- #1
Runei
- 193
- 17
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
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