# tài liệu bài giảng 4.2 bode diagram (cont )

The slides used the source PowerPoint files of Anthony Rossiter

• What is frequency response?
• How do I compute this efficiently?
• How do I represent frequency response
information in a helpful fashion?
• Why is this relevant to feedback loop analysis and
design?

• Previous topic showed how we can determine
expressions for frequency response gain and phase
based on a transfer function.
• These formulae are not insightful on their own.
• Humans often relate better to pictures and graphs.
• Hence it is useful to sketch the gain and phase in
graphical form and thus to judge whether such a
sketch would be helpful.

1. a transfer function object G
2. a frequency ‘w’
Then

[gain,phase]=bode(G,w);

3
3
1 w
st
take
GTo
( s )begin,

; Ga( simple
jw )  1 order;example.
G ( jw )  tan
2
2
s2
2
w 2
0

1.5

-20

Gain

Phase

1

-40
-60

0.5

-80
0
0

20

40

60

Frequency

80

100

-100
0

20

40

60

Frequency

80

100

Gain

1. For low frequencies gain is 1.5.
2. As frequency increases, gain
drops to zero.
3. For low frequencies phases is 0.
4. As frequency increases, phase
tends to -90.

1

0.5

0
0

20

40

60

80

100

Frequency
0
-20

Phase

k
G( s) 
sa

1.5

-40
-60
-80
-100
0

20

40

60

Frequency

80

100

4
G ( s)  2
;
s  5s  6
G ( jw)   tan 1

G ( jw) 

4
( w  2 )(w  3 )
2

2

2

2

;

w
w
 tan 1
2
3

0.7

0

0.6
-50

Phase

Gain

0.5
0.4
0.3
0.2

-100

-150

0.1
0
0

20

40

60

Frequency

80

100

-200
0

20

40

60

Frequency

80

100

0.7
0.6

4
G( s)  2
s  5s  6

Gain

0.4
0.3
0.2
0.1
0
0

20

40

60

80

100

Frequency
0
-50

Phase

1. For low frequencies gain is
0.67 and as frequency
increases, gain drops to zero.
2. For low frequencies phases is
0 and as frequency increases,
phase tends to -180.

0.5

-100

-150

-200
0

20

40

60

Frequency

80

100

s4
G( s)  2
;
s  2s  20

G ( jw) 

w2  4 2
(20  w2 ) 2  4w2

;

2w
G ( jw)   tan
20  w2
1

0.7

20

0.6

0

0.5

Phase

Gain

-20

0.4
0.3

-60

0.2

-80

0.1
0
0

-40

20

40

60

Frequency

80

100

-100
0

20

40

60

Frequency

80

100

s4
s 2  2s  20

0.5

Gain

1. Patterns are less obvious.
2. As frequency increases, gain
drops to zero and phase to 90.
3. For low frequencies overall
pattern is unclear although
there is a marked increase in
gain for a small frequency
range.

0.6

This
indicates a
resonance

0.4
0.3
0.2
0.1
0
0

20

40

60

80

100

Frequency
20
0
-20

Phase

G( s) 

0.7

-40
-60
-80
-100
0

20

40

60

Frequency

80

100

• It is straightforward to sketch the frequency response
parameters and hence get an overview of how gain and
phase change over a range of frequencies.
• However, all the focus is on the larger frequencies as
any notable changes in the low frequency range are
cramped into a small part of the graph domain.
Similarly very large frequencies are excluded.
• It is not obvious from formulae what causes the shapes
and asymptotes that arise and hence how changes in
poles and zeros will affect the overall shapes.

• The previous topic showed that simple sketches of
frequency response gain and phase are useful to gain
an overview of behaviour.
• However, the sketches could only focus on a relatively
limited frequency range, for example 1-10 rad/s or 10100 rad/s, etc.
• In practice, different decades need an equal spacing on
the graph in order to display effectively the changes in
behaviour that may occur in different frequency ranges.
• A similar comment can be applied to changes in gain.

1. Comprises 2 plots.
2. Both plots have log10w on the x-axis or w on a
log axis (same thing but this one displays actual
w which is helpful).
3. y-axis plots are either:
a. 20log10|G(jw)|
b. Arg(G(jw))

(denoted decibels)
(usually in degrees)

Bode Diagram

10

Magnitude (dB)

0
-10
-20

Note: use of
decibels for gain.

-30

Phase (deg)

-40
0

Note: frequency on
a log10 scale.

-45

-90
-1
10

10

0

1

10

Frequency (rad/s) (rad/s)

2

10

Bode Diagram

0

Magnitude (dB)

-10
-20
-30

s4
G 2
s  2s  20

-40
45

Phase (deg)

0
-45
-90
-135
-1
10

0

10

1

10

Frequency (rad/s) (rad/s)

2

10

It is straightforward to sketch the bode diagrams
using MATLAB and hence to get an overview of
how gain and phase change over a large range of
frequencies.
The use of log axis for frequencies enables us to
focus on several different decades in the same plot,
which is helpful, although we do not explicitly plot
values for w0,∞.
It is not yet obvious what causes the shapes and
asymptotes that arise and hence how changes in
poles and zeros will affect the overall shapes.

• There is a need to understand how different factors in
the transfer function relate to shapes in the Bode plot.
• This understanding will allow useful insight, especially for
control design which comes later.
• As will become apparent, we need to be comfortable
with rules of logarithms.

log(ab)=log(a)+log(b)
log(a/b)=log a- log(b)

How does a change in the decibel scale relate to changes
in the underlying gain?

20log10(10a)=
20log10 (√2 a)=
20log10 (a/√2 )=

20log10 (a/10)=

1. A change of 20dB is equivalent to a change in
gain of a factor of 10. (20dB up is multiply by 10
and 20dB down is divide by 10).
2. A change in gain of 3dB is equivalent to a change
in gain of a factor of √2 (approx).
Similarly one can see that 6dB is
equivalent to a factor of 2 (approx.).

It can be helpful, for mental arithmetic and insight,
to know logarithms for several key integers.
log10(1) = 0
log10(2) ≈ 0.3
log10(10) =1
log10(3) ≈ 0.48 (or 9.6dB)
log10(100) = 2
log10(10n) =n
log10(4) ≈ 0.6
(or 12dB)
log10(5) ≈ 0.7
(or 14dB)
log10(0.1) =-1
log10(6) ≈ 0.78
log
(0.01)
=-2
10
log10(8) ≈ 0.9
(or 18dB)

• The previous topic showed that MATLAB can be used for
forming exact Bode diagrams and reminded students of
core properties of logarithms.
• However, it is recognised that the ability to sketch is core
for developing insight and for use in design.
• This topic will use the definition of the Bode diagram,
and properties of logarithms, to sketch Bode diagrams
for:

1
1
G  s,
, ( s  a),
s
sa

Comprises 2 plots.
1. Both plots use log10w or w on a log axis
2. The other axis plots are either:
a. 20log10|G(jw)|
b. Arg(G(jw))

(denoted decibels)
(usually in degrees)

Next, apply these definitions to each factor in turn.

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