# Laplace Transform

By on April 23rd, 2012

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

## Latest Questions:

### Most Popular Notes:

Solar Electricity Handbook - 2015 Edition: ...
Michael Boxwell
Paperback - 204 pages
$16.12 Renewable and Efficient Electric Power ... Gilbert M. Masters Hardcover - 712 pages$112.96
Electric Power Distribution Engineering, ...
Turan Gonen
Hardcover - 1061 pages
$137.08 Electric Energy: An Introduction, Third ... Mohamed A. El-Sharkawi Hardcover - 606 pages$104.36
Electric Power Distribution System ...
Turan Gonen
Hardcover - 856 pages
Large-Scale Solar Power System Design ...
Peter Gevorkian
Hardcover - 704 pages
$98.34 Energy Systems Engineering: Evaluation and ... Francis Vanek, Louis ... Hardcover - 672 pages$60.99
Power System Relaying
Stanley H. Horowitz, ...
Hardcover - 398 pages
$122.01 Photovoltaic Systems Engineering, Third ... Roger A. Messenger, ... Hardcover - 527 pages$107.02
Power System Monitoring and Control
Hassan Bevrani, ...
Hardcover - 288 pages
$91.13 Renewable and Efficient Electric Power ... Gilbert M. Masters Hardcover - 680 pages Power Electronics and Renewable Energy ... Hardcover - 1607 pages$259.75
Power Systems and Renewable Energy: Design, ...
Gary D. Price
Paperback - 192 pages
$59.95 Large-Scale Solar Power Systems: ... Dr Peter Gevorkian Paperback - 400 pages$44.99
Alternative Energy Systems and Applications
B. K. Hodge
Paperback - 432 pages
Submarine Power Cables: Design, ...
Thomas Worzyk
Hardcover - 296 pages
$151.20 Design of Smart Power Grid Renewable Energy ... Ali Keyhani Hardcover - 592 pages$120.55
Power Quality: Problems and Mitigation ...
Bhim Singh, Ambrish ...
Hardcover - 600 pages
$131.65 Electric Power Substations Engineering, ... Hardcover - 536 pages$125.96
Wind Turbine Operation in Electric Power ...
Zbigniew Lubosny
Paperback - 262 pages
$170.11 Control of Power Inverters in Renewable ... Qing-Chang Zhong, ... Hardcover - 438 pages$115.20
Solar Engineering of Thermal Processes
John A. Duffie, ...
Hardcover - 936 pages
$99.04 Renewable Fuel Standard: Potential Economic ... Committee on Economic ... Paperback - 250 pages$68.00
Power Electronics for Renewable Energy ...
Haitham Abu-Rub, ...
Hardcover - 826 pages
$163.62 How to Solar Power Your Home Everything You ... Martha Maeda Paperback - 336 pages$18.20
Wind Power in Power Systems
Hardcover - 1120 pages
$150.52 Power Generation Handbook 2/E Philip Kiameh Kindle Edition - 736 pages Small Geothermal Energy Systems and ... U.S. Government, ... Kindle Edition - 226 pages Variable Renewable Energy and the ... Jay Apt, Paulina ... Hardcover - 360 pages$134.23
Principles of Sustainable Energy Systems, ...
Frank Kreith, Susan ...
Hardcover - 790 pages
\$119.16

## 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