THE BELL SYSTEM TECHNICAL JOURNAL 



(copyright 1952, American telephone and teleqkaph company) 



VOLUIME XXXI MAI{( II 1952 NUMBER 2 



Introduction to Formal Realizability 

 Theory — I 



By BROCKWAY McMILLAN 



(Manuscript received October 15, 1951) 



This paper offers a general approach to (he realizability theory of net- 

 worlxs with many accessible terminals. The rnethods developed are applied 

 to give a complete characterization of all finite passive networks. 



I. SUMMARY 



1.0 A principal result of this paper is to characterize those matrices 

 Z(p), functions of the frequency parameter p, which can be realized as 

 open-circuit impedance matrices of finite passive networks. This char- 

 acterization is provided by the following theorem: 



1.1 Theorem:* Let Z{p) be an n X n matrix whose elements are Zrs(p), 

 1 < >', -'^ ^ n, where 



(i) Each Zrsip) is a rational function 



(ii) Zrsip) = Zrsip) (the bar denotes complex conjugate) 



(iii) Zrsip) = Zsrip) 



(i\') For each set of real constants ki , • ■ • , /.'„ , the function 



^zip) = 2 Zrsip) krks 

 r,s = l 



has a non-negative real part whenever Re(p) > 0. 

 Then there exists a finite passive network, a 2n-pole, which has the 

 impedance matrix Zip). 



* Presented to the American Mathematical Society, April 17, 1948. Abstract 

 260, Bulletin of the A .M.S. No. 54, July, 194S. 



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