384 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 



the given function might be less than the sum of the variable terms for 

 these frequencies, with the result that the final resistance branch would 

 be negative for either the series or parallel type of network development. 

 To make the procedure as concrete as possible, particular forms of 

 networks are described in the section following with, explicit formulas for 

 computing their elements. 



NETWORK FORMULAS 



Simple forms of network branches are shoAMi in Figs. 1 and 2. Those 

 of Fig. 1, referred to as branches of "the first kind" are suitable for con- 

 nection in parallel where the given function F(p) is an admittance, F(p), 

 w^hile networks of "the second kind," shown in Fig. 2, are suitable for 

 connection in series to represent an impedance, F(p) = Z(p). The net- 

 works of Figs, la and 2a apply where the value P„(0) of the general term 

 is positive, while Figs, lb and 2b apply where P„(0) is negative. Figs. 3 

 and 4 illustrate, respectively, networks of the types of Figs, la and 2a 



Cn 



Rn Lr 



Ln 



J_ 



Gn 



Rn 



Cn 



'-WV-' 

 _1_ 



Gn 



;a) (b) 



Fig. 1 — General branches of the first kind. 



Fig. la 



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

 and F„(0) > 0) 



Ln = 



2a„ 





jr = — (cLnOCn — hn^n) 



Rn 1 



{ana„ + 6„/3„) 



Ln dn 



Go = F(0) - E F„(0) 



Ln — 



1 



Fig. lb 



(use where F(p) = Y(p) 

 and F„(0) < 0) 



^l{al + l3lY(al + //„) 



2M^ 



M^ 



LnCn Plial + bl) 



ttnOLn — bn^n 



CrnLn — — 



iin^n 



M 



N 



Go = F(0) 



(6) 



