144 BELL SYSTEM TECHNICAL JOURNAL 



Consider a line extending from x = to .t = =0 , there being no impressed 

 field. Viewing the line at .r = as an n terminal network shows that 

 there is a symmetrical matrix Zg such that v{o) = Zoi{o). Let this be 

 taken as the definition of the characteristic impedance matrix Zq. We shall 

 show from the differential equations of the line that 



1. The voltages and currents in the infinite line are given by 



v{x) — eT^ v{o) 



(1-44) 

 t[x) — e i{o) 



2. The matrix Zo satisfies the relations 



Zo = T~'Z = ZT'~' = TY~' = Y~'r' 



(1.45) 

 Zo = Z~'t - V'Z~^ = YT"' = r'"'F 



v{x) = Zoi{x) (1.46) 



3. The matrices Zg, Z, and Y obey the commutation rules 



^{T)Zo = z,<j>(r') 



$(r)z = z$(r') (1.47) 



F$(r) = 4>(r')r 



where ^(r) is any square matrix, such as e~'' , representable as a 

 convergent power series in T with scalar coefiicients. Furthermore, 

 the matrices $(r)Zo, $(r)Z, and F$(r) are symmetrical. 

 The differential equations of the transmission line are 



^=-Z.-, ^=-F:s ^. = ZF, (1.48) 



ax ax dx- 



the third following from the first two when / is eliminated. That v{x) = 

 e~^ v{o) is a solution of the third equation may be verified by direct substi- 

 tution and differentiation . Since this expression for v{x) approaches zero 

 as x — ^ 00 and reduces to v{o) at x = 0, it represents the voltages in an 

 infinite transmission line. Hence the first equation in (1.44) is true. Set- 

 ting it in the first differential equation of (1.48), putting .r = 0, replacing 

 v(o) by Zoi(o), and noting that i{o) may be regarded as an arbitrary column 

 gives 



TZo = Z (1.49) 



Since T was assumed to be non-singular, Zo is equal to r~ Z. Z is sym- 

 metrical and the reciprocity theorem for electrical networks requires that Zo 

 '' The differentiation of the exponential function is discussed in F.D.C. §2.7. 



