lecture 24 bode plot hung-yi lee. announcement 第四次小考 時間 : 12/24 範圍 : ch11.1, 11.2,...
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Lecture 24Bode Plot
Hung-yi Lee
Announcement
•第四次小考•時間 : 12/24•範圍 : Ch11.1, 11.2, 11.4
Reference
• Textbook: Chapter 10.4• OnMyPhD: http://www.onmyphd.com/?
p=bode.plot#h3_complex• Linear Physical Systems Analysis of at
the Department of Engineering at Swarthmore College: http://lpsa.swarthmore.edu/Bode/Bode.html
Bode Plot
• Draw magnitude and phase of transfer function
Magnitude
Phase
Angular Frequency100 10110-1
(log scale)
Deg
ree
dB
a
a
log20
log10 2
(refer to P500 of the textbook)
http://en.wikipedia.org/wiki/File:Bode_plot_template.pdf
g
Drawing Bode Plot
• By Computer• MATLAB• http://web.mit.edu/6.302/www/pz/• MIT 6.302 Feedback Systems
• http://www.wolframalpha.com• Example Input: “Bode plot of
-(s+200)^2/(10s^2)”• By Hand• Drawing the asymptotic lines by some simple
rules• Drawing the correction terms
Asymptotic Lines:Magnitude
Magnitude
21
21KHpsps
zszss
21
21KHpjpj
zjzjj
21
21|jH|)(pjpj
zjzjKa
21
21
log20log20
log20log20log20
)(log20
pjpj
zjzjK
adB
Draw each term individually, and then add them together.
21
21
log20log20
log20log20log20)(log20
pjpj
zjzjKa
Magnitude – Constant Term
Klog20
Magnitude – Real Pole
21
21
log20log20
log20log20log20)(log20
pjpj
zjzjKa
Suppose p1 is a real number
|| 1p
1p
If ω >> |p1|
log20log20
log20 1
j
pj
Decrease 20dB per decade
If ω = 10HzIf ω = 100Hz
dBMagnitude 20dBMagnitude 40
If ω << |p1|
11 log20log20 ppj Constant
Magnitude – Real Pole
21
21
log20log20
log20log20log20)(log20
pjpj
zjzjKa
If ω >> |p1|
log20log20
log20 1
j
pj
Decrease 20dB per decade
If ω = 10HzIf ω = 100Hz
dBMagnitude 20dBMagnitude 40
If ω << |p1|
11 log20log20 ppj Constant
Asymptotic Bode Plot
|p1|Constant
Decrease 20dB per decade
Mag
nitu
de
Magnitude – Real Pole
Cut-off Frequency(-3dB)
3||log20
2log20||log20
||2log20
log20
log20
1
1
1
11
1
p
p
p
pjp
pjIf ω = |p1|
If ω << |p1|
1log20 p
3dB lower
Magnitude – Real Zero
21
21
log20log20
log20log20log20)(log20
pjpj
zjzjKa
Suppose z1 is a real number
|| 1z
If ω >> |z1|
log20log20
log20 1
j
zj
Increase 20dB per decade
If ω = 10HzIf ω = 100Hz
dBMagnitude 20dBMagnitude 40
If ω << |z1|
11 log20log20 zzj Constant
1z
Magnitude – Real Zero
21
21
log20log20
log20log20log20)(log20
pjpj
zjzjKa
If ω >> |z1|
log20log20
log20 1
j
zj
Increase 20dB per decade
If ω = 10HzIf ω = 100Hz
dBMagnitude 20dBMagnitude 40
If ω << |z1|
11 log20log20 zzj Constant
Asymptotic Bode Plot
|z1|Constant
Increase 20dB per decade
Mag
nitu
de
Magnitude – Real Zero
• Problem: What if |z1| is 0?
|z1|M
agni
tude
1z
Asymptotic Bode Plot
If |z1|=0, we cannot find the point on the Bode plot
Magnitude – Real Zero
• Problem: What if |z1| is 0?
If |z1|=0
log20
log20
log20 1
j
zj
1
Mag
nitu
de (d
B)
srad /101.0If ω = 1Hz
If ω = 10Hz
If ω = 0.1Hz Magnitude=-20dBMagnitude=0dBMagnitude=20dB
Simple Examples
1z 1p
1p 2p
+
+
|| 1p
-20dB
|| 2p
-20dB
|| 1p|| 2p
-20dB
-40dB
-20dB
|| 1p
-20dB
|| 1p || 1z
+20dB
|| 1z
Simple Examples
1p
+-20dB
|| 1p
1z
1p
+20dB
1z
+-20dB
|| 1p
+20dB
|| 1p|| 1z
+20dB
|| 1z
+20dB
|| 1p
Magnitude – Complex Poles
20
02
1H
sQ
s
s 5.0Q The transfer function has complex poles
constant
2
002
1H
jQ
j
j
Qj 022
0
1
0log40||log20 jH
If 0
If 0
201 jH
21 jH
log40||log20 jH
-40dB per decade
Magnitude – Complex PolesThe asymptotic line for conjugate complex pole pair.
constantIf 0
If 0 -40dB per decade
The approximation is not good enough peak at ω=ω0
20
02
1H
sQ
s
s
2
002
1H
jQ
j
j
Qj 022
0
1
Qj
j 20
0
1H
0log40||log20 jH
log40||log20 jH
20
log20||log20
QjH
Qlog20log40 0
Magnitude – Complex Poles
20
02
1H
sQ
s
s
Height of peak:
Qlog20Q dB
constant
-40dB per decade
Only draw the peak when Q>1
Magnitude – Complex Poles
• Draw a peak with height 20logQ at ω0 is only an approximation• Actually,
20 2
11
Q
The peak is at
24
11Qlog20
Q
The height is
1Q 67.1Q
5.2Q
5Q 10Q
0
Magnitude – Complex Zeros
constant
+40dB per decade
Qlog20Q dB
Asymptotic Lines:Phase
Phase
21
21KHpsps
zszss
21
21KHpjpj
zjzjj
Again, draw each term individually, and then add them together.
