bstj13-4-532 EM theory of coax TL and shields 1934

TheElectromagnetic TheoryofCoaxialTransmission Lines andCylindrical Shields ByS.A.SCHELKUNOFF Aformofcircuitwhichisofconsiderable interest forthetransmission ofhighfrequency currents isoneconsisting ofacylindrical conducting tubewithinwhichasmaller conductor iscoaxially placed. Suchtubes havefoundapplication inradiostations toconnect transmitting andre· ceivingapparatus toantennre. Asapartofthedevelopment workonsuch coaxialsystems, ithasbeennecessary toformulate thetheoryoftrans­ missionoveracoaxialcircuitandoftheshielding againstinductive effects whichisafforded bytheouterconductor. Thispaperdealsgenerally withthetransmission theoryofcoaxialcircuits andextends thetheory beyond therangeofpresent application bothasregards structure and frequency. THEmathematical theoryofwavepropagation alongaconductor withanexternal coaxialreturnisveryold,goingbacktothe workofRayleigh, Heaviside and].J.Thomson. Muchimportant workhasbeendoneindeveloping andextending thistheory. Among theproblems dealtwithinthisdevelopment maybelistedthefollow­ ing:theextension ofthetheorytosystems consisting ofaplurality ofcylindrical conductors; theinvestigation ofshielding andcrosstalk incoaxialsystems andtheeffectsofeccentricity; theextension ofthe particular solution toincludethecomplementary modesofpropaga­ tion,etc.;andingeneraltheadaptation ofthemathematical theory toengineering uses,anditstranslation intotheconcepts andlanguage ofelectriccircuit theory. Inaddition totheauthor's contribution a substantial partofthismathematical workhasbeendonebythe groupofengineers associated withMr.JohnR.Carson, formerly of theAmerican Telephone andTelegraph Company, nowoftheBell Telephone Laboratories, Inc. Theproblem isideallyadapted tomathematical investigation, because theconductor shapefitsperfectly intothecylindrical system ofcoordinates, thereby makingitentirely feasible tocarryouta rigorous discussion onthebasisoftheelectromagnetic theory,instead ofusingordinary circuit theory. Thishasobvious advantages at ultrahighfrequencies, wheretheuncertainties ofthecircuittheoryare conspicuous andnoteasilycompensated for.Italsoprovestobe ofgreateradvantage atlowerfrequencies thanonemightatfirst assume. Fortunately, itturnsoutthatthefinalresultsobtained by meansoffieldtheorycanbeexpressed inafamiliarlanguage ofcircuit 532 ELECTROMAGNETIC THEORY OFLINES ANDSHIELDS 533 theory,thereby gainingallthesimplicity ofthelattercombined with alltheaccuracy oftheformer. CIRCULARLY SYMMETRIC ELECTROMAGNETIC FIELDS Inpolarcoordinates, Maxwell's equations ,assume thefollowing form: all,_all.(g+iw,)E,:'aE,_aE.-iW1J.Hp,pa<paz pa<paz all,all,(')EaE,aE,-iWJJ,H"",,/ --=g+1.WE 'P'azap azap(1) 1 [a(pIl.) all,]v------- (g+iw,)E"pap a<p ~[a(pE.)_aE,]= -iWjJ.Hz1pap a<p whereEandHarerespectively theelectromotive andmagnetomotive intensities.] Ingeneral, allsixfieldcomponents depend uponeachother.If, however, thesequantities areindependent ofeither 'Porz,thepartial derivatives withrespecttothecorresponding variable vanishandthe originalsetofequations breaksupintotwoindependent subsets, each involving onlythreephysical quantities. Eachofthesespecialfields hasimportant practical applications. Inthecircularly symmetric case,thatis,whenthequantities are independent of'P,oneoftheindependent subsetsiscomposed ofthe firstandthethirdequations ontheleftof(1),together withthesecond ontheright: (2)-(g+iw,)E" iWJ.llI\o"aH. az(g+iw,)pE" aE,_aE, apaz Thiscircularmagnetic field,withitslinesofmagnetomotive intensity 1Inthispaperwehaveadopted aunifiedpractical systemofunitsbasedupon thecustomary cgssystemaugmented byaddingonetypically electric unit.This systemhasthreeobviousadvantages: first,theoretical resultsareexpressed directly intheunitshabitually employed inthelaboratory; second,thedimensional character andphysical significance ofsuchquantities asiw~andg+iWEarenotobscured as inothersystems bysuppressing dimensions ofsomeelectrical unitsuchaspermea­ bilityordielectric constant; andthird,theformofelectromagnetic equations isvery simple. Inthissystem ofunitstheelectromotive intensity Eismeasured in volts/em., themagnetomotive intensity Hinamperes/em., theintrinsic conductance ginmhos/em., theintrinsic inductance ~inhenries/em., andtheintrinsic capacity E infarads/em. Thus,inemptyspace ~=411"10---9henries/em. orapproximately 0.OJ257/lit/em.andE=(l/361T) .1Q-llfarads/em. orapproximately 0.0884mmL/cm. 534 BELLSYSTEM TECHNICAL JOURNAL formingasystemofcoaxialcircles,isassociated withcurrents Bowing inisolated wiresas,forexample, inasingleverticalantenna andunder ordinary operating conditions itisalsofoundbetween theconductors ofacoaxialpair(Fig.1). z , /, I I,, I, ,,,, I,,,,,,, ----,--,,, ....""'"",~.../',-----:-_ ......------- x Fig.i-Therelativedirections ofthefieldcomponents inacoaxialtransmission line. Theremaining threeequations oftheset(1)formthesecondgroup: (3) describing thecircular electricfield.Uniformly distributed electric ELECTROMAGNETIC THEORY OFLINESANDSHIELDS 535 currentinacircular turnofwireissurrounded byafieldofthistype; inthiscase,thelinesofelectromotive intensity formacoa.xialsystem ofcircles. Two-DIMENSIONAL FIELDS Bydefinition, two-dimensional fieldsareconstant insomeone direction. Ifwetakethez-axisofourreference systeminthisdirec­ tion,allthepartialderivatives withrespecttozvanish,zdisappears fromourequations andwecanconfine ourattention toanyplane normaltothez-axis. Oncemorethesetofsixelectromagnetic equations breaksupinto twoindependent subsets. Oneoftheseis2 (4)1all, g+iWEap I -iw,J/;I:'E,= -1all, (g+iw.)poop, ![o(pE.)+oE,]pop oop Thecalculation ofwhatiscommonly knownas"electrostatic" cross­ talkbetween pairsofparallel wiresisbasedupontheseequations. Forthisreasonweshallnamethefielddefinedby(4)theelectric field. Similarly, theremaining threeequations definethemagnetic field: (5) (g+i",.)E,Il=__1_aE, PiwJ.l.pdrpI ![oCpIl.)+all,]pap aop andareusefulinthetheoryofwhatisgenerally knownas"electro­ magnetic" crosstalk. Thedistinction between electricandmagnetic fieldsispurelyprag­ maticandisbaseduponanecessary andvalidengineering separation ofgeneral electromagnetic interference intotwocomponent parts. InsomerespeC'ts thefirmlyentrenched terms"electrostatic crosstalk" andllelectromagnetic crosstalk" areunfortunate; itwouldbehopeless, however, totryachangeofterminology atthislatestageofengineering development. Further consideration oftwo-dimensional fieldswillbedeferred untiltheproblem ofshielding istakenuplaterinthispaper(page 567). 2Inpassingfromtheoriginal set(1)wereversed thesignofEpinordertomake thesetofequations symmetrical. ThepositiveEpisnowmeasured towardtheaxis. SIntheseequations, thesignofH'fwasreversed sothatthemagnetomotive intensity isnowpositive whenitpomtsclockwise. Withthisconvention, the flowofenergyisawayfromtheaxiswhenbothH",andE.arepositive. 536 BELLSYSTEM TECHNICAL JOURNAL EXPONENTIAL PROPAGATION Whileelectromotive forcescouldbeappliedinsuchawaythatthe fieldswouldbeofthekindgivenby(3),inthecoaxialtransmission lineasactually energized thefieldsareofthecircular magnetic type (2)whichwillclaimourspecialattention inthenextfewsections. Inordertosolveequations (2),wenaturally wanttoeliminate all variables butone.Thispurpose canbereadilyaccomplished ifE. andE,aresubstituted fromthefirstandthelastequations oftheset intothesecond. Thus,weobtainthefollowing equation forthe magnetomotive intensity: wherei.[!d(pH,)]+d'H,=q'H iJppop iJz2 '1"(6) q'=gw!,i-W'E!,. (7) Adopting theusualmethodofsearching forparticular solutions of (6)intheform N,=R(p)Z(z), (8) whereR(p)isafunction ofpalone,andZ(z)afunction ofzalone, weget !.d'Z=r'Zdz2 I !.!£[!d(pR)] =q'_r', Rdppdp(9) (10) whererissomeconstant aboutwhichwe have noinformation for thetimebeing. Equation (9)iswellknownintransmission linetheory;itsgeneral solution canbewrittenintheform (11) whereAandBarearbitrary constants. Thesolutions of(10)are Besselfunctions. Sinceequation (6)islinear,wemayinvokethe principle ofsuperposition andaddanynumber ofparticular solutions corresponding todifferent valuesofr.Thuswecanformaninfinite varietyofothersolutions soastosatisfythephysical conditions of variouspractical problems. Itisseenatoncefromthefirstandthelastequations oftheset(2) thattoeachH,oftheform(8)therecorrespond anE.andE,ofthe sameform;i.e.,thereexistcircularly symmetric electromagnetic (12)ELECTROMAGNETIC THEORY OFLINESANDSHIELDS 537 fields,allofwhosecomponents varyexponentially inthedirection of theaxisofsymmetry. Whether anyofthesefieldscanbeproduced individually bysomesimplephysical meansisimpossible todecideon theoretical grounds alone.Itmayhappen, ofcourse,thatthefield duetoanypractically reallzable sourceisalwaysacombination of severalsimpleexponential fields.Inanycase,however, wewantto knowtheproperties ofpureexponential solutions. Itisconvenient tomaketheexponential character ofthequantities Ep,E%andII.,explicitandwritethemrespectively intheformEpe-r" Eze-rzandl!rpe-fl•Thenewquantities Ep1E,andHrparefunctions ofponly.Ifthesuggested substitution ismadeinequations (2), thefactore-rlcancelsoutandwehave Ep=+r.Il.,iWJ.lIl.=ddE,+rEp>gIWf P d(pIl.)