1. Electrical Circuits Dr inż. Agnieszka Wardzińska Room: 105 Polanka
cygnus.et.put.poznan.pl/~award
Advisor hours:
Monday:
Wednesday:

2. Transient analysis The transients in electrical circuits occur when:
switching on/of power
changing the values of elements in the circuit
The first order circuits are the circuits where only one of the reactance element is unbalanced (only capacitance or inductance). When there are two elements unbalanced we talk about second order circuits. There can be more than two reactance elements in the circuits. For the lecture we will discuss only one method of analysing the transient circuit using Laplace transformation.

3. Laplace theorem
The Laplace transform is an integral transform, a linear operator that transforms time function (t > 0) f(t) to a function F(s) with complex argument s, given by:
The most common transforms for common function we can find derived and presented in tables in circuit theory books. In circuit theory we often need to calculate the inverse Laplace Transform .

4. Inverse Laplace Transform
In practice we can calculate the function f(t) with ressidue method:
where sk mean all poles of F(s)
we can also use Laplace Transform Table to find the function f(t)
Often requires partial fractions or other manipulation to find a form that is easy to apply the inverse

5. Partial fraction decomposition
Notice that the first and third cases are really special cases of the second and fourth cases respectively.

6. Laplace Transform Table

7. Common Transform Properties
f(t)
F(s)

8. Restrictions
There are two governing factors that determine whether Laplace transforms can be used:
f(t) must be at least piecewise continuous for t ≥ 0
If f(t) were very nasty, the integral would not be computable.
|f(t)| ≤ Meγt where M and γ are constants
If f(t) is not bounded by Meγt then the integral will not converge.

9. Example – calculatin Laplace Transform from definition

10. Laplace Transform for ODEs
Equation with initial conditions
Laplace transform
Apply derivative formula
Rearrange
Take the inverse

11. Examples
See:
for some more examples