NETWORKS FOR IMPEDANCE FUNCTIONS 



385 



coniuH'tcd to form the ('()in{)k'tiHl network witli Ihe liiuil non-reactive 

 hranch, (/„ or />'„ , in place. 



Formulas for the network elements are obtained by ecjuating the j)oles 

 and residues of the network impedance function to the ^iven poles and 

 residues of the general term of the series. Since botli pok^s and residues 

 occur in conjugate complex pairs, and since equality of real and imagi- 

 nary parts is involved, there are four equations, wliich are necessary and 

 sufficient to determine the four constants, R, L, G, C, of the network. 

 The formulas that are obtained by solving these equations are given 

 beneath Figs. 1 and 2. 



The values given for Go and Ro in each case assume that all the terms 

 of the series are of the type specified foi- that case. 



^" Ln 



OKKT 



-wv 



-\AA — OKKI^ 





' — V^ 



Rn Cn 



(a) (b) 



Fig. 2 — General branches of the second kind. 



Fig. 2a 



(use where F(p) = Z{p) 

 and Z„(o) > 0) 



C -± 



LnLn \a^ 



Fig. 2b 



(use where F(p) = Z{p) 

 and Zn(o) < 0) 



2ilf3 



^ = 1 



Ln ttn 



^ = 1 



(ttnan - bnl3n) 

 {ttnan + hn^n) 



00 



Ro = Z{Q>) - E ZM 



1 



ttnan — b„^n 



PC = — 



GnLin — 



M 



(7) 



N 



Ro = Z{0) 

 where M = Gni^l - al) + 2a„/3„6„ 



N = —ttnan + 3Q!„Mn + SttnOinlS n " ?>n/3„ 



