Laplace Transform Inverse by Partial Fraction Expansion

Cuthbert Nyack
Consider a Laplace Transform F(s) which can be represented as a ratio of 2 polynomials as shown below.
If the denominator can be factorised with simple roots at r1, r2 etc. then F(s) can be expanded as:-
With coefficients
If Q(s) has a multiple zero of order m (F(s) has a multiple pole of order m) then the expansion of F(s) must be written
with coefficients

Example


Consider finding the inverse f(t) of F(s) below
Since F(s) has a second order pole at the origin and complex conjugate poles at -1/2 j0.866, then it can be expanded to
The coefficient a1 is calculated
a2 is
a3 is
and a4 is
The Expansion for F(s) now becomes
with inverse
and simplifies to

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Copyright 1996 Cuthbert A. Nyack.