ln(b/a) (51) whereD1isasdefinedin(31).Bysetting ~==A~-InD1 2In(b/a) thefirsttermintheseriesformofCTisgivenby C·J:d+ba/2dT=-co ~W ba/2(52) (53) Since A~differsfromAonlyintheoccurrence ofb/2inplaceofb,it followswith(37a)to(37d)that,withd=lOb, •In(4d/b)-1 . 1.35CT=-bco2ln(b/a)=-beoIn(b/a) (54) Acurveoftheapproximate valueof-CT/bco forthenetwork inFig. 17.12isshowninFig.17.11indottedline.Sincethecorresponding approximate curvefortheantenna asendloadisingoodagreement with themoreexactvalueobtained numerically, itmaybetakenforgranted thattheapproximate curveforthefoldeddipoleisalsoasatisfactory representation, provided b/aexceeds10. 18.ChangeinSpacingofaTwo-wire Line.Thedistance between the parallelaxesoftheconductors ofatwo-wire lineisblfromz=0toa pointz=Sl.Fromz=Sztoz=Sl+Srthespacingisbr•Bothcon ductorsofbothsections ofline,aswellastheshortpiecesjoiningthe twosectionsatz=8t,havetheradiusa.Itisassumed thatthefollow inginequalities aregoodapproximations: Sf»bl»a2(1) AsectionoflinenearthechangeinspacingisshowninFig.18.1.In ordertouseuniform-line theorywiththeconstant parameters lOland COlatallpointstotheleftofthechangeincrosssectionandlorandCOr totheright,itisnecessary todetermine theactualquantities le(w)and 412 TRANSMISSION-LINE THEORY [Chap.V c(w)ontheleftandle(u)l:l,ndc(u)ontheright.Thevariables wandu havetheircommon originatthejunction ofthetwolinesofdifferent spacing, asshowninFig.18.1.Intermsofthecoordinate z,w=8z-z andu=z-8z.Theinductances le(w)andle(u)maybedetermined by assuming thelinesodrivenfrombothendsthatacurrentmaximum is atthejunction. Subjecttothecondition (2) Rb=Vw'-w-+-bl(4a) R12=V(u'-+-W)2+n2(4b) n=j(bz+br) (4c)thecurrent isapproximately constant inmagnitude overdistances w~lOb,andu~IObrnearthejunction. Thecapacitances c(w)and FIG.18.1.Linewithchangeinspacing. c(u)maybedetermined byreversing thegenerator atoneend,sothata currentminimum andvoltagemaximum aremaintained atthejunction. Subjectto(2),thescalarpotential difference intherangew~IObzand w~IObrisessentially constant. Inductive Correction. Thevectorpotential atdWI(Fig.18.1)onthe surfaceofconductor 1atadistancewtotheleftofthejunction isgivenby AI~(W) ==_I[I.colew')(~~_~)dw'+f00leu')(_1__1)dU'] 41rJl 0 RaRb RllR12 (3) wherelew')isthecurrentinconductor 1atadistance w'totheleftof thejunction andleu')isthecurrentinconductor 1atadistance u'to therightofthejunction. Itisassumed thatbothsections oflineare balanced sothatthecurrentinconductor 2isthenegative ofthecurrent inconductor 1foreachvalueofw'oru'.Theupperlimits 81and8r havebeenreplaced byinfinity, sincetheconditions (1)on8zand8rlead tofinalresultsthatarethesameasthoseobtained with8z==00==8r• Theexponential retardation factorse-ilJRhavebeenreplaced byunity, sinceitisassumed thattheconditionI~bl«Iissatisfied. Thefollow ingdistances occurin(3): Ra=yew'-W)2-+-a2 Rll=V(u'-+-W)2-+-m2-+-a2 where m=j(bz-br) Itisassumed thatbz~br• Sec.18] DISCONTINUITIES ANDNONUNIFORMITIES 413 Sincethecurrentisessentially constant inthejunction zone,thelead ingtermsintheTaylorexpansions are leu')==lew)==lew') Thesubstitution of(5)in(3)andthesubsequent integration give(5) le(w)==W2(w)=2Ab(w)=_1_(2In~_Inw+yw2+bf Ilew) lew) 2".11 a w+yw2+a2 +Inw+yw2+n2 )(6) w+yw2+m2+a2 Thelumpedinductance requiredtopermittheuseofl~l=(1/".11)In(b,/a) totheleftofthejunction isdefinedby (7) whered==lObi.Afterthesubstitution of(6)in(7),theintegration may beperformed withtheresult 1 ( m2 ) LTI==- -m-a+ym2+a2+-2".11 d ==-_1_(m_a+.ym2+a2) 2".11(8) Theexpression forl~(u)islike(6),withwreplaced byuandb,bybr. Theevaluation ofLTrisstraightforward andresultsin L~1mbr~0Tr----2".112d(9) Sinced~lOb,thetermswithdin(8)and(9)arenegligible. Theseriesinductance oftheshortconductors joiningthetwosections oflinewithdifferent spacingisdefinedby where1[(jb I!2j11./2) ]L"=1(0) -bl/2--br/2AJI(Y)dy 1[(jbl/2 jbr/2) dy' ] A(y)= - - JI4".-bl/2 -br/2y(y-y')2+a2(10) (11) Theseexpressions maybecombined togive _ 1[jbl!2jbl/2 dydy' jbr/2jbr/2dydy' LJI-4".11-bl/2-bl/2.y(y-y'p+a2+-11./2-11./2Y(y-y'p+a2 jbl/2jbr/2 dydy' ]- 2 (12)-bl/2-br/2y(y-y')2+a2 414 TRANSMISSION-LINE THEORY [Chap.V Thefollowing integrals areobtained, subjectto(1): LII=1:.-[bzln2bz-Vb;+a2+a+bIn2br-Vb2+a2+a211'" a ra r -2(nIn2:-vn2+a2-min2;;+vm2+a2)](13) where n=j-(bz+b·r)m=j-(bz-br)bz~br Thetotallumpedinductance required atthejunction inserieswith eachconductor isj-(LTl+£11)'asshowninFig.18.2.Withthislumped tLTlfTIlLy bUniformlineC -C-C-+-C-)--,..,u....,ni .....fo-rm--,.,.Iin-e T bl IeTIIJ TrTCCle,1ColI01 - or'or.i tLy-------tLTl FIG.18.2.Equivalent network forthejunction inFig.18.1. inductance, theinductance perunitlengthofeachlinemaybeassumed equaltotheconstant value l~zorlorcharacteristic ofuniform-line theory. Capacitive Correction. Thescalarpotential atdWlonthesurfaceof conductor 1atadistance 'Wfromthechangeinsplwingisgivenbythe following integrals: cPI(W)==~[r«Jq(w')(J__~)dw'+r«Jq(u')(_1-~)du' 411'"EJo RaRbzJo RllR12 +f.1blql(Y')dy'+j-1brq2(y')dY'](14) lbrRllz -lblRuz Asin(3),exponential factorsin(14)havebeenapproximated byunity. Thechargesperunitlengthareql=qonconductor 1andq2=-qon conductor 2,withvariables w',u',andY'appropriate tothethreeregions. In(14) (15) Sincethescalarpotential difference between conductors 1and2is essentially constant nearandatthejunction whenavoltagemaximum ismaintained acrossit,thechargesperunitlengthonthetwosidesof thejunction wouldberelatedasfollowsifuniform-line theorywere accurate: COr COl Coz=In(bz/a)whereq(u') ..!...q(w')----- 1I'"E COr=fIl(br/a)7rE(16) (17) Sec.18] DISCONTINUITIES ANDNONUNIFORMITIES 415 Indetermining thedeparture ofthescalarpotential fromthatpredicted byuniform-line theorynearthejunction, itisadequate tousethelead ingtermintheunperturbed distribution ofcharge. Thatis, whereq(w")==q(W) q(U')==q(W) COr=kq(w) COl k=COr=In(bl/a) -COlIn(b,Ja)(18) (19) Thechargeperunitlengthontheconnecting wiresmaybeassumed to varycontinuously andlinearly. Thus [k-1]ql(y')=q(y')==q(w)1+(jbl-y')---m- q2(y')=-q(y') ==-q(w)[1+(ibl+y')k~1] -ib:~y'~ Thefollowing shorthand notation isused:(20a) (20b) k-1p=-m(21) Notethat,sincekdoesnotdiffermuchfromunityinmostcasesand mequalsorexceeds1,pisusuallyquitesmall.Withthenotation intro ducedin(21)andwith(18)and(20a,b),thefollowing expression is obtained from(14): Yew)=2epl(W) =q(w)I{CO(.l_-.!.)dw'+k{co(_1__1)du' 21rEJoRaRbl JoRuR12 J:1bldy'j-lbr dY'}+[1+p(jbl-y')]-R- [1+p(jbl+y')]-R(22) Ibr II-lbl II Thismaybeintegrated usingstandard formulas. Theresultis whereYew)=q(w)[21n~-FI(W)] 21rE a Fl(w)=Inw+yw2+b'f_kinw+yw2+n2 w+yw2+a2w+yw2+m2+a2 -csch-1~+(1+pbl)(CSCh-1~b-CSCh-l~) Im l n -p(Vw2+m2-w+viw2+b'f-Vw2+n2) Thecapacitance perunitlengthontheleftofthejunctionis 21rE Cl(W)=21n(bl/a)-Fl(w)(23) (24) (25) Thecapacitance perunitlengthofaninfinitelineisobtained bysetting 416 TRANSMISSION-LINE THEORY [Chap.V W=00,forwhichF(w~00)=O.Hence COZ=In(bz/a)(26) (32)Inordertouseuniform-line theoryontheentirelinetotheleftofthe changeinspacing,itisnecessary tointroduce alumpedcapacitance CTl thatcompensates fortheerrormadeinusingCoinsteadofCl(W).This capacitance isgivenby (d) COL (d Fl(w) CTl=Jo[Cl(W-Co]dw==2ln(bl/a)Jo1 -Fl(w)/[2In (bl/a)]dw (27) whered==lOb.Thisintegralmaybeevaluated numerically. However, forsufficiently largevaluesofbl/atheratioFl(w)/[2In(bl/a)]issmall compared withunity,and(27)maybeexpanded inpowersofthisratio. Inparticular, CTlmaybeapproximated bytheleadingterminthis expansion. Thus Cn==2lnCt~l/a)f,dFl(w)dw (28) Theintegration in(28)C3,nbecarriedout.Iftermsoftheorderof magnitude blareneglected compared withd2==(10bl)2,thefollowing expression isobtained: Cn==2lnCt~z/a){bl-a-ken-vm2+-a2 )+-j(l+-k)mInm -(1+-i;pbl)blInbl+-[1+-{(3bl-br)]nInn}(29) Theevaluation ofCTrforthesectionoflinetotherightofthejunction parallels thatcarriedoutforCTl-Thepotential difference atadistance utotherightofthechangeinspacingisgivenby V(u)=2~I(U)=q2~If,co(~a-~b)du'+-~f,co(;11-;12)dw' +J.!bl[1-'E(:;'-i;br)]dy'-I-ill,.[1+-'E(Y'+-jbr)]dY'j(30) !brk Ryr -lbl k Rur where Ra=V(u'-U)2+a2 Rb=V(u'-U)2+-b~ Ru=v(u+-W')2+-m2+a2R12=V(u+-W')2+-n2(31) Rur=V(i;b,.-y)2+-u2 Theintegrations canallbecarriedout,andV(u)expressed asfollows: V(u)=q2<::[2In~-F,.(u)] Sec.18] DISCONTINUITIES ANDNONUNIFORMITIES 417 whereFr(u)=Inu+vu2+b~_!Inu+y~2_ U+Yu2+a2kU+yu2+m2+a2 -csch-1~+(1+Pir)(cSCh-1*-csch-1~) +~(yu2+m2-U-yu2+n2+yu2+b';) Thecapacitance perunitlengthontherightofthejunctionis 2'1l'"E cr(u)=2ln(br/a)-Fr(u) Thecapacitance perunitlengthofaninfiniteline(u=00)is 1f'E COr=In(br/a)(33) (34) (35) (37) (39)Thelumpedcapacitance CTrrequired tocorrectfortheuseofCoin(35) inplaceofcr(u)in(34)is fd Carfd Fr(u) CTr=Jo[cr(u)-COr]du=2ln(br/a)Jo1 -Fr(u)/[2In (br/a)]du (36) whered==10br•Corresponding to(28)forCn,theapproximate expres sionforCTrwhenbr/aissufficiently largeis CTr==2lnCCbr/a)J.dFr(u)du Iftermsofmagnitude b;orlessareneglected compared withd2==(10br)2, thefollowing resultisobtained from(37): CTr==21nCtbr/a)(2br-a-~(n+br-Vm2+a2)+k0c1minm +(1+~~)brInbr- [1+Ik(3br-bl)]nInnI(38) Inaddition tothecorrections CTlandCTrrequired topermittheuse ofuniform-line theory,thecapacitance between thetwoconductors, eachoflengthm=j(bl-br),mustbeincluded. Anapproximate for mulaforthecapacitance between coaxialcylinders isgivenbyKiipfmiiller (Ref.12,p.66).Intermsofthenotation ofthissection,theresultis C~ '1l'"Em Tc-In(!!!/2br+m ) a"J2br-3m Thecomplete lumpednetwork required topermittheuseofuniform linetheoryoneachsideofthejunction oftwosections oflinewithdif- 418 TRANSMISSION-LINE THEORY [Chap.V ferentaxialseparations isshowninFig.18.2.LprandL"aredefinedin (9)and(13),Cnin(29),CPrin(38),andCpcin(39). 19.Right-angle BendinthePlaneofaTwo-wire Line.Aright-angle bendintheplaneofatwo-wire lineisshowninFig.19.1.Sincesucha bendisnotsymmetrical withrespecttothetwoconductors ofthetrans missionline,itnecessarily hasanunbalancing effect.Anapproximately equivalent lumpednetwork forthebendisdetermined byseparating the "1'--- w'------l0l T h1u' 1 duz dUl FIG;19.1.Right-nngle bendintheplaneofatwo-wire line. voltageandcurrentintosymmetrical andantisymmetrical parts,as follows: Yew)=V(B)(W)+V(a)(w) lew)=I(B)(w)+l(a)(w)V(u)=V(B)(U)+V(a)(u) I(u)=I(B)(u)+l(a)(u)(1) (2) Thesymmetrical combination consists ofthevoltage, charge,andcur rentmaintained bytheequalandcodirectional generators showninFig. 19.2a;theantisymmetrical combination involves thevoltage,charge,and currentmaintained bytheequalandopposite generators showninFig. 19.2b.Intermsofthecoordinates illustrated inFig.19.1,thetwocases maybesummarized asfollows: Symmetrical Combination l(lr)(w)=i[l(w)+l(u)] (3a) V<lr)(w)=i[V(w)-V(u)] (3b) qW~)=i~~)-qM] ~~ Sec.19] DISCONTINUITIES ANDNONUNIFORMITIES 419 Antisymmetrical Combination l(a)(w)=-MI(w)-leu)] V(a)(w)=-MV(w)+V(u)] q(a)(w)=-Mq(w)+q(u)](4a) (4b) (4c) Symmetrical Case.Inthesymmetrical case(Fig.19.2a)thereisa maximum ofcurrentandminimum ofchargeperunitlengthatthe junction. Sincethejunction regionextendsonlyoveraverysmallfrac tionofawavelength, itmaybeassumed thatthecurrentisconstant in thejunction whenthelineisdrivensymmetrically andthatthecharge perunitlengthiszero. Thezcomponent ofthevectorpotential atanelementdwonthesur facesofthehorizontal conductors 1and2(Fig.19.1)isgivenby A1z(w)=-41f"11z(w')PL(w,w')dw'==141z(w)f"PL(w,w') dw'(5a) 1rJl-b< 1rJl-b A2z(w)=4J...-r"12z(w')PL(w,w')dw'==-~1z(W) (00PL(w,w') dw'(5b) 1rJl)0 1rJl)0 (b)(a) -V1o,+ FIG.19.2.Symmetrical andantisym metrical drivingconditions. (a)Sym metrical conditions. (b)Antisymmet ricalconditions.(6a) (6b)Notethatz=-wand, asinChap. II,Sec.1,Eq.(9a), +V( 8)- Sincetheconductor isverylongto+r-- --.I.,;.;;w_--,{Iw=Iu=O theleftofthejunction, theupper Via) Iwqw=qu=max limitintheintegrals maybemadein- finity.Intheapproximate expres- sionsontherightin(5a,b)theleading termintheexpansion ofthecurrent atw'aboutthepointatwissub stituted forthecurrent. Thisis justified inChap.II,Sec.1.In addition, ithasbeenassumed that 12z(w') ==-11z(w'). Actually theline mustbeslightlyunbalanced, sothat thecurrents inthetwowiresarenot exactlyequalandopposite. However, intheapproximate determination ofthelumpednetwork thatisresponsible forthisunbalance itissuffi- . , Clentlyaccurate toassumethecurrents equalandopposite. 420 TRANSMISSION-LINE THEORY [Chap.V (10)Theintegrals in(5a)and(5b)maybeevaluated directly. Theresults are Au(w)=Ih(w)[2In~_Inb+w+v'(b+W)2+b2](7a) 411"V ab+w+v'(b+wp+a 2 A2z(w)=-lh(w:~(2In~_Inw+v'w2+b2) (7b) 411"V aw+v'w2+a2 Intherangew~0thevectorpotential difference is Wz(w)==Alz(w)-A2z(w) =l1z(w)l41n~_In[w+v'w2+b2b+w+v'(b+W)2+b2]j(8) 4rv a w+v'w2+a2b+w+v'(b+wp+a2 Theinductance perunitlengthoflineintherangew~0isdefinedby 19(w)=Wz(w)/lz1(w).Theconstant value19for uniform-line theoryis givenby19(w)whenw2islargecompared withb2orwhenw~00.The lumpedseriesinductance LTrequired tocorrectfortheerrormadein using19(w)intherangew~0isgivenby LT=J.d[lg(w)-lo]dw 1f,d[W+v'W2+b2W+b+v'(W+b)2+ b2] =-- In dw 411"V0W+v'w2+a2w+b+v'(w+b)2+a2 wheredisoftheorderofmagnitude oflOb.Theintegration canbe carriedoutdirectlytogive LT= -~{b[v'2-In(1+v'2)+In2]-a}= -.l-(1.21b-a)47rv 411"V (9) NotethatthisvalueofLTisconsiderably smallerthanthatgivenby Sec.12,Eq.(5),foralineterminated inaconducting bridge. Sinceone oftheconductors continues beyondthepointw=0inthecaseofthe bendbutnotinthecaseofthebridge,thisistobeexpected. Thesectionofconductor 1extending fromw=0tothecornerat w= -bisnotapartofthetwo-wire line.Itconstitutes anadditional inductance inserieswithconductor 1.Itisdefinedby L1=(bA1z(w)dwJol1z(w) bfor->5awhereAlz(w)isasgivenby(7a).Theintegration resultsin b(b).b bL1= -In·--0.05= -In- 211"Va,211"v a(11) Sec.19] DISCONTINUITIES ANDNONUNIFORMITIES 421 Sincetheverticalsectionoflineisidentical withthehorizontal section inFig.19.1,theequivalent circuitforthesymmetrical caseisasshownin Fig.19.3.Owingtothepresence oftheseriesinductance 2L1inonecon ductorandnotintheother,theline isunbalanced. Ifthevoltagedrop 2jwL1ll:eisreplaced byanequivalent + UniformlineVIs) > w~generator withemfVI=-2jwL llb W=O andthisistreatedintwoparts,as indicated inFig.19.4,itisclearthat equalandopposite voltages{VIin thetwoconductors contribute tothe balanced transmission-line currents, whereasequalandcodirectional volt- ages{VImaintain currentsinthetwo wiresofthelinewhichareequal andinthesamedirection. Inthe+~*)open-wire linetheseareantennaFIG.19.3.Equivalent circuitforsym- currents thatradiatesignificantly. metrically drivenlinewithbend. Theycannotbedetermined bytrans- mission-line theory. (Inthecorresponding problem withashielded-pair line,thecodirectional currents areinparallel,withtheshieldasthereturn conductor, asdescribed inChap.III,Sec.14.) u-----------.Lumped , network I I I I I I I I 1I-Lor ~..J+..-----.,..,.----~I'l"'_.. +v(')- FIG.19.4.Equivalent circuitforsymmetrically drivenlinewithbend. Antisymmetrical Case.Theantisymmetrical distribution illustrated in Fig.19.2bischaracterized byavanishing currentandamaximum of chargeperunitlengthatthecorners. Thescalarpotentials onthesur- 422 TRANSMISSION-LINE THEORY [Chap.V facesoftheconductors attheelements dW1anddW2(Fig.19.1)are 1 [f00 e-i~R aJ.00 e-i~RbcP1(W)==-4 q1(W')-Rdw'+q2(W')-Rdw' 1T'E-b a 0 b f00 e-i~RllJ.00 e-i~R12]+Q1(U')-Rdu'+q2(U')-Rdu'-b 11 0 12 1 [f00 e-i~RbJ.00 e-i~RacP2(W)==-4 q1(W')-Rdw'+q2(W')-Rdw' 1T'E-bbOa f00 e-i~R21J.00 e-i~R22]+q1(U')-Rdu'+Q2(U')-Rdu'-b 21 0 22(12a) (12b) Theantisymmetrical potential difference intherangesw~0andu~0is Va(w)=cP1(W)-cP2(W) ==-41[f 00Q1(W')P L(w,w')dw'-J.00Q2(W')PL(W,W') dw' 1T'E-b 0 +J-:Q1(U,')P1(w,u')du'-J.00Q2(U')P2(w,u')dU'](13) wherePL(w,w') isasdefinedin(6a)andwhere (14a) (14b) RaandRbaredefinedin(Bb).Theotherdistances in(12a,b)and(14a,b) are Rll=y(w+b)2+(u'+b)2+a2 R12=yw2+(u'+b)2+a2 Asaconsequence ofsymmetry,R21=y(w+b)2+U'2+a2 (15a) R22=yw2+U'2+a2(15b) Q1(U')=Ql(W') Q2(U')=Q2(W') (16) Accordingly (13)maybeexpressed asfollows: Yew)==-41If 00Q1(W')[P L(w,w')+P1(w,w')]dw' 1T'E-b-J.00Q2(W')[PL(w,w')+P2(w,w')]dw'l(17) Owingtothefactthatconductor 1islongerthanconductor 2bya length2b,thecondition C12(W)=-Q1(W)(whichapplieswhenwandu arelargecompared withb)cannotbetrueatandnearthecornerifthe totalchargeonconductor 1istobethenegative ofthatonconductor 2, Sec.19] DISCONTINUITIES ANDNONUNIFORMITIES 423 sothatthelineasawholeiselectrically neutral. Sinceconductor 1is longerthanastraight transmission linewiththesamemeanlengthas thebentonebyanamount bandsinceconductor 2isshorterbythe sameamount, itfollowsthatthechargeperunitlengthonconductor 1 mustdecrease asthecornerisapproached, whereas thechargeoncon ductor2mustincrease. Roughsketches ofthenatureofchargedis tributions ql(W)andq2(W)andoftheelectricfieldintheplaneofthe q --------~aw$~l(W) I''1oI • t++++++ +++ + I I I I I I I I I I : I;J/ " /,/,/ IJI :l:I::1::I ,I/I" " " ",,'+I I I I I I I I I'"11'11'"!l:I II:lII : : :I//,','I~<//" +------L------- .---1-::~:::<:: Q2(W) ::..""-,--+ --_....:---+ =:-:---- + ------- + ------ + FIG.19.5.Chargeandelectric-field distributions nearabendinatwo-wire line(esti mated). conductors areshowninFig.19.5.Sincethetruedistributions ofcharge areunknown andaroughapproximation isadequate inordertodeter minethecapacitance CTforthelumpedcorrective network, theunper turbeddistributions thatwouldobtainifthelinewerestraight withthe samecentrallengthmaybeused.Thatis,theuniform chargeperunit lengthqisassumed tobeonconductor 1andtheuniform value-qon conductor 2,bothbeingequalinlength. Theseuniform distributions areshowninFig.19.5together with thevaryingonesql(W)andq2(W). Notethattheuniform chargesaredistributed overtwoconductors of equallength,sothattheconfiguration ofconductors actually assumed isthatshowninFig.19.6.Thetotalcapacitance ofthisstructure when uniformly chargeddoesnotdiffergreatlyfromthatofthebendwiththe 424 TRANSMISSION-LINE THEORY [Chap.V actualnonuniform distributions. Witheachconductor inFig.19.6uni formlycharged intherangefromw=-b/2totheendofthelineand fromu= -b/2totheotherend,thescalarpotential difference inthe rangew~0isgivenby Yew)==4qj00[2PL(w,w')+P1(w,w')+P2(w,w')]dw'(18) 1rE-b/2 Thisexpression maybeintegrated, withthefollowing results: Yew)==.!L(2In!!-A-B+c) (19a) 21rE a where A;=:Inw+b/2+V(w+b/2)2+b2(19b) w+b/2+V(w+b/2)2+a2 B==Inb/2+Vb2/4+(w+b)2+a2=csch-12V-(w-+-b-)2-+- a-2(19c) V(W+b)2+ a2 b C==Inb/2+Vb2/4+w2+a2=csch-12V'-w-2-+-a-2(19d)vw2+a2b Thecapacitance perunitlengthisgivenby q 21rE c(w)==Yew)=2ln(b/a)-A-B+C(20) (21)Thelumpedcapacitance CTrequired topermittheuseofuniform-line theoryisdefinedby fd-b/2CT== [c(w)-co]dw-b/2f-:~~J....)-----_...JI , I I I I (22)1rEwhereCo==In(b/a) With(20)and(22),itisclearthat A+B-C c(w)-Co=Co2in(b/a)-A-B+C (23) (24)FIG.19.6.Configuration ofIf(23)issubstituted in(21),CTmaybeevalu conductors usedtodetermineated.Sincetheintegration hasnotbeenc(w)andCT.carriedoutinclosedform,numerical methods mustbeused.Alternatively, iftheratiob/aissufficiently great,theinte grandin(21)maybeexpanded ininversepowersofthequantity 2ln(b/a), andonlytheleadingtermretained. Whensubstituted in(21),thisgives cjd-bl2 CT==2ln(~/a) ..cb/2(A+B-C)dw Sec.191 DISCONTINUITIES ANDNONUNIFORMITIES 425 Ifdislargecompared withb,thefollowing integrals areobtained: jd-b/2 Adw==b-a -b/2jd-b/2 b2 (B-C)dw==bsinh-11 -2d==O.88b -b/2(25a) (25b) va -va+ FIG.19.7.Equivalent circuitforanti symmetrically drivenlinewithbend.r-Tumped-' +r--_--:-:-:-:-_~ __ .....!.........."'""-networkI Uniformline CI w~o W.- T: I II , L__"__J U(26)Itfollowsthat C T==(1.88b-a)co 2In(b/a) Theequivalent circuitfortheanti symmetrical bendisshowninFig. 19.7,whereCTisgivenby(26)and uniform-line theoryappliestothe transmission lineoneachsideofthe bendbeginning atw=0andu=0 (Fig.19.1). Thesuperposition ofthesym- metrical andantisymmetrical prob lemsyieldsthegeneralcaseofaline drivenononesideofaright-angle bendbyagenerator Vo=Va+Va andloadedontheothersidebyanimpedance Zsthatsatisfiestherelation -lsZs=Va-Va.Theappropriate lumpedconstant network isshown inFig.19.8,withLTgivenby(9),L1by(11),andCTby(26). r1-------------, 11I"2LTLumped +_~~~~---- ..........rJT..........CT+netw~rk -----~ll.llJl~ -"2~ '"2LT+ +t I I I t , 1 1 I "2LT-Lr L________ ~.J Zs FIG.19.8.Equivalent circuitforright-angle bendintheplaneofatwo-wire line. 426 TRANSMISSION-LINE THEORY [Chap.V 20.BendsandTJunctions inBalanced Shielded-pair Lines.Bal ancedshielded-pair linesbehaveessentially likeopentwo-wire lineswith different lineconstants. Iftheshieldisnottooclosetotheinnercon ductorscompared withtheirseparation, thelumpedcorrective networks determined fortheopentwo-wire lineforachangeinradius(Sec.13), abend(Sec.15),a Tjunction (Sec.16),seriesbran.ches (Sec.17),ora changeinspacing (Sec.18)maybeadapted foruseincorresponding situations inthebalanced shielded-pair line.Iftheinductances and capacitances perunitlengthcharacteristic oftheshielded-pair linesare substituted forthecorresponding quantities fortheopen-wire line,rea sonableapproximations oftheappropriate corrective networks maybe obtained. 21.BendinaCoaxialLine;TJunction. Theanalysis ofabendina coaxiallinecannotbecarriedoutreadilyusingthequasi-one-dimensional integrals forthescalarandvectorpotentials owingtothecomplicated three-dimensional natureoftheproblem. However, sinceitispossible (seeChap.I,Sec.11)toconstruct acagetransmission linethathas essentially thesameelectrical properties asagivencoaxiallinewhen botharestraight,itmaybeexpected thatthebehavior ofthecagetrans missionlineataright-angle bendmustapproximate thatofacoaxialline. Evidently theapproximation improves asthenumber ofconductors in thecageisincreased. Itmaybeassumed, therefore, thatalumpednet workthatcorrects forjunction-zone errorsinacagetransmission line maybeappliedtoanequivalent coaxiallineasareasonable estimate. Consider thebendshowninFig.21.1,inwhichthecoaxial shield of innerradiusa2isreplaced byafour-conductor cageofthesameradius inaregionoflengthd=10a2ineachdirection fromtheright-angle bend atw=0,y=O.Thecagelineistobeusedonlyfordetermining the corrective network oflumpedelements. Thebendinthefive-wire cagelineisacombination (withsomeadded complications) ofthetwotypesofbendsinatwo-wire linedescribed in Sees.15and19.However, theunbalance resulting fromthedifferences inlengthbetween conductors 1and3(Fig.21.1)andthecentralcon ductormaybeignored, sincetheequivalent generator inconductor 1is equalandopposite tothatinconductor 3andthetwoconductors are connected inparallelatw=10d,U=10d.Thatis,thecoaxiallines beyond w=10d,U=10darenotunbalanced, since13issmallerthan -·Vby justtheamountthat11exceeds-iI,sothat11+13=-jI, where1isthecurrentinthecentralconductor. Intheanalysis ofthecoaxialline(Chap.I,Sec.6)thepotential of theshield(innerradiusa2)iszero,sothatthepotential difference is equaltothepotential ofthecentralconductor (radiusal).Letitbe assumed asanapproximation thatthesameistrueatthebendwhen thefour-wire cagereplaces theshield.Inthiscasethepotential differ- Sec.21] DISCONTINUITIES ANDNONUNIFORMITIES 427 enceismeasured between thecentralconductor andwires2and4inthe cage(Fig.21.1).Lettheradiusofthecirclearoundwhichthecon ductorsofthecageareplacedbethesameastheradiusa2ofthecoaxial line. Incalculating theaxialvectorpotential onthecentralconductor, it maybeassumed thatthecurrents inallfourconductors ofthecageare equal;thatis,11=12=13=14=-{.]incalculating thecorrective inductance LT.Thatis,theunperturbed currentisusedincalculating '3(w)/------lOa2-----.I w-; I@Idw' I Ty--.... Coaxialline FIG.21.1.Right-angle bendinacoaxiallineapproximated byacageline. theperturbation produced bythebend.Theapproximate vectorpoten tialatapointwonthecentralconductor ofradiusalisgivenby Wz(w)==Az(w)=14(w)[ko-F\(w)] (la) 7r1l whereko-Fl(w)=={d(l..__1_)dw'_(ddw'_fddw'(lb)JoRa2Rb Ja24Rb-a24Rb and Ra=V(w'-W)2+aiRb=V(w'-w)2+a~ (2) Thecorresponding expression forthescalarpotential involves thecom plication thatthechargeperunitlengthontheshorterconductor 1must begreater-and thatinthelongerconductor 3smaller-than thatin conductors 2and4.Account couldbetakenofthisdifference asin Sec.19,butthesimplerassumption ql==q2==qa==q4==-iqwillbe used.Theapproximate scalarpotential is . q(w)Yew)=tPl(W)=-4-[ko-Fl(w)+F2(w)] 7rE(3a) 428 TRANSMISSION-LINE THEORY [Chap.V whereko-FI(w)isasin(1b)and "focidy'focidy'J.cldy'fcldy'F2(w)==- - - - ----oROT 02R2T (124RIT -aa4R3TC3b) Thefollowing distances areinvolved: ROT=Vy'2+w2+afR2T=R4T=Vy'2+w2+a~2(4a) RlT=Vy'2+(w-a2)2 RaT=Vy'2+(w+a~)2 (4b) Theintegrals Fl(w)andF2(w)maybeevaluated without difficulty. Subjecttothecondition d2»a~,thefollowing resultsareobtained: ko=2ln~ (5)al F()-1.w+Vw2+a~+1 ( •h-lW-a2 lw-n 4sm-- w+vw2+af a2 +sinh-lw+a2_2sinh-l~) (6) a2 a2 F2(w)=i[In(w2+a~)+InIw2-a~1-2ln(w2+an Iw-a2\ w+a2] +csch-l . -csch-l-- (7)a2 a2 Thevariable inductance perunitlengthis w~O le(w)==Wz(w)=_1'ko-Fl(w)]I(w) 411"11 L Thevariable capacitance perunitlengthis c(w)==q(w)= 411"E Yew)ko-Fl(w)+F2(w)(8) (9) (10) (12)Theconstant valuesl8andCoareobtained byallowing wtoapproach infinityin(8)and(9).SinceF1(w~00)~0andF2(w~.00)~0,it followsthat ko1a2 411"E 211"Ele=-=-In- Co=-=;-----;,---,~o411"11 211"11al koIn(a2/al) Thelumpedinductance LTandcapacitance CTthatcorrectfortheuse ofl&andCoinplaceofle(w)andc(w)are LT=fodW(w)-19]dwCT=fod[c(w)-co]dw (11) Theevaluation ofLTissimple. With(8)and(10)andthecondition d2»a~,thefollowing isobtained: 1fod .a2-alLT= - - FI(w)dw= --- 411"110 411"11 Sec.21] DISCONTINUITIES ANDNONUNIFORMITIES 429 (13)Theevaluation ofCTinvolves theintegral in(11)with(9)and(10): CT=411"Ef.dFl(w)-F2(w)dw k501 -[Fl(w)-F2(w)]jko (14)Ingeneral,thisintegralmustbeevaluated numerically. However, when koissufficiently great,theintegrand in(13)maybeexpanded inpowers ofl/ko,andtheleadingtermusedasanapproximation. Thisis CT==4~Ef.d[Fl(W)-F2(w)]dw Subjecttothecondition d2»b2,thismaybeintegrated intothefollow ingsimpleformula: (15) r-----------, ---;-rUn:;:;if~orm=lin:':"e---+1 LumpednetworkI I I__ -=w'"-=Si:::,.:O:..-__--i' : Il -frJ (16)b-a -271'"V 211"Eo(b-a) 3k~ whereko=2In(b/a)andbisthedis tancebetween thetwoconductors, FIG.21.2.Lumped network forbendin coaxialline.eachofradiusa.Sincetheinduc- tanceperunitlengthofthetwo-wire lineis19=(1/1I"v)In(b/a),theratiosSincethelineisidentical inbothdirections fromthebend,thesame valuesofLTandCTapply. Thecomplete equivalent network isshown inFig.21.2. Itisinteresting tocompare theapproximate formulas (12)and(15) forLTandCTforthecoaxialline withthecorresponding onesforthe two-wire line.Theselatterareob tainedfromSec.15,Eq.(18a),with 8=11"/2,andSec.15,Eq.(27a),with M(8)=1.Theyare LT. b-a 19=-~ two-wireLT. a2-al lo= --k-o- coaxial(17) areessentially thesame.NotethatCTisnegative forthetwo-wire line andpositiveforthecoaxialline. TSectioninCoaxialLine.Theanalysis ofa Tsectioninacoaxialline maybecarriedoutapproximately bysubstituting acagefortheshield inthejunction region,asshowninFig.21.3,andproceeding inamanner paralleling thatfortheright-angle bend. 430 TRANSMISSION-LINE THEORY [Chap.V w~u r(u) ® FIG.21.3.ShuntTsectioninacoaxiallineapproximated byafive-wire cagelinein thejunction zone. Ground screenAntenna ~--" :~a2+ -JJt:.- ~1 FIG.22.1.Antenna overagroundscreen.22.EndCorrection foraCoaxialLineWhenDrivinganAntenna over aGroundScreen.127Consider acylindrical antenna thatistheexten sionoftheinnerconductor ofacoaxiallinethrough aholeinacon ducting plane.AsshowninFig.22.1,theradiusaloftheantenna is alsotheradiusoftheinnercon ductoroftheline;theradiusa2ofthe holeinthegroundscreenisalsothe innerradiusofthecoaxialsheath. Theadmittance apparently loading thelineatitsendw=0isYsa;with perfectconductors thisistheadmit tancelooking towardtheloadat tp='A/2. Conventional transmission-line formulas arebasedontheassump tionsthat(1)thechargesperunit lengthql(W)andq2(W)onthetwo conductors ofthelineatagivencross sectionwareequalandopposite, so thatq2(W)=-ql(W), andthat(2)thecapacitance perunitlength, definedasc(w)=V(w)/q(w) ,isequaltotheconstant capacitance per unitlength Co,characteristic ofaninfinitely longlineforallvaluesofw. Thesearegoodapproximations atdistances dfromw=0whicharelarge compared witha2-al.Nearw=0,q2(W)isnotexactlyequalto-ql(W), andc(w)isnotaconstant. Sinceprimary interestisintheantenna and innerconductor, c(w)mustbedefinedintermsofthechargeperunit Sec.22] DISCONTINUITIES ANDNONUNIFORMITIES 431 lengthontheinnerconductor. Changes inq(w)nearw=0andvari ationsinc(w)from Coareconsequences oftransmission-line endeffect andofthecapacitive coupling between theline,ontheonehand,and theantennaandgroundscreen,ontheother. Theidealtheoretical admittance Yo=I/Zooftheantenna assumes thattheantennaisdrivenbyadiscontinuity inscalarpotential atw=o. Therelations between Y.aandYomayberepresented approximately as follows: wherea2Y.a~Yoas-~1at Y.a==Yo+jwCT~>1at CT=fod[c(w)-co]dw(1) (2) (3) Evidently, ifuseistobemadeoftheidealadmittance Yoinconjunction withthemeasurable apparent admittance Y.a,aknowledge ofCTis necessary. Sincethevectorpotential atwisdetermined fromthecurrentinthe innerconductor (asshowninChap.I,Sec.6)andthisiscontinuous at w=0,itfollowsthatle(w)==19forallvaluesofw~0,sothat (4) Therefore thecorrective terminal-zone network consists oftheshunt capacitance CT,definedin(3). Theevaluation ofc(w)and,fromit,ofCTmaybeaccomplished inthe usualmannerbycalculating thescalarpotential difference Vatadis tancewfromtheendoftheline(Fig.22.1).SinceitisshowninChap.I, Sec.6,thatthepotential cl-2(W)calculated fromthechargesontheinner surfaceoftheshieldcontributes nothing tothepotential difference V(w)=cl-t(w)-cl-2(W),itissufficient tocalculate V(w)fromthecharges ontheinnerconductor, theantenna, andthegroundscreen. However, sincetheinnerconductor joinstheantenna withacontinuous distribu tionofcharge,thepotential difference VLa(w)forw~°whichismain tainedbythechargesontheinnerconductor andtheantenna isessen tiallythesameasifthecoaxiallinecontinued. Thatis, (5) It(ollowsthat,withV(w)=VLa(W)+Vg(w),whereVg(w)isthepoten tialdifference atwonthelineduetothechargesonthegroundscreen, V(w) 1 [ 1 Vg(W)] qL(W)=c(w)=C;;+qL(W)(6) 432 TRANSMISSION-LINE THEORY [Chap.V whereReferring toFig.22.1,thepotential difference Vg(w)is Vg(w)=_1(21rfcoq(r')(_1__1)r'dr'dO' 471"€)0)a2271"RlTR2T RlT=yr'2-2alr'cos0'+ai+w2 R2T=yr'2-2a2r'cos0'+a~+w2(7) (8) andwhereq(r')isthetotalchargeonaringofunitwidthatradiusr'. Sincethechargeperunitlengthonthelinemustbecontinuous, itmay beassumed thatq(r')==qL(W),sothat Vg(w)=q4L(W)Fg(w) (9) 7I"€ where FoCw)==!f1rfco(_11_)r'dr'dO' (10) 71")0)a2RlTR2T Thesubstitution of(9)in(6)gives c(w)-Co Fg(w) (11) Co= -Fg(w)+21n(a2/al) sothatthelumpedshuntcapacitance required tocorrectfortheuseof Coinplaceofc(w)is fa fa Fg(w) CT=)0[c(w)-Co]dw= -Co)0FoCw)+2ln(a2/al)dw(12) whered==10a2. Theexactevaluation ofFg(w),asdefinedin(10),leadstocompHcated integrals ofellipticintegrals. Anapproximate evaluation isaccom plishedbydividing therangeofintegration withrespectto0'intofour regions, ineachofwhichtheintegrand isassumed tobeconstant ata middlevalue.Bychoosing themiddlevaluesinthefullrangeof271"at o=0,71"/2,71",and371"/2,theintegral in(10)maybeapproximated as follows: Fg(w)=={-fco( 1 _ 1 ) dr' )a2vCr'-a2)2+w2vCr' or"al)2+w2 +.!.fco(. 