IEEE TANSACTIONS ON ELECTOMAGNETIC COMPATIBILITY, VOL 50, NO 3, AUGUST 2008 Clcultion of Elctricl Prmtrs of Two-Wir Lins in Multiconductor Cbls Boris M Lvin Abstrct A rigorous mthod for th clcultions of th chrctristics of two-wir lins twistd pirs) loctd in mtl shild is considrd In this ppr, it is first shown tht th mutul coupling btwn lins in multiconductor cbls rsults in th pprnc of lctromgntic intrfrnc crosstlk) in communiction chnnls; scond, th symmtry of xcittion nd lods rsults in th pprnc of common-mod currnts in th cbl Th voltg vlus intrfrnc) in th lods plcd t th bginning nd th nd of th djcnt lin r dtrmind t givn powr in th min lin Th ffct of lods connctd btwn wirs nd shild is xmind Th proposd mthod llows gnrliztion of th obtind rsults in th cs of multiconductor cbls with losss Indx Trms Cbls, communiction chnnls, lctromgntic EM) intrfrnc, lossy circuits, mutul coupling NOMENCLATUE dius of th wir b Distnc btwn wirs of two-wir lin c Vlocity of light C l Linr pr-unit-lngth) cpcitnc of conductor C ns0 Mutul cpcitnc btwn wirs n nd s pr unit of thir lngth d Distnc btwn xs of two twistd pirs D ns Distnc btwn th wirs n nd s D ns ) 0 Mn distnc btwn th wirs n nd s, EMF of th gnrtors G ns0 Lkg conductnc btwn wirs n nd s pr unit of thir lngth i n Currnt of th nth wir I n Currnt t th bginning of th nth wir k Propgtion constnt of wv in mdium L Lngth of lin wir L 0 Wir inductnc pr unit lngth M ns0 Mutul inductnc btwn wirs n nd s pr unit of thir lngth n, s Wir numbrs N Numbr of prlll wirs loctd insid mtl cylindr p ns Potntil cofficint btwn wirs n nd s dius of mtl cylindr shild of cbl) 0 Wir rsistnc pr unit lngth ns0 Loss rsistnc of wirs n nd s pr unit of thir lngth Mnuscript rcivd My 7, 2007; rvisd Sptmbr 30, 2007 nd Dcmbr 3, 2007 Th uthor is with th Holon Institut of Tchnology, Holon, Isrl Digitl Objct Idntifir 009/TEMC2008927924 u n Potntil of th nth wir U n Potntil t th bginning of th nth wir W Wv impdnc of lin W ns Elctrosttic wv impdnc btwn wirs n nd s z Coordint long wir Z, Z,Z 2,Z 3 Impdncs of th lin lods α Angulr displcmnt of points long th sction primtr β ns Cofficint of n lctrosttic induction btwn wirs n nd s ns Cofctor of th dtrminnt N,H Distncs by which wir is displcd N p ns N N Dtrminnt ε Prmittivity of th mdium insid th cbl γ Propgtion constnt of wv long wirs ρ ns Elctrodynmic wv impdnc btwn wirs n nd s I INTODUCTION THE DETEMINATION of th signl mgnitud t th nd of multiconductor cbl loctd insid mtl shild rquird th clcultion of th lctricl chrctristics of thos lins Th mutul coupling btwn two-wir lins twistd pirs) in multiconductor cbl rsults in th pprnc of lctromgntic EM) intrfrnc crosstlk) in communiction chnnls Th voltg vlus on lods plcd t th nds of n djcnt lin cn b usd s msur of such distortions []; for xmpl, [2] is dvotd to th qulittiv nlysis of mutul coupling btwn lins Howvr, th two-wir lin modl considrd in [2] is fr from n ctul twistd pir structur Th rigorous mthod of th clcultion of th mutul coupling btwn lins nbls to dvlop simpl nd ffctiv mthods of prvnting intrfrnc EM intrfrnc in communiction chnnls cbl unblnc) is cusd by th cbl symmtry nd lso by xcittion nd lod symmtry, which provoks