A METHOD OF IMPEDANCE CORRECTION 811 



A nalytical Relations between Polynomial Coefficients and Element Values 



Case I — w = 1. 



The general analysis shows that the conductance of the system must 

 be expressible in the form 



G = 



1 Vl 



X 



-2 



Zo 1 + Aix^ 

 A direct mesh computation of the network of Fig. S-a gives 



1 Vl 



Zo 1 - (1 - a{')x' 

 From which, by comparison of coefficients, 



^1 = - (1 - fli^) 



or 



ai = Vl + Ai. 



The susceptance characteristic is given by 



Zo 1 + Arx-" 

 It can be annulled exactly by the reactance 



iX = ^ Zo.r + -. — = I — Zr. -\ /.^k 



ax laix \ zfli / ai 



where Zik and Z^k are, as before, the series and shunt impedances of the 

 " constant-)^" filter. 



If the conductance and susceptance controlling portions of the 

 network are combined the resulting structure is identical with a half 

 section of the conventional "m-derived" type. We have merely to 

 replace Oi by m. Single branch conductance controlling networks 

 therefore contribute nothing new to filter impedance correction. 

 Multiple branch networks, which can be considered, if one pleases, as 

 natural extensions of the "m-derived" scheme, must be looked 

 to for the solution of impedance problems for which standard sections 

 are inadequate. 



Case II — n = 2. 



A direct computation of the network shown in Fig. S-h gives 



^_ 1 Vl 



Li — 



Zo 1 + (02^ - 2aia2).x-2 + a^\a,^ - l).v^' 



