082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

668 Chapter 21 The Laplace transform
Solutions
1 (a)
(c)
1
s 2 +1
1
2s 2 +3s+6
1
(b)
2s 2 +s−1
1
(d)
3s 3 +6s 2 +8s+4
(e)
(f)
3s+4
6s 2 +8s+4
4s 2 +2s+6
6s 3 +2s 2 −s+4
21.12 POLES,ZEROSANDTHEsPLANE
Mosttransferfunctionsforengineeringsystemscanbewrittenasrationalfunctions:that
is, asratiosoftwo polynomials ins, with a constantfactor,K:
G(s) =K P(s)
Q(s)
P(s) is of order m, and Q(s) is of order n; for a physically realizable system m < n.
HenceG(s) may bewritten as
G(s) = K(s−z 1 )(s−z 2 )...(s−z m )
(s−p 1
)(s−p 2
)...(s−p n
)
The values ofsthat makeG(s) zero are known as the system zeros and correspond to
the roots ofP(s) = 0, that iss =z 1
,z 2
,...,z m
. The values ofsthat makeG(s) infinite
are known as the system poles and correspond to the roots of Q(s) = 0, that is s =
p 1
,p 2
,...,p n
. As we have seen, poles may be real or complex. Complex poles always
occur incomplex conjugate pairs whenever the polynomialQ(s)has real coefficients.
Engineersfinditusefultoplotthesepolesandzerosonansplanediagram.Acomplex
planeplotisusedwith,conventionally,arealaxislabelof σ andanimaginaryaxislabel
of jω. Poles are marked as crosses and zeros are marked as small circles. Figure 21.18
shows ansplane plot for the transfer function
G(s) =
3(s −3)(s +2)
(s+1)(s+1+3j)(s+1−3j)
Thebenefitofthisapproachisthatitallowsthecharacterofalinearsystemtobedetermined
by examining thesplane plot. In particular, the transient response of the system
can easilybe visualized by the number and positions of the systempoles and zeros.
jv
3 3
s plane
3
–2–1 3 s
3–3
Figure21.18
Polesand zerosplotted forthe transferfunction:
G(s) =
3(s −3)(s +2)
(s+1)(s+1+3j)(s+1−3j) .