th pprnc of common-mod currnts in cbls Th rigorous mthod of th clcultion of th multiconductor cbl lctricl chrctristics nbls to dtrmin th common-mod currnts Compnstion of th common-mod currnts prmits to dcrs th EM rdition nd its suscptibility to xtrnl filds In this ppr, rigorous mthod for th clcultion is offrd for th chrctristics of two-wir lin loctd insid mtl shild nd scond for th mutul coupling btwn lins Th lins r considrd s uniform ons Th EM wvs r considrd s trnsvrs TEM) wvs, nd th cbl dimtr is considrd smll in comprison to th wvlngth 008-9375/$2500 2008 IEEE
2 IEEE TANSACTIONS ON ELECTOMAGNETIC COMPATIBILITY, VOL 50, NO 3, AUGUST 2008 Fig Fig 2 Two wirs insid cylindr Equivlnt circuit of singl lin insid shild gnrl cs, r dtrmind by 2U n N i n I n cos kz + j W nn u n U n cos kz + j U s W s ns ) sin kz N ρ ns I s sin kz ) s whr I n nd U n r, rspctivly, th currnt nd th potntil t th bginning of th nth wir t z 0), k is th propgtion constnt of wv in mdium, nd W ns nd ρ ns r th lctrosttic nd lctrodynmic wv impdncs btwn wirs n nd s In this cs { /cβns ), n s ρ ns p ns W ns 2) c /cβ ns ), n s whr p ns r th potntil cofficints, β ns r th cofficints of n lctrosttic induction, nd c is th vlocity of th light Th cofficints β ns nd p ns r linkd by th following rltionship: II TWO WIESINSIDE A SHIELD WITH A CICULA SECTION A Problm Sttmnt Asingl pir of wirs twistd pir) insid th mtl cylindr cn b modld s two wirs of rdius,loctd t distnc b from ch othr insid mtl cylindr of rdius nd lngth L Fig ) In multiconductor cbls, th wir rdius nd th distnc b r smll in comprison with th cylindr rdius, b ), soth chrctristic impdnc of th lin is constnt long its lngth whn th xis lins of th twistd pir nd th cylindr do not coincid, nd th givn inqution is not stisfid, th chrctristic impdnc vris long th lin) W ssum th wirs to b stright nd tk into ccount twisting by incrsing th lngth L of th quivlnt lin Sinc th pitch of th hlix followd by ch wir is lrgr thn hlix dimtr b, th inductnc L 0 pr unit lngth vris slightly with th rplcmnt of spirl wir by dirct on Th wir cpcitnc pr unit lngth lso vris only slightly, i, th wir twisting dos not chng th chrctristic impdncs of structur Th lin symmtry in rl cbl cn cus chng of th chrctristic impdnc nd chng of th two-wir lin input impdnc Anothr cus of th cbl symmtry is tht ch two-wir lin is md in th form of twistd pir hlix), dsign tht lds to diffrnc in th mn distncs btwn diffrnt wirs nd to th mutul coupling crosstlk) btwn two two-wir lins surroundd by singl shild, vn if both th xciting lctromotiv forc EMF) of ch lin nd th lin lod r symmtric Th input impdnc of two-wir lin insid mtl shild is dtrmind in th nxt sction B Clcultion of th Input Impdnc of th Two-Wir Lin Th quivlnt circuit of singl lin insid th shild is shown in Fig 2 Th two-wir lin is loctd bov th ground insid th mtl cylindr) Th thory of such lins hs bn workd out by Pistolkors [3] Th currnt nd th potntil of th nth wir of n symmtricl lin of N prlll wirs loctd bov th ground, in th β ns ns 3) N whr N p ns is th N N dtrminnt nd ns is th cofctor of th dtrminnt N For n symmtricl lin from two wirs, w cn writ N 2 W ρ 22 ρ ρ 22 ρ 2 2 W 22 ρ ρ ρ 22 ρ 