TRANSMISSION LINE EQUATIONS 145 



be symmetrical, hence 



Zo = T~^Z = ZV'~' 



The first group of equations in (1.45) follow from this together with the 

 expression VY for Z obtained from (1.43). The second group in (1.45) 

 is obtained from the first group. 



The commutation rule for Zo is obtained from (1.49) together with the 

 equation obtained from (1.49) by transposition. Since Z is symmetrical 



TZo = ZoT', Y"Zo = rZoV = ZoV'\ 



r"Zo = ZoV" 



and the first of equations (1.47) follow from this. The second and third 

 of equations (1.47) may be obtained similarly from the relations (1.45). 

 The matrix $(r)Zo is symmetrical since its transposed is Zo [^{V)Y and 

 this is equal to Zo^iV') = <J>(r)Zo. A similar argument applies to the 

 other matrices in (1.47). 



The expression for i{x) in (1.44) may be obtained by Maclaurin's ex- 

 pansion. Setting .V = in the second differential equation of (1.48), 



^) = -Yvio) = -YZoHo) = -T'i{o) 

 dx/o 



where we have used the equality between the first and last members of the 

 first equation of (1.45) and where the subscript denotes the value of the 

 derivative at .v = 0. Repeated differentiation gives 



ax- ax 



d i\ „,2 1 di \ „,3 



and so on. Hence 



..vrnu — "'-' 



i{x) =/- — + - 



2-n/2 



i{o) 



2! 



= e i{o) 



Equation (1.46) may now be obtained by using the commutation rule 

 for Zo : 



v(x) — eT"" v{o) = e~'^ Zoi{o) 



— Zoer"" i{p) — Zoi{x) 



This completes the proof of equations (1.44) to (1.47). 



