XETWORK SYN'THESIS 



615 



The "frequency variable" p represents, of course, ioj. The zeros p^ of 

 the rational fraction are those values of p at which there is infinite loss. 

 The poles p'J are the so-called natural modes,' or values of p at which 

 response V can exist in the absence of driving voltage E. The scale 

 factor K determines the average level of transmission. The zeros, poles, 

 and scale factor together determine the gain and phase completely. 



For a physical stable network, the zeros and poles must meet certain 

 well known restrictions, which are commonly stated in terms of loca- 

 tions in the complex plane for frequency variable p. Within these re- 

 strictions, the zeros and poles can bo subject to arbitrary choice, say 

 for purposes of network synthesis. 



eK 



LINEAR 



LUMPED 



ELEMENTS 



a+ l/3=LOG V/e 

 Fig. 1 — A general 4-pole. 



The symmetries required by the physical restrictions permit a and /3 

 to be represented separately as follows:! 



(p'l' - P^)(P2'^ - p^) ••• 



2a = log K 

 «2/3 = log 



/ "2 2n/ "2 2\ 



{Pl - p ){p-2 - p ) 



(p[ - p) • • ■ (p'l + p) • ■ 



(2) 



ip'i + ?>)••• (pi — p) ■• • 



These expressions hold at all real frequencies, but only at real fre- 

 quencies. 



3. TCHEBYCHEFF POLYNOMIALS 



It is functions of these special types which we are to synthesize with 

 the help of Tchebycheff polynomials. More generall}^, TchebychefT 

 polynomials come in various forms, and may be analyzed in various 

 ways. For our special purposes, however, they take somewhat special 

 forms (a little different from textbook definitions); and they are best 

 analyzed in quite special ways.J It will be simplest to start with arbi- 

 trary definitions, to be justified later on by demonstrations of usefulness. 



t The phase eciuation omits a possiljle 180° phase reversal, which is trivial for 

 present purposes. 



+ For discussions of Tchebycheff polynomials from other viewpoints, see 

 Courant and Hilbert^, and also a paper by Lanczos" on trigonometric interpola- 

 tion. 



