0195136047.pdf

3.3 LAPLACE TRANSFORM 145

The roots of the equation

N(s)= 0 (3.3.7)

are said to be thezerosofF(s); and the roots of the equation

D(s)= 0 (3.3.8)

are said to be thepolesofF(s).
The new functionF 1 (s) given byN 1 (s)/D(s) is such that the degree of the denominator
polynomial is greater than that of the numerator. The denominator polynomialD(s) is typically
of the form

D(s)=ansn+an− 1 sn−^1 +···+a 1 s+a 0 (3.3.9)

By dividing the numeratorN 1 (s) and the denominatorD(s)byan,F 1 (s) may be rewritten as
follows:

F 1 (s)=

N 2 (s)

D 1 (s)

=

N 2 (s)

sn+

an− 1

an

sn−^1 +···+

a 0

an

(3.3.10)

The particular manner of evaluating the coefficients of the expansion is dependent upon the nature
of the roots ofD 1 (s) in Equation (3.3.10). We shall now discuss different cases of interest. The
possible forms of the roots are (1) real and simple (or distinct) roots, (2) conjugate complex roots,
and (3) multiple roots.

Real and Simple (or Distinct) Poles

If all the poles ofF 1 (s) are of first order, Equation (3.3.10) may be written in terms of a partial-
fraction expansion as

F 1 (s)=

N 2 (s)

(s−p 1 )(s−p 2 )···(s−pn)

(3.3.11)

or

F 1 (s)=

K 1

s−p 1

+

K 2

s−p 2

+···+

Kn

s−pn

(3.3.12)

wherep 1 ,p 2 ,...,pnare distinct, andK 1 ,K 2 ,...,Knare nonzero finite constants. For anyk, the
evaluation of the residueKkofF 1 (s) corresponding to the poles=pkis done by multiplying
both sides by (s−pk) and lettings→pk,

Kk=lim

s→pk

[(s−pk)F 1 (s)] (3.3.13)

Equation (3.3.13) is valid fork=1, 2,... ,n. Once theK’s are determined in Equation (3.3.12),
the inverse Laplace transform of each of the terms can be written easily in order to obtain the
complete time solution.

Conjugate Complex Poles

It is possible that some of the poles of Equation (3.3.10) are complex. Since the coefficientsak
in Equation (3.3.9) are real, complex poles occur in complex conjugate pairs and will always be
even in number. Let us consider a pair of complex poles for which case Equation (3.3.10) may be
written as