318 THE BELL SYSTEM TECHNICAL JOURNAL, APRIL 1951 



For a finite network with lumped elements the ratio V /E is a rational 

 fraction and the transmission function may be represented by an expres- 

 sion of the form 



^(^) = iogit ;;-fiy-fi;-- 



\p - piAp - p2)--' (3) 



= log i^ + Z log {p - pi) -H log {p - pi), 



where K is a constant which may usually be ignored in the analysis since 

 its value merely alters the U'^el of gain or phase and does not affect their 

 variation with frequency. A/'e have introduced the complex oscillation 

 constant 



/> = ^ + tw (4) 



instead of the real frequency variable, co, and equation (3) defines the trans- 

 mission function in the complex />-plane. If we separate the real and imagi- 

 nary parts of (3) we find analytic expressions for the gain and phase: 



« = ao + Z^ log I /> - :^1 I - X log I /> - /n I , 



i8 = /3o + Z PKP - K) - Z PKP - p"n)' 



The significance of the parameters pm. and p'n is easily understood if we 

 note that when p — pm^^ have a: = — co and therefore V/E = 0. Hence 

 the zeros of the rational fraction in (3) represent points of infinite loss of 

 the network. Similarly if ^ = pn then a = oo and we may have a finite 

 value of V when E is zero. Thus the poles of the rational fraction are the 

 natural oscillation constants or natural modes of the network. For brevity 

 we shall refer to pm and ^n as the zeros and poles of F(p) though they are 

 really logarithmic singularities of the transmission function. 



The numerator and denominator of the rational fraction are finite poly- 

 nomials in p. If the network consists of real elements the coefficients in the 

 polynomials are real. Thus we have the first property of the transmission 

 function. The zeros and poles must be either real or conjugate complex. A 

 second essential property is that the real parts of the poles pn must be negative 

 if the network is to be stable. And the third property that concerns us is 

 that there must be at least as many poles as zeros, that is, as many finite natu- 

 ral modes as points of infinite loss. This condition insures the proper be- 

 havior of the transmission function at asymptotically high frequencies. 



Using these properties the gain and phase may be expressed in alterna- 

 tive forms. From the first property it follows immediately that the con- 

 jugate function [F(p)]* must be equal to the value of F when p = p*. 

 But p* — —p when p = ioi, hence in this case 



[F(p)]* = F(-p) = « - z/3. (6) 



