The Number of Impedances of an n Terminal Network 



By JOHN RIORDAN 



This paper gives the enumeration of impedances measurable at 

 the n terminals of a linear passive network. The enumeration 

 supplies background for the study of network representations and 

 the numerical results which are given up to ten terminals are 

 perhaps surprising in the rapidity of the rise of the number of 

 impedances with the number of terminals; almost 126,000,000 

 impedances, e.g., are measurable for ten terminals. 



A LINEAR passive network having n accessible terminals may be 

 completely represented by an equivalent direct impedance net- 

 work,^ consisting of branches, devoid of mutual impedance, connecting 

 the terminals in pairs. The number of elements (branches) in this 

 representation is equal to the number of combinations of n things 

 taken two at a time, i.e., |«(w — 1). Each of the elements is defined 

 by an impedance measured by energizing between one of the terminals 

 it connects and the remaining terminals connected together and taking 

 the ratio of the driving voltage to the current into the other terminal 

 it connects. The network then is represented by a particular set, 

 of ^n{n — 1) members, of impedances measurable at its terminals; 

 as will appear later, the set is of short-circuit transfer impedances. 



The direct impedance network is one among many network repre- 

 sentations; it is taken as illustrative of two aspects, (i) the necessity 

 of a certain number of elements ^n{n — 1) and (ii) the expression of 

 these elements in terms of measurable impedances. It is well known 

 that any linearly independent set, of |w(» — 1) members, of the 

 measurable impedances of an w-terminal network will serve as a 

 network representation; hence the enumeration of representations may 

 be taken in two steps, the first of which, the enumeration of measurable 

 impedances, is dealt with in the present paper. 



The number of measurable impedances for two to ten terminal 



linear passive networks is given in Table I, which lists the driving-point 



impedances, Dn, transfer impedances (open or short circuit), r„, 



certain additional transfer impedances to be described later, Un, and 



the total Nn- As mentioned below, this total counts once only 



1 Item (b) in the list of equivalent networks given by G. A. Campbell "Cisoidal 

 Oscillations," Trans. A.I.E.E. 30, pp. 873-909 (1911), p. 889; or p. 81, "Collected 

 Papers of George Ashley Campbell," Amer. Tel. & Tel. Co., New York, 1937. 



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