POTENTIAL ANALOGUE METHOD OF NETWORK SYNTHESIS 355 



to the accuracy of the final result, since a general theory of granularity errors 

 has not been developed. Hence in this section we shall consider the apphca- 

 tion of the method to some actual engineering problems. This should aid the 

 reader in using the method himself, and should also help to convince him 

 of its validity. 



Example 1. The Gaussian Filter 



It is required to design a low-pass filter whose voltage transfer ratio is 

 exp { — b(jy) and which has constant phase delay in the prescribed frequency 

 range. For convenience we choose our unit of frequency to make the cut-off 

 frequency equal to unity, and then we choose our contour C to be an ellipse 

 in the /)-plane passing through the points p = d=|, d=?". 



The assigned transmission function in the />-plane is 



Fiip) = bp'' - I3P, 

 and the transformation which maps C on the unit circle in the w-plane is 



p = Tiw) = - - — . 

 In the w-plane the transmission function is 



and the part of Fi analytic inside Ci is 



r. f \ 9Z> 2 3|S 3 , 



FaKw) = — w "" ^^ ~ 8^* 



Hence, by the separation theorem, the required continuous charge distribu- 

 tion on Ci is 



,W = ^ sin 2.-1^ sin. + f!^, 

 16t 47r ZTT 



where we have assumed a total charge Q on the circle. 



In practice the values of b and Q are usually assigned, while the magnitude 

 of the phase delay is at our disposal. Hence we choose /3 large enough to insure 



TT 



that ^'(^) is a mono tonic decreasing function for < ?? < - . This makes it 



possible to divide the continuous charge into a set of unit steps, such that 

 these steps are negative in the right half plane, and therefore correspond to 

 zeros of the transmission function. A typical set of numerical values is 



b = l, e = 3, 5 = ^'. 



