# Inverse Laplace Transform of a Transfer Function Using Matlab

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In this topic, we will find out how to calculate inverse Laplace of a transfer function using Matlab.

Let’s find out Inverse Laplace of the following function

$X(s)\frac{10{{s}^{2}}+20s+40}{{{s}^{3}}+12{{s}^{2}}+47s+60}=\frac{Numerator}{Denumerator}$

Let’s write a little code in Matlab now:

%% % Calculate Inverse Laplace of a function using Matlab
%Let's write numerator and denumerator from the given transfer function
num = [10 20 40]; % Numerator Coefficients
den = [1 12 47 60]; %Denumerator Coefficients

% "residue" command is used to do Partial Fraction Operation &;
% returns "residue", and "Poles" and direct term of the partial fraction
% expansion
% Write "help residue" in Maltab GUI to get better insight
[Residue,Poles,Direct_Term] = residue(num,den)

Results:

Here, we get the following results:

Residue =

95.0000

-120.0000

35.0000

Poles =

-5.0000

-4.0000

-3.0000

Direct_Term =

[]

Using above mentioned results, let’s write the partial fraction expansion of the function X(s)

$X(s)=\frac{95}{s+5}-\frac{120}{s+4}+\frac{35}{s+3}$

From partial fraction expression, we can easily write the inverse Laplace transform as:

$x(t)=95{{e}^{-5t}}-120{{e}^{-4t}}+35{{e}^{-3t}}$

In order to convert partial fraction expansion back to the original function, we can use the following Matlab command:

$[b,a]\text{ }=\text{ }residue(r,p,k)$

Here, we have b (coefficients for numerator) and a (coefficients for denumerator) to write the function like X(s).

You May Also Read: Laplace Transform: Introduction and Example

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