POTENTIAL ANALOGUE METHOD OF NETWORK SYNTHESIS 323 



If C is a closed contour, and if the above arc corresponds to passage from 

 00 to 2 in a counterclockwise direction around C, then ^i and ^2 in (24) cor- 

 respond respectively to the interior and exterior of C. 



On the other hand the potential is continuous across the line charge. 

 To prove this we note that the potential is the real part of the complex 

 potential W in (13), and is therefore given by 



V= - f e(rt log I z - f I I ^rl + constant. (25) 



The integral depends on the distance | z — f | between a t)^ical point f 

 on (C) and the given point z. For two points Zi and Z2 just on opposite sides 

 of (C) the distance is the same, so that F(z2) = F(zi). 



4. Analogy Between Transmission Functions and 

 Logarithmic Potentials 



Comparing equations (3) and (12) we see that the transmission function 

 F{p) in the complex />-plane may be identified with the complex potential 

 W oi a, system of discrete charges. If we assume that unit positive charges 

 are located at the natural modes, pn , of the network, and unit negative 

 charges at the infinite loss points, pm , the complex potential in the ^-plane is 



W = - 12 log ip - p"n) + Z log {p - pL) + constant. (26) 



The real part of this function is the potential and its imaginary part is the 

 stream function. Then, by the definition of gain and phase in equation (2), 

 the gain of the associated network is given by the potential on the imaginary 

 axis (the real frequency axis), and the phase by the corresponding stream 

 function. 



The zeros and poles of F(p) locate the charges producing the complex 

 potential W, and they form a discrete set of points. When F(p) corresponds 

 to practical problems these points are usually arranged along well-defined 

 lines in the complex />-plane and not distributed at random throughout a 

 whole area. The corresponding potential W should then be that of a discrete 

 set of charges arranged along corresponding fines in the charge plane. 

 When the potential function is given in analytic form, however, it is usually 

 simpler to use known methods of potential theory to determine a continu- 

 ous charge distribution over a convenient contour. This continuous dis- 

 tribution may then be approximated by a set of equal lumped units of 

 charge spaced on the same contour. The difference between the actual 

 'sources' of F(p) and W is usually small, and by using distributed charges 

 much of the algebraic complexity associated with the design of compHcated 

 networks may be avoided, at least in the earlier stages. 



