- #1
jegues
- 1,097
- 3
Homework Statement
Find the partial fractions expansion in the following form,
[tex]G(s) = \frac{1}{(s+1)(s^{2}+4)} = \frac{A}{s+1} + \frac{B}{s+j2} + \frac{B^{*}}{s-j2}[/tex]
Homework Equations
The Attempt at a Solution
I expanded things out and found the following,
[tex]1 = A(s^{2} + 4) + B(s^{2} + (1-j2)s -j2) + B^{*}(s^{2} + (1+j2)s + j2)[/tex]
From this I get the following equations,
[tex]A + B + B^{*} = 0[/tex]
[tex]B(1-j2) + B^{*}(1+j2) = 0[/tex]
[tex]4A - Bj2 + B^{*}j2 = 1[/tex]
This doesn't seem like a pleasant set of equations to solve.
I did another partial fractions expansion like so,
[tex]G(s) = \frac{1}{(s+1)(s^{2}+4)} = \frac{D}{s+1} + \frac{Es + F}{s^{2}+4}[/tex]
and found,
[tex]D = \frac{1}{5}, E = - \frac{1}{5}, F = \frac{1}{5}[/tex]
but this doesn't give me the form I need.
Any ideas? Any convenient ways to solve this?