112 BELL SYSTEM TECHNICAL JOURNAL 



When X — 0, i = io, and v = Vq. From (4) and (5) solving for A and 

 B we have 



A ^ io, Vi)/2 ^ B = ^ — '"^''^ 



2 [KTJ^/ 2 JR^ + jcL,. 



Gu + iw Cu \Gu + JCO Cu 



Substituting these values in (4) and (5), we have the equations 



Vq sinh Tx 



i = in cosh Vx — 



Ru H- 7'coL„ 

 Gu -\- jcoCu 



(6) 



V = Vo cosh r.v — ^'o^/ „" . . " sinh Fx. 



In this equation Tx = P, the propagation constant of the line, and 



Ru + JOiLu ^ y 



Gu -\- jooCu 



the characteristic impedance of the line. If we are interested in a 

 given length of line /, the parameters are 



P = ^J{R + JC.L) {G + jcC) ; Zo = ^j§^f^ ' (7) 



where R, L, G, and C are the total distributed constants for the length 

 of line considered. The characteristic impedance is the impedance 

 looking into a line of infinite length as can readily be seen from either 

 of equations (6) by letting x, the length, approach infinity. In this 

 case cosh Tx = cosh P approaches sinh P and both approach infinity. 

 Then from (6) 



Vo /cosh P — i/io\ V V 7 1 



Z — > Zo when x — > =0 , 



io \ sinh P 



since i/io can never be larger than 1. 



The physical significance of the propagation constant is that e~^ 

 represents the ratio of currents or voltages at the two ends of the line 

 when the line is connected to an infinite line of the same characteristic 

 impedance. To show this, suppose we terminate the section con- 

 sidered in an infinite line, which as we have seen above has an im- 

 pedance Zo. Then in equations (6), v = Vi the output voltage and 



