B

867

Appendix

Before we present MATLAB approach to the partial-fraction expansions of transfer

functions, we discuss the manual approach to the partial-fraction expansions of transfer

functions.

Partial-Fraction Expansion when F(s) Involves Distinct Poles Only. Consider

F(s)written in the factored form

form<n

wherep 1 ,p 2 ,p,pnandz 1 ,z 2 ,p,zmare either real or complex quantities, but for each com-

plexpiorzjthere will occur the complex conjugate of piorzj, respectively. If F(s)involves

distinct poles only, then it can be expanded into a sum of simple partial fractions as follows:

(B–1)

whereak(k=1,2,p,n) are constants. The coefficient akis called the residueat the pole

ats=–pk. The value of akcan be found by multiplying both sides of Equation (B–1)

byAs+pkBand letting s=–pk, which gives

=ak

+p+

ak

s+pk

As+pkB+p+

an

s+pn

As+pkBR

s=-pk

cAs+pkB

B(s)

A(s)

d

s=-pk

= c

a 1

s+p 1

As+pkB+

a 2

s+p 2

As+pkB

F(s)=

B(s)

A(s)

=

a 1

s+p 1

+

a 2

s+p 2

+p+

an

s+pn

F(s)=

B(s)

A(s)

=

KAs+z 1 BAs+z 2 BpAs+zmB

As+p 1 BAs+p 2 BpAs+pnB

,

Appendix B Partial-Fraction Expansion