146 BELL SYSTEM TECHNICAL JOURNAL 



1.8 Proof of Relations for Any Uniform Line — Impressed Field 



Here it is shown that if a matrix T satisfies the two conditions of §1.7 

 and if Zo is the characteristic impedance matrix defined there, then the 

 voltages and currents in any uniform line are given by the expressions (1.32). 

 If suitable conditions are fulfilled the relation (1.38) between v{o) and i{p) 

 for an infinite line may be obtained from (1.32). 



First of all, v{x) and i{x) reduce to the required values of v{o) and i{o) 

 at X — 0. All that remains to be shown is that v{x) and i{x) as given by 

 (1.32) are solutions of the transmission line equations (1.5). By substitut- 

 ing (1.32) in (1.5) and using the formulas 



^- cosh xV — T sinh xr = sinh xV T 

 dx 



-r- sinh xr = r cosh xT = cosh xT T 

 dx 



which follow immediately from the series definitions (1.26) of the hyper- 

 bolic functions, we obtain two matrix equations corresponding to the two 

 differential equations. The terms in these equations involving v(o) may 

 be canceled out provided 



r sinh xT = Z sinh xT' Z~^ 



(1.50) 

 r' cosh xT' Zo = V cosh xT 



and these are seen to be true from (1.45) and (1.47). The terms involving 

 i{o) may be canceled by a similar argument. The terms involving l(x) 

 may be canceled provided 



[ sinh (.T - ^)r r/(^) ^^ = [ Zsinh (.T -|)r'z:'/(^)j^ 



Jo Jo 



[ T' cosh {x - ^)T'Z-'m d^= [ Y cosh (x - ^)T l(^) d^ 

 Jo Jo 



and these are seen to be true when .v in (1.50) is replaced by (x — ^). The 

 terms involving t{x) may be similarly canceled. Thus we have verified 

 that v{x) and i(x) as given by (1.32) are solutions of the transmission line 

 equation provided that the commutation rules (1.47) and the relations 

 (1.45) involving Zo of §1.7 are satisfied. This is the case when T is such 

 that (a) r^ is equal to ZY and also (b) every element in e" "" approaches 

 zero as x ^ <x . 



In order to establish equation (1.38) for the T of §1.7 several assumptions 

 regarding the impressed field are required. Writing the hyperbolic func- 

 tions in the first of equations (1.32) in exponential form and premultiplying 



