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

670 Chapter 21 The Laplace transform
jv
s plane
Transient
response
3
3
s
t
Figure21.19
A polewith anegative real part leads to a decaying transientwhile a
pole with a zeroreal partleads to atransientthat doesnotdecay with
time.
jv
s plane
Transient
response
3
3
s
t
21.12.1 Rule1
Figure21.20
The furtherapole isfrom the imaginary axis, the quicker the decay of
its transient.
Thepolesmaybeeitherrealorcomplexbutforaparticularpoleitisnecessaryforthereal
parttobenegativeifthetransientcausedbythatpoleistodecaywithtime.Otherwisethe
transient response will increase with time and the system will be unstable, a condition
engineers usually design to avoid. In simple terms this means that the poles of a linear
system must all lie in the left half of the s plane for stability. Poles on the imaginary
axisleadtomarginal stabilityasthetransientsintroduced by suchpoles donotgrowor
decay. This isillustratedinFigure 21.19.
21.12.2 Rule2
21.12.3 Rule3
The further a pole is to the left of the imaginary axis the faster its transient decays (see
Figure 21.20). This is because its transient contains a larger negative exponential term.
Forexample,e −5t decaysfasterthane −2t .Thepolesneartotheimaginaryaxisaretermed
the dominant poles as their transients take the longest to decay. It is quite common for
engineers to ignore the effect of poles that are more than five or six times further away
from the imaginary axis than the dominant poles.
For a real system, poles with imaginary components occur as complex conjugate pairs.
The transient resulting from this pair of poles has the form of a sinusoidal term