_ (+.)E----cr;;--gIWEPz· Thequantity riscalledthelongitudinal propagation constant or simplythepropagation cO'11sllLnt whennoconfusion ispossible.4 Recalling theimplied exponential timefactor ei",t,weseethatthe complete exponential factorintheexpressions forthefieldintensities ise-I'J+i"'l. Thepropagation constantrisoftenacomplex number andcanberepresented intheform Cl+if3wheretherealpartis calledtheattenuation conslant andtheimaginary part,thephase constant. Thus,e-aJmeasures thedecrease intheamplitudes ofthe intensities and e-i(~J-",t), thechangeoftheirphasesintimeaswellas inthez-direction. Thelatterfactorsuggests thatwearedealing withawavemovinginthepositive direction ofthez-axiswithavelocity (w/f3).Awavemoving intheopposite direction isobtained byre­ versingthesignofr. PERFECTLY CONDUCTING COAXIAL CYLINDERS 5 Letusnowconsider oneofthesimplest problems which,though purelyacademic initself,willthrowsomelightonwhatislikelyto happen underlessidealconditions. Wesuppose thataperfect dielectric isenclosed between twoperfectly conducting coaxialcylinders (Fig.1)whoseradii 6arebanda(b<0).Ourproblem istofindthe symmetric electromagnetic fieldswhichcanexistinsuchamedium. 4"'-nother setofexponential solutions isobtained fromthisbychangingrinto-r. ~rorathorough discussion of"complementary" wavesincoaxialpairsthereader isreferred toJohnR.Carson[4J. •Onlytheouterradiusoftheinnerconductor andtheinnerradiusoftheouter conductor needbeconsidered because inperfectly conducting media'electric states areentirely surfacephenomena. 538 BELLSYSTEM TECHNICAL JOURNAL Inaperfectdielectric g=0andthepreceding setofequations becomes (13) Noforceisrequired tosustainelectriccurrent inperfectconductors andthetangential components oftheintensities arecontinuous across theboundaries between different media;therefore, thelongitudinal electromotive intensity vanishes wherepequalseitheraorb. Substituting E,fromthefirstequation intothesecond,solving thelatterforH.andinserting itintothethirdequation, wehave successively and rflE,dE,'E-pdp'+dp+mp,-0, where,forconvenience, weletr2+W2EJ1.=m2•Themost solution ofthelastequation isusuallywrittenintheform E,(p)=AJo(mp)+BYormp),(14) (15) general (16) whereJoandYoareBesselfunctions oforderzeroandAandBare\ constants 50farunknown.7 Theconstants AandBcanbedetermined fromthefactalready mentioned thatE.vanishes onthesurfaceofeitherconductor, Le., fromthefollowing equations: AJo(1Ilb)+BYo(mb) =0, and AJo(lIIa)+BYo(1Ila) ~O.(17) Theseequations arecertainly satisfied ifbothconstants areequal toO.If,however, theyarenotequalto0simultaneously, wecan determine theirratiofromeachequation oftheabovesystem. These ratiosshouldbethesame,ofcourse,andyettheycannotbeequalfor everyvalueofm.Thus,thepermissible valuesofmaretherootsof 7Forlargevaluesoftheargument theseBesselfunctions areverymuchlike slightly damped sinusoidal functions; infactJo(x)andYo(x)areapproximately equal,respectively, to-V2/TXcos(x-7f'/4)andV2/TXsin(x-T/4),provided xis largeenough. ELECTROMAGNETIC THEORY OFLINESANDSHIELDS 539 thefollowing equation: AY,(mb) -Ii=J,(mb)y,(ma) J,(ma)(18) Thisequation hasaninfinitenumber ofroots'whoseapproximate valuescanbereadilydetermined ifwereplaceBesselfunctions by theirapproximations intermsofcircularfunctions. Thus,wehave '''rmft=a-bI(n=1,2,3,...). (19) Thisisasurprisingly goodapproximation forallrootsiftheradiusof theouterconductor islessthanthreetimesthatoftheinner;andthe largerthen,thebettertheapproximation.' Thepropagation con­ stantsarecomputed fromthecorresponding values of m"bymeans ofthefollowing equation, (20) Firstofall,letusstudythesimplest solution inwhichbothAand Bvanish. Inthiscase,thelongitudinal electromotive intensity vanishes identically. Themagnetomotive intensity-and thetrans­ verseelectromotive intensity, aswetl-also vanishes unlessthede­ nominator m'inequation (14)equalszero.Ifallintensities wereto vanish,weshouldhavenofieldandtherewouldbenothing totalk about;hence,wetaketheotheralternative andlet i.e.,r=iw,r;j;. (21) thepositive signhavingbeenimplied inwritingequations (13).In air,Ell=(1/c2)wherecisthevelocity oflightinem.;hence,inair thisparticular propagation constant equals'iw/c.SinceE.equals zeroeverywhere, theelectromotive intensity iswhollytransverse; and theflowofenergybeing,according toPoynting, atrightanglestothe electromotive andmagnetomotive intensities, theenergytransfer is whollylongitudinal. Theabovemethod ofdetermining thepropagation constant may beopentosuspicion; besides, themethoddoesnottellhowtoobtain theactualvaluesoftheelectromagnetic intensities butmerelyleads toarelation compatible withtheexistence ofsuchintensities. There­ fore,letusobtainthewanted information directly fromthefunda- IA.GrayandG.B.Mathews, ..ATreatise onBesselFunctions" (1922),p.261. 'Itisstrictlyaccurate iftheradiiofthecylinders areinfinite, i.e.,ifwearedealing withadielectric slabbou~ded byperfectly conducting planes. 540 BELLSYSTEM TECHNICAL JOURNAL mentalequations (13)whichassumethefollowing simpleform: iwEE,.=rH.,. iwp.[f"=rE,.,d(pII,) =0 dp ,(22) ifE,vanishes identically. Eitherofthefirsttwoequations determines theratioof·theelectromotive intensity tothemagnetomotive: the tworatiosareconsistent onlyifthecondition (21)issatisfied. Then, wehavealso rj; E,=-.-II,=-H, 1WE EandAH,=-p (23) whereAissomequantity independent ofp.Thisconstant canbe readilycalculated fromAmpere's law.Themagnetomotive force actingalongthecircumference ofanyparticular cross-section ofthe innercylinder equals27rpli"amperes, i.e.,21iAisincethisM.M.F. shouldequalthetotalcurrent1flowingintheinnerconductor through thecross-section, thequantity Aequals1/21T.Reintroducing the impliedfactore-I'l,wehave IIi=--e-r:..27rp • E=...!- ~e-r, p27rp"\)~ ....(24) Inpractical measurements weareconcerned withthetotalpotential difference (V)between thecylinders, ratherthanwiththetransverse electromotive i.ntensity. Theformerismerelytheintegral ofthe intensity, v~l'E,dp=(:..!..I~logil:)Ie-r,.27T\}, b(25) ThisvoltageandthecurrentIvaryasvoltageandcurrentinasemi­ infinitetransmissio'n linewhosepropagation constant israndwhose characteristic impedance is (26) Atanypointztheintensities EpandH"havethesamevaluesaswould thevoltageandcurrentatthesamedistance zfromtheendofatrans­ missionlinewhosepropagation constantandcharacteristic impedance are respectively iwv-;~G'uiVl'/•. '".... \r ELECTROMAGNETIC THEORY OFLINES ANDSHIELDS 541 Theconnection between electromagnetic theoryandlinetheoryis soimportant that,riskingrepetition, wewishtoemphasize their intimate relationship byderiving thewcll·known differential equations ofthelinetheorydirectly fromtheelectromagnetic equations (2) combined withtheassu.mption thatJ.helongitudinal eteclromo/.ivc intensity vanishes everywhere. Wealready knowthatundertheassumed con­ ditionsthefirstequation ofthesystem(2)becomes IH'=-2-'..p(27) whereIisthetotalcurrent Rowingintheinnercylinder through a particular cross-section andissomefunction 10ofz.Wecantherefore rewritethelastt\\«lequations ofthesystemasfollows: _1_aI= _iw.E 21rpiJz P'(28) Wehavemerelytointegrate bothequations withrespecttopfrom btoaandsubstitute thepOtential difference Vfortheintegral ofthe transverse electromotive intensity toobtain aV (iw~a)-= --Iog- Taz 2..b 'aTaz=(29) whicharetheequations ofthetransmlSSlOn linewhosedistributed seriesinductance equals (~/2..)log(a/b)henries/em. andshuntcapacity 27l"./(log a/b)farads/em. Withthis.weconclude thespecialcaseinwhichthelongitudinal electromotive intensity vanishes everywhere, thepropagation constant equalsiwr.;;,andthevelocity oftransmission isthatoflight. Wenowturnourattention tothecaseinwhichAandBdonot vanish. Wehavealready notedthatthepropagation constants are givenbyequation (20).Since.inthiscase,weareinterested primarily inthenatureofthephenomena ratherthaninthedetailsoffield distribution, weshallsimplify ourmathematics bysupposing the radiiofthecylinders tobeinfinite. Thus,thecylinders become two planesperpendicular tothex-axis,distance aapart.The",·direction, then,coincides withthe)'-direction and,therefore, alltheintensities areindependent oftfey-coordinate. Letuschoosethez-axishalf­ waybetween theplanes. Theequations describing thistwo-dimen- 10Onthisoccasion, weshouldremember thataparticular typeofthisfunction hadnotyetbeen asce~tained atthetimetheequations (2)werearrivedat. 542 BELLSYSTEM TECHNICAL JOURNAL sionaltransmission lineare aH,._"ax='ZWt..C..u (30) aE._aE.= _iwp./l,.azax Ifnisanoddinteger,thesepossessthefollowing solutions: E Af".n7rX•=-.-sm--, 1WEa n1r n1rXE.=A-.-cos-twro.a' A.n7rXFIv=sm-;a andifnisaneveninteger. r" n7rXE~=A-.-cos--, 1-WE a n1l"•n1l"XEz=-A-.-sm--, 1wEa a mrxHII=Acos-,a where(31) (32) (33) and},isthewave-length corresponding tothefrequencyf. Letusnowdefinethelongitudinal impedance (Z.)astheratioof Ezto11M! Z~r... 'WE(34) andthetranstJerse impedance (theimpedance mthex-direction) as theratioofEztoHili Zn7r n7fX•=-.-cot--, 1wEa a n7r n1fXZz=--.-tan-- l.WEa aIifnisodd, ifniseven.(35) Itwillbeobserved that,depending onthefrequency, thelongitudinal ELECTROMAGNETIC THEORY OFLINESANDSHIELDS 543 propagation constantr"iseitherrealorpurelyimaginary; itvanishes ifa=n(X/2),thatis, ifthespacing between theplanesisawhole number ofhalfwave-lengths. Whenthepropagation constant is real,thelongitudinal impedance ispurelyimaginary, andviceversa, whenthepropagation constant ispurelyimaginary, thelongitudinal impedance isreal.Intheformercase,noenergyistransmitted longitudinally butmerelysurgesbackandforth,andinthelatter casewehaveatruetransmission line.Thetransverse impedance is purelyimaginary atallfrequencies and,hence,theenergymerely fluctuates toandfro. lfthefrequency issufficiently low,allofthesehigherorderpropaga­ tionconstants arerealandalltheenergyistransmitted intheprincipal modedescribed byequations (21)to(29).Therilleofthehigher propagation constants consists inredistributing theenergynearthe sending terminal,ll thatis;interminal distortion. Butasthefre­ quencygetshighenoughtomakethewave-length lessthan2a,the nexttransmission modemaybecome prominent, andsoforthupthe .infiniteladderoftransmission modes. IMPERFECT COAXIAL CONDUCTORS 12 Weshallnowsuppose thattheconductors arenotperfect; i.e., theconductivity insteadofbeinginfinite, ismerelylarge.Assuming thatoursolutions arecontinuous functions ofconductivity (thiscan beproved), weconclude: first,thereexistsaninfiniteseriesofpropaga­ tionconstants approaching thevaluesgiveninthepreceding section astheconductivity tendstoinfinity; second, oneofthesepropagation constants, namelythatapproaching iw,JE!.,isverysmallunlessthe conductivity istoosmall. Intheimmediately succeeding sections we shallbeconcerned onlywithelectromagnetic fieldscorresponding to thisparticular propagation constant. Letusnow'provethatthesimpleexpression forthemagnetomotive intensity inthedielectric between perfectly conducting cylinders is stilltrueforallpractical purposes, eveniftheconductors aremerely good,andevenwhentherearemorethantwoofthem.Sincethe linesofforcearecircles,coaxialwiththeconductors, andsinceIi.,is independent of<p,thetotalmagnetomotive forceactingalongany oneofthecirclesequalsH.timesthecircumference ofthecircle(2".p). ThisM.M.F. alsoequalsthetotalcurrentIpassingthrough thearea ofthecircle.Therefore, themagnetomotive intensity is(I/2".p) amperes/em. Thisexpression istrueatanypointintheconductors as UAndnearthereceiving tenninal aswell,ifthelineisfinite. 