A METHOD OF IMPEDANCE CORRECTION 



809 



the configuration of the resistance or conductance controlling network 

 be so chosen that the impedances of its series branches and the admit- 

 tances of its shunt branches are proportional to x. In other words 

 the series and shunt branches of the correcting network should be 

 similar physically to those of the "constant-^" filter. The complete 

 network is then that shown in Fig. 4. It is similar to that of Fig. 2 



REACTANCE 



CORRECTING 



NETWORK 



CORRECTED FILTER 

 IMPEDANCE 



^i-aaZoX 



lanX 



La,ZoX 



lZ|k(=lZoX) 



tapX 



FILTER IMPEDANCE 

 TO BE CORRECTED 



^-(=.1^x) 



Fig. 4 — Generalized schematic of first or "direct" type of filter terminations. 



except that the explicit introduction of the factor Zq into the expres- 

 sions for the series and shunt branches reduces the a's to constants of 

 proportionality which can be fixed, once for all, for all "constant-y^" 

 filters. Following the analogy of ordinary filter structures it will be 

 assumed that the first branch of the network is in series when the filter 

 proper is mid-series terminated, and vice versa. It is then easily 

 shown that the preceding general formula for the resistance ^ of the 

 system reduces, both for mid-series and mid-shunt terminated filters, 

 to 



R = 



ZoVT 



x^ 



1 + Aix^ + A2X' + 



^„.x2"' 



when w is the number of branches in the network. It will be observed 

 that odd powers of x are missing. 



The possibilities of manipulating this expression to secure desirable 

 resistance characteristics are obviously determined by the number, n, 

 of variable terms in the denominator of the expression. Since n is, 

 however, also equal to the number of branches of the resistance or 

 conductance controlling network, and therefore determines both the 

 cost of this network and the extent to which the resistance or conduct- 

 ance can be made to approximate a given curve, it offers a convenient 

 basis for differentiating between the various structures. The sim- 

 plest cases, and the only ones of practical importance in contemporary 

 filter design, are those for which w = 1, 2, or 3. They are illustrated 

 in Fig. 5 and will be taken up in order. Our first step will be the es- 

 tablishment of the algebraic relations between the element values 

 Oi . . . a„ and the parameters yli . . . ^„for each of these three cases. 



* Assuming that the final branch, IfianX is in shunt as in Fig. 4. When the analysis 

 is stated in terms admittances the results are precisely similar, except for an obvious 

 change from Zo to 1/Zo. 



