Unit 4: Laplace Transforms and their Applications#

Motivating example#

In Exercise 4.1: RC Circuit we presented the RC Circuit shown in Fig. 42

An RC Circuit

Fig. 42 An RC Circuit#

We showed that voltage across the capacitor \(v_c(t)\) is determined by the first-order differential equation

\[\frac{1}{RC}v_s(t) = \frac{d}{dt}v_c(t) + \frac{1}{RC}v_c(t)\]

Assuming that the input voltage is applied by operating a switch, that is \(v_s(t)\) is the step function \(V_s u_0 (t)\) what would the output \(v_c(t)\) look like?

We will sketch the solution in our OneNote notebook

About the Laplace Transformation#

The Laplace Transformation (named after Pierre-Simon Laplace) is a useful mathematical tool that is used in many branches of engineering including signals and systems theory, control theory, communications, mechanical engineering, etc.

Its principle benefits are:

  • it enables us to represent differential equations that model the behaviour of systems in the time domain as polynomials in \(s\) which facilitates their solution as we will now demonstrate.

  • it converts time convolution (which is how we determine the time-response of a system to a given signal) into a simple multiplication in the \(s\) domain.

  • it allows us to model linear time-invariant (LTI) system components using transfer functions and systems by block diagrams.

  • block diagram analysis allows us to readily compute system responses to complex signals and facilitates system simulation and system analysis.

The only downside is that time \(t\) is a real value whereas the Laplace transformation operator \(s\) is a complex exponential \(s = \sigma + j\omega\).

Table of Contents#

In this section of the course we will cover: