Differential Equations, Input Functions, Complex Exponentials, and Transfer Functions

Abstract

Differential system equations describe the dynamic relationship between an input driving the system, and one of the power variables within the energetic system. We simplify, or linearize, the individual energetic element equations, in order to derive a system equation which is an ordinary differential equation with constant coefficients, a form which we can solve for the output or response function. The method of undetermined coefficients superposes or sums the response of a system into the natural or homogeneous response of the system to a disturbance to its energetic equilibrium, and the steady-state or particular response to each input driving the system. Systems with two or more independent energy storage elements yield differential system equations which may describe oscillations or vibrations. Complex numbers, complex exponentials, and Euler’s equations simplify the solution and interpretation of the response of oscillatory systems. The Laplace transformation transforms differential equations into algebraic equations, which can be expressed as multiplicative dynamic operators called transfer functions. The chapter’s appendix introduces Mathcad and MATLAB to plot the solutions or response functions.