TRANSMISSION LINE EQUATIONS 175 



The following equations in which X is an arbitrary scalar multiplier may 

 be verified by equating coefficients of powers of X and using the relations 

 just given. 



(Z21 - XZn)(Fn + XI^2) = (XZ12 - Z22)(XF22 + F21) 

 (X2Zi2 - XZu - XZ22 + Z2l)(Fn + XF12) 



= (XZ12 - Z22)(X2I'i2 + XFix + XF22 + F21) (A4.5) 



r-F2i ifx^ -/ \B n r 



L Z21J \_ \C XD - r}~ \_ 



XF22 + F21 X/ 



\I XZ22 — Z2 



Sometimes it is of interest to obtain the elements of F12 , say, when 

 Zii , Z22 , Fii , F22 are known. Relations helpful in studying this problem 

 are 



1 11Z11F12 = I 12Z22I 22 > I iiZiiI 11 — I 11 = F12Z22F21 



1 12} 22 1 21 = I 11 ~ Zu Z12 = ~ZiiFi2l 22 



F21I 11 J 12 = I'22 ~ Z22 Z21 = — Fi2(FiiZii — /) 



When the typical section is symmetrical some simplification takes place 

 and we have 



(A4.6) 



APPENDIX V 



Properties of the Matrix G\^ -}- H\ -\- G' 

 The matrix 



fix) = GX' + //X 4- G' (A5.1) 



which entered the discussion of the case of unsymmetrical sections has a 

 number of interesting properties which are given below. G and // are 

 square matrices with m rows each, and H is required to be symmetrical. 

 As before, we shall denote by Xi , • • • Xm , Xr\ • • • X^' the 2 m roots of the 

 determinantal equation 



|/(X) I = 



