bstj14-2-179 cabe crosstalk Hunter Booth

TheBellSystem Technical Journal Vol. XIV April, 1935 No.2 Cable Crosstalk—Effect ofNon-Uniform Current Distribution inthe Wires ByR.N,HUNTER and R.P.BOOTH “a te elt caghsb heyaepl, themti ‘quencies.Becauseofthenon-uniform andnon-symmetrical distribution Sen ea daa ahi a ahurea he meee ote conto gees muawithfrequency. "Thispaperdiscusses theresults ofcertain measurements THIN meiATigceeeeeane risupdlceger eearearrange FOR manyyearsithasbeenrecognized thatnon-uniform distribu-tion ofcurrent over the cross-section ofaconductor reduces the efficiency oftransmission ineither power orcommunication circuits. With direct current orwith alternating current ofvery lowfrequency the current isdistributed almost uniformly. Asthe frequency in- creases, the current distribution becomes more and more non-uniform. Ifthe two conductors ofacircuit are remote from each other the high-frequency current distribution ineither conductor ispractically symmetrical with respect toitscenter, thedensity ofthecurrent being lowest inthecenter ofthe conductor and highest near the surface of the conductor. If,however, the conductors are close together, the high-frequency current distribution ineither conductor isunsym- metrical due totheproximity oftheother conductor. This isknown astheproximity effect. Itisprobably notwell known that this proximity effect may have animportant bearing onthecrosstall between communication circuits. While theeffect isnegligible inopen-wire circuits, itisquite marked in cable circuits. This paper describes aninvestigation ofthe influence“ThiseflectwamentionedintheCarsonHoyepaperon“Propagation ofPratiCurrentsOveraSystemofParallelWires,”intheBellSystemTeeknicalJournalof July, 1927. 9 180 BELLSYSTEM TECHNICAL JOURNAL ofthe proximity effect oncrosstalk between long non-loaded cable circuits which arebeing studied inconnection with thedevelopment of high frequency carrier systems suitable fortoll telephone cables. More specifically, thepaper covers tests made todetermine thein- fluence ofthe proximity effect onthe mutual inductance between circuits; data aregiven both forthecase oftwo isolated non-twisted pairs and forthecase ofpairs inaquadded 19-gauge cable. Incable carrier systems itisnot practicable tooperate like fre- quency bands inopposite directions ondifferent pairs inthesame cable without heavy shields between the pairs. The relatively large level differences that may exist between pairs transmitting inopposite directions would result inexcessive crosstalk ofthe near-end type. Like carrier frequency bands are, therefore, transmitted inthesame direction inacable and thecrosstalk between pairs used forcarrier systems isofthefar-end type. Tthas been shown? that far-end crosstalk atcarrier frequencies between long non-loaded cable pairs can beconsiderably reduced by theuseofsimple networks connected between thetwo pairs atone point intheir length. The crosstalk balanced outbysuch networks is ofthe“transverse” type. Crosstalk oftheinteraction type varies inacomplicated way with frequency,andcannot,therefore,beannulled byasimple network. Foranytwosimilar circuits alltheelements of transverse crosstalk, due tothe unbalances occurring atvarious points along theline, arrive atthesame time atthefarendoftheline. The crosstalk currents due tounbalances ofthesame type such as capacitance unbalances arriveinthesameoropposite phase(ifthe . circuits are perfectly smooth). Itwill beseen, therefore, that a properlydesignednetworkconnected atonepointinthelinemaybe . used topractically annul thefar-end transverse crosstalk. Inorder todesign themost effective type ofnetwork forbalancing transverse crosstalk itisnecessary toknow the manner inwhich the crosstalk coupling inany elementary length varies with frequency. ‘Thecrosstalkcouplingbetweentwopairsinanelementarylength :may berepresented byamutual admittance and amutual impedance. { Ttcanbeconsidered that thevoltage between thetwowires ofthe