36 BELL SYSTEM TECHNICAL JOURNAL 



where 



4Zo'Zo" 



^ (Zo"+Zo')(Zo"+Z,) 



_ {Z," - Zn'){Z," - Zt) 

 ^ iZo" +Zo'){Zo" -^Zt)' 



In the special case when Zi = Zq', we have 



If PQ is electrically long, (17) becomes simply 



Tr = pe-^"\ Ty = ^, Tj. 



An interesting physical interpretation of (17) will follow if we ex- 

 pand the factor in parentheses into a series 



Ti = pe-^"' + pqe-^^"^ + ^g^g-sr-z + . . .. 



The first term represents what remains of the original wave on the 

 first passage through PQ. A part of the original wave is reflected 

 back at Q and then partially re-reflected from P; the second term 

 represents that fraction of the re-reflected wave which is transmitted 

 beyond Q. The following terms represent succeeding reflections. In 

 making this analysis, we must remember that p = pipi where pi and 

 p2 are respectively the transmission coefficients across the first and 

 the second boundaries on the supposition that the inserted piece is 

 infinitely long. Similarly, q = qiq^, the product of the two reflection 

 coefficients. 



Let us now consider a non-uniform transmission line. The propa- 

 gation of a disturbance is no longer exponential and we introduce the 

 ratios k+ = V+(x2)/V+(xi) and k~ = V~(xi)/V~(x2) for the voltage 

 ratios in the waves moving in opposite directions. In what follows 

 Xi and X2 are the coordinates of the beginning and the end of the 

 inserted piece. The transmission coefficient T across the insertion, 

 that is, the ratio of the total quantity at x = X2 to the impressed 

 quantity at x = Xi, is then 



T = piK+p2 + (piK+)(q2K~qiK+)p2 + (piK+){q2K-qiK+){q2K-qiK+)p2 + ■ ■ '. 



This can be rewritten as follows : 



P ..4- 



r = ^[1 + g/c + (qKY + {qK^ +•■•]«+ = 



I — qx 



