898 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1953 



(1) 



(2) 



where: 

 Zs is the impedance looking into the network with a thermistor value 



R 



Zu is the impedance looking into the network with a thermistor value 

 kR 



Zl is the impedance looking into the network with a thermistor value 

 R/k 

 For this design R = 500 ohms and fc = 3 



We may solve equations (1) and (2) and obtain expressions for Zi 

 and Z2. However, for synthesis Zi and Z2 should be expressed in terms 

 of singularities in the p-plane. This may be accomplished by approxi- 

 mating the wanted gain change with a Tchebycheff series as developed 

 by S. Darlington. The steps in the process are: 



1. The coefficients of a Tchebycheff series that matches the desired 

 gain change over the frequency band of interest are found: 



a = Co + C2 cos 2<t> + Ci cos 40 • • • C2n cos 2n(f> (3) 



where the relation between w and <f> is given by the band-pass trans- 

 formation 



2 ^ fci + sm^ 

 kz — 8ur 4> 



K = C0clC0c2 



03ei - K ^ lower cutoff frequency (.2 mc) 



«c2 = K ~ upper cutoff frequency (8.35 mc) 



K2 — 1 



and CjA = , — Xj «r cos 2k<^r , ' , - 



n \- \ fzi ^ r = l,2....n -f 1 



(Match points) 



