Analysis of nonlinear coupled circuits - Part II

IN A paper presented in 1954,1 the analysis of nonlinear coupled circuits was made by the method of high order phase planes or phase space. The simultaneous differential equations are first combined into one higher order differential equation and then solved by a method outlined in the previous paper and illustrated by two examples. However, when there are nonlinearities existing in both loops of, say, a 2-loop coupled circuit, it would be difficult to apply the procedure given previously. An alternative is here presented by working with the set of differential equations as originally given and solving several phase plane equations simultaneously. The new method may be called the method of solving simultaneous phase plane equations, in contrast with the method of solving one higher order phase space equation. This method has the advantage that nonlinearities may exist in two or more original differential equations. In a 2-loop circuit for example, there may be a nonlinear resistor in the primary loop and a nonlinear inductor in the secondary loop. Using examples I and II in the previous paper, the relations between the primary and secondary currents and between their first derivatives are shown. Example III gives the results of phase plane trajectories for a circuit with nonlinearities in both the primary and the secondary loops. Example IV shows that for a coupled circuit with a nonlinear capacitor in the secondary loop (the nonlinear capacitance being a function of voltage) the primary and secondary currents can be correlated by one equation in the phase space and solved simultaneously.