2 2 ρ 2 W 2 ρ ρ 22 ρ 2 2 4) Th boundry conditions for th currnts nd potntils in th circuit shown in Fig 2 r i 0) + i 2 0) 0 u 0) u 2 0) + i 0)Z i L)+i 2 L) 0 u L) + u 2 L) 5) Hr, Z is th impdnc of th lin lod s Fig 2) Substituting xprssions ) in th first nd scond qutions of systm 5), w find I 2 I U 2 U I Z From th third qution of systm 5), tking into ccount xprssions 4), w find tht U I Z W 22 W 2 ) W + W 22 2 ρ ) ρ 2 I Z ρ + ρ 22 2ρ 2 W 2 And from th fourth qution, w obtin I [Z cos kl + jρ + ρ 22 2ρ 2 )sinkl] Th input impdnc of two-wir lin insid mtl shild th lod impdnc of gnrtor ) isqul to Z l /i L) Substituting th vlu of i L) from xprssion ) nd using th rltionships btwn, I,I 2,U, nd U 2,wfind tht Z l W Z + jwtgkl 6) W + jztgkl whr W ρ + ρ 22 2ρ 2
LEVIN: CALCULATION OF ELECTICAL PAAMETES OF TWO-WIE LINES IN MULTICONDUCTO CABLES 3 It is rdily sn tht th xprssion 6) coincids with th xprssion for th input impdnc of losslss two-wir-long lin tht is loctd in fr spc, chrctrizd by wv impdnc W, nd lodd by impdnc Z Thlin symmtry rsults in th diffrnc of wirs lctrosttic ρ ρ 22 ) nd lctrodynmic W W 22 ) chrctristic impdncs Th clcultion of currnts i z) nd i 2 z) shows tht th currnts in two-wir lin r idnticl in vlu nd opposit in sign i z) i 2 z) I cos kz + ji Z sin kz W In wir pir, thr r only diffrntil-mod currnts A common-mod currnt in th wirs is bsnt bcus th EMF nd lod impdnc r plcd btwn th lin wirs Th pprnc of common-mod currnt cn b cusd by th connction of n dditionl EMF or n dditionl lod btwn on wir of lin nd th shild C Clcultion of th Chrctristic Impdnc For th dtrmintion of th potntil cofficints p ns,itis pproprit to mk us of [4] It givs, in prticulr, formuls for th clcultion of linr pr-unit-lngth) cpcitnc C l of conductors in th form of n indfinitly long closd nvlop of circulr sction In this cs, th potntil cofficints clcultd with considrtion for th mirror img in prfctly conducting cylindricl surfc qul p ns C l 7) To s this, if systm consists of two idnticl conductors wir nd its img) nd this structur is lctriclly nutrl, th mutul prtil cpcitnc coincids with th intrconductor cpcitnc s [4, xprssion B-4)]) Th mutul prtil cpcitnc quls C 2p p 2 ) whr p is th slf-potntil cofficint nd p 2 is th potntil cofficint of th img Th conductor-to-ground cpcitnc is twic s much s th cpcitnc btwn two conductors C l 2C p p 2 p For two wirs ofrdius, loctd t distnc b from ch othr, symmtriclly loctd insid th mtl cylindr of rdius to th cylindr xs s Fig ) in ccord with 7) nd [4, xprssion 4) 20)], w cn writ p p 22 2πε ch 2 + 2 b 2 /4 2 Hr, ε is th prmittivity of th mdium insid th cbl If th wir rdius nd th distnc b r smll in comprison with th cylindr rdius,thn p p 22 2πε ln nd in th ir ρ ρ 22 60 ln 8) Fig 3 Offst wirs insid cylindr Similrly, in ccord with 7) nd [4, xprssion 4) 20)], w find ρ 2 60 ln 9) b i, th chrctristic impdnc W 0 ρ + ρ 22 2ρ 2 60 ln b 0) of losslss two-wir lin, symmtriclly loctd insid th mtl cylindr, is hlf of th chrctristic impdnc of th sm lin in fr spc In uniform lin lodd by its chrctristic impdnc, th rflctd wv is null, so th signl trnsmission in th bsnc of losss quls nd dos not dpnd on frquncy In this rgrd, two-wirlin