Can You Always Factor the Denominator in Partial Fraction Expansion?

[Runei]

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

But when doing partial fraction expansions of the type

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

[itex]\left(s-s_{1}\right)\left(s-s_{2}\right)[/itex]

where

[itex]s_{1} = -\zeta\omega_{n}+\omega_{n}\sqrt{\zeta^{2}-1}[/itex] and
[itex]s_{2} = -\zeta\omega_{n}-\omega_{n}\sqrt{\zeta^{2}-1}[/itex]

And thus being able to make the following expansion:

[itex]\frac{A}{s-s_{1}}+\frac{A}{s-s_{2}} = \frac{K}{\left(s-s_{1}\right)\left(s-s_{2}\right)}[/itex]

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

 

[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 I am using partial fraction expansions, and for example right now I am trying to do the partial fraction expansion of

[itex]\frac{K}{s\cdot\left(s^{2}+2\zeta\omega_{n}s + ω_{n}^{2} \right)}[/itex]

And I am 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:

[itex]\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)}[/itex]

 

[Mark44]

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:

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

 

[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 :)

 

[CellCoree]

Yes, you are correct. The partial fraction expansion is a method used to decompose a rational function into simpler fractions. In this case, we are decomposing the rational function into two simpler fractions with denominators of linear terms (s-s1 and s-s2). This is possible because the denominator of the original function can be factored into the linear terms (s-s1)(s-s2), where s1 and s2 are the roots of the polynomial.

It is important to note that the roots may be real and distinct (as in this case), repeated, or complex conjugates. The method for finding the coefficients A and B in the expansion may vary depending on the type of roots, but the general idea remains the same.

I hope this helps clarify the concept of partial fraction expansion. Thank you for your question and for making sure you are not making any mistakes. As scientists, it is important to always double check our work and make sure we are using the correct methods and formulas. Keep up the good work!