TRANSMISSION LINE EQUATIONS 101 



Premultiplicalion by B and use of BA' = AB gives 



{^r-A)BN'(^) = B\m\/\m 



1(1) 



Hence, the third equation in (2.39) holds except possibly for ^ = i",. , and 

 from the concept of continuity it holds there also. The fourth equation in 



(2.39) may be proved in the same manner. 



The expression (2.20) for Zo may be obtained by letting n become very 

 large in the expression (2.23) for v(n). v(o) and i(o) must be related so that 

 v(>i) remains finite. Since | X,- | < 1 and the X,'s are unequal the coefficients 

 of X7" must vanish. This requires 



Xr — Xr 



Summing r from 1 to m and using (2.38) gives the required expression for Zo . 

 2.8 The Unsymmetrical Section Line 



The method used here is analogous to those described in Appendices I and 

 II for the uniform line and the symmetrical section line. The other methods 

 apparently do not lead to the simplification which occurs in the symmetrical 

 case. 



Equations (2.2) and (2.1) lead to the difference equations 



YM'^ + 2) + [Tn + y22]v{n + 1) + Yo,v(u) = -i°(n + 1) -j-f(n) (2.40) 



Zui(n + 2) - [Zn + Z22]i(n + 1) + Z^An) = v°{n + 1) - u°(n) (2.41) 



Both of these equations are of the form 



Gx(n + 2) + Hx{n + 1) + G'xin) = g(n) (2.42) 



in which G and H are square matrices of order m, H being symmetrical and 

 G' being the transposed of G. Wlien the sections are passive equations 



(2.40) and (2.41) becom.e 



YMn + 2) + [I'll + Y^oHn + 1) + y^An) = (2.43) 



Zui(n + 2) - [Zn + Z2Mn + 1) + Z2ii(n) = (2.44) 



In the passive, unsymmetrical case the expressions for v(n) and i(n) are 

 of the form 



v(n) = PA"a + Pi\r"a 



(2.45) 

 i(n) = ()A»a - QA-»a 



Comparison with (A2.8) shows that in the symmetrical case P — P and 

 Q = Q. The minus signs over P, Q, and a indicate that they are associated 

 with propagation in the negative direction. The propagation constants of 



