CHAPTER 10. Consider the transmission lines for voltage and current as developed in Chapter 9 from the distributed equivalent circuit shown below.

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1 CHAPTER 1 1. Sinusoidal Stady Stat in Transmission ins 1.1 Phasor Rprsntation of olta and Currnt Wavs Considr th transmission lins for volta and currnt as dvlopd in Chaptr 9 from th distributd quivalnt circuit shown blow. I r I 1.1 z t I - - c z t I I z I r z l z z cz z z z - - z 1. Dfinin phasors, (z) and Î(z) and assu a harmonic tim variation of th form jt th tim instantanous volta and currnt can b xprssd as follows. z, t j t R z, t j t RÎ 1. I 1.4 To obtain th phasor from of th transmission lin quations, rplac by, I by Î, z r j Î ẑ Î by j. t 1.5 whr Î z j c ŷ sris impdanc ẑ r j shunt admittanc ŷ j c. 1.6 Elctromantic Wavs and Transmission ins By Dr. Jayanti nkataraman 1

2 1. Wav Equations for volta and currnt To obtain th volta wav quation, tak drivativ of (1.5) with rspct to z and Î substitut for from (1.6). Follow a similar procdur to obtain th currnt wav quation z (1.7). Th solutions to th wav quations ar ivn in (1.8) Î z z Î ẑ Î ẑ z z Î ẑ ŷ ẑ ŷ Î whr th propaation factor, z z Propaation Factor, In ordr to compar with th uniform plan wav in a lossy dilctric, considr th transmission lin with prfct conductors (r=) and a lossy dilctric () btwn th two conductors. j j c j j c ŷ ẑ 1 j c j 1 j j j j ˆ j Thrfor, TEM wavs in transmission lins ar idntical to UPW in unboundd rions. Charactristic Impdanc z Substitut th forward travlin wavs z and Î z z Î in (1.5) z z ẑ Î ẑ ẑ ẑ Rarranin, I ẑ ŷ ŷ Charactristic Impdanc, This is similar to th rsult obtaind for in th tim domain. 11 Elctromantic Wavs and Transmission ins By Dr. Jayanti nkataraman 1.1 I Î

3 1. Rflction and Transmission in Cascadd Transmission ins ~ in 1 in in, 1 1,, // // In vry transmission lin th total volta and currnt can b writtn as follows Î z z z z 1 ˆ z z 1 z z z 1 ˆ z 1.1 Whr th complx rflction cofficint, ˆ (z), at any point z is th ratio of th backward travlin wav to th forward travlin wav. ˆ z 1.14 z Dfin complx impdanc at any point z, as th ratio of th total volta to th total currnt. z Dfin a convrs quation for rfction cofficint, 1 ˆ z ˆ z ˆ z z ˆ (z), in trms of wav impdanc, (z) 1.16 z In ordr to dfin a rlationship btwn rflction cofficints at z and z, th rflction cofficint at anothr location z can b writtn as ˆ z' Dividin this by th rflction co-fficint at z ˆ Ê Ê - z' z' z z' ˆ z 1.17 Finally th boundary conditions at th junction btwn two transmission lins can b statd as follows. (i) z is continuous (ii) ˆ z is discontinuous. Elctromantic Wavs and Transmission ins By Dr. Jayanti nkataraman 1

4 1.4 Standin Wav Pattrn and Standin Wav Ratio Th suprposition of two wavs travlin in opposit dirctions in a losslss rion ivs ris to a standin or stationary wav. Th nvlop of th volta ivn by 1.7 plottd vrsus distanc is th Standin Wav (SW) Pattrn, which contains ima and ima. Th ratio of to is dfind as Standin Wav Ration (SWR). It should b notd that SW pattrn and SWR ar masurabl quantitis. Standin Wav Pattrn: z 1 ˆ z Considr a transmission lin of charactristic impdanc tratd by a impdanc. vs z 1.18 ~ whr z Elctromantic Wavs and Transmission ins By Dr. Jayanti nkataraman ˆ ˆ ˆ j z 1 - j z j 1. ˆ. It can b sn that as z chans varis btwn th valus and ±. Whn = ± ( occurs at z=z ) j z ˆ and z 1 ˆ z Whn = ( occurs at z=z ) 1 j z and z 1 ˆ z Standin Wav Ratio (SWR) z = - l ˆ SWR Maxima and Minima in SW pattrn Phas diffrnc btwn points of imum and imum is ivn by z Solvin for z, th distanc btwn conscutiv ima (or ima) is ivn by th followin. z = z 1.6

