POTENTIAL ANALOGUE METHOD OF NETWORK SYNTHESIS 359 



In practice we soon find that we must use an unnecessarily large number N 

 of zeros and poles to get good accuracy from the simple trick just described. 

 A better plan is to keep just those zeros and poles which lie within some more 

 moderate distance from the origin, say R = 2. Then the remaining gain and 

 phase f(p) must be approximated by other means. This offers no special 

 difficulty; the disagreeable p^^- type singularity at the origin has already been 

 produced, leaving f(p) a relatively slowly varying function over the band 

 I CO I < 1. One way of approximating /(^) by the log of a rational function 

 with the desired number of zeros and poles is first to find a polynomial 

 approximation to /^^^ and then pick the rational function which has the 

 same first few terms in its power series as the polynomial. In the design 

 carried out at BTL the polynomial approximation was performed by a 

 method using Tchebycheff polynomials. This method will be the subject of 

 a later paper. For purposes of illustration we may equally well imagine f(p) 

 to be produced by placing charge on an eUiptic contour surrounding the 

 interval | co | < 1. 



The following numerical example will give the reader some idea of how well 

 the method works in actual practice. The cable had a loss of 5.368 nepers 

 (46 db) at CO = 1 and it was required that the cable be equahzed to within 

 .005 db from co = .02 to co = 1 . Using zeros only on the negative real axis, 

 the granularity error would have been much too high. Sufficiently low 

 granularity error was obtained by putting poles at 



p = -.0068498 (2n - 1)^ 

 and zeros at 



p = -.0034948 (2n - 1)2. 



This choice of position of zeros and poles makes every seventh zero cancel 

 every fifth pole. In the final design only 6 of these zeros and 6 of the poles 

 were used. The remaining gain and phase were produced, to the desired 

 accuracy, by a pair of real poles at /> = —1.5 and four pairs of conjugate 

 complex poles lying close to an elliptic contour about the frequency band 

 of interest. 



Example 3. Delay Equalizer 



A problem of frequent occurrence is that of ''delay equalizing" a given 

 network with known singularities. From the potential analogue point of 

 view the problem is, given the location and sign of certain lumped charges, 

 to find a distribution Qi(s) of charge on a contour C which produces no other 

 effect on the real frequency axis in the range of interest but to cancel the 

 transverse component of the electric field of the given charges. The distribu- 