1 _ 1 ) dr' 2 )a2yr'2+a~+w2yr'2+ai+w2 +{-fco(1 _1.) dr'(13) )a2vCr'+a2)2+w2vCr'+al)2+w2 Thisexpression integrates into F( )=.!.[ .h-1a2-al+1w2+a~ gW 4SIn n2+2W Wa1 _2lna2+y2a~+w2_In2a2+y4a~-w2](14) a2+ya~+ai+w2a2+al+y(a2+al)2+w2 Sec.22] DISCONTINUITIES ANDNONUNIFORMITIES 433 Sincethelasttwotermsin(14)differonlyslightlyfrom1evenwhen w=0,theymaybeneglected. Theremaining termsmaybearranged asfollows: Thefunction [c(w)-coJicoobtained bysubstituting (15)in(11)is represented graphically inFig.22.2,withw/a2astheindependent varia bleanda2/alasparameter. Although curvesareshownforawiderange 1.0r----,--~--.,---__r_-_.,---, 0.81----+--__+---f---+-----t----i 0.6t-----t---+----"1r---+---j-----t 0.41\----t---+----lr---+---j-----t 02 ~I0.1'i~o0.081---4l~~,--1----+----+---+---1 ,"(;,0.061--~~~~1----+----+---+---1 I0.041----'\~........,.~~...,-7I---, ---k--'--+-~ 0.011----+---1- 0.0081----4---+-- 0.0060L.--...L--...J----'--~----L-.;:::w wja2 FIG.22.2.Thefunction -[c(w)-col/coforacoaxiallinedrivinganantenna overa groundscreen. ofvaluesofadal,thoseforwhicha2/alissmallerthanabout7arenot accurate fordetermining CT.Thisfollowsfromthefactpointedoutin thefootnote relatingtoChap.I,Sec.3,formulas (30a,b),thattheinte gralforthescalarpotential onthesurfaceofacylindrical conductor, whenexpressed intermsofthechargeperunitlength,isagoodapproxi mationonlywhentheintegration isextended overdistances thatareat least5alineachdirection fromw.Sincethedifference c(w)-Cois actually significant onlyoverarangeofwfromzerotoverysmall integral multiples ofa2-aI,itisclearthat,inordertohavea2-al greaterthan5al,adalmustbeatleastasgreatas7.Thevalidity of thisargument maybequestioned onthegrounds that,inevaluating the partofthepotential determined bythechargesontheinnerconductor andontheantenna, theintegration isactually carriedoutoververy 434 TRANSMISSION-LINE THEORY [Chap.V muchgreaterdistances than5al.Moreover theratiooftheuniform chargedistribution ontheinnerconductor andtheantennatothepoten tialdifference duetothisdistribution yieldsonlytheconstant co.Butthis argument overlooks thefactthatinthisapproximate analysis theratio ofpotential difference tochargeisobtained byassuming aconstant chargeanddetermining theresultant potential difference asafunction ofw.Actually itisthepotential difference whichisconstant andthe chargewhichisafunction ofw;andthenonzero valueofc(w)-Co withinadistance oftheorderofmagnitude ofa2-atoftheendofthe lineproperly corresponds toachargedistribution, andnotavoltagedis tribution, whichisnonuniform overthisdistance. Therefore thecorrect calculation ofpotential difference fromthepotential integrals usingthe truechargedistribution wouldinvolvethedetermination ofaneffect produced byavariation inthechargeperunitlengthwhichisconfined toarangeoftheorderofmagnitude ofa2-at.Thecontribution to thepotential obtained fromanintegration oversuchadistance isnot accurate unlessthedistance isatleast5atineachdirection. Sincethisis approximately trueonlywhena2/at ~7,itfollowsthatthecurvesin Fig.22.2areusefulfordetermining CTonlyoverthisrange. Bydetermining theareasunderthecurvesinFig.22.2(whendrawnto alinearscale)intherangew/a2=0tow/a2=10bynumerical methods, CTasdefinedin(12)maybeevaluated. ItisshowninsolidlineinFig. 22.3asafunction ofadatfortwovaluesofat/X.Aspointedoutabove, onlytherangesadat>7aresatisfactory approximations. Thedimen sionlessquantity -CT/a2cO isshowninsolidlineinFig.22.4foradat greaterthan7. Inordertoobtainanexpression forCTforvaluesofa2/atnear1,it maybenotedthat,whena2-atissmallcompared withatanda2,the potential atanypointinthecoaxiallineisdetermined principally by thechargesontheadjacent parallelsurfaces, sothattheymayberepre sentedapproximately byparallelplanes,i.e.,cylinders ofinfiniteradius insteadofsections ofcircularcylinders. Moreover, since(a2-at)/Xois necessarily smallcompared with1,thedistribution ofchargenearw=0 mustcorrespond closelytoanelectrostatic one.Thissuggests thedeter mination ofCTforacoaxiallinewitha2-atverysmallbyrepresenting thecoaxiallinebyaparallel-plate regionextending fromw=0tow=00, withtheplatesseparated adistance a2-at.Oneoftheplatesextends overtherange-00~w~00;theothermakesaright-angle bendat w=0andthencontinues toinfinity. Thedistribution ofsurfacechargeperunitarea 1'Jonboththestraight andthebentplatesmaybedetermined byconformal transformations usingtheSchwarz-Christoffel formula. Thepartofthesolution ofinter estintheproblemathandisthedistribution ofchargeonthestraight plateintherangefromw=0tow=00,whereitisparalleltothe Sec.22] DISCONTINUITIES ANDNONUNIFORMITIES 435 --Approx.theoryfora2/aJ>7 .------Approx.theoryfora2/aJ<1.2 -'-'-Extrapolated theory0.20 0.15 ~,3: tS I 0.10 0.05"/i././/., I/.. II . ,//..//../,..I ".·7 --Theory: a2/aJ~7 ..••.•Theory: a2Ial~1.2 _.-Extrapolated theory.........-.....:---...../' ..........r--, /- / / V ~,~...~0.3 ~ ~ C,)I0.20.4 0.1 o11.52 3 4 6 8 10152030 a2/al FIG.22.4.Capacitive endcorrection forantenna drivenfromacoaxiallineovera groundscreen.o~-..................L.-"",,- ........-.....--L...,......-""""'- ........--..L-,- ............o...-""""" 1 5 10azlaJ15 20 FIG.22.3.Theoretical endcorrection foracoaxial linedrivinganantenna overa groundscreenat'"=60cm. 0.5 secondplate.Itisthisrangewhichcorresponds totheinnerconductor ofthecoaxiallinewhenthishasaninfiniteradius. Thestaticdistribu tionofchargemaybeobtained byapplying apotential Vacrossthe plates. Thetransformations involved inthesolution forthesurface densityofcharge 1'/aredescribed intheliterature.1,30.32Thepertinent resultistheratioofchargedensity 7J(w)onthestraight plateatadis tance Wfromtheplanew=0,wherethesecondplateisbenttothe 436 TRANSMISSION-LINE THEORY [Chap.V chargedensity 1](00)sufficiently farintotheparallel-plate regionsothat aconstant valueisreached. Thisratiois wheretherelation1](W) 1 1](00)=VI-t(16) -~[2VI-t-In(VI-t+1)+In(vT="t-1)] 7r t~0(17) obtains. Aplotof1](w)/1]( 00)asafunction ofw/(a2-al)isgivenin Fig.22.5forthepartofthestraightplatewhichformshalfoftheparallel plateregionnearW=O.Thechargedensityonthisplatedecreases as 1](W) 1](OQ) 0.8O~--'----'---'--L-.-..I.--"---"---..L.-.o---'---'--'-....L..-.L-...o.---' 0.5 1.0 1.5 w/(aZ-al) FIG.22.5.Chargedistribution onplaneAB. Wdecreases towardzero,whereasitisreadilyshownthat,onthecorre sponding partofthebentplate,thedensityofchargeincreases asW=0 isapproached fromwithintheparallel-plate region. Noteparticularly thattheentiresignificant variation in1](w)occursinadistance ofmagni tudea2-alfromw=O.Arepresentation oftheelectric-field linesand theequipotentials isgiveninFig.22.6.Itissignificant tonotethatthe electriclinesarecurvednearw=0,sothattheelectriclinethatendsat A(Fig.22.6)onthestraight plateoriginated atapointonthesecond platewhichisnotatw=o. Sec.22] DISCONTINUITIES ANDNONUNIFORMITIES 437 SinceVistheconstant potential difference, itfollowsthat 27ral1](W) =c(w) 27ral1](00)Co(18) --, ,/'"~....--Equi~otential _'/, hne /.........//\,.X'" /\.-'{. "~.,.--"-..-Electric :-/"~-- \ \ fieldline I' III'-1+1--:-1+:-_I_r--I--']:-r-l-+']:-r-:-.-,.1-:-/+: 1-,..1 I I I-v+-Ia2-atl.- FIG.22.6.Equipotential linesofelectric field.lengthofaninfinitely longline. (19) withalinmeters. Forthetwo valuesofalforwhichcurvesare showninFig.22.3,CT= -0.0068 p.p.f forthevalueofal=0.179,and CT=-0.0090 p.p.fforal=0.238cm.Thesearethelimiting constant valuesthataregoodapproximations fora2/al<1.2.Theyareshown dottedinFig.22.3. Withapproximate theoretical valuesavailable fora2/al>7andfor a2/al<1.2,itispossibletoconstruct smooth, continuous curvesforthe rangeofa2/albetween 1.2and7.Suchextrapolated curvesareshownin Fig.22.3.Fromthesetheratio-CT/a2cO maybecomputed, andthe singleextrapolated curveshowninFig.22.4determined. Sincebothof theextrapolated curvesinFig.22.3mustyieldthesinglecurveinFig. 22.4,acheckontheshapesofthecurvesinFig.22.3isprovided, and theaccuracy ofthesinglecurveinFig.22.4istherebyenhanced. Itmaybeassumed thatthecombined curveforthedimensionless ratio -CT/a2co inFig.22.4isasatisfactory approximation ofthelumpedcor rectivecapacitance CTrequired ifuniform-line theoryisusedtodeter minetheproperties ofanantenna asendloadinthearrangement in Fig.22.1.Satisfactory experimental verification ofFig.22.4hasbeen obtained byHartig.120.l27where Coisthecapacitance perunit HenceFig.22.4isalsoameasure of thechangeincapacitance perunit areaaswapproaches zerofrom infinity. Inordertodetermine thelumped capacitance CTrequired tocompen satefortheerrormadeinusingCoin steadofc(w)nearw=0,itisneces sarytoform(3)andtochoosed sufficiently greatsothatc(d)==co. Theevaluation ofCTfrom(3),using (18)andFig.22.5,gives CT==-0.0683(2 7ralE) ==-3.8al p.p.f PROBLEMS 1.Theinputterminals 11ofasymmetrical two-terminal-pair network arecon nectedasloadtoamaintransmission line.Theoutputterminals 22ofthenetwork 438 TRANSMISSION-LINE THEORY [Chap.V areconnected toanauxiliary short-circuited sectionoflineofvariable lengthi.The complex reflection coefficientr=reN'oftheloadterminating themainlineat11is determined frommeasurements onthemainlinetohavethefollowing valuesforthe indicated lengthsioftheauxiliary line: i/A........... ....01 1 3 8" 4" 8" r................. 0.52 0.52 0.85 0.93 1/1,deg............. 62 127 214 235 Determine thepowerdissipated inthenetwork andthepowertransmitted toa matched loadconnected acrossterminals 22. 2.Determine theimpedance elements ofa Tsectionthatisequivalent tothenet workbetween terminals 11and22,asdescribed inProb.1. 3.Thedielectric constant ofasampleofmaterial ismeasured usingthemaximum shiftmethod. Thethickness ofthesampleis2em;themeasured maximum shiftis 20ematawavelength of150em.Whatisthedielectric constant ofthesample? 4.Whatwouldbetheminimum shiftinProb.3ifthesamplewere4emthick? Howisthisrelatedtothemeasurement ofdielectric constant usingacoaxialcavity andamethodinwhichtheresonant lengthofthecavityisdetermined whenthe cavityisemptyandagainwhenthesampleisplacedagainstthepistonterminating oneendofthecavity? 5.Foragivensampleofthickness 0.2emtheminimum shiftis4em,andthe maximum shiftis10ematawavelength of1m.Whataretherelativedielectric constant andpermeability ofthesample? 6.Twopolystyrene beads(E,=2.6)eachoflength0.25in.arespacedadistance v between adjacent edgesinanair-filled coaxiallineterminated inamatched wideband load. (a)Determine vsothatthestanding-wave ratio8=cothponthemainline between thegenerator andthenearerbeadisunityat3,000Me/sec. (b)Withvasin(a)andassuming theloadtobematched continuously, determine p and8asthefrequency isvariedfrom1,500to9,000Me/sec. Plot8asafunction offrequency onsemilogpaper.(Itissufficient todetermine 8at1,500,2,000,3,000, 6,000,and9,000Me/sec.) 7.Twosectionsofflexiblecoaxiallinearejoinedbyaconnector. Thelineisfilled continuously withpolystyrene ofdielectric constant E,=2.6.Theconnector has innerandouterconductors ofthesamesizeastheline,butitisfilledwithairfora lengthof1em.Thejoinedlinefeedsamatched load.Determine thestanding-wave ratiointhesectionoflineoneachsideoftheconnector whentheoperating frequency is3,000Me/sec. 8.Atwo-wire linemadeofNo.9copperwirespaced2embetween centersistobe matched toitsloadwithamovable open-ended singlestub.Determine thelumped elements ofallnecessary junction- andterminal-zone networks iftheapparent imped anceoftheloadisknown. 9.Averticalantenna isdrivenoverahorizontal groundscreenbyacoaxialline. Theinnerconductor ofthelineanditsextension astheantenna areofcopperwitha radiusof0.4em.Thecoaxiallinehasaninnerdiameter of4em.Whatisthe magnitude ofthelumpedcapacitance CTrequired tocorrectforendeffect? 10.Thetheoretical impedance ofacenter-driven antenna atantiresonance is Z=840+jOohms.Whatapparent impedance ismeasured onatwo-wire line drivingthisantenna ifthelinespacingis2em,theradiusoftheconductors 0.1em, andthefrequency 150Me/sec? CHAPTER VI TRANSMISSION-LINE OSCILLATORS ANDCOUPLED SECTIONS OFTRANSMISSION LINE 1.Frequency Characteristics ofSimpleTriodeOscillators withTrans mission LinesasTankCircuits.143,146,147,166 Atultrahigh frequencies coilsandcapacitors areoftenunsatisfactory ascircuitelements, andsec tionsoftransmission lineareusedtoreplacethem,especially inthetank circuitsofoscillators. Variouscircuitsareincommon use,anumberof whichareanalyzed below. Single-tube Quarter-wave. Oscillator. Oneofthesimplest vacuum-tube oscillators makesuseofasectionofcoaxialortwo-wire lineconnected to theplateandgridterminals ofatriode,asshowninFig.1.1a.Since (a)Transmission line_--------~.-Choke Blocking condenser Choke + (b)~O!ZO=No-jfWCo z-O{Zs=iX S Z=8 Iz------------------------f :IXs~O -------------------------W+{e} FIG.1.1.Triodeoscillator withtransmission-line sectionastankcircuit. theanalysis ofthetriodeasanegative-resistance deviceisbeyondthe scopeofthisbook,itisrepresented simplyasaseriescombination ofa negative resistance No= -Roandtheoutputcapacitance ofthetube. Foratriodethiscapacitance is Co=C1C2+C2C3+C3Cl C2+C3 439(1) 440 TRANSMISSION-LINE THEORY [Chap.VI (2a)whereC1istheplate-grid capacitance, C2isthegrid-filament capacitance, andC3istheplate-filament capacitance. Thus,insofarastheattached transmission lineisconcerned, thetriodeisreplaced bytheimpedance Zo=No+jXo=No-.i-wCo attheend(z=0)oftheline,asshowninFig.LIb.Asaconsequence of terminal-zone effects,theimpedance thatmustbeusedinconjunction withuniform-line theoryistheapparent impedance ZOa=Noa+jXoa (2b) where ZOaconsists ofanappropriate lumpednetwork combined withZoo Inmostcasesitisadequate toassume ZOa==Zoo Attheotherend(z=s)thelineisterminated inanimpedance Z8=R8+jX8whichusuallyconsists ofametalpistoninthecaseofa coaxiallineandofaconducting wirebridgeoflengthbinthecaseofa two-wire line.Theapparent impedances tobeusedinconjunction with uniform-line formulas isZsa=R8a+jXsa. Theterminal functions atthegenerator endofthelinemaybeapproxi matedbyformulas appropriate toapredominantly reactive termination. Theseare nl0 Po=x2+1107r <Po=2"+tan-1XI0 (3) wherenlO=N0/ReandXIO=X0/Re,ifitisassumed thatnroissmall compared withxio+1.Attheotherendthecorresponding functions are p••J0 (jabforpiston forbridgeforpiston forbridge(4) wherek.aisasevaluated inChap.II,Sec.20. Thecondition forresonance fortheentirecircuitdetermines thenatural frequencies in=wn/27ratwhichthecircuitmayoscillate. Itis nintegral (5) where (3n=wn/V=27r/An•With(3)and(4)expressed fortheWlre bridge,(5)maybewrittenasfollows: (6) (7)Forapistonk8a=O.Itfollowsthat,withRe=Vl/cand{3=wv'lC, Xo 1 C XIO=Re= -wCoRe= -{3Co Sec.1] OSCILLATORS ANDCOUPLED SECTIONS OFLINE 441 sothat(6)becomes (3ntanf1ns'=~0 (8) wheretheeffective lengthiss'=s+ksaforatwo-wire lineands'=8 foracoaxiallinewithpiston. Thisequation determines thefrequencies generated. Ingeneral, thefundamental orlowestfrequency fohasthe largestamplitude, butoneormoreofthehigherharmonics maybe maintained aswell. Inpractice, itisoftenconvenient todetermine thegenerated funda mentalwavelength Aoasafunction oftheeffective lengths'ofthesec tionoftransmission line.Thislengthisgivenby sothats'=1-tan-l_c_=~_.ltan-l(3oCo (30 (3oC04(30 c ~=s'+.ltan-l(3oC0 4 (30 c(9) (10) Fortherangethatsatisfiestheinequality (10)becomes(11) (12) Thisisthelong-wavelengthlimit. ItisseenthatCo/cisalengthcharac teristicoftheoutputcapacit.ance ofthegenerator. Itrepresents the shortening effectofthetubecapacitance. Intherangeofshortlengths s'definedby «(3oS') 2«1 thetangentin(8)maybeapproximated byits argument, sothat '..:... C_:!C S-(32C-Ao42Co0 1r/0(13) (14a) Itfollowsthatthegenerated wavelength intheshort-wavelength limitis (14b) Aplotofthegeneralformula (9)isgiveninFig.1.2.Thelimitingranges givenby(12)and(14b)areindicated, asisthelengthCo/c.Notethat thegenerator hasbeenrepresented byalumpedimpedance. Frequently theleadsfromthelinetotheelectrodes areappreciable inlength. When 442 TRANSMISSION-LINE THEORY [Chap.VI (15)I.(w)=1maxsin({jw+ipsa) =1maxcos(jew+ksa) o0's'I AoI41 IA.&I.&I':\'o=s'+1.tan-l(l.Co) 4fJ C ATfV.[S -co-c-thisistrue,theoriginfors'liesatapointsuchas0'totherightofthe idealvaluebyadistance 00'equaltotheeffective lengthoftheleads. Thedistribution ofcurrent forthe fundamental frequency is wherew=s-zismeasured fromthe terminating reactance Xs•Asketch ofthecurrentdistribution isgivenin Fig.1.Ic. Two-tube Half-wave Oscillator. A two-tube circuitwithatriodeateach FIG.1.2.Theoretical wavelength endofthesectionoftransmission line characteristics oftransmission-line oscillator ofeffective lengths'. isshowninFig.1.3.Sincebothends ofthelinearethesameasatz=0in Fig.1.1,thecondition forresonance issimply nintegral (16) With(6)and(7),theequation corresponding to(8)is lQ C {jntan"2fJnS=Co (17) Itisseenthat,bysettings'=s/2,(17)isidentically (8),sothatthe entirediscussion following (8)andFig.1.2appliestothepush-pull cir cuitifitshalflengths'=s/2isusedasthevariable. @_T~r.nsm!:onlin_e® (a) NO! ~Ns=No CO'1"~i---------- __~Tcs=cO • i z=o (b) z=s 1T______ ~L _-------- -------• ..L T----------------------------- (c) FIG.1.3.Push-pull oscillator withtransmission-line tankcircuit. Sec.1] OSCILLATORS ANDCOUPLED SECTIONS OFLINE 443 Thedistribution ofcurrentalongthetransmission lineismostcon veniently expressed intheform (18) wherevismeasured fromthecenterofthelinesection. Tuned-plate Tuned-grid CircuitforPush-Pull Operation. Anadap tationofthewell-known tuned-plate tuned-grid circuitforusewith transmission-line sections isshowninFig.1.4a,wheretheplatesoftwo Transmission line CoNo Zg=jXg)I--__---1::r- ......lzpmjXp CoNo I+---Sg----t I-----sp-----t (b) FIG.1.4.Tuned-plate tuned-grid circuits. tubesoperated inpush-pull areconnected toonesectionoftransmission lineoflength Spandthetwogridsareconnected toasecondsectionof lineoflengthso.Forpresentpurposes eachtubeisrepresented byits outputcapacitances Coinserieswithanegative resistance No.Eachof thetransmission linesisterminated inashortcircuitorasmallinductive reactance, sothat 7r 4'p=<1>/1=2+{3ksa (19) Lets+ksa=8'.Forashortcircuitksa=0ands=8'. Sinceallresistances aresmall,thenaturalfrequency ofthesystem maybedetermined byneglecting lossesintheformulas forthereactances. Theinputreactance ofasectionoflineoflength 8is (20) Thesumofthereactances intheseriescircuitconsisting ofthetwo capacitances Coandtheinputimpedances ofthetwolinesmustvanish atthegenerated frequency fn=wn/27r.Thatis, 444 TRANSMISSION-LINE THEORY [Chap.VI AlternativelyWith(19)thisexpression becomes tanJ3ns~+tanJ3ns~=J3~~0 sinJ3n(S~+s~)_2c cosJ3ns~cosJ3s~-J3nC0(22a) (22b) Sincephaserelations inatrioderequirethatthealternating voltage acrossthegridbe1800outofphasewiththevoltageacrosstheplate, onesideofComustbemaximum positive whentheothersideismaxi mumnegative. Thisoccurswhenthestanding-wave patternonthecom pletetransmission lineconsisting ofbothsectionshasavoltagemaximum acrosseachCo.Thismeans nintegral (23) With(23)in(22a),thisreducesto J3ntanJ3ns~=~0 (24) Thisisthesameinformas(8),sothatthefrequency characteristics of thetuned-plate tuned-grid oscillator areessentially thesameasthose showninFig.1.2forthequarter-wave oscillator, exceptthatcondition (23)alsomustbesatisfied. Ifn=0in(24)ands~determines thefunda mental,asinFig.1.2,thegridlinemayexceedtheplatelineinlength, sothatthegridcircuitisactually oscillating ataharmonic. Thisisoften convenient incoupling aloadtotheoscillator. Alternatively SpandSg maybothbemadelongenoughsothateachoperates onahigherhar monicfortheparticular lengths SpandSg,withn=1,2,...,in(24). Inthiscasethefundamental frequency generated isdetermined bythe requirement (25) Ingeneral,thecircuitoscillates atthelowestpossiblefrequency. Sinusoidal distributions ofcurrentonplateandgridlinesaresketched inFig.1.5forthreepossible combinations oflengths. InFig.1.5athe plateandgridlinesareofequallength,andeachisoscillating initsfunda mental. InFig.1.5bthelengthofthegridlineexceedsthatoftheplate linebyahalfwavelength, sothatthelatteroscillates initsfundamental andtheformerinafirstharmonic. InFig.1.5cthegridandplatelines havelengthsthatmaketheproperphaserelations acrosstheplate-grid capacitance ofthetriodeimpossible iftheplatelineoscillates initsfunda mental. Thelowestfrequency atwhichtherequisite 1800phaserela tionsaremaintained occurswhentheplatelineoscillates inafirsthar monicandthegridlineinasecondharmonic, withSg-Sp="/2,wherp "isthegenerated wavelength. Sec.1] OSCILLATORS ANDCOUPLED SECTIONS OFLINE 445 Double-ended TubeswithTransmission-line TankCurrents. Sometri odesdesigned foruseatultrahigh frequencies haveplateandgridcon nectionsthatpasscompletely throughtheglassenvelope asashorttwo wireline,sothattransmission-line sectionsmaybeattached ontwosides, asshowninFig.1.6a.Ifthetriodeisrepresented byacapacitance Co (a)Iz X.-oi:-------:-------jx.-o ---------------- J+-Sg=Sp-f+-Sp--l It../4=sp+C o/c x.=oE--o:.:::::-----:----~x.;.o (b) ~sg=~;~Aii-----III-~;~ A/4=8p+CojtJ Xg=Oi:--~"- :::--~!=.:::=: ----~---.~---:----::....::::3Xp~O ---~' '--------,~--------,' (c)I- Sg=sp+7t./2 "I' S .1 3Aj4=sp+C o/c FIG.1.5.Possible distributions ofcurrentintuned-plate tuned-grid circuit. ~r---_~C (26)(a) (b) FIG.1.6.Circuitfordouble-ended tube. insofarasitsreactive contribution tothecurrentisconcerned, thenet workofFig.1.6bisobtained (terminal-zone effectsarenotconsidered). Ineffect,twoidentical sections oftransmission lineareinparallelwith thetriode. Theinputreactance ofeachsectionisgivenby(20),sothat forthetwoinparalleltheinputreactance isone-half of(20),or Xin= -~ccot({3s+<pa)= -~ctan{3s' wheres'=8+kBa•Theexpression ontherightisforterminating reactances attheendsofthelinesconsisting ofconducting bridgesof equivalent lengthkBc• 446 TRANSMISSION-LINE THEORY [Chap.VI Thegenerated frequency isthenaturalfrequency ofthetriodein parallel withthetwolinesections.Itisobtained byequating the reactance (26)tothereactance -i/wCoofthetriode. Thus tR' 2 2canfJnS=-0R=R0Wn0cfJn0(27) s'ASingle·ended"'),/....~./ 4tubewithCo..../' '>" ....~Double·ended ............../tubewithCo ........Thisequation islike(8)exceptforthefactor2.Itindicates thatthe capacitance Co/2playsthesamepartindetermining thegenerated wave lengthinthecircuitofFig.1.6bas CodoesinthecircuitofFig.LIb. Itfollowsthat,fortriodeswiththe sameCo,thewavelength character isticofthedouble-ended tubeliesbe lowthatforthesingle-ended tUbe,as showninFig.1.7. Lighthouse-tube Reentrant Oscillator. Animportant typeofultrahigh frequency oscillator makesuseof theso-called lighthouse, ordisk-seal, tube,inwhichtheconnections tothe parallel planeelectrodes consistof circularcopperdisksofgraduated sizes.Thisconstruction permitsthe triodetobeinserted intwoconcentric coaxiallinesinsuchamanner thattheinnerandoutersurfaces ofthecylinder incontactwiththegrid-Co0 ~ FIG.·1.7.Comparison ofwavelength characteristics ofsingle-anddouble endedtubesintransmission-line os cillator. ~Sl----+/ (a) Lpf Idealopen Cpg--~___________ end ,...1...1-----sl-----'Jo~1 (b) FIG.1.8.Reentrant oscillator (a)withapproximate tank-circuit equivalent (b). diskserve,respectively, asouterconductor oftheplate-grid lineand innerconductor ofthecathode-grid line.Thisconstruction isshownin Fig.1.8inatypicallighthouse-tube circuitknownasthereentrant oscil lator.Various othercircuitsareinuse,butallarecharacterized by separate plate-grid andcathode-grid transmission-line circuitswitha coupling linkoraperture connecting them.Owingtothesections of Sec.2] OSCILLATORS ANDCOUPLED SECTIONS OFLINE 447 (28)16 °O~~-'~~~8-~~-""- 81 FIG.1.9.Wavelength characteristics of reentrant oscillator.radiallinewhicharebounded bythedisksandcontaintheelectrodes, noneofthecircuitscanberepresented accurately bysections ofcoaxial lineandlumpedelements. However, inthecaseofthereentrant circuit inwhichthefrequency generated isprimarily determined bytheplate gridsectionoflinewhichisofrelatively smallcross-sectional size,a reasonable approximation maybeachieved intermsofcoaxialandlumped elements. ThecircuitinFig.1.8aconsists essentially ofasectionofbifurcated coaxiallinewhichisequivalent tothreesections oftransmission linecon nectedinseries.AsgiveninChap.V,Sec.14,therequired junction zonenetwork oflumpedelements intheplaneofbifurcation consists of positive capacitances acrosstheendsofthetwolines1and2ofsmaller cross-sectional areaandanegative capacitance acrossthelargeline.The plateleadextending beyondtheendoftheinnermost coaxialconductor andtheplate-grid spacemaybeapproximated byaninductance Lpin serieswithacapacitance Cpg.The properphaserelations foroscillation areobtained whenthelength S3of thelargelineisadjusted sothatthe terminating impedance fortheplate gridlineintheplaneofbifurcation isessentially thatofanopencircuit. Asaconsequence thefrequency of oscillation isprimarily determined bytheplate-grid linetreatedasifit hadanidealopenend.Thiscannot beshownanalytically without an elaborate analysis oftheentiretriode circuit,whichisbeyondthescope ofthisbook.Forpresentpurposes, then,theeffective tankcircuitis thatshowninFig.1.8b.Theresonant frequency isobtained fromthe solution oftheequation 1wLp--C=Reicot(jSIwpg whereRei=60In(adal)ohms.Thenumerical solution144of(28)for X=21r/(jforaparticular circuitisshowninFig.1.9,whichisrepresenta tiveofthewavelength characteristic ofareentrant lighthouse-tube oscillator. 2.Frequency Characteristics ofaTransmission-line Oscillator with Coupled Secondary)62,166 Transmission-line oscillators aresometimes operated withacloselycoupled secondary thatmodifies thefrequency characteristics. Various arrangements areavailable, butthegeneral 448 TRANSMISSION-LINE THEORY [Chap.VI (Ib)(Ia)properties ofmostoftheseresemble thoseofthesimplecoupled-circuit oscillator showninFig.2.1.Thisconsists ofaprimary circuitwhichis identical withthesingle-circuit oscillator inFig.1.laandwhichmaybe represented bytheequivalent circuitinFig.LIb.Thesecondary con sistsofanaddedsectionoftransmission lineextending beyondthebridge terminating theprimary. Letthelengthoftheprimary be81andthatofthesecondary, 82,so thattheover-alllengthis8=81+82.Theimpedance andemfofthe generator atz=0arerepresented bytheseriescombination ofan ~o*L..-_pr_im_:_ry__I._sl..!_Z_s_al_=_jX--,-l _·_s_e_co_n_da_ry_---.!Zsa=jX'G z=o 81 S=S1+S2 51 W+0~u 82 FIG.2.1.Equivalent circuitoftransmission-line oscillator withbridge-coupled secondary. apparent negative resistance NOaandreactance XOa;theapparent imped anceatthejunction z=81ofprimary andsecondary isZsla,andthatof thetermination forthesecondary atz=8isZsa.Thecorresponding apparent terminal functions are ll.+';F.,th-lNoa+jXoa U'Oa=pOaJ'J!Oa=co Zc I _+';F.,_h-1ZsaGsa-psaJ'J!sa-cot Zc Theadmittances lookingtowardthetwoendsfromthejunction plane atz=81are Y1=Yctanh(i81+GOa) Y2=Yctanh(i82+Gsa) (2) Anequation representing thefrequency anddamping properties ofthe circuitinFig.2.1maybederivedbynotingthatthevoltage across thecoupling impedance (whichisassumed tobeasmallinductive reactance) is VsI=jwLs1[/1(81)-12(81)] (3) andthatthecurrents intheprimary andsecondary wherethesejoinare 11(81)=Vs1Y112(81)= -Vs1Y2 (4) Thenegative signfor12(8)isaconsequence ofthefactthatthevoltage induced inthesecondary is1800outofphasewiththatintheprimary. Bysubtracting 12(81)from11(81)in(4)andusing(3),thefollowing equation isobtained: 1=jwLs1(Yl+Y2) SinceYc==Gc==I/vl,itfollowsthat(5) k=LSI - l(6) Sec.2] OSCILLATORS ANDCOUPLED SECTIONS OFLINE 449 Accordingly, with(2)and(6),(5)maybeexpressed asfollows: j~k=tanh(AI+jFl)+tanh(A2+jF2) where,with"(=a+j{3,(7) Al==aSl+POa A2==aS2+P8aFl=={3S1+tf>oa F2==(3S2+tf>8a(8a) (8b) (9a) (9b)Byseparating realandimaginary partsthefollowing pairofequations is obtained: tanhAlcos2Fl1+tanh2Altan2Fl tanhA2cos2F21+tanh2A2tan2F2 1 tanFl -13k=cosh2Al(1+tanh2Altan2Fl) tanF 2+cosh2A2(1+tanh2A2tan2F2) Equations (9a)and(9b)maybesolvedforAland{3,andthesevalues usedtodetermine thenegative resistance NOaandthegenerated fre quency. SinceA1andA2maybeassumed small,thefollowing inequal itiesaregoodapproximations: Ai«1A~«1 (10) (lla)Eqs.(9a)and(9b)maybesimplified. Theresultsare -AIcos2F11+Aitan2F1 -:::r;-=cos2F21+A~tan2F2 1 tanFl tanF2 -13k=1+Aitan2F1+1+A~tan2F2 (lIb) With(10)itfollowsthat,exceptforvaluesofFlorF2verynearanodd integral multiple of7r/2,theseequations maybeapproximated by -AI. cos2F1 ---;r;-=cos2F2 -1-==tanF1+tanF213k(12a) (12b) (13)Evidently (12b)determines thegenerated frequency intermsofthe lengths SIandS2andtheterminal functions tf>oaandtf>8a.Similarly (12a) specifies thecriticalvalueof-AI(orNoa)atwhichoscillations canbe established. Thesetwoequations maybeexpressed directlyintermsof thetwounknowns. AsshowninSec.1, , + tf>oa.+GOaSl=Sl----;f=SlC 450 TRANSMISSION-LINE THEORY [Chap.VI whereCOaistheapparent outputcapacitance ofthetriodeandcisthe capacitance perunitoftheline.Let {{382foranidealopenendatz=8 F1= 1l"1l". (14) (3(82+kBa)+2={38~+2forabrIdgedendatz=8 With(13)and(14),(12b)hasthefollowing forminthetwocasesspecified in(14): (15b)(15a) openend bridgedend-;k=tan{38~+tan{382 1 , ,-13k=tan{381-cot{382 ForanopenendPBa=0;forabridgedendPsa==ab/2.Since POais small, POa==~o: (16) Witha=r/2Rc=ra/Rc(wherera=r/2istheresistance ofaunitlength ofoneconductor) and(16),thedamping equations derivedfrom(12a) andassociated with(15a)and(15b)are NOa+ra8~cos2{38~ ra82=cos2{382 NOa+raS~cos2{3s~ raS~=sin2{3s~openend bridgedend(17a) (17b) Theseequations mayberearranged intothefollowing formsconvenient fornumerical evaluations: OpenEndatz=8 BridgedEndatz=81-KS 1=tan81+tann8t Noa/R1+1 _cos281 n-cos2n81 1---=tan81-cotn81KS1 NoafR1+1cos2S1 n=sin2nSI(18a) (18b) (19a) (19b) whereS=={38~=21l"8~/'Aistheelectrical lengthoftheprimary, K=k/8~is thefractional equivalent lengthofthecoupling bridge,R1=ra8listhe totalresistance intheprimary, andn=82/8~or8~/8~.NotethatNoais negative. Equation (18a)hasbeensolvedgraphically for81asafunction of therationofthelengthofthesecondary tothatoftheprimary, with Sec.2] OSCILLATORS ANDCOUPLED SECTIONS OFLINE 451 ~o.---....,......,~:---r.,.---\ ":":n'=""I.,......,..,":":n---'2 ..,'" FIG.2.2.Theoretical wavelength char acteristics (solidlines),damping curves (dashed lines),andexcitation curve forNoa/R 1=-2(dotted curve)for coupled-circuit oscillator withopen endedsecondary. Thecurvesarefor krls~=0.1.Kasparameter. Instead ofplotting 81againstn,theratioofthe quarter wavelength generated tothelength 8~oftheprimary, namely, A/48~=7r/281,isplottedagainstn=82/8~insolidlineinFig.2.2.Of theinfinityofpossible solutions, thefirstthreeareshownnumbered in Roman. Theparticular branches thatgivepossible generated wave lengthsdependontheassociated damping curves. Theseareobtained byplotting therightsideof(18b) againstn,asshownindashedlinein Fig.2.2.Oscillations canbemain tainedatawavelength specified bya-i1.0r---t==~Il:F=~::::t:;~r:I~n l.r particular pointonawavelength -<0.5 .....\.I ~ curveifNOaissufficiently greatsothat ;- theleftsideof(18b)atleastequals ~":::';""....L.-~n==..J.:2-IS""""; ":::'-..1:.:3==..