disturbing circuit drives crosstalk currents into thedisturbed circuit through the mutual admittance. The currents inthe disturbing circuit acting through the mutual impedance also cause crosstalk +Asdiscussed inthe Clark-Kendall paper on“Carrier inCable" intheBell ‘System Technical Journal ofJuly, 1933.Thevariouspenofcrsstaliareicanineepaperon"OpenWireCrossggbyfeGsUpma intheBalSytemFeet Jourmat ofJanay and CABLE CROSSTALK 181 currents. The mutual admittance isduealmost entirely tocapacitive coupling, theleakance ordinarily being negligible initseffect oncross- talk coupling. This capacitive coupling varies but little with fre-quencyanditseffectoncrosstalk maybebalanced outbymeansofasimple condenser. Iftheproximity effect were negligible, themutual impedance would besubstantially that ofasimple mutual inductance constant with frequency. The crosstalk due tothis coupling would, therefore, bebalanceable bymeans ofasimple inductance coil. If, however, theproximity effect isnotnegligible themutual impedance isdue toacomplex mutual inductance both ofwhose components vary considerably with frequency. This isthecase incable circuits and acomplex balancing unit must bedesigned ifthecomplex magnetic coupling istobeaccurately simulated. ‘The mutual impedance, Z, between twocircuits isbydefinition the negative ratio ofthe induced series voltage*(e)inthedisturbed circuit tothecurrent (J)inthedisturbing circuit. Thus, e Zu=5. Since theinduced voltage isproportional tothetime variation ofthe magnetic field setupbythedisturbing current, itisimportant to visualize how thisfield may bealtered bychanges inthedistribution of the current, I,over the cross section ofthe disturbing conductor. Four types ofcurrent distribution willbeconsidered and theeffects onZu noted. Inorder tosimplify the following qualitative explanation ofthe effect ofcurrent distribution onmutual impedance itwill beassumed that inallcases the disturbed wire isafilament. When the disturbed wire isfinite incross section theeffect isgenerally similar, but more complicated. Case T—CurRENT CONCENTRATED INAFILAMENTARY DisturBinG WIRE Inthecase ofawire ofinfinitely small cross section themagnetic field due toasinusoidal current, 7,induces avoltage inanother filamentary wire located inthis field asexpressed bythe familiar equation ¢= —joM!. ‘The mutual impedance isapure reactance equal tojwMf, where M, thecoefficient ofmutual inductance, isapure number and independent offrequency. motiveforcenach2tobogthetacurenitheeturbedcretose 182 BELL SYSTEM TECHNICAL JOURNAL Case [Current Untrormty Distrinurep 1x ASoup Cyzinpricat. DistursinG Wire Consider next the case where the total disturbing current isuni- formly distributed over the cross section ofasolid cylindrical wire, Such adistribution exists exactly with direct current only, but is closely approximated atvery low frequencies. Since the magnetic field outside ofaconductor carrying auniformly distributed current isthe same aswould exist ifthe total current were concentrated inthe center filament, thetotal induced voltage inafilamentary wire located inthis field isagain equal to—jwMJ, where Misthesame asinthe case oftwo filaments similarly located inspace. Case T1I—Current Syaoerricatty DistrrButep INASoLID CyzinpricaL DisturBinc Wire ‘The a.c. distribution inasolid cylindrical wire isnot uniform. However, when thewire isataconsiderable distance from itsreturn, thecurrent distribution ispractically symmetrical about theaxis of thewire although itsdensity varies from aminimum value atthecenter toa maximum value atthesurface. Such adistribution iscaused by the fact that the counter-electromotive force induced inafilament near the center ofthe wire due tothe current inallofthe other fila- ments isgreater than that induced inafilament atthesurface. This isthe well-known skin effect. Inthis case thetotal current, I,may beconsidered asdistributed in infinitely thin concentric rings inany one ofwhich thecurrent isthe same inphase and magnitude