12Thegeneral theoryofwavepropagation inamultiple systemofimperfect coaxialconductor's isamplycovered byjohnR.Carsonandj.j.Gilbert[2,3]. 544 BELLSYSTEM TECHNICAL JOURNAL wellasinthedielectric between them.Inaconductor thetotal currentIpassingthrough theareaofthecircleisafunction ofpsince thecurrent isdistributed throughout theentirecross-section ofthe conductor. Strictly speaking, thesameistrueofanycircleinthedi­ electric. Thereisoneimportant difference, however; theconduction currentpassingthrough suchcirclesisthesameandthedisplacement current isusually sosmallthatitcanbelegitimately neglected. Thus,inthedielectric, wehavetoanextremely highdegreeofaccuracy unless pisverylarge IH.=-2-'1fp(36) whereIismerelyaconstant, namely, thetotalconduction current passing1:hrough theareaofthecircleofradiusp. Thatthelongitudinal displacement currentcanbeneglected, unless theconductivity oftheconductors issmall,hasbeenalreadyindicated intheopening paragraph. Thefollowing comparison isanaidtothe mathematical argument. Thedensityofthelongitudinal conduction currentisgEandthatofthedisplacement current isiwEE.Nearthe boundary, Eissubstantially thesameintheconductor andinthedi­ electric. Incopper, g=(1/1.724)10' andinair.=(1/361f)IO-u. Thus,evenatveryhighfrequencies, thedensityofthedisplacement current isverysmallcompared tothatoftheconduction current. Ontheotherhand,theconduction current isordinarily distributed overasmallareawhilethedisplacement current mayflowacrossa largearea.Thelatterareawouldhavetobeverylarge,however, beforeitcouldevenbegintocompensate fortheextremely lowcurrent density. ELECTROMOTIVE INTENSITIES INDIELECTRICS Withtheaidofequations (12)and(36),wecannowcalculate the electromotive intensities inthedielectric between twoconductors. Thus,thetransverse intensity is E=fl ,21f(g+iw.)p(37) Substituting thisinthesecondequation oftheset(12),weobtain thefollowing differential equation forthelongitudinal intensity: dE.[.f']I dp="W~-g+iWf27rp'(38) whereI'isthepermeability ofthedielectric. Integrating withrespect ELECTROMAGNETIC THEORY OFLINES ANDSHIELDS 545 toP,wehave '[.f'] p E,~2>rtW/i-g+iWE!logb'+A, (39) whereAisaconstant tobedetermined fronitheboundary conditions,u THEPOTENTIAL DIFFERENCE BETWEEN TwoCOAXIAL CYLINDERS Equation (36)relatesthetransverse electromotive intensity toth~ totalcurrent flowing intheinnerconductor. ]npractice, however, weareinterested inthedifference ofpotential between theconductors, thatis,inthetransverse electromotive forceratherthantheelectro­ motiveintensity. Thispotential difference Visobtained atonce fromequation (37)byintegration: an 1," 1'" f!log-V=Edp~rI dp= b'..,' h(g+iWE).'p2".(g+iWE)(40) Thistransverse E.M.F. produces atransverse electriccurrent which ispartlyaconduction current-if thedielectric isnotquiteperfect­ andpartlyadisplacement (or"capacity") current. Now,thetotaltransverse currentpercentimeter lengthoflineis J,=2,,-p(g+iWE)E,. Then,byequation (37),wehave J,~fJ. Therefore, equation (40)becomes anlogy V=~~~~J2,,-(g+iWE),.(41) (42) Theratioofacurrent totheelectromotive forcethatproduces it iscalledadmittance. Hence, thedistributed radialadmittance per UThefollowing systemofnotation willbeadhered tothroughout theremainder ofthepaper:Theinnerradiusofanycylindrical conductor isdenoted bya,andits outerradiusbyb.Whenseveralcoaxialconductors areused,theyaredifferen­ tiatcdbysupcrscripts; a',a",aUl,...referring totheirinnerradii,forcxample, andb',b",blJI•.•.totheirouterradii.Thisconvention alsoappliestoconduc­ tivities, permeabilities, andotherphysical constants oftheconductors inQuestion. Forconvcnience, wehavewrittentheratioofptotheouterradiusoftheinner conductor inplaceofp;thischangeaffectsonlythearbitrary constant Awhichwill eventually beassigned thevaluerequired bytheboundary conditions. When written inthisform,thefirsttermofEzvanishes onthesurfaceoftheinnercon­ ductorwhichisaconvenience indetermining thevalueofA. 546 BELLSYSTEM TECHNICAL JOURNAL unitlengthbetween twocylindrical conductors is y=21r(g+iw.)=G+.Ca"- ~w I !Ogll(43) thesymbols GandCbeingusedin distributed radialconductance and rately,we haveth.eusual capacity.waytodesignate the Writing thesesepa- C=~a"· log17(44) Returning to(40),wefindthatVcanbewrittenintheform V~!Iy' Buttheratioofthetransverse electromotive forceVtothelongitudinal current Iisknownasthelongitudinal characteristic impedance ofthe coaxialpair.Itsvalueisobviouslyr/Y. THEEXTERNAL INDUCTANCE Indealing withparallel wiresitiscustomary tousetheterm "external inductance" forthetotalmagnetic fluxinthespacesur­ rounding thepair.14Weshalladoptthesameusageinconnection withcoaxialpairs.Strictly speaking, wel11usttherefore consider it asbeingcomposed oftwoparts:onebeingthefluxbetweenthecylinders, theotherthefluxinthespacesurrounding them.Butthelongi­ tudinaldisplacement currentisnegligible bycomparison withthecon­ ductioncurrent, andeffectsduetoithavebeenconsistently ignored throughout thispartofourstudy.'Tothesameorderofapproxima­ tion,thefluxoutsidethepairisnegligible bycomparison withthat between them,whencewefindthe"external inductance" tobe L.IJ.r'"H,.dp ".1.. IJ.Iah . /I=271"og71ennesem. (45) 14Whilethisdefinition isverydescriptive, itisnotstrictlyaccurate unlessthe wiresareperfectly conducting. Thecorrectdefinition shouldreadasfollows: Theextemal inductance ofaparallelpairisthemeasure (perunitcurrent) ofmag­ neticenergystoredinthespacesunounding thepair.Thereasonthesimpler definition failsforimperfectly conducting parallelwiresisbecause someofthelines ofmagnetic fluxliepartlyinsideandpartlyolltsidethewires.Thisdoesnothappen inconnection withcoaxialpairsevenwhentheyarenotperfectly conducting. Hencewearewarranted inusingthesimpleridea. (46)ELECTROMAGNETIC THEORY OFLINESANDSHIELDS 547 Comparing thiswithequation (44),wehavethefollowing relation between theexternal inductance andthecapacity CL.~'1'. PROPAGATION CONSTANTS OFCOAXIAL PAJRS Sincetherelation between electromotive intensity andcurrent is linear,wearejustified inwriting theintensities attheadjacent surfaces ofthepairintheform E.(b')=Z,'I, (47) (48)whereZ,'andZ."dependonlyuponthematerial oftheconductors andthegeometry ofthesystem. Thesequantities willbecalledsur­ faceimpedances oftheinnerandouterconductors, respectively. Inserting (47)in(39)weobtain A=Z,'I, 1[.r']. a" "h''''I'-g+i",.Ilogy+A= -Z.I, bymeansofwhichAandrmaybeexpressed intermsofZ,'andZ.". IfwesolvethefirstoftheseforAandsubstitute thevaluethusderived inthesecondweget,byvirtueof(45), f2 a" "" .h(g+i",.)logy=Z.+Z.+,,,,L.. or,by(43) r'~YZ, whereforbrevitywehavewritten Z=Za'1+Zb'+iwL.,.(49) (50) (51) DIRECT CONVERSION OFTHECIRCULARLY SYMMETRlC FIELD EQUA­ TIONS INTOTRANSMISSION LINEEQUATIONS Asthepractical applications ofMaxwell's theorybecome more numerous, it~comes increasingly important toformulate itsexact connection withtransmission linetheory. Withthispurpose inmind, letusattempt tothrow(2)intotheformofthetransmission line equations. Theobvious planofattackistointroduce into(2)thetransverse voltageVandthelongitudinal currentI,inplaceoftheintensities EandH.Thetotalcurrentisintroduced bysubstituting (I/2rrp)for H.,andthetotalvoltagebyintegrating thesetofequations (2)in thetransverse direction. Thefirstequation givesusnothing of 548 BELLSYSTEM TECHNICAL JOURNAL importance." Thesecondandthirdequations, ontheotherhand, give iw~IIa"=E."(a)_E.'(b)av~og71 -az'(52) a" logT7aI V=-"2..-('--g-+"'--'i-w'.) az But,uponsubstituting (45),(47)and(51)inthefirstoftheseequations and(43)inthesecond,weget clV~_ZI az'aI az-YV, (53) whereZandYaretobeinterpreted respectively asthedistributed seriesimpedance andshuntadmittance. CURRENT DISTRIBUTION INCYLINDRICAL CONDUCTORS Sofar,wehavebeendealing withelectromagnetic intensities in dielectrics. WenowturnOUfattention toconductors anddetermine theircurrentdistributions withtheultimate viewofcalculating their surfaceimpedances. Oneofoursourcesofinformation isthefamiliar setofequations (12).Intheseequations, however, wenowlet.=0 sincethedisplacement current inconductors isnegligibly smallby comparison withtheconduction current. Fromtheseequations, we eliminate electromotive intensities andthusobtainadifferential equation forthemagnetomotive intensity. Thelatterisinfact equation (6)withonlyonedifference: theexponential factor.-r.has beenexplicitly introduced andcancelled sothattheequation has become or whered'H.+!dH._H.=(u'_r')H dp'l pdp p2 ".(54) ,,'=gw~i~2..g~fi. This"willbecalledtheintrinsic propagation constant ofsolidmetal. IiOurstandard practice ofneglecting thelongitudinal displacement currents hasgivenusthegeneralrulethath'pH",=1isindependent ofp.Usingthisrelation inthefirstofequations (2.2),weget (g+iw.)E.