The Potential Analogue Method of Network Synthesis 



By SIDNEY DARLINGTON 



(Manuscript Received Sept. 27, 1950) 



A general method is developed for designing networks with assigned gain 

 or phase characteristics. It is based on the analogy between the gain and phase 

 of linear networks and two-dimensional potential and stream functions, pro- 

 duced by charges corresponding to the network singularities. These analogies 

 exist because the gain and phase functions are the real and imaginary parts of 

 analytic functions of a complex frequency variable. Potential 3ieory is used 

 here to determine charge arrays which correspond to physical network singu- 

 larities and also yield approximations to assigned potential or stream functions. 



1. Introduction 



THE problem of network synthesis is the inverse of the much simpler 

 problem of network analysis. If an exponential input voltage, E exp (/>/), 

 is applied to a given network consisting of a finite number of lumped linear 

 elements, we can always calculate the corresponding output voltage, 

 V exp {pt), in terms of the network constants. Then we define a transmission 

 function F{p) as the logarithm of the ratio V/E. In general F{p) is an ana- 

 lytic function in the complex ^-plane. Its value on the real frequency axis, 

 p = io), defines the gain and the phase shift of the network. 



In the inverse problem we start with an assigned transmission function 

 F{p) and are required to find a network for which F{p) is the transmission 

 function. More frequently we have to design a network with assigned gain 

 or phase characteristics over a prescribed frequency range. Obviously, there 

 will be certain restrictions on the assigned transmission function if the 

 network is to be physically realizable. Further, the solution will not be 

 unique, though certain solutions may be more convenient than others. 

 Engineering and cost requirements usually impose severe limitations on 

 the number of elements that may be used in constructing a physical net- 

 work, hence it may not be possible to match the given function exactly 

 even within the prescribed range of frequencies. Thus from the practical 

 design point of view the problem of network synthesis may be formulated 

 as follows : To design a network with a reasonable number of lumped elements 

 such that its transmission function approximates a given transmission func- 

 tion to a prescribed tolerance in a given frequency range. 



The potential analogue method of network synthesis is a method of 

 approximating to the prescribed transmission function by considering charge 

 distributions in a complex plane and their associated potential and stream 

 functions. In other words, the fact that the prescribed function is usually 

 analytic means that its real and imaginary parts are potential functions 



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