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[SOLVED] Laplace Transform

dwsmith

Well-known member
Feb 1, 2012
1,673
\[
\frac{(s+1)^2}{s^2 - s + 1}
\]
I have simplified it down to
\[
\frac{s - \frac{1}{2} + s^2 + s + \frac{3}{2}}{(s - 1/2)^2 + \frac{3}{4}} =
e^{1/2t}\cos\Big(t\frac{\sqrt{3}}{2}\Big) + \sqrt{3}e^{1/2t}\sin\Big(t\frac{\sqrt{3}}{2}\Big) + \frac{s^2 + s}{(s - 1/2)^2 + \frac{3}{4}}
\]
but I can't figure out the last transform.
 

chisigma

Well-known member
Feb 13, 2012
1,704
\[
\frac{(s+1)^2}{s^2 - s + 1}
\]
I have simplified it down to
\[
\frac{s - \frac{1}{2} + s^2 + s + \frac{3}{2}}{(s - 1/2)^2 + \frac{3}{4}} =
e^{1/2t}\cos\Big(t\frac{\sqrt{3}}{2}\Big) + \sqrt{3}e^{1/2t}\sin\Big(t\frac{\sqrt{3}}{2}\Big) + \frac{s^2 + s}{(s - 1/2)^2 + \frac{3}{4}}
\]
but I can't figure out the last transform.
With symple steps You obtain...

$\displaystyle H(s) = \frac{(s+1)^{2}}{s^{2} - s + 1} = 1 + \frac{3\ s}{s^{2} - s + 1} = 1 + 3\ \frac{s - \frac{1}{2}}{(s - \frac{1}{2})^{2} + \frac{3}{4}} + 3\ \frac{\frac{1}{2}}{(s - \frac{1}{2})^{2} + \frac{3}{4}}\ (1) $

... and from (1) You derive...

$\displaystyle h(t) = \delta (t) + 3\ e^{\frac{t}{2}}\ \cos \frac{\sqrt{3}}{2}\ t + \sqrt{3}\ e^{\frac{t}{2}}\ \cos \frac{\sqrt{3}}{2}\ t\ (2)$

Kind regards

$\chi$ $\sigma$