170 BELL SYSTEM TECHNICAL JOURNAL 



the matrix multiplication may be avoided. Taking choice 1 as an example, 

 we may determine the rth row of Q from the expression ^rtr ■ To determine 

 jSr only one element in the rth column of — Y^S'L need be known, for ^r is 

 the quotient obtained by dividing this element by the corresponding element 

 in tr . The product P'Q must be a diagonal matrix, and the sam.e is true of 

 S'T. This may serve to check computations. 



That the expressions for v{n) and i{n) given by (A2.8) and (A2.10) satisfy 

 the transmission equations (A2.1), (A2.6) and (2.10) may be verified by 

 direct substitution and use of 



5(A + A~') = 2AS T(A + A"') = 2A'T (MM) 



These relations follow from the properties of the individual columns of 

 5 and T. 



When the system extends to » = oo a must be zero in order that the 

 voltages and currents may remain finite. This is true because Xr is chosen 

 so that I Xr I < 1. From equations (A2.8) it follows that 



v(n) = PA"a = PA"P~'i'(o) 



i(n) = <2A"a = QA"Q-'i(o) (A2.12) 



v{n) = PQ~ i(n) i(n) — QP" v(n) 



the reciprocal matrices Q~ and P~ always exist when the sections are 

 symmetrical and the roots f i , f 2 , • • • Tm distinct. The last equations in 

 (A2.12) suggest the introduction of the characteristic impedance and 

 adrriittance matrices Zg and Vo : 



v{n) = Zoi{n), i{n) = !>(«), Zo = Y~^ . 



Zo = PQ~' = Zn - ZuQAQ~' = -Zn + ZnQA~'Q~' 



= sir^s~^B = szs~^Zi2 



= ZnTJ.T~^ = BTX~^T~^ (A2.13) 



Yo = QP'' = Yn + YnPAP-' = -Yn - YioPA-'p-' 



= - Yi2SZS~' = C5S~'5"' 



= Tir^T~^C = -TZT~^Y 12 



Not all of the expressions for Zo and Yo obtainable from (A2.10) have been 

 included in (A2.13). Zo and Yo are symmetrical matrices. Although P and 

 Q are arbitrary to some extent the same is not true of Zo and Yo . Com- 

 puted values of Zo and Yo may be checked by use of the relations 



A' - I = (ZoZV2'f = iYT2Yof 



ZoCZo = B, YoBYo = C (A2.14) 



YoZn = - YnZo 



