324 THE BELL SYSTEM TECHNICAL JOIIRNAL, APRIL 1951 



When the assigned gain or phase is represented in analytic form it is 

 sometimes possible to determine a distributed charge over a suitably chosen 

 contour which matches the desired characteristic exactly. Then the only 

 approximations involved in obtaining a finite network are those which 

 arise from replacing the continuous charge distribution by a set of lumped 

 charges. The errors are easy to calculate and can usually be adjusted to 

 meet the allowable network tolerance. 



It is important to stress that for physical networks the complex poten- 

 tial W must be generated by unit charges. Hence, if we have determined 

 a continuous charge distribution over a given contour in the complex 

 />-plane, we must choose our unit of charge to make the total charge on the 

 contour equal to an integral number of charge units. Then the contour can 

 be divided into segments each carrying a unit charge, and the lumped 

 charge distribution is obtained by locating one unit of charge at some 

 convenient point on each segment, usually at or near the center. The total 

 charge determines the number of lumped charges that may be used. This 

 limitation is not so restrictive as it might appear at first sight, since the 

 assigned transmission function frequently involves a constant parameter 

 in terms of which the unit of charge may be defined. It is also possible, 

 as we shall see later, to increase the total charge on the contour by special 

 devices, appropriate to different types of problem. 



We assume that the gain, a, corresponds to the real potential, F, and 

 the phase, jS, to the stream function '^; but it would be equally permissible 

 to interpret a as the strea n function of another complex potential, iW, 

 and then /8 would be the negative of the potential. It is usually more con- 

 venient to equate gain and potential, in network synthesis problems, and 

 we shall confine our analysis to this interpretation. 



The desired for n of gain and phase may be given as a condition on their 

 variation with frequency. Since the electric intensity is the gradient of the 

 potential, we see from equations (17) that da/doo is analogous to the elec- 

 tric intensity in the direction of the negative frequency axis. Similarly, the 

 variation of ^ with frequency is analogous to the electric intensity in the 

 direction of the negative real p-Sixis, that is, at right angles to the frequency 

 axis. Thus we may summarize the analogies we shall use most frequently: 



a) Transmission function and complex potential 



b) Gain and potential 



c) Phase and stream function 



d) — -;- and field along real frequency axis 



t) — -f- and field across real frequency axis. 

 do) 



