Lecture 24 Bode Plot
Download
Report
Transcript Lecture 24 Bode Plot
Lecture 24
Bode 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_com
plex
• 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
dB
Degree
g
10 log a
20 log a
(refer to P500
of the textbook)
Phase
10-1
100
Angular Frequency
(log scale)
2
101
http://en.wikipedia.org/wiki/File:Bode_plot_template.pdf
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
H s K
s z1 s z 2
s p1 s p 2
a ( ) | H j | K
H j K
j z1 j
j p1 j
z 2
p 2
j z1 j z 2
j p1 j p 2
dB 20 log a ( )
20 log K 20 log j z 1 20 log j z 2
20 log j p 1 20 log j p 2
Draw each term individually, and then add them together.
Magnitude – Constant Term
20 log a ( ) 20 log K 20 log j z 1 20 log j z 2
20 log j p 1 20 log j p 2
20 log K
Magnitude – Real Pole
20 log a ( ) 20 log K 20 log j z 1 20 log j z 2
20 log j p 1 20 log j p 2
If ω >> |p1|
Suppose p1 is a
real number
p1
20 log j p 1
20 log j 20 log
| p1 |
If ω = 10Hz Magnitude 20 dB
If ω = 100Hz Magnitude 40 dB
Decrease 20dB per decade
If ω << |p1|
20 log j p 1 20 log p 1
Constant
Magnitude – Real Pole
20 log a ( ) 20 log K 20 log j z 1 20 log j z 2
20 log j p 1 20 log j p 2
Magnitude
Constant
If ω >> |p1|
20 log j p 1
|p1|
Decrease
20dB per
decade
20 log j 20 log
If ω = 10Hz Magnitude 20 dB
If ω = 100Hz Magnitude 40 dB
Decrease 20dB per decade
If ω << |p1|
Asymptotic Bode Plot
20 log j p 1 20 log p 1
Constant
Magnitude – Real Pole
If ω << |p1|
20 log p 1
If ω = |p1|
20 log j p 1
20 log jp 1 p 1
Cut-off Frequency
(-3dB)
20 log
2 | p1 |
20 log | p 1 | 20 log
20 log | p 1 | 3
3dB lower
2
Magnitude – Real Zero
20 log a ( ) 20 log K 20 log j z 1 20 log j z 2
20 log j p 1 20 log j p 2
Suppose z1 is a
real number
20 log j z 1
20 log j 20 log
z1
| z1 |
If ω >> |z1|
If ω = 10Hz Magnitude 20 dB
If ω = 100Hz Magnitude 40 dB
Increase 20dB per decade
If ω << |z1|
20 log j z 1 20 log z 1
Constant
Magnitude – Real Zero
20 log a ( ) 20 log K 20 log j z 1 20 log j z 2
Magnitude
20 log j p 1 20 log j p 2
Constant
Asymptotic
Bode Plot
If ω >> |z1|
20 log j z 1
Increase
20dB per
decade
|z1|
20 log j 20 log
If ω = 10Hz Magnitude 20 dB
If ω = 100Hz Magnitude 40 dB
Increase 20dB per decade
If ω << |z1|
20 log j z 1 20 log z 1
Constant
Magnitude – Real Zero
Magnitude
• Problem: What if |z1| is 0?
z1
Asymptotic
Bode Plot
|z1|
If |z1|=0, we cannot find
the point on the Bode plot
Magnitude – Real Zero
If |z1|=0
20 log j z 1
20 log j
20 log
If ω = 0.1Hz
If ω = 1Hz
If ω = 10Hz
Magnitude=-20dB
Magnitude=0dB
Magnitude=20dB
Magnitude (dB)
• Problem: What if |z1| is 0?
0 .1
1
rad / s
10
Simple Examples
-20dB
p1
+
p2
| p1 |
-20dB
-20dB
| p 2 | | p1 |
| p2 |
-40dB
+20dB
z1
p1
| z1 |
-20dB
-20dB
+
| p1 |
| p1 |
| z1 |
Simple Examples
-20dB
p1
+20dB +
z1
| z1 |
| p1 |
+20dB
| z1 |
| p1 |
+20dB
p1
z1
-20dB
+
| p1 |
+20dB
| p1 |
Magnitude – Complex Poles
1
H s
s
2
0
s
Q
If 0
1
H j
j
Q 0 .5
2
0
2
0
2
0
j
Q
1
2
The transfer function
has complex poles
j
0
Q
H j 1 0
2
2
0
20 log | H j | 40 log 0
If 0
constant
H j 1
2
20 log | H j | 40 log
-40dB per decade
Magnitude – Complex Poles
H s
s
2
0
Q
s 0
2
If 0
j
2
0
2
0
j
Q
1
2
1
H j 0
0
2
j
Q
constant
20 log | H j | 40 log 0
1
H j
The asymptotic line for
conjugate complex pole pair.
