ECE317 : Feedback and Control

Lecture :Bode plots

Dr. Richard Tymerski

Dept. of Electrical and Computer Engineering

Portland State University

1

Course roadmap

2

Laplace transform

Transfer function

Block Diagram

Linearization

Models for systems

• electrical

• mechanical

• example system

Modeling Analysis Design

Stability

• Pole locations

• Routh-Hurwitz

Time response

• Transient

• Steady state (error)

Frequency response

• Bode plot

Design specs

Frequency domain

Bode plot

Compensation

Design examples

Matlab & PECS simulations & laboratories

Frequency response (review)

• Steady state output • Frequency is same as the input frequency

• Amplitude is that of input (A) multiplied by

• Phase shifts

• Frequency response function (FRF): G(jw)

• Bode plot: Graphical representation of G(jw)

3

Gain

G(s)

y(t)Stable

Phase shift (review)

4

G(s)

Bode plot of G(jw) (review)

• Bode diagram consists of gain plot & phase plot

5

Log-scale

Sketching Bode plot

• Basic functions• Constant gain

• Differentiator and Integrator

• First order system and its inverse

• Second order system

• Product of basic functions1. Sketch Bode plot of each factor, and

2. Add the Bode plots graphically.

6

Bode plot of a constant gain

7

10-2

10-1

100

101

102

19

19.5

20

20.5

21

10-2

10-1

100

101

102

-1

-0.5

0

0.5

1

(for all)

Old method:

8

𝐻(𝑠) = 𝐴

New method:

Sketching Bode plot

• Basic functions• Constant gain

• Differentiator and Integrator

• First order system and its inverse

• Second order system

• Product of basic functions1. Sketch Bode plot of each factor, and

2. Add the Bode plots graphically.

9

Bode plot of a differentiator

10

10-2

10-1

100

101

102

-40

-20

0

20

40

10-2

10-1

100

101

102

89

89.5

90

90.5

91

Old method:

11

𝐻(𝑠) = 𝐴𝑠

New method:

Bode plot of an integrator

12

10-2

10-1

100

101

102

-40

-20

0

20

40

10-2

10-1

100

101

102

-91

-90.5

-90

-89.5

-89

Mirror image of the

Bode plot of G(s)=s

with respect to w-axis.

Old method:

13

𝐻(𝑠) =𝐴

𝑠

New method:

Sketching Bode plot

• Basic functions• Constant gain

• Differentiator and Integrator

• First order system and its inverse

• Second order system

• Product of basic functions1. Sketch Bode plot of each factor, and

2. Add the Bode plots graphically.

14

10-2

10-1

100

101

102

-50

-40

-30

-20

-10

0

10-2

10-1

100

101

102

-100

-80

-60

-40

-20

0

Bode plot of a 1st order system

15

Corner frequencyStraight-line approximation

Old method:

16

𝐻(𝑠) =𝐴

1 +𝑠𝜔𝑜

New method:

Bode plot of an inverse system

17

Mirror image of the

original Bode plot with

respect to w-axis.10

-210

-110

010

110

20

10

20

30

40

50

10-2

10-1

100

101

102

0

20

40

60

80

100

Old method:

18

Bode plot: Zero at

𝐻(𝑠) = 𝐴 1 +𝑠

𝜔𝑜

𝜔𝑜

New method:

Sketching Bode plot

• Basic functions• Constant gain

• Differentiator and Integrator

• First order system and its inverse

• Second order system

• Product of basic functions1. Sketch Bode plot of each factor, and

2. Add the Bode plots graphically.

19

Bode plot of a 2nd order system

20

10-1

100

101

-60

-40

-20

0

20

10-1

100

101

-200

-150

-100

-50

0

resonance

Resonant freq.

Peak gain

Old method:

21

Bode plot of a 2nd order system

𝐻(𝑠) =𝐴

1 +𝑠

𝑄𝜔𝑜+

𝑠𝜔𝑜

2

𝑄 =1

2𝜍

New method:

Sketching Bode plot

• Basic functions• Constant gain

• Differentiator and Integrator

• First order system and its inverse

• Second order system

• Product of basic functions1. Sketch Bode plot of each factor, and

2. Add the Bode plots graphically.

22

Main advantage of Bode plot!

An advantage of Bode plot

• Bode plot of a series connection G1(s)G2(s) is the addition of each Bode plot of G1 and G2.• Gain

• Phase

• Later, we use this property to design C(s) so that G(s)C(s) has a “desired” shape of Bode plot.

23

Short proofs

• Use polar representation

Then,

Therefore,

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Example 1

• Sketch the Bode plot of a transfer function

1. Decompose G(s) into a product form:

2. Sketch a Bode plot for each component on the same graph.

3. Add them all on both gain and phase plots.

25

Example 1 (cont’d)

26

dB

deg

-20

Example 2

27

dB

deg

-20

Example 3

28

dB

deg

-20

-40

Remark

• You can use MATLAB command “bode” to obtain the precise magnitude and phase responses.

29

Summary

• Sketches of Bode plots • Basic transfer functions

• Products of basic transfer functions

• The new approach to sketching Bode plots is useful:• With simplified annotations we’re able to quickly obtain

good approximations to magnitude and phase values.

• This approach will be particularly useful in the process of compensator design where simplified design expressions will be derived from the sketched plots. (After a design is completed, it may be verified with MATLAB using the exact transfer function expressions).

• Next, practice sketching Bode plots

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