316 THE BELL SYSTEM TECHNICAL JOURNAL, APRIL 1951 



which satisfy Laplace's equation in two dimensions. Hence they may be 

 interpreted as potential and stream functions (interchangeably) of certain 

 charge distributions. In potential theory the problem of network analysis 

 corresponds to the problem of determining the potential of a given charge 

 distribution, while the problem of network synthesis corresponds to the 

 problem of determining an appropriate charge distribution when the poten- 

 tial is given. 



This is one of the fundamental problems of potential theory, and it has 

 been widely discussed in the mathematical hterature of the subject. The 

 usefulness of the potential analogue method of network synthesis derives 

 primarily from the fact that we may use the whole background of our knowl- 

 edge of potential theory and of the properties of electrostatic fields in for- 

 mulating the solution of the charge distribution problem. A general solu- 

 tion is obtained in terms of a continuous distribution of charge over a 

 contour (C) in the complex plane. This is the mathematical part of the 

 problem. Thereafter, the design problem is to approximate the continuous 

 distribution by means of a set of lumped charges which will have approxi- 

 mately the same potential function. The solution of this problem involves 

 a certain amount of ingenuity, and may at times seem to be more of an art 

 than a science. Once the lumped charge distribution has been determined, 

 the locations of the charges are interpreted as corresponding locations of 

 poles and zeros of the transmission function. Well-known methods of de- 

 signing a network with assigned poles and zeros may then be used, and 

 the problem regarded as solved. 



We may note that neither the lumped charge distribution nor the con- 

 tour (C) is tmiquely determined by a given transmission function. Physical 

 restrictions on the type of distribution which will lead to a realizable net- 

 work usually impose sufficient limitations on the charge distribution, but 

 the contour (C) remains to some extent at our disposal. If our first choice 

 of contour proves unsatisfactory we can always try another contour which 

 may give more suitable results. This introduces another important char- 

 acteristic of the potential analogue method, namely that we may use the 

 properties of conformal transformations to simplify the choice of contour. 

 Thus any simple closed contour in the complex />-plane may be mapped 

 on a unit circle in a second complex plane. The solution of the charge dis- 

 tribution problem on the unit circle is particularly simple, but it may not 

 lead to the most suitable network design formula. However, we may use 

 the inverse transformation to map the unit circle on some more convenient 

 contour and locate equivalent charges at corresponding points of the two 

 contours. 



From the mathematical standpoint the use of continuous charge dis- 

 tribution instead of lumped charges corresponds to the use of integrals 



