Electrical Circuits as Dynamical Systems

Abstract

An electrical circuit containing at least one dynamic circuit element (inductor or capacitor) is an example of a dynamic system. The behavior of inductors and capacitors is described using differential equations in terms of voltages and currents. The resulting set of differential equations can be rewritten as state equations in normal form. The eigenvalues of the state matrix can be used to verify the stability of the circuit. The most fitted numerical methods to integrate electrical circuit differential equations are the Euler Method (Forward and Backward), the Trapezoidal Rule, and the Gear Method of second to sixth degree, for circuits having stiff equations. These methods are implemented, with adjustable time-step integration, in the majority of circuit simulation software, such as SPICE. The analytical solution can also be computed, for small-size circuits, applying the Laplace Transform. It is interesting to compare the graphical presentation of numerically and analytically obtained solutions. While the numerical methods can be used for both linear and nonlinear circuits, the Laplace Transform is mostly used for linear circuits. A method of using it for nonlinear circuits is also presented.