-..140 therightside.Thedottedcurvein Fig.2.2isthemagnitude ofthe leftsideof(18b),withNoa/R1=-2 plottedagainstn.Onlywavelengths forwhichthecorresponding point ontheassociated dampingcurve (dashed line)doesnotlieabovethe dottedexcitation curvemaybegenerated. Inparticular, asn=82/8~is increased fromzero,thegenerated quarterwavelength isgivenbywave lengthcurveIfromn=0ton=1,wheredamping curveIrisesabove thedottedexcitation curve.Fromn=1tonverynearly3(where damping curveIIrisesabovetheexcitation curve)thewavelength of curveIImaybegenerated; beyond n==3,wavelengths oncurveIIare damped out.Beginning withnsomewhat lessthan3,damping curveIII dropsbelowtheexcitation curve,sothatwavelengths ofcurveIIImay begenerated. Evidently thegenerated wavelength behaves asfollows asthesecondary isincreased inlengthfromzeroto4s~:Atn=0the generated wavelength isthatcorresponding toasingle-circuit oscillator with 8~=A/4andaterminating inductance Ls1•Itisthisinductance which'makes thegenerated quarterwavelength with82=0greaterthan1. Asn=82/8~isincreased, thequarter wavelength increases slowly,fol lowingcurveIuntiln=1.Atn=1thequarter wavelength drops abruptly tocurveII,whichitfollowsasnisincreased untiln=3is almostreached; thenitdropstocurveIII,whichitfollowston=4. Ifnisnowdecreased, wavelength curveIIIisfollowed untilapointis reached whichliesdirectly abovetheintersection ofdamping curveIII andtheexcitation curve. Herethegenerated quarterwavelength rises tocurveII,whichisfollowed ton=1,whereitrisestocurveI.Ina smallrangeofnbelow3thegenerated wavelength maybegivenby eithercurveIIorcurveIII.Suchanoverlapping rangeofinstability is calledadragloop,sincethegenerated wavelength tendstocontinue ona 452 TRANSMISSION-LINE THEORY [Chap.VI 20---.-10---...1oO.5r---H--z0....---.....,.--~,.. 1.5t-----I--fl---f 1 ,"'n=s2/St FIG.2.4.Theoretical wave lengthcharacteristics ofthe idealized coupled-circuit oscil latorwithopen-ended second aryforthreedifferent values oftheratiok/s:(solidlines). Atypicalexperimental curve following observed pointsis showndotted.~1- ~...~ ~, LlI'"" ..... o1.5 ..."..::t1.0 t<0.5givenwavelength curveuntilitsdamping curverisesabovetheexcitation 1.5curve.IfthevalueofNOa/R1is "'".-morenegative than-2,thedotted 1.0.::t 0.5«excitation curveishigher,theextent ofthedragloopnearn=3isin creased, andadragloop(narrower 1.5......thanthatnearn=3)occursnear 1.0.:tn=1.IfNoafR1islessnegative 40;5Kthan-2,oscillations breakoffover 2I3 arangenearn=1,andthedragn=8z81loopnearn=3isreduced.If FIG.2.3.Theoretical wavelength charac- teristics ofacoupled-circuit oscillator NoafR1issufficiently lessnegative withopen-ended secondary. A-small than-2,oscillations mayalsobreak excitation, INoa/R1\<2;B-critical ex-offnearn=3.Thesethreepos c~tation, INoa/R11=2;C-strong excita-siblecasesareillustrated inFig.2.3. tlOn,INoa/R1\>2. Inordertoillustrate theeffectof varyingthecoupling inductance, thelefthalfofthewavelength curvein Fig.2.2isshowninFig.2.4together withtwo othertheoretical curvesforsmallervaluesof K=k/8~.Experimental pointsforatypical caseforwhichk/8~<0.03arealsoshown. Whenthesecondary isclosedatz=8in steadofopen,sothat(19a)and(19b)apply, analogous resultsareobtained. Wavelength anddamping curvesdetermined from(19a,b), .(0'"E;:::::::~~~~~=d withk/8~ =0.1,areshowninFig.2.5together ~1.°r withthesingleexcitation curveNoafR1= -2. Experimental wavelength characteristics foranoscillator inwhich k/8~isverysmall areshowninFig.2.6awhenthesecondary hasabridgedend(curveA)andanopenend (curveB).Thewavelength characteristics ofadifferent bridged-end oscillator witha somewhat greatervalueofk/8~areshownin Fig.2.6b.Thedragloopsarenoteworthy. Aplotofthetheoretical wavelength charac teristicdetermined fromFig.2.5,withthe experimental curveofFig.2.6bproperly scaled,isgiveninFig.2.7.Thevalueof kI/8~fortheexperimental curveisreadily computed tobek/8~=0.044.Ifthisvalue ofk/8~isusedtocalculate curveslikethose inFig.2.5,thetheoretical andexperimental curvespractically coincide. Itisnotdifficult toexplainthewavelength characteristics ofthe Sec.2] OSCILLATORS ANDCOUPLED SECTIONS OFLINE 453 1.5:8 '-... e 0) 1.~ ~1.00.~ Eco 0.5c 1234567 n=s~/s; FIG.2.5.Theoretical curvesforwavelength (solidlines),damping (dashed lines), andexcitation forNoa/R 1=-2(dottedline)forcoupled-circuit oscillator with bridged-end secondary. Thecurvesarefork/8~=0.1. 5 4 l!? ~3 E 5 t( 4-' ..... ~ 3o 1 2 3 lengthofsecondary, meters4 (a) 91011Scale,meters 1413 12 1516174.6r--""'T"--r----r--......,~-...,..---,--..,....-_ ~4.4t:--==1:=::::;;f'ti--::::;t===:::p"fir-t:===lP1N--t"":::::::::::=t CI>1/' EIf<4.2t-+--;--...p----+---t-.:...---+----1f---L-f----1 4.0........_"""-_--1-_----' __"'--_......._......__01----' o0.81.82.83.84.85.86.87.8 lengthofsecondary, meters (b) FIG.2.6.Experimental wavelength characteristics ofcoupled-circuit oscillators. (a)Oscillator withbridged-end secondary inuppercurve,open-end secondary in lowercurve. (b)Different oscillator withbridged-end secondary. 454 TRANSMISSION-LINE THEORY [Chap.VI (la)"~m•.0__.. ~ ~_ ~M-_i~.•....-;.,./,i... 0.8.23456 FIG.2.7.Combined theoretical (dotted,k/s;=0.1)andexperi mental (dashed,k/s;=0.044) wavelength characteristics of coupled-circuit oscillator with bridged-end secondary.coupled-circuit oscillator inaqualitative manner. Thegenerated wave lengthisdetermined bythecomplete primary-secondary circuit. In effect,theprimary sectionoftransmission lineisterminated inaparallel combination ofthelowimpedance oftheinductive bridgeandthevaria bleinputimpedance ofthesecondary transmission line.Whenthelength ofthesecondary lineissuchastoprovide aninputimpedance thatishighcompared withthatofthecoupling bridge,thepri marybehavesessentially asifthesecondary werealumpedhighimpedance inparallel. Practically theentirecurrent entersthe bridge;verylittleentersthesecondary. As thesecondary ischanged inlengthsoasto reduceitsinputimpedance, theeffective im pedance ofthebridgeinparallelwiththe secondary departs moreandmorefrom thatofthebridge,andanincreasing cur- rententersthesecondary. Finally, whentheinputimpedance ofthe lineissmallerthanthatofthecoupling bridge,thecurrentintothe secondary exceedsthatintothebridge,andthewavelength increases almostlinearly withincreasing lengthofthesecondary, muchasifthe coupling bridgewereabsent. Butasthesecondary isfurtherincreased inlength,itsinputimpedance increases andsoonexceedstheimpedance ofthebridge. Whenthisoccurs,mostofthecurrententersthebridge, andthewavelength isagaindetermined bythelengthoftheprimaryand itseffectively lumpedtermination. Inoscillators ofthepositive-grid, orBarkhausen-Kurz, typeandinthe magnetron, theprimary circuitconsists ofacloudofelectrons, andthe secondary includes allelectrodes andattached conductors. Sincethepri marycannotexistwithout thesecondary, itfollowsthatthefrequency characteristics ofsuchanoscillator aredetermined bothbythenatural frequency oftheprimary cloudofelectrons andbythenaturalfrequency ofthesecondary circuit. 164,166 3.Electric FieldofaConductor withSinusoidally Distributed Cur rent.10Inordertodetermine thedistributions ofcurrentandvoltage alongatwo-wire linewhenthis'isdrivenbyatransmission-line oscillator ofatypedescribed inSec.1(orbyanequivalent coupling element) coupledatanarbitrary location alongtheline,itisnecessary toderivea formula fortheelectricfieldmaintained bythecurrents intheoscillator. Sincetherelevant element ofthiscircuitconsistsofasectionoftwo-wire linecarrying anessentially sinusoidally distributed current, afirststepis toobtainaformula fortheelectromagnetic fieldofasinglewireofarbi trarylengthhwithacurrentofthegeneralform I(')1(0)sin(3o(g-z')I.R(')Z= . R=mSIn!JOg-zsm!Jog Sec.3] OSCILLATORS ANDCOUPLED SECTIONS OFLINE 455 (lb) g==h+k where1m==.1(0) sm(jo{l Thisdistribution isillustrated inFig.3.1.Byasuitablecombination of conductors ofthistypewithappropriate valuesofkandh,itispossible torepresent mostoftheusualtransmission-line oscillators andcoupling elements. TheelectricfieldatthepointQ(r,u)ismostconveniently determined fromthegeneralintegral [Chap.I,Sec.3,Eq.(30b)]forthevector z Q(r.z) _.-._._.-._._.-._._._._~ /'/. R/'•I y./.. . I.//RAjr--r----- ,/...i~ 1'(~) ~r'".[14k·fdr~I z=O z' hg-h+k. FIG.3.1.Conductor withsinusoidally distributed current. potential. Inthenotation ofthissection(inwhichthedielectric is assumed tobeperfect,sothat ~=Eand~=(j)itisgivenby where1(h e-ifJR A=iA,(z) A,(z)=~JoI,(z')IIdz' R=V(z'-Z)2+r2(2a) (2b) Theelectricandmagnetic fieldsmaybecalculated from(2a)using Chap.I,Sec.3,Eqs.(7),(Sa,b),forms(18),and(19).Therelevant are E= -%(graddivA+(j2A)= -%curl curl A (3a) B=curlA (3b) If(la)isexpanded inexponential formandsubstituted in(2a),this maybeexpressed asfollows: (4)-jl[ihe-ifJ(R+,')ihe-ifJ(R-Il') ]A,(z)=__meifJu dz'-e-ifJu dz' ~v 0R 0R Inordertoobtainamoreconvenient formoftheintegrals, letthefirstbe multiplied bye-ifJlIinfrontofthesignandbyeifJlIunderthesign.Simi larlyletthesecondintegralbemultiplied byeifJlIinfrontofthesignand bye-ifJlIunderit.Withtheabbreviations u==R+z'-z v==R-z'+z (5) 456 TRANSMISSION-LINE THEORY [Chap.VI (6)thefollowing expression isobtained: -'1[Ioh e-i(ju 10k ei{j'IJ ]Az(z)=~ ei(j(g-z)--dz'-e-i{j(g-z) -dz' 87rV 0R 0R InordertoexpressRandz'inthetwointegrals intermsofuandvand thuschangethevariables ofintegration, itisnecessary todifferentiate (5)toobtain auu az'R(7) (9) (12)(lOa) (lOb) (10c)(8a) (8b) (8c)Thelimitsofintegration forthenewvariables uandv,whichmaybe introduced atthispoint,are Forz'=0:u=Uo=Ro-z v=Vo=Ro+z Forz'=h:u=UI=RI+h-z v=VI=RI-h+z where Ro=~ RI=V(z-h)2+r2 With(7)to(8b),(6)becomes -'1[lu1e-i{ju j,'lJie-i{j'IJ ]Az(z)=~ ei{j(g-z)--du+e-i{j(g-z) -dv 87rv UOU 'lJ0v Although theintegrals inthisformula canbeexpressed intermsoftabu latedsineandcosineintegrals, itisunnecessary todosoforthepresent purpose ofevaluating theelectricfield,Thisisdoneasfollows: Incylindrical coordinates withAr=0andAo=0androtational sym metry,sothatalao=0,itfollowsthat Br=curlrA=~aa~r=0 B0=curloA= _aAz ar Bz=curlzA=0 sothat '1[alUie-i(ju aj,Vie-i{j'IJ ]B0=~ej{j(g-z) ---:---du+e-i{j(g-z) ---dv(11) 87rV ar UOu ar Vov Sinceroccursinbothlimitsofintegration, thegeneralformula26for differentiating adefinite integral withrespecttoaparameter mustbe used.Notethattheintegrands donotinvolver.Usingthefollowing derivatives ofthelimitsofintegration: auoavoraUIaVIar=ar=Roarar theexpression (11)maybeintegrated into JIm[ .(re-i{jUi re-j{j1to)Bo= - e1{j(g-z) ---- --- 87rV IIIUIRo'lto.(re-i:3'IJ1re-ifJ'lJO )J +e-1{j(g-z) ---- ---RIVIRoVo(13) Sec.4] OSCILLATORS ANDCOUPLED SECTIONS OFLINE 457 (16a) (16b) (16c)Considerable rearrangement andsimplification of(13)arepossible. The finalresultis B.~1;:r[e-i'"(cosK+jZ~lhsinK)-.-.;,••(COSG+j~osinG)] (14) where K=={jkG==H+K==(j(h+k) (15) Thecomponents oftheelectricfieldarenowevaluated from(3a)with (3b)and(14).Since,withBr=0andBz=0, 1acurlrB= ---a(rBtJ)r z curltJB=0 1acurlzB= --a(rBe)r r itisclearthatErandEzdifferfromzero,whereas EtJvanishes. differentiations ofrBewithrespecttozandrareelementary. resultsare Er=jtolm!e-ifJR1[z-hcosK+j(z-2h)2sinK-~sinK] 471'"r RI RI {jRI -e-ifJRo(~cosG+jz 2sinG-~sinG)jRo R5 (jRg _jtolml-ifJRl[cosK+j(z-h).K+z-h.K]Ez= 471'"e RIRism {jR~sm OR(cosG+jz.G+z.G)j -e-1l'Ro-- 2sm-3sInRoRo {jRoThe The (17) (18) Incalculating theelectricfieldmaintained tangenttotheconductors of atransmission linebythecurrents inaparallel oscillator, onlyEzas givenby(18)issignificant. 4.TheElectric FieldofaDrivenSectionofTwo-wire Line.148,16IA two-wire lineoflength 8andspacing bisexcitedasasecondary bya coupled shortersectionoftransmission lineoflengthhandspacingbo, inwhichacurrentismaintained byanappropriate negative resistance. Thedistance between theplanescontaining thetwoparallellinesisd. Forsimpl.icity, letallconductors inthelonglineandthecoupledsection havetheradiusa.Thustheproblem isthedetermination oftheelectric fieldsatpointsQIandQ2inFig.4.1paralleltotheconductors 1and2of thelongline.Thefieldismaintained bythebalanced currents incon ductors3and4oftheoscillator. Itisassumed that (1) 458 TRANSMISSION-LINE THEORY [Chap.VI ItfollowsthattheelectricfieldsatQlandQ2areequalinmagnitude andopposite indirection. Adistribution ofcurrentwhichmaybeadapted torepresent thecur rentsindifferent oscillators andvariously terminated coupledsectionsof ~---- ------- %-----~ ..~', Cb) FIG.4.1.Transmission linewithcoupled drivensectionofline.(a)Sideview;(b) endview.---------------......](0) ]3(U') ....... =========-I-=======~-------~ ====~1-======------7 -]3(U') _".",------------ I I I u'=O u' hg=h+k FIG.4.2.Distribution ofcurrentalongdrivensectionofline. wherelineisshowninFig.4.2andgivenanalytically by -14(u')=13(u')=13(0)sin(j(g-u') g=h+ko~u'~h(2a) (2b) Thecoordinate u'locatingthecurrentismeasured fromtheleftendof theoscillator, whichisatadistance Xofromtheleftendofthemain transmission line.Thisdistance satisfiestheinequality Xo»bo (3) Sec.4] OSCILLATORS ANDCOUPLED SECTIONS OFLINE 459 Itisdesiredtodetermine theelectricfieldatpointsQlandQ2atadis tancexfromtheendofconductors 1and2. ItfollowsfromSec.3,Eq.(18),withchanges innotation, thatthe axiallytangential component oftheelectricfieldmaintained atapoint suchasQlonconductor 1byadistribution ofcurrentoftheform(2a)in conductor 3isgivenby (4)I'R[cosKj(u-h).K+u-h.KJe-1"RhU--+ 2sm--a-smRh13 Rh13 {3Rh13 'R(cosGju.G+u.G)I -e-1"R018--+-2-sm -3-smR013R013 {3R013 Thefieldmaintained atthesamepointQ1bythecurrentinconductor 4 oftheoscillator isgivenby(4),withallsubscripts 3replaced by4and theentirerightsidepreceded byaminussign.Thefollowing notation isinvolved: N==13kG==H+K=={3(h+k) (5) Rh13=y(u-h)2+T~3=~(u-h)2+T3+~-boTocos()(6a) ROl3=yu2+Tf3=~U2+T3+~-borocos() (6b) Rh14=y(u-h)2+Tf4=~(U-h)2+T3+~+borocos()(6c) R014=yu2+T~4=~U2+T3+~+borocos() (6d) Thedistances T13, T14,androandtheangle()aredefinedinFig.4.1b. Theoriginofuandu'isinthetransverse planethrough theleftendof theoscillator orcoupledsectionofline.Thatis,E_-jroI3(O) 2113-471'sinG u=x-xo (7) (8)Theresultant axiallytangential electricfieldatQ1is -jroI3(O)I(e-ifJRh13 e-ifJRh14) .Ed=EZ13+Ez14=4 .G-R--RcosK+J(u-h) 71'SIn h13 h14 (e-ifJRhU e-ifJRh14). (e-ifJRh18 e-ifJRh14). X-R2--R2smK+(u-h){3R3-{3R3smK h13 h14 hl3 h14 [(e-;fJROU e-ifJR014) .(e-ifJR013 e-ifJR014).------cosG+Ju-2- --2-smGROl3 R014 R013 R014 (e-;fJROll e-;fJR014).]I+U{3R3-{3R3smG 013 014 Thisisthegeneralexpression fortheelectricfieldatanypointQlalong conductor 1ofthetransmission line.ThefieldatQ2onconductor 2is thenegative of(8). 460 TRANSMISSION-LINE THEORY [Chap.VI Considerable simplification in(8)ispossibleunderthepractically most important circumstances inwhichtheangle8doesnotdiffergreatlyfrom 1r/2,sothatcos8issmall.Inparticular, witht/;==1r/2-8, cos8=sint/;==t/; With(9)andthenotation b2 V5==u2+T5+i itfollowsthatt/;2«3 (9) (10) where EvidentlyRh13==VI-!It/; ROl3==Vo-Jot/; Rh14==VI+!It/;R014==Vo+Jot/; !I==boTofo==boTo 2VI 2vo boTo!IVI=fovo=2(lla) (lIb) (12a) (12b) Since!IandJoarelessthan1,itfollows,with(9),that e-1Mhll==e-ipv1(1+j(3!It/;) e-iMh14==e-iPv1(1-j(3JIt/;) (13a) e-iMoli==e-iPvo(1+j(3Jot/;) e-iMo14==e-iPvo(1-j(3Jot/;) (13b) With(9)to(13b),(8)reducestothefollowing approximate formifall termswitht/;2andstillhigherpowersoft/;asafactorareneglected: Ed(u) ==roI3~0)t/;(!Ie-iPV1[_j1+j(3VIcosK+(u_h)(2+j(3Vl 21rsmG vi vf j3+j(3VI).K]f-iPv[ •1+j(3voG- - SIn-oe°-J cos ~vt V5 +u(2~f(3vo-~3~t(3vo)sinG])(14) Thisistheelectricfieldmaintained atQIinconductor 1(Fig.4.1)by theequalandopposite currents andchargesinconductors 3and4ofthe oscillator orcoupledsectionofline. Inordertorepresent theelectricfieldoftransmission-line oscillators ofevenandoddsymmetries withrespecttotheircenters,itisadvan tageoustodetermine thefieldmaintained atQIbytwosections oftrans missionline,ofwhichthefirstisthatshowninFig.4.1andthesecondis similarbutextended intheopposite direction, asshowninFig.4.3.The currentinthissecondsectionisgivenby -14(u')=13(u')=±13(O)sin(3(g+u')- h~u'~0(15) wheretheuppersignappliestoanoscillator inwhichthecurrentiseven, I(-u')=I(u),andthelowersignappliestoanoscillator inwhichthe currentisodd,I(-u')=-leu). Thefieldmaintained bythecurrentinthesecondsectionislikethat givenin(14)ifthepropersignisprefixed anduisreplaced by-u. Sec.4] OSCILLATORS ANDCOUPLED SECTIONS OFLINE 461 Thisinvolves thesubstitution ofV2andh,asdefinedby b2 v~==(u+h)2+r~+4 (16) Oscillator or couplingelement (18)forVIand11,asdefinedin(10)and(12a).Theresulting expression is Ez1(u)==+=jsola.(O)1/;(/2e-ifJV2[1+j{3V2cosK 211"smG v~ _j(u+h)(2~t{3V2+3~V{{3V2) sinK] I"8[1+j{3vo.G.(2+j{3vo+3+j{3vo).G]l(17) -oe-1vo cos-JU SIn . V5 vg {3v~ Thesumof(14)and(17)givesthecomplete electricfieldmaintained at Qlbyatransmission-line oscillator oflength2hwhichhasaneven(upper Mainline Q1~ ------.----:::-'l':~?4'7f4 ..--- .~/.'l/ ././~.//.}i/./#R~13/RiRh14<22./~/L;;J \ hu=-h!Ro14 %=xo-h %0 %o+h FIG.4.3.Transmission linewithcoupleddrivensectionoflineoroscillator consisting of twopartseachoflengthh. sign)orodd(lowersign)distribution ofcurrentwithrespecttoitscenter atu=0orx=xo. EvenCurrents andFields. Thesimplest specialcaseisthefieldofa half-wave sectionoflinewithvanishing currents atbothends.Inthis caseK=0,andG=H=11"/2.Sincethecurrent iseven,theupper signin(17)applies. Thecomplete fieldatQlis Ez1(u)=-jsola(O)t/t (fte-ifJVl1+j{3Vl+he-ifJv21+j{3V2) 211" VI V2 Numerical computations havebeenmadeforanactualoscillator of thetypeshowninFig.1.2.Thefollowing constants apply: 211" {3=X=0.333cm-1 2h=20emb=bo=2cmro=10cm 1/;=sin-10.1==0.1 K=1.2~7 H=={3oh=0.333 462 TRANSMISSION-LINE THEORY [Chap,VI ("hbI I"'F1'0.-.:,1Q~ bo Conductor no.1 Ez1(u)ofmainline r-------\1tIV-1\---cur-renf--ro!lO~ Oscillator line \\/ '\ VV2.02.5 -1.0§1.5 ~ QJa. ~1.0... QJa. ~ ~0.5 ~0 ~ :! ~"-0.5'" -1.5 -40-30-20-10 010203040 u,em FIG,4.4.Electricfieldmaintained bytransmission-line oscillator withevencurrent. -20 -10 0 10 20 u(arbitrary scale) FIG.4.5.Sketchoftypicalelectricfieldmaintained bytransmission-line oscillator withoddcurrent. Sec.5] OSCILLATORS ANDCOUPLED SECTIONS OFLINE 463 Agraphofthequantity ofjEzl(u)/lz(0)1/I asafunction ofuisgivenin Fig.4.4. OddCurrents andFields. Thesimplest specialcaseisthefieldofa half-wave sectionoflinewithvanishing currentsatthecenter.Inthis caseK=1r/2,H=1r/2,andG=K+H=1r.Sincethecurrentat u=0vanishes, itisconvenient tosubstitute la(h)=la(O)/sin Gforthe indeterminate form0/0.Theresultant fieldisthesumof(14)and(17), withlowersign.Thus EleU)=tola(h)[(U_h)!le-ifJv1(2+j{3Vl_t3+j{3Vl) z 21r vf {3v1 +(u+h)/2e-ifJv2(2~t{3V2_~3~({3V2)] (19) Asketchofthetypeoffieldrepresented by(19)isgiveninFig.4.5. Asymmetrical Currents andFields.Ifthecurrents areneitheroddnor evenwithrespecttothecenteroftheoscillator orcoupling element, theymayberepresented asthesumofevencurrents andoddcurrents. Accordingly theelectricfieldisthesumofafieldoftheeventypeshown inFig.4.4andtheoddtypeshowninFig.4.5withappropriate amplitudes andphases.Iftheprimary sectionoflineisterminated initscharacter isticimpedance atu=hinFig.4.3,thecurrentdistribution is -14(u')=la(u')=1a(0)(cos(3u'-jsin(3u') (20) (2)(1)Sincethisisthesumoftheevencurrents and-jtimestheoddcurrents, itfollowsthattheelectricfieldofatraveling waveofcurrentonahalf wavelength oflineisEzl(u)in(18)minusjtimesEzl(u)in(19). 5.Current andVoltageinaLineDrivenbyaCoupled Sectionof Transmission Line;Directional Coupler.l49Thecurrentandvoltageata pointzalongatransmission linewhichextendsfromz=0toz=sand whichisdrivenbyapairofequalandopposite pointgenerators with emfsjV~atz=xaregivenbyChap.IV,Sec.2,Eqs.(5)and(6).They maybeexpressed asfollows: lz=i:~:sinh(yx+80) Vz=V;~wsinh(yx+80) where andSw=sinh(yw+8.)S.=sinh(ys+80+8.) Cw=cosh(yw+8.) Asusual,w==8 -z. Ifthelineisdrivenbyacontinuous distribution ofgenerators overa rangeextending fromz=Xo-Utoz=Xo+u,itfollowsfromthe principle ofsuperposition thatthecurrentandvoltagearethesuper- 464 TRANSMISSION-LINE THEORY [Chap.VI (4)(3)positions ofthecurrents andvoltages maintained byallpairsofgener atorslocalized ineachelementdxbetween Xo-g'andXo+g.Thus, iftheemfmaintained bythedistribution ofgenerators perunitlengthin conductor 1isE~(x)andthatinconductor 2is-E~(x),thetotalcurrent andvoltagemaintained bytheentiredistribution are Sj,xo+g Is=ZSW E;(x)sinh(yx+00)dx C8xo-g' Cj,xo+g Vs=SW E;(x)sinh(yx+00)dx 8XO-g' Theelectricfieldmaintained alongtheconductors ofthelinebythecur rentinacoupled sectionisequivalent toadistribution ofgenerators. Inthiscasethedistribution ofemfsperunitlengthE;(x)isreplaced by Es1(u),asgivenbythesumofSec.4,Eqs.(14)and(17),ifthecurrents intheoscillator haveevenoroddsymmetry withrespecttothecenterof theoscillator atu=0orx=Xo.Notethatu=x-xo.Inthegeneral case,inwhichthecurrentintheoscillator ischaracterized bynopar ticularsymmetry, itmaybeexpressed asthesumofoddandevenparts, andEs1(u)appropriate toeachpartmaybeobtained usingSec.4,Eqs. (14)and(17),withsuitable relativeamplitudes andphases. Theelectricfieldmaintained alongthelinebythecurrents inthe oscillator maybeexpressed inaFourierseriesofthefollowing type: 00 Ez1(x-xo)=L[amcosm(3'(x-xo)+bmsinm(3'(x-xo)](5) m-l intherangeofxbetween Xo-gandXo+g.Theam'sandbm'sare complex constants. Forusein(3)and(4)itismoreconvenient to express(5)intermsofhyperbolic functions. Thus,withy'=j(3', 00 Ez1(x-xo)=L[amcoshmy'(x-xo)-jbmsinhmy'(x-xo)](6) m=l If(6)issubstituted in(3)and(4),thefollowing integrals mustbe evaluated: 11==l~~:~sinh(yx+(0)coshmy'(x-xo)dx (7a) J2==(xo+gsinh(yx+00)sinhmy'(x-xo)dx (7b) }XO-g' Byusingfamiliar formulas (e.g.,Dwight651.03and651.05)theprod uctsoftwohyperbolic functions maybetransformed intosumsofhyper bolicfunctions thatarereadilyintegrated. Theresultsare 11=(P+q)sinh(yxo+00)+(p'+q')cosh(yXO+(0)(8a) 12=(P'-q')sinh(yxo+00)+(p-q)cosh(yxo+00)(8b) Sec.5] OSCILLATORS ANDCOUPLED SECTIONS OFLINE 465 I I I, \ \ \. \where p=sinho(r+mr')+sinho'(r+mr') (9a) - 2(r+mr') P'=cosho(r+mr')-cosho'(r+mr') - 2(r+mr') (9b) _sinhO(r-mr')+sinh0'(r-mrf ) (9c) q= 2(r-mr') , _coshO(r-mr')-coshOf(r-mr') (9d) q= 2(r-mr') Bynowintroducing thenotation n Vx==l[am(p+q)-jbmCP'-q')] (lOa) m=l n Wx==L[am(p'+q')-jbm(P-q)] (lOb) m=l Eqs.(3)and(4)reduceto I:e=[Vxsinh(rxo+00)+Wxcosh(rxo+00)]~SB (11) Vz=[Vxsinh(rxo+00)+Wxcosh(rxo+00)]~: (12) Theseexpressions areidentical withChap.IV,Sec.4,Eqs.(5)and(6), forthecurrentandvoltagemaintained bythreepairsofequalandoppo sitepointgenerators, asillustrated inChap.IV,Fig.4.1.Notethatthe Mainline,--. ---_-_-_~__ Load"'----------------- ._-~ ------ -1-------- f~-cOUPling elementwithfasymmetrical currents, /Resonant line Togeneratort FIG.5.1.Transmission-line coupling element withopenendandasymmetrical current distribution andelectricfield. coordinate Xo,whichlocatesthepointonthelineopposite thecenterof theoscillator, replacesthecoordinate x,whichlocatesthemiddlepairof pointgenerators. Acoupling unitthatrequires thisgeneralrepresen tationisshowninFig.5.1.Theappropriate electricfieldscanbecon structed fromcombinations ofSec.4,Eqs.(14)and(17),orfromSec.4, Eq.(14),alone. EvenCurrents. Ifthedistribution inthedrivensectionoflineis evenwithrespecttoitscenter,thecoefficients bmofthesinetermsin(5) 466 TRANSMISSION-LINE THEORY [Chap.VI vanish,andthefieldmaintained bythesesymmetrical currents isalso even.Ifthedrivensectionisfarenoughfromtheendsofthelinesothat thecoordinates z=Xo-g'andXo+gwherethefieldisvanishingly smallarepointsactually onthelineandnotbeyondoneortheotherend, theeffectiveexciting fieldoftheevencurrents isalsoeven,andg'=g. Mainline_________ -_-~~~~Load (a)Oscillator with evencurrent 'e:.-- (b)Mainline_________ -_-~~::~ Load ",.,--------- ......... ",---------- A/2-coupling element withevencurrent Resonant line Togenerator+ FIG.5.2.Transmission-line oscillator andcoupling element withevencurrents and electricfields. Itfollowsfrom(9b,d)thatP'=q'=O.Theeffective drivingvoltage oftheevenexcitation isgivenby(lOa,b),withbm=0andp'=q'=O. Itis n VI:=I[amCP+q)] m=l(13) Thiscorresponds toasinglepairofequalandopposite pointgenerators atz=xo.Anoscillator andacoupling unitwiththistypeofsymmetry areshowninFig.5.2.Theappropriate electricfieldisgivenbythesum ofSec.4,Eqs.(14)and(17),withtheuppersign.Itisillustrated in Fig.4.4foroneparticular oscillator. OddCurrents. Ifthedistribution inthedrivensectionisoddwith respecttoitscenter,thecoefficients amofthecosinetermsin(5)vanish, andthefieldmaintained bytheoddcurrents isitselfodd.If,inaddition, thedrivensectionisfarfrombothendsoftheline,sothatthecoordinates z=Xo-g'andz=Xo+gareontheline,theeffectiveexciting fieldof theoddcurrents isalsoodd,andg'=g.Itfollowsfrom(9b,d)that p'==q'=O.Theeffective driving voltage oftheoddexcitation is givenby(10a,b),witham=0andp'=q'=O.Itis n Wz=l[-jbm(P -q)] m=l(14) Thiscorresponds totwopairsofequalandopposite pointgenerators Sec.5] OSCILLATORS ANDCOUPLED SECTIONS OFLINE 467 locatedatxo-gandxo+g,asshowninChap.IV,Fig.3.1.Anoscil latorandacoupling unitwiththistypeofsymmetry areshowninFig. 5.3.Theappropriate electricfieldisgivenbythesumofSec.4,Eqs. (14)and(17),withthelowersign.Atypicalexample issketched in Fig.4.5. Traveling Waves;Transmission-line Directional Coupler. Asindicated attheendofSec.4,atraveling-wave distribution ofcurrentinthedriven ~--- (a) '=---Mainline__________ -_--~~-~~ Load \----- !is---=~- Oscillator with_!"-_____-- oddcurrents Mainline__________ --=~Load A/2-coupling element -=-="t~f--'.'=-=-- withoddcurrents (b) Togeneratort FIG.5.3.Transmission-line oscillator andcoupling element withoddcurrents and electricfields. lineisequivalent toasuperposition ofevenandoddcurrents intime quadrature. Itfollowsthattheexcitation inacoupled linemustbe equivalent toasuperposition ofonepairofequalandopposite point generators torepresent theevencurrent,andtwopairsofsuchgenerators torepresent theoddcurrent. Theappropriate generalformulas forthe currentareEq.(11)orChap.IV,Sec.4,Eq.(5),whenx~z~sand Chap.IV,Sec.4,Eq.(7),when0~z~x.Ifthecoupled secondary lineisterminated atbothendsinitscharacteristic impedance sothat Zo=Zc,Z8=Zc,orPo=P8=00,theseequations reduceto lz=VI:~cw:c eY(:C-z) x~z~s (15a) V-Wlz=:c2Zc:ccY(:C-z) 0~z~x (15b) wherexisthefixedlocation ofthepointgenerators andzisthepointalong thelinewherethecurrentismeasured. Thefollowing areimportant specialcases: ForW:c=V:c: lz={V.eY(~" x~z~s(16a) Zc 0 O~z~x {~xe-Y('-"x;£z~s ForW:c=-V:c: lz=O~z~x(16b) Zz 468 TRANSMISSION-LINE THEORY [Chap.VI Thus,whenWz=Yz,atraveling waveofcurrentisexcitedonthecoupled transmission linethatmovesinthepositivezdirection towardZs=Zc, whereasthelineinthenegative zdirection isnotexcited. Alternatively, whenWz= -Yz,atraveling waveofcurrentmovestowardZo=Zcin thenegative zdirection, whereasthelineinthepositivezdirection isnot excited. SincefromChap.IV,Sec.4,Eq.(7),Wz==j{3gye,wherejyeis theemfofeachofthefourpointgenerators, itfollowswiththesign convention inChap.IVthatthecondition Wz=yzcorresponds to yB=-jYz/{3gandtothecurrent 13(u')=13(O)e-ifJ(h-u.'),whichisa traveling waveinthenegative zdirection. Similarly thecondition Wz= -Yzcorresponds toye=jYz/{3gand13(u')=13(O)eifJ(h-u'),which isatraveling waveinthepositivezdirection intheprimary circuit.It followsthatatraveling waveinthepositivezdirection intheprimary sectionofline(terminated coupling element) inducesatraveling wavein theopposite direction inthecoupled transmission line.Thisbehavior corresponds tothatdescribed inChap.III,Sec.15,forthehybridjunc tionwhenusedasadirectional coupler.Itfollowsthatasectionofline thatisterminated atbothendsinitscharacteristic impedance andis coupledtoasecondlinecanbeusedasadirectional coupler. Clearly, energythatreaches Zo=Zcmustoriginate inatraveling waveinthe positivezdirection intheprimary line,andenergythatreachesZs=Zc originates inawavetraveling inthenegative zdirection intheprimary line.Detailsontheconstruction andoperation ofsuchdirectional couplers aregivenintheliterature. 142,166 Theexpressions [Sec.4,Eqs.(8),(14),and(17)]involvenorestriction ontheperpendicular distancedbetween themainlineandtheoscillator. Thatis,anydegreeofcoupling ispermissible. However, ifdissmall enoughsothatthefieldmaintained bythecurrents inthelineinduces asignificant voltageintheoscillator, thefrequency generated ismodified bythepresence oftheline,andcoupled-circuit effectsresembling those described inSec.2areobserved. Thesemaybeanalyzed byintroducing ineachcircuitappropriate generators maintained bythecurrentinthe othercircuitandexpressing theseintheform Y=-IZm Inthismannertwosimultaneous equations, eachinvolving thecurrents inbothcircuits, areobtained, justasinSec.2,wherethemutualimped anceconsistssimplyofaninductance incommon. IfthecQupling between theoscillator andthemaintransmission lineis loose,sothatthereisnosignificant modification ofthecurrentinthe oscillator orofthefrequency generated byit,theequivalent induced emfsVzandWzmaybetreatedasconstant pairsofpointgenerators. 6.Coupled Transmission Lines.Whentwotransmission linesare sufficiently closeandsoorientedthatthecurrents andchargesintheone Sec.6] OSCILLATORS ANDCOUPLED SECTIONS OFLINE 469 inducesignificant currents andchargesintheother,amutualinterfer enceofthesignalstransmitted alongthelinesoccurs. Atlowfrequencies thisisknownascrosstalk.Ataudiofrequencies theinduced voltages maybederivedfromrelatively simpleelectrostatic andmagnetostatic analyses involving thedetermination ofmutualcapacitances andinduct ances.Thegeneralorhigh-frequency problem ofdetermining theinter actionbetween twotransmission linesismuchmorecomplicated. A specialcaseofthegeneralproblem isanalyzed inSees.4and5,where itisassumed thatthetwolineslieinparallelplanesatthecornersofa regulartrapezoid, sothatbothlinesremainbalanced. Inamoregeneralcasethetwolinesmaybeassumed parallelbutwith arbitrary relativepositions. Thecurrents arethoseoftransmission lines witharbitrary loads.Ingeneral,theyarenotsinusoidal butarederived !Vle II <DQI -~~"::3Zsl + 12®Q2!vt !v2e :'<ID -~:02 3Za2 + -14®!v2e FIG.6.1.Coupled two-wire lines. fromahyperbolic sineorcosineofcomplex argument, asgivenby Chap.IV,Sec.2,Eq.(7).Theelectricfieldofsuchadistribution in asingleconductor maybedetermined byusingthiscurrentinsteadof Sec.3,Eq.(la),intheintegral [Sec.3,Eq.(2a)].Thefieldofthecur rentsandchargesinthetwoconductors (e.g.,3and4inFig.6.1)ofone lineatapointQIonconductor 1ofthesecondlineisobtained bysuper imposing thefieldofthecurrents andchargesineachoftheconductors 3and4atQl.Sincethepositions ofthetwoparalleltransmission lines arearbitrary, thefieldatQ2onconductor 2isnotthenegative ofthat atQl.However, itmaybedetermined bysuperimposing thefieldsof conductors 3and4atQ2.Thefieldsalongeachoftheconductors 1and 2maythenbetreatedasadistribution ofpointgenerators. However, theyarenotequalandopposite inthetwoconductors, sothatthelineis unbalanced. Thebalanced partofthecurrent, whichcontributes tothe balanced currentmaintained inthelinebyitsdrivinggenerators, maybe determined inamanneranalogous tothatusedinSec.5.Theunbal ancedcurrentmustbeanalyzed bythemethods ofantenna theory.lO Ifeachofthecoupledlinesconsists ofasingleconductor paralleland closetoalargemetalplane,eachconductor withitsimageisequivalent 470 TRANSMISSION-LINE THEORY [Chap.VI toatwo-wire lineandmaybeanalyzed assuchfollowing thegeneral methodusedinSecs. 4and5. 