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Bode Diagrams
Chapter VI
Frequency-Response Analysis
Bode diagrams
2
Gain, K
Integral and derivative factors
First-order factors
Quadratic factors
The gain, K KjG
0Im;Re K
KjG 22
ImRe0
Re
Imtan 1
3
Case 1
The logarithmic magnitude of G(j) in decibels is
constantlog20log20 KjG
where
The phase angle of G(j) is
The gain, K
4
Integral Factor
5
j
jG1
Case 2
The logarithmic magnitude of G(j) in decibels is
dB log201
log20
j
1Im;0Re
1
12jG
90Re
Imtan 1
where
The phase angle of G(j) is
Integral Factor
0
1
Re
Im
cos
sintan
cosImsinRe
cos10cosIm0
901
0cos 1
6
From
90
Integral Factor
7
Derivative Factor
8
jjG Case 3
The logarithmic magnitude of G(j) in decibels is
dB log20log20 j
Im;0Re 2
jG
90Re
Imtan 1
where
The phase angle of G(j) is
Derivative Factor
Re
Im
cos
sintan
cosImsinRe
cos0cosIm0
900
cos 1
9
From
90
Derivative Factor
10
First Order Factors real pole
11
Tj
jG
1
1Case 4
The logarithmic magnitude of G(j) in decibels is
dB 1log201
1log20 22T
Tj
2222 1
Im;1
1Re
T
T
T
2222
2
221
1
11
1
TT
T
TjG
where
First Order Factors real pole
TT
TT
22
22
11
1
Re
Imtan
12
From
T 11 tanRe
Imtan where
First Order Factors real pole
13
First Order Factors real zero
T 11 tanRe
Imtan
14
TjjG 1Case 5
The logarithmic magnitude of G(j) in decibels is
dB 1log201log20 22TTj
T Im;1Re 221 TjG where
The phase angle of G(j) is
First Order Factors real zero
15
Second Order Factors complex poles
22
2
2 sssG
nn
n
n
ujuuj
jG
;
21
12
16
2
21
1
nn
jj
jG
Case 6
or
or
Second Order Factors complex poles
22
2
2
221log20
21
1log20
nn
nn
jj
2
22
221log20
21
1log20 uu
juuj
17
The logarithmic magnitude of G(j) in decibels is
or
Second Order Factors complex poles
222222
2
2
2
2
22
21
2
21
1
21
21
21
1
21
1
21
1
uu
uj
uu
u
uju
uju
uju
ujujuujjG
2
22
2
21
1Re
uu
u
222 21
2Im
uu
u
18
Conjugate
where
Second Order Factors complex poles
2
1
2
1
1
1
2tan
1
2
tan
Re
Imtan
u
u
n
n
19
The phase angle of G(j) is
Second Order Factors complex pole
20
Second Order Factors complex pole
21
Second Order Factors complex pole
22
Ex: Plotting Bode diagram
11.0
10
sssG
11.0
10
jjjG
11.0
1log20
1log2010log20
11.0
10log20log20
jj
jjjG
23
The log magnitude is
Consider the following transfer function:
Ex: Plotting Bode diagram
2010log2010log20
log20log201
log20 jj
rad/s
10-2 10-1 100 101 102 103
Y,dB 40 20 0 -20 -40 -60
24
Term 1:
Term 2:
Ex: Plotting Bode diagram
2
212
1.01log10
1.01log2011.0
1log20
j
rad/s
10-2 10-1 100 101 102 103
Y,dB 0 0 0 0 -20 -40
25
Term 3:
Bode Diagram asymptote magnitude plot
26
1
2
3
Bode Diagram exact magnitude curve
27
Ex: Plotting Bode diagram
90Re
Imtan 1
0Re
Imtan 1
28
The phase angle is
Term 1: G(j)=K=10
Term 2:
j
jG1
Ex: Plotting Bode diagram
1.0tanRe
Imtan 11
rad/s
10-2 10-1 100 101 102 103
P degree
0 0 0 -45 -90 -90
29
11.0
1
jjGTerm 3:
Bode Diagram asymptote phase plot
30
1
2
3
Bode Diagram exact phase curve
31
Plotting Bode Diagrams with MATLAB
32
bode(num,den) bode(num,den,w) bode(sys) bode(A,B,C,D) bode(A,B,C,D,w)
Plotting Bode Diagrams with MATLAB
33
[mag,phase,w]=bode(num,den) [mag,phase,w]= bode(num,den,w) [mag,phase,w]= bode(sys) [mag,phase,w]= bode(A,B,C,D) [mag,phase,w]= bode(A,B,C,D,w)
W=logspace(a,b)
Plotting Bode Diagrams with MATLAB
ss
s
ss
s
ss
ssG
12.0
105.04
12.05
105.020
5
20
jj
jjG
12.0
105.04
34
เมือ่ระบบมฟัีงก์ช่ันถ่ายโอนดังนี ้
หรือ
Plotting Bode Diagrams with MATLAB
12.0
1 .4
1 .3
105.0 .2
4 .1
jjG
jjG
jjG
jG
35
ซ่ึงเราสามารถแบ่งไดท้ั้งหมด 4 เทอม
Plotting Bode Diagrams with MATLAB
ss
s
ss
s
ss
ssG
12.0
105.04
12.05
105.020
5
20
jj
jjG
12.0
105.04
36
เมือ่ระบบมฟัีงก์ช่ันถ่ายโอนดังนี ้
num=[1 20];den=[1 5 0]; bode(num,den)
MATLAB command
หรือ
Plotting Bode Diagrams with MATLAB
37
rad/sec 20z
rad/sec 5p
Plotting Bode Diagrams with MATLAB
38
rad/sec 5p
12.0
1
jjG
