Transcript Slide 1

Chapter 13
Bode Diagrams
• A special graph, called the Bode diagram or Bode plot,
provides a convenient display of the frequency response
characteristics of a transfer function model. It consists of
plots of AR and φ as a function of w.
• Ordinarily, w is expressed in units of radians/time.
Bode Plot of A First-order System
Recall:
AR N 
1
ω2 τ 2  1
and φ   tan 1  ωτ 
 At low frequencies (ω  0 and ω
1) :
AR N  1 and   
 At high frequencies (ω  0 and ω
1) :
AR N  1/ ωτ and   
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Chapter 13
Figure 13.2 Bode diagram for a first-order process.
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Chapter 13
• Note that the asymptotes intersect at ω  ωb  1/ τ, known as the
break frequency or corner frequency. Here the value of ARN
from (13-21) is:
AR N  ω  ωb  
1
 0.707
11
(13-30)
• Some books and software defined AR differently, in terms of
decibels. The amplitude ratio in decibels ARd is defined as
AR d  20 log AR
(13-33)
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Chapter 13
Integrating Elements
The transfer function for an integrating element was given in
Chapter 5:
Y s K
G s 

(5-34)
U s s
K
K
AR  G  jω  

jω ω
φ  G  jω   K       90
(13-34)
(13-35)
Second-Order Process
A general transfer function that describes any underdamped,
critically damped, or overdamped second-order system is
K
G s  2 2
(13-40)
τ s  2ζτs  1
4
Substituting s  jω and rearranging yields:
AR 
K

Chapter 13
1 ω τ
2 2

2
(13-41a)
  2ωτζ 
 2ωτζ 
φ  tan 1 

1  ω 2 τ 2 
2
(13-41b)
Figure 13.3 Bode diagrams for second-order processes.
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Chapter 13
Time Delay
Chapter 13
Its frequency response characteristics can be obtained by
substituting s  jω ,
G  jω   e jωθ
(13-53)
which can be written in rational form by substitution of the
Euler identity,
G  jω   e jωθ  cos ωθ  j sin ωθ
(13-54)
From (13-54)
AR  G  jω   cos 2 ωθ  sin 2 ωθ  1
(13-55)
 sin ωθ 
φ  G  jω   tan 1  

cos
ωθ


or
φ  ωθ
(13-56)
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Chapter 13
Figure 13.6 Bode diagram for a time delay, e θs.
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Chapter 13
Chapter 13
Figure 13.7 Phase angle plots for e θs and for the 1/1 and 2/2
Padé approximations (G1 is 1/1; G2 is 2/2).
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Process Zeros
Chapter 13
Consider a process zero term,
G  s   K ( sτ  1)
Substituting s=jw gives
G  jω   K ( jωτ  1)
Thus:
AR  G  jω   K ω2 τ 2  1
φ  G  jω    tan 1  ωτ 
Note: In general, a multiplicative constant (e.g., K) changes
the AR by a factor of K without affecting φ .
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Chapter 13