# Laplace Transform

By on

Laplace transforms and their inverse are a mathematical technique which allows us to solve differential equations, by primarily using algebraic methods. This simplification in the solving of equations, coupled with the ability to directly implement electrical components in their transformed form, makes the use of Laplace transforms widespread in both electrical engineering and control systems engineering.

This note is a recap/review of Laplace theory and reference which can be used while carrying out day to day work.  It is not an introduction or tutorial and does assume some prior knowledge of the subject.

## Theory

The Laplace transform, named after Pierre-Simon Laplace who introduced the idea is defined as:

$F(s)=L{ f(t) }= ∫ 0 ∞ e −st f(t)dt$

where: S = σ + iω,  (σ and ω as real numbers)

the result is a function of s, and the transform can be written as:

$F(s)=L{ f(t) }$

Give the transform, the inverse can be found from:

$f(t)= L −1 { F(s) }= 1 2πi lim T→0 ∫ γ−iT γ+iT e st F(s) ds$

While we can use the above equations to find the Laplace transform (or it's inverse) for a given function, in practice the use of table is more common.  The following table lists the more common transforms and their inverse:

Time Function
f(t)
Laplace Transform
F(s)
, unit impulse 1
, unit step

where

### Useful Operations

Linearity
Multiplication (by constant)
Real Shift Theorem
Complex Shift Theorem
Scaling
Derivative (1st)
Derivative (nth)
Integral
Convolution
Heaviside (step) Function

### Partial Fractions

In finding the inverse transforms, it is helpful if the equation to be solved is expressed as a sum of fractions.  Partial fractions are a way to achieve this and the process is called resolving partial fractions.

To be able resolve partial fractions, the numerator needs to be of lesser degree than the denominator and depending on the type of partial fraction:

Linear factors in the denominator:

Repeated linear factors in the denominator:

Example: the following example illustrates how to use the method of partial fractions to resolve a simple equation using linear factors. Ffind the partial fractions of:

which can be resolved into

From the above, it can be seen that:

By choosing values of s to make A or B equal to zero (s = -2, s = –1), the above can be solved to give A (=4) and B (=-3), giving us the final solution:

## Application

One of the great things about Laplace Transforms is that the core electrical quantities (resistance, inductance and capacitance) can be easily represented in their Laplace form; simplifying the solving of circuits.  The table below summaries the time and Laplace representation of each quantity:

Time domain Laplace domain
v(t) i(t) F(s) V(S) I(S)

Resistor

R

Inductor

L
$L= di(t) dt$

Capacitor

C

Example:  consider a series circuit consisting of a resistor and capacitor as shown. A step voltage v(t) equal to V0 is applied at time zero.  What is the form of the voltage across the capacitor, vc(t).

To represent the input voltage, we can use the Heaviside (step):

From the above table, the Laplace transform of the current, i through the circuit can be written directly as:

and the Laplace voltage across capacitor as:

Rearranging the equation and taking the inverse Laplace transforms, gives:

More interesting Notes:

Steven has over twenty five years experience working on some of the largest construction projects. He has a deep technical understanding of electrical engineering and is keen to share this knowledge. About the author

myElectrical Engineering

Getting Started with Patents

If you have a great idea or invent something the last thing you want is someone to steal the idea. One of the things you can do is protect the intellectual...

Windows Live Writer and myElectrical

When making adding a Note to our site we have a great online WYSIWYG editor and things are pretty simple.  However, if you prefer you can write, manage...

110 or 230 Volts

I've been considering a blog on the 110 or 230 Volt issue for a while.  While browsing the Internet I came across a great summary by Borat over at  engineering...

Standard Cable & Wire Sizes

IEC 60228 is the International Electrotechnical Commission's international standard on conductors of insulated cables. Among other things, it defines a...

9 power supply issues solved by using a UPS

Installation of a UPS can help in reducing problems due to issues with the power supply.  A lot of people relate this to nine key issues.  Depending on...

Voltage Levels to IEC 60038

The standard aims to consolidate AC and traction voltages within the industry and defines the following bands: band 1 - A.C. systems 100 V to 1...

Photovoltaic (PV) Panel - Performance Modelling

In an earlier note on the site [Photovoltaic (PV) - Electrical Calculations], the theory of solar (PV) cell calculations was introduced.  In particular...

Michael Faraday (the father of electrical engineering)

Famed English chemist and physicist Michael Faraday was born on September 22, 1791, in Newington Butts, a suburb of Surrey just south of the London Bridge...

How to Size Current Transformers

The correct sizing of current transformers is required to ensure satisfactory operation of measuring instruments and protection relays. Several methods...

Capacitors - Energy Storage Application

Capacitors have numerous applications in electrical and electronic applications.  This note examines the use of capacitors to store electrical energy....

## Have some knowledge to share

If you have some expert knowledge or experience, why not consider sharing this with our community.

By writing an electrical note, you will be educating our users and at the same time promoting your expertise within the engineering community.

To get started and understand our policy, you can read our How to Write an Electrical Note