*0; butthismerelyreflectsthefactthatg+iw~isverysmall. ELECTROMAGNETIC THEORY OFLINESANDSHIELDS 549 Theattenuation andthephaseconstants areeachequalto,J'lrgllf. Theintrinsic propagation constants ofmetalsarelargeQuantities exceptatlowfrequencies astheaccompanying tableindicates. PROPAGATION CONSTANT OFCOMMERCIAL COPPER g=5.800lOimhos/em. JJ.=0.01257ph/em. f o I 10 100 10,000 1,000.000 100,000,0000.0 0.1513 0.4785 1.513 15.13 151.3 1513. Ontheotherhand,risverysmall;ifairisthedielectric between the conductors, risoftheorderof(1/3)iw10-10•Hence,evenathigh frequencies r1isnegligibly smallbycomparison with (12andwecan rewrite(54)asfollows: d[1d ]- --(pH)dppdp•(55) ThisisBessel's equation and itssolution canhewritten downat once 16as (56) (57)wherethefunctions [I(U)andKla)arethemodified Besselfunctions ofthefirstorderandrespectively ofthefirstal1dsecondkind,Forlarge valuesoftheargument we have approximately [I(U)=";:/t(1 -8~), KI(u)~~e-"(1+8~') IIItisinteresting tonotethatinthecaseofafairlythinhollowconductor whose innerradiusisnottoosmallthereexistverysimpleapproximate solutions of(55). Underthesecircumstances pvariesoversuchasmallrangethatnoseriouserroris introduced intreating thefactors(tIp)andpin(55)asconstants, andtheequation becomes . d'-H", ~lldp2=rr"', whichissatisfied bytheexponential functions e'"ande-"'.Thelargerthevalue ofpandthefasterthechangeinH"withP.thebetteristheapproximation. 550 BELLSYSTEM TECHNICAL JOURNAL whileforsmallvalues 111.UK,(u)=-+-Iog-.u2 2(58) Thefunction I,(u)becomes infiniteandKI(u)vanishes whenuis infinite.17Whenuiszero,I1(u)vanishes andK1(u)becomes infinite. Thelongitudinal electromotive intensity iscalculated fromthe thirdequation (12)withtheaidofthefollowing rulesfordifferentiation ofmodified Besselfunctions ofanyordern: Thus, whereddx(X"!n.)=x"In_1, ddx(x"K.) -x"K._,. E.=~[Alo(up) -BKo(up)], CT'f,WjJ. ~= -=-.gu(59) (60) (61) Forreasonswhichwillappearlater,thisquantity ~willbecalledthe intrinsic impedance ojsolidmetal. Thecurrentdensityismerelytheproduct oftheintensityE.and theconductivity g. Inageneralwaythebehavior ofthefunctions ofzeroorderis similartot.hatofthefunctions whoseorderisunity.Thus,forlarge valuesoftheargument, Io(u)=v;~,(1+8~')' Ko(u)=~e-'(1-8~'), andforsmallvalues u'Io(u)=1 -'4' Ko(u)= -logu+0.116.(62) (63) 17Thisstatement iscorrectonlyaslongastherealpartof'Uispositive. Thisis soinourcasebecause outoftwopossible valuesofthesquarefootrepresenting tr wecanalwayschoosetheonewiththepositive realpart. ELECTROMAGNETIC THEORY OFLINES ANDSHIELDS 551 SURFACE IMPEDANCE OFASOLIDWIRE Onpage.547wedefinedthesurfaceimpedances ofacoaxialpairas theratiosofthelongitudinal electromotive intensities ontheadjacent surfaces ofthecylinders tothetotalcurrents flowingintherespective conductors. Inthatplace,however, wewereunabletogiveexplicit formula> fortheimpedances sodefinedbecause wedidnotyethavea precisevalueforE~.Nowthatthisomission hasbeensupplied, we areprepared tocompute Zb'andZa". Weconsider thecaseofasolidinnercylinder surrounded byany coanalreturn,andseektodetermine theconstants AandBin(60). SincetheE.M.I.mustbefinitealongtheaxisofthewirewemust makeB=0,because theK-function becomes infinitewhen p=O. Onthesurfaceofthewirethemagnetomotive intensity isI/27rbif Iisthetotalcurrentinthewire.Byequation (56)thisintensity equalsAI,(ub); hence, IA27rbI,(ub) andthefinalexpression fortheelectromotive intensity withinthe wireis (64) Thus,wehavethefollowing expression forthesurfaceimpedance of thesolidwire: E,(b) ~lo(ub) Z,=-l-=27rbl,(ub) 'ohms/em. (65) Astheargument increases, themodified Besselfunctions ofthe firstkind(theI-functions) becomemoreandmorenearlyproportional totheexponential functions ofthesameargument. Thus,ifthe absolute valueofubexceeds SO,theBesselfunctions inthepreceding equation canceloutandthefollowing simpleformula holdswithin 1percent: (66) Zb=--'!..---lW(!+')h /21rb-2b\J-:;:g ~,01115em. Thissurfaceimpedance consists ofaresistance representing the amount ofenergydissipated inheat,andareactance duetothemag­ neticfluxinthewireitself.Separating (66)intothesetwoparts, wehave,approximately, • 552 BELLSYSTEM TECHNICAL JOURNAL However, mostoftheerrorin(66)occursintherealpart. accurate approximations forBesselfunctions areused,thenIfmore (67)1fiJ 1 Rb=2b\/;g+41rgb'' wLb=-!.II'!; 2b\/.-g thesearecorrectwithin1percentifIubI>6.Thesurfaceinductance Lbequals(l/4.-b)..JIl/.-g! henries/cm.; itdecreases asthefrequency increases. Ifthewireissothinorthefrequency issolowthatIubI<6, equation (65)hastobeused.Itsuseincomputations isquitesimple, however, because theargument ubisacomplex number oftheform ltft;andthenecessary functions havebeentabulated. LordKelvin introduced thesymbols berItandbeiItfortherealandtheimaginary partsof10(1"'1.),sothatwenowwrite [.(",';i) =berIt+ibeiIt. Differentiating, wehave(68) ,[<[,(It,[<) ~ber'It+ibei'It, andtherefore her'1l+ibei'1t ,1"1.(69) (70)Ifweinsertthesevaluesin(65),andrecallthatthed.-c.resistance ofasolidwireis1/trgb2•andthat CT=g71.weobtainatonce 1rb2Z"=!!:her1lbei'1t-bei11.ber'u g 2 (ber'11)'+(bei'It)' +.ttheru.ber'u.+bei1their1t ,-2(ber'It)'+(bei'It)' whereItistheabsolute valueofub.Theaccompanying graph illustrates therealandimaginary partsofthisequation 18(Fig.2). THESURJ;ACE IMPEDANCES OFHOLLOW CYLINDRICAL SHELLS" Inthecaseofahollowconductor whoseinnerandouterradiiarc respectively equaltoaandb,thereturncoaxialpathforthecurrent 18Forequation (70)andvariousapproximations seeE.JahnkeandF.Emde. "Inthecaseofself-impedances themoregeneral equations ortwoparallel cylindrical shellswerededuced byMrs.S.P.Mead. Forthespecialrormulz concerning selr·impcdances orcoaxialpairsseeA.Russell. ELECTROMAGNETIC THEORY OFLINES ANDSHIELDS 553 maybeprovided eitheroutsidethegivenconductor orinsideitor partlyinsideandpartlyoutside. \Nedesignate byZoothesurface impedance withinternal return,andbyZbb.thatwithexternal return. Theseimpedances areequalonlyatzerofrequency; butifthecoo- 4.0,... w>« 3.5a ow "3.0g o,••4 o0 ./,, /•V", 0V,, /,• ~,,, 0V,, // ,•V,,, .0 ,,", 5",", 0/ I"o4. w ~a3- oo Sl3. ~2. •u Fig.2-Theskineffectinsolidwires.Theuppercurverepresents theratioo( thea-c.resistance ofthewiretoitsd-c.resistance andthelowercurvetheratioof theinternal reactance tothed-c.resistance. ductoristhin,theyarenearlyequalatallfrequencies. Ifthereturn pathispartlyinternal andpartlyexternal, wehaveineffecttwo transmission lineswithadistributed mutualimpedance ZlIbdueto themingling ofthetwocurrents inthehollowconductor common to bothlines.However, sincethisquantity Zabisnotthetotalmutual impedance between thetwolinesunlessthehollowconductor isthe onlypartoftheelectromagnetic fieldcommon tothem,itisbetterto callZnbthetransferimpedance fromonesurfaceoftheconductor to theother. Inordertodetermine theseimpedances, letussuppose thatofthe totalcurrentI.+Ibflowing inthehollowconductor, thepartI. returnsinsideandtherestoutside. Sincethetotalcurrentenclosed bytheinnersurfaceofthegivenconductor is-la,andthatenclosed bytheoutersurfaceislb.themagnetomotive intensity takesthevalues -(l./h-a) and(h/hb), respectively, atthesesurfaces. Thisinfor­ mationissufficient todetermine thevaluesoftheconstants AandB intheequation (59)governing current distribution. Infact,we 554 have andtherefore whereBELLSYSTEM TECHNICAL JOURNAL (71) (72) (73) Substituting theseintothesecondequation oftheset(59).weobtain thelongitudinal electromotive intensity atanypointoftheconductor. Weareinterested, however, initsvaluesatthesurfaces sincethese valuesdetermine thesurface impedances. Equating psuccessively toaandb,weobtain .where 20E,(a) ~Z"I.+ZobIb. E,(b) ~Zb.Io+ZbbIo,(74) Zo.=2'1r~D[Io(ua)K,(ub)+Ko(ua)I,(ub)]. Zbb~2'1r~D[Io(ub)K,(ua)+Ko(ub)I,(ua)], (75) 1 Zub=Zba=27rgabD Theresultsembodied inequation (74)canbestatedinthefollowing twotheorems: Theorem 1:Ifthereturnpathiswhollyexternal (Ia~0)orwhollyin­ ternal(Ib=0).thelongitudinal electromotive intensity onthat surfaceofahollowconductor whichisnearesttothereturnpath equalsthecorresponding surfaceimpedance perunitlengthmultiplied bythetotalcurrent flowing intheconductor; andtheintensity onthe othersurfaceequalsthetransferimpedance perunitlengthmultiplied bythetotalcurrent. 20Toobtainthelastequation, itisnecessary tousetheidentity 1l"(x)K,(x)+Ko(x)I,(x) ~x. ELECTROMAGNETIC THEORY OFLINES ANDSHIELDS 555 Theorem 2:Iftheretn",pathispartlyexternalandpartlyinternalthe separate components oftheintensity dnetothetwopartsofthetotal wrrentarecalcnlated bytheabovetheoremandthenaddedtoobtain thetotalintensities. Athighfrequencies, orwhentheconductors areverylarge,(75)can bereplaced bymuchsimplerapproximate expressions,21 If,however, wearecompelled tousetherigorous equations innumerical computa­ tions,itisconvenient toexpress theBesselfunctions intermsof Thomson functions. Twoofthese,theberandbeifunctions, or Thomson functions ofthefirstkind,havealready beenintroduced. Thefunctions ofthesecondkindaredefinedinanentirelyanalogous fashionas Ko(x{i) ~kerx+ikeix. Differentiating, wehave(76) {iKo'(x{i) = -{iK,(x{i) =ker'x+ikei'x, (77) sotha't K(r.) ker'x+ikei'x Ix\}~= - ..vi Allthesesubsidiary functions havebeentabulated;" but processofcomputing theimpedances islaborious nevertbeless.(78) the THE'COMPLEX POYNTING VECTOR 23 Inthepreceding sectionswehavebeenabletodetermine thesurface impedances ofthecoaxialconductors byreducing thefieldequations totheformoftransmission lineequations, andinterpreting various termsaccordingly. However, iftheconductors areeccentric orof irregular shape,theeffect'ivesurfaceimpedances aremoreconveniently calculated bytheuseofthemodified Poynting theorem. Thistheorem statesthat,ifEandHarethecomplex electromotive andmagnetomotive intensities atanypoint,andifE*andIi*are theconjugate complex numbers, then,. JJ[EH*JdS =gJJJ(EE*)d.