twisting) insid th mtl shild is diffrnt from nonuniform lin considrd in [5], th spirl two-wir lin loctd long mtl pln D sons of th Chrctristic Impdnc Chng ) ltiv Displcmnt of th Wirs: Lt us considr possibl rsons tht lin s chrctristic impdnc chngs insid th shild If wirs insid th mtl cylindr of rdius r loctd symmtriclly, for xmpl, thy r displcd to th right by distnc Fig 3) p 2πε ch 2 + 2 b/2 ) 2 2 so t, b p { [ ]} 2πε ln b ) + 2 [ ln ] b ) + 2πε 2 p 22 2πε [ ln b + ) 2 ] Thn, th chrctristic impdnc of th lin is W W 0 20 2 2 ) 2) Incrs of th Distnc Btwn Wirs: If th distnc btwn wirs is incrsd by vlu, thn if th distnc is smll in comprison with th distnc btwn th wirs b), thn ρ 2 60 ln 60 ln b + ) ) b 2b
4 IEEE TANSACTIONS ON ELECTOMAGNETIC COMPATIBILITY, VOL 50, NO 3, AUGUST 2008 Fig 4 Equivlnt circuit of two coupld lins plcd insid shild Fig 5 Distnc btwn wirs nd 4 ) Usul winding of wir 4 b) Countr winding of wir 4 Hnc W 60ln b + W 0 +60 b 2) As cn b sn from ) nd 2), chng of distnc btwn wirs hs mor ffct on th chrctristic impdnc of th lin thn th wir displcmnt rltiv to th cylindr xis III TWO WIES PAISINSIDE A SHIELD WITH A CICULA SECTION A Problm Sttmnt Th quivlnt circuit of two-coupld two-wir lins insid shild is shown in Fig 4 On of ths lins is xcitd by th gnrtor nd lodd by complx impdnc Z,th lods Z 2 nd Z 3 r connctd to both nds of th othr lin It is ncssry to mphsiz tht such circuit hs th most gnrl chrctr If, for xmpl, gnrtor is connctd t th nd of th scond lin in th point z L), th currnts nd voltgs crtd by th gnrtor cn b dtrmind by rplcing th vlu Z 3 by th input impdnc of th gnrtor Lt us considr tht b d, hr, d is distnc btwn xs of twistd pirs) In mny css, th dimtr of wir bunch is smll in comprison with th dimtr of th cbl mtl shild Whn thr r mny wirs in th bunch, its dimtr is clos to th shild dimtr Howvr, it is ncssry to tk into ccount tht th mximum mutul coupling xists btwn djcnt lins Thrfor, nlyzing mutul coupling btwn thm is possibl by considring s first pproximtion tht d In th nxt sction, it is shown tht twisting is th rson of symmtry B Twisting s th son of Asymmtry As ws sttd in Sction I, th cbl symmtry rsults in mutul coupling crosstlk) btwn two two-wir lins Th rson of such symmtry is th fulfillmnt of ch lin s twisting spirl) Th plcmnt of th lins conductors t th diffrnt vrints of winding is shown in Fig 5 If, t th cbl s initil cross sction, th lds of spirls nd 3r loctd in th sm point of thir sction w shll nm it s initil on) nd th lds of spirls 2 nd 4 r shiftd long th cross-sction primtr by π from this point, it mns tht th distnc btwn wirs nd 3 nd lso btwn wirs 2 nd 4) long ll thir lngth quls D 3 D 24 d, whrs th distnc btwn wirs nd 4 nd lso btwn wirs 2 nd 3) vris long wirs from d + b to d bforxmpl, th distnc btwn wirs nd 4 [s Fig 5)] is qul to D 4 d + b cos α) 2 + b 2 sin 2 α d + b cos α + b2 sin 2 α 2d hr, α is th ngulr displcmnt of points nd 4 long th cross-sction primtr), i, th mn distnc btwn ths wirs D 4 ) 0 π Ddα d + b2 3) π 0 4d diffrs from th vlu d Th potntil cofficints nd lso th lctrodynmic nd lctrosttic wv impdncs vry ccordingly If, t th cbl s initil cross sction, th lds of spirls 3 