7.Admittance ofBridge-coupled Sections ofLow-loss Transmission Line;Coupled-circuit EffectsInvolving Minima andDoublePeaks.lSO.lS8 Consider theproblem ofdetermining theinputadmittance Ylofthe transmission lineshowninFig.7.1.Itconsists ofasectionoflineof lengthw(tobecalledtheprimary), whichisterminated inaninductive reactance wLinparallelwiththeinputimpedance Z2ofasecondsection oflineoflengthu(tobecalledthesecondary), whichisterminated inan 1-+----- w----*'"------- u------~ z=o z=w z=u+w (a) ivoBl B2 - + .lot+Yl IL IL + I14 lv/I'w u -I20 (b) FIG.7.1.Bridge-coupled sections oftransmission line(a)witharbitrary termination Z",and(b)withinductive bridgeLastermination andapairofpointgenerators at theinputterminals. arbitrary impedance Zu.Notethatthecoupled-circuit oscillator, for whichthenaturalfrequencies aredetermined inSec.2,consists essen tiallyofthissamecircuit,butwithanegative resistance connected across theinputterminals andwithZuspecialized tobeashortcircuitoran opencircuit.ItisshowninChap.III,Sec.1,thatthenormalized input admittance Yl=Yl/Yeofasectionoflineoflengthw,whenterminated inanarbitrary normalized admittance Ylw=Yw/Ye,is Ylw+tanhywYlwcoshyw+sinhyw Yl=Ylwtanhyw+1=Ylwsinhyw+coshyw (1) Ye=Ge(1+jcPe)isthecharacteristic admittance ofthelineand 'Y=a+j(3isitspropagation constant. TheNormalized Terminal Admittance ofthePrimary. Inthecircuit showninFig.7.1thenormalized terminal admittance oftheprimary is 1 Ylw=j(3k+Y2(2) whereY2isthenormalized inputadmittance ofthesecondsectionofline Sec.7] OSCILLATORS ANDCOUPLED SECTIONS OFLINE 471 andwhere,withvthephasevelocity andltheinductance perloopunit length, wL{1k= - =wWcvl(3) (5b)(5a)Thenormalized inputadmittance ofthesecondsectionoflinemaybe expressed inthecompletely hyperbolic form,asfollows: Y2=coth(A+jF') (4a) where A:=au+PuF':={1u+4>~ (4b) Theterminal functions pand4>'arediscussed inChap.II,Sec.15.Using Chap.III,Sec.1,Eqs.(16b)and(17b),theinputconductance andsus ceptance definedbyY2=g2+jb2are sinhAcoshA. A g2=sinh2A+sin2F'=A2+sin2F' b_ -sinF'cosF'...:...-sinF'cosF' 2-sinh2A+sin2F'-A2+sin2F' Theexpressions ontherightin(5a)and(5b)applytoalinewithlow over-allattenuation whichsatisfiesthefollowing inequalities: (6) (7b)(7a)With(5a)and(5b)substituted in(2),Ylw=glw+jb1wisobtained. Thus . . A glw=g2=A2+sin2F' b==_(1+sinF'cosF') lw\flkA2+sin2F' Thesearethefinalexpressions forthenormalized conductance andsus ceptance ofthetermination ofline1. Forlaterreference itisconvenient toexamine certainrangesofF'for which(7a)and(7b)assumesimpleforms. Consider firsttheprincipal ranges,whichincludeallvaluesofF'={1u+4>~whichsatisfythefollow inginequality: sin2F'»A2 (8) Intheseranges(7a,b)become glw==Acsc2F'b1w==-(tk+cotF') (9) Forsufficiently smallvaluesoftheattenuation functionA,asrequired by(6),therangesofF'excluded by(8)departonlyslightly fromthe points (10) (11)nintegral 1 b1w= -13kF':={1u+4>~=n1f' 1glw=A sothat 472 TRANSMISSION-LINE THEORY [Chap.VI Inordertoexamine smallrangesnearthepointsdefinedin(10),let F'=={3u+cI>~=n1r+{3dn=1,2,. . . (12) wheredisasmallpositiveornegative quantity thatisassumed tosatisfy theinequality ({3d)2«1.With(12)itfollowsthat (14b)(13a) (13b) (14a)sinF'cosF'=jsin2F'=jsin2{3d=={3d sin2F'=sin2{3d=({3d)2 AHenceglw=A2+({3d)2 b1w= -[(3lk+A2:d({Jd) 2] Themagnitude ofthenormalized admittance atF'=n7r+{3dis where A=~{3u+Pu=~(n7r-~+(3d)+Pu==An+ad==An(15b) where An=a;7r (15e) Since{3dmaybepositiveornegative, itisevidentthatb1wmayvanish. Therequired valuesof{3dare Forb1w=0:pd=-P;[1±~1-e~n)2]={=p~ (16a) Thecorresponding valuesofglware ~An •An 2An •{32k2 glw= -{3d{3h=({Jk)2[1±VI-(2An/{3k)2]=1-.(16b) An Bydifferentiating b1win(14b)withrespectto{Jdandequating the derivative tozero,themaxima andminima ofb1warelocated. They occurat (17a) andhavethevalues b1w= -;k(1±:~~) Thecorresponding valuesofglware 1±a/{3 glw=--2A-:---(17b) (17e) Sec.7] OSCILLATORS ANDCOUPLED SECTIONS OFLINE 473 Themaximum valueofglwmaybelocatedinthesamemanner. Its locationandmagnitude aregivenby {3d==0 (18) Theminimaofglwoccurintherangespecified in(8)where(9)applies. Theyoccuratvaluesofusomewhat smallerthanthosedefinedby I I I Ol=3.93x10-3nep/m Y1w-4L IL fJ=Tf/2rad/m;A.=4mi4---U---~ k=L/I=O.04m l'iw~=Ylw= glw+jb1UJ 1'~rst ~idgealone.-J 7 V-'f-Ir-~ I I I-GO-40b1UJ -2060 o2040 -80 o 025 0.50 0.75 1.00 1.25 FIG.7.2.Normalized susceptance bI",ofinductance inparallelwithasectionoflineof lengthuterminated inanequalinductance. F'=={ju+4>~=[(2n+1)/2]11'andhavemagnitudes closetoA=aU. Extreme valuesofthemagnitude Ylwmaybeobtained bydifferentiating (15a)withrespectto{3d.Withnintegral, theyarelocatedat {3d=0 {jd==-{3hor or{ju=n1l'-4>~ {3u=n1l'-{jk-4>~forthemaxima (19) fortheminima (20) ItisseenthattheminimaofYlwvirtually coincide withoneofthesets ofvaluesforwhichb1w=O.Theextreme magnitudes ofYlware ().A; Ylwmin=13k (21) Graphsofthenormalized inputsusceptance b1w,theconductance glw, andthemagnitude oftheadmittance Ylw=Vgrw+brwareshownin Figs.7.2to7.4foraspecialcaseinwhichthetermination ofthesecond sectionofthelineisalsoaninductive reactance wL.Thatis,Zu==jwL, sothat 4>~=13kandPu==o.ItfollowsthatF'=(j(u+k).The numerical valuesoftheseveralconstants giveninthefiguresarethose ofaparticular apparatus. Numerical dataaregiveninTable7.1. 474 TRANSMISSION-LINE THEORY [Chap.VI AstudyofFigs.7.2and7.3revealsthatwiderangesofnormalized terminal conductance andsuscep tanceareavailable. Thesuscep tanceremains nearthevaluechar acteristic ofthefirstbridgealone, exceptnearF'=(j(u+k)=n7r, whereitvariesrapidlybetween high positiveandnegative values. The conductance isquitesmallexcept nearthepointsF'=n7r,whereit risest<,>highmaxima. Themagni tudeofthenormalized admittance showninFig.7.4ischaracterized byadjacent minima andmaxima nearF'=n7r.Asindicated in(19) and(20),themaxima occurat {ju=n7r-{jkandtheminimaat {ju=n7r-2{jkforthespecialcase represented inFig.7.4,inwhich ~={jk.Thelatterformula is convenient touseintheexperi mentaldetermination ofkfora 0.0010~J....&-~"""""~:7-''''''''''''~'''''''''~'''''''~ conducting bridge,asdiscussed later 0.25 0.50 0.75 1.01.25inthissection. F/211'=(u+k)/l\.TheInputAdmittance ofthePriFIG.7.3.Normalized conductance associ- atedwiththesusceptance inFig.7.2. maryintheBridge-coupled Section. Therealandimaginary partsof(1) maybeseparated byintroducing Ylw=glw+jb1w•Thefollowing results I I ~=3.93xlO-3nep/m Ylw-.!L IL fJ=1f/2rad/m-t-----u---~ k=L/I=O.04 mY1wZ'C=Yl...=gl...+jb1w ",\100120 204080 1.11...1 60 00 0.25 0.50 0.75 1.00 1.25 F121f=(u+k)/i\ FIG.7.4.Magnitude ofnormalized inputadmittance associated withthesusceptance andconductance ofFigs.7.2and7.3. Sec.7] OSCILLATORS ANDCOUPLED SECTIONS OFLINE 475 TABLE7.1.NUMERICAL VALUES OFb1wANDglwFORTHEFOLLOWING PARAMETERS: /3=iradians/m; k=0.04m;a=3.93X10-3neper/m F'//3= F'//3= u+k,b1w gltl1 Ylwu+k, b1w glw Ylw m m 0.05 -28.6 0.0068 28.6 2.50 -16.9 0.019316.9 0.08.......... 0.0099 3.00-15.9 0.011615.9 0.10 -22.2 0.0096 22.2 3.50-14.9 0.027214.9 0.20 -19.0 0.0066 19.0 3.80-12.8 0.15512.8 0.50 -16.9 0.0036 16.9 3.90-9.6 0.6169.6 1.00 -15.9 0.0038 15.9 3.95-3.2 2.3974.0 1.50 -14.9 0.0115 14.9 3.96 -0.99 3.6663.8min 1.80 -12.8 0.0726 12.8 3.963 0.0 4.1784.2 1.90 -9.6 0.297 9.6 3.97 +3.18 6.2557.0 1.95 -3.2 1.200 3.41 3.98 +9.54 12.5415.7 1.96 -0.2 1.885 1.9min3.99+15.9max 31.4335.2 1.9606 0.0 1.940 1.94 3.9973 0.036.3936.4 1.97 +4.7 3.231 5.7 4.00-15.9 63.0365.0 1.98 +14.1 7.273 15.9 4.01-47.7min31.5957.2 1.995 +47.7max 62.23 78.5 4.02-41.4 12.6743.3 1.9994 0.0123.3 123.3 4.03-35.0 6.35035.6 2.00 -15.9 124.8max 125.8max 4.04-30.9 3.74031.1 2.005 -79.5min62.55 101.2 4.05-28.6 2.45828.7 2.01 -66.7 25.08 71.3 4.10-22.2 0.64822.2 2.03 -36.6 3.331 36.8 4.20-19.0 0.17219.0 2.05 -28.6 1.264 28.6 4.50-16.9 0.035116.9 2.10 -22.2 0.329 22.2 5.00-15.9 0.019515.9 2.20 -19.0 0.0891 19.0i I areobtained: Dcos({3w+0)+jEsin({3w+e) Yl=Ecos({3w+e)+jDsin({3w+0)(22a) (23c)(22b) (22c) (23b) (23d)(23a)gl=D2sin2({3w+0)+E2cos2({3w+e) b 1=([E2sin2({3w+e)-D2sin2({3w+0)] D2sin2({3W+0)+E2cos2({3w+e) 1=[D2cos2({3w+0)+E2sin2({3w+e)J! YD2sin2({3w+0)+E2cos2({3w+e) ()_ t-1E2sin2({3w+e)-D2sin2({3w+0) 1 -an2DEros(0-e)where D==[(glw+aw)2+biwa2w2]1 E==[(1+glwaw)2+biw]l o==tan-1b1waw e==tan-1 b1w glw+aw 1+glwaw Itfollowsthat,withYl=gl+jb1=Ylej8" DEcos(0-e) 476 TRANSMISSION-LINE THEORY [Chap.VI SpecialCasewithSelf-resonant Primary. Aninteresting specialcaseis obtained whentheprimary circuitisadjusted tobeself-resonant before thesecondary isconnected. Inputresonance forasectionoflineof lengthw,whenterminated inaninductive bridgeofequivalent lengthk, isdefinedby(j(w+k)=m1r,wheremisaninteger. Withthissetof valuesof{jw,themagnitude Ylin(23c)maybetransformed into =[F2cos2(jk-2bl~(1-a2w2)sin{jkcos{jk+G2sin2{jk]l24 YlG2cos2(jk+2b1w(1-a2w2)sin{jkcos{jk+F2sin2(jk() where F2==biw+(glw+aw)2 G2==biwa2w2+(1+glwaw)2 (25) Inmostpractically interesting casesthefollowing inequalities are satisfied: (aw)2«1({jk)2«1 (26) With(26)themagnitude (24)reducesto . [ (blw-{jk)2+(glw+aw)2 ]1 YI=(biw+groo)(a2w2+(j2k2)+2(glwaw+b1oo{jk)+1(27) Thegeneralbehavior ofYIin(27)asafunction oftheelectrical length {juofthesecondary maybedetermined byconsidering theordersof magnitude ofb100andgloo.Thus,intheprincipal rangesspecified in (8),glwissmallcompared withbloo,whichremains almostconstant near -1/{jk,asisclearfrom Figs.7.2and7.3.Withthisvalueofb1wsubsti tutedin(27),theleadingterminthenumerator isbiw=11{j2k2.The terms bi~{j2k2+2b1w{jk+1inthedenominator cancel,andtheleading termisbiwa2w2=a2w21{32k2.ItfollowsthatYIremains approximately constant atthevalue . 1Yl=aw(28) intherangedefinedin(8).However, thisisthevaluecharacteristic of theself-resonant primary alone,withthesecondary removed. Thusitis tobeexpected thatYlissensibly constant atthelargevaluel/aw,except whenthesecondary lengthissuchthatF'={3u+4>~isnearn1r,where bothb1wandgloobecomelarge. IntherangesofuforwhichF'=(3u+cf>~isnearn1r,glwisverylarge compared withaWand13k,andblwvariesbetween largepositive and negative values.Itfollowsthatintheserangesthenumerator in(27) reducestobioo+groo:=:yroo'Thus Ylw YI=[Yioo(a2w2+(32k2)+2(glwaw+b1w{3k)+1]1 (29) Although, with(26),awand13karebothsmall,Ylwmaybesufficiently greatsothatalltermsinthedenominator contribute significantly. Nevertheless, sinceYlwistheentirenumerator, itmaybeexpected that Sec.7] OSCILLATORS ANDCOUPLED SECTIONS OFLINE 477 thevariation ofYlwithuwillresemble thatofYlw.Thatis,thereshould be'maxima near{3u=n-tr-4>~andminima near{3u=n7l"-{3k-4>~. SincetheseminimaofYlwcoincide withzerovaluesofb1w,andglwissuf ficiently smallsothattheleadingterminthedenominator of(29)is1, itfollowsthattheminimainYlmustoccuressentially atthesamevalues of{3uastheminimainYlw.Graphsofthemagnitude Ylasfunctions of thelengthuofthesecondary aregiveninFigs.7.5and7.6forvarious conditions. Figure7.5hasbeenevaluated foraratherlongprimary circuitforareasonthatisdiscussed below. Asaconsequence, thehigh maxima inYlwshowninFig.7.4havecorresponding maxima inYlwhich 1.0 0.7512',__ _ _ _ ":'_ I18a=3.93X10-3nep/rri\-" r+-~r----+--+---+t+--t )/1=11'/2rad/m •4.\=0.04m +--+I--~---t---+--t+---I ow+k-5i\. o 0.25 0.50 u/ll FIG.7.5.Theoretical normalized magnitude oftheinputadmittance ofaself-resonant sectionoflineofelectrical length (:3(w+k)=10,whenterminated inaninductance L=klinparallelwithasecondsectionoflineofelectrical lengthualsoterminated in L=kl. areverymuchreduced. Ontheotherhand,theminimaofY1arecom parablewiththoseofYlw.InFig.7.6,Ylisshownforseveralvaluesof w,a,andk.Itisseenthat,whereasthemaximamaybegreatlyaltered, theminimaareallsharpandactually occurat {3u=n7l"-2{3k (30) inallcaseswhen 4>~={3k. TheDetermination oftheEquivalent LengthkofaConducting Bridge. Usemaybemadeofthesimpleformula (30)todetermine theequivalent lengthkofeachoftwoidentical conducting bridgesbydirectexperimental measurement. Clearly, ifthesharpminima (inFig.7.5or7.6)inthe magnitude ofthenormalized inputadmittance canbelocated, thedis tancebetween thefixedbridgeterminating theprimary andtheidentical bridgeterminating theextensible secondary, whenthishasalengthto givethefirstminimum inYl,is u=~-2k (31) andthedistances between adjacent successive minimaareXj2.Thus, bylocating thefirstandsecondminimum relativetothefixedbridge (theseareshown,forexample, inFig.7.5),bothXj2-2kandXj2may bemeasured. Fromthesemeasurements 2kisdetermined. Itisnotnecessary actually tomeasure theinputadmittance Y1ofthe bridge-coupled circuit,showninFig.7.1(withZu=jwL)asafunction 478 TRANSMISSION-LINE THEORY [Chap.VI ofu,inordertolocatetheminimainYI=YI/Yc==YI/Gc•Sincethe normalized inputadmittance isdefinedby 10YI=VOYc(32) where10isthecurrentandVoisthevoltageattheinputterminals z=0 6Or--------;---oooo:---:----------. a=3.93x10-3nep/m;p=7T/2 rad/m;,\,...4m 50~k=O.Olm;w+k='\'" "'...........----_-.1 .....-..'-_ I I40- \:~ ~ ~ 30i- IJI. II·· 20::.~::~:~~:.~+k="-) ~ . 10::-.-.-.-.~.:::; ::.:::.::.::~~:~:: ...t·_·_·-.- k=0.04.m; w+k=5ji\ '.~o I 'Io 0.125 0.250 0.375 0.500 0.625 ulll (a) 150,..------:-----:'---------- k=O.OIm;p=rr/2 rad/m;A=4m;w+k=i\ I I 100i- III I 1)'11.. ka=1.31 x10-31\ J,;-", .... ":l" '..------ .....1.------~,I-------,1 50~ 't··· 'J..... ,I......."'1~~3.93·~·i;;:;T1 '" '""""1. 0'"---""'- ,__ .&.''1-'1-__"--,__......' o0.250.50 0.751.00 1.251.50 u(meters) (b) FIG.7.6.Magnitude ofnormalized admittance (a)fordifferent valuesofL=kland wand(b)fordifferent valuesoftheattenuation constant Cl'. oftheprimary, itisclearthatIYIIisproportional to1101.Hence,ifan impedanceless ammeter isconnected inserieswithanimpedanceless generator acrosstheinputterminals oftheprimary inFig.7.1,andthe magnitude ofthecurrent10isobserved asuischanged (withZu=jwL), theminimaof10coincide withthoseofYI. Inpractice, thecombination ofanimpedanceless ammeter inseries withanimpedanceless generator isunavailable, andafinitevalueofZo isunavoidable. However, anactualcircuitconsisting ofaprimary of lengthw',drivenbyagenerator inserieswithanammeter, andtheir Sec.7] OSCILLATORS ANDCOUPLED SECTIONS OFLINE 479 combined impedance Zoatz=0maybeapproximated byanidealcir cuitconsisting ofaprimary oflengthw>w',drivenbyanimpedance lessgenerator inserieswithanimpedanceless ammeter. Therequire mentthattheinputcurrents inthetwocasesbethesameissimply (33a) ,----r---- ""I\, I \' "\ I "\' t ~~15coo ~10 of! co - 5~or,intermsofnormalized values, coth("'('w'+Ow)+Z10=coth("'(w+00) (33b) In(33b)ithasbeenassumed thatthecharacteristic impedances ofthe 20 o6.86.5 6.0 5.55.04.54.03.5 3.0 2.5 B1 PositionofB2onscale FIG.7.7.Experimentally determined magnitude ofcurrententering primary aslength ofbridge-coupled secondary isincreased. B1isthelocation ofthebridgeterminating theprimary; B2isthelocation ofthebridgeterminating thesecondary. Lineand bridgesareofNo.12copperwire;A=4.3m. actualandtheequivalent sections oflinearethesame,butthepropa gationconstants "'('and"'(maydifferintheirrealparts.Thatis, "'('=a'+j{3,and"'(=a+j{3. Intheprincipal rangesoftheinputimpedances, Chap.III,Sec.2, Eqs.(32a,b),maybeusedin(33b)toseparate therealandimaginary parts.Thisleadstothefollowing pairofequations fordetermining w andaoftheequivalent sectionofline: XI0-cot({3w'+tPw)= -cot({3w+tPw) (34a) 2rl0+sinh2(a'w'+Pw)csc2({3w'+tPw) =sinh2(aw+ww)csc2({3w+tPw)(34b) From(34a) {3w=cot-1[cot({3w'+tPw)-XI0]-tPw-n7rnintegral (35) Ifw,asgivenby(35),issubstituted in(34b),'avalueofamaybedeter minedforeachchoiceoftheintegern.Ingeneral,itisconvenient to selectthatvalueofnwhichwillmakeaanda'asnearlyequalaspossible. Actually, sincethelocation oftheminimainYlisnotsensitive tothe precisevalueofa,itisoftenadequate (especially ifwislarge)toselect thevalueofnwhichmakesaasneara'aspossibleandthentoapproxi mateabya'. InFig.7.7isshownanexperimentally determined curveofanampli tudeproportional toloinacurrentindicator acrosstheinputterminals 480 TRANSMISSION-LINE THEORY [Chap.VI ofabridge-coupled line.Thesecondary circuitandthecoupling and terminating bridgesarethosedescribed inconnection withthenumerical valuesinTable7.1andthecurvesofFigs.7.2to7.4.Byobserving the successive amplitudes ofsixresonance maxima intheprimary circuit whenwwasvariedwiththesecondary absent,theapproximate length ofanequivalent circuitwithimpedanceless detector andgenerator was determined. Thetotalequivalent lengthwoftheprimary turnedout tobe5A(tothenearesthalfwavelength indetermining a,whichwas thenmadeequaltoa').ThusthecurveofYlinFig.7.5istheapproximate theoretical equivalent ofFig.7.7.Theagreement isseentobeexcellent. Bydetermining thevaluesofuforthetwominimainFig.7.7,the wavelength ofthegenerated signalandtheequivalent lengthkofthe identical bridgesmaybedetermined. Inthisparticular casethedis tancebetween thetwominimais2.15m,sothatA=4.30m;thedis tancebetween thecoupling bridgeB1andB2whenatthefirstminimum is2.07m,sothatk=(2.15-2.07)/2=0.04m. Double-hump Phenomena. Interesting coupled-circuit effectssome timesknownas"double-hump phenomena" areeasilyobtained with thecircuitofFig.7.1.Inprinciple, doublehumpsdependonasuper position oftheconventional resonance peaksdescribed inChap.IV, Sec.9,andcoupled-circuit minimaofthetypesshowninFigs.7.5to7.7. Depending ontheparticular tuningadjustments, theminima maybe madetooccurexactlyatthecenterofasimpleresonance maximum or displaced towardoneortheotherside.Theappropriate conditions may bedescribed inconjunction withthecircuitofFig.7.1b. Inordertoobtainconventional resonance maxima ofloinFig.7.1b orofnormalized inputadmittance inFig.7.1abymovingtheconducting bridgeB1(Fig.7.1b),itisnecessary thatthesecondary circuithavea negligible reaction ontheprimary when(3wisnearandatitsresonant values,definedby(3w+<Po+<Pw=m1r,wheremisanintegerwith(3w positive. InthecircuitofFig.7.1b,<Po=1r/2,and<Pw=1r/2+(3k,so thatthecondition forresonance inthiscaseissimply(3(w+k)=m7r, withm=1,2,. ...Thereaction ofthesecondary isnegligible ifits electrical effective length(3u+<Pu=(3(u+k)=n1r+7r/2,wherenis aninteger. Thus,iftheelectrical length (38=(3(u+w)isnearN7r+ 7r/2,whereNisaninteger,theresonant valuesof(3wcoincide withvalues of(3uforwhichtheeffectofthesecondary isnegligible. Hence,ifB1in Fig.7.1aorbismovedwithB2fixed,sothat(38==N1r+7r/2,thenormal izedinputadmittance YlinFig.7.1aorthemagnitude ofthecurrentin Fig.7.1bwillhavemaxima when(3(w+k)=m7r.Atheoretical curve ofYl,whichisproportional to10,iscurvefinFig.7.8.Thiscurvewas computed forthelongprimary circuitpreviously described. Similarbut sharpercurvesareobtained forsmallervaluesofw,asdescribed inChap. IV,Sec.8. Sec.7] OSCILLATORS ANDCOUPLED SECTIONS OFLINE 481 Themaxima showninFig.7.8maybesplitbyaminimum ofthetype illustrated inFig.7.5iftheover-allelectrical length {1sisadjusted toa valuesuchthat{1u=m7r-2{1kwhen{1w=n7r-{1k.Thatis,when {1s=N7r-3{1k,thereaction ofthesecondary shouldsplittheprimary maximum symmetrically asB1inFig.7.1ismovedthrough anappro priaterange. 14 12 10 4 2 oI I II"Curvt'sAI-«=3.93x10-3nep/m'h 13:1'/2 rad/m N~\f5.7400 e5.9875 I-k-O.04mliP, a5.9575 1\=4m " ' c5.9725.,~ I1\\J..t IA',:c,...\ I!\J: ~I'..;,,~\,. ~ ~V r--r-- 4.90 4.95 5.00 !S.05 5.10 Lengthofprimary, wi7\, FIG.7.8.Theoretical normalized inputadmittance ofprimary asafunction ofthe lengthwoftheprimary withbridge-coupled secondary oflengthU;8=W+u. 12e ~..Curve sli\. 10a5.9575 b5.9650 8 c5.9725 IYtl d5.9800 6 e5.9875 4 2 0'"-..... .1--......._........_01---'_-'-_01---'_ ... 0.85 0.90 0.95 1.00 1.05 1.10 Lengthofsecondary, u/l\. FIG.7.9.Theoretical normalized inputadmittance ofprimary oflengthwasafunction ofthelengthuofabridge-coupled secondary; 8=W+u. Theoretical curvesofYtasafunction ofthelengthwoftheprimary, with{1ssetatthreevaluesnear{1u=27r-2{1k,aremarked a,c,andein Fig.7.8.Thesamecurvesandtwoadditional onesplottedasafunction ofthelengthuofthesecondary areshowninFig.7.9.Itisclearfrom Figs.7.8and7.9thattheminimum resulting fromtheinteraction with thesecondary occursat{3u=27r-{1kandthatitslocation withrespect totheprimary maximum isdetermined by{3s.Theexperimental verifi cationofthedoublehumpsisshowninFigs.7.10and7.11,where10is plottedasafunction of{1wand{3u.Thesecurveswereobtained with thesamelineasusedforthecurvesinFig.7.7.Asingleresonance peak 482 TRANSMISSION-LINE THEORY [Chap.VI withthesecondary detuned isshownincurve!ofFig.7.10.Theremain ingcurvesinFig.7.10andallthecurvesinFig.7.11havetheelectrical length{3uofthesecondary near211'"-2{3k,asindicated. Thecorre spondence between Figs.7.8and7.10isevident, asisthatbetween Figs. 7.9and711. CurveB2atr't~'\~=4.3m a0.73 I \;{ b0.685Ib'l#,..'. c0.67'cf-' -:b~\, d0.61,..'!y't~.\ /'..'~tII\~''.. -~...;;:;."V"....:~~'. ~¢.:.1" ....~~~80 20100 GI '560II) .ti ~40 ~ o 3.8 4.0 4.2 4.4 4.6 Lengthuofsecondary, meters FIG.7.11.Experimentally determined 10 asafunction ofthelengthofthesecondary whenvariedbymovingB1,withB2fixed attheindicated scalepoints.A=4.3m fCurveBzatHi (J'f!I.1df1.70 \(J0.73 Iib0.67 ~.\d0.61 P:~~III,.,::,'. ~~,::~:,t,';...... ~~...--- ~.220120 100 -;;80 iiiu ~60 011 ~40 5.0 4.8 4.6 PositionofB1onmeterscale FIG.7.10.Experimentally determined 10asafunction oftheposition ofthe bridgeB1inFig.7.1,withB2fixed attheindicated scalepoints. 8.Transmission-line Measurements withaMultiple-frequency Source; FilterSections.79.149,162 Throughout thepreceding sectionsandchapters ithasbeenassumed thattheoscillators drivingatransmission linegener ateonlyasinglefrequency. Ifoneormoreharmonic frequencies are generated together withthefundamental, thecurrentandvoltagedis tributions alongthelinearethesuperpositions oftheindividual currents andvoltages oftheseveralfrequencies. Anoscillator oracoupling elementthatmaintains anelectricfieldthat isevenwithrespecttoitscentermayinvolveafundamental withathird harmonic. Inthiscasetheequivalent pairofequalandopposite point generators atz=xhasacombined emf V~lforthefundamental fre quency11andanemf V~3fortheharmonic frequency fa.Foraline extending fromz=0toz=8,theresultant voltageacrossthelinein therangex~z~8isobtained usingChap.IV,Sec.2,Eq.(6).Itis v,=Vesinh("(IX+(01)cosh("(lW+(81) xlsinh("(18+001+(08) +Vesinh("(3X+(03)cosh("(3W+(83) x3sinh("(38+003+(83)(1) where W=8 -z,thesubscript 1referstothefundamental, andthesub script3referstotheharmonic. Anoscillator orcoupling elementthatmaintains anelectricfieldthat isoddwithrespecttoitscentermayinvolveafundamental andasecond Sec.8] OSCILLATORS ANDCOUPLED SECTIONS OFLINE 483 harmonic. Inthiscasetwopairsofequalandopposite pointgenerators symmetrically placedwithrespecttotheircenteratz=xarerequired, andthevoltageatapointzintherange0~z~xalongthelineis obtained fromChap.IV,Sec.3,Eq.(6).Itis Vz=Wecosh('Y1Z+001)cosh('Y1Y+Osl) xlsinh('Y18+001+Osl) +Wecosh('Y2Z+802)cosh('Y2Y+Os2) 202sinh('Y28+002+Os2)(2) whereY=8 -xandthesubscripts Iand2refer,respectively, tothe fundamental andthesecondharmonic. 100 I80d1~60 (a) ~~g~~~ 2I , ,,ji,ll,,t,,,if .560DOl 211;g40BridgedI 20lend ..f,4iII(b) olitItIiiiI I I I I I i t-T-r1oDOl 2 3 4 Positionofbridge(meters) FIG.8.1.Experimental resonance curvesofoscillator withsecondharmonic. Peaks ofthefundamental arenumbered inRoman; thoseoftheharmonic, inArabic. D locatesthevoltagedetector, 0locatesthecenteroftheoscillator withoddfielddis tribution. (a)Openendatleft;(b)bridgedendatleft. Asanexplicitapplication of(2),consider atransmission lineexcited byalooselycoupledoscillator ofthetypeshowninFig.1.6awhichmain tainsanelectricfieldthatisoddwithrespecttothepointxopposite its centerandincludes afundamental andasecondharmonic. Suchan oscillator inducesamaximum voltageinaresonant linewhencoupled atapointofmaximum voltage. Avoltagedetector islooselycoupled totheopenendofthelineatz=0(DinFig.8.Ia),andtheoscillator is nearit(at0inFig.8.Ia).Whenalargeconducting diskismovedalong thelineasashort-circuiting termination, thevoltageatz=0varies, asshowninFig.8.la.Theobserved curveconsists oftwofamilies of resonance curvesofthetypedescribed inChap.IV,Sec.7.Onefamily ischaracteristic ofthefundamental (numbered inRoman), andthe otherfamilyischaracteristic ofthesecondharmonic (numbered in Arabic). Ineachfamilythedistances between successive maxima are halfwavelengths. Theformula (2)maybespecialized toapplytothespecificcaserepre sentedinFig.8.labysetting 801=002==0,8s1=082=J1r/2,and 484 TRANSMISSION-LINE THEORY [Chap.VI P2=2P1.Theconditions forresonance forthefundamental andsecond harmonic are,respectively, -rrP28+<1>02+<l>B2=(328+2=n2-rr Theseareequivalent tonl=1,2, n2=1,2,(3a) (3b) n1X1Xl8=---2 4 nzXzX2n2X1Xl8=---= ---2 4 4 8(4a) (4b) Thusresonances forthefundamental occurwhen 8=XI/4,3XI/4,5XI/4, . . . ,andthoseforthesecondharmonic when 8=XI/8,3XI/8,5X1/8, . ...Theseconclusions areinagreement withFig.8.la.Iftheattenu ationisneglected inthenumerator, (2)reducestothefollowing simple form,asappliedtotheconditions characteristic ofFig.8.la: Yo=.~yesin(31(8-x)+·Wesin(32(8-x) Jzlcosh"(18 Jx2cosh"(28(5) Iftheendofthelineatz=0isterminated inaconducting bridgewith anequivalent lengthko,theterminal phasefunction is<1>0=-rr/2+pko• Inthiscasethemaxima inthevoltagedistributions ofthefundamental andtheharmonic havenolocations incommon, sothatitisnotpossible toplaceavoltagedetector oranoscillator withafieldofoddsymmetry inaposition whereitdetectsorinduces amaximum. (Ontheother hand,acurrentdetector andanoscillator withafieldofevensymmetry couldbecoupledtothebridgeatz=0orathalfwavelengths ofthe fundamental fromit.)Actually, adequate voltages ofbothfrequencies werereceived whenthedetector waspl~cednear(31(Z+k)=-rr/8(Din Fig.8.lb),andtheoscillator asclosetoitasphysical conditions per mitted. Thisturnedouttobeat0(Fig.8.lb),whichisratherclose toavoltage nodeoftheharmonic. With601=602=j(-rr/2+Pdco), 6B1=6B2=j-rr/2,and(32=2(31,theconditions forresonance forthefunda mentalandsecondharmonic are,respectively, 8+ko=!1'zX2=n2XI 2 4(6) Thustheresonances forthefundamental occurwhen 8+k=XI/2,Xl, 3X1/2,etc.,andthosefortheharmonic at8+k=XI/4,XI/2,3X1/4,Xl, etc.Clearlyeveryotherresonance maximum oftheharmonic coincides witharesonance maximum ofthefundamental. Anexperimental curve Sec.8] OSCILLATORS ANDCOUPLED SECTIONS OFLINE 485 (7)v.==illustrating thisisshowninFig.8.lb.Theequation forthesecurvesis _Wecos[{31(Z+k)Jsin(32(8-x) :z:1sinhh'l(S+k)J _Wecos[{32(Z+k)Jsin(32(S-x) :z:2sinh[i2(S+k)J Itisclearfrom(4)thatbyaproperchoiceofeitherzorxthecon tribution toVzatresonance byeitherthefundamental ortheharmonic maybemadeverysmall.Thismeansthattheoscillator maybeso located(byachoiceofx)thatitinduceseithernofundamental orno harmonic voltageintheline,orthatthedetector issosituatedthateither thefundamental ortheharmonic voltageissmallatthepointofcoupling. Notethatthelocations fornegligible response ofasymmetrical current indicator areaquarterwavelength fromthecorresponding locations for avoltageindicator. Similarly thelocations fornegligible induced volt agebyanoscillator withanevenelectricfield(onepairofpointgener ators)areshiftedaquarterwavelength fromthelocations fornegligible induced currentbyanoscillator withanoddelectricfield(twopairsof pointgenerators). Itisseenthat,byanappropriate location oftheoscillator (orcoupling element) andofthedetector probeandbyaproperchoiceofthetermi nations, aresonance maximum ofthefundamental orofthesecond harmonic maybelocatedattheminimum oftheother.Inthismanner resonance-curve measurements maybemadeusingeitherthefunda mentalortheharmonic ifthelossesinthecircuitaresufficiently low andtherelative amplitudes arenotsodifferent thattheminimaofthe onefrequency arenotnegligible compared withthemaxima oftheother. Coupled-circuit measurements involving thesharpminimainresonance curvesdescribed inSec.7maybemadewiththefundamental orthe secondharmonic. Thisisillustrated inFig.8.2aandbbyasetofexperi mentalcurves. ThecurvesinFig.8.2acorrespond closelytothosein Fig.8.la.BelowtheminFig.8.2bisarelatedsetofcoupled-circuit curveslikethoseinFig.7.7.Theuppercurveisobtained byobserving thevoltageatDinaself-resonant primary circuitformedbylocating aninductive bridgeatB1andthenmovingaconducting diskbeyondit toincrease thelengthofthesecondary. Inthiscasetheprimary is tunedtothefundamental. ThelowercurveinFig.8.2bisobtained in thesamemannerastheupperone,butwiththeprimary circuittuned totheharmonic bylocating theconducting bridgeatB2• The.separation ofafundamental frequency fromaharmonic byproper location alongthetransmission lineofthepointsofcoupling anddetection isusefulonlywhentheover-alllossonthelineislowandsharpresonance curvescanbeobtained. Amoregenerally usefulmethodofsuppressing aharmonic without interfering withthefundamental, orviceversa,isto 486 TRANSMISSION-LINE THEORY [Chap.VI arrangeapairofshuntstubssothattheypresentalowimpedance tothe frequency tobesuppressed andahighimpedance tothefrequency tobe transmitted. Formaximum effectiveness thestubsareseparated adis tanceequaltoaquarterwavelength ofthefrequency tobesuppressed. 100r------r--;;-----,r------r-----~---___. 80 -;60n; ~40 .ci ~20gOt"r=-...L..--J..:~~Sl.:::::t.........L..::.~~~:t:::::.J.~~;Ib,,~~~..:=.I!:ll:l..lIl.oo~::f:_....l:::~.c-L__I ~100r------t------t------+--:------+-."......--~ '0 ~80 ~60 ~40 20 oO~,.""---"--,/---'--~....L..r...I..-..J.-L-:L-.....L2..--l..-.L--l-L...;!:l-....L..--l..-L-l~l..-""'--..J......J I 4.0Meters iJB2 B) FIG.8.2.(a)Resonance curvesofasourcewithasecondharmonic. (b)Coupled circuitcurveswithconducting bridgeatB1(foruppercurve)orB2(forlowercurve). Togenerator_-.;.-._+-~--4-~:_".._--_ .......;;...Toload 2i, I I ~_k=i\2_k4,2· i.t FIG.8.3.Combination ofstubstosuppress thesecondharmonic fromthelinetothe rightofthestubswithout interfering withthefundamental. Acircuitforsuppressing allevenharmonics andpassingthefunda mentalisshowninFig.8.3.Ontheotherhand,thefundamental is suppressed andthesecondharmonic passedtotherightofthejunction 22inFig.8.4. ltisslightlymoredifficulttosuppress thethirdharmonic whilethe fundamental ispassed. Onemethodusingtwostubsinparallelisillus tratedinFig.8.5.The>-3/4openstubandthe>-3/2closedstubboth presentalowimpedance at11forthethirdharmonic, sothatthisis suppressed fromthelinetotherightof22.Neglecting losses,theadmit tanceoftheparallelcombination ofstubsforthefundamental frequency Sec.9] OSCILLATORS ANDCOUPLED SECTIONS OFLINE 487 isapproximately Yin==-jYc(cot {JSO-tan(JSc)=-jYc(cot~-tani)=0 sothatthefundamental ispassedwithoutattenuation. Ineffect,thestubcombinations inFigs.8.3to8.5arespecialtypesof filtercircuitsusingsectionsoftransmission lineastheirelements. They areeffective insuppressing onlyparticular frequencies, suchasmaybe Allr----4----;1 Togenera,.:;:;to:.:..r"';-~--I~-.:- ~I--~--r "'...:;.Toload 24 I, I &_"24-2 Toloadl: ~~Al 4,12 t1 2 rator ---. 12t I I &=~26 I I I I I "'""'-It...TogeneI I I I ..t FIG.8.4.Combination ofstubstosuppress thefundamental fromthelinetothe rightofthestubswithoutinterfering withthesecondharmonic . ...-.~--~4 FIG.8.5.Combination ofstubstosuppress thethirdharmonic fromthelinetothe rightofthestubswithout interfering withthefundamental. encountered whensimplegenerators ofthetypesdescribed inSec.1are usedunmodulated. Whenhigh-frequency generators aremodulated, variousfrequencies areintroduced, andmorecomplicated filtersare required. Athighfrequencies theseareadvantageously constructed of sections oftransmission line(usually coaxial). Sincetheirtransmission andsuppression properties aredescribed intermsofthegeneraltheory ofwavefilters,theiranalysis isbeyondthescopeofthisbook. 