+iw!,JJJ(HH*)dv. (79) !ISeeportionofthistextundertheheading "Approximate Formulre forthe SurfaceImpedance ofTubular Conductors," page557. "British Association Tables,1912,PP.57-68;1915,pp.36-38;1916,pp.108-122. ~Foranearlyapplication oftheComplex Poynting vectorseeAbraham v. Fiippl,Vol.1(Ch.3,Sec.3). NThebrackets signifythevectorproductandtheparentheses thescalarproduct ofthevectorssoenclosed. Theinwarddirection ofthenormaltothesurfaceis chosenasthepositivedirection. Thedivision by4...doesnotoccuriftheconsistent practical systemofunitsisusedasitisdoneinthispaper. 556" BELLSYSTEM TECHNICAL JOURNAL Togetaninsightintothesignificance ofthisequation, letuscon­ sideraconductor whichispartofasingle-mesh circuit,andextendour integrals overtheregionoccupied bythisconductor. Thenthefirst integral ontherightof(79)represents twicethepowerdissipated in heatintheconductor, whilep.fff(HH*)d!J isfourtimestheaverage amount ofmagnetic energystoredinit. Ontheotherhand,whenwelookattheconductor fromthestand­ pointofcircuittheory,thesetwoquantities arerespectively Rl'and LP;RandLbeingbydefinition the"resistance" and"inductance" oftheconductor. Hencewehavetheequation,JJ[EH*]"dS =(R+iwL)I'=ZI', (80) fromwhichtheimpedance Zcanbecomputed whenthefieldintensities areknownatthesurfaceoftheconductor. If,ontheotherhand,theconductor ispartofatwo-mesh circuit andI,andI,aretheamplitudes ofthecurrents inmeshes1and2 respectively, theaverage amount ofenergydissipated inheatper secondcanberegarded asmadeupofthreeparts,twoofwhichare proportional tothesquares oftheseamplitudes, whilethethirdis proportional totheirproduct. Thefirsttwoofthesepartsbeing dependent onthemagnitude ofthecurrentflowinginonemeshonly areattributed totheself-resistance oftheconductor tothecorre­ sponding current; thethirdpartisattributed tothemutualresistance oftheconductor. Designating theself-resistances byR"andR"and themutualresistance byRl2,werepresent theenergydissipated in heatintheform1/2(RllI,'+2Rl2I,I,+R"I,'). Similarly, the average amount ofenergystoredintheconductor canberepresented intheform1/4(L"I)'+2Ll2I,I,+L"I,'),whereL"andL"are calledrespectively self-inductances andLl2m"t"alind"ctance. Inthis case,equation (79)canbewrittenasfollows:JJ[EH*]"dS =Z"I,'+2Zl2I,I,+Z"I,', (81) wherethequantities Z",Z"andZ"arerespectively theself-im­ pedances andthem"tualimpedance oftheconductor. Ingeneral, iftheconductor ispartofak-meshcircuit,wecan obtainallitsself-and mutualimpedances byevaluating theintegral ff[EH*]"dS overitssurface, andpickingoutthecoefficients of variouscombinations of['s. Weshallhaveanoccasion toapplytheseresultsincomputing the efI;ectofeccentricity upontheresistance ofparallelcylindrical con­ ductors. ELECTROMAGNETIC THEORY OFLINESANDSHIELDS 557 ApPROXI'~TE FORMULE FORTHESURFACE IMPEDANCE OF TUBULAR CONDUCTORS Theexactformul", (75)fortheinternal impedances ofatubular conductor arehardtousefornumerical computations, butsimple approximations canbeeasilyobtained ifthemodified Besselfunctions arereplaced bytheirasymptotic expansions andthenecessary division performed asfarasthesecondterm.Thus,wehave Zbb=2:blcothcrt+;cr(~+i)], z••=~[cothcrt--"=-(~+!)],21ra 2crb a Z..=~,_cschcrt, 21rvab(82) (83)wheretisthethickness ofthetube.Separating therealandimaginary parts,we have R",_!-[jJsinh"+sin"+a+3b -2b'\j;gcosh"-cos11161rgab'' R~~[jJsinh"+sin11_b+3a 002a\J:;:gcosh1tcosti,1671'"gba2' 'h" Uh"'u_1_11'1sm"2cos"2+cos"2Sin"2 Rab=...jab\J7rgcoshucos1t ' wL bb=..!.fiifsinhu-sin1l 2b\J;gcoshu-cos1t• wL=!-[jJsinhU-sin" aa2a\J-;gcosh'It-cos11.I .1/ 1t U.U1$esmh-COS- -cosh-sm- wL..=~42 2 2 2 ...jab7rg cosh1tcosu IZobl~ -.r;;j .;l1rgab(cosh" casu) whereu=t-J2gwl" Itisobvious thatintheequations fortheself-resistances, thesecond termsrepresent thefirstcorrections forcurvatureand vanishaltogether iftheconductors areplane.Although theseformul", werederivedby usingasymptotic expansions whicharevalidonlywhentheargument islarge,i.e.,athighfrequencies, theresultsaregoodevenatlow 558 BELLSYSTEM TECHNICAL JOURNAL frequencies, provided thetubularconductor isnottoothick.Thus, ifthefrequency is0,thefirsttermintheaboveexpression forR" hecomes lj2"gbtwhichisthed.-c.resistance ofthetuheifitscurvature isneglected. Thesecondtermonlypartially corrects forcurvature, theerrorbeingoftheorderofl'j8b'.Hence,ifthethickness ofthe tubeisnotmorethan25percentofitshigh-frequency radius,thatis, theradiusofthesurfacenearestthereturnpath,theerrorislessthan 1percent.Theformula forthemutualimpedance isexceedingly gooddowntozerofrequency forallordinary thicknesses. Ifthefrequency isveryhigh,furtherapproximations canbemade andtheformul,., simplified asfollows: (84) Iftheratioofthediameters ofthetubeisnotgreaterthan4j3, thenwehavethefollowing formulaforthesurfacetransferimpedance: IZ"I 1t Rd.-e.=-Vcoshu.-cosIf.I(85) whichiscorrecttowithin1percentatanyfrequency. Thisratiois illustrated inFig.3.Theratiosofthemutualresistance andthe mutualreactance tothed.-c. resistance areshowninFig.4. Inthecaseofself-resistances, welet 1 Ro=2-t'..-gr(86) ELECTROMAGNETIC THEORY OFLINESANDSHIELDS 559 whereristhehighfrequency radiusofthetube.ThusRoisthed.-c. resistance ofthetubeifthecurvature isneglected. Thenwehave approximately R 11.sinh11.+sin1tt-=-. ±-Ro2cosh1t-cos11.2r'(87) 0 ~•1\ 8\ , \ \• r\ \ 4 3"" 2 I""~"'- .I 0 0 12345676910 u Fig.J-Thetransfer impedance fromonesurfaceofacylindrical shelltotheother. Thecurverepresents itsratiotothed-c.resistance.iftbetubeisfairlythin.Thecurvature correction ispositive ifthe I. oQ'08 1"'"x 8v o.o. returnpathisexternal, andnegative ifitisinternal. Thegraphof thefirsttermisshowninFig.5. Aninteresting observation canbemadeatoncefromtheformulre (83)fortheself-resistances ofatubularconductor. Ifthefrequency iskeptfixedandthethickness oftheconductor isincreased from0, itsresistance (witheitherreturn)passesthrough asequence ofma.xima and rninima.~5 Thefirstminimum occurswhenu=11"',i.e.,when !iThe~eneral fluctuating character ofthisfunction wasnotedbyMrs.S.P. Mead[12j. ~ ~"" ~ ~ "'N X~ ~~o0u u, , :sIN:I ~x u8 ~IN X Z ~ ~This ensity ayers. Iusor ndrical rded, nitely con-BELLSYSTEM TECHNICAL JOURNAL /(Z"Jg/lf); thefirstmaximum occurswhen" =Z..,et ioninresistance isduetothephaseshiftinthecurrend roceedfromthesurfaceoftheconductor todeepeI ptimum" resistance isRo«"/Z) tanh../Z)=1.44Rop 0'0"• ••\8 .8\ 7 .7 • .•\ , \., \0.' 3 03 0.2 1\ ,0.'\- "/ " ... , 0 , ,/'/ ,0.' \\// \ "--0.2 \ J/'\-0.3 \ I-0.4 \/ I \/.-o.~ \ -0.6 \/ /,, -0.7 -0.& 0,23•,•78•10" u. heratiosofthetransfer resistance andtransfer reactance ofacyli shelltoitsd-c.resistance. ecurvature correctio.n t/2r.Ifcurvature isd5rega oftheoptimum resistance totheresistance ofthinfi nductorwiththesamemtemal diameter astheollowo. 0.' -0.,o. -0.7 -0.6o.o. -0.4o.o.,.560 ;;;; Z0.2 in "iN :x::io. ~~o0u u+' 0 ::tIN:I 8~-0. u0 U ~N-0.2 ~ ~-0.3 ~t=.,r; f1uctuat aswep The lto minusth theratio thickcoFig.4-T ELECTROMAGNETIC THEORY OFLINES ANDSHIELDS 561 ductoristanh.../2=0.92.When,,=2....theratioreaches its firstmaximum coth...=1.004.At1megacycle theoptimum thick­ nessofacopperconductor isabout0.1038mm. Byamethodofsuccessive approximations. H.B.Dwighthas'ob- /'V ./V ..-V I---V~~ ~S2.0 "'u+1 :1:i1.5 %% ~8 Ulu1.0 ~IN3.0 2.' 0.' oo ~5 1.01.5 2~2.5 3~15 4~ 4~~ ~M u. Fig.5-Theskineffectincylindrical shells.Thecurverepresents theratioofthe a·c.resistance ofatypicalshelltoitsd-c.resistance. (88)tainedtheimpedance ofatubularconductor withanexternal coaxial return." Hisfinalresultsappearastheratiooftwoinfinitepower series,whichconverge forallvaluesofthevariables involved. though theycanbeused'advantageously innumerical computations only whenthefrequencies arefairlylowandtheconvergence israpid. Weshallmerelyindicate howDwight's formula andothersimilar formul", canbeobtained directlyfromtheexactequations (75). Letusreplacetheouterradiusbo((75)bya+I.whereIisthe thickness ofthewall,andreplacethevariousBesselfunctions bytheir TaylorseriesinI: on(ul)'l,(ub) ~l,(ua+uI)=I:-,-l,'·'(ua). 11_0n. on(ut).K,(ub)=K,(ua+ut)=I:-,-K,("(ua). n_On. l,(ub)=l,'(ub)=f:(ull'1,'·+I)(ua). RaOn. -K,(ub)=Ko'(ub)=f:(ull'K,'·+II(ua). ".0n. ~."SkinEffectinTubular andFlatConductors," A.I.E. E.Journal, Vol.37 (1918).p.1319. ---,,------ --- 562 BELLSYSTEM TECHN[CAL JOURNAL Wethusobtain ~(ut)o 2LAo-,-Z_1111",,0n. bb-b~(ut)",LAo+1-,- 11_0 1S. whereA11isdefinedas A_II.'(ua) I.(O'(ua)1 • -K.'(ua) K.(o'(ua) • Inspiteofthecomplicated appearance of(90)theA' verysimplefunctions ofua,astheaccompanying list(9(89) (90) areinreality )willshow.27 1Ao=-,ua1A,~­ua'A,=1 q%a2• (91) 1 9 60A.=-+-+_·uau3a3q5a5 Theformula (90)canbemademorerapidlyconverge bypartially summing thenumerator andthedenominator bymeansofhyperbolic functions. Thus,thenumerator becomes ~coshul+sinhut[1\Iii 2ua andthedenominator_3t+...]4a3t'--+a' (O..h+3(utcoshut-sinhut)+\Iiismut 8(ua)' Thereaderwillreadilyseethattherewouldbenodifficulty inusing thismethod toobtainotherexpansions somewhat similarto(89). Forexample, wemightwritea=b-tin(75)andexpress our resultsintermsoftheouterradius. Inthisrespectthemethodthat wehaveusedhasgreaterflexibility thanDwight's; butthereseemsto belittleadvantage gainedfromit,sincethesimpleformulre (82)are sufficient formostpractical purposes. HThevaluesgivenin(91)areexact,notapproximate. Oneofthem,namely, A,~I[,'(.a) [.(.a)1_~, X.'