POTENTIAL ANALOGUE METHOD OF NETWORK SYNTHESIS 341 



This is the value of the transformed function as we approach the circle from 

 points just outside C, corresponding to the value Vi + i'^i for points just 

 inside C. Thus the potential is continuous across C and consequently we have 

 proved that for all points outside C, the function 



FeiP) = Fi(0,l/P) = Ve(P) + i^ifeip) (60) 



is the exterior function for the circle. 



We have just seen that on the circle ^e = —^i^ hence equation (24) 

 for the integrated charge reduces to 



qW = -[^i(t?) - ^iiM + Qo , (61) 



TT 



where Qo is a constant charge density, and the charge is measured from t?o . 



12. CONTORMAL TRANSFORMATIONS 



From the network point of view, unfortunately, the simple solution for 

 a circular contour does not always lead to the best solution of the design 

 problem. Hence we must also consider more general contours. The potential 

 analogue method requires an ab initio choice of contour on which the zeros 

 and poles of the approximating transmission function are to be located. 

 Small changes in the contour shape should not be of great importance, but 

 it may happen that our initial choice leads to a very complicated network 

 when a much simpler one would satisfy the physical requirements. Experi- 

 ence is required to make the most effective use of the method, and various 

 simplifications may frequently be available. For instance, it may be possible 

 to split the assigned gain or phase functions into components, for some of 

 which the zeros and poles may be located by inspection. Then the potential 

 analogue is used to synthesize the remaining components. 



There is, however, one limitation on the choice of contour which is inherent 

 in the potential interpretation, namely, that the transmission function must 

 be finite and analytic inside the contour. This is because the value of the po- 

 tential on C defines its values at all points inside C only if these values, and 

 their derivatives, are finite throughout the interior. We can see this intui- 

 tively when we remember that for a given charge distribution the potential 

 and its derivatives are finite at all points not occupied by the charges. 



It happens that the type of contour most frequently used up to the present 

 has been the ellipse, and we shall discuss this contour in more detail later. 

 For the present we shall consider more generally any simple closed contour 

 in the complex p-p\a,ne, surrounding the frequency band of interest, 

 I CO I < coo . The contour must be symmetric in the real />-axis, as we have 

 seen, but we shall not impose any other restrictions except the fundamental 

 one that the given complex potential must be analytic inside C. 



