A Reactance Theorem

T h e theorem gives t h e most general form of t h e driving-point impedance of a n y n e t w o r k composed of a finite n u m b e r of self-inductances, m u t u a l inductances, a n d capacities. T h i s impedance is a pure reactance with a n u m b e r of resonant a n d a n t i - r e s o n a n t frequencies which a l t e r n a t e with each other. A n y such impedance m a y be physically realized (provided resistances can be made negligibly small) by a network consisting of a n u m b e r of simple resonant circuits (inductance a n d capacity in series) in parallel or a n u m b e r of simple anti-resonant circuits (inductance a n d capacity in parallel) in series. F o r m u l a s are given for t h e design of such networks. T h e variation of t h e r e a c t a n c e with f r e q u e n c y for several simple circuits is shown by curves. T h e proof of t h e t h e o r e m is based upon t h e solution of t h e analogous dynamical problem of t h e small oscillations of a system a b o u t a position of equilibrium with no frictional forces acting.