Modelling and control by Anthony Rossiter

# LAPLACE TRANSFORMS

This is a section in the chapter on mathematical skills. Some is revision of pre-university mathematics and some is syllabus mostly covered in year 1 of engineering programmes. Use the left hand toolbar to access the other chapters and themes. It is implicit that for many engineering topics, students have core competence in some mathematical topics such as polynomials, roots, complex numbers, exponentials, logarithms and Laplace. This section contains the following topics. Under each topic there are hardcopy (pdf) notes, a video talk through of key derivations with example problems and also a tutorial sheet for users to test themselves.

## Laplace 1- introduction and basic functions

Gives the definition for a Laplace transform and from there derives the transforms for exponentials, steps and simple power functions.

• A summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.

#### Quick test question

Which is correct Laplace transform for exp(-0.2t)?
A. 1/(0.2s+1)
B. 0.2/(s+0.2).
C. 5/(5s+1)
D. None of the above.

## Laplace 2 - sinusoids and shift rule

Extends video 1 by introducing sinusoidal signals and the shift theorem, thus also considering sinusoids scaled by an exponential. Note obvious typo at 9.40 where write '4' instead of '42'.

• A summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.

#### Quick test question

Which is correct Laplace for sin(4t)?
A. 4/(s2+16).
B. s/(s2+16)
C. 4/(16s2+1).
D. None of the above.

## Laplace 3 - derivatives

Shows have the definition of Laplace leads to the transform for an nth order derivative and thus how one can apply Laplace to determine the solution for differential equations.

• A summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.

## Laplace 4 - summary and tutorial

Covers some topics that may be useful from time to time such as the Laplace of an integral, a delayed signal and impulse function. Also summarises content from 1st 4 videos and gives some self test tutorial questions.  