TRANSMISSION LINE EQUATIONS 



147 



+ e 



f e-''-''^m-\-Zotmd^ 



Jo 



When .V -^ x equation (1.38) is obtained provided that the impressed field 

 and the terminal conditions at the far end are such that (a) v(x) remains 

 finite, (b) the integral in (1.38) converges, and (c), the last expression on 

 the right in the equation above approaches zero as .v — ^ oo . 

 1.9 Derivation of Equations {1.25) 



Although equations (1.25) may be obtained by setting l{x) = t{x) = 

 in §1.8, it is of some interest to derive them directly. By repeated dififer- 

 entiation of the equations 



dv 

 dx 



= —Zi, 



dj. 



dx 



-Yv 



the second, third and higher order derivatives may be obtained, 

 these in Maclaurin's expansion about .r = gives 



4 



(1.48) 

 Using 



v(x) = 



/ + |;ZF + |j(ZF)^ + 



v{o) 



3 ..5 



X „,, , X 



f!^ + i!^^ + "5!^^''^'' + 



Zi(o) 



i{x) = - 



h'^h'^^^h^^'^'^ 



(1.51) 



Yvio) 



+ 



^ + ll ^'^ + 1] ^^'^^'' + 



i(o) 



These series converge for all values of .v and could be used for computation 

 were it not for the unfortunate fact that in most problems a great many 

 terms would be required for a satisfactory answer. For the time being, 

 let r be any matrix whose square is ZV. The definitions (1.26) of the 

 hyperbolic functions enable us to write (1.51) as 



v{x) = cosh .vT v(o) — sinh .\T F" Zi{o) 



(1.52) 

 i(x) = -sinh xT' F' ^Yvio) + cosh .\T' i(o) 



If in addition to being a matrix whose square is ZY, T is also such that 

 every element in e""" approaches zero as x -^ go , then we may use the 

 relations (1.45) for Zo and obtain (1.25). 



