TRANSMISSION LINE EQUATIONS 135 



the first method converge too slowly to be of value. However, it requires 

 the solution of an wlh degree equation and the determination of the m 

 modes of propagation where rn is the number of circuits. For ni = 2 

 this is no handicap and the method is quite convenient. In this case 

 the method is closely related to one described by John Riordan in an un- 

 published memorandum. 



First Method: Multiply the matrices Z and I' together to obtain ZY . 

 Choose the number 7- in 



ZY = It + R, (1.12) 



where / is the unit matrix, so that the elements of R are small in comparison 

 with 7-. For many transmission lines it is possible to do this. V may be 

 obtained by using the binomial theorem to expand the square root in the 

 formula 



r = VzF = 7(/ + i--Rf, (1.13) 



where 7 is that square root of 7- whose real and imaginary parts are non- 

 negative. In carrying out the work it is convenient to introduce the matrix 

 5" whose elements are small in comparison with unity. 



r = 7(/ + s) (1.14) 



To compute .S', first compute the matrix R/2y- and then use the power 

 series 



V27V 2V27V 2V2t7 8\27V 



,T(RY_2l(RX. 

 ^ 8 V2tV 16 \2tV 



This series will usually converge rapidly. The matrix c ^ is given by 



—xT —z —zS /< ^ jr\ 



e = e -e (1.16) 



where s is a number, 2 = 7.r, and e~'^ is to be computed from 



-ZS J. zs (zsy (zSY , . 



' ^ ^"r! + -2r"^r + -" ^^-^^^ 



e ' is obtained from e "" by interchanging the rows and columns. The 

 characteristic impedance matrix may be obtained from (1.11), 



Zo = TY-\ 



after computing F from S as in (1.14). 



