Question

1. Show that for a lossless N-port network, REAL([Z]) = -REAL([Z]T).(The real matrix is anti-symmetric)

Answer 1

For a lossless N-port Network n = vottun Io In + Is of the microwave network The impedance Matrix CZ] then relates the voltages N-port For and corrents Zil Z12 Zız V2 ZIN V 1 온치 11 AT TIM ZzI ZNI ZNN Hea IN We now show how the scattering matrix the [z] (or[y]) can be determined from matrix and vice versa. First we most that the characteristic impedance Zon of all the ports are identical assume The total current Voltage and be re-written at the nth port. can as Vo = vottun Int of In Ja VO 20 ง ZO using the definition of [z] Zo TZ]CIJE I [[+][ wt] - [7] [v]) [2] [ut] [Z][v-] - (vt]+[v-] Zo Zo ZICIJ [ut] + [v-]

(I) = (t]+(5-) zo (Cut] - [v]) [z][1] = [ [+]vt] - [Z][v]) - (Cw*) ) (2x3.3(v1) = to{w} 4tv) [v-] = [s] [v+] (w] = Identity Matrix [w] = ( [7] + Zolu]]" (.CZ] - Zo[u]][v+] (v) (vtj =[s] [s] = (c) + zo[u] ) (EJ =20[u]) Vo = vit tvo In = In't t. In unt zo zo -> ха voo vnts (vntzoin) Ź cun - Zoon) [v+] = {( UJ + Zo[7]) TV): [2][1] [-] Il cv] - Zo [I]] [v]= [s I wt] [ut] (12]+zołuj(1) [w] = ₃ ( [2] = ZO E UJ) [I] - Zo[g) (12]+ żołu])?? [s] = ([23

88] = [[z] - Zo[u]) C[Z] + zo co]) -1 From this [sj=[ST [Z] = [z]T then real part (2) Real [z] = -(real (zj]