Csl10 9 f15 extra
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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
CONTROL SYSTEMS
B.Madhuri
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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of a second order control system?
Second Order SystemSystem with two poles
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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Second order system
• Two independent energy storage elements– Exchange of energy through mechanical mass and
fluid capacitance (tank) through piston– Exchange between electrical inductance and
mechanical inertia in electric motor
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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of a second order control system
Over-damped Time Response
a
acbb
2
42
2,1
146.1
854.7
2
459
)1(2
)9)(1(499
2
1
2
2,1
tt eKeKKtc 21321)(
General solution
2
3
1
21
21 ))((
9)(
s
K
s
K
s
K
ssssC
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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of a second order control system?
Under-damped Time Response
a
acbb
2
42
2,1
81
81
2
322
)1(2
)9)(1(422
2
1
2
2,1
j
j
j
)cos()( d
tdAetc
General solution
nd j
j
2,1
2,1 81
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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of a second order control system
Under-damped Time Response
)cos()( d
tdAetc
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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of a second order control system
Un-damped Time Response
a
acbb
2
42
2,1
3
3
2
36
)1(2
)9)(1(400
2
1
2
2,1
j
j
j
)cos()( Atc
General solution
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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of a second order control system
Critically-damped Time Response
a
acbb
2
42
2,1
3
2
6
)1(2
)9)(1(466
2,1
2
2,1
tt teKeKtc 11
21)(
General solution
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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system - Summary
Over-damped
Under-damped
Un-damped
Critically-damped
Poles are real and distinct
Poles are real and imaginary
Poles are imaginary
Poles are real and repeated
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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system - Summary
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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system - Example
a
acbb
2
42
2,1
23.1352,2
)175(410
)1(2
)200)(1(41010
1
2,1
2
2,1
j
j
The poles are real and imaginary.Therefore UNDER-DAMPED.
)cos()( d
tdAetcGeneral solution of under-damped
)23.13cos()( 5 tAetc
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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system - Example
2
2)(
s
sG
What are poles and zeroes?Plot on s-planeWrite the general expression of the step response?State the nature of response
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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system - Example
)6)(3(
5)(
sssG
What are poles and zeroes?Plot on s-planeWrite the general expression of the step response?State the nature of response
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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system? - Example
)20)(10(
)7(10)(
ss
ssG
What are poles and zeroes?Plot on s-planeWrite the general expression of the step response?State the nature of response.
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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system
So far, we based on the poles of the second order system
a
acbb
2
42
2,1
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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system
We are going to learn two quantities that will help us
analyze second order system
Natural Frequency
nDamping
Ratio
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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system- Natural Frequency
Natural frequency of a second order system is the frequency of oscillation of the system without
damping
Natural Frequency n
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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system- Natural frequency
bass
bsG
2)( Natural frequency of a
second order system is the frequency of
oscillation of the system without damping
bass
bsG
2)(
bs
bsG
2)( Poles:
njs
bjs
bs
bs
02Therefore,
2n
n
b
b
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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system- Damping Ratio
Damping Ratio
(rad/sec)Frequency Natural
FrequencyDecay lExponentia
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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system- Damping Ratio
Damping Ratio
n
a
2
bass
bsG
2)(
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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system
bass
bsG
2)(
22
2
2)(
nn
n
sssG
General Form
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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system
362.4
36)(
2
sssG Find damping ratio and
the natural frequency?
6
362
n
n
Natural Frequency
35.0)6(2
2.4
2.42
n
Damping Ratio
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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system
22
2
2)(
nn
n
sssG
a
acbbs
2
42
2,1
Find poles in general form.
1
2
422
22,1
22
2,1
nn
nnn
s
s
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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system.
122,1 nns
Case 1: 0 njs 2,1
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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system
122,1 nns
Case 2: 10
22,1 1 nn js
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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system
122,1 nns
Case 3: 1 ns 2,1
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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system
122,1 nns
Case 4: 112
2,1 nns
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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system
1
1
10
0
OVERDAMPED
CRITICALLY DAMPED
UNDER DAMPED
UNDAMPED
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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system- Example
40012
400)(
2
sssG
90090
900)(
2
sssG
22530
225)(
2
sssG
625
625)(
2 s
sG
3.0)20(2
12
20400
nUNDERDAMPED
5.1)30(2
90
30900
nOVER-DAMPED
1)15(2
30
15225
nCRITICALLY-DAMPED
0)25(2
0
25625
nUN-DAMPED
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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system. Special Case: Performance of under-damped System
Under-damped:
10 Four performance parameters:1.Rise time2.Peak time3.Percent overshoot4.Settling time
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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system. Special Case: Performance of under-damped System
Rise TimeGeneral
definition: The time required for the waveform to go from 0.1 of the final value to 0.9 of the final value
rnT
Underdamped case: The time taken to reach the final value the first time.
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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system. Special Case: Performance of under-damped System
Peak Time
The time required to reach the first maximum peak
21
n
pT
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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system. Special Case: Performance of under-damped System
Percent Overshoot
The amount that the waveform overshoots the steady-state
21
100%
eOS
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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system. Special Case: Performance of under-damped System
Settling Time
The time required for the transient damped oscillations to reach and stay within 2% of the steady-state value
nsT
4
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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system. Special Case: Performance of under-damped System – Example
36116
361)(
2
sssG Find: OSTTT rpsn ,%,,,,
19361 n
421.0)19(2
16
5.0)19(421.0
44
nsT
182.0421.01191 22
n
pT
%3.23100100%22
421.01
421.0
1
eeOS
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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system. Special Case: Performance of under-damped System – Example
36116
361)(
2
sssG
Find:rT
19n
421.0
079.019
501.1
501.1
r
nr
T
T
Or interpolate
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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system? –Lessons
April 14, 2023 37
11
10
0
OVERDAMPED
CRITICALLY DAMPED
UNDER DAMPED
UNDAMPED
WHERE DO WE GET
22
2
2)(
nn
n
sssG
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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system? -
April 14, 2023 38
What Interpretations?
Performances of second order under-damped control system are determined by two quantities called natural frequency and damping ratioFour performance parameters:1.Rise time2.Peak time3.Percent overshoot4.Settling time
21
n
pT
nsT
421100%
eOS
nr
fT
)(
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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system? -
April 14, 2023 39
a
acbb
2
42
2,1
22
2
2)(
nn
n
sssG
22,1 1 nn js
dd js 2,1
For under-damped system
)cos(
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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system? -
April 14, 2023 40
nsT
4
Line of constant settling time
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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system?
April 14, 2023 41
21
n
pT
Line of constant peak time
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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system?
April 14, 2023 42
21100%
eOS)cos(
Line of constant %OS
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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system?
April 14, 2023 43
Constant Settling Time
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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system?
April 14, 2023 44
Constant Peak Time
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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system?
April 14, 2023 45
Constant Percent Overshoot