(fTa} K,(a-a) era isoneofthefundamental identities foundinallbooksonBesselfunctions. The restareconsequences ofanalogous, though essfamiliar, identities. Thegeneral expressions forthecoefficients A.wereobtained byH.Pleijel[20]. ELECTROMAGNETIC THEORY OFLINES ANDSHIELDS 563 INTERNAL IMPEDANCES OFLAMINATED CONDUCTORS Sofarwehavesupposed thatallconductors werehomogeneous. Weshallnowconsider asomewhat moregeneralconductor composed ofncoaxiallayersofdifferent substances. Asbefore,weareinterested infindingexpressions fortheinternal impedances jbesides, wemay wishtoknowhowthetotalcurrentisdistributed between thedifferent layersoftheconductor. Tobeginwith.letussuppose thatacoaxialreturnpathisprovided outsidethegivenconductor. Wenumber ourlayersconsecutively and calltheinnerlayerthefirst.LetZ,,(m)andZ,,(m)bethesurface impedances ofthemthlayer,thefirstwhenthereturnisinternal, the otherwhenitisexternal; andletZa,,(III)bethetransfer impedance fromonesurfacetotheother.Formul", fortheseimpedances have already beenobtained inthesectionunder"TheSurface Impedances ofHollowCylindrical Shells," page552.Also,let",,(m)bethes"rface impedance ofthefirst",layerswithexternal return;thatis,theratio ofthelongitudinal electromotive intensity attheoutersurfaceofthe 1/Ithlayertothetotalcurrent1minall'"layers." Byhypothesis, thereisnoreturnpathinsidethelaminated con­ ductorasawhole. Hence, \vhenwefixourattention onanyonc layeralone,saythe",th,wemaysaythatthecurrentinthislayer returnspartlythrough the1/1-1layerswithinit,andpartlyoutside. Inthem-1innerlayers,however, thecurrent isassumed tobe 1m_Iintheoutward direction---or whatamounts tothesamething -1m_Iinthereturndirection. Henceweconclude that,ofthe current1m-1m_Iinthelayerunderdiscussion, 1mreturns outside and-!m-Ii11side. Substituting thesevaluesinTheorem 2on page555,wefindthattheelectromotive intensity alongtheinner surfaceofthelayerisZab(m)!m.- Zaa(ml!m_l . .Buttheiunersurfaceofthemthlayeristheoutersurface ofthe composite conductor comprising them-1innerlayers,andby Theorem 1,theelectromotive intensity onthisoutersurface is Z/Ib(m-l)!m_I' Asthetwomustbeequal,weobtainanequation from whichwecandetermine theratioofthecurrent flowing inthefirst ?It-1layerstothatflowing in?Itlayers. Thisis Zllb(m) Zllll(ml+Zbb'm 1)(92) Inthisformula fortheeffectofanextralayeronthecurrent dis- 28Inthisnotation, thecurrentflowinginthemthlayeris1m-/"'_1.Itshould alsobenotedthatZbl\(I)=Zb,,{l). 564 BELLSYSTEM TECHNICAL JOURNAL tribution, itwillbenotedthatthedenominator istheimpedance (withinternal return)oftheaddedlayerplustheoriginalimpedance. Wenowconsider theelectromotive intensity ontheoutersurface ofthemthlayer,whichiszoo(m)Imontheonehand,and(Zoo(m)Im -Zab(m)Im~l) ontheother.Thus,wehavethefollowing equation, [Z'b(m)]'(93) expressing theeffectofanadditional layerupontheimpedance ofthe conductor. Thisequation isaconvenient reduction formula. Starting with thefirstlayer(forwhichZoo(l)~Zoo(l)),weaddtheremaining layers onebyoneandthusobtaintheimpedance ofthecomplete conductor intheformofthefollowing continued fraction: (94) Wecanalsogetareduction formula forthetransfer impedance between theinnerandoutersurfaces ofthecomposite conductor formedbythefirstmlayers. Todoso,itisonlynecessary tonote that,sincetheinnersurfaceofthefirstm-1layersisalsotheinner surfaceofthefirstmlayersaswell,theelectromotive intensity on thatsurface canbeexpressed eitherasZab(m-l)[m_l orasZab(m)!mo Thus,wehave (95) Bynotingthat Zabel)=Zab{l) ,wecandetermine successively the transfer impedances acrossthefirsttwolayers,thefirstthree,and soon.Thisformula isnotquiteassimpleas'(94),owingtothe presence ofZbb(m-l)initsdenominator, anditistherefore notexpedient toevaluate z",,(m)explicitly; butitisnotprohibitively cumbersome fromthenumerical standpoint whenthecomputations aremadestep bystep. Although indeducing equations (93)and(95)wesupposed thatthe addedlayerwashomogeneous, theequations arecorrectevenifthis layerconsists ofseveralcoaxiallayers,provided Zaa(m+l) andZab{m+l) areinterpreted astheimpedances oftheaddednon-homogeneous layerintheabsence oftheoriginal coreofmlayers. Theselatter ELECTROMAGNETIC THEORY OFLINESANDSHIELDS 565 impedances themselves havetobecomputed bymeansofequations (94)and(95). Ifthereturnpathisinsidethelaminated conductor, insteadof outside, formulre (92)and(93)stillhold,provided weinterchange aandb,andcountlayersfromtheoutsideinsteadoftheinside,so thatm=1istheoutermost, ratherthantheinnermost, layer. Thebasicrulefordetermining thesurfaceimpedances oflaminated conductors canbeputintothefollowing verbalform: Theorem 3:Lettwoco,uiuctors, bothofwhichmaybe,oodeupofcoaxial layers,fittightlyoneinsidetheother.Anysurfaceself--impedance ofthecompourul coruluctor equalstheirulirtidual impedance ofthe coruluctor nearesttothereturnPOll.diminished bythefraction whosenumerator isIhesquareofthetransferimpedance acrossthis cond1lctor andwhosedenominator isthe$1tmofthesurfaceimpedames ofthetwocomp01..nlcoruluctors ifeachisregarded asthereturn Pathfortheother.Thetransferimpedance ofthecompound con­ ductoristhefractionwhosenumerator istheproductofthetransfer impedances ofthei,uiirtidual c01uiuctors a,uiwhosed...omi'llltor is thatoftheself-impedance. Iftwocoaxialconductors areshort-circuited atintervals, sbort compared tothewave-length, theabovetheorem holdsevenifthe conductors donotfittightlyoneovertheother,provided weaddin thedenominators athirdtermrepresenting theinductive reactance of thespacebetween theconductors. DISKS ASTERMINAL IMPEDANCES FORCOAXIAL PAIRS SOfarwehavebeenconcerned onlywithinfinitely longpairs.We nowtakeupaproblem ofadifferent sort;namely, thedesignofa diskwhich,whenclappedontheendofsuchapair,willnotgiverise toareflected wave. Thelineofargument willbeasfollows:Tobeginwith,weshall assumeadiskofarbitrary thickness I.,compute thefieldwhichwill besetupinit,andthenadjustthethickness soastomakethisfield matchthatwhichwouldexistinthedielectric ofaninfiniteline. Thefieldinthediskhastosatisfyequation (2)whereiw,canbe disregarded bycomparison withg.Thus,wehave aH.~_gE,!(pH.)=gE dz Jpdp :, aE.iJE,.H-a--iJ..='f,WJ.l fl·P•(96) 566 BELLSYSTEM TECHNICAL JOURNAL Inthedielectric between thecoaxialconductors, thelongitudinal displacement currentdensity isverysmall;infact,itwouldbezero iftheconductors wereperfect. Thiscurrent density iscontinuous acrossthesurfaceofthediskand,therefore, gE.isexceedingly small. Hence,thesecondoftheaboveequations becomes approximately sothata(pH.) ~o. ap ,(97) PH.=-, (98)p wherePisindependent ofpbutmaybeafunction ofz.Underthese conditions, theremaining twoequations are aE, .Iiaz= -'J,wp.-'PIaaH•= -gE,.z(99) Fromtheformoftheseequations andfrom(98),weconclude that thegeneralexpressions fortheintensities inthediskare u[Be-"-A...·JE,=--=-------=gp ,(100) whereu=.Jgw~i. Ontheoutsideflatsurfaceofthedisk(givenbyz=hwherehis thethickness oftheplate),themagnetomotive intensity isverynearly zero; 29therefore, Fromthisweobtain AA..'+Be-"=O. B=C....(101) (102) whereCissomeconstant. Thusequations (100)canbewrittenas follows: H=Csinhu(h-z) • p • E=uCcoshu(h-z). P gp I(103) andattheboundary between thediskandthedielectric ofthetrans­ missionline(z=0),wehave E,=~cothuh. (104)H.g HOnaccountofthenegligibly smalllongitudinal currentinthedisk. ELECTROMAGNETIC THEORY OFLINES ANDSHIELDS 567 Ontheotherhand,ifthereistobenoreflection ..[pJ;byequation (24).Hence .":cothulL=~.gW IfulLissmall,cothulLequalsapproximately lfulL,andthismustequal (105) 1I' IL~g,~cm. (106) Undertheseconditions, thegeneralized fluxofenergyacrossthe innersurfaceofthediskis,inaccordance withthetextunderlIThe Complex Poynting Vector," page555,andequation (14), 1"1'" 1r;;" •"E,H..pdpd<p~27r\I;log~,I'. (lOi) Thus,theimpedance ofthisdiskisapureresistance equaltothe characteristic impedance ofthecoaxialpair. CYLINDRICAL WAVES ANDTHEPROBLEM OFCYLINDRICAL SHIELDS 3. Itiswellknownthatwhentwotransmission linesaresidebyside, toagreater orlesserextenttheyinterfere witheachother.This interference isusuallyanalyzed into jjelectromagnetic crosstalk" and IIelectrostatic crosstalk." Thus,electriccurrents inapairofparallelwiresproduce amagnetic fieldwithlinesofforceperpendicular tothewires.Theselinescut theotherpairofwiresandinduceinthemelectromotive forces,thereby producing whatisusuallycalledthe..electromagnetic crosstalk"; thiscrosstalk isseentobeproportional tothecurrent flowing inthe firstpair.The"electrostatic crosstalk/' ontheotherhand,iscaused byelectriccharges induced onthewiresofthesecondsystem; these charges areproportional tothepotential difference existing between thewiresofthe"disturbing" transmission line. Thedistinction between twotypesofcrosstalk isvalid,although theterminology issomewhat unfortunate; theword"electromagnetic" isusedintoonarrowasenseandtheword"electrostatic" isacon­ tradiction intermssinceelectriccurrents andchargesinatransmission linearevariable. Theterms"impedance crosstalk" and"admittance crosstalk" wouldbepreferable because theformerisduetoadis­ tributed mutualseriesimpedance between twolinesandthelatteris produced byadistributed mutualshuntadmittance. ~oSincethispaperwaswritten, arelatedpape.rhasbeenpublished byLouisV. King[18J.However, thephysical pictureheredeveloped appears tobenew. Theearliest writerwhotreated theproblem ofelectromagnetic shielding isH. Pleijel[21J. . 