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Time Response AnalysisTime Response Analysis
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ContentsContents
• Introduction
• Influence of Poles on Time Response
• Transient Response of First-OrderSystem
• Transient Response of Second-Order
System
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IntroductionIntroduction• The concept of poles and zeros, fundamental to the
analysis of and design of control system, simplifiesthe ealuation of system response!
• The poles of a transfer function are"
i! #alues of the $aplace Transform aria%les s, thatcause the transfer function to %ecome infinite!
ii! &ny roots of the denominator of the transfer functionthat are common to roots of the numerator!
• The zeros of a transfer function are"
i! The alues of the $aplace Transform aria%le s, thatcause the transfer function to %ecome zero!
ii! &ny roots of the numerator of the transfer functionthat are common to roots of the denominator!
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Influence of Poles on TimeInfluence of Poles on TimeResponseResponse
• The output response of a system is a sum of i! Forced response
ii! 'atural response
a) System showing an input and anoutput
b) Pole-zero plot of the system
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Influence of Poles on TimeInfluence of Poles on TimeResponseResponse
c) Evolution of a system response. Followthe blue arrows to see the evolution ofsystem component generated by the
pole or zero
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Influence of Poles on TimeInfluence of Poles on TimeResponseResponse
a) First-order system
b) Pole plot of thes stem
Eect of a real-ais pole upon transient
response
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First-Order SystemFirst-Order System• (eneral form"
• Pro%lem" )erie the transfer function for the
follo*ing circuit
1)(
)()(
+==
s
K
s R
sC sG
τ
1
1)(
+= RCs
sG
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First-Order SystemFirst-Order System• Transient Response" (radual change of output from
initial to the desired condition!
• +loc diagram representation"
• +y definition itself, the input to the system should %ea step function *hich is gien %y the follo*ing"
C(s)R(s) 1+ s
K
τ
s s R
1)( =
!here"
# $ %ainτ $ &ime constant
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First-Order SystemFirst-Order System• (eneral form"
• Output response"
1)(
)()(
+==
s
K
s R
sC sG
τ
1
1
1)(
++=
+
=
s
B
s
A
s
K
s sC
τ
τ
τ
τ
t e B
At c −+=)(
)()()( s R sG sC =
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First-Order SystemFirst-Order System• Pro%lem" Find the forced and natural responses for
the follo*ing systems
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First Order SystemFirst Order System
First-order system response to a unit step
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Transient ResponseTransient ResponseSpecificationsSpecifications
• Time constant, τ The time for e-at to decay ./0 of its
initial alue!
• Rise time, tr
The time for the *aeform to go
from 1!2 to 1!3 of its final alue!
• Settling time, ts
The time for the response to reach,
and stay *ithin 40 of its final alue!
a
1=τ
at r
2.2=
at s
4=
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Transient ResponseTransient ResponseSpecificationsSpecifications
• Pro%lem" For a system *ith the transfer functionsho*n %elo*, find the releant responsespecifications
i! Time constant, τ
ii! Settling time, ts
iii! Rise time, tr
50
50)(
+= s
sG
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Second-Order SystemSecond-Order System• (eneral form"
•Roots of denominator"
( )22
2
2nn
n
s s
K sG
ω ςω
ω
++=
!here"# $ %ain' $ (amping ratioωn $ ndamped natural
fre*uency
02 22
=++nn
s s ω ςω
12
2,1 −±−= ς ω ςω nn
s
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Second-Order SystemSecond-Order System• 'atural fre5uency, ω
n
Fre5uency of oscillation of the system *ithoutdamping!
• )amping ratio, 6 7uantity that compares the e8ponential decay
fre5uency of the enelope to the natural fre5uency!
(rad/s)frequency Natural
frequencydecaylExponentia=ς
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Second-Order SystemSecond-Order System• Pro%lem" Find the step response for the follo*ing
transfer function
• &ns*er"
( )22530
2252 ++
= s s
sG
( ) t t teet c
1515 151 −−−−=
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Second-Order SystemSecond-Order System
Pro%lem" For each of the transfer function, find thealues of 6 and ωn, as *ell as characterize the nature
of the response!
a9
%9
c9
d9
( )40012
4002 ++
= s s
sG
( )000
002 ++
= s s
sG
( )
22530
2252
++
=
s s
sG
( )!25
!252 +
= s
sG
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Second-Order SystemSecond-Order System
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Second-Order SystemSecond-Order System
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Second-Order SystemSecond-Order System• Step responses for second-order system damping
cases
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Second-Order SystemSecond-Order System• Pole plot for the underdamped second-order system
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Second-Order SystemSecond-Order System• Second-order response as a function of dampingratio
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Second-Order SystemSecond-Order System• Second-order response as a function of dampingratio
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Second-Order SystemSecond-Order System• :hen 1 ; 6 ; 2, the transfer function is gien %y thefollo*ing!
• Pole position"
( )( ) ( )d nd n
n
j s j s
K sG
ω ςω ω ςω
ω
−+++=
2 !here"2
1 ς ω ω −=nd
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Second-Order SystemSecond-Order System• Second-order response components generated %ycomple8 poles
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Second-Order SystemSecond-Order System• Second-order underdamped responses for dampingratio alue
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Transient ResponseTransient ResponseSpecificationsSpecifications
• Second-order underdamped response specifications
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Transient ResponseTransient ResponseSpecificationsSpecifications
• Rise time, Tr
The time for the *aeform to go from 1!2 to 1!3 of itsfinal alue!
• Pea time, Tp
The time re5uired to reach the first
or ma8imum pea!
• Settling time, Ts
The time re5uired for the transient<s
damped oscillation to reach and stay
*ithin =40 of the steady-state alue!
2
1 ζ ω
π
−=n
pT
n
sT ζω
4=
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Transient ResponseTransient ResponseSpecificationsSpecifications
• Percent oershoot, 0OS The amount that the *aeform oershoots the steady-
state, or final alue at pea time, e8pressed as apercentage of the steady-state alue!
"100" )1/( 2
×= −− ζ ζπ eOS
)100/("ln
)100/ln("
22 OS
OS
+
−=
π ζ
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System PerformanceSystem Performance• Percent oershoot ersus damping ratio
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System PerformanceSystem Performance• $ines of constant pea time Tp, settling time Ts andpercent oershoot 0OS
&s+ , &s
&p+ , &p
/S , /S+
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System PerformanceSystem Performance• Step responses of second-order underdampedsystems as poles moe
a) !ith constantreal part
b) !ith constantimaginary
part
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System PerformanceSystem Performance• Step responses of second-order underdampedsystems as poles moe
c) !ith constant dampingratio