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

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

http://web.mit.edu/6.302/www/pz/

http://www.wolframalpha.com/



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


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


21

21

log20log20


pjpj

zjzjKa

If ω >> |p1|

log20log20

log20 1

j

pj




If ω << |p1|

11 log20log20 ppj Constant

Asymptotic Bode Plot

|p1|Constant


Mag

nitu

de


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


pjpj

zjzjKa

Suppose z1 is a real number

|| 1z

If ω >> |z1|

log20log20

log20 1

j

zj

Increase 20dB per decade



If ω << |z1|

11 log20log20 zzj Constant

1z


21

21

log20log20


pjpj

zjzjKa

If ω >> |z1|

log20log20

log20 1

j

zj




If ω << |z1|

11 log20log20 zzj Constant


|z1|Constant


Mag

nitu

de


• Problem: What if |z1| is 0?

|z1|M

agni

tude

1z


If |z1|=0, we cannot find the point on the Bode plot



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


20

02

1H

sQ

s

s

Height of peak:

Qlog20Q dB

constant

-40dB per decade

Only draw the peak when Q>1


• 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


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


1z 90

Phase – Complex Poles

If 0

If 0

If 0

1p

2p

0

0

180

20

02

1H

sQ

s

s

90



(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


Exercise 11.52

10 40

80

8dB

Is 8dB the maximum value?


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


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

Documents

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

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