5 Summary for SW Pattrn Th followin calculations ar rquird in ordr to plot th SW pattrn. vii. Find ˆ oad at th usin (1.8) viii. Writ xprssions for ˆ, SWR, ˆ z, and ˆ z usin (1.5, 1.1, and 1. ) ix. ocat th first imum, z, from th usin ( 1.17). x. Sinc th distanc btwn conscutiv ima (or ima) is, th standin wav pattrn can b drawn. xi. Calculat, and z for all valus of z usin (1.4, 1., and 1.18) Th followin is notd 1. 1 SWR 1. Th smallr th rflction, th smallr th valu of SWR and th flattr th SW pattrn.. Ran of rflction: (no rflction) 1(total rflction) Corrspondin ran of SWR: 1 SWR 4. Distanc btwn conscutiv imum (or imum) in a SW pattrn is. Elctromantic Wavs and Transmission ins By Dr. Jayanti nkataraman 14

6 Exampl-1.1: Rflction in a transmission lin from ˆ 1 Thrfor a volta imum occurs at th. To prov that a volta imum dos occur at th, considr th followin. That is, which is xpctd. j z z ˆ ˆ j z z 1 SWR z z ˆ 1, 1 1 ˆ z 1, z 1 1 Elctromantic Wavs and Transmission ins By Dr. Jayanti nkataraman 15

7 Exampl-1. Rflction in a transmission lin from a purly rsistiv ( =1, =5) = 5 = ˆ ˆ z 1 5 Thrfor a imum occurs at th. This can b provd as follows. j 1 1 j ˆ z j z ˆ z ˆ z 4 z j 1 j 1 4 z which implis that occurs at SWR 1 1 (z) z/ ˆ 1 1 z, z 1 ˆ z, z Chck: 1 4 SWR Elctromantic Wavs and Transmission ins By Dr. Jayanti nkataraman 16

8 Exampl-1. Rflction in a transmission lin from a purly ractiv ( j5, 5 ) 5 j5 This implis = Equat phas, ˆ j.678 j5 5 j.464 j SWR 1 11 ˆ j z 1 j j.14 j 4 z z.14 4 z.46 j.14 This indicats that th occurs at z =-.176 (z) z/ ˆ ˆ z 1, z 1 1 ˆ z 1, z 1 1 j j.14, Elctromantic Wavs and Transmission ins By Dr. Jayanti nkataraman 17

9 Exampl1.4 Rflction in a Transmission lin from a complx = 5 () = 5 j 75 () ˆ 5 j j j SWR j z Thrfor j j1.18 j4 z Equat phas z Solv.16 z z z ˆ ˆ z.745, z z.745, z ˆ j j , Chck SWR Elctromantic Wavs and Transmission ins By Dr. Jayanti nkataraman 18

10 To plot th currnt SW pattrn, not th followin. I occurs at and I occurs at Î (z) z ˆ ˆ ˆ z.745 I z.745 I j1.18 I j Summary From th xampls abov, not th followin 5) = and =R whr R<, volta imum (and currnt imum) occurs at th. 6) = and =R whr R<, volta imum (and currnt imum) occurs at th. 7) j X and R j X 8) j X and R j X, (and I ) occurs first from th., (and I ) occurs first from th. Elctromantic Wavs and Transmission ins By Dr. Jayanti nkataraman 19

11 Exampl-1.5 Th masurd SWR =. and th distanc to th first volta imum in th SW pattrn is (.1 ) from th. Find = 5 SWR 1 SWR 1 1 ˆ j d.5 ˆ oad ˆ d 4 j 4 d j j.4 1 ˆ 1 ˆ oad oad j.4 j.4 (5) (.78 j.9 ) j.55 ( ) Elctromantic Wavs and Transmission ins By Dr. Jayanti nkataraman 14

12 CHAPTER 1 1. Input Impdanc 1.1 Analytical Dvlopmnt Î(- ) Î () ~ (-), () z = - l Fiur 1.1 Transmission lin z = A transmission lin of charactristic impdanc,, in nral lossy, and tratd by a impdanc,, is shown in Fiur 1.1. An analytical xprssion for th input impdanc can b drivd by th mthodoloy dscribd in Chaptr 1, usin (1.15, 1.16, and 1.17) ˆ ˆ ˆ 1. ˆ ˆ in in 1 1 in l l l l l l l l l l cosh l sinh l 1.4 cosh l sinh l For a losslss lin j and th charactristics impdanc is ral. Thrfor in cos j sin 1.5 cos j sin cosh ( ) l l sinh ( ) l l coshj cos j jsin sinh. Elctromantic Wavs and Transmission ins By Dr. Jayanti nkataraman 165