Unit 4: Laplace Transforms and their Applications

Unit 4: Laplace Transforms and their Applications#

Motivating example#

In Exercise 4.1: RC Circuit we presented the RC Circuit shown in Fig. 41

We showed that voltage across the capacitor $v_{c} (t)$ is determined by the first-order differential equation

\frac{1}{R C} v_{s} (t) = \frac{d}{d t} v_{c} (t) + \frac{1}{R C} 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.

The principle benefits of the Laplace transformation:#

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.

Downside of the Laplace transformation#

The only downside is that time $t$ is a real value whereas the Laplace transformation operator $s$ is a complex exponential $s = σ + j ω$ .

Table of Contents#

In this section of the course we will cover: