Inverse Laplace Transform: Partial fraction decomposition
To find the inverse Laplace transform of a complicated function, we can convert the function to a sum of simpler terms for which we know the Laplace transform of each term. The result is called a partial-fraction expansion.
Given:
where the order of
If the order of
We want to expand
Case 1. Roots of the Denominator of F(s) Are Real and Distinct
Let’s suppose we have all distinct poles:
we want to find the coefficient
Multiplying for
we obtain:
and finally:
For example:
which means that:
and finally:
Case 2. Roots of the Denominator of F(s) Are Real and Repeated
If we have muliple poles the decomposition is similar.
Let’s consider, as an example
The roots of
We can write the partial-fraction expansion as a sum of terms, where each factor of the denominator forms the denominator of each term.
In addition, each multiple root generates additional terms consisting of denominator factors of reduced multiplicity.
In our case
We obtain
as before. In this caseWe obtain
multiplying the previous equation by :
When
- We obtain
differentiating the previous equation with respect to :
From which
In this case then:
and the inverse transform is:
If the denominator root is of higher multiplicity than 2, successive differentiation would isolate each residue in the expansion of the multiple root.
In general, given a
We can find the general expression for
Case 3. Roots of the Denominator of F(s) Are Complex or Imaginary
The technique used for the partial-fraction expansion of
However, the residues of the complex and imaginary roots are themselves complex conjugates.
In this case, the resulting terms can be identified as:
and
For example:
To find
which we can inverse transform to obtain:
where
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from http://www.dii.unimo.it/~zanasi/didattica/Fondamenti%20CA_Mec/Luc_CA_06_Fratti_semplici.pdf