568 BELLSYSTEM TECHNICAL JOURNAL Thecrosstalk between twoparallel pairs(thisappliestotwisted pairsaswell)canbereduced byenclosing eachpairinacylindrical metallic shield.Itistheobjectofthisandthefollowing twosections todevelopatheoryforthedesignofsuchshields. Thistheoryisbaseduponanassumption thatinsofarastheradial movement ofenergytowardandawayfromthewiresisconcerned we candisregard thenon-uniform distribution ofcurrents andcharges alongthelengthofthewires.Noseriouserrorisintroduced thereby aslongastheradiusoftheshieldissmallbycomparison withthewave­ length. Thefieldaroundthewiresisconsidered, therefore, asdueto superposition oftwotwo-dimensional fieldsofthetypesgivenby equations (4)and(5). Theactualcomputation oftheeffectiveness ofagivenshieldwill bereduced toananalogous problem inTransmission LineTheory. Equations (4)and(5)aretoogeneral astheystand.Strictly speaking theeffectofashielduponanarbitrary two-dimensional fieldcannotbeexpressed byasinglenumber. Thefieldatvarious pointsoutsidetheshieldwillbereduced byitindifferent ratios. However, anysuchfieldcanberesolved intol'cylindrical waves," eachofwhichisreducedbytheshieldeverywhere inthesameratio. Moreover, toallpractical purposes thefieldproduced byelectric currents (orelectric charges) inapairofwiresisjustsuchapure cylindrical wave. SincebothEandIIareperiodic functions ofthecoordinate <p, theycanbe.resolved intoFourier series.Thename"cylindrical waves"willbeappliedtothefieldsrepresented bytheseparate terms oftheseries. Asthenameindicates thewavefrontsofthesewaves arecylindrical surfaces, although owingtorelatively lowfrequencies andlongwave-lengths usedinpractice theprogressive motion of thesewavesisnotclearlymanifested exceptatgreatdistances from thewires. Turni,og ourattention specifically tomagnetic ~ylindrical waves ofthenthorder,andwritingthefieldcomponents tangential tothewave frontsintheformEcosn<pandIIcosn<p,wehavefromequations (5): dE= _iwp.IIdp • Fromtheseweobtaind(pII)~_ [(g+iw.)p+-:!-]E. dp 1W~p(108) d'E.dE p'dp'+pdp=[iwp.(g+iw.)p'+n'JE. (109) ELECTROMAGNETIC THEORY OFLINES ANDSHIELDS 569 Thisequation, beingofthesecondorder,possesses twoindependent solutions: onefordiverging cylindrical wavesandtheotherforreflected waves. TheratioofEtoHinthefirstcaseanditsnegative inthe secondwillbecalledtheradialimpedallce offeredbythemedium to cylindrical waves. Inthenextsectionweshalldetermine radialimpedances indi­ electrics andmetalsandshowthatforallpractical purposes the attenuation ofcylindrical wavesinmetalsisexponential. Thesig­ nificance oftheradialimpedance is'thesameasthatofthecharac­ teristicimpedance ofatransmission line.V\lhenacylindrical wave passesfromonemedium intoanother, areflection takesplaceunless theradialimpedances arethesameinthetwomedia.ThusifEo andHoaretheimpressed intensities (attheboundary between thetwo media),E,andH,thereflected andEIandH"thetransmitted intensities, wehave Eo+E,~EIandHo+H,=lit, (110) sincebothintensities mustbecontinuous. Ontheotherhand,ifkis theratiooftheimpedance inthefirstmedium tothatinthesecond, thenequations (110)become kHo-kIl,=HI Solvingweobtain 2k HI=k+1Hoand andIlo+Il,=Ih (111) (112) Thereflection losswillbedefinedas (113) Whenawavepassesthrough ashield,itencounters twoboundaries andiftheshieldiselectrically thick,thatis,iftheattenuation ofthe waveintheshieldissogreatthatsecondary reflections canbedis­ regarded without introducing aseriouserror,thetotalreRection loss isthesumofthelossesateachboundary. Thefirstlosscanbecom­ puteddirectly from(112)andthesecondfromthesameequation if wereplacekbyitsreciprocal. Thus,thetotalreflection lossfor electrically thickshieldsis Ik+11'R~20log,,'41kldecibels. (114) 570 BELLSYSTEM TECHNICAL JOURNAL Whentheratiooftheimpedances isverylargebycomparison with unity,theformula becomes IklR~2010g!OT' (115) andwhenkisverysmall,then 1R~20log!041kl . (116) Inthenextsectionweshallseethattoallpractical purposes, the waveintheshieldisattenuated exponentially. Ifaistheattenuation constant innepersandiftisthethickness oftheshield,thenthe attenuation lossis A=8.686"tdecibels (117) andthetotalreduction inthemagnetomotive intensity duetothe presence oftheshieldiss~R+A. (118) Theelectromotive intensity isreduced inthesameratio. Butiftheshieldisnotelectrically thick,acorrection termhasto headdedtothereflection loss.Thiscorrection termcanbeshown tobe31 C201I(k-l)',r'ld'bl~og!O1 -(k+1)'e-eCIes, (119) andifkisverylargeorverysmallbycomparison withunitythen C'"3 -8.686at+1010g!O(cosh2at-cos2{3t). (120) Equation (120)doesnotholddowntot=0;whenrtisnearlyzero, then I(k-1)'IC~2010g!O1 -(k+1)'. (121) Sofarwesupposed thattheshieldswerecoaxialwiththesource. IfthisisnotSOtitisalwayspossible toreplaceanygivenlinesource withintheshieldbyanequivalent systemoflinesourcescoaxialwith theshieldandemitting cylindrical wavesofproperorders. Mathe­ matically thisamounts toachangeoftheoriginofthecoordinate system. Inthenextsectionweshallseethattheshielding effective­ nessisnotthesameforallcylindrical waves. Thismeans,ofcourse, thatiftheshieldisnotcoaxialwiththesource,thetotalreduction in 31Here,r=a+ifjisthepropagation constant intheshield. ELECTROMAGNETIC THEORY OFLINESANDSHIELDS 571 thefielddepends upontheposition ofthemeasuring apparatus. The variation isverysmall,however, unlessthesourceisalmosttouching theshieldanditcanbestatedthatapproximately theshielding effectiveness isindependent oftheposition ofthesource. Itisinteresting toobserve fromtheaccompanying tablesthat whiletheattenuation lossisgreaterinironthanincopper,thereflection lossisgreateratacoppersurface. Infact,atsomefrequencies the impedances ofironandairnearlymatchandpractically noreRection takesplace.Hence,athincoppershieldmayhemorceffective than anequally thinironshield. Andifacomposite shieldismadeof copperandiron,theshieldwillbemoreeffective ifcopperlayersare placedonlheoutsidetotakeadvantage oftheaddedreflection. TABLE 1 THEABSOLUTE VALUE OFTHERADIAL IMPEDANCE OFFERED BYAIRTOCYLINDRICAL MAGl\'ETIC \\lAVES OFTHEFIRST ORDER <INMICROHMS} f.R"ldius -0.5em. Iem. 2em. 1cycle... 0.0395 0.07896 0.1579 10cycles..... 0.395 0.790 1.58 100cycles........... 3.95 7.90 158 1kilocycle ... 39.5 79.0 158. 10kilocycles ... 395. 790. 1,580. 100kilocycles ........3,950. 7,900. 15,800. 1megacycle........39,500. 79,000. 158,000. 10megacycles. 395,500. 790,000. 1.58ohms 100megacycles ... 3.95ohms 7.9ohms 15.8ohms TABLE II THEINTRINSIC IMPEDANCE OFCERTAIN METALS (T/)f(-vi) INMICROHMS Copper Le,d Aluminum lron g=5.8005X10'g=4.S0i7X101g-}I()lIJg..l~mhos/em. f mhos/em. mhos/em. mhos/em. ,.-1.257,.h/em. ,.-0.01257 ",h/em.,.-0.01257 ",h/em.,.-0.01257 ,.h/em. ""(100relative tocopper) 1cycle....... 0.369 1.28 0.487 8.88 10cycles..'". 1.17 4.05 1.54 28.1 toOcycles... 3.69 12.8 4.87 88.8 1kilocycle.... 11.7 40.5 15.4 281. 10kilocycles ... 36.9 128. 48.7 888. 100kilocycles ...117. 405. 154. 2,810. 1megacycle ..369. 1,280. 487. 8,880. 10megacycles .1,170. 4,050. 1,540. 28,100. 100megacycles .3,690. 12,800. 4,870. 88,800. TABLE III THEINTRINSIC PROPAGATION CONSTANT OFCERTAIN METALS Copper !..<ad Aluminum Iron g=5.8005X!OImhosem.f-4.8077X10'mhos/em. g-~l()lmhos/em. Il-lQ1mho~cm. I.l...0.01257 ,..II/em. I.l-0.01257 ,..h/cm. ,..-0.01257 ,..h/em. ,..-1.257,..hem. f..(100relativetocOPDer) t7•nepen..!-indb/em.u.ne~rs..!"indb/em.u.nepers..!!....indb/em.fT•nepers•.I -m-- -m-- -m-- -m-- "',Indbem.viem.,Ii -Viem.Vi Vi.em.Vi ..Jiem. 1cycle.................. 0.214 1.86 0.0616 .535 0.162 1.41 0.888 7.72 10cycles.. . . . . ......... 0.677 5.88 0.195 1.69 0.513 4.46 2.81 24.4 100cycles................... , .2.14 18.6 0.616 5.35 1.62 14.1 8.88 77.2 1kilocycle ..........•....... 6.77 58.8 1.95 16.9 5.13 44.6 28.1 244. 10kilocycles.............. .... 2\.4 186. 6.16 53.5 16.2 141. 88.8 772. 100kilocycles .................. 67.7 588. 19.5 169. 51.3 446. 281. 2,440. 1megacycle ................ 214. 1,860. 61.6 535. 162. 1,410. 888. 7,720. 10megacycles ............ 677. 5,880. 195. 1690. 513. 4,460. 2810. 24.400. 100megacycles ............ 2140. 18,600. 616. 5350. 1620. 14,100. 8880. 77,200.'"~ N ~ v, CJ t;J i!; t;J @ ~ t-< C3 ~ ~ t-< (122)ELECTROMAGNETIC THEORY OFLINESANDSHIELDS 573 CYLINDRICAL WAVES INDIELECTRICS ANDMETALS Ingooddielectrics gissmallbycomparison with WEandthefirst termontherightin(109)verynearlyequals(hpjX)' whereXisthe wave-length. Butweareinterested inwave-lengths measured in milesandshieldswithdiameters measured ininches;thusweshall write(109)inthefollowing approximate form: . d'EdE p'dp'+pdp-n'E.=o. Whenn+0therearetwoindependent solutions andwhen 11=0, E,=logpand andE,~I.(123) (124) Thecorresponding expressions forHare,by(108), inthefirstcase,andand H~= (125) 1=-.- ~W}J.pandH,=0, (126) inthesecond. ThesecondcaseinwhichE1and1-11aretheelectromotive andmag­ netomotive intensities intheneighborhood ofanisolatedwirecarrying electriccurrentisofinteresttousonlyinsofarasithelpstointerpret (123)and(125).Ifweweretoconsider 2ninfinitesimally thinwires equidistributed uponthesurfaceofaninfinitely narrowcylinder, the adjacent wirescarrying equalbutoppositely directed currents of strength sufficient tomakethefielddifferent fromzero,andcalculate thefield,weshouldobtaineq)ressions proportional toE,andH,. Anactualclusterof2".wiresclosetogether wouldgenerate principally acylindrical waveoforderfl.;thestrengths ofothercomponent waves oforder3n,5n,etc.rapidlydiminish asthedistance fromthecluster becomes largebycomparison withthedistance between theadjacent wiresofthecluster. Forthepurposes ofshielding designwecan regardapairofwiresasgenerating acylindrical waveofthefirst order(n=1).Theradialimpedance ofannthorderwaveis E1iwJ.lp Zp=Ii l=-n-' (127) 574 BELLSYSTEM TECHNICAL JOURNAL andthatofthecorresponding reflected wavehasthesamevalue. Itshouldbenotedthatbythe"reflected" cylindrical waveinthe spaceenclosed byashield,wemeanthesumtotalofaninfinitenumber ofsuccessive reft.ections. Eachofthelatterwavescondenses onthe axisanddiverges againonlytobere-reflected back;inasteadystate allthesereflected wavesinterfere witheachotherandformwhat mightbecalleda"stationary reflected wave." Notbeinginterested inanyotherkindofreflected waveswetookthelibertyofomitting thequalification. Inconductors theattenuation ofawaveduetoenergydissipation ismuchgreater(exceptatextremely lowfrequencies) thanthatdue tothecylindrical divergence ofthewave.Hence,intheshieldwe canregardthewaveasplaneandwrite(108)inthefollowing approxi­ mateform: ~:= -iwpH,dHd;=-gE. (128) Inform,theseareexactlylikeordinary transmission lineequations. Hence,inashieldtheradialimpedance issimplytheintrinsic im­ pedance ofthemetal, Zp=71=~Ohms, andthepropagation constant, u=-.!iwl'g=-.J'lrfl'g(1+i)nepers/cm.