nd 4r shiftd long th cross-sction primtr by π/2 nd 3π/2 from th initil point ccordingly, thn th distnc btwn wirs nd 3 or wir 4) quls D 34) d + b b2 cos α ± sin α)+ 2 8d sin α cos α)2 Hr, th top sign pplis to wir 3, nd th lowr sign to wir 4 From this qution, th vrg distnc btwn ths wirs is D34) )0 d ± b π + b2 4) 8d i, th shift of th spirl lds of th cbl by π/2 ssntilly chngs th vrg distnc btwn wirs Diffrnc btwn D 3 ) 0 nd D 4 ) 0 incrss from vlu b 2 /4d to 2b/π, ttht b d In ordr to mk th vrg distnc D 0 btwn wirs nd 4 undistinguishbl from d, itisncssry to wind wir 4 countr to th othr wirs In this cs [s Fig 5b)] D d + b cos α D 0 d 5) C Clcultion of Elctricl Prmtrs Th lctrodynmic chrctristic impdncs of this structur t usul winding bcom ρ ρ 22 ρ 33 ρ 44 ρ 60ln ρ 2 ρ 34 ρ 2 60ln ρ 3 ρ 24 ρ 3 60ln ρ 4 ρ 23 ρ 4 60ln b d d + b2 π/4d) 6)
LEVIN: CALCULATION OF ELECTICAL PAAMETES OF TWO-WIE LINES IN MULTICONDUCTO CABLES 5 In th cs of lins rrngd t finit distnc H from cbl xis in ccord with formul 7) nd [4, xprssion 4 20)], w find ρ 60ln H 2 / 2) Th xprssions for othr mgnituds ρ n rmin vlid ons This mns tht th wv impdnc of losslss two-wir lin loctd insid th mtl cylindr t distnc H from its xis in ccord with 0) is W 60ln b H 2 / 2) 2 i, s rsult of lin displcmnt from cbl xis, its chrctristic impdnc dcrss Whn H is smll nd qul to, wrriv t xprssion ) According to 2) nd 3), w find th lctrosttic chrctristic impdncs N, n s ns W ns 7) N, n s ns whr N ρ ns is th N N dtrminnt nd ns is th cofctor of th dtrminnt N Forstructur md up of four wirs in ccord with 6) nd 7) W W 22 W 33 W 44 W 4 / Knowing ll prmtrs in xprssions ), it is possibl to clcult th loding impdnc of th gnrtor Z l i L) Z +2j[ρ ρ 2 + Aρ 3 ρ 4 )]tgkl 2 2+j[Z /W +/W 2 ) AZ 2 /W 3 /W 4 )]tgkl 2) nd th currnts in th wirs of th scond unxcitd) lin i 3 z) I A cos kz + j I 2 [AZ 2/W +/W 2 ) Z /W 3 /W 4 )] sin kz i 4 z) i 3 z) 22) Th sum of th currnts qul zro, i, s wll s t plcmnt of on lin in th shild, th common-mod currnt is bsnt sinc th EMF nd th loding impdncs r connctd only btwn wirs of ch lin Th voltgs cross pssiv lods r qul to V i 0)Z I Z V 2 i 3 0)Z 2 I AZ 2 V 3 i 3 L)Z 3 I Z 3 {A cos kl + j 2 [AZ 2 /W +/W 2 ) } Z /W 3 /W 4 )] sin kl 23) W 2 W 34 W 2 4 / 2 W 3 W 24 W 3 4 / 3 W 4 W 23 W 4 4 / 4 8) Th currnt nd potntil of th nth wir of n symmtric lin from N prlll wirs loctd bov th ground r dtrmind from xprssion ) Th boundry conditions for th currnts nd voltgs in th circuit shown in Fig 4 r i 0) + i 2 0) 0 i 3 0) + i 4 0) 0 u 0) u 2 0) + i 0)Z u 3 0) u 4 0) + i 3 0)Z 2 i L)+i 2 L) 0 i 3 L)+i 4 L) 0 u L) + u 2 L) u 3 l)u 4 L)+i 3 L)Z 3 9) Substituting xprssions ) in th qutions of systm 9), w find nlogously to Sction II) I Z cos kl +2j[ρ ρ 2 +ρ 3 ρ 4 )A]sinkL I 3 AI 20) whr A 4ρ 3 ρ 4 )+Z Z 3 /W 3 /W 4 ) 4ρ ρ 2 )+Z 2 Z 3 /W +/W 2 )+j2z 2 Z 3 )ctgkl If ρ 3 ρ 4 nd ccordingly W 3 W 4 ), thn A 0,thcurrnt t th bginning of th scond lin is null In this cs, th prsnc of th scond two-wir lin hs no ffct on th first lin This rsult obviously corroborts tht th cbl symmtry rsults in mutul coupling crosstlk) btwn th two two-wir lins D Numricl sults As n xmpl, w considr th structur