POTENTIAL ANALOGUE METHOD OF NETWORK SYNTHESIS 361 



theorem to Fa{w) directly. We could modify the theorem without difficulty 

 but we may also solve the problem by using the well-known electrostatic 

 method of images, f The complex potential for the required shielding dis- 

 tribution is 



, 1 w — Wq 1 w — w 

 WW = - log j- - - log Y y 



Wo d w* 



and the shielding charge distribution is obtained by evaluating the imaginary 

 part of W\ If the contour C is symmetric with respect to the real frequency 

 axis, symmetry considerations in the w-plsine will show that w = —wq; 

 then the charge distribution may be written in the expUcit form 



^.W=-tan [ ^(^.+ ,)_2ig2 3i,, J> 



where Wq = —A-\-iB and R^ = A^ -\- B^. This (integrated) charge distri- 

 bution, when mapped back on the original ^-plane contour C, becomes the 

 shielding distribution qx{s) sought. If the singularity were a zero instead of a 

 pole, q\ would be given by the same expression with a factor — 1. 



The procedure for delay equalizing a group of singularities can be out- 

 Hned as follows: 



(1) Find a conformal mapping of the outside of C on the outside of the 

 unit circle. 



(2) Compute 



Qi=H qii 



i 



as a function of d. Here the sum runs over all the given singularities and 

 qu is the distribution which equalizes the i-th singularity (computed 

 from an expression like that for qi given above). 



(3) Since Qi puts some poles in the right half-plane, compute 



Q = Qi- Dsin d, 



choosing the constant D large enough to make all the poles of the distribution 

 Q he in the left half-plane. The only effect of the distribution D sin d is to 

 add flat delay. 



(4) Approximate Q by a, function with unit steps, say at ^i , ^2 , • • • , ^at . 



(5) Map the singularities found in (4) on the p-plane to obtain the equalizer 

 singularities. 



Figures 23, 24, and 25 illustrate a delay equalizer design taken from 

 actual practice. Figure 23a shows the p-plane locations of the singularities 



t L. Page, Introduction to Theoretical Physics, D. Van Nostrand an^ Co., New York, 

 1935, p. 404. 