(129) (130) Theexactvalueoftheradialimpedance inmetalscanbefoundby solving(108).Thus,wecanobtain fordiverging waves,andKn(up) Zp~-~""'K;;'n-,;,T( u'-'p~) z_In(up) p-~In'(up)(131) (132) forthereflected waves. Cylindrical wavesoftheelectrictypecanbetreatedinthesame manner. Itturnsoutthatthetransmission lawsinmetalsareidentical withthoseformagnetic waves. Theradialimpedance inperfect dielectrics, ontheotherhandisgivenby "Zp=-.-. ~WEP(133) ELECTROMAGNETIC THEORY OFLINESANDSHIELDS 575 Thisisenormous bycomparison withtheimpedance inmetals, therebyexplaining analmostperfect" electrostatic" shielding offered bymetallicsubstances. Evenwhenthefrequency isashighas100kc. theradialimpedance ofair1em.fromthesourceisabout36X10' ohmswhiletheimpedance ofacoppershieldisonly117X10-'ohms. Thereflection lossisapproximately 220db. POWER LOSSES INSHIELDS Aswehaveshowninthetextunder"TheComplex Poynting Vector," page555,theaverage powerdissipated inaconducto~ isthe realpartoftheintegral <I>=1/2ff[EH*J.dS takenoverthesurface oftheconductor. Ifthesourceofenergyisinsideashield,the integration needbeextended onlyoveritsinnersurface, because theaverage energyflowingoutward through thissurfaceisalmost entirely dissipated intheshield,theradiation lossbeingaltogether negligible. Ifacylindrical wavewhoseintensities attheinner surfaceoftheshieldofradius IIa"are Ii"=Hocosntp,IIp=Hosinnip, (134) ~="iwl'lJ/"beingtheradialimpedance inthedielectric, isimpressed upontheinnersurfaceoftheshield,areflected waveissetup.The resultant ofthemagnetomotive intensities inthetwoisreadilyfound tobe(2k/k+l)Ho,wherekistheratiooftheradialimpedance of thedielectric columninsidetheshieldtotheimpedance Zlooking intotheshield.Iftheshieldiselectrically thick,theimpedance Zis obviously theradialimpedance oftheshield;otherwise itismodified somewhat byreflection fromtheoutsideoftheshield.Theaverage powerlossintheshieldpercentimeter oflength IS,then,thereal partof _27rakk*Z * <I>-(k+l)(k*+1)HoHo. Thisbecomes simply <I>=2"aZHoll o*,(135) (136) ifthefrequency is sohighthatkislargeascompared withunity. Ifthesourceoftheimpressed fieldisapairofwiresalongtheaxis oftheshield,themagnetomotive intensity onthesurfaceoftheshield canbeshowntobe IH.=-2,Jcos'1',"a(137) 576 BELLSYSTEM TECHNICAL JOURNAL where1istheseparation between theaxesofthewires.Therefore, kk*I'Z <l>~2.-a'(k+1)(k*+1)I'. (138) RESISTANCE OFNEARLY COAXIAL TUnULAR CONDUCTORS Whentwotubularconductors arenotquitecoaxial, aproximity effect 32appears whichdisturbs thesymmetry ofcurrentdistribution andtherefore somewhat increases theirresistance. Thiseffectcan beestimated bythefollowing method ofsuccessive approximations. Tobeginwith,weassumeasymmetrical currentdistribution inthe innerconductor. Themagnetic fieldoutsidethisconductor istben tbesameasthatofasimplesourcealongitsa.xisandcanbereplaced byanequivalent distribution ofsourcessituated alongtheaxisoftbe outerconductor. Theprincipal component ofthis distribution isa simplesourceoftbesamestrength astbeactualsourceanddoesnot enterintotheproximity effect.Thenextlargestcomponent isa doublesourcegivenby iwplIEl=-2--C05 0,7rr IIHs=-22C058,"7(139) whereIistheinteraxial separation, risthedistance ofatypicalpoint ofthefieldfromtheaxisoftheouterconductor, and9istheremaining polarcoordinate. Thisfieldisimpressed upontbeinnersurfaceoftheouterconductor 33 andtheresulting powerlossequals,byequation (136),there,,1partof (I)'I'eI>=2'1l'"a'1 21l'"a212 =2.".a'Tjl", (140) whereathighfrequencies ~~-Jiwp./gissimplytbeintrinsic impedance oftheouterconductor.3f Thislossincreases theresistance ofthe outertubebytheamount, l1R.=~/4 (141)a3'J1f'g' UForpromixity effectinparallelwiresexternal toeachother,thereaderis referred tothefollowing papers:JohnR.Carson[1],C.Manneback [9],S.P. Mead[12J. IITheradiusofthissurfaceisdesignated bya. NAtlowfrequencies" hastobereplaced bytheradialimpedance lookinginto theshield. (142)ELECTROMAGNETIC THEORY OFLINESANDSHIELDS 577 inexcessoftheconcentric resistance R.=(1/2a)Vp.f/Tg givenby (84).Therelativeincrease is,therefore, tJ.R.21'R.=a'• Themagnetic field(139)ispartially reflected frointheoutertube, impressed upontheinnerconductor, partially refracted intoitand dissipated there.Using(110)and(111)wecanshowthatthereflected fieldis IIJI,=?:icos 0,."a iwp.1IEr=-2..pcosO."a'(143) Thisfieldconverges totheaxisoftheouterconductor. Inorderto estimate itseffectupontheinnerconductor, itisconvenient toreplace itbyanequivalent fieldconverging towardtheaxisoftheinner conductor. Byproperly changing theoriginofthecoordinate system thisequivalent fieldcanbeshowntobe iwp.1IE.= --2•(I+pcos",),"a' IIH.= --2,cos'",."a(144) Applying oncemore(138)(replacing thereabytheradiusbofthe innerconductor), wefindthatthepowerlossduetothisfieldisgiven bytherealpartof (145) sothattheabsolute increase inresistance oftheinnerconductor is (146) whichmustbeaddedtotheconcentric resistance oftheinnercon­ ductorR,=(1/2b),Jp.f/"g. Therelativeincrease istherefore (147) Itisunnecessary tocarrytheprocessfurther. 578 BELLSYSTEM TECHNICAL JOURNAL Considering thepairasawhole,theresistance whenconcentric is R=R.+R"andtheincrease duetoeccentricity isdR=dR.+tiR, thusgivingapercentage increase, ~=2(~)(~r(148) Itisobvious that,solongasbandIaresmallcompared witha, thispercentage increase isverysmall. Fromthewell-known formul", fortheinductance andthecapacity between parallelcylindrical conductors, wefindthatthecharacteristic impedance ofanearlycoaxialpairisgivenintermsofthecharacteristic impedance ofthecoaxialpairby (k'e'k' ] 1)logk '(149) wherethe"eccentricity" eisdefinedastheratiooftheinteraxial separation totheinnerradiusoftheouterconductor andkastheratio oftheinnerradiusoftheouterconductor totheouterradiusofthe innerconductor. Combining (149)and(148)wehavefortheattenua­ tionofthenearlycoaxialpair: ex=aD[12e2 +k+(k'e2k1.]. 1)logk(ISO) REFERENCES Papers 1.JohnR.Carson, "Wave Propagation OverParallel Wires:TheProximity Effect,"Phil.Mag.,Vol.41,Series6,pp.607-{)33, April,1921. 2.JohnR.CarsonandJ.J.Gilbert, "Transmission Characteristics oftheSub­ marineCable,"Jour.Franklin Instituk, p.705,December, 1921. 3.johnR.Carsonandj. j.Gilbert, "Transmission Characteristics oftheSub­ marineCable,"BellSys.Tuh.Jour.,pp.88-115,july,1922. 4.johnR.Carson,"TheGuidedandRadiated EnergyinWireTransmission," A.I.E.E. J014r.,pp.908-913. October. 1924. 5.johnR.Carson,"Electromagnetic TheoryandtheFoundations oftheElectric CircuitTheory," BellSys.Tuh.Jour.,january, 1927. 6.S.Butterworth, ..EddyCurrent LossesinCylindrical Conductors, withSpecial Applications totheAlternating Current Resistances ofShortCoils,"Phil. Trans.,RoyalSoc.ojLondon, pp.57-100,September, 1921. 7.H.B.Dwight, ..SkinEffectandProximity EffectinTubular Conductors," A.I.E.E. Jour.,Vol.41,pp.203-209, March,1922. . 8.H.B.Dwight, "SkinEffectandProximity EffectinTubular Conductors," A.I.E.E. Trans.,Vol.41,fP.189-195.1922. 9.C.Manneback, "AnIntegra Equation forSkinEffectinParallelConductors," Jour.ojAfath.andPhysics, April,1922. 10.H.B.Dwight, "APreciseMethod ofCalculation ofSkinEffectinIsolated Tubes,"A.I.E.E. Jour.,Vol.42,pp.827-831, August, 1923. 11.H.B.Dwight, "Proximity EffectinWiresandThinTubes," A.I.E.E. Jour., Vol.42,pp.961-970. September, 1923:Trans.,Vol.42,pp.85(}-859, 1923. ELECTROMAGNETIC THEORY OFLINES ANDSHIELDS 579 12.Mrs.S.P.Mead,"WavePropagation OverParallelTubular Conductors: The Alternating Current Resistance," BellSys.Tech.JOflr.,pp.327-338, April, 1925. 13.Chester Snow,"Alternating Current Distribution inCylindrical Conductors," Scie,dijic Pa~rsof'heBureauofStandards, o.509,1925. 14.johnR.CarsonandRayS.Hoyt,"Propagation ofPeriodic Currents overa SystemofParallelWires,"BellSys.Tech.Jour.,pp.495-545, July,1927. 15.JohnR.Carson,"Rigorous andApproximate Theoric3 ofElectrical Transmission AlongWires,"BellSys.Tech.Jour.,January, 1928. . 16.JohnR.Carson,"WireTransmission Theory," BellSys.Tech.Jour.,April,1928. 17.A.Ermolaev, "DieUntersuchung desSkineffektes-Drahten mitComplexer Magnetischer Permeabilitate," Archiv.f.Elektrolech"ik, Vol.23,pp.101-108, 1929. lB.LouisV.King,"Electromagnetic Shielding atRadioFrequencies," Phil.},{ag., Vol.15,Series7,pp.201-223, February, 1933. 19.E.].SterbaandC. B.Feldman, "Transmission LinesforShort·WaveRadio Systems," Proc.I.R.E.,]uly,1932,andBellSys.Tech.Jour.,July,1932. 20.H.Pleijel,"Berakning afMotstand ochSjaHinduktion," Stockholm, K.L. Beckmans Boktryckeri, 1906. 21.J-f.Pleijel,"Electric andMagnetic Induction Disturbances inParallelConducting Systems," 1926,Inge"iorsvelenskapsakadem;ens Handli"gar NR49. 22.].Fisher," DieaJlseitige inzweiKreiszylindrishen, konaxial geschichteten Stoffen beiax..ialerRichtung desWechselstromes," ]ahrbuch derdrahtlosen Tele­ graphieundTelephol,lie, Band40,1932,pp.207-214. Books J.ClerkMaxwell, "Electricity andMagnetism," Vols.1and2. O.Heaviside, "Electrical Papers." SirWilliam Thomson, ..Mathematical andPhysical Papers." LordRayleigh, "Scientific Papers." SirJ.]_Thomson, "RecentResearches inElectricity andMagnetism." A.Russell,"ATreatise ontheTheoryofAlternating Currents." JohnR.Carson,"Electric CircuitTheoryandtheOperational Calculus." R.W.Pohl,"Physical Principles ofElectricity andMagnetism." MaxAbraham andR.Becker,"TheClassical TheoryofElectricity andMagnetism." E.JahnkeandF.Emde,"Tables ofFunctions," B.G.Teubner, 1933. Note:Thislistofreferences isbynomeanscomplete. Onlythemorerecent papersdealingwithsomephaseofthesubjecttreatedhereareincluded.