220 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 



ance. Except for the trivial limiting case of an open circuit, every two- 

 pole has a well-defined impedance, Z{p), a function of the complex 

 frequency parameter p, which describes its performance in a way which 

 is by now well understood. Dually, except for the Hmiting case of a 

 short circuit, every two-pole has a well-defined admittance function 

 F(p). Even the limiting cases are tractable: every open circuit has the 

 admittance function Y{p) = and every short circuit the impedance 

 function Z{p) = 0. 



In other words, by exercising his option to speak in terms either of 

 impedance or of admittance, one can always describe the performance 

 of a two-pole by using a single function of frequency. 



The descriptive simplicity and practical importance of two-poles led 

 early to a fairly complete realizability theory for them. In 1924 R. M. 

 Foster^ gave a function-theoretic characterization of the impedance 

 functions of finite passive two-poles containing only reactances. The 

 corresponding problem for two-poles which are not at all limited as to 

 structure, beyond being finite and passive, was solved by 0. Brune in 

 1931. The effects of various structural limitations have since been 

 studied by several writers (cf. Darlington, Bott and Duffin ). 



2.4 Technology, and the promptings of conscience, have meanwhile 

 urged the study of devices with more than two accessible terminals. 

 Here, however, Nature has been less kind, in that no uniquely simple 

 method is available for describing the performance of such devices as 

 viewed from their terminals. 



Indeed, basic network theory has been remiss here, in not even mak- 

 ing available a mode of description which is generally applicable — 

 whether simple or not. 



W. Cauer^ showed that, when one admits ideal transformers among 

 his network components, it is sufficient to study networks which are 

 natural and direct generalizations of two-poles, namely, 27i-poles,* for 

 arbitrary values of n. The corresponding natural generalization of the 

 impedance function Z(p) of a two-pole is the impedance matrix of a 

 2w-pole: just as one multiplies a scalar current by a scalar impedance to 

 get a scalar voltage, one multiplies a vector current by an impedance 

 matrix to get a vector voltage. 



2.41 Not all descriptive difficulties are resolved, however, by consider- 

 ing 2n-poles and their impedance or admittance matrices. For the 

 moment, a simple example will suffice to show this: the 2 X 2-pole which 

 consists simply of one pair of short-circuited terminals and one pair of 



* Defined in Cauer,* and also later here. 