K)(
21 pjpj
21 zjzj
Phase - ConstantK)( 21 pjpj 21 zjzj
K K0K 0K
Two answers
Phase – Real PolesK)( 21 pjpj 21 zjzj
p1 is a real number
1p
If ω >> |p1|
If ω << |p1|
?1 pj 0
?1 pj 90
If ω = |p1|
?1 pj 45
Phase – Real PolesK)( 21 pjpj 21 zjzj
1pj
0 |p1|
0.1|p1|
10|p1|
p1 is a real number
If ω >> |p1|
If ω << |p1|
?1 pj 0
?1 pj 90
If ω = |p1|
?1 pj 45
Phase – Real Poles
Asymptotic Bode Plot
ExactBode Plot
||1.0 1z
|| 1z
||10 1z
Phase – Real ZerosK)( 21 pjpj 21 zjzj
z1 is a real number
If ω >> |z1|
If ω << |z1| ?1 zj 0
90
If ω = |z1|45
1z
If z1 < 0
?1 zj ?1 zj
1zj
|z1|0
45
90
Phase – Pole at the Origin
• Problem: What if |z1| is 0?
1z 90
Phase – Complex Poles
If 0
If 0
If 0
1p
2p
0
0
180
20
02
1H
sQ
s
s
90
Phase – Complex Poles
Phase – Complex Poles
(The phase for complex zeros are trivial.)
The red line is a very bad approximation.
Correction Terms
Magnitude – Real poles and zeros
Given a pole p
|P| 2|P|0.5|P|0.1|P| 10|P|
Magnitude – Complex poles and Zeros
1p
2p
0
20
02
1H
sQ
s
s
Q20
Computing the correction terms at 0.5ω0 and 2ω0
Phase – Real poles and zeros
Given a pole p
|P| 2|P|0.5|P|0.1|P| 10|P|
0 。
(We are not going to discuss the correction terms for the phase of complex poles and zeros.)
Examples
Exercise 11.58
• Draw the asymptotic Bode plot of the gain for H(s) = 100s(s+50)/(s+100)2(s+400)
100K 50,0,z 21 z
400,100,100,,p 321 pp
K 40dB|K|20log
1p 2p 3p
100 100 400
40dB- 40dB- 52dB-
If ω >> |p| Decrease 20dB per decade
If ω << |p| plog20
Exercise 11.58
K40dB 1p
100
40dB-2p
100
40dB-3p
400
52dB-
1z
dBHz 0,1
dBHz 20,10
dBHz 40,100
If ω >> |z1| Increase 20dB per decade
If ω << |z1| 1log20 z
34dB50
2z
100K 50,0,z 21 z
400,100,100,,p 321 pp
Exercise 11.58
K40dB 1p
100
40dB-2p
100
40dB-3p
400
52dB-
1z
34dB50
2z
e20dB/decad
e40dB/decad e20dB/decad-
100 40050
?Compute the gain at ω=100
Exercise 11.58
K40dB 1p
100
40dB-2p
100
40dB-3p
400
52dB-
1z
34dB50
2z
Compute the gain at ω=100
dBHz 0,1
dBHz 40,100
100
40dB
6dB
40dB40dB52dB40dB40dB40dB dB12
Exercise 11.58
K40dB 1p
100
40dB-2p
100
40dB-3p
400
52dB-
1z
34dB50
2z
e20dB/decad
e40dB/decad e20dB/decad-
100 40050
-12dB
Exercise 11.58
• MATLAB
Exercise 11.52
• Draw the asymptotic Bode plot of the gain for H(s) = 8000s/(s+10) (s+40)(s+80). Add the dB correction to find the maximum value of a(ω)
8000K 0z1
80,40,10,,p 321 pp
K78dB
1p
10
20dB-2p
40
32dB-3p
80
38dB-
1z
10 40
80
8dB
• Draw the asymptotic Bode plot of the gain for H(s) = 8000s/(s+10) (s+40)(s+80). Add the dB correction to find the maximum value of a(ω)
Exercise 11.52
10 40
80
8dB
Is 8dB the maximum value?
• Draw the asymptotic Bode plot of the gain for H(s) = 8000s/(s+10) (s+40)(s+80). Add the dB correction to find the maximum value of a(ω)
Exercise 11.52
K78dB
1p
10
20dB-2p
40
32dB-3p
80
38dB-
1z
Correction 5 10 20 40 80 160
P1 -1dB -3dB -1dB
p2 -1dB -3dB -1dB
p3 -1dB -3dB -1dB
Total -1dB -3dB -2dB -4dB -4dB -1dB
• Draw the asymptotic Bode plot of the gain for H(s) = 8000s/(s+10) (s+40)(s+80). Add the dB correction to find the maximum value of a(ω)
Exercise 11.52
10 40
80
8dB
Correction 5 10 20 40 80 160
P1 -1dB -3dB -1dB
p2 -1dB -3dB -1dB
p3 -1dB -3dB -1dB
Total -1dB -3dB -2dB -4dB -4dB -1dB
Maximum gain is about 6dB 6dB20loga
210a 20
6
Homework
• 11.59, 11.60, 11.63
Thank you!
Answer
• 11.59
Answer
• 11.60
Answer
• 11.63
• http://lpsa.swarthmore.edu/Bode/underdamped/underdampedApprox.html
Examples
• http://lpsa.swarthmore.edu/Bode/BodeExamples.html
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