9.Radiation fromOpen-wire Lines.141,167Thederivation ofthedif ferential equations forthecurrentandvoltagealongatwo-wire linein Chap.Iinvolves neglecting termsoftheorderofmagnitude l~bI2.Itis thesetermswhichspecifyradiation fromtheline.Accordingly neither thedistribution ofcurrentalongasectionoflinenoritsinputimped- 488 TRANSMISSION-LINE THEORY [Chap.VI anceasdetermined fromsolutions oftheconventional equations includes ortakesaccount ofradiation. Sincethepowerradiated fromacircuit depends ontheentireconfiguration ofconductors andcannotbeassigned piecewise totheseveralparts,itisnecessary toconsider thecomplete circuitconsisting ofthelineandtheterminations atbothendsifthe radiated poweristobeevaluated. Consider atwo-wire lineextending fromz=0toz=Sinair.The generator isatz=0;ithasanimpedance ZooTheloadatz=8isZ,. Theimpedance seenbythegenerator is Z=Zo+Zctanh("(s+8~)+Rg (1) where,asusual,"(=a+j/1and 6~=Ps+jCl>~=tanh-1(Zs/Zc)and whereRoistheexternal orradiation resistance referredtothecurrent10• ltisdefinedby (2) wherePistheaverage radiated power. Thedistribution ofcurrentas determined bylinetheoryisobtained fromChap.IV,Sec.2,Eq.(5). ltis I=Icosh["(s-z)+8~] z 0cosh("(s+8~)(3) (5)Thereareseveralmethods ofevaluating Ro,butsincealldependon antenna theory,theyarenotdescribed here.Theresultisasfollows: Re=so/12b2cosh(as+2ps)(COShas_sin2/1S) (4) o411"0Icosh("(s+8~)12 2/1s Inthederivation of(4)itisassumed thattheterminal impedances Zo andZ,areeffectively lumped, sothatitmaybeassumed thatthecur rents10andIsareconstant inamplitude fromoneofthelinewiresto theother.Exceptforhigher-order termstheseterminations areequiva lenttofilaments ofcurrent10andI,oflengthbintheircontribution toRo' Thegeneralformula (4)issimplified whenappliedtoimportant special cases. Nonresonant Line.Anonresonant lineischaracterized byp,=00 However, thesameresultsobtaintoanexcellent approximation when Psexceeds3.Theappropriate formof(4)is "0 ( sin2/1S)Re==~/12b2coshas---e-as o211" 2/18 Usually asissufficiently smallsothatthefollowing formula isagood approximation oftheradiation resistance, referredtotheinputcurrent Sec.9] OSCILLATORS ANDCOUPLED SECTIONS OFLINE 489 (6)ofalow-loss lineterminated initscharacteristic impedance: Re==ro{32b2(1_sin2{3S) o211" 2{3s Thisformulaisfurthersimplified if2{3s=n1l"orif{3sissufficiently great. Inthesecases ohms (7) 60CIJ :Ec 40~ ~eco 20z (8)Agraphof(6)isgiveninFig.9.l. Ohms Ohms 40....---r----..,.--~-...,._--r--~-..,_-...,80 N.() ~30...e ~~ C1J=y- lijlij20tic "iii~ ~~ :s10.. IV'0 IV 0:::00"---'------&.--""-:---'--- ........~-'------'-:-- ......0 1T/2 1T31T/22rr 31T /38 FIG.9.1.Theoretical radiation resistance oflow-loss resonant andnonresonant lines. Low-loss LinewithLow-loss Terminations. Ifthelossesintheline anditsterminations aresufficiently smallsothattheinequality (as+ 2p,)<0.1issatisfied, thefollowing simplification of(4)applies: Re==~0{32b2sec2({3s+«p')(1_sin2{3S) o411" ' 2{3s Sincetheapproximations implicit inthisformula areequivalent to neglecting alllossesindetermining thedistribution ofcurrent, thisis purelysinusoidal. Accordingly theinputcurrent10vanishes when {3s+«P~=(2n+1)11"/2andRobecomes infinite.Ifadifferent reference currentischosen,inparticular, themaximum 1mofthesinusoidal dis tribution, aradiation resistance R~referredtoImmaybedefined. With (8),itis ohmsRe==2P==ro{32b2(1_sin2{3S) mI;" 411" 2{3s When2{3s=n1rorwhen{3sissufficiently long,(9)reducesto Re==~{32b2==30{32b2 m411"(9) (10) Agraphof(9)isgiveninFig.9.1.Itisseentodifferfrom(6)forthe 490 TRANSMISSION-LINE THEORY [Chap.VI nonresonant low-loss lineonlybyafactor2.Notethat(9)and(10) applytolineswithopenorshort-circuited endsortolineswithlumped reactive terminations ofanytype. Although theradiation resistance ofatwo-wire linewithcloselyspaced conductors isverysmall,itisnotnecessarily negligible incomparison withthecorresponding ohmicresistance. Ifthetwoconductors ofthe lineareofcopperandofsufficiently largediameter andthelineistermi natedinessentially reactiveimpedances, suchasmetaldisks,conducting bridges,oropenends,thepowerdissipated inheatmaybeassmallas orsmallerthanthepowerradiated. Owingtothesmallover-allattenu ation,standing-wave ratiosmaybeenormous. Thisistrue,inparticu lar,ofheavysingleconductors placedclosetoandparalleltoalarge highlyconducting plane.Inthiscaseohmiclossesintheconducting planeareusuallynegligible, andtheradiation resistance isone-halfthat givenin(9)or(10)ifbisthedistance between theaxisoftheconductor andtheaxisofitsimageortwicethedistance between theaxisofthe conductor andtheconducting plane.Ifthesingleconductor issuf ficiently largeindiameter andthedistance bisnottoosmall,the powerradiated maybemadetoexceedgreatlythepowerdissipated in heat.Inthiscasethesectionofresonant lineisareasonably efficient antenna. Ifthedistance.b between theaxesofthetwoconductors ofatwo-wire line(orasingleconductor anditsimageinaconducting plane)isnot largecompared withtheradiusaoftheconductors, theeffective centers ofthecurrents inthetwoconductors aremovedclosertogether thanb. Inthiscasetheeffective separation (11) maybeusedinplaceofthedistancebbetween axes. Anexperimental determination oftheradiation resistance ofresonant sections oftwo-wire linehasbeencarriedoutbyChipman141andhiscol laborators. Theprocedure wastodetermine thewidthsoftheresonance curvesatthehalf-power pointsofasectionoftwo-wire lineprojecting vertically aboveahighlyconducting metalplanewhenenclosed ina metalshieldandwiththeshieldremoved. Thelineconsisted ofsilver conductors ofdiameter 2a=0.100in.separated adistance b=1.00in. between centers. Itwasadjusted toresonance withopenandshort circuited endsforarangeoflengthsextending fromA/4to4Aatfre quencies of342to1,420Me/sec. Fortheshort-circuited lineabrass shield4in.ininsidediameter wasused;fortheopenlinetheshieldwas ofaluminum 4.5in.indiameter. ThewidthWoftheresonance curvebetween half-power pointsis Sec.9] OSCILLATORS ANDCOUPLED SECTIONS OFLINE 491 relatedtotheover-allattenuation bytheformula fjW2=a8+Po+P, (12) (13)where 8isthelengthofthelineandpoandP8aretheattenuation func tionsoftheterminations atthetwoendsz=0andz=8.Thisformula doesnotincludeattenuation duetoradiation, butitmaybegeneralized toincluderadiation byaddingafunction pr'Thus fjW2=a8+po+P8+pr Sincethecharacteristic impedance and,withit,theattenuation constant ofthelinechangewhentheshieldisremoved quiteirrespective ofradi ation,itisadvantageous tomultiply (12)and(13)throughbythechar acteristic impedance appropriate, respectively, fortheshielded and unshielded line. L~tthesubscript 1beusedfortheformerandthe subscript 2forthelatter.Thetwoequations are fjRelW 1()--2-=rl8+RetPo+P8 fjRe2W2 ( )--2-=r28+Re2Po+P8+Pr(14) (15) whererlandr2aretheresistances perloopunitlengthoftheshielded pairandtheopentwo-wire linesandReIandRe2arethecharacteristic impedances. SinceZoisalow-impedance essentially reactive termi nation, po==rIO=Ro/ Re.ItfollowsthatRc1po==Re2Po==Ro.Simi larlyZ8iseitherashortcircuitoranopencircuitforwhich P8==O. Itfollowsthat,if(15)issubtracted from(14),thefollowing resultis obtained (sinceprrepresents asmallpureresistance, itislogicaltoset Pr==Re/Re2,sothatRelPr==R~==R:r,,): (16) Thedifference inohmicresistance perunitlengthoftheshielded and unshielded linesarisesfromthesmalleddycurrents intheshield,when thisispresent, andasmallchangeintransverse distribution. Actually, withaslargeashieldasusedintheexperimental measurements, these char.gesareinsignificant, anditisagoodapproximation tosetrl==r2. ItfOllowsthat R:n==jfj(Re2W2-RetW 1) (17) Bymeasuring thehalf-power widthsWIandW2,Chipman etal.deter minedR:n.Foreachof13different frequencies, measurements were madeusingeitheropen-end orclosed-end lines.Ateachfrequency a 492 TRANSMISSION-LINE THEORY [Chap.VI Theoretical curve - Re=15p2b2 +Exp.range} closedendChipman J.Exp.range et.a!. fopenend6 410r----,r- .......-..,---r---r--,.....,........,...,., 8 2.5 0.4 0.150.25 0.2E2.0 ~1.5 <II ~ .~1.0t------+......:--------I ~ c0.8o ~0.6 IVa::numberofdifferent resonant lengthswereused.Themeanvalueofthe measured R:nandtherangesfortheseverallengthsoflineareshownin Fig.9.2.Thetheoretical curveR:n=15~2b2isalsoshown. (Notethat thefactor15occursinplaceof30 whenthelineisoveraninfiniteimage planeandradiates intoahalfspace.) Although afewoftheexperimental pointslieonthetheoretical curve,most ofthemareconsiderably belowit.In general,theyliemorecloselyalonga linedefinedbyR:n=13.75~2b2. Itis difficulttoachievehighexperimental accuracy inradiation problems, and thetheoretical formulaisalsoapproxi mateinthesensethatitdepends ona distribution ofcurrentwhichisvalid strictlyonlywhenthereisnoradiation. Therefore an8percentdifference in theconstant fj,ctormultiplying ~2b2 andtheexcellent agreement withwhich Reobeysasquarelawin~barequite acceptable. Radiation fromatwo-wire lineis greatlyincreased ifthelineiseven slightlyunbalanced. Inthiscasethe partofthecurrentinthetwoconduc torswhichisgivenby11+12isin thesamedirection inthetwowires, sothattheseactlikeanantennacarry ingthiscurrent. Radiation fromacoaxiallinewith smallcrosssectionmaybeignoredsolongasnocurrents existontheout sideoftheshield. However, ifacoaxiallineisusedwithabalanced load, e.g.,acenter-driven symmetrical antenna, currents maybeexcitedon theoutsideoftheshield,andthisthenbehaves likeanantenna.0.1'-~"--""--'--J.._'---'--""""'-J..""" 0.10.150.20.250.30.4 0.50.6 0.8 1.0 fib=2TTb/1\. FIG.9.2.Radiation resistance of resonant two-wire lineoveranimage plane.Theoretical curveinsolid line;experimental pointsbyChip manetal. PROBLEMS 1.Atriodeofthedouble-ended typehasthefollowing interelectrode capacitances: gridplate,2#L#Lf;gridfilament, 2#L#Lf;platefilament, 0.4#L#Lf.Determine andplotthe generated frequency asafunction ofthelengthoftwoidentical effectively short circuited sections oftwo-wire lineattached tothegrid-plate leads,oneoneachside ofthetube. 2.Whatarethewavelength characteristics ofthetriodeoscillator inthepreceding problem ifonlyonesectionofthetwo-wire lineisused? Chap.VI]OSCILLATORS ANDCOUPLED SECTIONS OFLINE 493 J.Aringoscillator consists ofaclosedsquareoftwo-wire line.Adouble-ended tubeisconnected atthecenterofeachsideofthesquare.Iftheequivalent capaci tanceofeachtubeisCo,whatarethewavelength characteristics oftheoscillator asa function ofthelengthofonesideofthesquare? (Neglect theeffectofthecorners.) 4.Determine theeffectofthecornersintheringoscillator inProb.4inmodifying thenaturalfrequency generated. 5.Thecoupled-circuit oscillator described inSec.2ismodified bycuttingthe coupling bridgeinthemiddleandinserting asmalleffectively lumpedcapacitance in serieswithit.Determine theeffectofsuchacapacitance onthefrequency orwave lengthgenerated asafunction ofthemagnitude ofthecapacitance. 6.Atransmission lineconsists ofasinglesilvered brasstubeiin.indiameter placedparallelto,andwithitscenteratadistance ofiin.from,averylargehighly conducting plane(assumed perfectly conducting andinfiniteinextent). Theline isdrivenatoneendbyagenerator atafrequency of750Mc/sec; theotherendis terminated inalargemetaldisk(whichmaybeassumed equivalent toaperfectshort circuit). (a)Compare thepowerdissipated asheatinthelinewiththepowerradiated as thelengthofthelineisincreased fromone-quarter totwowavelengths. (b)Whatarethesuccessive maximum andminimum valuesoftheinputresistance ofthelineinthisrangeoflength? BIBLIOGRAPHY BOOKS 1.Bewley, L.V.:"Two-dimensional FieldsinElectrical Engineering," The Macmillan Company, NewYork,1948. 2.Everitt, W.L.:"Communication Engineering," 2ded.,McGraw-Hill Book Company, Inc.,NewYork,1937. 3.Guillemin, E.A.:"Communication Networks," vol.II,chaps.I-III,John Wiley&Sons,Inc.,NewYork,1935. 4.Jackson, Willis:"HighFrequency Transmission Lines,"3ded.,Methuen &Co.,Ltd.,London, andJohnWiley&Sons,Inc.,NewYork,1951. 5.Johnson, W.C.:"Transmission LinesandNetworks," McGraw-Hill Book Company, Inc.,NewYork,1950. 6.Karakash, JohnJ.:"Transmission LinesandFilterNetworks," The Macmillan Company, NewYork,1950. 7.Kennelly, A.E.:"ChartAtlasofComplex Hyperbolic andCircular Func tions,"3ded.,Harvard University Press,Cambridge, Mass.,1924;"Tables ofComplex Hyperbolic andCircular Functions," 2ded.,Harvard Uni versityPress,Cambridge, Mass.,1921. 8.King,D.D.:"Measurements atCentimeter Wavelength," D.VanNostrand Company, Inc.,NewYork,1952. 9.Kmg,R.W.P.:"Electromagnetic Engineering," vol.I,McGraw-Hill BookCompany, Inc.,NewYork,1945. 10.King,R.W.P.:"Theory ofLinearAntennas," Harvard University Press, Cambridge, Mass.(inpress). 11.King,R.W.P.,H.R.Mimno, andA.H.Wing:"Transmission Lines, Antennas, andWaveGuides," McGraw-Hill BookCompany, Inc.,New York,1945. 12.Kupfmuller, K.:"Theoretische Elektrotechnik," Springer-Verlag OHG, Berlin,1932. 13.Marcuvitz, N.:"Waveguide Handbook," McGraw-Hill BookCompany, Inc.,NewYork,1951. 14.Montgomery, C.G.:"Technique ofMicrowave Measurements," McGraw HillBookCompany, Inc.,NewYork,1947. 15.Montgomery, C.G.,R.H.Dicke,andE.M.Purcell:"Principles ofMicro waveCircuits," McGraw-Hill BookCompany, Inc.,NewYork,1948. 16.Moreno, T.:"Microwave Transmission DesignData,"McGraw-Hill Book Company, Inc.,NewYork,1948. 17.Pierce,G.W.,"Electric Oscillations andElectric Waves," McGraw-Hill BookCompany, Inc.,NewYork,1920. 18.Pipes,L.A.:"Applied Mathematics forEngineers andPhysicists," Mc Graw-Hill BookCompany, Inc.,NewYork,1946. 19.RadioResearch Laboratory: "VeryHigh-frequency Techniques," vols. I-II,McGraw-Hill BookCompany, Inc.,NewYork,1947. 20.Ragan,G.L.:"Microwave Transmission Circuits," McGraw-Hill Book Company, Inc.,NewYork,1948. 494 BIBLIOGRAPHY 495 21.Ramo,S.,andJ.R.Whinnery: "FieldsandWavesinModern Radio," JohnWiley&Sons,Inc.,NewYork,1944. 22."Reference DataforRadioEngineers," 3ded.,Federal Telephone and RadioCorporation, NewYork,1949. 23.Ryder,JohnD.:"Networks, LinesandFilters," Prentice-Hall Inc.,New York,1949. 24.Skilling, H.H.:"Electric Transmission Lines,"McGraw-Hill BookCom pany,Inc.,NewYork,1951. 25.Slater,J.C.:"Microwave Transmission," McGraw-Hill BookCompany, Inc.,NewYork,1942. 26.Sokolnikoff, I.S.andE.S.:"Higher Mathematics forEngineers and Physicists," McGraw-Hill BookCompany, Inc.,NewYork,1934. 27.Sommerfeld, A.:"Lectures onTheoretical Physics," vol.III,"Electro dynamics," Academic Press,Inc.,NewYork,1952.("Vorlesungen iiber Theoretische Physik. III.Elektrodynamik," especially pp.198-211, Dieterich'sche Verlagsbuchhandlung, Wiesbaden, 1948.) 28.Stratton, J.:"Electromagnetic Theory," McGraw-Hill BookCompany, Inc.,NewYork,1941. 29.Terman, F.E.:"RadioEngineers' Handbook," McGraw-Hill BookCom pany,Inc.,NewYork,1943. 30.Thomson, J.J.:"Recent Researches inElectricity andMagnetism," Clarendon Press,Oxford,1893. 31.Ware,L.A.,andH.R.Reed:"Communication Circuits," 3ded.,John Wiley&Sons,Inc.,NewYork,1949. 32.Weber,E.:"Electromagnetic Fields," vol.I,JohnWiley&Sons,Inc., NewYork,1950. 33.Weissfloch, A.:"Schaltungstheorie undMesstechnik desDezimeter- und Zentimeter-Wellengebietes," VerlagBirkhauser, Basel,1954. PAPERS References Primarily forChaps.IandII 34.Angelakos, D.J.:ACoaxialLineFilledwithTwoNon-concentric Dielec trics,Technical ReportNo.102,SeriesNo.60,University ofCalifornia, Division ofElectrical Engineering, Berkeley, Calif.,November, 1953. 35.Arnold,A.H.M.:Proximity EffectinSolidandHollowRoundConductors, J.Inst.Elec.Engrs.London,pt.II,88:349(1941). 36.Assadourian, F.,andF.Rimai:Simplified TheoryofMicrostrip Trans missionSystems, Proc.IRE,40:1651(1952). 37.Brick,D.:Radiation ofaHertzian DipoleoveraCoatedConductor, Proc. Inst.Elec.Engrs.London, pt.C(1955);Monograph 113,RadioSection(1954). 38.Brown,G.H.:Characteristics ofUnbalanced Overhead Transmission Lines, Broadcast Rev.,May,1941. 39.Carson,J.R.:ThePresentStatusofWire-transmission TheoryandSome ofItsOutstanding Problems, BellSystemTech.J.,7:268(1928). 40.Carson,J.R.:WavePropagation overParallel Wires.TheProximity Effect,Phil.Mag.,ser.6,41:607(1921). 41.Carson,J.R.,andR.S.Hoyt:Propagation ofPeriodic Currents overa SystemofParallelWires,BellSystemTech.J.,6:495(1927). 42.Colebrook, F.M.:Transmission-line Theory,WirelessEng.,21:167(1944). 43.Frankel, S.:Characteristic Impedance ofParallel WiresinRectangular Troughs, Proc.IRE,30:(1942). 496 TRANSMISSION-LINE THEORY 44.Fubini,E.,W.Fromm, andH.Keen:NewTechniques forHigh-QStrip Microwave Components, Convention RecordoftheIRE,1954National Con vention,pt.8:91(1954). 45.Goubau, G.:SurfaceWavesandTheirApplication toTransmission Lines, .T.Appl.Phys.,21:1119(1950). 46.Hikosaburo, A.:EllipseDiagram ofaLecher-wire System,Proc.IRE,21: 303(1933). 47.Jackson, W.,andL.G.H.Huxley: TheSolution ofTransmission-line Problems bytheUseoftheCircleDiagram ofImpedance, .T.Inst.Elec. Engrs.London,pt.III,91:105(1944). 48.King,R.:TheTelegraphist's Equations atUltra-high Frequencies, Physics, 6:121(1935). 49.King,R.,andK.Tomiyasu: Terminal Impedance andGeneralized Two-wire LineTheory,Proc.IRE,37:1134(1949). PartIinTechnical ReportNo. 74,CruftLaboratory, Harvard University, 1949. 50.Laport,E.A.:Open-wire Radio-frequency Transmission Lines,Proc.IRE, 31:271(1943). 51.Levin,S.A.:Electromagnetic WavesGuidedbyParallel Wires,Trans. AIEE,46:983(1927). 52.Mie,G.:Elektrische Wellenanzweiparallelen Drahten, Ann.Physik,2: 201(1900). 53.Namiki, M.,andH.Takahashi: SomeVariational Principles forProblems inTransmission Lines,.T.Appl.Phys.,23:1056(1952). 54.Pipes,L.A.:MatrixTheoryofMulticonductor Transmission Lines,Phil. Mag.,ser.7,24:97(1937). 55.Pipes,L.A.:Steady-state Analysis ofMulticonductor Transmission Lines, .T.Appl.Phys.,12:782(1941). 56.Rice,S.0.:Steady-state Solutions ofTransmission-line Equations, Bell SystemTech..T.,20:131(1941). 57.Sim,A.C.:NewHigh-frequency Proximity EffectFormula, WirelessEngr., 30:203(1953). 58.Smith,H.P.:AnImproved Transmission-line Calculator, Electronics, 17:130(1944). 59.Tai,C.T.:High-frequency Polyphase Transmission Lines,Proc.IRE,36: 1370(1948). 60.Walker, L.R.,andN.Wax:Non-uniform Transmission LinesandReflec tionCoefficients, .T.Appl.Phys.,17:1043(1946). 61.Zinke,0.:Grundlagen derStrom-u.Spannungsverteilung aufAntennen, ArchivElektrotech., 35:67(1941). References Primarily forChaps.IIIandIV 62.Andrews, H.W.:Image-plane andCoaxial-line Measuring Equipment at 600Mc/s,Technical ReportNo.177,CruftLaboratory, Harvard Uni versity,1953. 63.Bloch,A.,F.J.Fisher,andG.J.Hunt:NewEquipment forImpedance Matching andMeasurement atVeryHighFrequencies, Proc.Inst.Elec. Engs.,London, pt.III,100:93(1953). 64.Bruckmann, H.:Widerstandsmessungen mitderParalleldrahtleitung, Hochfrequenztech., 51:128(1938). 65.Cafferata, H.:TheCalculation ofInputorSending-end Impedances of Feeders andCablesTerminated inComplex Loads,Marconi Rev.,6:12 (1937). BIBLIOGRAPHY 497 66.Carter,P.S.:ChartsforTransmission-line Measurements andComputa tions,RCARev.,3:355(1938-1939). 67.Chipman, R.A.:AResonance-curve MethodfortheAbsolute Measurement ofImpedance atFrequencies oftheOrder300Mc/s,J.Appl.Phys.,10: 27(1939). 68.Clayton, R.J.,J.E.Houldin, H.R.L.Lamont, andW.E.Willshaw: Radio Measurements intheDecimetre andCentimetre Wavebands, J.Inst. Elec.Engrs.London,pt.III,93:97(1946). 69.Cox,C.R.:DesignDataforBeadedCoaxialLines,Electronics, 19:130(1946). 70.Diamond, J.M.:ShortedStubsofHighResonant Impedance, Proc.IRE, 40:188(1952). 71.Duffin,W.J.:Three-probe Method ofImpedance Measurement, Wireless Engr.,29:317(1952). 72.Essen,L.:TheMeasurement ofBalanced andUnbalanced Impedances at Frequencies near500Mc/sandItsApplication totheDetermination ofthe Propagation Constant o,fCables,J.Inst.Elec.Engrs.London, pt.III,91: 84(1944). 73.Essen,L.,andA.C.Gordon-Smith: TheMeasurement ofFrequencies inthe Range100Mc/sto10,000Mc/s,J.Inst.Elec.Engrs.London, pt.III, 92:291(1945). 74.Frankel, S.:Characteristic Functions ofTransmission Lines,Communica tions,March,1942. 75.Fubini,E.G.,andP.S.Sutro:AWide-band Transformer fromanUn balanced toaBalanced Line,Proc.IRE,35:1153(1947). 76.Hoag,J.B.:Measurement oftheFrequency ofUltra-radio Waves,Proc. IRE,21:29(1933). 77.Kaufman, H.:Scheinwiderstandsmessungen imDezimeterwellengebiet, Hochfrequenztech., 53:61(1939). 78.King,D.D.:Impedance Measurement onTransmission Lines,Proc.IRE, 35:509(1947). 79.King,R.:Electrical Measurements atUltra-high Frequencies, Proc.IRE, 23:885(1935). 80.King,R.:General Amplitude Relations forTransmission LineswithUn restricted LineParameters, Terminal Impedances, .andDriving Point, Proc.IRE,29:640(1941). 81.King,R.:Transmission-line TheoryandItsApplication, J.Appl.Phys., 14:577-600 (1943). 82.Klopfenstein, R.W.:DesignofTransmission-line TuningElements for Minimum Dissipation, Proc.IRE,39:1089(1951). 83.Kostrize, J.A.:Microstrip Components, Proc.IRE,40:1658(1952). 84.Labus,J.W.:Measurement ofResistances andImpedances atHighFre quencies, Proc.IRE,19:452(1931). 85.Macalpine, W.W.:Computation ofImpedance andEfficiency ofLine withHighStanding-wave Ratio,Elec.Commun., 30:238(1953). 86.Matthews, E.W.:AShielded Two-wire HybridJunction andItsUseasan Ultra-high Frequency Impedance Bridge,Ph.D.Thesis,Harvard Univer sity,Division ofAppliedScience, 1954. 87.Medhurst, R.G.,andS.D.Pool:Correction FactorsforSlottedMeasuring LinesatVeryHighFrequencies, Proc.Inst.Elec.Engrs.London, pt.III, 97:223(1950). 88.Meinke, H.H.:Fortschritte derDezimeterwellen-Messtechnik, Frequenz, 2: 41(1948). 498 TRANSMISSION-LINE THEORY 89.Morita,T.:Measurement ofCurrent andChargeDistributions onCylin dricalAntenna, Proc.IRE,38:898(1950). Technical ReportNo.66, CruftLaboratory, Harvard University, 1949. 90.Morita,T.,andL.Sheingold: ACoaxialMagic-T, Technical ReportNo. 162,CruftLaboratory, Harvard University, 1952. 91.Nergaard, L.S.:ASurveyofUltra-high Frequency Measurements, RCA Rev.,3:156(1938-1939). 92.Nergaard, L.S.,andB.Salzberg: Resonant Impedance ofTransmission Lines,Proc.IRE,27:579(1939). 93.Paine,R.C.:Graphical Solution ofVoltageandCurrentDistribution and Impedance ofTransmission Lines,Proc.IRE,32:686(1944). 94.Roder,Hans:Graphical Methods forProblem Involving Radio-frequency Transmission Lines,Proc.IRE,21:290(1933). 95.Roosenstein, H.0.:TheConduction ofHigh-frequency Oscillatory Energy, Proc.IRE,19:1849(1931);Hochfrequenztech., 36:81,120(1930). 96.Russell, A.:Effective Resistance andInductance ofConcentric Mains, Phil.Mag.,ser.6,17:524(1909). 97.Salinger, H.:TheQuarter-wave Step-up Transformer, Proc.IRE,32: 553(1944). 98.Schmidt, 0.:DasParalleldrahtsystem alsMessinstrument, Hochfrequenz. u.Elektroakustik, 41:1(1933). 99.Smith,P.H.:Optimum CoaxDiameters, Electronics, 23:111(1950). 100.Sterba,E.J.,andC.B.Feldman: Transmission LinesforShort-wave Radio, Proc.IRE,20:1163(1932). 101.Tai,C.T.:ShuntandSeriesSections ofTransmission LineforImpedance Matching, J.Appl.Phys.,17:44(1946). 102.Terman, F.E.:Resonant LinesinRadioCircuits, Elec.Eng.,53:1046 (1934). 103.Tomiyasu, K.:Antennas andOpen-wire Lines.PartII.Measurements on Two-wire Lines,J.Appl.Phys.,20:892(1949). 104.Tomiyasu, K.:Terminal Impedance andGeneralized Two-wire Line Theory. PartII.EffectofaBend,Technical ReportNo.74,Cruft Laboratory, Harvard University, 1945. 105.Tomiyasu, K.:Unbalanced Terminations onaShielded-pair Line,Techni calReportNo.86,CruftLaboratory, Harvard University, 1949. 106.Tomiyasu, K.:TheUnbalance Squelcher, Rev.Sci.Instr.,19:675(1948). 107.Wheeler, H.A.:Transmission-line Impedance Curves,Proc.IRE,38:1400 (1950). 108.Wholey, W.B.,andW.H.Eldred:ANewTypeofSlotted-line Section, Proc.IRE,38:244(1950). 109.Wing,A.H.,andJ.Eisenstein: SingleandDouble-stub Impedance Match- ing,J.Appl.Phys.,16:615(1944). 110.Witt,B.J.:Concentric TubeLines,MarconiRev.,6:20(1936). References Primarily forChap.V 111.Angelakos, D.J.:Current andChargeDistributions onAntennas and Open-wire Lines,J.Appl.Phys.,22:910(1951). 112.Arditi,Maurice: Experimental Determination oftheProperties ofMicro stripComponents, Elec.Commun., 30:283(1953). 113.Brennecke, C.G.:Equivalent TandPiSections fortheQuarter-wave lengthLine,Proc.IRE,32:15(1944). 114.Comes,R.W.:ACoaxialLineSupportfor0to4000Mc,Proc.IRE,37:94 (1949). BIBLIOGRAPHY 499 115.Crosby, D.R.,andC.H.Pennypacker: Radio-frequency Resistors as Uniform Transmission Lines,Proc.IRE,34:62(1946). 116.Deschamps, G.A.:Application ofNon-Euclidean Geometry totheAnalysis ofWaveguide Junctions, ReportURSI-IRE SpringMeeting, 1952. 117.Deschamps, G.A.:NewChartfortheSolution ofTransmission-line and Polarization Problems, Trans.IRE,Professional GrouponMicrowave TheoryandTechniques, 1:5(1953). 118.Felsen,L.B.,andA.A.Oliner:Determination ofEquivalent CircuitPara metersforDissipative Microwave Structures, Proc.IRE,42:477(1954). 119.Hartig,E.0.:AStudyofCoaxialLineDiscontinuities UsingaVariational Method, Technical ReportNo.108,CruftLaboratory, Harvard University, 1950. 120.Hartig,E.0.:Circular Apertures andTheirEffectsonHalf-dipole Imped ances,Technical ReportNo.107,CruftLaboratory, Harvard University, 1950. 121.Hollway, D.L.:AnInstrument forDielectric Measurements intheFre quencyRange100-300 Mc/s,Proc.Inst.Elec.Engrs.London,pt.III,99: 364(1952). 122.Kaden,H.,andG.Ellenberger: Reflexionsfreie Sttitzscheiben inKoaxialen Leitungen, Arch.elektr.Ubertrag., 3:313(1949). 123.King,R.:AnAbsolute MethodforMeasuring Dielectric Constants ofFluids andSolidsatUltra-high Frequencies, Rev.Sci.Instr.,8:201(1937). 124.King,R.:Antennas andOpen-wire Lines.Part1.TheoryandSummary ofMeasurements, J.Appl.Phys.,20:832(1949). 125.King,R.:Capacitance atUltra-high Frequencies, Phil.Mag.,ser.7,25: 339(1938). 126.King,R.:EineZuzammenfassende Untersuchung tiberstehende elektrische Drahtwellen, Ann.Physik,ser.5,7:805(1930). 127.King,R.:End-correction forCoaxial LineDriving anAntenna overa Ground Screen,Technical ReportNo.174,CruftLaboratory, Harvard University, 1953(Trans.IRE,AP-3,April,1955). 128.Kohn,C.T.:TheDesignofaRadioFrequency CoaxialResistor, Proc. Inst.Elec.Engrs.London,pt.IV,Monograph 83(1953). 129.Lamont, H.R.L.:TheoryofResonance inMicrowave Transmission Lines withDiscontinuous Dielectric, Phil.Mag.,ser.7,29:521(1940). 130.Lamont, H.R.L.:TheUseoftheWaveGuideforMeasurement ofMicro waveDielectric Constants, Phil.Mag.,ser.7,30:1(1940). 131.Matsumoto, K.:OntheTurning PointandBranching PointofLecher Wires,J.Inst.Elec.Commun. Engrs.Japan,26:203(1951). 132.Miles,J.W.:PlaneDiscontinuities inCoaxialLines,Proc.IRE,35:1498 (1947). 133.Oliver,M.H.:Discontinuities inConcentric-line Impedance-measuring Apparatus, Proc.Inst.Elec.Engrs.London,pt.III,97:29(1950). 134.Powles,J.G.,andW.Jackson: Measurement oftheDielectric Properties ofHigh-permittivity Materials atCentimetre Wavelengths, Proc.Inst. Elec.Engrs.London,pt.III,96:383(1949). 135.Storer,J.E.,L.S.Sheingold, andS.Stein:ASimpleGraphical Analysis of Waveguide Junctions, Proc.IRE,41:1004(1953). 136.Weissfloch, A.:EinTransformationsglied fUrDezimeter- undZentimeter wellenmitgeringer Frequenz Abhangigkeit, Elek.Nachr.Tech.,20:189 (1943). 137.Weissfloch, A.:EinTransformationssatz tiberVerlustlose Vierpole und seineAnwendung aufdieexperimentelle Untersuchung vonDezimeter- und Zentimeterwellen-Schaltungen, Hochfrequenztech., 60:67(1942). 500 TRANSMISSION-LINE THEORY 138.Weissfloch, A.:Kreisgeometrische Vierpoltheorie undIhreBedeutung fUr Messtechnik undSchaltungstheorie desDezimeter- undZentimeterwellen gebietes, Hochfrequenztech., 61:100(1943). 139.Whinnery, J.R.,andH.W.Jamieson: Equivalent Circuits forDiscon tinuities inTransmission Lines,Proc.IRE,32:98(1944). 140.Whinnery, J.R.,H.W.Jamieson, andT.E.Robbins: Coaxial-line Dis continuities, Proc..IRE,32:695(1944). References Primarily forChap.VI 141.Chipman, R.A.,E.F.Clark,N.A.Hoy,andM.Yurko:Radiation Resist anceofResonant Transmission Lines,J.Appl.Phys.,23:613(1952). (Thispapercontains anextensive listandadiscussion ofearlierworkon radiation fromlines.) 142.Firestone, W.L.:Analysis ofTransmission-line Directional Couplers, Proc. IRE,42:1529(1954). 143.Gavin,M.R.:TriodeOscillators forUltra-short Wavelengths, Wireless Engr.,16:287(1939). 144.Gureivitsch, A.M.,andJ.R.Whinnery: Microwave Oscillators Using Disk-seal Tubes,Proc.IRE,36:462(1947). 145.Hansen, W.W.:OntheResonant Frequency ofClosedConcentric Lines, J.Appl.Phys.,10:38(1939). 146.King,R.:AContinuously Variable Oscillator forParallelLineMeasure mentsat100to2000Megacycles perSecond,Rev.Sci.Instr.,11:270(1940). 147.King,R.:AGeneralReciprocity Theorem forTransmission LinesatUltra highFrequencies, Proc.IRE,28:223(1940). 148.King,R.:AGeneralized Coupling Theorem forUltra-high Frequency Cir cuits,Proc.IRE,28:84(1940). 149.King,R.:AVariable Oscillator forUltra-high Frequency Measurements, Rev.Sci.Instr.,10:325(1939). 150.King,R.:Amplitude Characteristics ofCoupled Circuits Having Dis tributed Constants, Proc.IRE,21:1142(1933). 151.King,R.:Application ofLow-frequency CircuitAnalysis totheProblem ofDistributed Coupling inUltra-high Frequency Circuits, Proc.IRE,27: 715(1939). 152.King,R.:BeamTubesasUltra-high Frequency Generators, J.Appl. Phys.,10:638(1939). 153.King,R.:Coupled Antennas andTransmission Lines,Proc.IRE,31:626 (1943). 154.King,R.:Parasitic Electronic Oscillations andCoupling Frequencies ina PowerTube,J.Appl.Phys.,11:615(1940). 155.King,R.:Wavelength Characteristics ofCoupled Circuits HavingDis tributed Constants, Proc.IRE,20:1368(1932). 156.Oliver,B.M.:Directional Electromagnetic Couplers, Proc.IRE,42:1686 (1954). 157.Storer,J.E.,andR.King:Radiation Resistance ofTwo-wire Line,Proc. IRE,39:1408(1951). 158.Takagishi, E.:OnaDouble-hump Phenomenon ofCurrentthroughaBridge acrossParallelLines,Proc.IRE.18:513(1930). 159.Tomiyasu, K.:Loading andCoupling EffectsofStanding-wave Detectors, Proc.IRE,37:1405(1949). 