atallpoints. Since thefield outside of onesuch ring isthesame aswould exist ifallofthering current were concentrated inafilament atthecenter, thetotal field due tothesum ofthecurrents inalltheconcentric rings isthesame aswould exist ifthe total current were concentrated atthe center ofthe wire. Thus, the total voltage induced inafilamentary wire bythe field setup byasymmetrically distributed current inthe disturbing wire is again expressed by—jwMZ, where Misagain apure number asin the case oftwo filaments. Case TV—CurRent UNsYMMETRICALLY DistRIBUTED INASoLID Cytinpricat, Distursinc Wire ‘ Ifasolid wire and itsreturn areplaced close together, asinacable pair, the a.-c. distribution isneither uniform nor symmetrical about theaxis ofthewire. Inthis case themagnetic field setupbythe current inthe return wire contributes tothe counter-electromotive CABLE CROSSTALK 183 force acting ineach filament oftheother conductor and causes a further redistribution ofthe current inthat conductor over and above that due tothe above mentioned skin effect. This additional altera- tion incurrent distribution isknown astheproximity effect. The resultant current distribution can nolonger besymmetrical about theaxis ofeither wire. ‘The current inthereturn wire sets up greater back-electromotive forces inthefilaments oftheother wire which areclose toitthan inthemore remote filaments. These back- electromotive forces tend toactinopposition tothose setupbythe current inthewire itself since thecurrent inthereturn wire isopposite insign. Hence, theproximity ofthereturn wire reduces thecounter- electromotive force acting inthefilaments closest toitintheother wire more than itdoes inthe filaments farther away. This results in higher current density inthesides ofthewires adjacent toeach other. The current distribution due tothe combined action ofskin and proximity effects isshown forapairofround copper wires inspace in Figs. 1-Aand1-B.$ The wires areNo.19A.W.G. andareseparated a distance equivalent tothat between wires in19-gauge cable pairs. ‘The current distribution at56kilocycles isshown inFig. 1-Aand at 112kilocycles inFig. 1-B. Itisseen that thetendency atthehigher frequencies isforthecurrent toconcentrate onthesides ofthewires adjacent toeach other. With perfect conductors thecurrent would allbeonthesurface ofthewires and forthis wire spacing would be distributed asshown inFig. 1-C.* With actual conductors this distri- bution isapproached asthefrequency increases toward thehighest conceivable wire communication frequency. Inaddition tothisunsymmetrical distribution ofcurrentwith—*respect tomagnitude thecurrents invarious filaments intheconductor may beconsiderably out ofphase with the current atthe center. This phase shift may bequite unsymmetrical asindicated forthree wire diameters inFigs. 2-A and 2-B. While similar phase shifts occur when only skin effect ispresent, such shifts aresymmetrical about thecenter ofthe wire sothat thecurrents atallpoints inathin concentric ring have thesame phase. Figure 2-A shows thephase shift at56kilocycles and Fig. 2-B thephase shift at112 kilocycles. Itisseen that thetendency atthehigher frequencies isforthecurrents atdifferent points onthesurface tobecome inphase with each other. Atinfinite frequency thesurface currents would beinphase. . “The current distribution and phase change at56and 112 kilocycles were com potefomformula givenbyHarveyCtsinBurenofStandardScenic MerNovtirented “AnIntegration Meth ofDeriving theAlkemating Current Resistance and InductanceofConductors. "This distribution was calculated byRay’ S.Hoyt. “ee ;; ses BELLSYSTEMTECHNICAL JOURNAL ‘ (eo) ij Zz \S ‘ comaanon seer Fig. 1-Currentdissibton inall 14 wes. Ifa wire iscarrying atotal current, /,distributed unsymmetrically inphase and magnitude asinFigs. 