Page 334 - Mechanical Engineers' Handbook (Volume 2)

5 Approaches to Linear Systems Analysis 325

Inversion by Partial-Fraction Expansion
The partial-fraction expansion theorem states that a strictly proper rational function F(s)
with distinct (nonrepeated) poles p , i 1, 2,..., n, can be written as the sum
i
n
A 1 A 2 A n
1
F(s) A i (3)
s p 1 s p 2 s p n i 1 s p i
where the A , i 1, 2,..., n, are constants called residues. The inverse transform of F(s)
i
has the simple form

n
ƒ(t) Ae pt 1 Ae pt 2 Ae pt n Ae p t i
1
n
2
i
i 1
The Heaviside expansion theorem gives the following expression for calculating the
residue at the pole p ,
i
A (s p )F(s) s p i for i 1, 2,..., n
i
i
These values can be checked by substituting into Eq. (3), combining the terms on the right-
hand side of Eq. (3), and showing that the result yields the values for all the coefficients b , j
j 1, 2,..., m, originally specified in the form of Eq. (3).
Repeated Poles
When two or more poles of a strictly proper rational function are identical, the poles are
said to be repeated or nondistinct. If a pole is repeated q times, that is, if p p i 1
i
p i q 1 , then the pole is said to be of multiplicity q. A strictly proper rational function with
a pole of multiplicity q will contain q terms of the form
A i1 A i 2 A iq
(s p ) q (s p ) q 1 s p i
i
i
in addition to the terms associated with the distinct poles. The corresponding terms in the
inverse transform are

1 (q 1) 1 (q 2) A pt i
i 2
i1
(q 1)! At (q 2)! At iq e
The corresponding residues are
A (s p ) F(s) s p i
q
i1
i
A
d [(s p ) F(s)]

q
i 2
ds i
s p i

1
d (q 1) [(s p ) F(s)]
A i q
iq
(q 1)! ds (q 1)
s p i
Complex Poles
A strictly proper rational function with complex-conjugate poles can be inverted using par-
tial-fraction expansion. Using a method called completing the square, however, is almost
always easier. Consider the function