808 BELL SYSTEM TECHNICAL JOURNAL 



later recommend that one in which the filter is taken to be the load 

 impedance for most designs. This approach will therefore be con- 

 sidered first and in greatest detail. We will, moreover, limit ourselves 

 to image impedances of the "constant-^" type. Practical filter 

 designs of course are usually composite structures containing several 

 types of sections. The image impedances at the junctions between 

 the sections are however, nearly always of the "constant-^" type and 

 our restriction to image impedances belonging to this class does not, 

 therefore, seriously limit the field of application for the network. 



Notation 



The image impedance of a mid-series terminated "constant-^" 

 filter is usually written as 



V 



1 + ^'' 



4Z2, 



that of a mid-shunt terminated filter as 



1+ ^'^ 



4Z: 



2*; 



where Zu and Zo^ represent in each case the series and shunt impedances 

 of the " constant-y^ " filter, and Zo( = VZu-Z2fc) is a constant which can 

 be chosen arbitrarily to fix the impedance level of the circuit. 



We will find it convenient to represent the way in which the various 

 branches vary with frequency by a new quantity x, defined by the 



relation 



Zifc .„ 

 — = tZnX. 



In a low pass filter, for example, x = f/fc, in a high-pass filter x = fjf, 

 and in a band-pass filter 



X 



Upon making use of the relation ZiuZ^k = Z^ the formulae for mid- 

 series and mid-shunt "constant-^ " image impedances can be written as 

 ZoVl - x2 and Zo/Vl - x^.^ 



This method of representing the image impedances suggests that 



* In terms of the usual filter notation this x = V — Uk- 



