Electrical circuits wyklad 9

Dr inż. Agnieszka Wardzińska

Dr inż. Agnieszka Wardzińska

Room: 105 Polanka

Room: 105 Polanka

agnieszka.wardzinska@put.poznan.p

agnieszka.wardzinska@put.poznan.p

l

l

cygnus.et.put.poznan.pl/~award

cygnus.et.put.poznan.pl/~award

Transient analysis

Transient analysis

 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.

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

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

Partial fraction decomposition

Partial fraction decomposition

Notice that the first and third cases are really special cases of the second and fourth cases respectively.

Laplace Transform Table

f(t) F(s) ) ( 1 ) ( ) ( ) ( ) 0 ( . . . ) 0 ( ' ) 0 ( ) ( ) ( ) ( ) ( ) ( [ 0 ), ( ) ( ) ( 0 ), ( ) ( 0 1 0 2 1 0 0 0 0 0 s F s d f ds s dF t tf f f s f s f s s F s dt t f d a s F t f e t t f L e t t t u t f s F e t t t u t t f t n n n n n n at s o t s o t                     

Common Transform

Properties

Restrictions

Restrictions

There are two governing factors that

There are two governing factors that

determine whether Laplace transforms can

determine whether Laplace transforms can

be used:

be used:



f(t) must be at least piecewise continuous for

f(t) must be at least piecewise continuous for

t ≥ 0

t ≥ 0

 If f(t) were very nasty, the integral would not be If f(t) were very nasty, the integral would not be

computable.

computable.



|f(t)| ≤ Me

|f(t)| ≤ Me

where M and

where M and

γ

γ

are constants

are constants

 If f(t) is not bounded by MeIf f(t) is not bounded by Meγγtt then the integral will not then the integral will not

converge.

Example – calculatin Laplace

Example – calculatin Laplace

Transform from definition

Transform from definition

Laplace Transform for ODEs

Laplace Transform for ODEs

•Equation with initial conditions •Laplace transform

•Apply derivative formula •Rearrange

Examples

Examples

See:

See:

http://tutorial.math.lamar.edu/Classes/D

http://tutorial.math.lamar.edu/Classes/D

E/InverseTransforms.aspx#Laplace_InvTran

E/InverseTransforms.aspx#Laplace_InvTran

s_Ex1a

s_Ex1a

for some more examples