TRANSMISSION LINE EQUATIONS 137 



Applying (1.22) to F, e~'' and Zo even though they arc not polynomials 

 in ZY gives results which may be verified to be true. 



r = V2F = E N{y;)yr 



Zo = (ZV)h'-' = 23 N{yl)yrY~' 



e-'''Zo= ZNiyDyrC-'^'^y-' 



where the summations extend from r = 1 to r = m and 7i , 72 , • • • 7m 

 are the square roots of 71 , 72 , • • • 7^ respectively whose real parts are non- 

 negative. 7i , 72 , • • • 7m are also the propagation constants of the "normal 

 modes" of propagation. Some light is thrown on the physical significance 

 of the matrix N(yr) by supposing that only the rth normal mode is being 

 propagated on the transmission line. N{yr) is such that it can be expressed 

 as a column matrix times a row matrix. The voltages in circuits 1,2, ■ ■ • m 

 are proportional to the first, second, • • • wth elements, respectively of the 

 column matrix. The currents in circuits 1,2, • • • m are proportional to the 

 corresponding elements in the row matrix. 

 1.3 Results for Any Uniform Line — No Impressed Field 



When the length of the line is finite the voltages and currents may be 

 expressed as 



v{x) = cosh xT v(o) — sinh xF Zoi(o) 



_i (1.25) 



i(x) = — sinh xF' Zo v(o) + cosh xT' i(o) 



where Zo and T have the same meaning as before. The matrices sinh xTZo 

 and sinh xT'Z~ are symmetrical. The square matrices cosh xT and sinh xT 

 are defined by the series 



2 p2 4 p4 



cosh XT = 1 + ^- + ^^ + •.. 



smh ^r = — -f — ^ + • • • 



cosh xV is obtained by interchanging the rows and columns of cosh xT 

 and sinh xT' is obtained similarly from sinh xT. Solving (1.25) for v(o) 

 and i{o) gives 



v{o) = cosh xT v(x) + sinh xT Zoi{x) 



i(o) = sinh xF' Z~ vix) -|- cosh xV i{x) 



As in the case of the infinite line, we have two ways of computing the 

 coefficients of v{o) and ^(o) in the expressions (1.25) for v{x) and iix). 



