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

21.12 Poles, zeros and thesplane 671
3
jv
s plane
Transient
response
3
s
t
Figure21.21
Apairofstable complex poles gives riseto a decaying sinusoid
transient.
multiplying an exponential term. For a pair of stable poles, that is negative real part,
the transient can be sketched by drawing a sinusoid confined within a ‘decaying exponentialenvelope’(seeFigure21.21).Thereasonforthisisthatwhenthesinusoidalterm
has a value of 1 the transient touches the decaying exponential. When the sinusoidal
term has a value of −1 the transient touches the reflection in thet axis of the decaying
exponential. The larger the imaginary component of the pair of poles, the higher the
frequency ofthe sinusoidal term.
It should now be clear how useful a concept thesplane plot is when analysing the
response of a linear system. A complex system may have many poles and zeros but by
plotting them on the s plane the engineer begins to get a feel for the character of the
system. The form of the transients relating to particular poles or pairs of poles can be
obtainedusingtheaboverules.Themagnitudeofthetransients,thatisthevaluesofthe
coefficientsA 1
,A 2
,...,A n
, depends on the systemzeros.
Itcanbeshownthathavingazeroneartoapolereducesthemagnitudeofthetransient
relatingtothatpole.Engineersoftendeliberatelyintroducezerosintoasystemtoreduce
the effect of unwanted poles. If a zero coincides with the pole, it cancels it and the
transient corresponding tothatpole iseliminated.
Engineeringapplication21.8
Asymptotic Bode plot of a transfer function with real poles
andzeros
Recall that the Bode magnitude plot of a simple linear circuit was examined in
Engineeringapplication2.14.Itispossibletoconstructaplotofamorecomplicated
transferfunctionby following somesimple steps.
Hereweconsideratransferfunction with only real polesand zeros
( ) ( )···( )
s −z1 s −z2 s −zm
G(s) =K
s ( ) ( )···( )
s −p 1 s −p2 s −pn
wherez 1
,z 2
... are the systemzeros and p 1
,p 2
... are the system poles.
Thefirststageistoobtainanexpressionforthefrequencyresponseofthesystem
by substitutings = jω
( )( ) )
jω −z1 jω −z2 ···(
jω −zm
G (jω) =K
jω ( )( ) )
jω −p 1 jω −p2 ···(
jω −pn
➔