160.King,R.W.P.,C.W.Harrison, Jr.,andD.H.Denton, Jr.:Trans mission-Line MissileAntennas, Trans.IRE,AP-8:88(1960). Symbol A As,Aw a 01(W) B b C Cw Cx,Cz CT C c(w) E Fs,Fw f(h) g g(h) g(w) he hm Iz IzL(w) IZT I~(y) I~(y) i,i ko ko(w),k1(w) L Ls Lsa LT Le l h,2 Ie le(w) It nINDEX OFSYMBOLS Definition vectorpotential over-allattenuation function radiusofconductors oftwo-wire line ratiofunction magnetic vector distance between conductors oftwo-wire line amplitude function amplitude function amplitude functions lumpedshuntcapacitance capacitance perunitlength capacitance perunitlength electricvector over-all phasefunctions realpartofcomplex radical conductance perunitlength imaginary partofcomplex radical conductance perunitlength losstangentofdielectric material losstangentofmagnetic material currentintransmission line currentinline currentinload antisymmetrical orequalandopposite currents symmetrical orcodirectional currents volumedensityofcurrent-real, complex constant parameters powerloss inductance oftermination apparent terminal inductance inductance duetoendeffect external inductance inductance perunitlength inductance perunitlengthofconductor 1,2 external inductance perunitlength external inductance perunitlength internal inductance perunitlength indexofrefraction 501Page 8 87 13 63 8 13 86 87 250 401 7 63 8 87 92 7 92 63 8 9 5 60 60 213 213 11 17 16 244 121 122 121 121 6 5 17 63 18 334 502 Symbol PL Ps Pz Po P p P(w),P'(w),Po(w) Q q qL(W) qT(U) R Ra Rb RS1 R1T,R2T Ro R~ r rl,2 ri rL2 S S S,SWR f:h Sv So,Ss,Sw Sx,Sy S11, S12, S21, S22 81,2 U V(w) VL(w) Vs VT(w) V~ Vz Va VS Vg Vp Vz VO,V,V VoTRANSMISSION-LINE THEORY Definition powerinline powertoload powertolinebeyondpointz powertoline parameter inmethodofsymmetrical components ratiofactor qualityfactor chargeperunitlength chargeperunitlengthonline chargeperunitlengthonload distance between twopoints distance toconductor distance toconductor distance between pointsincoaxialline distances frompointonlinetopointsontermina- tion radiation (external) resistance oftwo-wire line radiation resistance resistance perunitlengthofline resistance perunitlengthofconductor 1,2 internal resistance perunitlengthofline internal resistance perunitlengthofconductor 1,2 amplitude function scattering matrix standing-wave ratio currentstanding-wave ratio voltagestanding-wave ratio amplitude functions amplitude functions elements ofscattering matrix distance between pointsincoaxialline unitmatrix scalarpotential difference scalarpotential difference duetochargesonline modified amplitude scalarpotential difference duetochargesonload generator voltageatxalongline potential difference acrosstransmission line antisymmetrical orequalandopposite voltages symmetrical orcodirectional voltages groupvelocity phasevelocity instantaneous realvoltageatdistance z velocity ofelectromagnetic wavesinfreespace, dielectric medium, lossymedium instantaneous potential differencePage 252 243 251 243 41 266 63 270 7 60 60 11 15 15 21 61 488 489 6 5 18 18 86 305 260 261 261 87 250 305 21 305 15 60 86 60 245 5 213 213 54 52 87 10 50 Symbol W W Wz(w) WzL(W) WzT(w) W~ W W xi xi,2 Ycya ys y y(W),Yo(W) YT(W) Yl Zc Zt Zo,Zs Zoa,Zsa Zll,Z22 Z12,Z21za Zs Z z(w) Zi zi,2 ZI ,"€,€ €r €o €w €:z; €zINDEX OFSYMBOLS Definition powerratio widthofresonance curve vectorpotential difference vectorpotential difference duetocurrents inline vectorpotential duetocurrents inload voltageofequalandopposite doublet distance measured fromloadendofline distance measured fromloadendofline internal reactance perunitlengthoflin~ internal reactance perunitlengthofconductor 1,2 characteristic admittance antisymmetrical inputadmittance symmetrical inputadmittance admittance perunitlengthofline admittance perunitlength admittance perunitlength normalized admittance characteristic impedance transfer impedance terminating impedances apparent terminating impedances self-impedances mutualimpedances antisymmetrical inputimpedance symmetrical inputimpedance impedance perunitlengthofline impedance perunitlength internal impedance perunitlengthofline internalimpedance perunitlp.ugthofconductor 1,2 normalized impedance attenuation constant complex wavenumber wavenumber infreespace,dielectric medium, lossymedium voltagereflection coefficients ofterminations apparent terminal reflection coefficient ofZsa complex propagation constant ofline complex propagation constant angleofamplitude function absolute permittivity realandimaginary partsofcomplex permittivity relativepermittivity permittivity offreespace angleofamplitude function angleofamplitude function angleofamplitude function503 Page 252 491 15 60 60 247 13 80 1818 49 214 215 7 63 63 103 49 216 74 74 216 216 214 215 7 66 18 18 103 52 9 10 76 280 7 68 86 8 8 9 9 87 250 250 504 Yl 8 8,8' 80,8s /-",/-''' /-'0 v Vr Vo; p po,ps pOa,psa PI,2 a ao,as,aw ax,"a,a <P,<1>' (Po,<Ps <Poa,<Psa WI(W) C/> C/>c tl'saTRANSMISSION -LINETHEORY Definii'ion Page characteristic resistance offreespace,dielectric medium 10 surfacedensityofcharge-real, complex 11 complex constant (terminal function) 49 complex terminal functions 84 complex terminal functions ofZo,Zs 84 realandimaginary partsofcomplex permeability 9 permeability offreespace 9 reluctivity (reciprocal permeability) 9 relativereluctivity 9 reluctivity offreespace 9 complex dielectric factor 8 terminal attenuation function 84 terminal attenuation functions ofZo,Zs 86 apparent terminal attenuation functions forZoa, Zsa 276 constant parameter indefinition ofcirclediagram 26 angleofamplitude function 86 anglesofamplitude functions 37 angleofamplitude function 250 realandimaginary partsofcomplex conductivity 9 terminal phasefunctions 84 terminal phasefunctions ofZo,Zs 86 apparent terminal phasefunctions forZOa,Zsa 273 ratiofunction 63 scalarpotential 8 distortion factor 93 angleofcomplex apparent reflection coefficient ~~ ~ anglesofcomplex reflection coefficients 76 INDEX ABeDconstants, 84,194,289,290 Activenetworks (seeOscillators) Admittance, ofbridge-coupled sections of line,470-482 characteristic, definition, 49 input,oflinesection, 147-152, 161-169 graphs, 162-169 schematic diagrams, 162 symmetrical, 196 ofloadedlinesection, 168,169 oflow-loss line,criticalvalues,160, 161 normalized input,oflinesection, 133 135 determination usingcircledia gram,108 perunitlength,7 ofcoaxialcageline,45 ofcoaxialline,22 offour-wire line,20 generalized, interminal zone,63,69 ofshielded eccentric line,33 ofshielded-pair line,circular, 36,38 rectangular, 36,39 ofsinglelineoverimageplane,29, 46 ofstripline,46 ofthree-phase cable,43 ofthree-wire line,41 oftwo-wire line,closelyspaced,28, 29 withunequal conductors, 28 widelyspaced, 17 terminal-zone, definition, 73 discussion, 72,73 Admittance matrix, 198,199 Antenna, end-loading two-wire line,junc tionnetwork for,407-410 foldeddipole,210 junction network for,410-411 overgroundscreen,junction network for,430-437 shielded loop,223,224 stub-supported, junction network for, 405-407 Antiresonance, inputoflinesection, 149, 150 low-loss, 150 normalized input,150-152Attenuation, circlesofconstant, forline section, 137 Attenuation constant oftransmission line,definition, 52 generalformulas, 91-93 withhighattenuation, 99 withlargeleakage, 100 withlowattenuation, 95-98 withlowdistortion, 98,99 measurement of,275,276 andwidthofresonance anddistribu tion..curves,268,269 Attenuation function, ofconducting bridge,125,126 definition, 84 description, 102-104 oflinesection, 135 measurement, 279,280 negative, 128-130 forreactive termination, 118 forresistive termination, 114 significance, 86 andwidthofresonance anddistribu tioncurves,268,269 Attenuator, lossy,98,99 Balun,220-224 Beadsinline,346-351 impedance transformation through use of,351-358 standing-wave ratio,duetosinglebead, 324 duetotwobeads,349 Bend,incoaxialline,426-430 inshielded-pair line,426 intwo-wire line,382-389 Bilinear transformation, 109 Boundary conditions, atdiscontinuity in dielectric, 320 atterminations ofline,73-76 Bridge,conducting, attenuation function, 123 equivalent length,124,125 phasefunction, 123 reactance, 122 resistance, 122 resistive wire,127 tandem, 127 505 506 TRANSMISSION-LINE THEORY Bridge,conducting, transmission-line, usinghybridjunction, 230 wire,120 Cable,ocean,100,101 three-phase, 41 Cageline,43 constants of,45 Capacitance, lumped,tocorrectopen end,365-367 perunitlength,forcoaxialcageline,45 ofcoaxialline,22 inequivalent circuitofline,3,4 offour-wire line,20 generalized, interminal zone,63 ofshielded eccentric line,33 ofshielded-pair line,circular, 36,38 rectangular, 38,39 ofsinglelineoverimageplane,29,46 ofstripline,46 ofthree-phase cable,43 ofthree-wire line,41 oftwo-wire line,closelyspaced,28, 29 withunequal conductors, 28 widelyspaced,17 Characteristic admittance, definition, 49 Characteristic impedance, definition, 49 discussion ofsignificance, 76 generalforms,93 ofline,ofdielectric medium, 10 offreespace,10 ofimperfect dielectric medium, 10 matrixfor,305 measurement, 282-284 Characteristic resistance, definition, 49 Chargeperunitlength,definition, 7 discontinuity in,acrossboundary, 321 nearopenendofline,367,368 Circlediagram, applications, 107, 108, 147 construction, 104-110 equations for,derivation, 104-106 forequipotentials oftwo-wire line,27 polarform,111 rectangular form,106,107 Smithchart,111 Circuit,equivalent, ofinfiniteline,4,5 Closedendofline,364-368 Coaxialline,analysis, 20-23 bendin,426-430 bifurcation, 381,382 changeofradiusin,377-381 constants,22 discontinuous dielectric in,23 electricfieldin,23 end-correction for,whendrivingan tenna,430-437Coaxial line,magnetic fieldin,22 Coaxialmodesuppressor, 207 Coefficient ofreflection (seeReflection coefficient) Conditions forinfinitelineequations to applytofiniteline,49 Conductance, input,oflinesection, 147 153,162,166-169 extreme, 153-157 graphsof,162,166-169 ofloadedlinesection,graphsof,168, 169 leakage (seeLeakage conductance per unitlength) normalized input,oflinesection, 135 Conductivity, complex, 8 measurement, 285,286 realeffective, 9 Constants oflineinequivalent circuit,3,4 (Seealsospecificconstants) Continuity equation, 7 Coupled-circuit phenomena, 480-482 Coupler, directional (seeDirectional coupler) Coupling effectsattermination, general description, 68-71 Crosstalk,469 Curl,8 Currentandscattering matrix,305 Current density, 11 Currents online,balanced, 6 codirectional, 6 continuity acrossboundary, 321 drivenbycoupledsectionofline,463- 468 evenandodd,195-197 nonresonant, 243-244 withonepairofgenerators, 244-246 nearopenend,367,368 onoutsideofshield,6 polarform,249-251 resonant distribution, 262-265 symmetrical andantisymmetrical, 196 withthreepairsofgenerators, 248-249 withtwopairsofgenerators, 246-248 unbalanced, 6 Deschamps' methodfordetermining scattering matrix,304-314 Dielectric, discontinuous, incoaxialline, 23 losstangent for,8 Dielectric constant, complex, 8 measurement of,byDrude's method, 285-286 byshiftmethod, 329-341 relative, 9 realeffective, 9 INDEX 507 Dielectric factor,complex, 8 Dielectric andmagnetic slabinmatched line,323 Dielectric slabinline,generalanalysis, 317-328 atopencircuit,326 atshortcircuit,327,328 Dielectric slabsorbeadsinline,346-351 Differential equation, generalized first order,forcurrent, 66 second-order, forvoltage, 48 Differential equations, first-order, for currentandvoltage, 7,48 generalized, fortransmission lines,64 68 ofline,derivation of,conventional, 3-7 electromagnetic, 13-19 restrictions on,4-5 forpotential functions, 14-15,24-25 solution of(seeSolution ofdifferential equations) Directional coupler, hybridjunction as, 235-237 transmission line,467 Discontinuities (seespecificdiscontinui ties) Disk,conducting, astermination, 127, 128 resistive, 358,359 Dispersion, anomalous, 54 definition fortransmission line,53 normal, 54 Distortion factor,definition, 93 Distortionless line,98,99 Distribution ofvoltagealonginfinite line,50-56 Distribution curve,currentorvoltage, 257-259 definition, 257-259 formovinggenerators, 258,259 widthathalf-power points,266-269 Distribution-curve method formeasur- ing,275, 279, 280 Distribution-curve ratio,259-262 Div,8 Double-hump phenomena, 480-482 Double-slug tuner,351-353 Double-stub tuner,184-190 Eccentric line,31 constants, 33 Eccentricity incoaxialline,33 Effective spacingforcloselyspacedtwo wireline,29 Efficiency oftransmission, maximum, 253 generalformula, 253 withnonresonant line,244Electric field,8 ofasymmetrical currents, 463 boundary conditions atdiscontinuous dielectric, 320 incloselyspacedtwo-wire line,31 incoaxialline,23 ofconductor withsinecurrent,454- 457 ofevencurrents, 461,462 ofoddcurrents, 463 ofsectionoftwo-wire line,457-463 atsurfaceofconductors ofline,17 oftwo-wire line,31,457-463 Endcorrection forcoaxiallinedriving antenna, 430-437 Endeffects,generaldescription, 68-71 Equations, current, for'Il"-network, 291 differential (seeDifferential equations) Helmholtz, forpotentials, 14 transmission-line, 18 conditions for,19 voltage, forTnetwork, 290 Equipotentials, oflinewithunequal con- ductors, 25,26 ofshielded-pair line,37 oftwo-wire line,25,26 ofunbalanced shielded-pair line,37 Equivalent circuits (seeJunction net works) Equivalent lengthofline,136 Equivalent IInetwork oflinesection, 198, 199 Equivalent pointgenerators, foreven currents incoupled section, 465,466 foroddcurrents incoupled section, 466,467 fortraveling wavesincoupled section, 468,469 Equivalent Tofjunction, seriesele ments,202 shuntelement, 202 Equivalent Tnetwork oflinesection, 195-198 Exponential solutions, 73-83 f(h),definition, 92 Field(seeElectric field;Magnetic field) Filter,transmission-line, 482-487 Flatline(seeMatched line) Foldeddipoleantenna, 210 Four-terminal network, 194-203 Four-wire line,withconductors atcor- nersofsquare,19 constants of,20 withconductors inoneplane,34 Freespace,constants of,9,10 Frequency measurement, 275 508 TRANSMISSION-LINE THEORY g(h),definition, 92 Generators, point,244-249 distributed equivalents, 463-467 Grad,8 Groupvelocity, definition, 54 significance, 54,56 Guidedmodeinstripline,47 Gyrator, ideal,scattering matrixfor,2Jl Half-power pointsinresonance anddis tribution curves,268,269 Half-wave transformer, 326 Harmonics, indriving generator, 482-487 measurements onlinewith,482-486 suppression, 485-487 Hybridjunction, 225-241 admittance matrix,228,229 analysis ofequivalent circuit,227-230 asbridge,230 circuitforcoaxialline,226, 237, 238 circuitfortwo-wire orshielded-pair line,225 asdirectional coupler, 235-237 equivalent circuit,227 withhigh-impedance stubsfortwo wireline,238 aslinestretcher, 230-232 formeasuring phase,balanced-detector method, 233-235 ratiomethod, 232,233 ring-circuit form,239-241 withshielded loop,237 specialcases,229,230 without transformer, 236-239 Hyperbolic functions, definition, 83 inpolarform,86,87 Hyperbolic solutions, 83-91 Imageincylinder, 34 Image-plane line,29,34,41,45 constants, 29,36, 38,39,46 Impedance, apparent terminal, defini tion,71 determined frommeasurements, 75, 280-282 ofwirebridge,122 andcirclediagrams, 106-112, 146,147 characteristic (seeCharacteristic im pedance) input,oflinesection, 147-152 antisymmetrical, 196 maximum, 164-172 schematic diagrams of,161,163 ofloadedlinesection,graphs,163, 168 oflow-loss line,criticalvalues,160, 161Impedance, measurement, 280-282 through junction, 314-317 normalized input,oflinesection, 133 135 perunitlength,6 inequivalent circuitofline,3,4 external, 17,18 generalized, forterminal zone,66 internal, ofcoaxialcageline,45 ofcoaxialline,22 offour-wire line,20 ofshielded-pair line,circular, 36, 38 rectangular, 38,39 ofsinglelineoverimageplane,20 ofstripline,46 ofthree-phase cable,43 ofthree-wire line,41 oftubularconductors, 30 oftwo-wire line,closelyspaced,30 unequal conductors, 30 widelyspaced,18 terminal, apparent, 71 terminal-zone, definition, 73 discussion, 72,73 transfer, forseriessectionsofline,216- 219 Impedance matching (seeMatching) Impedance matrix,197,198,304 Impedance transformation (seeMatch- ing) Incident waveonline,80 Inductance, apparent, ofrectangle, 122 correction, forrectangle, 121 forwirebridge,364,365 perunitlength,ofcoaxialcageline,45 ofcoaxialline,22 complex, fortwo-wire line,17,18 inequivalent circuitofline,3,4 .offour-wire line,20 generalized interminal zone,63,69 ofshielded eccentric line,33 ofshielded-pair line,circular, 36,38 rectangular, 38,39 ofsinglelineoverimageplane,29,46 ofstripline,46 ofthree-phase cable,43 ofthree-wire line,41 total,48 oftwo-wire line,closelyspaced,28, 29,121 withunequal conductors, 28 widelyspaced,17,121 ofrectangle, 121 Infiniteline,generalsolution for,49 exponential form,49 hyperbolic forms,49 interpretation of,50-56 withpowercoefficients, 49 INDEX 509 Input-output equations, 289 Insertion loss,253 Insulator, linesectionas,164-172 Junction, oftwocoaxiallineswithdif ferentinnerconductors, 377-381 oftwo-wire linesofdifferent dimen sions,368-377, 411-417 Junction networks, forantenna, asend load,407-411 overgroundscreen,430-437 withstubsupport, 405-407 forbend,incoaxialline,426-430 inplaneoftwo-wire line,418-425 forbendsandTjunctions inshielded pairlines,426 forbifurcation ofcoaxialline,381,382 forchange,inradiusofcoaxialline, 377-380 inspacingoftwo-wire line,411-417 forfoldeddipole,410,411 foropenend,366,367 forseriesbranches intwo-wire line, 397-411 forTjunction intwo-wire line,389 397 fortwo-wire linesofdifferent radii, 368-377 forwirebridge,120-127, 364,365 ko(w),definition, 16 forinfiniteline,17 k1(w),definition, 16 forinfiniteline,17 Kirchhoff's lawsappliedtoequivalent circuitofline,5 Laplace's equation fortransverse prob. leminlines,25 Laplacian operator, 10 Leakage conductance perunitlength, ofcoaxialcageline,45 ofcoaxialline,22 inequivalent circuitofline,3,4 offour-wire line,20 generalized, interminal zone,63 ofshielded eccentric line,33 ofshielded-pair line,circular, 36,38 rectangular, 38,39 ofsinglelineoverimageplane,29 ofthree-phase cable,43 ofthree-wire line,41 oftwo-wire line,closelyspaced,28,29 withunequal conductors, 28 widelyspaced,17 Lecherwires(seeTwo-wire line)Length, equivalent, ofline,forattenua tion,136 forphaseshift,136 Linestretcher, hybridjunction as,230- 232 Load,reactive, 117-120 Loopantenna, 223,224 Lorentz condition, 8,12,13 Loss,insertion, 253 measurement of,fordielectric andmag neticmaterials, 341-346 transmission, onnonresonant line,244 Losstangent, 8 Lossyattenuator, 98,99 Lossyline,360 Lossyterminations, 358-364 Lumped equivalent oflinesection,194 203 Magnetic field,8 incloselyspacedtwo-wire line,30 incoaxialline,22 onterminating disk,128 Magnetization, timelagsin,9 Matched line,149 withdielectric andmagnetic slab, 323-327 half-wave slab,326 quarter-wave slab,325 thinslab,325 loadedwithlumpedcapacitances or inductances, 328,329 unaffected bydielectric slab,condi tionsof,323 Matching, 172-194 double-stub, 184-190 examples, 187-190 generalformulation, 172-174 quarter-wave transformer, 176 seriestransformer, 174-177 limitations, 176,177 singlemovable stub,178-184 examples, 180-184 shuntsections, 190-193 examples, 192-193 stubwithresistive load,curves,180 Matrices, impedance, admittance, and scattering, relations between, 201 203 Matrix,admittance, 198,199 ofhybridjunction, 229 impedance, 197,198,304 scattering (seeScattering matrix) Matrixelements, scatterinll:. determina tionof,304-314 interpretation of,201,305 ofsymmetrical Tsection,293 510 TRANSMISSION-LINE THEORY Measurements, withhybridjunction, 23~236 methods, Chipman's (resonance-curve ,method), 273, 279, 280 Deschamps', 304-317 distribution-curve, 274,275,278-280 maximum-minimum-shift, 329-345 resonance-curve, 273, 276, 278-280 three-probe, 281,282 Weissfloch tangent, 294-304 withmultiple-frequency source,482 487 transmission-line, theoryof,272-286 Methods ofanalysis, 1 basedonelectric-circuit theory,1,3 electromagnetic, 2 Microstrip, 46,47 Modulated voltageappliedtoinfinite line,53 Multiple-frequency source,measure mentswith,482-487 Nablaoperator, 10 Natural frequencies ofoscillation (see Oscillators) Network, four-terminal, forline,84 two-terminal pair,288-294 Nonreciprocal elements inscattering matrix,201 Nonresonant sectionofline,148 Openendofline,currentandvoltage near,367,368 equivalent circuitfor,364-366 Open-wire line(seeTwo-wire line) Optimum termination forline,253 Oscillators, withcoupledsecondary, 447- 454 withdouble-end triode,445,446 withlighthouse tube,446,447 push-pull, withtriode,442,443 reentrant, 446,447 withsingletriode,439,442 tuned-plate tuned-grid, 443-445 Parallel-strip line45,46 Parallel-wire line(seeTwo-wire line) Permeability, 9 complex, 9,18 measurement of,285,286 byshiftmethod, 329-341 Permittivity (seeDielectric constant) Phase,circlesofconstant, 137 comparison of,284,285 measurement of,balanced-detector method, 233-235Phase,measurement of,withhybrid junction, 233-235 ratiomethod, 232,233 ofvoltageoninfiniteline,51 Phaseconstant, ofdielectric medium, 10 offreespace,10 generalformsof,92 generalized, int~rminal zone,63 ofimperfect dielectric, 10 measurement of,273-275 oftransmission line,definition, 53 generalformulas, 91-93 withhighattenuation, 99 withlargeleakage, 100 withlowattenuation, 95-98 withlowdistortion, 98,99 withnoattenuation, 98 Phasefunction, ofconducting bridge,123 definition, 84 description, 102-104 ofdisk,127 oflinesection, 135 measurement, 276-278 ofpiston,127 ofreactive termination, 118,119 ofresistive termination, Xl=0,114 X=0,117 significance, 86 tabulation fordifferent terminations, 104 Phasemeasurement (seePhase) Phasevelocity (seeVelocity) IInetwork asloadfortwolines,288-294 Pistonastermination, 127 Potential, scalar,8 separation intopartsduetolineand termination, 13,65 vector,8 separation intocomponents dueto lineandtermination, 12,65 Potential difference, duetochargesand currents, online,59,60 ontermination, 59,60 generalized forterminated two-wire line,61 scalar,15 vector,15 Potentials andpotential differences, for coaxialcageline,44 forcoaxialline,21,22 forcylindrical conductors, 11,13 forfour-wire line,20 Helmholtz integrals, 11 ratiofunctions, definitions, 63 forshielded eccentric line,32,33 forshielded-pair line,35,37 forterminated line,58,59 forthree-phase cable,42 forthree-wire line,40- INDEX &11 Potentials andpotential differences, for two-wire line,closelyspaced,25-29 withunequal conductors, 25,26 widelyspaced,11,15,16 Powerfactor,129n. (SeealsoLosstangent) Powertransfer alongline,251-254 Propagation constant, complex, 7,25,49 generalforms,91-93 generalized, forterminal zone,68 intermsofseparation constant, 25 Proximity effect,internal impedance with,30 Pulseappliedtotransmission line,55 Qofline,269-272 external, 271 loaded,271 unloaded, 271 Quarter-wave transformer, 176 Radiation fromtwo-wire line,487-492 condition tomakenegligible, 16,17 nonresonant line,488,489 resonant low-loss line,489,490 Ratrace,239-241 Ratio,distribution-curve, 259-262 resonance-curve, 259-262 standing-wave, 260-263 Ratiomethodformeasuring phase,232, 233 Reactance, input,oflinesection, 147- 152 extreme, 157-160 graphs,161,163-165, 168 loaded,graphs,163,168 nonresonant, 148 schematic diagrams of,161 normalized, 102 contours ofconstant, 137 oflinesection,135 tables,142-146 Reactive load,117-120 Reciprocal elements inscattering matrix, 201 Reflected wave,amplitude, 201 online,80 Reflection coefficient, definition, 76 discussion, 78 elements ofscattering matrix,201 formulas for,134 greaterthanunity,128-130, 252n. measurement, 280 through junction, 314-317 asratiosofcurrents andvoltages, 83 relationtoterminal functions, 101,280 Reflections onterminated line,77-80Reluctivity, 9 Resistance, characteristic, 49 input,oflinesection,147-152 extreme, 153-157, 164-172 graphs,161,163-165, 168 loaded,graphs,163,168 maximum, 164-172 nonresonant, 148 schematic diagrams, 161 normalized, 102 contours ofconstant, 136 oflinesection, 135 tables,138-141 perunitlength,inequivalent circuit ofline,3,4 generalized toincludemagnetic losses,19 (SeealsoImpedance perunitlength, internal) ofwirebridge,120-127 Resistive disktermination, 358,359 Resistor, coaxial,361-364 Resonance, conditions for,255,256 input,oflinesection,149 low-loss, 149,150 normalized input,150-153 Resonance-curve methodformeasuring, 273, 279, 280 Resonance-curve ratio,259-262 Resonance curves,definition, 254-257 withmultiple-frequency source,483 widthathalf-power points,266-269 Resonant sectionofline,148,149 Ringcircuit,239-241 Sandwich line,46 Scalarpotential, 8,13, 15,65 (SeealsoVoltage) Scattering matrix,199-203, 305 determination ofelements, 304-314 interpretation, 307 Separation constant, 24 Seriesbranchintwo-wire line,397-401 Seriesexpansions ofchargeandcurrent inline,15 Seriessectionoftwinline,209-223 antisymmetrical problem for,213,214 coaxialmodein,218 equaltwinandcoaxialmodesin,219 symmetrical problem for,214,215 twinmodein,217 Seriesstubs,209-223 on.balanced line,211,212 Shielded linewitheccentric innercon ductor,31 constants, 33 Shielded loopantenna, 223-224 Shielded-pair line,34-39 512 TRANSMISSION-LINE THEORY Shielded-pair line,balanced, 35,36 withcircularshield,34-37 constants, 36-39 withrectangular shield,38,39 seriessection (seeSeriessectionof twinline) unbalanced, 37, 38,203-224 Short-circuiting barorwire,120 Shuntsections, matching with,190-193 Signalvelocity, 56 Single-stub tuner,177-184 Single-wire lineoverimageplane,con stants,29,46 Smithchart,derivation ofequations for, 108-112 withscalesofP,r,<P,112 Solution ofdifferential equations, forin finiteline,48 forterminated line,exponential, 73-76 hyperbolic forms,83-86. instantaneous, 86-91 incident- andreflected-wave form, 79-83 infinite-series form,77-79 Squelcher, unbalance, 207-209 Standing-wave ratio,260-263 Standing waves(seeDistribution curve) Stripline,45,46 Stub,singlemovable, formatching, 177 184 Stubmatching, 177-193 Stubsupport, 164-172 forantenna, 405-407 Suppressor, ofcoaxialmodeinshielded pairline,207-209 offundamental whilepassingeven harmonics, 487 ofsecondharmonic whilepassingfun damental, 486 ofthirdharmonic whilepassingfunda mental, 487 Susceptance, input,oflinesection, 147- 153 extreme, 157-160 graphs,162,166-169 loaded,graphs, 168,169 schematic diagram, 162 normalized, 135 Symmetrical networks, 293 Tjunction intwo-wire line,389-397 Tnetwork asloadfortwolines,288-294 Tangent relation, Weissfloch, 296,297 TEMmodeinstriplines,47 Terminal function, complex, compared withreflection coefficient, 84 definition, 84,85 graphical representations,104-107 relation toreflection coefficient, 101Terminal impedance, apparent, 71 idealized, 72 Terminal zone,definition, 68-71 length,68 Terminal-zone network, description, 71-73 Terminated line,58-130 Termination, lossy,358-364 open-end, 365,366 reactive, 117-120 resistive, 112-117 wire-bridge, 364,365 Three-phase line,39-43 Three-probe method, 281,282 Three-wire line,39-43 Transfer impedance forseriessections of line,216-219 , Transformer, double-slug, 351-358 equivalent, experimental determina tionof,298-304 fortwo-terminal pairnetwork, 294 304 withhybridjunction, 227 elimination of,236,237 impedance, 172-174 quarter-wave, 176 series,174-177 Transmission coefficient, 305 element ofscattering matrix,201 Transmission lines,changeincrosssec tion,317-328 coaxial (seeCoaxial line) coaxialcage,43 constants, 45 coupled, 457-470 distortionless, 98,99 eccentric, 31 constants, 33 equivalent uniform, interminal zone, 71 flat(seeMatched line;Matching) four-wire, 19 constants, 20 withhighattenuation, conditions and constants,99 image,29,34,41,45 constants, 29,36, 38, 39, 46 withlargeleakageandnegligible re sistance, 100 lossless, 98 formulas forcurrentandvoltage, 86 lossy,360 withlowattenuation, negligible leak ageconductance, 98 perunitlength,95-96 factorsofhigherorderfor,96,97 withlowdistortion, 98 matched (seeMatched line;Matching) n-phase, 39,43 INDEX 513 Transmission lines,n-wire,39,43 nonresonant, 243,244 resonant, currentandvoltage, 262-265 sandwich, 46 shielded, 31, 34,43 shielded-pair (seeShielded-pair line) singleconductor overimageplane,29, 45 strip,45 constants, 46 terminated, 58 three-phase, 41 constants, 43 three-wire, 39 constants, 41 twin(seeShielded-pair line) two-wire (seeTwo-wire line) Transmitted wave,amplitude of,201 Transverse problem, equation for,25 Traveling waveonline,52 Tuner,double-slug, 351-353 double-stub, 184-190 single-stub, 177-184 triple-stub, 241' Twinline(seeShielded-pair line) Twb-terminal pair,equivalent trans former,294-304 Two-wire line,analysis, 13-20,23-31 bendin,382-389 changeinspacing, 411-417 closelyspaced,23-31 effective spacingfor,29 constants, 17,28,29 electricfield,31 magnetic field,30 planeof,bendin,418-425 radiation from(seeRadiation from two-wire line) seriesbranches in;397-411 shielded, 34-39 Tjunction in,389-397 unbalanced, 224,397-399 Unbalance squelcher, terminated, 209 Tomiyasu's, 207 Unbalanced generator, 208 Unbalanced line,open-wire, 209,210 shielded-pair, 203-209 Unbalanced load,difference impedance for,203 equivalent generators for,204 terminating shielded-pair line,203-209 Variables, separation of,24 Vectorpotential, definition, 8 discontinuity acrossboundary inline, 321 nefi,ropenend,367,368Vectorpotential, forterminated line,66 fortwo-wire line,30 (SeealsoPotentials andpotential dif ferences) Velocity, characteristic, ofdielectric medium, 10 offreespace,10 ofimperfect dielectric medium, 10 group,definition, 54 significance, 54,56 phase,definition, 52,53 generalforms,94 inhyperbolic formofsolution, 88-91 inillcident- andreflected-wave form ofsolution, 80 ininfinite-series formofsolution, 77 lowerfrequency limit,94 measurement, 275 inoceancable,100 significance, 52-54 upperfrequency limit,94 varying, forfiniteline,89 signal,56 Voltage, continuity acrossboundary, 32~321 distribution, onnonresonant line,243, 244 onresonant line,262-265 instantaneous, alongfiniteline,77,87 alonginfiniteline,50 asinfiniteseries,77 online,drivenbycoupled sectionof line,463-468 withonepairofpointgenerators, 244-246 withthreepairsofpointgenerators, 248,249 withtwopairsofpointgenerators, 246-248 nearopenend,367,368 polarformforline,249-251 andscattering matrix,305 Voltages online,evenandodd,195-197 symmetrical andantisymmetrical, 196, 29~391,39~39~41~419 Wave,incident, 80 reflected, 80 Wavelength, definition forinfiniteline, 51,52 measurement, 273-275 Waves,ofcurrentoninfiniteline,run ning,52 traveling, 52 ofvoltage, 52 Weissfloch tangent relation, 296,297 Zone,terminal, 68-71 CATALOGUE OFDOVERBOOKS Catalogue ofDoverBooks PHYSICS Generalphysics FOUNDATIONS OFPHYSICS, R.B.Lindsay&H.Margenau. Excellent bridgebetween semi. popul~rworks&technical treatises. A.~iscus~ion atmethods ofphysical description, con structIOn oftheory;valuable torphysIcist withelementary calculus whoisinterested in ideasthatgivemeaning todata,toolsofmodernphysics.Contents includesymbolism, math ematical equations; space&timefoundations ofmechanics; probability; physics&con tinua;electron theory;special&generalrelativity; quantum mechanics; causality. "Thor oughandyetnotoverdetailed. Unreservedly recommended," NATURE (London). Unabridged, corrected edition. Listofrecommended readings. 35illustrations. xi+537pp.5%x8. S377Paperbouna ~2.75 FUNDAMENTAL FORMULAS OFPHYSICS, ed.byD.H.Menzel.HighlyusefUl,fullyinexpensive reference andstudytext,rangingtromsimpletohighlysophisticated operations. Mathematics integrated intotext-each cnapterstandsasshorttextbook otfielarepresented. VOl.1: Statistics, Physical Constants, SpecialTheoryofRelativity, HydrodynamiCs, Aerodynamics, Boundary ValueProblems inMath.Physics; Viscosity, Electromagnetic Theory,etc.Vol.2: Sound,Acoustics, Geometrical Optics,Electron Optics,High-Energy Phenomena, Magnetism, Biophysics, muchmore.Index.TotalofSOOpp.5318xIS. Vol.1S595Paperbound $2.00 Vol.2S596Paperbound ~2.00 MATHEMATICAL PHYSICS, D.H.Menzel.Thorough one-volume treatment ofthemathematical techniques vitalforclassicmechanics, electromagnetic theory,quantum theory,andrela tivity.Written bytheHarvard Protessor ofAstrophysics forjUnior,senior,andgraduate courses,itgivesclearexplanations ofallthoseaspectsoffunctIOn theory,vectors, matrices, dyadics, tensors, partialdifferential equations, etc.,necessary tortheunderstanding ofthe variousphysical theories. Electron theory,relativity, andothertopicsseldompresented appearhereinconsiderable detail.Scoresofdefinitions, conversion factors, dimensional constants, etc."Moredetailed thannormalforanadvanced text...excellent setofsec tionsonDyadics, Matrices, andTensors," JOURNAL OFTHEFRANKLIN INSTITUTE. Index. 