Theorems Regarding the Driving- Point Impedance 

 of Two-Mesh Circuits* 



By RONALD M. FOSTER 



~^\^ul•^l>: The ncccssiir>- anil siifticient conditions that a drivinK-pciint 

 irii|>t'<lan(-e In.' rc-alizabli- by moans of a two-niesh circuit consistiuK "f rc- 

 sislanci-s, caivicitics. and inductances are stated in terms of the four roots 

 and four poles (incluilinj; the |m)Ics at zero aiul infinity i of the impedance. 

 The rixits ami the poles are the time coefficients for the free osiillations of 

 the circuit with the driving branch closed and opened, respectively. For 

 assigned values of the rcwis, the poles are restricted to a certain domain, 

 which is illustrated by ligiires for several typical cases; the case of real 

 jxjies which are not continuously transformable into complex poles is of 

 special interest. .Ml driving-point impedances satisfying the general 

 conditions can lie realized by any one of eleven networks, each consisting 

 of two resistances, two capacities, and two self-inductances with mutual 

 inductance between them; these are the only networks without superlluous 

 elements by which the entire range of possible impedances can be realized: 

 the three remaining networks of this type give special cases only. Kor 

 each of these eleven networks, fornuilas are given for the calculation of the 

 \alues of (he elements from (he assignefl values of the r<K)(s and poles. 



1. St.\ti:mi:nt or Ricsi i.ts 



TIIK object of this paper is, first, to determine tiie necessiir\- ■irul 

 sufficient conditions thai a driving-point impedance ' l)e realizal)le 

 !)>• means of a two-niesh circuit consisting t)f resistances, capacities, 

 .ind inductances, and second, to determine the networks- realizing 

 ,iny s[X*cified driving-point impedance staisfying these conditions. 



These necessary and sufficient conditions are stated in the form of 

 the following theorem: 



Theorem I. Any driving-point impedance S of a two-niesh circuit 

 consisting of resistances, capacities, and inductances is a function of 

 the time coefficient X = ip of the form 



c- „( X-ai)(X-a;)(X-a3)(X-«4) 

 ^-'^ X(X-0,)(X-^,) *'•" 



_fli)X^-f-aiX'+a3X-+a3X+a4 



• I'resenteil by title at the International Mathematical Congress at Toronto. 

 .\ugust I lit), l'J24, as "Two-mcsh Electric Circuits realizing any specific<l Driving- 

 point Impedance." 



' The driving-|M)int im|>cdancc of a circuit is the ratio of an impressed electro- 

 motive force at a point in a liranch of the circuit to the resulting current at the same 

 point. 



■ The networks considered in this paper consist of any arrangement of resistances, 

 capacities, and inductances with two accessible terminals such that, if the two 

 terminals are short-circuited, the resulting circuit has two independent meshes. 

 Thus the impedance measureiJ lietween the terminals of the network is the sanv; 

 as the driving-point impe<lance of the corresponding two-mesh circuit. Throughout 

 the fuifXT this distinction will lie made in the use of the terms "network" and 

 "ciniiil." 



6.SI 



