# 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.

The dynamical behaviour of many (physical) systems are described with a PDE in this class (like the inverted pendulum). This is no class where you learn to model systems with PDE’s. We just give some examples.

In a later chapter on Control Theory the Laplace transform and its use in solving these type of differential equations will be used quite a lot.