193problems, withanswers. x+412pp. 5318x8. S56Paperbound $2.00 THESCIENTIFIC PAPERS OFJ.WILLARD· GIBBS.AllthepUblished papersofAmerica's outstand ingtheoretical scientist (exceptfor"Statistical Mechanics" and"Vector Analysis"). VolI (thermodynamics) contains oneofthemostbrilliantofall19th-century scientific papers-the 300-page "OntheEquilibrium ofHeterogeneous Substances," whichfounded thescienceof physical chemistry, andclearlystatedanumberofhighlyimportant naturallawsforthefirst time;8otherpaperscomplete thefirstvolume. VolIIincludes 2papersondynamics, 8on vectoranalysisandmultiplealgebra,5ontheelectromagnetic theoryoflight,and6miscella neouspapers.Biographical sketchbyH.A.Bumstead. Totalofxxxvi+718pp.55/8x8318. S721VolIPaperbound $2.00 S722VolIIPaperbound $2.00 Theset$4.00 BASICTHEORIES OFPHYSICS, PeterGabrielBergmann. Two-volume setwhichpresents a criticalexamination ofimportant topicsinthemajorsubdivisions ofclassical andmodern physics. Thefirstvolume isconcerned withclassical mechanics andelectrodynamics: mechanics ofmasspoints,analytical mechanics, matterinbUlk,electrostatics andmagneto statics, electromagnetic interaction, thefieldwaves,specialrelativity, andwaves.The secondvolume(HeatandQuanta)contains discussions ofthekinetichypothesis, physicsand statistics, stationary ensembles, lawsofthermodynamics, earlyquantum theories, atomic spectra, probability waves,quantization inwavemechanics, approximation methods, and abstract quantum theory.Avaluable supplement toanythorough courseortext. HeatandQuanta:Index.8figures.x+300pp.53/8X81/2. S968Paperbound $1.75 Mechanics andElectrodynamics: Index.14figures.vii+280pp.53/8x81/2. S969Paperbound $1.75 THEORETICAL PHYSICS, A.S.Kompaneyets. Oneoftheveryfewthorough studiesofthe subjectinthispricerange.Provides advanced students withacomprehensive theoretical background. Especially strongonrecentexperimentation anddevelopments inquantum theory.Contents: Mechanics (Generalized Coordinates, Lagrange's Equation, Collision of Particles, etc.),Electrodynamics (VectorAnalysis, Maxwell's equations, Transmission of Signals,TheoryofRelativity, etc.),Quantum Mechanics (theInadequacy ofClassical Mechan ics,theWaveEquation, MotioninaCentralField,Quantum TheoryofRadiation, Quantum Theories ofDispersion andScattering, etc.),andStatistical Physics(Equilibrium Distribution ofMolecules inanIdealGas,Boltzmann statistics, BoseandFermiDistribution, Thermodynamic Quantities, etc.).Revisedto1961.Translated byGeorgeYankovsky, author izedbyKompaneyets. 137exercises. 56figures.529pp.53/8x81/2.S972Paperbound $2.50 ANALYTICAL ANDCANONICAL FORMALISM INPHYSICS, AndreMercier. Asurvey,inonevol· ume,ofthevariational principles (thekeyprinciples-in mathematical form-from which thebasiclawsofanyone branchofphysicscanbederived) oftheseveralbranches of physical theory,together withanexamination oftherelationships amongthem.Contents: theLagrangian Formalism, Lagrangian Densities, Canonical Formalism, Canonical Formof Electrodynamics, Hamiltonian Densities, Transformations, andCanonical FormwithVanishing Jacobian Determinant. Numerous examples andexercises. Foradvanced students, teachers, etc.6figures.Index.viii+222pp.53/8x81/2. S1077Paperbound $1.75 Catalogue ofDoverBooks Acoustics, optics,electricity andmagnetism, electromagnetics, magneto hydrodynam ics THETHEORY OFSOUND,LordRayleigh. Mostvibrating systems likelytobeencountered in practice canbetackledsuccessfully bythemethods setforthbythegreatNobellaureate, LordRayleigh. Complete coverage ofexperimental, mathematical aspectsofsoundtheory. Partialcontents: Harmonic motions, vibrating systems ingeneral, lateralvibrations ofbars, curvedplatesorshells,applications ofLaplace's functions toacoustical problems, fluid friction, planevortex-sheet, vibrations ofsolidbodies,etc.Thisisthefirstinexpensive editionofthisgreatreference andstudywork.Bibliography. Historical introduction byR.B. Lindsay. Totalof1040pp. 97figures. 5=¥sx8. S292, S293, Twovolumeset,paperbound, $4.70 THEDYNAMICAL THEORY OFSOUND,H.Lamb.Comprehensive mathematical treatment ofthe physicalaspectsofsound,covering thetheoryofvibrations, thegeneraltheoryofsound,and theequations ofmotionofstrings,bars,membranes, pipes,andresonators. Includes chap tersonplane,spherical, andsimpleharmonic wavesandtheHelmholtz TheoryofAudition. Complete andself-contained development forstudentandspecialist; allfundamental differ entialequations solvedcompletely. Specificmathematical detailsforsuchimportant phenom enaasharmonics, normalmodes,forcedvibrations ofstrings,theoryofreedpipes,etc.Index. Bibliography. 86diagrams. viii+307pp.5%x8. S655Paperbound $1.50 WAVEPROPAGATION INPERIODIC STRUCTURES, L.Brillouin. Ageneralmethodandapplica tiontodifferent problems: purephysics, suchasscattering ofX-raysofcrystals, thermal vibration incrystallattices, electronic motioninmetals;andalsoproblems ofelectrical engineering. Partialcontents: elasticwavesinI-dimensional lattices ofpointmasses. Propagation ofwavesalongI-dimensional lattices. Energyflow.2dimensional, 3dimensional lattices. Mathieu's equation. Matrices andpropagation ofwavesalonganelectric line. Continuous electriclines.131illustrations. Bibliography. Index.xii+253pp.5%x8. 534Paperbound $1.85 THEORY OFVIBRATION~, N.W.McLachlan. Basedonanexceptionally successful graduate coursegivenatBrownUniversity, thisdiscusses linearsystems having1degreeoffreedom, forcedvibrations ofsimplelinearsystems, vibration offlexiblestrings,transverse Vibra tionsofbarsandtUbes,transverse Vibration ofcircularplate,soundwavesoffiniteampli tUde,etc.Index.99diagrams. 160pp.53/sx8. S190Paperbound $1.35 LIGHT:PRINCIPLES ANDEXPERIMENTS, GeorgeS.Monk.Coverstheory,experimentation, and research. Intended forstudents Withsomebackground ingeneralphysicsandelementary calculus. Threemaindivisions: 1)Eightchapters ongeometrical optics-fundamental con· cepts(therayanditsopticallength,Fermat's principle, etc.),lawsofimageformation, apertures inopticalsystems, photometry, opticalinstruments etc.;2)9chapters onphysical optics-interference, diffraction, polarization, spectra, theRayleigh refractometer, the wavetheoryoflight,etc.;3)23instructive experiments baseddirectly onthetheoretical text."Probably thebestintermediate textbook onlightintheEnglishlanguage. Certainly, itisthebestbookwhichincludes bothgeometrical andphysical optics," J.RudNielson, PHYSICS FORUM.Revisededition.102problems andanswers. 12appendices. 6tables.Index. 270illustrations. xi+489pp. 5=¥sx8lf2. S341Paperbound $2.45 PHOTOMETRY, JohnW.T.Walsh.Thebesttreatment ofboth"bench" and"illumination" photometry inEnglishbyoneofBritain's foremost expertsinthefield(President ofthe International Commission onIllumination). Limitedtothosematters, theoretical andprac tical,whichaffectthemeasurement oflightflux,candlepower, illumination, etc.,and excludes treatment oftheusetoWhichsuchmeasurements maybeputaftertheyhavebeen made.Chapters onRadiation, TheEyeandVision,Photo-Electric Cells,ThePrinciples of Photometry, TheMeasurement ofLuminous Intensity, Colorimetry, Spectrophotometry, Stellar Photometry, ThePhotometric Laboratory, etc.Thirdrevised(1958)edition.281illustrations. 10appendices. xxiv+544pp.5lf2x 91/4. S319Clothbound $10.00 EXPERIMENTAL SPECTROSCOPY, R.A.Sawyer.Cleardiscussion ofprismandgratingspectro graphsandthetechniques oftheiruseinresearch, Withemphasis onthoseprinciples and techniques thatarefundamental topractically allusesofspectroscopic equipment. Begin ningwithabriefhistoryofspectroscopy, theauthorcoverssuchtopicsaslightsources, spectroscopic apparatus, prismspectroscopes andgraphs,diffraction grating, thephoto graphicprocess, determination ofwavelength,spectral intensity, infrared spectroscopy, spectrochemical analysis, etc.Thisrevisededitioncontains newmaterial ontheproduction ofreplicagratings, solarspectroscopy fromrockets, newstandard ofwavelength,etc. Index.Bibliography. 111illustrations. x +358pp.53/8x81f2. 51045Paperbound $2.00 FUNDAMENTALS OFELECTRICITY ANDMAGNETISM, L.B.Loeb.Fotstudents ofphysics, chem istry,orengineering whowantanintroduction toelectricity andmagnetism onahigherlevel andinmoredetailthangeneralelementary physicstextsprovide.Onlyelementary differential andintegral calculus isassumed. Physical lawsdeveloped logically, frommagnetism to electriccurrents, Ohm'slaw,electrolYSIS, andontostatic'electncity, induction, etc.Covers anunusualamountofmaterial; onethirdofbookonmodernmaterial: solutionofwaveequa tion,photoelectric andthermionic effects,etc.Complete statement ofthevariouselectrical systems ofunitsandinterrelations. 2Indexes. 75pagesofproblems withanswers stated. Over300figuresanddiagrams. xix+669pp. 53/8x8. S745Paperbound $2.75 Catalogue ofDoverBooks MATHEMATICAL ANALYSIS OFELECTRICAL ANDOPTICAL WAVE-MOTION, HarryBateman. Written byoneofthiscentury's mostdistinguished mathematical physicists, thisisapractical introduction tothosedevelopments ofMaxwell's electromagnetic theorywhicharedirectly connected withthesolution ofthepartialdifferential equation ofwavemotion.Methods of solvingwave-equation, polar-cylindrical coordinates, diffraction, transformation ofcoordinates, homogeneous solutions, electromagnetic fieldswithmovingsingularities, etc.Index.168pp. 53/8x8. S14Paperbound $1.75 PRINCIPLES OFPHYSICAL OPTICS, ErnstMach.Thisclassical examination ofthepropagation oflight,color,polarization, etc.offersanhistorical andphilosophical treatment thathas neverbeensurpassed forbreadthandeasyreadability. Contents: Rectilinear propagation of light.Reflection, refraction. Earlyknowledge ofvision.Dioptrics. Composition oflight. Theoryofcoloranddispersion. Periodicity. Theoryofinterference. Polarization. Mathematical representation ofproperties oflight.Propagation ofwaves,etc.279illustrations, 10por traits.Appendix. Indexes. 324pp.53/8x8. S178Paperbound $1.75 THETHEORY OFOPTICS, PaulDrude.Oneoffinestfundamental textsinphysical optics, classicoffersthorough coverage, complete mathematical treatment ofbasicideas.Includes fullesttreatment ofapplication ofthermodynamics tooptics;sinelawinformation of images, transparent crystals, magnetically activesUbstances, velocity oflight,apertures, effectsdepending uponthem,polarization, opticalinstruments, etc.Introduction byA.A. Michelson. Index.110illus;567pp.53/8x8. S532Paperbound $2.45 ELECTRICAL THEORY ONTHEGIORGISYSTEM, P.Cornelius. Anewclarification ofthefunda mentalconcepts ofelectricity andmagnetism, advocating theconvenient m.k.s.systemof unitsthatissteadilygainingfollowers inthesciences. Illustrating theuseandeffectiveness ofhisterminology withnumerous applications toconcrete technical problems, theauthor hereexpounds thefamous Giorgisystemofelectrical physics. Hislucidpresentation andwell-reasoned, cogentargument fortheuniversal adoption ofthissystemformoneof thefinestpiecesofscientific exposition inrecentyears.28figures.Index.Conversion tables fortranslating earlierdataintomodernunits.Translated from3rdDutcheditionbyL.J. Jolley.x+187pp.5112x 83/4. S909Clothbound $6.00 ELECTRIC WAVES: BEINGRESEARCHES ONTHEPROPAGATION OFELECTRIC ACTION WITH FINITEVELOCITY THROUGH SPACE,Heinrich Hertz.Thisclassicworkbringstogether the original papersinwhichHertz-Helmholtz's proteg~ andoneofthemostbrilliant figures in19th-century research-probed theexistence ofelectromagnetic wavesandshowedexperi mentally thattheirvelocity equalled thatoflight,research thathelpedlaythegroundwork forthedevelopment ofradio,teleVision, telephone, telegraph, andothermoderntechnological marvels. Unabridged republication oforiginal edition.Authorized translation byD.E.Jones. PrefacebyLordKelvin.Indexofnames.40illustrations. xvii+278pp.53/8x8112. S57Paperbound $1.75 PIEZOELECTRICITY: ANINTRODUCTION TOTHETHEORY ANDAPPLICATIONS OFELECTRO· MECHANICAL PHENOMENA INCRYSTALS, WalterG.Cady.Thisisthemostcomplete andsys tematiccoverage ofthisimportant fieldinprint-now regarded assomething ofscientific classic.Thisrepublication, revisedandcorrected byProf.Cady-one oftheforemost con tributors inthisarea-contains asketchofrecentprogress andnewmaterial onFerro electrics. TimeStandards, etc.Thefirst7chapters dealwithfundamental theoryofcrystal electricity. 5important chapters coverbasicconcepts ofpiezoelectricity, including com parisons ofvariouscompeting theories inthefield.Alsodiscussed: piezoelectric resonators (theory, methods ofmanufacture, influences ofair-gaps, etc.);thepiezooscillator; the properties, history, andobservations relatingtoRochelle salt;ferroelectric crystals; miscel laneousapplications ofpiezoelectricity; pyroelectricitYi etc."Agreatwork,"W.A.Wooster, NATURE. Revised (1963)andcorrected edition. NewprefacebyProf.Cady.2Appendices. Indices. Illustrations. 62tables.Bibliography. Problems. Totalof1+822pp.53/8x8112. S1094Vol.IPaperbound $2.50 S1095Vol.IIPaperbound $2.50 TwovolumesetPaperbound $5.00 MAGNETISM ANDVERYLOWTEMPERATURES, H.B.G.Casimir. Abasicworkintheliterature oflowtemperature physics. Presents aconcisesurveyoffundamental theoretical principles, andalsopointsoutpromising linesofinvestigation. Contents: Classical TheoryandExperi mentalMethods, Quantum TheoryofParamagnetism, Experiments onAdiabatic Demagnetiza tion.Theoretical Discussion ofParamagnetism atVeryLowTemperatures, SomeExperimental Results, Relaxation Phenomena. Index.89-itembibliography. ix+95pp.53/8x8. S943Paperbound $1.25 SELECTED PAPERS ONNEWTECHNIQUES FORENERGY CONVERSION: THERMOELECTRIC METHODS; THERMIONIC; PHOTOVOLTAIC ANDELECTRICAL EFFECTS; FUSION, EditedbySumner N.Levine.Bringstogether inonevolumethemostimportant papers(1954-1961) inmodern energytechnology. Included amongthe37papersaregeneralandqualitative descriptions ofthefieldasawhole,indicating promising linesofresearch. Also:15papersonthermo electricmethods, 7onthermionic, 5onphotovoltaic, 4onelectrochemical effect,and2on controlled fusionresearch. Amongthecontributors are:Joffe,MariaTelkes,Herold,Herring, Douglas, Jaumot, Post,Austin,Wilson, Pfann,Rappaport, Morehouse, Domenicali, .Moss, Bowers, Harman, VonDoenhoef. Preface andintroduction bytheeditor.Bibliographies. xxviii+451pp. 6118x 91/4. S37Paperbound $3.00 Catalogue ofDoverBooks SUPERFLUIDS: MACROSCOPIC THEORY OFSUPERCONDUCTIVITY, Vol.I,FritzLondon. The majorworkbyoneofthefounders andgreattheoreticians ofmodernquantum physics. Consolidates theresearches thatledtothepresentunderstanding ofthenatureofsuper conductivity. Prof.Londonhererevealsthatquantum mechanics isoperative onthemacro scopicplaneaswellasthesubmolecular level.Contents: Properties ofSuper~onductors andTheirThermodynamical Correlation; Electrodynamics ofthePureSuperconductmg State; RelationbetweenCurrentandField;Measurements ofthePenetration Depth;Non-Viscous Flow vs.Superconductivity; Micro-waves inSuperconductors; RealityoftheDomainStructure; andmanyotherrelatedtopics.Anewepilogue byM.J.Buckingham discusses developments inthefieldupto1960.Corrected andexpanded edition.Anappreciation oftheauthor's lifeandworkbyL.W.Nordheim. Biography byEdithLondon.Bibliography ofhispublica tions.45figures.2Indices.xviii+173pp.55/8x8%. S44Paperbound $1.45 SELECTED PAPERS ONPHYSICAL PROCESSES INIONIZED PLASMAS, EditedbyDonaldH. Menzel,Director, HarvardCollegeObservatory. 30important papersrelatingtothestudyof highlyionizedgasesorplasmasselected byaforemost contributor inthefield,withthe assistance ofDr.L.H.Aller.Theessaysinclude18onthephysical processes ingaseous nebulae, covering problems ofradiation andradiative transfer, theBalmerdecrement, electron temperatives, spectrophotometry, etc.10papersdealwiththeinterpretation of nebularspectra, byBohm,VanVleck,Aller,Minkowski, etc.Thereisalsoadiscussion oftheintensities of"forbidden" spectral linesbyGeorgeShortley andapaperconcern ingthetheoryofhydrogenic spectrabyMenzelandPekeris.Othercontributors: Goldberg, Hebb,Baker,Bowen,Ufford,Liller,etc.viii+374pp.61/8x91/4. S60Paperbound $2.95 THEELECTROMAGNETIC FIELD,MaxMason&WarrenWeaver. Usedconstantly bygraduate engineers. Vectormethods exclusively: detailedtreatment ofelectrostatics, expansion meth ods,withtablesconverting anyquantityintoabsolute electromagnetic, absolute electrostatic, practical units.Discrete charges, ponderable bodies,Maxwellfieldequations, etc.Introduc tion.Indexes.416pp.5%x8. 5185Paperbound $2.00 THEORY OFELECTRONS ANDITSAPPLICATION TOTHEPHENOMENA OFLIGHTANDRADIANT HEAT,H.Lorentz. Lectures delivered atColumbia University byNobellaureate Lorentz. Unabridged, theyformahistorical coverage ofthetheoryoffreeelectrons, motion, absorption ofheat,Zeemaneffect,propagation oflightinmolecular bodies,inverseZeeman effect,opticalphenomena inmovingbodies,etc.109pagesofnotesexplainthemore advanced sections. Index.9figures.352pp.5%x8. S173Paperbound $1.85 FUNDAMENTAL ELECTROMAGNETIC THEORY, RonoldP.King,Professor AppliedPhysics,Harvard University. Original andvaluable introduction toelectromagnetic theoryandtocircuit theoryfromthestandpoint ofelectromagnetic theory.Contents: Mathematical Description ofMatter-stationary andnonstationary states;Mathematical Description ofSpaceandof SimpleMedia-Field Equations, Integral FormsofFieldEquations, Electromagnetic Force, etc'iTransformation ofFieldandForceEquations; Electromagnetic WavesinUnbounded RegIons; SkinEffectandInternal Impedance-in asolidcylindrical conductor, etc.;and Electrical Circuits-Analytical Foundations, Near-zone andquasi-near zonecircuits, Balanced two-wire andfour-wire transmission lines.Revisedandenlarged version. Newprefaceby theauthor.5appendices (Differential operators: VectorFormulas andIdentities, etc.). Problems. Indexes. Bibliography. xvi+580pp.53/8x81/2. 51023Paperbound $2.75 Hydrodynamics ATREATISE ONHYDRODYNAMICS, A.B.Basset.Favoritetextonhydrodynamics for2genera tionsofphysicists, hydrodynamical engineers, oceanographers, shipdesigners, etc.Clear enoughforthebeginning student,andthorough sourceforgraduate students andengineers on th~workofd'Alembert, Euler,Laplace, Lagrange, Poisson, Green,Clebsch, Stokes,Cauchy, Helmholtz, J. J.Thomson, love,Hicks,Greenhill, Besant,Lamb,etc.Greatamountofdocu mentation onentiretheoryofclassical hydrodynamics. VolI:theoryofmotionoffrictionless liquids,vortex,andcyclicirrotational motion,etc.132exercises. Bibliography. 3Appendixes. xii+264pp.VolII:motioninviscousliquids,harmonic analysis, theoryoftides,etc.112 exercises, Bibliography. 4Appendixes. xv+328pp.Twovolumeset.5%x8. 5724VolIPaperbound $1.75 8725VolIIPaperbound $1.75 Theset$3.50 HYDRODYNAMICS, HoraceLamb.Internationally famouscomplete coverage ofstandard refer enceworkondynamics ofliquids&gases.Fundamental theorems, equations, methods, solutions, background, forclassical hydrodynamics. Chapters includeEquations ofMotion, Integration ofEquations inSpecialGases,Irrotational Motion,MotionofLiquidin2Dimen sions,MotionofSolidsthroughLiquid-Dynamical Theory,VortexMotion,TidalWaves,Surface Waves,WavesofExpansion, Viscosity, Rotating Massesofliquids.Excellently planned, ar ranged;clear,lucidpresentation. 6thenlarged, revisededition.Index.Over900footnotes, mostlybibliographical. 119figures.xv+738pp.61/8x91/4. S256Paperbound $3.25 Catalogue ofDoverBooks ENGINEERING ANDTECHNOLOGY Generalandmathematical ENGINEERING MATHEMATICS, Kenneth S.Miller.Atextforgraduate students ofengineering tostrengthen theirmathematical background indifferential equations etc.Mathematical stepsveryexplicitly indicated. Contents: Determinants andMatrices, Integrals, LinearDif feren~ial Equations, FourierSeriesandIntegrals, LaplaceTransform, Network Theory,Random Functl~n :. .allvitalrequisites foradvanced modernengineering studies. Unabridged republicatIOn. Appendices: BorelSets;Riemann-Stieltjes Integral; FourierSeriesandIntegrals. Index.References atChapterEnds.xii+417pp.6 x8lf2. S1121Paperbound $2.00 MATHEMATICAL ENGINEERING ANALYSIS, RufusOldenburger. Abookdesigned toassistthe research engineer andscientist inmakingthetransition fromphysical engineering situations t~thecorresponding mathematics. Scoresofcommon practical situations foundinallmajor fieldsofphysicsaresupplied withtheircorrectmathematical formulations-applications to automobile springsandshockabsorbers, clocks,throttletorqueofdieselengines, resistance n.etworks, capacitors, transmission lines,microphones, neontUbes,gasoline engines, refrigera tioncycles,etc.Eachsectionreviewsbasicprinciples ofunderlying variousfields:mechanics ofrigidbodies,electricity andmagnetism, heat,elasticity, fluidmechanics, andaerodynamics. Comprehensive andeminently useful.Index.169problems, answers. 200photosanddiagrams. xiv+426pp.5%x8lf2. S919Paperbound $2.00 MATHEMATICS OFMODERN ENGINEERING, E.G.KellerandR.E.Doherty. Writtenforthe Advanced CourseinEngineering oftheGeneralElectricCorporation, dealswiththeengineer· inguseofdeterminants, tensors, theHeaviside operational calculus, dyadics, thecalculus ofvariations, etc.Presents underlying principles fully,butpurposeistoteachengineers to dealwithmodernengineering problems, andemphasis isontheperennial engineering attack ofset-upandsolve.Indexes. Over185figuresandtables.Hundreds ofexercises, problems, andworked-out examples. References. Twovolumeset.Totalofxxxiii+623pp.5%x8. 8734VolIPaperbound $1.85 8735VolIIPaperbound $1.85 Theset$3.70 MATHEMATICAL METHODS FORSCIENTISTS ANDENGINEERS, L.P.Smith.Forscientists and engineers, aswellasadvanced mathstudents. Fullinvestigation ofmethods andpractical description ofconditions underwhicheachshouldbeused.Elements ofrealfunctions, differential andintegral calculus, spacegeometry, theoryofresidues, vectorandtensor analysis, seriesofBesselfunctions, etc.Eachmethodillustrated bycompletely-worked-out examples, mostlyfromscientific literature. 368gradedunsolved problems. 100diagrams. x+453pp.55/8x8%. S220Paperbound $2.00 THEORY OFFUNCTIONS ASAPPLIED TOENGINEERING PROBLEMS, editedbyR.Rothe,F.Dilen dorff,andK.pohlhausen. Aseriesoflectures giv~natthe.Berlin Ins~itut~ ofTechnol.ogy ~hat showsthespecificapplications offunction.theory Ine.lectnc~1 andalliedfle.ldsofenglneefl!lg. Sixlecturesprovidetheelements offunction t~eory InaSimpleandprac.tlcal form,covenng complex quantities andvariables, integration Inthecomplexplane,reslguetheor.ems, etc. Then5lecturesshowtheexactusesofthispowerful mathematical tool,Withfulldiscussions ofproblemmethods. Index.Bibliography. 108figures.x+189pp.5%x8. d$1358733Paperboun • Aerodynamics andhydrodynamics AIRPLANE STRUCTURAL ANALYSIS ANDDESIGN, E.E.SechlerandL.G.Dunn.Systematic authoritative bookwhichsummarizes alargeamountoftheoretical andexperimental work onstructural analysis anddesign.Strongonclassical subsonic material stillbasictomuch aeronautic design. . .remains ahighlyusefulsourceofinformation. Coverssuchareas aslayoutoftheairplane, appliedanddesignloads,stress-strain relationships forstable structures, trussandframeanalysis, theproblem ofinstability, theultimate strength of stiffened flatsheet,analysis ofcylindrical structures, wingsandcontrolsurfaces, fuselage analysis, enginemounts, landinggears,etc.Originally published ~spartoftheCALCIT Aeronautical Series.256Illustrations. 47studyproblems. Indexes. Xl+420pp.5%x8lf2. S1043Paperbound $2.25 FUNDAMENTALS OFHYDRO· ANDAEROMECHANICS, L.PrandtlandO.G.Tietjens. Thewell knownstandard workbaseduponPrandtl's lecturesatGoettingen. Wherever possible hydro dynamics theoryisreferred topractic~1 c.onsiderations inhydraulics, wi~h t~evie~of unifyingtheoryandexperience. Presentation ISextremely clearandthoughpnmarlly physical, mathematical proofsarerigorous andusevectoranalysis toaconsiderable extent.An Enginering SocietyMonograph, 1934.186figures.Index.xvi+270pp.5%x8.5374Paperbound $1.85 Catalogue ofDoverBooks FLUIDMECHANICS FORHYDRAULIC ENGINEERS, H.Rouse.Standard workthatgivesacoherent pictureoffluidmechanics fromthepointofviewofthehydraulic engineer. Basedoncourses giventocivilandmechanical engineering students atColumbia andtheCalifornia Institute ofTechnology, thisworkcoverseverybasicprinciple, method, equation, ortheoryof Interesttothehydraulic engineer. Muchofthematerial, diagrams, charts,etc.,inthis self-contained textarenotduplicated elsewhere. Coversirrotational motion,conformal map ping,problems inlaminarmotion,fluidturbulence, flowaroundimmersed bodies,transporta tionofsediment, generalcharcteristics ofwavephenomena, gravitywavesinopenchannels, etc.Index.Appendix ofphysicalproperties ofcommonfluids.Frontispiece+245figuresand photographs. xvi+422pp. 5=M1x8. 5729Paperbound $2.25 WATERHAMMER ANALYSIS, JohnParmakian. Valuable exposition ofthegraphical methodof solvingwaterhammer problems byAssistant ChiefDesigning Engineer, U.S.Bureauof Reclamation. Discussions ofrigidandelasticwatercolumntheory,velocityofwaterhammer waves,theoryofgraphical waterhammer analysis forgateoperation, closings, openings, rapidandslowmovements, etc.,waterhammer inpumpdischarge causedbypowerfailure, waterhammer analysisforcompound pipes,andnumerous relatedproblems. "Withaconcise andlucidstyle,clearprinting, adequate bibliography andgraphsforapproximate solutions attheprojectstage,itfillsavacantplaceinwaterhammer literature," WATERPOWER. 43problems. Bibliography. Index.113illustrations. xiv+161pp. 5=M1x8112. 51061Paperbound $1.65 AERODYNAMIC THEORY: AGENERAL REVIEW OFPROGRESS, WilliamF.Durand,editor-In-chief. Amonumental jointeffortbytheworld'sleadingauthorities prepared underagrantof theGuggenheim FundforthePromotion ofAeronautics. Intended toprovidethestud.ent andaeronautic designer withthetheoretical andexperimental background ofaeronauttcs. Neverequalled forbreadth, depth,reliability. Contains discussions ofspecialmathematical topicsnotusuallytaughtintheengineering ortechnical courses.Also:anextended two-part treatise onFluidMechanics, discussions ofaerodynamics ofperfectfluids,analyses of experiments withwindtunnels,appliedairfoiltheory,thenon-lifting systemoftheairplane, theairpropeller, hydrodynamics ofboatsandfloats,theaerodynamics ofcooling, etc. Contributing expertsincludeMunk,Giacomelli, Prandtl.Toussaint, VonKarman, Klemperer, amongothers.Unabridged republication. 6volumes boundas3.Totalof1,012figures,12 plates.Totalof2,186pp. Bibliographies. Notes.Indices. 5=M1x8. S328-S330 Clothbound, TheSet$17.50 APPLIED HYDRO-ANDAEROMECHANICS, L.PrandtlandO.G.Tletjens. Presents, forthemost part,methods whichwillbevaluable toengineers. Coversflowinpipes,boundary layers, airfoiltheory,entryconditions, turbulent flowinpipes,andtheboundary layer,determining dragfrommeasurements ofpressure andvelocity, etc."Willbewelcomed byallstudents ofaerodynamics," NATURE. Unabridged, unaltered. AnEngineering SocietyMonograph, 1934. Index.226figures,28photographic platesillustrating flowpatterns. xvi+311pp.5%x8. S375Paperbound $1.85 SUPERSONIC AERODYNAMICS, E.R.C.Miles.Valuable theoretical introduction tothesuper sonicdomain,withemphasis onmathematical toolsandprinciples, forpracticing aerody namicists andadvanced students inaeronautical engineering. Coversfundamental theory, divergence theorem andprinciples ofcirCUlation, compressible flowandHelmholtz laws,the Prandtl-Busemann graphicmethodfor2·dimensional flow,obliqueshockwaves,theTaylor Maccollmethodforconesinsupersonic flow,theChaplygin methodfor2·dimensional flow,etc. Problems rangefrompractical engineering problems todevelopment oftheoretical results. "Rendered outstanding bytheunprecedented scopeofitscontents..•hasundoubtedly filled aVitalgap,"AERONAUTICAL ENGINEERING REVIEW. Index.173problems, answers. 106dia grams.7tables.xii+255pp.5%x8. S214Paperbound $1.45 HYDRAULIC TRANSIENTS, G.R.Rich.Thebesttextinhydraulics everprintedinEnglish••• byone.o!America's foremost engineers (formerChiefDesignEngineer forT.V.A.).Provides a .transl.tlo~ from.thebasicd!fferentia! equations ~fhydra~lic transient theorytothe anthmetlc Intergratlon computatiOn reqUired bypracticing engineers. Sections coverWater Hammer, TurbineSpeedRegulation, Stability ofGoverning, Water-Hammer Pressures inPump Discharge Lines,TheDifferential andRestricted OrificeSurgeTanks,TheNormalized Surge Tank Ch~rtsofCalameandGaden,NaVigation Locks,SurgesinPowerCanals-Tidal Harmonics, etc.ReVisedandenlarged. Author's prefaces. Index.XIV+409pp.53/8x8112. S116Paperbound $2.50 HYDRAULICS ANDITSAPPLICATIONS, A.H.Gibson.Excellent comprehensive textbook forthe ~tu~entandthorough practical manualfortheprofessional worker,aworkofgreatstature InItsarea.HalfthebookISdevotedtotheoryandhalftoapplications andpractical prob lemsn:'etinthefi~ld.Coversmodes ~fmotion ~fafluid,criticalvelocity, viscousflow,eddy formation, Bernoulli'S theorem, flowInconverging passages, vortexmotion,formofeffluent streams, n.otchesandweirs,skinfriction, lossesatvalvesandelbows,siphons, erosionof channels, Jetpropulsion. wavesofoscillation, andover100similartopics.Finalchapters (nearly400pages)covermorethan100kindsofhydraulic machinery: PeltonWheel,speed regulators, thehydraulic ram,surgetanks,thescoopwheel,theVenturimeter,etc.A specialch<.:ptertreatsmethods oftestingtheoretical hypotheses: scalemodelsofrivers tidal.estuaries, siphonspillways, etc.5threvisedandenlarged (1952)edition.Index. Ap~ pendIX.427photographs anddiagrams. 95examples, answers. xv+813pp.6 x9. 5791Clothbound $8.00 Catalogue ofDoverBooks FLUIDMECHANICS THROUGH WORKEG EXAMPLES, D.R.L.SmithandJ.Houghton. Advanced text.covering principles andapplic~tions topractical situations. Eachchapterbeginswith con~lses.ummanes offundamental Ideas. 16~fullyworkedoutexamples applying principles outimed 10thetext.275otherproblems, withanswers. Contents; ThePressure ofliquids onSurface~; Floati~g Bodie~;FlowUnder Con~tan.t HeadinPipes;Circulation; Vorticity; The ~otentlal Function; Lammar FlowandLubncatlon; ImpactotJets;Hydraulic Turbines; Centnfugal andReciprocating Pumps;Compressible Fluids;andmanyotheritems.Total of438examples. 250lineillustrations. 340PP.Index.6 x8~&. 5981Clothbound $6.00 THEORY OFSHIPMOTIONS, S.N.Blagoveshchensky. Theonlydetailed textinEnglishin arapidlydeveloping branchofengineering andphysics,itisthe·workofoneofthe world's foremost authorities-Blagoveshchensky ofLeningrad Shipbuilding Institute. A senior-level treatment writtenprimarily forengineering students, butalsoofgreatimportance tonavalarchitects, designers, contractors, researchers inhydrodynamics, andotherstudents. Nomathematics beyondordinary differential equations isrequired forunderstanding the text.Translated byT.&L.Strelkoff, undereditorship ofLouisLandweber, IowaInstitute ofHydraulic Research, underauspices ofOfficeofNavalResearch. Bibliography. Index. 231diagrams andillustrations. Totalof649pp.5%x81/2.Vol.I:S234Paperbound $2.00 Vol.II:S235Paperbound $2.00 THEORY OFFLIGHT,RichardvonMises.Remainsalmostunsurpassed asbalanced, well-written accountoffundamental fluiddynamics, andsituations inwhichaircompressibility effects areunimportant. Stressing equallytheoryandpractice, avoiding formidable mathematical structure, itconveysafullunderstanding ofphysical phenomena andmathematical concepts. Contains perhapsthebestintroduction togeneraltheoryofstability. "Outstanding," Scientific, Medical, andTechnical Books.Newintroduction byK.H.Hohenemser. Bibliographical, histor icalnotes.Index.408illustrations. xvi+620pp. 5=0/'&x8%. S541'Paperbound $2.95 THEORY OFWINGSECTIONS, I.H.Abbott,A.E.VOnDoenhoff. Concisecompilation ofsubsonic aerodynamic characteristics ofmodernNASAwingsections, withdescription oftheirgeom etry,associated theory.Primarily reference workforengineers, stUdents, itgivesmethods, dataforusingwing-section datatopredictcharacteristics. Particularly valuable: chapters on thinwings,airfoils; complete summary ofNACA'sexperimental observations, systemof construction families ofairfoils. 