Graphic Representation of the Impedance of Net- 

 works Containing Resistances and Two Reactances 



By CHARLES W. CARTER. Jr. 



AiisTHACT: The driving-jtolnt linpcdanrc of an electrical network com-i 

 jiosttl of any nunifier of resistances, arrangcil in any way, and two pure 

 reactances, of any degree of complication within themselves but not related 

 to each other by mutual reactance, inserted at any two points in the resist- 

 ance network, is limited to an eccentric annul.ir region in the complex plane 

 which is <lctermincd by the resistance network alone. 



The Iwundaries of this region are non-intersecting circles centered on 

 the a.xis of reals. The tliameter of the exterior boundary extends from the 

 value of the impedance when both reactances are short-circuited to its 

 value when Ixtth are o|>cn-circuited. The diameter of the interior boundary 

 extends from the value of the impedance when one reactance is short- 

 circuited and the other o|x.-n-cirouited to its value when the first reactance 

 is open-circuitc<l and the second short-circuited. 



VV'hcn either reactance is fixed and the other varies over its complete 

 range, the locus of the driving-point impedance is a circle tangent to both 

 boun<iaries. By means of this grid of intersecting circles the locus of the 

 driving-point impedance may be shown over any frequency range or over 

 any variation of elements of the reactances. This is most conveniently 

 done on a doubly-sheeted surface. 



The paper is illustrated by numerical examples. 



iNTROnLCTION 



SIPPOSK that any numljer of resistances are combined into a 

 network of any sort and provided with three pairs of terminals, 

 nimibered (1) to (3) as in Fig. 1. The problem set in this paper is 

 to in\estigate the driving-point impedance' of such a network at 



Fig. 1 — The Network to be Discussed 



terminals (1) when variable pure reactances, Z; and Z,i, are connected 



to terminals (2) and (3), respectively. Zj and Z3 are forined of 



capacities, self and mutual inductances. They are not connected 



to each other by mutual reactance, but they may be of any degree 



of complication within themselves. 



The problem is dealt with in terms of the complex plane: that is, 



the resistance components of the impedance, 5, measured at terminals 



' The driving-point impedance of a network is the ratio of an impressed electro- 

 motive force at a point in a branch of the network to the resulting current at the 

 same point. 



387 



