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Modern Control Systems
Bode plot
1
Emam FathyDepartment of Electrical and Control Engineering
email: [email protected]://www.aast.edu/cv.php?disp_unit=346&ser=68525
Introduction
• Frequency response is the steady-state response of asystem to a sinusoidal input.
r(t) c(t)
R
C)sin(r 0 tA
jssGjG
)()(
22
21
)( sin
)())((
)(
)(
)()( :
s
AsRtAr(t)and
pspsps
sM
sR
sCsGassume
n
))(()())((
)(
)()()(
21
jsjs
A
pspsps
sM
sRsGsCthen
n
))(sin()( )( jGtjGAtc
Frequency Domain Plots
• Bode Plot
• Nyquist Plot
• Nichol’s Chart
Bode Plot
• A Bode diagram consists of two graphs:
– One is a plot of the logarithm of the magnitude ofa sinusoidal transfer function.
– The other is a plot of the phase angle.
– Both are plotted against the frequency on alogarithmic scale.
Decade
Definitions
• The decibel (dB) is a logarithmic unit used to express the ratio of two values of a physical quantity.
• A logarithmic scale is a nonlinear scale used when there is a large range of quantities.
– Log–log plots If both the vertical and horizontal axis of a plot is scaled logarithmically, the plot is referred to as a log–log plot.
– Semi logarithmic plots If only one axis is scaled logarithmically, the plot is referred to as a semi logarithmic plot.
)sG(M |)(|log20
Basic Factors of a Transfer Function
• The basic factors that very frequently occur inan arbitrary transfer function are
1. Gain K
2. Integral and Derivative Factors (jω)±1
3. First Order Factors (jωT+1)±1
4. Quadratic Factors
))((
)()(
251
13202
ssss
ssG
)1)2/5(2/)(1(
)13(102
ssss
s
Basic Factors of a Transfer Function
1. Gain K
• The log-magnitude curve for a constant gain K is a horizontalstraight line at the magnitude of 20 log(K) decibels.
• The phase angle of the gain K is zero.
• The effect of varying the gain K in the transfer function is thatit raises or lowers the log-magnitude curve of the transferfunction by the corresponding constant amount, but it has noeffect on the phase curve.
-15
-5
5
15
Mag
nit
ud
e (d
ecib
els)
Frequency (rad/sec)0.1 1 10 100
5KIf db)((K) 1452020 loglog Then
103 104 105 106 107 108 109
-90o
-300
30o
90o
Ph
ase
(deg
rees
)
Frequency (rad/sec)0.1 1 10 100
5KIf 0 )5
0(tan)
Re
Im(tan Then 1-1-
103 104 105 106 107 108 109
0o
Basic Factors of a Transfer Function
2. Integral and Derivative Factors (jω)±1
jswheressG ,)(
)log()( 20jG
900
1 )(tan)(
jG
Derivative Factor
Magnitude
Phase
ω 0.1 0.2 0.4 0.5 0.7 0.8 0.9 1
db -20 -14 -8 -6 -3 -2 -1 0
Slope=20db/decade
-30
-10
10
30
Mag
nit
ud
e (d
ecib
els)
Frequency (rad/sec)0.1 1 10 100
decadedb20
103 104 105 106 107 108 109
0
-20
-180o
-600
60o
180o
Ph
ase
(deg
rees
)
Frequency (rad/sec)0.1 1 10 100
90 )0
(tan 1-
103 104 105 106 107 108 109
0o
900
Basic Factors of a Transfer Function
2. Integral and Derivative Factors (jω)±1
• When expressed in decibels, the reciprocal of a numberdiffers from its value only in sign; that is, for the number N,
)log()log(N
N1
2020
)log()(
201
j
jGMagnitude
• Therefore, for Integral Factor the slope of the magnitude line wouldbe same but with opposite sign (i.e -6db/octave or -20db/decade).
