Partial Fraction Decomposition with complex polesI have a function which I'd like to perform partial...

Janet Forbes

Janet Forbes

Answered

2022-07-03

Partial Fraction Decomposition with complex poles
I have a function which I'd like to perform partial fraction decomposition on, to allow easier inverse laplace transform.
F ( s ) = 1 s ( s 2 + 140 s + 10 4 )
I begin with finding the poles
s = 0 , s = 70 ± j 10 51
To which I then try putting F ( s ) in this form:
F ( s ) = A s + B s 70 + j 10 51 + B s 70 j 10 51
Because one unknown ( B ) is just the complex conjugate of B, I only need to find out what A, and B is.
A = F ( 0 ) = 10 4
B = F ( 70 + j 10 51 ) 6.31514 10 9 + j 6.44274 10 9
Where the last step was done in Mathematica.
Answer, according to several other people, is supposed to be B = 70 + j 71.41, which looks a lot nicer, but I'm not sure HOW they got to that answer.

Answer & Explanation

Perman7z

Perman7z

Expert

2022-07-04Added 13 answers

Note that the partial fraction expansion of F ( s ) is:
F ( s ) = A s + B s s 0 + B s s 0
where s 0 = 70 + i 10 51 and s 0 = s 0 = 70 i 10 51 are the complex-conjugate roots of s 2 + 140 s + 10 4
Note that we have
A = lim s 0 s F ( s ) = 10 4
and
B = lim s s 0 ( s s 0 ) F ( s ) = 1 s 0 ( s 0 s 0 ) = 70 i 10 51 i 20 51 10 4 = 51 + i 7 51 1020000

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