TABLE OF CONTENTS Frequency-domain analysis techniques, introduced in Chapter 11 to find the steady-state ac response and generalized in Chapter 14 to find, in addition to the steady-state, the transient and, hence, the complete response, allow us to avoid differential equations in favor of algebraic equations. However, the techniques of Chapter 14 succeed in predicting only the functional form of the response. The unknown coefficients appearing in the transient component must subsequently be found on the basis of the initial conditions in the circuit. This can be a tedious and time-consuming task.
In the present chapter we present a powerful analytical tool known as the Laplace transform. This not only retains the advantage of transforming differential equations into algebraic equations, it also takes the initial circuit conditions into account automatically. Since these conditions are an inherent part of the transform process, the Laplace method provides the expression for the complete response explicitly.
After introducing the Laplace transform, we investigate functional and operational transforms as well as the inverse Laplace transform. We then demonstrate the application of Laplace methods to the solution of differential circuit equations, both for the case in which the initial conditions are specified in terms of the response itself and the case in which they are specified in terms of the initial stored energies in the reactive elements of the circuit. Next, we illustrate the use of convolution to predict circuit responses in those situations...
