POTENTIAL ANALOGUE METHOD OF NETWORK SYNTHESIS 325 



The conditions imposed on the zeros and the poles of the transmission 

 function to make it physically realizable have their counterparts which 

 must be imposed on the charge distribution associated with the complex 

 potential if it is to be equivalent to a realizable network. Using the above 

 analogies they may be summarized as follows: 



1) The charge distribution must be symmetrical about the real 

 axis in the complex plane. 



2) The positive charges must be in the negative haK of the plane. 



3) The net charge must be non-negative. 



4) If the contour is made up of disjoint curves in the plane there 

 must be an integral number of units of charge on each segment. 



(28) 



The first three conditions correspond exactly to the zero and pole li nita- 

 tions, while the last is a corollary of the unit charge limitation we have 

 already discussed. 



5. Condenser Delay Networks 



As a simple example of the potential analogy we shall consider the design 

 of a network with constant phase delay in a prescribed frequency range. 

 Analytically the condition is that d^/doi should be constant for | co | < coo , 

 where c^o has an assigned value. The corresponding function in the potential 

 plane is the field transverse to the imaginary axis. This suggests the field 

 between the plates of a parallel plate condenser, and we construct im ne- 

 diately the analogy illustrated in Fig. 5. The distributed charge is shown 

 in Fig. 5a, where we assume a constant charge density on each plate of the 

 condenser, the plates being parallel to the real frequency axis. The positive 

 charge is placed on the left-hand plate to satisfy the second condition of 

 the set (28). 



As long as the distance between the plates is small compared with their 

 width the field between the plates is transverse, and substantially con- 

 stant, except for an edge effect which will diminish as the dimensions of 

 the plates are increased. If we could use infinite plates the field would be 

 exactly constant, and the continuous charge distribution on the plates 

 would match the network stipulation exactly. In practice we must use a 

 finite number of lumped charges; hence we choose the charge points shown 

 in Fig. 5b, where the crosses represent unit positive charges, the natural 

 modes of the network; and the circles represent unit negative charges, the 

 infinite loss points. To keep the end effects small it is desirable to extend 

 the plates considerably beyond the frequency coo . 



We note that for the lumped charge distribution the field along the real 

 frequency axis vanishes, since each unit positive charge contribution is 



