(A3. 11) 



TRANS^fISSION LINE EQUATIONS 173 



given by 



= UM'^ir'Zn = UM^U~'l7i 



= ZnWM~^W~' = YuWM^W' 



(A3.9) 

 r„ = QP~' = YnPM''P~' = QM~'Q~'Zn 



= YnUM'^ir' = ZnUM^U"' 



= WM~Hv~'Yn = WM^W'Zn 

 The matrices Zg and To may be checked by means of the relations 



Zoi 11 = Z\i\ o , ZqI \\Zo = Zn , I oZiii o = 1 11 (A3. 10) 

 Another set of solutions may be based upon the equations 

 2v(n) = -Zu[i(n + 1) - i(n - 1)] 



2i{n) = YiMn + 1) - v{n - 1)] 



which are derivable from (A2.1) and (A2.2). Combining these equations 

 gives, upon using YY2ZY2 = —BC, 



v(n + 2) - 2v(n) + v(n - 2) - ABCv(n) 



(A3. 12) 

 i{n + 2) - 2i(n) + i(n - 2) = 4CBi(n) 



However, we shall not consider these equations here beyond pointing out 

 that they lead to 



P = ZuQ^, Q = - VuP^ 



PIT = BCP, ()2:' = CBQ 



which may also be derived from (A2.10). 



APPENDIX IV 



Relations Between the Square Matrices of 

 A Multi-Terminal Section 



When the reciprocal theorems of network theory are applied to equations 



(2.1) and (2.2) it is found that Zu , Z22 , I^ii , 1^22 are symmetrical and 



Z2i= Zi'2, F2i= Y[2 (A4.1) 



i.e., Z21 and r2i are the transposed matrices of Z12 and ri2 , respectively. 

 Solving equations (2.1) for the currents and comparing the result with 



(2.2) shows that 



Zii Z12 

 Z21 Z22 



[Yn Ynl r i°(n) 1 ^ _rFn F12I h°(w: 



LF21 F22J' l-An)] lYn Y22]lu°{n 



