Graphic Representation of the Impedance of Networks Containing Resistances and Two Reactances

Abstract

The driving-point impedance of an electrical network composed of any number of resistances, arranged 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 resistance network, is limited to an eccentric annular region in the complex plane which is determined by the resistance network alone. The boundaries of this region are non-intersecting circles centered on the axis of reals. The diameter of the exterior boundary extends from the value of the impedance when both reactances are short-circuited to its value when both are open-circuited. The diameter of the interior boundary extends from the value of the impedance when one reactance is short-circuited and the other open-circuited to its value when the first reactance is open-circuited and the second short-circuited. When 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 boundaries. 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.