Chapter 6 - Linear Systems Analysis in the Complex Frequency Domain: The Laplace Transform and the Analysis of Transients

Abstract

This chapter presents a discussion of linear systems in a complex frequency domain. The chapter focuses on Laplace transforms and the analysis of the transients. With the Laplace transform, all of the analysis tools can be applied to systems exposed to a broader class of signals. Transfer functions written in terms of the Laplace variables serve the same function as frequency domain transfer functions, but to a broader class of signals. The Laplace transform can be viewed as an extension of the Fourier transform where complex frequency s is used instead of imaginary frequency jω . Considering this, it is easy to convert from the Laplace domain to the frequency domain by substituting jω for s in the Laplace transfer functions. Bode plot techniques can be applied to these converted transforms to construct the magnitude and phase spectra. Thus the Laplace transform serves as a gateway into both the frequency domain and the time domain through the inverse Laplace transform. An approach of how to convert the Laplace transform to the frequency domain is discussed in the chapter. It also provides information about the Laplace transfer function, time-delay theorem, constant gain element, and second-order elements.