1and 2,the magnetic field sur- rounding thewire cannolonger bethesame aswould beproduced by the same current flowing inthe center filament ofthe wire. The voltage induced inadisturbed filamentary wire located inthis field must therefore bedifferent from that induced bythe field setupby any ofthe preceding types ofcurrent distribution inthe disturbing wire. Ineach ofthose cases the induced voltage, e,was exactly in CABLE CROSSTALK 18s BsOxi Foiszsoed Seaman eeie [}yt] aesAEE NESS SGREDiDeny, INETANYApLNUZ| LNUZ) EXALY| gLLFE) CEPT) LE eA]TT] CTTT]eATTd Ne a [\i|7] z : ¥ ‘aan HAH Fig. 2—Phas shift inparal! 1g, wires phase quadrature with thetotal disturbing current, 7,and themutual impedance was equal tojwM, apure imaginary. For unsymmetrical current distribution inthe disturbing conductor, the following dis- cussion shows that theinduced voltage can nolonger beexactly in. phase quadrature with thedisturbing current and that themutual impedance iscomplex. The total voltage induced inadisturbed filamentary wire bythe total current flowing inasolid wire isthevector sum oftheinduced voltages due toallofthe currents inthe various filaments ofthe disturbing wire. Thus, ifi:,i2,is,--+i,arethevector currents inthe various filaments ofthedisturbing wire and ifmy,ma, ms, +++m,are the corresponding coefficients ofmutual inductance between each of these filaments and thedisturbed filamentary wire, the total induced 186 BELL SYSTEM TECHNICAL JOURNAL voltage inthedisturbed circuit is €==jolts +main +mis ++++main). ‘The mutual impedance, Zar, may therefore bewritten Ly=—2=Gels +mas+msiy+++main)wens 7 . where I= i,+i2+ is++++im, This isageneral expression and holds forany type ofcurrent distribution inthedisturbing conductor. Inthe case ofsymmetrical current distribution (Case III) all filamentary currents having thevalue ilieinaring concentric with thecenter ofthewire. The voltage induced inadisturbed filamentary wire due toallthecurrents inone such ring isthesame asiftheir total value, J,was concentrated inthe center ofthe ring. This voltage isequal to~jwMili where Myisthecoefficient ofmutual inductance between thecenter filament ofthedisturbing wire and the disturbed filamentary wire. The same reasoning holds forcurrents having values is,is,+++i,and themutual impedance may bewritten By=—6=ie(Mal +Mala +Maly ++++Mls)i a. as But My=Mz=My=+++My=Mince allarecomputed from the center filament inthe disturbing wire tothe disturbed filamentary wire. Then Zu=jouBABEBAIn) =joM since I,+Is+Is++++In=I.This isthesame expression forZr asgiven inthediscussion onsymmetrical current distribution. However, when thecurrent distribution inthedisturbing wire is unsymmetrical inphase and magnitude itisimpossible tomake the above simplifications. Inthegeneral expression Ly=—£adeolensis +main +mais++++mai) dum—7 T there isnocorrespondingly simple way toseparate them’sfrom the #’sinthe complex expression inbrackets and the phase angle ofthe expression may bequite different from that ofI.Therefore, ¢cannot beinphase quadrature with respect to/.Inorder toputtheequation CABLE CROSSTALK 187 forZyinthesame form asinpreceding cases thebracketed expression may bearbitrarily rewritten as myiy +mais +mals ++++main =1Ma +5M). ‘Then, Zu=jo(Me +Mi) =—oMy +joMe where themutual inductance isnow considered complex and ashaving two components such that M=M.+jM. ‘The total current inthe disturbing circuit acting through the com- ponent AM,ofthemutual inductance sets upaninduced voltage inthe disturbed circuit inquadrature with theinduced voltage due toMs, and inthesame oropposite phase asthetotal current inthedisturbing circuit. Ordinarily thephase willbeopposite and theactual values of M,will benegative with respect toMy. Both M,and M,vary with frequency. While thetotal current in thedisturbing wire isassumed constant, the unsymmetrically dis- tributed currents inthevarious filaments change inrelative magnitude and phase asthe frequency changes. Atvery low frequencies the current isdistributed nearly uniformly inphase and magnitude over the cross-section ofthe wire. The mutual inductance between this wire and thedisturbed filamentary wire isnearly thesame asthed.