TRANSMISSION LINE EQUATIONS 



167 



and obtain the square matrix Q from P: 



Q = Z^'PG = YPG-' 

 where G is the diagonal matrix 



"71 



G = 



72 

 ,0 73 



The voltages and currents at any point x are 



v{x) = PM{x)a + PM(-x)d 

 i(x) = QM(x)a - QM(-x)d 



(A1.2) 



(A1.3) 



(A1.4) 



where a and a are arbitrary column matrices associated with propagation 

 in the positive and negative directions of .v and M{x) is the diagonal matrix 



M(x) = 



(A1.5) 



The values of a and a are to be determined from the boundary conditions. 

 When the line extends to x = 00 



v(x) = PM{x)P \{o) = Zoi{x) 

 i{x) = QM{x)Qr\{o) 

 where the characteristic impedance matrix Zo is given by 

 Zo = P()~' = PG^^P'^Z = PGP~^V~^ 

 - ZQC'^Q"' = V~'QGQ~' 



(A1.6) 



(A1.7) 



Since v = pre'^'' and i — qr^"'' , where qr is the rth column of Q, are solutions 

 the differential equations give 



{lyl - ZY)Pr = 0, {hi - YZ)qr = 

 and from these it follows that 



P~'ZYP = Q'^YZQ = G' 



(A1.8) 



(A1.9) 



The relations (A1.8) may be used to prove the following: 



1. The elements in the rth column of Q are proportional to those in the 

 non-zero rows of F(yr). 



2. The matrix P'Q is a diagonal matrix and from this it follows that if \f/ 

 is any diagonal matrix 



(PrpP"')' = Qxl^Q~' (ALIO) 



