csl10 9 f15 extra
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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
CONTROL SYSTEMS
B.Madhuri
ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of a second order control system?
Second Order SystemSystem with two poles
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
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
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
ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of a second order control system
Under-damped Time Response
)cos()( d
tdAetc
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
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
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
ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system - Summary
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
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
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
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.
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
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
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
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
ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system- Damping Ratio
Damping Ratio
(rad/sec)Frequency Natural
FrequencyDecay lExponentia
ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system- Damping Ratio
Damping Ratio
n
a
2
bass
bsG
2)(
ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system
bass
bsG
2)(
22
2
2)(
nn
n
sssG
General Form
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
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
ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system.
122,1 nns
Case 1: 0 njs 2,1
ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system
122,1 nns
Case 2: 10
22,1 1 nn js
ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system
122,1 nns
Case 3: 1 ns 2,1
ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system
122,1 nns
Case 4: 112
2,1 nns
ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system
1
1
10
0
OVERDAMPED
CRITICALLY DAMPED
UNDER DAMPED
UNDAMPED
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
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
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.
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
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
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
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
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
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
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
)(
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(
ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system? -
April 14, 2023 40
nsT
4
Line of constant settling time
ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system?
April 14, 2023 41
21
n
pT
Line of constant peak time
ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system?
April 14, 2023 42
21100%
eOS)cos(
Line of constant %OS
ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system?
April 14, 2023 43
Constant Settling Time
ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system?
April 14, 2023 44
Constant Peak Time
ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Time response of second order control system?
April 14, 2023 45
Constant Percent Overshoot
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