Lecture 20. Transmission Lines: The Basics

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1 Lcu 0 Tansmissin Lins: Th Basics n his lcu u will lan: Tansmissin lins Diffn ps f ansmissin lin sucus Tansmissin lin quains Pw flw in ansmissin lins Appndi C 303 Fall 006 Fahan Rana Cnll Univsi Guidd Wavs S fa in h cus u hav bn daling wih wavs ha ppagad in infini si mdia F man applicains i is dsiabl hav lcmagnic ng b guidd in much h sam wa as wa flw is guidd b having i flw in pips Tansmissin lins a h simpls sucus ha guid lcmagnic wavs Tansmissin lin: n his cus a ansmissin lin wuld b an w abia shapd mal cnducs ha a v lng and unifm in a las n dimnsin A ansmissin lin mad up f w mal cnducs ha a v lng and unifm in h dicin C 303 Fall 006 Fahan Rana Cnll Univsi

2 Cmmn Tansmissin Lin Sucus b W a ε ε d Caial Cabl π ε C b lg a µ b L lg π a PaalllPla Tansmissin Lin W C ε d L µ d W LC µ ε LC µ ε Capacianc and nducanc p uni lngh f ach sucu a shwn C 303 Fall 006 Fahan Rana Cnll Univsi Cmmn Tansmissin Lin Sucus adius a d d/ ε adius a Paalll Mal Wis Mal Wi Ov a Gund Plan F d >> a : F d >> a : π ε π ε C C d d lg lg a a µ d µ d L lg L lg π a π a LC µ ε LC µ ε Capacianc and nducanc p uni lngh f ach sucu a shwn C 303 Fall 006 Fahan Rana Cnll Univsi

3 Tansmissin Lin lags Supps h pnial diffnc bwn h w cnducs f a ansmissin lin a lcain is () hn fild lin ingal in a plan paalll h plan a h lcain is indpndn f h cnu C an and is lad h pnial diffnc () as: C ( ). ds. ds c c Th chag p uni lngh Q() n h ansmissin lin a lcain is: Q ( ) C ( ) Capacianc p uni lngh Tw pins wih diffn valus can hav diffn pnials: ( ). ds. ds ( ) c c ( ) C ( ) C Th cnducs a n lng quipnials C 303 Fall 006 Fahan Rana Cnll Univsi Tansmissin Lin Cuns Supps h al cun in h upp cnduc in h dicin a lcain is () and in h lw cnduc is () Th fild flu p uni lngh λ() nclsd bwn h w cnducs a h lcain is lad h cun () as: λ L nducanc p uni lngh ( ) ( ) Tw pins wih diffn valus can hav diffn cuns ( ) ( ) C 303 Fall 006 Fahan Rana Cnll Univsi 3

4 C () Faada s Law () ( ) Us Faada s law f h cnu C:. ds µ. da c ( ) ( ) λ( ) ( ) ( ) L ( ) ( ) ( ) L () λ( ) L ( ) C 303 Fall 006 Fahan Rana Cnll Univsi CunChag Cninui quain Q() () () ( ) Us h pincipl f cnsvain f chag (cunchag cninui quain): f cun is vaing in spac h mus b chag ih piling up piling dwn smwh Q( ) ( ) ( ) Q ( ) C ( ) ( ) ( ) C ( ) ( ) ( ) C () C 303 Fall 006 Fahan Rana Cnll Univsi 4

5 5 C 303 Fall 006 Fahan Rana Cnll Univsi Tansmissin Lin quains Th fllwing w quain dscib h ppagain f guidd lcmagnic wavs n ansmissin lins (als calld h Tlgaph s quains): L C Wav quains: () () () L () C (3) (4) (3) and (4) LC quain f a wav avling wih a vlci LC v LC A simila quain can b divd f h cun: C 303 Fall 006 Fahan Rana Cnll Univsi Nau f Guidd Wavs in Tansmissin Lins Th guidd wav cnsiss f filds and filds gh wih chags and cuns n h cnducs ha all mv gh in snc wih a vlci givn b: λ Th chags saisf h bunda cndiins f h filds LC LC LC v

6 Nau f Guidd Wavs in Tansmissin Lins ( ) ( ) ( ) ( ) LC LC Th guidd wav cnsiss f filds and filds gh wih chags and cuns n h cnducs ha all mv gh in snc wih a vlci givn b: v LC Th chags saisf h bunda cndiins f h filds λ Th cuns saisf h bunda cndiins f h filds Th wav is calld a TM wav sinc bh h fild and fild pin in a dicin ansvs h dicin f ppagain C 303 Fall 006 Fahan Rana Cnll Univsi Filds and filds f Cmmn Tansmissin Lins Caial Cabl PaalllPla Tansmissin Lin Nic ha a ach pin ng flw in h dicin pins in h dicin indicaing C 303 Fall 006 Fahan Rana Cnll Univsi 6

