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

T h e driving-point impedance of an electrical network composed of a n y n u m b e r of resistances, arranged in a n y way, a n d t w o pure reactances, of a n y degree of complication w i t h i n themselves but not related to each other by m u t u a l reactance, inserted at a n y t w o points in the resistance network, is limited to an eccentric a n n u l a r 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. T h e diameter of the exterior b o u n d a r y extends from the value of the impedance when both reactances are short-circuited to its value when both are open-circuited. T h e diameter of the interior b o u n d a r y extends from the value of the impedance when one reactance is shortcircuited and the other open-circuited to its value when the first reactance is open-circuited and the second short-circuited. W h e n 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 m a y be shown over a n y frequency range or over a n y variation of elements of the reactances. This is most conveniently done on a doubly-sheeted surface. The paper is illustrated by numerical examples.