1
j
0
Q
2
0
If 0 -40dB per decade
20 log | H j | 40 log
The approximation is not good
peak at ω=ω0
enough
20 log | H j | 20 log
Q
0
2
40 log 0 20 log Q
Magnitude – Complex Poles
1
H s
s
2
0
Q
s 0
2
Height of peak:
Q dB 20 log Q
Only draw the
peak when Q>1
constant
-40dB per decade
Magnitude – Complex Poles
• Draw a peak with height 20logQ at ω0 is only an
approximation
• Actually,
Q 1 . 67
Q 2 .5
The peak is at
0 1
Q 5
1
2Q
Q 10
2
The height is
20 log Q
Q 1
1
1
4Q
2
0
Magnitude – Complex Zeros
+40dB per decade
constant
Q dB 20 log Q
Asymptotic Lines:
Phase
Phase
H s K
s z1 s z 2
s p1 s p 2
H j K
j z1 j
j p1 j
( ) K
j z 1 j z 2
j p 1 j p 2
Again, draw each term individually,
and then add them together.
z 2
p 2
Phase - Constant
( ) K j z 1 j z 2 j p 1 j p 2
Two answers
K
K 0
K
K 0
Phase – Real Poles
( ) K j z 1 j z 2 j p 1 j p 2
p1 is a real number
If ω << |p1|
j p1 ?
0
If ω = |p1|
p1
j p 1 ? 45
If ω >> |p1|
j p 1 ? 90
Phase – Real Poles
( ) K j z 1 j z 2 j p 1 j p 2
p1 is a real number
If ω << |p1|
j p1
j p1 ?
0.1|p1|
0
|p1|
10|p1|
0
If ω = |p1|
j p 1 ? 45
If ω >> |p1|
j p 1 ? 90
Phase – Real Poles
Exact
Bode Plot
0 .1 | z1 |
Asymptotic
Bode Plot
| z1 |
10 | z 1 |
Phase – Real Zeros
( ) K j z 1 j z 2 j p 1 j p 2
z1 is a real number
If z1 < 0
If ω << |z1| j z 1 ?
If ω = |z1| j z 1 ?
If ω >> |z1| j z 1 ?
j z1
z1
90
45
0
0
45
90
|z1|
Phase – Pole at the Origin
• Problem: What if |z1| is 0?
z1
90
Phase – Complex Poles
1
H s
If 0
s
2
0
Q
0
s 0
2
If 0
p1
180
0
If 0
90
p2
Phase – Complex Poles
Phase – Complex Poles
The red line is a very bad approximation.
(The phase for complex zeros are trivial.)
Correction Terms
Magnitude – Real poles and zeros
Given a pole p
0.1|P|
0.5|P| |P| 2|P|
10|P|
Magnitude
– Complex poles and Zeros
1
H s
s
2
0
Q
s 0
2
p1
0
0
2Q
p2
Computing the correction
terms at 0.5ω0 and 2ω0
Phase – Real poles and zeros
Given a pole p
0.1|P|
0.5|P| |P| 2|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
K 100
z 1 , z 2 0 , 50
p 1 , p 2 , p 3 100 , 100 , 400
• Draw the asymptotic Bode plot of the gain for H(s)
= 100s(s+50)/(s+100)2(s+400)
K
20log | K | 40dB
If ω << |p|
20 log p
If ω >> |p|
Decrease 20dB per decade
p 1- 40dB
100
p 2 - 40dB
100
p 3- 52dB
400
Exercise 11.58
K
40dB
p 1- 40dB
K 100
z 1 , z 2 0 , 50
p 1 , p 2 , p 3 100 , 100 , 400
p 2 - 40dB
100
z1
100 Hz , 40 dB
10 Hz , 20 dB
1 Hz , 0 dB
p 3- 52dB
100
400
If ω << |z1| 20 log z 1
If ω >> |z1| Increase 20dB
per decade
z2
34dB
50
Exercise 11.58
z1
z2
34dB
50
K
40dB
p 1- 40dB
Compute the
gain at ω=100
p 2 - 40dB
100
100
400
?
- 20dB/decad e
40dB/decad e
20dB/decad e
50
p 3- 52dB
100
400
Compute the
gain at ω=100
Exercise 11.58
K
40dB
p 1- 40dB
p 2 - 40dB
100
z1
100 Hz , 40 dB
1 Hz , 0 dB
p 3- 52dB
100
400
40dB
z2
6dB
34dB
50
40dB 40dB 40dB 52dB 40dB 40dB
100
12 dB
Exercise 11.58
z1
z2
34dB
50
K
40dB
p 1- 40dB
p 2 - 40dB
100
100
p 3- 52dB
400
-12dB
- 20dB/decad e
40dB/decad e
20dB/decad e
50
100
400
Exercise 11.58
• MATLAB
K 8000
z1 0
Exercise 11.52
p 1 , p 2 , p 3 10 , 40 , 80
• 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)
K
78dB
z1
p 1- 20dB
p 2 - 32dB
10
40
10
p 3- 38dB
80
8dB
40
80
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)
8dB
10
40
80
Is 8dB the maximum value?
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)
K
78dB
p 1- 20dB
p 2 - 32dB
10
40
z1
Correction
5
10
20
P1
-1dB
-3dB
-1dB
p2
-1dB
p3
Total
-1dB
-3dB
-2dB
p 3- 38dB
80
40
80
160
-3dB
-1dB
-1dB
-3dB
-1dB
-4dB
-4dB
-1dB
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)
8dB
Maximum gain is about 6dB
20loga 6dB
6
a 10 20 2
Correction
5
10
20
P1
-1dB
-3dB
-1dB
p2
-1dB
p3
Total
-1dB
-3dB
-2dB
40
10
40
80
160
-3dB
-1dB
-1dB
-3dB
-1dB
-4dB
-4dB
-1dB
80
Homework
• 11.59, 11.60, 11.63
Thank you!
Answer
• 11.59
Answer
• 11.60
Answer
• 11.63
• http://lpsa.swarthmore.edu/Bode/underdamped/u
nderdampedApprox.html
Examples
• http://lpsa.swarthmore.edu/Bode/BodeExamples.h
tml