# Matlab for Laplace Transform Inversion / Partial Fraction Expansion

## Contents

The code that generated this page is available at: http://lpsa.swarthmore.edu/LaplaceXform/InvLaplace/PFE1Matlab/PFE1.m

## Background: Matlab and polynomials

Define the polynomial

F=[1 3 2]

F = 1 3 2

Display it

```
poly2str(F,'s')
```

ans = s^2 + 3 s + 2

Evaluate it as s=1

polyval(F,1)

ans = 6

Find roots, Note that since roots are at -1 and -2 the polynomial is (s+1)(s+2)

roots(F)

ans = -2 -1

Define a second polynomial by its roots The polynomial is

```
G=poly([-1+2j -1-2j -1 0 0]);
poly2str(G,'s')
```

ans = s^5 + 3 s^4 + 7 s^3 + 5 s^2

Multiply polynomials H=FG (use convolution command)

H=conv(F,G)

H = 1 6 18 32 29 10 0 0

## First Example - Simplest case - distinct real roots

To see this example worked out manually go to: http://lpsa.swarthmore.edu/LaplaceXform/InvLaplace/InvLaplaceXformPFE.html#Distinct_Real_Roots_

```
disp('Example 1: PFE with distinct real roots');
```

Example 1: PFE with distinct real roots

This case considers only distinct real roots.

Define numerator and denominator polynomial.

n=[1 1]; %n=s+1 d=conv([1 0],[1 2]); %Use "conv" to multiply polynomial disp(['Numerator = ' poly2str(n,'s')]); disp(['Denominator = ' poly2str(d,'s')]);

Numerator = s + 1 Denominator = s^2 + 2 s

Now use "residue" command to do inverse transform. r = magnitude of expansion term p = location of pole of each term k = constnat term (k=0 except when numerator and denominator are same order (m=n)).

[r,p,k]=residue(n,d)

r = 0.5000 0.5000 p = -2 0 k = []

Note that the function is implicitly defined only for t>=0. Some texts show the time domain function multiplied by the unit step. We will keep our expressions simpler by making that relationship implicit.

## Second Example - Repeated roots at origin

To see this example worked out manually go to: http://lpsa.swarthmore.edu/LaplaceXform/InvLaplace/InvLaplaceXformPFE.html#Repeated_Real_Roots_

```
disp('Example 2: PFE with repeated real roots (at origin in this case)');
```

Example 2: PFE with repeated real roots (at origin in this case)

Define numerator and denominator polynomial.

n=[1 0 1]; %n=s^2+1 d=[1 2 0 0]; %d=s^2(s+2)=s^3+2s^2 disp(['Numerator = ' poly2str(n,'s')]); disp(['Denominator = ' poly2str(d,'s')]); [r,p,k]=residue(n,d)

Numerator = s^2 + 1 Denominator = s^3 + 2 s^2 r = 1.2500 -0.2500 0.5000 p = -2 0 0 k = []

Note, first order term (1/s) comes before the 2nd order term (1/s^2) in the Matlab results.

## Third Example - Complex Conjugate roots

To see this example worked out manually go to: http://lpsa.swarthmore.edu/LaplaceXform/InvLaplace/InvLaplaceXformPFE.html#ExampleRepeat

```
disp('Example 3: PFE with complex conjugate roots.');
```

Example 3: PFE with complex conjugate roots.

Define numerator and denominator polynomial.

n=[5 8 -5]; %n=s^2+1 d=[1 2 5 0 0]; %d=s^2(s^2+2s+5)=s^4+2s^3+5s^2 disp(['Numerator = ' poly2str(n,'s')]); disp(['Denominator = ' poly2str(d,'s')]); [r,p,k]=residue(n,d)

Numerator = 5 s^2 + 8 s - 5 Denominator = s^4 + 2 s^3 + 5 s^2 r = -1.0000 - 1.0000i -1.0000 + 1.0000i 2.0000 + 0.0000i -1.0000 + 0.0000i p = -1.0000 + 2.0000i -1.0000 - 2.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i k = []

Note, first two roots are complex conjugate roots.

Get magnitude and phase from magnitude and phase of pfe Refer to manual solution on web page listed above.

M=2*abs(r(1)) %Magnitude of cosine phi=angle(r(1)) %Phase of cosine in radians

M = 2.8284 phi = -2.3562

Get frequency and decay rate from location of pole

```
omega=abs(imag(p(1))) % omega has to be positive
alpha=-real(p(1))
```

omega = 2 alpha = 1

Plot the response

t=linspace(-1,4,1000); f=(M*exp(-alpha*t).*cos(omega*t+phi) + r(3) + r(4)*t) .* heaviside(t); plot(t,f,'Linewidth',2); xlabel('Time'); ylabel('f(t)'); grid;

## Fourth Example - Order of numerator=order of denominator

To see this example worked out manually go to: http://lpsa.swarthmore.edu/LaplaceXform/InvLaplace/InvLaplaceXformPFE.html#Order_of_numerator_polynomial_equals_order_of_denominator_

```
disp('Example 4: PFE: order of num=order of den');
```

Example 4: PFE: order of num=order of den

If the numerator and denominator have the same order, we get a constant as part of the partial fraction expansion.

Define numerator and denominator polynomial.

n=[3 2 3]; %n=3s^2+2s + 3 d=[1 3 2]; %d=s^2+3s+2=(s+1)(s+2) disp(['Numerator = ' poly2str(n,'s')]); disp(['Denominator = ' poly2str(d,'s')]); [r,p,k]=residue(n,d)

Numerator = 3 s^2 + 2 s + 3 Denominator = s^2 + 3 s + 2 r = -11 4 p = -2 -1 k = 3

Note this time that k is not empty, k=3.