ABSTRACT
The Laplace transform (LT) provides an extremely powerful tool for analyzing linear systems. It will be recalled that the power and usefulness of the phasor approach is in transforming ordinary, linear differential equations to algebraic equations for analyzing the sinusoidal steady state. The LT extends this approach by transforming ordinary, linear differential equations to algebraic equations for deriving the complete response, that is, steady-state plus transient, to any arbitrary excitation that has an LT. The LT can be derived from the Fourier transform, which is derived in turn as a limiting case of the Fourier series expansion as the period becomes infinitely large. The characteristic equation of a linear differential equation is a polynomial in complex frequency obtained by taking the LT of the equation with zero forcing function, zero initial conditions, and assuming a nonzero value of the variable of the equation. The poles of an overdamped second-order circuit lie on the negative real axis of the s-plane.
