TRANSMISSION LINE EQUATIONS 159 



using the definition of Zg , and regarding i{o) as an arbitrary column gives 



(A' - e"^>""'' = Ce-'^^'Zo 



Replacing A' — e~ by CZo, as follows from the case n — 0, and pre- 

 multiplying by C gives 



Z— 71 r ' — n r 7 



and this leads to the first of equations (2.v35). From (2.34) and the rela- 

 tions AB = BA', CA = A'C we have 



sinh T B = B sinh T' cosh T B = B cosh T' 



C sinh r = sinh T' C C cosh T = cosh F' C 



Addition and subtraction leads to 



e B = Be Ce = e C 



from which the last two of equations (2.35) follow. Since each of equations 

 (2.35) expresses the equality of a matrix and its transposed, it follows that 

 the matrices are symmetrical. 



Equation (2.33), which is almost self-evident on physical grounds, fol- 

 lows from 



v{n) = e~" v{o) = e"" Zoi{o) 



= Zoc"" i{o) = Zoi(n). 



2.7 Proof of Relations for Any Symmetrical Section Line 



The expressions (2.24) for v(n) and i(n) in a line whose sections contain 

 generators may be verified to satisfy the difference equations (2.9). The 

 expressions (2.34) for Zo and the commutation rules (2.35) for B and C 

 are used in the verification. Setting n = 1 in the expressions for v(h) and 

 i(n) gives the difference equations (2.9) and hence v{n) and i(n) are the 

 solutions which correspond to the initial values v(o) and i(o). 



In order to derive the relation (2.28) between v(o) and i(o) for an infinite 

 line we put the hyperbolic functions in the expression (2.24) for v{n) in 

 exponential form and multiply through by 2e~" 



n 



2e-"^.(w) = v{o) - Zoi{o) + E e-^^lBi^p - 1) - ZoCv^'ip - 1)] 

 + e-'^'Mo) + Zoi{o)] 

 + e-^ i: e-'''-''^[Bi%p - 1) + ZoCv'^ip - 1)] 



