4.1. The S-Domain

The Laplace transform takes a continuous time signal and transforms it to the \(s\)-domain.

The Laplace transform is a generalization of the CT Fourier Transform. Let \(X(s)\) be the Laplace transform of \(x(t)\), then the Fourier transform of \(x\) is found as \(X(j\omega)\).

For most engineers (and many fysicists) the Laplace transform is just a mathematical trick to easily solve a class of partial differential equations. The Laplace transforms a constant coefficient linear differential equation relating the input \(x(t)\) with the output \(y(t)\) into an algebraic polynomial in \(s\). We will look at this class of constant coefficient linear partial differential equations (PDE’s) in the context of analog filter design and in the context of control theory.

The dynamical behaviour of many (physical) systems is described with a PDE in this class (like the inverted pendulum). This is not a course where you learn to model systems with PDE’s. We just give some examples. Either as examples of constructing an analog filter or constructing a control system.

• 4.1.1. The Laplace Transform
  • 4.1.1.1. Definition
  • 4.1.1.2. Eigenfunctions of an LTI system
  • 4.1.1.3. The Laplace Transform and the Fourier Transform
• 4.1.2. Properties of the Unilateral Laplace transform
• 4.1.3. Pairs of the (Unilateral) Laplace Transform
• 4.1.4. Differential Equations and the Laplace Transform
  • 4.1.4.1. Poles and Zeros in the S-plane
  • 4.1.4.2. An Electronic Example
  • 4.1.4.3. A Mechanical Example
• 4.1.5. Exercises