ELECTRIC CIRCUIT THEORY 83 



disposal to satisf\- an\ lioundarx' conditions, proxided T is so chosen 

 that 



cosh r = ~^„-^^" 



2s2 (221) 



= l+2p 

 where p = Si/4z2- 



Now by reference to equation (219) it is easily seen that V is the 

 propagation constant of the artificial line, precisely analogous to the 

 propagation constant 7 of the smooth line. In terms of the im- 

 pedances Zx and Go, the propagation constant of the artificial line is 

 determined by (221). This equation may either be regarded as an 

 operational equation or a symbolic equation, depending on whether 

 the impedances are expressed in terms of the operator p or in terms 

 of iw, where w is 27r times the frequency. 



Now suppose in (221) we write f^' = .v; the equation becomes 



.r+l/.r = 2(l + 2p) 

 and sol\-ing for .v we get 



x- = er = (l + 2p) + V(l + 2p/-^-l 



„ __ _ (222) 



= (\/l+P + Vp)- = f\/l+p-\/p) 2 



which is an explicit formula for F. 



Now return to equation (219) and let us assume that the line is 

 either infinitely long, or, what amounts to the same thing, that it is 

 closed by an impedance which suppresses the reflected wave. We 

 assume also that a voltage Vo is impressed at mid-series position of the 

 zero" section (» = ). Equation (219) becomes 



and the currents in the zero"' and P' sections are 



Now, by direct application of Kirchhoft"'s law to the zero"' section, 

 we have 



l^,= (|Si+S2)/o-=2/l, 



whence 



A]hz,+z,{l-e-n'^=Vo. (223) 



But 



K 

 " K • 