from two pirs of wirs insid th shild with sizs in millimtrs): 02, b 05, d 2, 2 For th idnticl lods Z Z 2 Z 3 00 Ω, thrtio A of th currnts t th bginning of th scond unxcitd) nd first lin quls 03 If th vlus of th lods r qul to th chrctristic impdnc of th singl twowir lin insid th mtl shild, i, in ccordnc with 0) Z Z 2 Z 3 55 Ω, thrtio of th currnts is ssntilly incrsd A 076) Th bsolut vlus of th currnts s functions of kz r plottd in Fig 6 Hr, k is th propgtion constnt of wv in mdium, z is th coordint long lin s Fig 4) IV LOADS BETWEEN THE WIES AND THE SHIELD Lt us considr th ffct of th lods plcd btwn th wirs nd th shild using two-wir lin s n xmpl Fig 7) It diffrs from th circuit shown in Fig 2 by conncting its wirs t lin nd nr th gnrtor) with shild vi complx impdncs Z nd Z 2,whos vlus dpnd on th circuit of lin xcittion In rl circuit, th scondry winding of th trnsformr cn ct s th EMF, xciting two-wir lin In this cs, prsitic cpcitis of this winding to ground to cbl shild) ct s impdncs Z nd Z 2 Th currnt nd potntil of th nth wir of n symmtricl lin of N prlll wirs loctd bov th ground r dtrmind by xprssions ) Th boundry conditions for th currnts nd
6 IEEE TANSACTIONS ON ELECTOMAGNETIC COMPATIBILITY, VOL 50, NO 3, AUGUST 2008 Fig 6 Absolut vlus of th currnts in th xcitd nd unxcitd wirs Fig 7 Equivlnt circuit of th singl lin with th lods connctd btwn th wirs nd th shild potntils in th circuit shown in Fig 7 r i L)+i 2 L)+ u L) Z i 0) + i 2 0) 0 u 0) u 2 0) + i 0)Z + u 2L) 0 Z 2 u L) + u 2 L) 24) Substituting xprssions ) in th qutions of systm 24), w find nlogously to Sction II) th input impdnc of th two-wir lin, givn in 25), shown t th bottom of th pg, nd th sum of th currnts in th lin wirs s i s z) i z)+i 2 z) ji [Z /W 2 /W 22 ) + U /I /W +/W 22 2/W 2 )] sin kz 26) i, th lods connction rsults in th pprnc of th common-mod currnt in th wirs nd th currnt long th innr surfc of th cbl shild, qul in vlu but opposit in dirction For two wirs of th idnticl rdius loctd symmtriclly to th cylindr xs, givn in 27), shown t th bottom of th pg, nd C sin kz i s z) j Z cos kl + j2ρ ρ 2 )sin kl 28) whr 2ρ ρ 2 ) jzctgkl C 2Z Z 2 Z Z 2 jρ + ρ 2 ) Z +Z 2 Z Z 2 ctgkl It is not difficult to mk sur tht t /Z /Z 2 0, th mgnitud C is zro nd th xprssions for U nd Z l coincid with th similr xprssions for th circuit without lods btwn wirs nd shild From th prsntd rsults, it is lso sy to obtin xprssions for th css whn thr is only on from lods, for xmpl /Z 0 Th nlysis dscribd bfor vrifis tht th rson of th pprnc of th common-mod currnts in lin wir is th symmtry of its xcittion cusd by th connction of complx impdncs, for xmpl, prsitic cpcitncs of scondry trnsformr winding to th ground to cbl shild) A lods symmtry t th lin nd distnt from th gnrtor t z 0) givs th similr rsults Th common-mod currnts in th xcitd lin induc th common-mod currnts in th wirs of th djcnt unxcitd lin, vn if it is symmtric compltly bout th ground nd th xcitd lin) At tht, rmovl of th xcittion nd lod symmtry in th xcitd lin rsults in th dispprnc of th common-mod currnts in wirs of both xcitd nd unxcitd lin In ordr to dcrs or limint th common-mod currnts, it is ncssry