338 THE BELL SYSTEM TECHNICAL JOURNAL, APRIL 1951 



The total charge is Q = 2w, and the transmission function for the lumped 

 charge distribution is 



2m-l 



Fiip) = constant - 2 log(p - pk) . (53) 



Since 



{P - PoXp - Pl) '"{p- p2m-l) = f"" - 1, (54) 



this is equivalent to 



Fi{p) = constant - log {p^"" - 1), (55) 



at all points inside the circle \ p\ = 1. This is the transmission function 

 for the Butterworth "maximally-flat" filter.^ As m increases Fi is more 

 and more nearly constant inside C. But the objection to this solution is 

 that it involves poles (or positive charges) in the right half of the ^-plane. 

 If the phase is of no importance we may use the gain invariant trans- 

 formation to transfer these poles to the left half of the plane, which is 

 equivalent to using only the left half of the contour, and doubUng the 

 charge at each charge point. Then we have a physically realizable charge 

 distribution such that 



For integral values of Q we locate charge points at ^& = ^* * , where 



TT IT ^ ,» , TT 



''»=i-2^' ..,. = ^. + ^. (57) 



The shape of the gain characteristic for small values oi Q = 2m is illus- 

 trated in Fig. 12. It approximates zero gain at frequencies inside the circle, 

 and the approximation improves as m increases, or as the frequency de- 

 creases. At frequencies outside the circle the gain becomes a high loss, 

 and the filter is of the low-pass type. 



The transfer of poles from the right to the left half of the />-plane leaves 

 the gain unaltered, but it changes the phase delay, since the sign of the 

 phase contribution from each transferred charge is reversed. It is possible to 

 compensate for this change by adding a simple charge distribution such as 

 that shown in Fig. 13. Here the positive charges on the left are matched by 

 the negative charges on the right, so that the electrostatic field is zero along 

 the real frequency axis and the charges merely add a constant gain. The 

 contribution to the phase delay from each negative charge equals that from 

 the corresponding positive charge. Just as in the condenser plate analogue 



