Network Synthesis Using Tchebycheff 

 Polynomial Seriest 



By SIDNEY DARLINGTON 



(Manuscript received April 17, 1952) 



A general method is developed for finding functions of frequency which 

 approximate assigned gain or phase characteristics, within the special class 

 of functions which can he realized exactly as the gain or phase of finite 

 networks of linear lumped, elements. The method is based, upon manipula- 

 tions of two Tchebycheff polynomial series, one of which represents the 

 assigned characteristic, and the other the approximating network function. 

 The ivide range of applicability is illustrated with a number of examples. 



1. IXTRODUCTION 



Network synthesis is the opposite of network analysis — namely, the 

 design of a network to have assigned characteristics, as opposed to the 

 evaluation of the characteristics of an assigned network. In general, 

 there are specifications on the internal constitution of the network, as 

 well as requirements relating to its external performance. A common 

 form of the general problem is the design of a finite network of linear 

 lumped elements, to produce an assigned gain or phase characteristic 

 over a prescribed interval of useful frequencies. The present paper re- 

 lates to this particular form. 



In general, the restrictions on the network are such that the assigned 

 performance cannot be matched exactly. This gives rise to an approxi- 

 mation or interpolation problem. For present purposes, the problem is: 

 to choose a function of frequency w^hich matches the assigned gain or 

 phase to a satisfactory accuracy, from that special class of functions 

 which can be realized exactly with physical finite networks of linear 

 lumped elements. The function of frequency may be defined in terms 

 of network singularities (natural modes and infinite loss points). The 



t Presented orally, in briefer form, at the 1951 Western Convention of the 

 Institute of Radio Engineers, and at the Sj-mposium on Modern Network Syn- 

 thesis sponsored by the Polytechnic Institute of Brooklyn and The Office of 

 Naval Research, New York City, April, 1952. 



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