to nnihilt this symmtry, for xmpl, to nutrliz ffct of prsitic cpcitncs to th ground to th cbl shild) To this gol in [6], it is offrd to cncl th currnt through prsitic cpcitnc with th currnt qul in vlu nd opposit in dirction, which is crtd by th dditionl trnsformr winding V COUPLED LINES WITH LOSSES In th prvious sctions, for th clcultions of th lctricl chrctristics of two-wir lins twistd pirs) in multiconductor cbls, th thory of symmtricl lctriclly coupld lins dvlopd by Pistolkors is usd This thory is bsd on th tlgrph qutions nd on th rltions btwn wirs potntil cofficints nd cofficints of n lctrosttic induction For ch wir of th structur, on cn writ two tlgrph qutions On of thm procds from th fct tht th potntil drop long th sction dz of givn wir is rsult of suprposition of EMF inducd by its own nd othr currnts Scond qution is bsd on lctrosttic xprssions conncting chrgs to potntils, with considrtion for th continuity qutions Th Z l Z cos kl + j ρ + ρ 22 2ρ 2 ) i L)+u L)/Z +U /I [/Z + j /W /W 2 ) tgkl]+j [Z/W 2 +ρ ρ 2 )/Z ] tgkl 25) Z l Z +2jρ ρ 2 )tgkl +j[ Zρ 2 /ρ 2 ρ 2 2)]tgkL + j/[2ρ + ρ 2 )][Z +ρ + ρ 2 )C]tgkL +/2Z )[Z +ρ + ρ 2 )C +2jρ ρ 2 )tgkl] 27)
LEVIN: CALCULATION OF ELECTICAL PAAMETES OF TWO-WIE LINES IN MULTICONDUCTO CABLES 7 coordint xis z is slctd in prlll to wirs s, for xmpl, Fig 2), nd th dpndnc of currnt on coordint z is ccptd s xp γz),whr γ is th propgtion constnt of th wv long th wirs In th bsnc of losss in th wirs nd in th mdium, in which thy r plcd, th lctrosttic W ns nd lctrodynmic ρ ns chrctristic impdncs btwn wirs n nd s r rl quntitis dtrminbl by qulitis 2), nd γ jk is purly imginry k is th propgtion constnt of th wv in th mdium) At tht, th currnt nd potntil of th nth wir of n symmtricl lin of N prlll wirs loctd bov th ground r clcultd from xprssions ) As follows from sid, th lctrodynmic wv impdncs ρ ns r proportionl to th slf nd mutul inductncs of wirs sctions, i, r proportionl to rctncs connctd in sris with wirs circuits Th lctrosttic wv impdncs W ns r proportionl to th mutul cpcitncs btwn wirs, i, to suscptncs btwn thm Thrfor, it is nturl to connct in circuit th rsistnc of losss in wir for xmpl, th skin ffct losss) in sris with th inductnc, nd th lkg conductnc in prlll with th mutul cpcitncs W shll tk into ccount th losss in mdium nd in wirs considring tht chrctristic impdncs W ns nd ρ ns nd th propgtion constnt k r complx vlus If th inductnc of th nth wir pr unit of its lngth is L 0 nd its ctiv rsistnc is 0 thn its impdnc pr unit lngth is jρ nn jωl 0 + 0, i, th slf-lctrodynmic chrctristic impdnc of th wir with losss is qul to ρ nn ρ 0 j ) 0 29) ρ 0 whr ρ 0 ωl 0 is th lctrodynmic chrctristic impdnc in th bsnc of losss nd 0 is th totl rsistnc of losss in th nth wir nd in mtl shild pr unit lngth For th mutul lctrodynmic chrctristic impdnc btwn wirs n nd s, wshll obtin ρ ns ρ ns0 j ) ns0 30) ρ ns0 whr ρ ns0 ωm ns0, M ns0 is th mutul inductnc btwn wirs n nd s pr unit lngth, nd ns0 is th loss rsistnc in both wirs pr unit lngth Similrly, for th dmittnc btwn wirs n nd s pr unit lngth, w find: jw ns jωc ns0 + G ns0, i, th lctrosttic chrctristic impdnc in mdium with losss quls W ns