46 BELL SYSTEM TECHNICAL JOURNAL 



impedance and shunt mutual admittance jXn and jByi . If we consider a 

 wave which varies as exp(— jTz) in the z direction we have 



r/i - jB^Vi - jBnV. = (3.65) 



TVi - jXJy - jXnh = (3.66) 



r/2 - JB0V2 - jBnVi = (3.67) 



TV2 - jX2h - 7X12/1 = (3.68) 



If we solve (3.65) and (3.67) for /i and A and eUminate these, we obtain 



Fo -(r ^ XrBi + XnBr^ 



Vi Xi B12 + B2 X12 



(3.69) 

 (3.70) 



(3.71) 



V2 X2 B12 4- Bi X12 



Multiplying these together we obtain 



r^ + (Xi B, + Xo B2 + 2X12 Bu)r 



+ (Xi Xo - X^2) (Bi B2 - B^2) = 



We can solve this for the two values of F- 



P = _i(Xi 5i + Xo B2 + 2X12 ^12) 



± I [(Xi 5i - Xo ^2)-^ + 4 (Xi B, + X2 ^o) (X12 Bu) (3.72) 



+ 4 (Xi Xo Bu' + Br B2 Xio2)]i/2 



Each value of T'~ represents a normal mode of propagation involving both 

 transmission lines. The two square roots of each F- of course indicate waves 

 going in the positive and negative directions. 



Suppose we substitute (3.72) into (3.69). We obtain 



- (Xi Bi - X2 B^ ± [(Xi Bx - X2 B.f 

 V2 ^ + 4(Xi Bi + X2 ^2)(Xi2 ^12) + 4(Xi X2 Bil + Bi B2 XA)]"- . . 

 V, 2{X,Bn + B2Xu) ^ - '^^ 



We will be interested in cases in which Xi^i is very nearly equal to X2B2. 

 Let 



^Tl = Xi/^i - X2B2 (3.74) 



and in the parts of (3.73) where the difference of (3.74) does not occur use 



Xi = .Y2 = X 



(3.75) 

 B^^ B2 =^ B 



I 



