Extension of the dual-input describing-function technique to systems containing reactive non-linearity

Kochenburger´s describing function and also the dual-input describing function due to West, Douce and Livesley are well known. These techniques enable the graphical solution of nonlinear differential equations in which the non-linear coefficient is associated with the first-order differential term to be obtained. The paper describes an extension of these methods, allowing the non-linear coefficient to be associated with any term in the differential equation. It is shown that the typical non-linearity presented by an iron-cored inductor or by a ferro-electric capacitor may be resolved into a simple feedback system containing a single non-linear element which is independent of frequency, together with elements possessing linear transfer functions. The resulting equivalent block diagram is suitable for analysis by either of the describing-function techniques. As an example, the dual-input technique is used to predict accurately jump phenomena in an iron-cored ferroresonant circuit, and the validity of the equivalent block diagram suggested is also checked by simulation on an analogue computer.