W ns0 j G ) ns0 3) W ns0 whr W ns0 ωc ns0, C ns0 is th mutul prtil cpcity btwn wirs n nd s pr unit lngth, nd G ns0 is th lkg conductnc pr unit lngth Thus, t th clcultion of th lctricl prformncs of th coupld lins with losss it is possibl to us th rsults obtind for th losslss lins by substitution of th complx chrctristic impdncs into xprssions obtind bfor in ccord with 29) 3) At tht, both losss in wirs nd losss in n imprfctly conducting mtllic tub shild) r tkn into ccount VI CONCLUSION A rigorous mthod for th clcultion of th chrctristics of two-wir lins insid th mtl shild llows us in rfining th mchnism of mutul coupling btwn lins in multiconductors cbls It prmits to dtrmin th voltg vlus intrfrncs) long impdncs plcd t th bginning nd th nd of th djcnt lin t th givn powr on th min lin As it is shown in Sction III, th rson of th crosstlk is th symmtry of th wir rrngmnt diffrnc in th vrg spc btwn diffrnt wirs), nd ccordingly, symmtric chrctristic impdncs Th limintion of this symmtry will rduc th crosstlk in multiconductor cbls, i, will nbl to incrs th chnnl crrying cpcity Tht is lso vlid for multiconductor connctors Th rson of th pprnc of common-mod currnts in th lins of th multiconductor cbl is th symmtry of xcittion nd lods Compnstion of th common-mod currnts offrs to dcrs th EM rdition nd to rduc its suscptibility to xtrnl filds Th rigorous mthod for th clcultion of th currnts nd voltgs on th wirs of multiconductor cbl nbls to find mthods to rduc th crosstlk nd common-mod currnts nd to lbort mthods to compnst thm EFEENCES [] C Vlnti, NEXT nd FEXT modls for twistd-pir North Amricn loop plnt, IEEE J Sl Ars Commun, vol 20, no 5, pp 893 900, Jun 2002 [2] X Xu, S Nitt, A Mutoh, nd S Jyrm, Study of lctromgntic intrfrnc of multiconductor twistd-pir wir circuit: Th cs of two-cord twistd-pir wirs, Elctron Commun Jpn,vol80,pp9 6, Dc 997 [3] A Pistolkors, Antnns Moscow, ussi: Svyzizdt, 947 in ussin) [4] Y Y Iossl, E S Kotchnnov, nd M G Strunsky, Clcultion of n Elctricl Cpcitnc Lningrd, ussi: Enrgoizdt, 98 in ussin) [5] J A B Fri nd M V G Nvs, Anlysis of th hlicl twistd-wir pir running bov ground: Trnsfr function vlution, IEEE Trns Elctromgn Compt, vol 45, no 2, pp 449 453, My 2003 [6] D Cochrn, Pssiv cnclltion of common-mod lctromgntic intrfrnc in switching powr convrtrs, MS thsis, Virgini Polytchnic Inst Stt Univ, Blcksburg, 200 Boris Lvin ws born in Srtov, ussi, in Jnury 937 H rcivd th Grdut dgr from Lningrd Polytchnic Institut, Sint Ptrsburg, ussi, in 960, th PhD dgr in rdio physics from th Cntrl srch Institut of Automtic Dvics, Lningrd, ussi, in 969, nd th Doctor of Scincs dgr in physics nd mthmtics from Sint Ptrsburg Polytchnic Univrsity, Sint Ptrsburg, in 993 From 963 to 998, h ws with th Dsign Offic Svyzmorproykt of ussi Shipbuilding Dprtmnt From 2000 to 2002, h ws with MAS, Holon, Isrl H hs uthord or couthord thr books, 76 originl pprs in tchnicl journls, 38 pprs in procdings of intrntionl scintific confrncs, nd 37 bstrcts of confrnc rports H is th holdr of 44 ptnts His currnt rsrch intrsts includ lctromgntic thory, th thory of linr ntnns nd ntnn optimiztion, nd nlysis, dsign, nd dvlopmnts of nw ntnns