350PP.oftablesonBasicThickness Forms,MeanLines, AirfoilOrdinates, Aerodynamic Characteristics ofWingSections. Index.Bibliography. 191 illustrations. Appendix. 705pp. 5=0/'&x8. S558Paperbound $3.25~ WEIGHT-STRENGTH ANALYSIS OFAIRCRAFT STRUCTURES, F.R.Shanley. Scientifically sound methods ofanalyzing andpredicting thestructural weightofaircraftandmissiles. Deals directlywithforcesandthedistances overwhichtheymustbetransmitted, makingitpossible todevelopmethods bywhichtheminimum structural wei~htcanbedetermined forany material andconditions ofloading.Weightequations forwingandfuselage structures. In cludesauthor'soriginalpapersoninelastic buckling andcreepbuckling. "Particularly success fulinpresenting hisanalytical methodsforinvestigating variousoptimum designprinciples," AERONAUTICAL ENGINEERING REVIEW.Enlarged bibliography. Index.199figures.xiv+404pp. 5%x8%. S660Paperbound $2.45 Electricity TWO-DIMENSIONAL FIELDSINELECTRICAL ENGINEERING, L.V.Bewley.Ausefulselection of typicalengineering problems ofinteresttopracticing electrical engineers. Introduces senior students tothemethods andprocedures ofmathematical physics. Discusses theoryof functions ofacomplexvariable, two-dimensional fieldsofflow,generaltheorems ofmathe maticalphysicsandtheirapplications, conformal mapping ortransformation, methodof images,freehand fluxplotting, etc.Newprefacebytheauthor.Appendix byW.F.Kiltner. Index.Bibliography atchapterends.xiv+204pp.5%x81/2. S1118Paperbound $1.50 FLUXLINKAGES ANDELECTROMAGNETIC INDUCTION, L.V.Bewley. Abrief,clearbook whichshowsproperusesandcorrects misconceptions ofFaraday's lawofelectromagnetic induction inspecificproblems. Contents: Circuits, Turns,andFluxlinkages; Substitution of Circuits; Electromagnetic Induction; GeneralCriteriaforElectromagnetic Induction; Appli cationsandParadoxes; Theorem ofConstant FluxLinkages. NewSection:' Rectangular Coli inaVaryingUniformMedium. Valuable supplement toclasstextsforengineering students. Corrected, enlarged edition. Newpreface. Bibliography innotes.49figures, xi+l06pp. 5%x8. S1103Paperbound $1.25 INDUCTANCE CALCULATIONS: WORKING FORMULAS ANDTABLES, Frederick W.Grover. An invaluable booktoeveryone inelectrical engineering. Provides simplesingleformulas to coverallthemoreimportant casesofinductance. Theapproach involves onlythosepara metersthatnaturally enterintoeachsituation, whileextensive tablesaregiventopermit easyinterpolations. Willsavetheengineer andstudentcountless hoursandenablethem toobtainaccurate answers withminimal effort.Corrected republication of1946edition. 58tables.97completely workedoutexamples. 66figures,xiv+286pp.5%x81/2. S974Paperbound $1.85 Catalogue ofDoverBooks GASEOUS CONDUCTORS: THEORY ANDENGINEERING APPLICATIONS, J.D.Cobine. Anindis pensable textandreference togaseous c~nd~ction phenomena, wi!h.the. engine~rir:tg view pointprevailing throughout. Studiesthekinetictheoryofgases,Ionization, emISSion phe nomena; gasbreakdown, sparkcharacteristics, glow,anddischarges; en~ineering applica tionsincircuitinterrupters, rectifiers, lightsources, etc.Separate detaIled treatment of highpressure arcs(Suits);lowpressure arcs(Langmuir andTonks).Muchmore."Well organized, clear,straightforward," Tonks,ReviewofScientific Instruments. Index.Bibliog raphy.83practice problems. 7appendices. Over600figures. 58tables.xx+606pp. 5%x8. S442Paperbound $2.95 INTRODUCTION TOTHESTATISTICAL DYNAMICS OFAUTOMATIC CONTROL SYSTEMS, V. V.Solo dovnikov. firstEnglishpublication oftext·reference covering important branchofautomatic controlsystems-random signals; initsoriginaledition,thiswasthefirstcomprehensive treatment. Examines frequency characteristics, transfer functions, stationary randomproc esses,determination ofminimum mean,sQuared error,oftransferfunction forafiniteperiod ofobservation, muchmore.Translation editedbyJ.B.Thomas, L.A.Zadeh.Index.Bibliog raphy.Appendix. xxii+308pp.53/8x8. S420Paperbound $2.25 TENSORS FORCIRCUITS, GabrielKron.Aboldlyoriginalmethodofanalyzing engineering prob lems,atcenterofsharpdiscussion sincefirstintroduced, nowdefinitely provedusefulin suchareasaselectrical andstructural networks onautomatic computers. Encompasses a greatvarietyofspecificproblems bymeansofarelatively fewsymbolic equations. "Power andflexibility...becoming morewidelyrecognized," Nature.Formerly "AShortCourse inTensorAnalysis." Newintroduction byB.Hoffmann. Index.Over800diagrams. xix+ 250pp.53/8x8. S534Paperbound $2.00 SELECTED PAPERS ONSEMICONDUCTOR MICROWAVE ELECTRONICS, editedbySumnerN.Levine andRichard R.KurzrOk. Aninvaluable collection ofimportant papersdealingwithoneof themostremarkable devolopments insolid-state electronics-the useofthep-njunction toachieve amplification andfrequency conversion ofmicrowave frequencies. Contents: GeneralSurvey(3introductory papersbyW.E.Danielson, R.N.Hall,andM.Tenzer); Gen eralTheoryofNonlinear Elements (3articlesbyA.vanderZiel,H.E.Rowe,andManley andRowe);DeviceFabrication andCharacterization (3piecesbyBakanowski, Cranna, and Uhlir,byMcCotter, WalkerandFortini,andbyS.T.Eng);Parametric Amplifiers andFre QuencyMultipliers (13articles byUhlir,HeffnerandWade,Matthaei, P.K.Tien,vander Ziel,Engelbrecht, CurrieandGould,Uenohara, LeesonandWeinreb, andothers);andTunnel Diodes(4papersbyL.[saki,H.S.Sommers, Jr.,M.l::.Hines,andYarivandCook).Intro duction. 295Figures. xiii+286pp.61J2x 91/4. S1126Paperbound $2.25 THEPRINCIPLES OFELECTROMAGNETISM APPLIED TOELECTRICAL MACHINES, B.Hague.A concise, butcomplete, summary ofthebasicprinciples ofthemagnetic fieldanditsappli cations,withparticular reference tothekindofphenomena whichoccurinelectrical ma chines.PartI:GeneralTheory-magnetic fieldofacurrent, electromagnetic fieldpassing fromairtoiron,mechanical forcesonlinearconductors, etc.PartII:Application oftheory tothesolution ofelectromechanical problems-the magnetic fieldandmechanical forces innon·salient polemachinery, thefieldwithinslotsandbetween salientpoles,andthe workofRogowski, Roth,andStrutt.Formery titled"Electromagnetic Problr:ms inElectrical Engineering." 2appendices. Index.Bibliography innotes.115figures.xiv+359pp.5%x81J2. S246Paperbound $2.25 Mechanical engineering DESIGN ANDUSEOFINSTRUMENTS ANDACCURATE MECHANISM, T.N.Whitehead. Forthe instrument designer, engineer; howtocombine necessary mathematical abstractions with Independent observation ofactualfacts.Partialcontents: instruments &theirparts,theory oferrors,systematic errors,probability, shortperioderrors,erraticerrors,designprecision, kinematic, semikinematic design,stiffness, planning ofaninstrument, humanfactor,etc. Index.85Photos,diagrams. xii+288pp.53/8x8. S270Paperbound $2.00 ATREATISE ONGYROSTATICS ANDROTATIONAL MOTION: THEORY ANDAPPLICATIONS, Andrew Gray.Mostdetailed, thorough bookinEnglish, generally considered definitive study.Many problems ofallsortsinfulldetail,orstep-by-step summary. Classical problems ofBour Lottner, etc.;lateronesofgreatphysical interest. Vibrating systems ofgyrostats earth asatop,calculation ofpathofaxisofatopbyellipticintegrals, motionofunsymmetrical top,muchmore.Index.160illus.550pp.53/8x8. S589Paperbound $2.75 MECHANICS OFTHEGYROSCOPE, THEDYNAMICS OFROTATION, R.F.Deimel,Professor of Mechanical Engineering atStevens Institute ofTechnology. Elementary generaltreatment ofdynamics ofrotation, withspecialapplicationofgyroscopic phenomena. Noknowledge of~ectorsneeded.Velocityofamovingcurve,acceleration toapoint,generalequations of mO~lon, gyro~copic h~rizon, freegyro,motionofdiscs,thedamped gyro,103similar tOPICS.ExerCises. 75figures.208pp. 5318x8. 566Paperbound $1.65 Catalogue ofDoverBooks STR~NGTHofMATERIALS, J.P.DenHartog.Distinguished textprepared forM.I.T.course,ideal asIn!roductlO!1, refre~her, refe.rence, orself-study text.Fullcleartreatment ofelementary matenal (tenslO~, torsion, ~end!ng, compound stresses, deflection ofbeams,etc.),plusmuch advanced matenal onenglneenng methods ofgreatpractical value:fulltreatment ofthe ~ohrcircle,lucidelementary discussions ofthetheoryofthecenterofshearandthe"Myoso tiS"!ll~thodofcalculating. beamdeflections, rei~forced concrete, plasticdeformations, photo elasticIty, etc.Inallsections, .bothgeneralpr.lnclples andconcrete applications aregiven. Index.186.~gures(160othersInproblemsection). 350problems, allwithanswers. Listof formulas. VIII+323pp.5%x8. S755Paperbound $2.00 PHOTOELASTICITY: PRINCIPLES ANDMETHODS, H.T.Jessop,F.C.Harris.Fortheengineer, forspecificproblems ofstressanalysis. Latesttime-saving methods ofchecking calcula tionsin2-dimensional designproblems, newtechniques forstresses in3dimensions, and luciddescription ofopticalsystems usedinpractical photoelasticlty. Usefulsuggestions andhintsbasedonon-the-job experience included. Partialcontents: strained andstress strainrelations, circulardiscunderthrustalongdiameter, rectangular blockwithsquare holeunderverticalthrust,simplysupported rectangular beamundercentralconcentrated load,etc.Theoryheldtominimum, noadvanced mathematical trainingneeded.Index.164 illustrations. viii+184pp.6¥ax9¥4. S720Paperbound $2.00 APPLIED ELASTICITY,. J.Prescott. Provides theengineer withthetheoryofelasticity usually lackinginbooksonstrength ofmaterials, yetconcentrates onthoseportions usefulfor immediate application. Develops everyimportant typeofelasticity problemfromtheoretical principles. Coversanalysisofstress,relations betweenstressandstrain,theempirical basis ofelasticity, thinrodsundertensionorthrust,SaintVenant's theory,transverse oscillations ofthinrods,stability ofthinplates,cylinders withthinwalls,vibrations ofrotatingdisks, elasticbodiesincontact, etc."Excellent andimportant contribution totheSUbject, not merelyintheoldmatterwhichhehaspresented innewandrefreshing form,butalsointhe manyoriginalinvestigations herepublished forthefirsttime,"NATURE. Index.3Appendixes. vi+672pp.5%x8. S726Paperbound $2.95 APPLIED MECHANICS FORENGINEERS, SirCharlesInglis,F.R.S.Arepresentative surveyof themanyandvariedengineering questions whichcanbeanswered bystaticsanddynamics. Theauthor,oneoffirstandforemost adherents of"structural dynamics," presents distinc tiveillustrative examples andclear,concisestatement ofprinciples-directing thedis cussionatmethodology andspeCificproblems. Coversfundamental principles ofrigid-body statics,graphicsolutions ofstaticproblems, theoryoftautwires,stresses inframeworks, particledynamics, kinematics, simpleharmonic motionandharmonic analysis, two-dimen sionalrigiddynamics, etc.437illustrations. xii+404pp.5¥ax8¥2.S1119Paperbound $2.00 THEORY OFMACHINES THROUGH WORKED EXAMPLES, G.H.Ryder.Practical mechanical engineering textbook forgraduates andadvanced undergraduates, aswellasagoodrefer enceworkforpracticing engineers. Partialcontents: Mechanisms, Velocity andAccelera tion(including discussion ofKlein'sConstruction forPistonAcceleration), Cams,Geometry ofGears,Clutches andBearings, BeltandRopeDrives,Brakes,InertiaForcesandCouples, GeneralDynamical Problems, Gyroscopes, LinearandAngularVibrations, Torsional Vibrations, Transverse Vibrations andWhirling Speeds(Chapters onvibrations considerably enlarged fromprevious editions). Over300problems, manyfullyworkedout. In~ex.195lineillus trations. Revisedandenlarged edition.viii+280pp.50/ax8¥4.S980Clothbound $5.00 THEKINEMATICS OFMACHINERY: OUTLINES OFATHEORY OFMACHINES, FranzReuleaux. Theclassicworkinthekinematics ofmachinery. Thepresentthinking aboutthesubject hasallbeenshapedingreatmeasure bythefundamental principles statedherebyReuleaux almost90yearsago.Whilesomedetailshavenaturally beensuperseded, hisbasicviewpoint hasendured; hence,thebookisstillanexcellent textforbasiccoursesInkinematics and astandard reference workforactiveworkersinthefield.Coverssuchtopicsas:thenature ofthemachine problem, phoronomic propositions, pairsofelements, incomplete kinematic chains,kinematic notation andanalysis, analyses ofchamber-crank trains,chamber-wheel trains,constructive elements ofmachinery, complete machines, etc.,withmainfocuson controlled movement inmechanisms. Unabridged republication oforiginaledition,translated byAlexander B.Kennedy. Newintroduction forthiseditionbyE.S.Ferguson. Index.451 illustrations. xxiv+622pp.5¥ax81/2. S1124Paperbound $3.00 ANALYTICAL MECHANICS OFGEARS,EarleBuckingham. Provides asolidfoundation upon whichlogicaldesignpractices anddesigndatacanbeconstructed. Originally arisingout ofinvestigations oftheASMESpecialResearch Committee onWormGearsandtheStrength ofGears,thebookcoversconjugate gear-tooth action,thenatureofthecontact,andresult inggear-tooth profilesof:spur,internal, helical,spiral,worm,bevel,andhypoidorskew bevelgears.Also:frictional heatofoperation anditsdissipation, frictionlosses,etc., dynamic loadsinoperation, andrelatedmatters. Familiarity withthisbookisstillregarded asanecessary prerequisite toworkinmoderngearmanufacturing. 263figures.103tables. Index.x+546pp.5%x8¥2. S1073Paperbound $2.75 Catalogue ofDoverBooks Opticaldesign,lighting THESCIENTIFIC BASISOFILLUMINATING ENGINEERING, ParryMoon,Professor ofElectrical Engineering, M.I.T.Basic,comprehensive study.Complete coverage ofthefundamental theoretical principles together withtheelements ofdesign,vision,andcolorwithwhich thelightingengineer mustbefamiliar. Valuable asatextaswellasareference source tothepracticing engineer. Partialcontents: Spectroradiometric Curve,Luminous Flux, Radiation fromGaseous-Conduction Sources, Radiation fromIncandescent Sources, Incandes centLamps,Measurement ofLight,Illumination fromPointSourcesandSurfaceSources, Elements ofLighting Design.7Appendices. Unabridged andcorrected republication, with additions. Newprefacecontaining conversion tablesofradiometric andphotometric con cepts.Index.707-item bibliography. 92-itembibliography ofauthor's articles. 183problems. xxiii+608pp.5%x8¥2. S242Paperbound $2.85 OPTICS ANDOPTICAL INSTRUMENTS: ANINTRODUCTION WITHSPECIAL REFERENCE TO PRACTICAL APPLICATIONS, B.K.Johnson. Aninvaluable guidetobasicpractical applications ofopticalprinciples, whichshowshowtosetupinexpensive workingmodelsofeachofthe fourmaintypesofopticalinstruments-telescopes, microscopes, photographic lenses,optical projecting systems. Explains indetailthemostimportant experiments fordetermining their accuracy, resolving power,angularfield.ofview,amounts ofaberration, allothernecessary factsabouttheinstruments. formerly "Practical Optics." Index.234diagrams. Appendix. 224pp.5%x8. S642Paperbound $1.65 APPLIED OPTICSANDOPTICAL DESIGN, A.E.Conrady. Withpublication ofvol.2,standard workfordesigners inopticsisnowcomplete forfirsttime.OnlyworkofitskindinEnglish; onlydetailed workforpractical designer andself-taught. Requires, forbulkofwork,no mathabovetrig.Step-by-step exposition, fromfundamental concepts ofgeometrical, physical optics,tosystematic study,design,ofalmostalltypesofopticalsystems. Vol.1:allordi naryray-tracing methods; primaryaberrations; necessary higheraberration fordesignof telescopes, low-power microscopes, photographic equipment. Vol.2:(Completed fromauthor's notesbyR.Kingslake, Dir.OpticalDesign,Eastman Kodak.)Specialattention tohigh-power microscope, anastigmatic photographic objectives. "Anindispensable work,"J.,OpticalSoc. ofAmer."Asapractical guidethisbookhasnorival,"Transactions, OpticalSoc.Index. Bibliography. 193diagrams. 852pp.61fax9¥4. Vol.1S366Paperbound $2.95 Vol.2S612Paperbound $2.95 Miscellaneous THEMEASUREMENT OFPOWERSPECTRA FROMTHEPOINTOFVIEWOFCOMMUNICATIONS ENGINEERING, R.B.Blackman, J.W.Tukey.Thispathfinding work,reprinted fromthe"Bell SystemTechnical Journal," explains variouswaysofgettingpractically usefulanswers in themeasurement Dfpowerspectra, usingresultsfrombothtransmission theoryandthe theoryofstatistical estimation. Treats:Autocovariance Functions andPowerSpectra; Direct AnalogComputation; Distortion, Noise,Heterodyne Filtering andPre-whitening; Aliasing; Rejection Filtering andSeparation; Smoothing andDecimation Procedures; VeryLowFre quencies; Transversal Filtering; muchmore.Anappendix reviewsfundamental Fouriertech niques.Indexofnotation. Glossary ofterms.24figures.XIItables.Bibliography. General index.192pp.5%x8. S507Paperbound $1.85 CALCULUS REFRESHER FORTECHNICAL MEN,A.AlbertKlaf.ThisbookisuniqueinEnglish asarefresher forengineers, technicians, students whoeitherwishtobrushuptheir calculus ortoclearupuncertainties. Itisnotanordinary text,butanexamination of mostimportant aspectsofintegralanddifferential calculus 10termsofthe756questions mostlikelytooccurtothetechnical reader.Thefirstpartofthisbookcoverssimplediffer entialcalculus, withconstants, variables, functions, increments, derivatives, differentiation, logarithms, curvature ofcurves,andsimilartopics.Thesecondpartcoversfundamental ideasofintegration, inspection, substitution, transformation, reduction, areasandvolumes, meanvalue,successive andpartialintegration, doubleandtripleintegration. Practical aspectsarestressed ratherthantheoretical. A50-pagesectionillustrates theapplication ofcalculus tospecificproblems ofciVilandnautical engineering, electricity, stressand strain,elasticity, industrial engineering, andsimilarfields.-756 questions answered. 566 problems, mostlyanswered. 36pagesofusefulconstants, formulae forreadyreference. Index.v+431pp.5:}8x8. T370Paperbound $2.00 METHODS INEXTERIOR BALLISTICS, ForestRayMOUlton. Probably thebestintroduction to themathematics ofprojectile motion.Theballistics theories propounded werecoordinated withextensive provinggroundandwindtunnelexperiments conducted bytheauthorand othersfortheU:S.Army.Broadinsc~peandclearinexposition, itgivesthebeginnings ofthetheoryusedformodern-day proJectile, long-range missile, andsatellite motion.Six maindivisions: Differential Equations ofTranslatory Motionofaprojectile; GraVityandthe Resistance function; Numerical Solution ofDifferential Equations; TheoryofDifferential Variations; ValidityofMethodofNumerical Integration; andMotionofaRotating Projectile. Formerly titled:"NewMethods inExteriorBallistics." Index.38diagrams. viii+259pp. 5%x8¥2. S232Paperbound $1.75 Catalogue ofDoverBooks LOUDSPEAKERS: THEORY, PERFORMANCE, TESTING ANDDESIGN, N.W.McLachlan. Mostcom. prehensive coverage oftheory,practice ofloudspeakerdesign,testing;classicreference studymanualinfield.First12chapters dealwiththeory,forreadersmainlyconcerned with math.aspects; last7chapters willinterestreaderconcerned withtesting,design.Partial contents: principles ofsoundpropagation, fluidpressure onvibrators, theoryofmoving coilprinciple, transients, drivingmechanisms, response curves,designofhorntypemoving colispeakers, electrostatic speakers, muchmore.Appendix. Bibliography. Index.165illustra tions,charts.411pp.53/8x8. S588Paperbound $2.25 MICROWAVE TRANSMISSION, J.C.Slater.Firsttextdealingexclusively withmicrowaves, bringstogether pointsofviewoffield,circuittheory,forgraduate student inphysics, electrical engineering, microwave technician. Offersvaluable pointofviewnotinmost laterstudies. UsesMaxwell's equations tostudyelectromagnetic field,important inthis area.Partialcontents: infinitelinewithdistributed parameters, impedance ofterminated line,planewaves,reflections, waveguides,coaxialline,composite transmission lines, impedance matching, etc.Introduction. Index.76illus.319pp.5%x8. S564Paperbound $1.50 MICROWAVE TRANSMISSION DESIGN DATA,T.Moreno. Originally classified, nowrewritten an~enla~ged (14newchapters) forpUbli~ rele~seunderauspices ofSperryCorp.Material ofImmediate valueorreference usetoradiOengineers, systemsdesigners, appliedphysicists, etc.Ordinary transmission linetheory;attenuation; capacity; parameters ofcoaxialIinesj hi~hermodes;flexiblecables;obstacles, discontinuities, andinjunctions; tunable wave gUideimpedance transformers; effectsoftemperature andhumiditYj muchmore."Enough theoretical discussion isincluded toallowuseofdatawithout previous background," Electronics.. 324circuitdiagrams, figures,etc.Tablesofdielectrics, flexiblecable,etc., data.Index.ix+248pp. 5318x8. S459Paperbound $1.65 RAYLEIGH'S PRINCIPLE ANDITSAPPLICATIONS TOENGINEERING, G.Temple&W.Bickley. Rayleigh's principle developed toprovideupperandlowerestimates oftruevalueoffunda mentalperiodofaVibrating system,orcondition ofstability ofelasticsystems. Illustrative examplesj rigorous proofsinspecialchapters. Partialcontents: Energymethodofdiscussing Vibrations, stability. Perturbation theory,whirling ofuniformshafts.Criteriaofelasticsta bllity.Application ofenergymethod. Vibrating systems. Proof,accuracy, successive approxi mations, application ofRayleigh's principle. Synthettt: theorems. Numerical, graphical methods. Equilibrium configurations, Ritz'smethod.Bibliography. Index.22figures.ix+156pp.5%x8. 5307Paperbound $1.50 ELASTICITY, PLASTICITY ANDSTRUCTURE OFMATTER, R.Houwink. Standard treatise on rheological aspectsofdifferent technically important solidssuchascrystals, resins,textiles, rubber,clay,manyothers.Investigates generallawsfordeforn~ationsj determines. divergences fromtheselawsforcertainsubstances. CoversgeneralphYSical andma'thematlcal aspects ofplasticity, elasticity, viscosity. Detail~d exam!nation of.deformati?ns, internal struct,,!re ofmatterinrelationtoelasticandplastiCbehaVIOr, formation ofsolidmatterfromaflUid, conditions forelasticandplasticbehavior ofmatter.Treatsglass,.asphalt, gutta pe.r~ha, balata,proteins, baker'sdough,lacquers, sulphur, others.2ndrevised, enlarged e.dlllon. Extensive revisedbibliography inover500footnotes. Index.Tableofsymbols. 214figures. xviii+368pp.6 x91/4. S385Paperbound $2.45 THESCHWARZ-CHRISTOFFEL TRANSFORMATION ANDITSAPPLICATIONS: ASIMPLEEXPOSITION, MilesWalker. Animportant bookforengineers showing how.this val~able toolcanbeem ployedinpractical situations. Verycareful,clear presenta~lOn coverln~ numerous ~oncrete engineering problems. Includes athorough accountofconjugate functIOns fore~glneers usefulforthebeginner andforreview. A.ppIiC'ltio~s tosuch.problems as:Stream-lines round acorner,electricconductor inair-gap,dynamoslot.s,magnetized poles,muchml:!re.Formerly "Conjugate Functions forEngineers." Preface. 92figures,severaltables.Index. IX+116pp. 5318x~1/2. S1149Paperbound $1.25 THELAWSOFTHOUGHT, GeorgeBoole.Thisbookfoundedsymbolic logicsomehundredyears ago.Itis.the1stsignificant attempttoapply logi~toall~spects.of.hU!Y1an ende~v~ur. Partialcontents: derivation oflaws,signs&laws,interpretations, eliminations, conditIOns ofaperfectmethod, analysis, Aristotelian logic,probability, andsimilartopics.xviii+ 424pp.53/8x8. S28Paperbound $2.00 SCIENCE ANDMETHOD, HenriPoincare. Procedure ofscientific discovery, methodology, experi ment,idea-germination-the intellectual processes bywhichdiscoveries comeintobeing. Mostsignificant andmostinteresting aspectsofdevelopment, application ofideas.Chapters coverselection offacts,chance,mathematical reasoning, mathematics, andlogic;Whitehead, Russell,Cantor;thenewmechanics, etc.288pp.5318x8. S222Paperbound $1.35 FAMOUS BRIDGES OFTHEWORLD, D.B.Steinman. Anup-to-the-minute revisededitionofa bookthatexplainsthefascinating drama.ofho~thew~rld's grea~bridgescameto~ebuilt. Theauthordesigner ofthefamedMackinac bridge,discusses bridges ~romallperiodsand allpartsoftheworldexplaining theirvarioustypesofconstruction, anddescribing the problems theirbuilders'faced.Although primarily foryoungsters, thiscannotfailtointerest readersofallages.48illustrations inthetext.23photographs. 99pp.61/8x91/4. TI61Paperbound $1.00 Catalogue ofDoverBooks Technological, historical ADIDEROT PICTORIAL ENCYCLOPEDIA OFTRADES ANDINDUSTRY, Manufacturing andthe Technical ArtsInPlatesSelected from"L'Encyclopedie ouDictlonnaire Raisonne desSciences, desArts,etdesMetiers" ofDenisDiderot.EditedwithtextbyC.Gillispie. Thisfirstmodern selection ofplatesfromthehighpointof18thcenturyFrenchengraving isastorehouse ofvaluable technological information tothehistorian ofartsandscience. Over2000 illustrations on485fUll-page plates,mostofthemoriginalsize,showthetradesand industries ofafascinating erainsuchgreatdetailthattheprocesses andshopsmight verywellbereconstructed fromthem.Theplatesteemwithlife,withmen,women,and childrenperforming allofthethousands ofoperations necessary tothetradesbeforeand duringtheearlys'tagesoftheindustrial revolution. Platesareinsequence, andshow generaloperations, closeups ofdifficultoperations, anddetailsofcomplex machinery. Such important andinteresting tradesandindustries areillustrated assowing,harvesting, bee keeping, cheesemaking, operating windmills, millingflour,charcoal burning, tobaccoprocess ing,indigo,fishing,artsofwar,saltextraction, mining,smelting, castingiron,steel, extracting mercury, zinc,SUlphur, copper, etc.,slating, tinning, silverplating, gilding, makinggunpowder, cannons, bells,shoeinghorses,tanning, papermaking, printing, dyeing, andmorethan40othercategories. Professor Gillispie, ofPrinceton, supplies afullcom mentary onalltheplates,identifying operations, tools,processes, etc.Thismaterial, pre sentedinalivelyandlucidfashion,isofgreatinteresttothereaderinterested inhistory ofscienceandtechnology. Heavylibrarycloth.920pp.9 x12.T421Twovolumeset$18.50 CHARLES BABBAGE ANDHISCALCULATING ENGINES, editedbyP.Morrison andE.Morrison. Babbage, leading19thcenturypioneerinmathematical machines andheraldofmodern operatiohal research, wasthetruefatherofHarvard's relaycomputer MarkI.HisDifference EngineandAnalytical Enginewerethefirstmachines inthefield.Thisvolumecontains a valuable introduction onhislifeandwork;majorexcerpts fromhisautobiography, revealing hiseccentric andunusualpersonality; andextensive selections from"Babbage's Calculating Engines," acompilation ofhard-to-find journalarticlesbyBabbage, theCountess ofLovelace,l.F.Menabrea, andDionysius Lardner. 8illustrations, Appendix ofmiscellaneous papers. Index.Bibliography. xxxviii+400pp.5%x8. Tl2Paperbound $2.00 HISTORY OFHYDRAULICS, HunterRouseandSimonInce.Firsthistoryofhydraulics andhydro dynamics available inEnglish.Presented inreadable, non-mathematical form,thetextismade especially easytofollowbythemanysupplementary photographs, diagrams, drawings, etc. Coversthegreatdiscoveries anddevelopments fromArchimedes andGalileotomoderngiants vonMises,Prandtl,vonKarman,etc.Interesting browsing forthespecialist; excellent intro ductionforteachers andstudents. Discusses suchmilestones asthetwo-piston pumpof Ctesibius, theaqueducts ofFrontius, theanticipations ofdaVinci,Stevinandthefirstbook onhydrodynamics, experimental hydraulics ofthe18thcentury,the19th-century expansion of practical hydraulics andclassical andappliedhydrodynamics, theriseoffluidmechanics In ourtime.etc.200illustrations. Bibliographies. Index.xii+270pp. 5=¥'4x8. 51131Paperbound $2.00 BRIDGES ANDTHEIRBUILDERS, DavidSteinman andSaraRuthWatson.Engineers, historians, everyone whohaseverbeenfascinated bygreatspanswillfindthisbookanendless sourceofinformation andinterest. Dr.Steinman, recipient oftheLouisLevymedal,was oneofthegreatbridgearchitects andengineers ofalltime,andhisanalysisofthegreat bridgesofhistoryisbothauthoritative andeasilyfollowed. GreekandRomanbridges, medieval bridges, Oriental bridges, modernworkssuchastheBrooklyn Bridgeandthe GoldenGateBridge,andmanyothersaredescribed intermsofhistory,constructional prin ciples,artistry, andfunction. Allinallthisbookisthemostcomprehensive andaccurate semipopular historyofbridgesinprintinEnglish. New,greatlyrevised,enlarged edition. 23photographs, 26linedrawings. Index.xvii+401pp.53/sx8.T431Paperbound $2.00 Pricessubjecttochangewithout notice. 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Doverpublishes over125newbooks andrecordseachyouronsuchfieldsasmathematics. physics, explaining science, art,langullges, philosophy, classical records, andothers. ~1• Transmission-line Theory BYRONOLD w.P.KING Professor ofAppliedPhysics,HarvardUnivtrsfty Within thelastd("C;ldc. c:Iectrical science hallbecome theprovince not10 muchofrheph,·sids!. butoftheengineer. forthexpeople.aknowlnlge of c:l«tromagnctic theoryis.of(ourlll:. c~ntial. Itisthecontention ofProfl'S8Or King,awcll-knowll authority whohutauglHatHar\'ard formanyyeanand hasmadeaignificam contribllliOllS 10applied plll"sics, thatoneofthebe5t appl"Oilches toelectromagnetic theoryforlhe';engina:r ilthrough the(Om"en· (KlnaltTansmiJlion line.for.inthisWI)',itiJpossible tointroduce fun<b ~ntalconapu without btowmillg in,'Oh'ed in111the:COI1Iplic.alions of'~IOT linetheory. ni,book.nowrqJrinto:d inacorrttted «lilianwithanewindexofs)'1llbols, isthe$landard Englillh-Ianguage rdcrctlu workontransmitsion·liJH' theory Inilsfirstchaplcr, theInnsmi!lllion·line e(luadoll' foraninfinit~ Iin~are deduced bothillIheC(Illn~ntion31 mann~r andfromd«tromagnetic funda· mClltal5 for\'arious CTOM·sections. Inchapu:r IWO.lhederh'alioll ofthese t.'(lualions isspeciali~cd tolinesoffinitelenglh, andthebasicmethod of trcaliug terminated lil1t.'Iisfomlulatt.'(I. Chapter Ihrt.'CiJconcerned wilhthe impedance o(seclions oftransrnission lincandtheir U~a.shuntandseriet elcmr'flt•.ThenextIWOchapters discll!Sdistributions ofcu",ent and\"Ol!aF, thelransfer ofpennT,dislribution arK! r('l()nallC~ curves,disconlinuitiC$ and r>(Kluniformilie1 alollll:.ffiOIHh linea, d.~\\'cissftoc:h lanlCni rdalion, Des champs'. method fordclCTl11ining lheprope'rdes ofjunctions, Tjunctions, endcorreaions. ClC.Thefinalchapterisdc\'Oted touansmiuion·linc MCiI· atoTl,coupled-circuit pocnomc:na onlines,andradiation. DesiplCd aaafundamental introduclion tomo~teriouJworkinwa,~guidel andamennas. thisbookisprimaril)' analytic inIreatmCllI, WillemoreIhan aIOlmdknOWledge ofdill:ert:lltial andimegl'1l1akulul andelemenlary dill:c",,"lial equations ilwppoeed, hOIl'e,'er, f..achehapter concJudC$ wilha seleclion ofproblems, milking thebookusefulasacJaNroom texlandfor testingcomprehension. t:leclrical cuginL'Crs andstudcnt., andphysicists in. terested illanimportant modcm approach 10tht.-.etoricalI'illfindtbe bookim'aluable asatCXIorareference. "'m'llJuab1e toan)"Onc ,,'110wilhe. 101I5Ctr.tnllnission·line theoT)'e1lecti,'Cly ,..shouldpUIthen:adeTinaposition 10tacklenewproblems withconfi· dence:'T.Teichmann, PHYSICS TODAY. "Utedasalexlorreference:, this bookundoUbtedly isoneoflhemostcomplCle In:atmellt. ofth~subjecl to date;'J.OffRAi\'KLlN INSTITUTE. Comelet! (1964),unabridged republication, Authors prc:face. 8iblio(raphy, Subjcc:t index.Newillde.of.ymbols. 255figurQl.9lables,62probl~..; chapler elllh,xiv+Sl3pp..5~xSy!. SI285PaperboUndJ>+, "....!p... ~,..,!! ADOVElEDITION DESICNEO FOIYUIISOFUSE! WehIVesparednopainstornalttthisthebestbookpossiblt. OUrPIP«is..... withminimalshow-through; itwillnotdiscolororbecomebrittlewithace.PlIes.. sewnif!siiNtures, inthemethodtraditionally usedforthebestboob.IIoob... latforeasyrelereace. Pageswillnotdropout,asoftlllIaIppMswitItPIIIII1tIdls heldtoeetberwithglut.ThebindingwillnotcrldlandsplitThisbI...... boo..,.., III ::J Ul 3_. Ul Ul o ::J I--