TRANSMISSION LINE EQUATIONS 165 



show that ZuQAQ'^ and Z2iQA~^Q~^ are s)anmetrical, Zo and ¥„ satisfy the 

 relations 



ZoCZo + ZoD - AZo- B = Y^Yo -j- Y^A - DYo - C = 



{Z,, + Zo)Z:^{Zn - Zo) = Z21 , (F22 + Yo)Y^^{Yn - Yo) = Y21 (2.60) 



ZoQAQ'^' - PAF~'Zo YoPkP~' = QAQ~^Yo 



The characteristic and admittance matrices Zo and Yo associated with 

 propagation in the negative direction are introduced in a similar way. 

 Suppose the system extends to w = — co . Then a — and 



v{n) = PA-"d = PA~"P"'2)(o) 

 iin) = QA""a = -QA""Q~'i(o) (2.61) 



vin) = -PQ~\{n), i(n) = -QP~\{n) 

 Hence we write 



v{n) = —Zoi{n), i{n) = —Yov(n) 

 Zo = PQ-^' = -Zn + ZuQA~'Q~' = Z22 - Z21QAQ"' (2.62) 

 Yo = QP~' = - Yn - YuPA~'P~' = F22 + YnPAP-' 

 Zo and Yo satisfy the relations 



ZoCZo -ZoD-\-AZo-B = YoBY, - YoA + DYo - C ^ 



(2.63) 

 (Zn + Z<,)Z2-/(Z22 - Zo) - Zio (Fu + Fo)F^/(F22 - Yo) = F12 



The fact that Q'iZo + Zo)Q = P'{Yo + Yo)P is a diagonal matrix may 

 be used as a check on computations. 



When the expressions (2.45) for v(^n) and i(n) are placed in (2.3), /"(w) and 

 u^(n) being zero, we obtain the relations 



PA~^ = AP -\- BO PA = AP - BQ 



(2.64) 

 QA'' = CP + DQ QA = -CP + DQ 



When the typical section contains generators the difference equation to 

 be solved is of the form (2.42) 



Gx{n + 2) + Hx{n + 1) + G'.t(w) = gin) (2.42) 



This is true for symmetrical as well as unsymmetrical sections, G being a 

 symmetrical matrix in the former case so that G' = G. The expressions for 

 v{n) and i{n) are those of (2.45) with the particular solutions added: 



v{n) = PA"a + PA~"a + u{n) 



(2.65) 

 i{n) — ()A"a — QA "o + j{n) 