-c. value since M,cannot beappreciably changed from thed.-c. value and Mymust bevery nearly zero. Atvery high frequencies themajor part ofthecurrent flows unsymmetrically onthesurface ofthedis- turbing wire but thefilamentary surface currents arepractically in phase with each other. This results again inalow value ofMs because thetotal induced voltage will bepractically inphase quadra- ture with the total disturbing current. However, due tothe un- symmetrical current distribution, the value ofMa isconsiderably altered from itsd.-c. value. Atintermediate frequencies thecurrent distribution lies between these two extremes and produces corre- sponding values ofMand My. Since Myiszero forboth zero and infinite frequency itisevident that amaximum value must bereached atsome intermediate frequency. As noted atthe outset ofthis discussion, the disturbed circuit is assumed tobeafilament. Inallpractical casesthewiresinvolved arefinite incross section, and the reasoning outlined above must be applied toeach filament ofthe disturbed conductor inorder toget the total effect. ie pel ate Discussion or Test RESULTS Inorder tostudy thevariation with frequency ofthemutual in- ductance between cable circuits, measurements were made onvarious. combinations ofpairs ina55-foot length ofNo. 19A.\W.G. tollcable. Toobtain information ontheperformance ofthemeasuring apparatus, measurements were also made onthe calculable case oftwo non-twisted pairs, sixfeetinlength (approximately). Various separations between thetwowires ofapair and three different wire gauges were used to show thechange inmutual inductance forvarious degrees ofproximity effect. The measurements were made with a“crosstalk bridge" orad- mittance unbalance measuring setwhich permits themeasurement of crosstalk inboth phase and magnitude. Although the mutual in- ductance between two pairs may bedetermined from either near-end_ orfar-end crosstalk tests, itwas found that greater accuracy inMy could beobtained from far-end tests. The computation ofM,from. near-end tests involves two terms ofopposite sign and ofnearly the same magnitude. Consequently asmall error inthereading ofthe crosstalk bridge may result inaconsiderably greater error inMy. The results ofthetests onthesix-foot non-twisted pairs areshown inFigs. 3to9. The data cover arange of1to1000 kilocycles. InFigs. 3and 4thevariation with frequency ofM,and M,isshown Soooieee eIos He eadECCT HeSoo xeransKASSzl|BeesI het ool—pene]ATH HPberm ct 2 in aoeso THtiliccOS |coo!|_| TPEe134 ceauewey wmmiLaevctes Persecono >99700 CABLE CROSSTALK 189 aaTE 7 A] cog|oe! | TT TIT eed iEee§oop—|LTTTT TTTgL Tean eee TIT TTT3coool|BE-ove—KP |TTttt[TTT 3Ls geucewmes Beans £ooost|“meme ATTTT[emPtH 2|TTyi ST) 5coat |ETI 5a | A3oot} |TTTEee rt TTT a a eee -oooel__| TTT TTT TTT) Fig.{Mutual inductance between pairs ofparallel wires. fortwoarrangements ofpairs inahorizontal plane. Inboth cases the axial separation ofthetwowires ofapairwasabout 0.075 inch, butin Fig.3theaxial separation between thenearest wires ofthetwopairs was 0.075 inch and inFig. 4itwas0.312 inch. The wires were No. 20A.W.G. cotton-covered and were pulled taut tomaintain accurate spacing. ‘Two plots areshown forMy,oneactual andtheother after multiplying by—10toshow thevalues more clearly. ‘The above data arereplotted inFig. $toshow thefrequency varia- tion ofM,and Myinterms ofthevalues ofM,atonekilocycle. The factthatthefrequency characteristics forthetwocases aresonearly alike despite thedifference inthemagnitude ofthecoupling indicates that theeffect depends primarily onthespacing between thewires ofapairandnotsomuch ontherelative positions ofthepairs. ‘Theproximity effect may bereduced byseparating thewires ofeach pairasshown inFig.6.Inthiscase thefrequency characteristic of M,isnearly flatandM,issosmall that itcould notbeplotted onthe same scale asM,; thecurves shown areM,and 100M,. Acomparison al v0 BELL SYarmas TRCHNICAL JOURNAL Sono a a SU oeCOa 0 oe SI i a 0 er es aa 08°CCT apersepeoTSC EE oo densi CUT eetspeerin “seaettecr® rrrReeTo I a osa aa TTa IaEES osse rmctr Bnet ve ow ofthecurves ofM,and M,inFigs. 