CXVII. Some general theorems for non-linear systems possessing reactance

Abstract

Summary The theorems and concepts presented here are generally applicable to circuits of electrical or mechanical elements with non-linear characteristics. The quantity called the co-energy (the dual of energy) is shown to possess stationary properties (maximum or minimum) and superposition properties ; this is sufficient to establish the concept of an “ equivalent element ” for any 2-terminal system of like elements (all-inductor, all-capacitor, all-springs, etc.). The unfamiliar “ rectangle representation ” of a circuit of linear resistors is explained and extended to the non-linear case, including reactive elements. It is shown that the equations of motion of a non-linear system, possessing reactance, may be expressed in Lagrangian form, thus emphasizing the importance of co-energy and also showing that the Principle of Duality is applicable. Finally, systems are considered possessing mutual inductance and moving magnetic circuits (as in rotating machines). This paper is intended as a companion paper to the accompanying one by W. Millar.