900
1 )(tan)(
jGPhase
-30
-10
10
30
Mag
nit
ud
e (d
ecib
els)
Frequency (rad/sec)0.1 1 10 100
decadedb20
103 104 105 106 107 108 109
0
20
-180o
-600
60o
180o
Ph
ase
(deg
rees
)
Frequency (rad/sec)0.1 1 10 100
90 )0
(tan 1-
103 104 105 106 107 108 109
0o
-900
Basic Factors of a Transfer Function
2. First Order Factors (jωT+1)
– For Low frequencies ω<<1/T
– For high frequencies ω>>1/T
)log()( TjM 120
)log()( 22120 TM
0120 )log()(M
)log()( TM 20
)()()( 13
13 sssG
T
T
1
Basic Factors of a Transfer Function
2. First Order Factors (jωT+1)
)(tan-1 T )(
000 )(tan when -1)(,
4511
)(tan when 1-)(, T
90 )(tan when -1)(,
-30
-10
10
30
Mag
nit
ud
e (d
ecib
els)
Frequency (rad/sec)0.1 1 10 100 103 104 105 106 107 108 109
0
20
)()()( 13
13 sssG
ω=3
6 db/octave
20 db/decade
-90o
-300
30o
90o
Ph
ase
(deg
rees
)
Frequency (rad/sec)0.1 1 10 100 103 104 105 106 107 108 109
0o
45o
Basic Factors of a Transfer Function
3. First Order Factors (jωT+1)-1
– For Low frequencies ω<<1/T
– For high frequencies ω>>1/T
)log()( TjM 120
)log()( 22120 TM
0120 )log()(M
)log()( TM 20
)()(
3
1
ssG
Basic Factors of a Transfer Function
3. First Order Factors (jωT+1)-1
)(tan-1 T )(
000 )(tan when -1)(,
4511
)(tan when 1-)(, T
90 )(tan when -1)(,
-30
-10
10
30
Mag
nit
ud
e (d
ecib
els)
Frequency (rad/sec)0.1 1 10 100 103 104 105 106 107 108 109
0
-20
)()(
3
1
ssG
ω=3
-6 db/octave
-20 db/decade
-90o
-300
30o
90o
Ph
ase
(deg
rees
)
Frequency (rad/sec)0.1 1 10 100 103 104 105 106 107 108 109
0o
-45o
Example
)1100/(
)1(10)()(
2
ss
ssHsG
0dB1001010.1
)(log
20dB
-20dB
-40dB
40dB
-60dB
－40dB/dec
－20dB/dec
－40dB/dec
)1100/(
)1(10)()(
2
ss
ssHsG
①
②
③
④
0o
1001010.1
)(log
)(
45o
-45o
-90o
90o
-180o
-135o
)1100/(
)1(10)()(
2
ss
ssHsG
①
②
③
④
Basic Factors of a Transfer Function
4. Quadratic Factors
– For Low frequencies ω<< ωn
– For high frequencies ω>> ωn
2
2
2
2120 )()(log)(nn
M
0120 )log()(M
decdbMn
/)log()( 4040
Example
)(
)()(
1
123
s
ssG
8
10
12
14
16
Magnitu
de (
dB
)
10-2
10-1
100
101
0
5
10
15
20
Phase (
deg)
Bode Diagram
Frequency (rad/sec)
Example
))((
))(()(
151
171310
ss
sssG
20
25
30
35
Magnitu
de (
dB
)
10-2
10-1
100
101
102
0
10
20
30
40
Phase (
deg)
Bode Diagram
Frequency (rad/sec)
Example
))((
)()(
1513
446 2
ss
sssG
-10
0
10
20
30
Magnitu
de (
dB
)
10-2
10-1
100
101
102
-135
-90
-45
0
Phase (
deg)
Bode Diagram
Frequency (rad/sec)
Example
)(
)()(
1
123
s
ssG 8
10
12
14
16
Magnitu
de (
dB
)
10-2
10-1
100
101
102
0
45
90
135
180
Phase (
deg)
Bode Diagram
Frequency (rad/sec)
Example
)(
))(()(
1682
121510
sss
sssG
0
10
20
30
40
Magnitu
de (
dB
)
10-2
10-1
100
101
102
-90
-45
0
45
90
Phase (
deg)
Bode Diagram
Frequency (rad/sec)
Relative Stability
3/15/2020 47
Gain cross-over point
ωg
Phase cross-over point
ωp
3/15/2020 48
ωp
Gain Margin
Unstable Stable
Stable
Unstable Stable
Stable
ωg
Phase Margin
Example#3
• Obtain the phase and gain margins of thesystem shown in following figure for the twocases where K=10 and K=100.
End of Lecture
Example
0dB, 0o
1001010.1)(log
)( ),( L
20dB, 45o
-20dB, -45o
-40dB, -90o
40dB, 90o
-80dB,-180o
-60dB.-135o
-100dB,-225o
-120dB,-270o
－60dB/dec
－20dB/dec
)101.001.0)(11.0(
)1(10)(
22
ssss
ssG
－20dB/dec
