# Partial fraction expansion

1. Jun 2, 2012

### Runei

Now this is a pretty straight forward question. And I just want to make sure that Im not doing anything stupid.

But when doing partial fraction expansions of the type

$\frac{K}{s^{2}+2\zeta\omega_{n}s+\omega_{n}^{2}}$ Shouldnt I always be able to factor the denominator into the following:

$\left(s-s_{1}\right)\left(s-s_{2}\right)$

where

$s_{1} = -\zeta\omega_{n}+\omega_{n}\sqrt{\zeta^{2}-1}$ and
$s_{2} = -\zeta\omega_{n}-\omega_{n}\sqrt{\zeta^{2}-1}$

And thus being able to make the following expansion:

$\frac{A}{s-s_{1}}+\frac{A}{s-s_{2}} = \frac{K}{\left(s-s_{1}\right)\left(s-s_{2}\right)}$

Since s1 and s2 are the roots of the polynomial?

These roots may ofcourse either be real and distinct, repeated or complex conjugates.

Thank in advance,
Rune

2. Jun 2, 2012

### Runei

The reason for the question is that I am reading for an exam for Control Systems. And I am using LaPlace transforms to solve the differential equations.

To get back to the time-domain im using partial fraction expansions, and for example right now im trying to do the partial fraction expansion of

$\frac{K}{s\cdot\left(s^{2}+2\zeta\omega_{n}s + ω_{n}^{2} \right)}$

And Im trying to determine whether I am actually doing it wrong when factoring, or whether I can actually solve the problem by expanding to the following:

$\frac{A}{s}+\frac{B}{s-s_{1}}+\frac{C}{s-s_{2}} = \frac{K}{s\cdot\left(s-s_{1}\right)\cdot\left(s-s_{2}\right)}$

3. Jun 2, 2012

### Staff: Mentor

This factorization is fine, as long as there are no repeated real roots in the quadratic.

Consider the case where the denominator is s(s2 + 4s + 4). Here is the partial fraction decomposition:

$$\frac{K}{s(s^2 + 4s + 4)} = \frac{A}{s} + \frac{B}{s + 2} + \frac{C}{(s + 2)^2}$$

4. Jun 2, 2012

### Runei

Thank you Mark!

I actually found an error further back in my work and that was because I didn't get the correct result. But thank you for clarifying and making me sure :)

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