218 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 



Conversely, if a finite passive 2n-pole has an impedance matrix Z{p), 

 that matrix has the properties (i), (ii), (iii), (iv). 



A formally identical dual theorem holds for open-circuit admittance 

 matrices Y(p). 



1.2 A general realizability theorem, applicable to and characterizing 

 completely all finite passive networks, whether having impedance ma- 

 trices or not, is also proved. 



1.3 An effort is made to lay a foundation adequate for the realizability 

 theory of both active and passive multi-terminal devices. To this end, 

 a large part of the paper is devoted to the scrutiny of fundamental 

 properties of networks. 



II. INTRODUCTION AND FOREWORD 



2.0 Network theory provides direct means for associating with an 

 electrical network a mathematical description which characterizes the 

 behavior of that network. Typically, this results in shifting engineering 

 attention from a detailed, possibly quite intricate, electrical structure 

 to a mathematical entity which succinctly describes the relevant be- 

 havior of that structure. An essential feature of this shift in focus is 

 emphasized by the word "relevant": only those terminals of the net- 

 work which are directly relevant to the problem at hand are considered 

 in the mathematical description. Design work can then be done in 

 terms of constructs relating explicitly to these accessible terminals, the 

 effect of the internal structure being felt only by imphcation. 



The physical origins of these mathematical constructs, and the im- 

 plications of the internal structure upon them, cannot however be en- 

 tirely forgotten, for they have mathematical consequences which are 

 not always immediately evident. Until he knows these limitations — 

 imposed upon him by the physical nature or the necessary structural 

 form of the networks he is designing — a design engineer cannot make 

 free use of the mathematical tools that network theory has provided. 



We give the name "realizability theory" to that part of network 

 theory which aims at the isolation and understanding of those broad 

 limitations upon network performance, i.e., upon the mathematical 

 constructs which describe that performance — which are imposed by 

 limitations on the network structure. One may also include in the 

 province of realizability theory some of the converse questions: the 

 study of those structural features common to all networks whose per- 

 formance is limited in some specified way. 



Reahzability theory would have little content were it not that "per- 



