134 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



The transmission line equations may be written: 



r/i - jB,V, + jB„y2 = 

 rFi - iXi/i + jXJo = 



r/o - JB0V2 + jB„yi = 



TV2 - jXJa + jXJ, - 



where 



B, - 5io + 5« 



Bo = B20 + Bm 



X2 = X20 -f" Xm 



1 1 and 1 2 are eliminated from the (2.2.1) and we find 



F2 ^ + (r- + XiBi + x^Bj 



Fi 



F2 



(2.2.1) 



X\Bm + B%Xm 



+ (r- + X2S2 + x^Bj 



XlBm + 5lX„ 



(2.2.2) 



(2.2.3) 



These two equations are then multipUed together and an expression for 

 r of the 4th degree is obtained : 



r' + (XiBi + X2B2 + 2Z,„Bjr' 



+ (X1Z2 - Xj){B,B2 - Bj) = 

 We now define a number of dimensionless quantities: 



(2.2.4) 



B, 



BiB. 



Xm 



= h' = (eapacitive coupling coefficient)' 



= X = (inductive coupling coefficient) 



XiXo 



B\Xi = ^1, B2X2 = (82' 

 X1B1X2B2 = 13^ = (mean phase constant) 

 With these substitutions we obtain the general equation for T~ 



T' = 13' 



2 \(3-r ^ I3{' ^ 



y 4v^2'^^/3i^ 



_ (2.2.5) 



+ 26.r - (1 - .r-)(l - U') 



