UNIVERSITY OF NOTTINGHAM
School of Electrical and Electronic Engineering
Control 1: H5BCT1
Inverse Laplace Transforms
Useful to be able to go from the s-domain to the time domain as well as the other way
– will allow time domain solutions to be found.
f (t ) =
∞
F (s ).e st .ds
−∞
Where t is treated as a constant for the purpose of integration.
However, since ‘s’ is a complex variable a good knowledge of complex variable
theory is required to solve the integral – engineers do not normally attempt this.
Inverse Laplace transforms are usually found by inspection from known transforms –
see the Laplace Transforms table with your examples sheets.
Eg. If:
F (s ) =
1
s+2
On tables look up: F (s ) =
f (t ) = e −αt
1
s +α
Therefore:
f (t ) = e −2t
To analyse a systems we can:
1. Find a system model – block diagram
2. Find an input model – Laplace transform of input signal
3. Find the output in terms of ‘s’
4. Use Partial Fractions to enable the use of the Laplace Transform tables
5. Use the Inverse Laplace transforms to find the output in the time domain
Example 1.
(if you can solve this you can solve anything you will encounter on this course)
If: F (s ) =
(
)
10 s 2 + s + 2
s (s + 1)(s + 2) s 2 + 2s + 5
(
)
find f(t)