POTENTIAL ANALOGUE METHOD OF NETWORK SYNTHESIS 



337 



For the unit circle the exterior potential and charge equations take very 

 simple forms. Corresponding to the interior potential 



Fiip) = ao + Z OmP-, (49) 



we find 



F.{p) = -Q log (/./U) + E a„/.-", 



Q^ 



gW = ^ + - 2^ dm sin »«? , 



(50) 



where Q is the total charge on the unit circle, Q = ao/log | coo | . The coeffi- 

 cients in all three series are identical. 



. x--^^ 2a: = -LOG[i + (a?/a;o)^"] 



Fig. 11 — Unit charges arranged symmetrically on a circle for the maximally flat filte 

 approximation. 



As a simple example let us determine the charge distribution on the unit 

 circle which corresponds to a constant gain for | co | < wo , and to a phase 

 shift independent of co. By equation (49) this requires that a^ = when 

 m 7^ Oj and hence the continuous charge distribution on the circle is simply 



,«, = g 



(51) 



where Q is the total charge on the circle. Equal increments in ?> give equal 

 increments in the accumulated charge round the circle. If we ignore the 

 requirements of reahzability this distributed charge may be approximated 

 by simply dividing the unit circle into 2m equal parts, and placing a unit 

 positive charge at each point of division, Fig. 11, 



p^ = g»*'/-, yfe = 0, 1, . . . 2m - 1. (52) 