3and 4with thecurves ofFig. 6 illustrates therelative importance oftheproximity effect onmagnetic crosstalk incable pairs and inopen-wire pairs. Incable pairs the wires areclose together asinFigs. 3and4,while inopen wire thesepa- ration ismuch greater than that shown inFig. 6. The effect ofwire gauge onthevariation ofM,and M,isshown in Figs. 7,8,9-A and 9-B. Two. gauges ofwire (No. 10and No. 18 A.W.G.) were used with centers located atthecorners ofa0.14-inch square, The mutual impedance between thevertically adjacent pairs was measured. The corresponding values ofM,and Myareshown in oof LT TT TTT] Sea OO Fe a | i en oeLL ‘oi | *Con Zot |TTT —ssuansrustmmes osarelTTTToe | mani ESC eM ¢mes gtCOBT boner TTT Co 2 CeeCI eee ee ll ; Sea FBS emeQuency inxKDeveLes PERsecono“°°M25021090 Fig. 7forthe10-gauge wires and inFig. 8forthe18-gauge wires.7 The values of—10M, arealso plotted inFig. 7and —100M, in Sooogee TTT eC)Oe osa2potCA ST OO" PSEC SCT eeepealCeeNTT] Sere gosetase eee Pea TS ONfofCee?JSSS ET Te PA N29 45crequency mvKiLOCYcLEs PERsecono“°°905001000 wn DELL SYSTEM TECHNICAL JOURNAL oea LI Htool O""Or mail Po Soe8 ERPS8oso|mentener AEESNFeea)eee SSmegeo PO SES eeeetETvol TTT fT Fig, 8Mutal inductance between pir ofparal wires Fig. 8toshow theshapes more clearly. Acomparison ofthefre- quencycharacteristics ofM,andM;forthetwowiregaugesisshown | inFigs. 9-Aand 9-Binterms oftheone-kilocycle value ofMzforeach case, InFig. 9-Athefrequency scale islogarithmic and inFig. 9-Bis linear. Aswould beexpected, theuseofsmaller wires (No. 18gauge) decreases theeffect ofproximity onthemagnetic coupling and shifts thefrequency atwhich M,reaches amaximum value. ‘The results oftests onthe55-foot length ofNo. 19A.W.G. quadded cable areshown onFigs. 10-A and 10-B. The data cover arange of 10to480kilocycles. ‘These figures show thevariation with frequency oftheaverage values ofM,and M;,interms oftheten-kilocycle average value ofM,which istaken asunity. InFig. 10-A the fre- quency scale islogarithmic andinFig. 10-B itislinear. Itwillbeseen that M,isnegative with respect toM,asinthe case oftwo pairs in space. AsinFigs. 3,4and 7,curves aregiven both forM,and for —10M, the purpose ofthe latter curve being toshow theshape of ; ‘Mymore clearly. The value ofM,decreases with frequency, becoming. nearly constant above 300 kilocycles atavalue 22percent less than the value at10kilocycles. The component Myisofnegative sign and at56kilocycles reaches amaximum value which is13.4 percent ofM,atthis frequency. cane crossrane ws ooo=ERS EES||Hh SC St HH goo] TT St ec: sammie, iBoa—|mmentensre” FTTTT COTJal Preo eet>t—EEHEHot -onA eee 2 PSH Py a ee tit TyTTT ajyeEt ttTt araneTT#HR +-HH ‘LETT berePPPere ey WO200300400-500 600-700-800 9001000 The frequency characteristics ofM,and M,forindividual pair combinations areabout thesame asshown onFig. 10,although there areoccasional differences such asapositive value ofM,orachange in sign inM,orMyatsome frequency. Values forpairs intheoutside layer did not appear tobemuch affected byeddy currents inthe sheath. Asinthecase ofmeasurements onparallel wires inspace the values ofM,and M,arevery small. For example, between two pairs ws BELL SYSTEM TECHNICAL JOURNAL .Lou NG De TTT goaWeTPH} a7 |TENS wk TTA PaI2o4Ae“TOCCOA ‘pel | A ee ‘slhe ale> sk — oe elt TyTTT Per ‘yLtt yet et ‘|00 3 sges80 a80 Pao mio en te Fig. 10-Mutual inductance between cable pas interms ofvale forAfatten Heyes inthesame quad theaverage values at10kilocycles are0.056 and —0,0030 microhenries. For non-adjacent pairs the values are, of course, much smaller, Acknowledgment Inthis work thewriters received much help from Ray S.Hoyt inthe matter ofgeneral circuit theory. Aspreviously noted, the accuracy ofthemeasurements was established bythework ofSallie Pero Mead indeveloping acalculation formula forthecase ofstraight wires.