Kindly simplify this equation with partial fraction and tell me procedure as well

3/[s(s+1)(s+3)]

Thanks

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- Jun 19th 2010, 06:21 AMmoonnightingaleHelp for Partial Fraction
Kindly simplify this equation with partial fraction and tell me procedure as well

3/[s(s+1)(s+3)]

Thanks - Jun 19th 2010, 06:28 AMsimplependulum
$\displaystyle \frac{3}{s(s+1)(s+3) } $

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

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

$\displaystyle = \frac{1}{2} \left( 3~\frac{s+2 }{ (s+1)(s+2) - 2 } - \frac{3}{ s+1} \right) $

$\displaystyle = \frac{1}{2} \left( 3 ~ \frac{s+2}{ s(s+3) } - \frac{3}{ s+1} \right) $

$\displaystyle = \frac{1}{2} \left( ( \frac{2}{s} + \frac{1}{s+3} ) - \frac{3}{ s+1} \right) $

$\displaystyle = \frac{1}{s} + \frac{1}{2 } \frac{1}{s+3} - \frac{3}{2} ~ \frac{1}{s+1} $ - Jun 19th 2010, 06:43 AMmoonnightingale
Thanks a lot for prompt reply.

How did u perform third step

1/2( 3 ----------)

thanks - Jun 19th 2010, 06:44 AMAckbeet
Or, since all the poles are simple, you could use the Heaviside cover-up method.

- Jun 19th 2010, 06:53 AMsimplependulum
I regard $\displaystyle s+2 $ in $\displaystyle (s+1)(s+2) - 2 $ as a constant $\displaystyle a$ and let $\displaystyle X = s+1$ so we have

$\displaystyle \frac{2}{X(aX-2 )} = \frac{a}{aX-2} - \frac{1}{X} $

$\displaystyle \frac{1}{X(aX-2)} = \frac{1}{2}( \frac{a}{aX-2} - \frac{1}{X} )

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