7 Filds and filds f Cmmn Tansmissin Lins Mal Wi Ov a Gund Plan Paalll Mal Wis Nic ha a ach pin ng flw in h dicin pins in h dicin indicaing C 303 Fall 006 Fahan Rana Cnll Univsi Cnv phass: Cun and lag Phass ω [ ] ω [ ] ( ) R ( ) ( ) R ( ) Tansmissin lin quains in phas nain: ( ) ( ) L ( ) ( ) C ( ) ( ) ω L ω C ( ) ( ) Wav quains in phas nain: ( ) ( ) LC ( ) ( ) LC ( ) ω LC ( ) ω LC ( ) ( ) C 303 Fall 006 Fahan Rana Cnll Univsi 7

8 Sluins f Tansmissin Lin quains Sa wih h cmpl wav quain: ( ) ω LC ( ) Assum a sluin f h fm f a avling wav: ( ) C 303 Fall 006 Fahan Rana Cnll Univsi A wav avling in h dicin Subsiu in h cmpl wav quain: ( ) ω LC ( ) ω LC ω LC ω LC Dispsin lain f a wav avling wih a vlci v LC mpdanc f a Tansmissin Lin lag is: ( ) Find h cun fm h ansmissin lin quain: Wh Z givn b: ( ) ω L ( ) ω L ( ) ω L ( ) Z L Z ω L C ( ) is calld h chaacisic impdanc f h ansmissin lin S a vlagcun wav ppagaing in h dicin n a ansmissin lin is spcifid cmpll b: ( ) C 303 Fall 006 Fahan Rana Cnll Univsi Z ( ) 8

9 9 C 303 Fall 006 Fahan Rana Cnll Univsi Bacwad Wavs n a Tansmissin Lin ( ) A vlagcun wav ppagaing in h dicin n a ansmissin lin is spcifid cmpll b: A vlagcun wav ppagaing in h dicin n a ansmissin lin is spcifid cmpll b: ( ) Nic h v sign ( ) Z ( ) Z λ C 303 Fall 006 Fahan Rana Cnll Univsi Fwad and Bacwad Wavs n a Tansmissin Lin n gnal vlag n a ansmissin lin is a suppsiin f fwad and bacwad ging wavs: ( ) Th cspnding cun is als a suppsiin f fwad and bacwad ging wavs: ( ) Z Z

10 PaalllPla Tansmissin Lins: Filds lags and Cuns W d f h vlag and h cun wavs a: ( ) Z ( ) PaalllPla Tansmissin Lin hn h fild and h fild a (igning h finging filds): ( ) ˆ d W ( ) ˆ Givn h ampliud(s) f h vlag and/ cun wavs h fild and h fild assciad wih h wav can b fund C 303 Fall 006 Fahan Rana Cnll Univsi ng Flw and Pw n a Tansmissin Lin Cnsid a ansmissin lin wih a vlagcun wav ging in h dicin: w much is h al imavag pw (n pw p uni aa) caid b h wav in h dicin? P () S( ). ˆ dd Th aa ingal is v h ni plan ( an plan paalll h plan) R [ S( )]. ˆ dd * can b shw ha his ingal quals: () [ ] P R R * Z And if h is a als bacwad wav hn: () [ ] * * P R R * Z Z * C 303 Fall 006 Fahan Rana Cnll Univsi 0

11 Wavs in F Spac and Wavs in Tansmissin Lins F Spac ( ) ω µ ( ) ( ) ω ε ( ) ω ( ) µ ε ( ) ω ( ) µ ε ( ) ( ) ˆ ( ) ˆ η Tansmissin Lins ( ) ( ) ω L ω C ( ) ( ) ( ) ω LC ( ) ω LC ( ) Z ( ) ( ) ( ) C 303 Fall 006 Fahan Rana Cnll Univsi Appndi: ng Flw and Pw n a Tansmissin Lin This appndi ffs a pf f h pw flw fmula f abia ansmissin lins () S( ). ˆ dd R [ S( )]. ˆ dd * R [ ( ) ( )]. ˆ dd P B assumpin bh and filds hav nl ansvs cmpnns (i.. n cmpnn in h dicin) Fm Faada s Law: ˆ ˆ ˆ ˆ ω µ T ˆ ω µ C 303 Fall 006 Fahan Rana Cnll Univsi T

12 Appndi: ng Flw and Pw n a Tansmissin Lin ˆ T ω µ 0 T Thf n ma wi h fild as h ansvs gadin f a scala pnial: T φ Wh b assumpin h pnial saisfis: φ s φ nd cnduc cnduc Fm Amps s Law: ˆ J ω ε T ˆ ˆ J ω ε T ˆ J C 303 Fall 006 Fahan Rana Cnll Univsi Appndi: ng Flw and Pw n a Tansmissin Lin s cnduc J dd nd cnduc J dd P * () R [ ]. ˆ dd * R [ Tφ ]. ˆ dd 0 * R T [ φ ]. ˆ dd * R [ φ T ]. ˆ dd * R [ φ J ] dd * R[ ( φ s φ nd ) ] cnduc cnduc * R[ ] C 303 Fall 006 Fahan Rana Cnll Univsi

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