B_lecture13 Bode Diagram of Open-loop System and Minimum-phase Transfer Function Automatic control...
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Transcript of B_lecture13 Bode Diagram of Open-loop System and Minimum-phase Transfer Function Automatic control...
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1G 2G 3G nG
R c
Bode diagram of the open loop system
Plotting methods of the Bode diagram of the open-loop system
sGsGsGsGsG n321
s replaced by j
n
i
iGjn
i
i eGjG1
1
n
i
iGjG1
n
iiGjG
1
magnitude characteristic
phase characteristic
open-loop transfer function
n
ii
n
i
i GGG11
lg20lg20lg20Logarithmic magnitude characteristic
That is, Bode diagram of a open loop system is the superposition of the
Bode diagrams of the typical elements.
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Example
Constant gain K =10 (20lg10=20dB)
A pole at origin
First break frequency: 2 rad/s
)15.0(
10)(
sssG 10(s)G1
ssG
1)(2
15.0
1)(3
ssG
-
-20dB/dec -40dB/dec
Example
)15.0(
10)(
sssG
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Facility method to plot the magnitude response
of the Bode diagram
Summarizing example, we have:
1) Mark all break frequencies in the -axis of the Bode diagram.
2) Determine the slope of the L() of the lowest frequency band (before the first break frequency) according to the number of the integrating
elements:
[-20] dB/dec for 1 integrating element
[-40] dB/dec for 2 integrating elements
3) Continue the L() of the lowest frequency band until to the first break frequency, afterwards change the the slope of the L()
The slope of the L() should be increased 20dB/dec for the break frequency of the 1th-order differentiating element .
The slope of the L() should be decreased 20dB/dec for the break frequency of the Inertial element
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Example: Plot the L() of the G(s)
)101.001.0)(11.0(
)1(10)(
22
ssss
ssG
2
111
)01.0(1
01.0 )1.0(90)(
tgtgtgo
100 9.174
10 5.56
1 3.51
)(
o
o
o
The Bode diagram
Facility method to plot the magnitude response
of the Bode diagram
0dB, 0o
100 10 1 0.1
)( ),( L
20dB
-45o
-90o
40dB
-180o
-135o
-225o
-270o
60dB/dec
20dB/dec
1.25dB
r
20dB/dec
There is a resonant peak Mr at:
7.705.021100 21 22 nr
dBMr 25.1154.1
12
1
2
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Minimum-phase transfer function
A transfer function is called a minimum phase transfer function if its zeros and poles all lie in the left hand of s-plane.
A transfer function is called a non-minimum phase transfer function if it has any zero or pole lie in the right hand of s-plane.
For the minimum phase systems we can affirmatively determine the relevant transfer function only from the magnitude
response of the Bode diagram.
For the non-minimum phase system we must combine the magnitude response and phase response together to determine
the transfer function.
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Minimum phase transfer function versus
Non-minimum phase transfer function
phase minimum-non 1
1 )2(
phase minimum 1
1 )1(
Ts
Ts
[-20]
)(L
1/T
)(
1/T
090)1(
)2(
)2)(1(
phase minimum-non 1 )4(
phase minimum 1 )3(
s
s
[+20]
)(L
)(
090
/1
/1
)4)(3(
)3(
)4(
The magnitude responses are the same.
But the phase responses are different when vary from 0 to infinite.
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2 20 200 0dB
100 10 1 0.1
)(L
40dB/dec
40dB/dec
20dB/dec
)1005.0(
)15.0()(
:
2
ss
sKsG
diagramthe Bodefrom the G(s) we can get
40
0)5.0log(20log20log20)(20
2
K
KL
Example 1#
Determine the transfer function of the minimum phase
systems from the magnitude response of the Bode diagram
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0dB 100 10 1 0.1
)(L
0.5 200
20dB/dec 20dB/dec 20dB
05.0 0)log(20)2.0log(20log202log20)(
2.0 20log202log20)(
2 0log20log20)(
:
22002
1/1
5.0
1
TTL
TdBL
KKL
and
T
Determine the transfer function of the minimum phase
systems from the magnitude response of the Bode diagram
)1)(1()(
:
21
sTsT
KssG
diagramodefrom the B the G(s) we can get
Example 2#
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2.0 136.812
1log20
100 20log20log20)(
0.1T 10T
1
2
10
KdBKL
0dB
100 10 1 0.1
)(L
20dB/dec
20dB/dec 60dB/dec
8.136 dB
20 dB 1)0.04ss(0.01s
1)100(0.01sG(s) :
2
then
)12(
)101.0()( :
2
2
TssTs
sKsGdiagramodefrom the B the G(s) we can get
Determine the transfer function of the minimum phase
systems from the magnitude response of the Bode diagram
Example 3#