Nguyễn Công Phương
CONTROL SYSTEM DESIGN
Frequency Response Methods
Contents
Introduction
I. II. Mathematical Models of Systems III. State Variable Models IV. Feedback Control System Characteristics V. The Performance of Feedback Control Systems VI. The Stability of Linear Feedback Systems VII. The Root Locus Method VIII.Frequency Response Methods IX. Stability in the Frequency Domain X. The Design of Feedback Control Systems XI. The Design of State Variable Feedback Systems XII. Robust Control Systems XIII.Digital Control Systems
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2
Frequency Response Methods
1. Introduction 2. Frequency Response Plots 3. Performance Specifications in the Frequency
Domain
4. Frequency Response Methods Using Control
System Software
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3
Introduction (1)
• The frequency response of a system: the steady – state response of the system to a sinusoidal input signal.
• The sinusoid is a unique input signal, and the resulting output signal for a linear system, as well as signals throughout the system, is sinusoidal in the steady state.
• It differs from the input waveform only in
amplitude and phase angle.
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4
Introduction (2)
m s ( )
=
=
T s ( )
n
m s ( ) q s ( )
+
(
s
p
)
i
= 1
i
=
w
(cid:213)
r t ( )
A
sin
t
= ( ) R s
2
2
w A +
w
s
a
fi
= ( ) Y s
= ( ) ( ) R s T s
2
k 1 +
k +
b + s w + 2
s
+ n p
s
s
+ + ... p 1
n
a
fi
1
+ np t
= ( ) y t
+ + p t ... 1
L
k e 1
k e n
2
b + s w + 2
s
a
- - - fi
1
=
=
y
t ( )
L
steady state
2
y t lim ( ) t
lim t
b + s w + 2
s
=
+
= — w
w A T j (
w ) sin(
f t
f ),
T j (
)
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5
- - fi ¥ fi ¥
Frequency Response Methods
1. Introduction 2. Frequency Response Plots 3. Performance Specifications in the Frequency
Domain
4. Frequency Response Methods Using Control
System Software
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6
Frequency Response Plots (1)
=
+
=
w G j (
)
G s ( )
Re[
w ( G j
)]
Im[
w G j (
)]
=
w
j
s
+
=
w (
R
)
w jX
(
)
f w ( j
)
=
=
w G j (
)
e
w ( G j
fw )
[ (
)]
—
2
w
w 2
=
+
R
(
)
X
(
w 1 ) {tan [ X
(
w ) /
R
(
)]}
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7
- —
Frequency Response Plots (2)
Ex. 1
+
+
=
=
=
V
s ( )
I s ( )
R
.
V s ( ) 2
capacitor
1 Cs
1 Cs
+
R
1( )V s
2( )V s
V s ( ) 1 1 Cs
C
= ( ) G s
+
1 RCs
1
( ) V s = 2 V s ( ) 1
- - fi
0.5
= w G j ( )
G s ( )
=
= w j
s
Positive w Negative w
1 w j RC (
+ ) 1
0.4
0.3
1
=
=
w
,
0.2
1
1 RC
j
(
w w /
+ ) 1
1
0.1
)
(w
0
X
-0.1
w w /
1
=
fi
j
2
2
-0.2
1 w w /
+ 1 (
)
w w + 1 (
/
)
1
1
-0.3
-0.4
-
=
-0.5
1 tan (
w w /
)
1
2
1 w w /
+ 1 (
)
1
0
0.2
0.4
0.6
0.8
1
R(w
)
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8
- — -
Frequency Response Plots (3)
Semilog plots of the magnitude (in decibels) and phase (in degrees) of a transfer function versus frequency
=
20log
T
10
f
T=T
dBT f
=
=
f
f
f
...
...
fi
T T T T 3
1 2
T 1
3
=
f
) +
( (
( 2T )
)1 ...
...
T T T 1 2 3
( )2 3T f+ f + 2
1
3
+
+
+
20log
20log
...
T 1
T 2
10
T 3
10
= +
20log 10 + ...
20 log 10 f f f + = 1
T f 2
3
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9
fi
Frequency Response Plots (4)
2
j
1
1
+
+
w K j (
)
1
...
zw 2 j w
w + w
w j z 1
k
k
1
=
w
–
T (
)
2
j
2
+
+
1
...
z w 2 j w
w + w
w j p 1
n
n
1
K
: gain
1
1
: quadratic pole
: pole at the origin
: simple pole
2
1 jw
j
j
+
+
1
1
z w 2 2 w
w j p 1
w + w
n
n
2
j
j
+
+
1
: simple zero
jw
: zero at the origin
1
: quadratic zero
zw 12 w
w j z 1
w + w
k
k
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10
Frequency Response Plots (5)
=
20 log
K
10
w
=
fi
T (
)
K
0
dBT = f
f
dBT 20 log K
10
0
0.1
10
100 w
10
100 w
1
0.1
1
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11
Frequency Response Plots (6)
= -
w
20 log
10
w
=
fi
T (
)
o
1 w j
90
dBT = - f
f
dBT 20
0
o0
w
w
10
0.1
1
10
1
0.1
20
-
o90
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12
-
Frequency Response Plots (7)
=
w
20 log
10
w
=
w
fi
T (
j
)
o
90
dBT = f
f
o90
dBT 20
0
o0
w
w
10
0.1
1
10
1
0.1
20
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13
-
Frequency Response Plots (8)
= -
T
+ 20log 1
dB
10
1
w j p 1
w
=
fi
T (
)
w
+
1
1
=
tan
w j p 1
-
f
p 1
dBT 5
-
0
-5
-10
-15
w
-20
0.1p 1
1p
10 p 1
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14
-25
Frequency Response Plots (9)
= -
T
+ 20log 1
dB
10
1
w j p 1
w
=
fi
T (
)
w
+
1
1
=
tan
w j p 1
-
f
p 1
f
0.1p 1
1p
10 p 1
-
100 p 1 w
0
-10
-20
-30
-40
-50
-60
-70
-80
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15
-90 -100
Frequency Response Plots (10)
=
+
T
20log 1
dB
10
w j z 1
= +
w
fi
T (
) 1
w
w j z 1
1
tan
= f
z 1
dBT 25
-
20
15
10
5
w
0
0.1z 1
1z
110z
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16
-5
Frequency Response Plots (11)
=
+
T
20log 1
dB
10
w j z 1
= +
w
fi
T (
) 1
w
w j z 1
1
tan
= f
z 1
f
-
100
90
80
70
60
50
40
30
20
w
10
0.1z 1
1z
110z
100z 1
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17
0
Frequency Response Plots (12)
2
j
j
= -
T
+ 20log 1
dB
10
z w 2 2 w
w + w
n
n
1
=
w
fi
T (
)
2
j
j
/
j
1
+
+
1
= -
tan
z w 2 2 w
w w
-
n
n
z w w 2 2 2 w w /
1
n 2 n
f
-
0.05
z = 2
dBT
20
0.2
z = 2
0.4
z = 2
10
0
0.707
z = 2
z = 2 1.5
-10
-20
w
-30
w
w 0.1 n
n
w 10 n
w 100 n
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18
-40
Frequency Response Plots (13)
2
j
j
= -
T
+ 20log 1
dB
10
z w 2 2 w
w + w
n
n
1
=
w
fi
T (
)
2
j
j
/
j
1
+
+
1
= -
tan
z w 2 2 w
w w
-
n
n
z w w 2 2 2 w w /
1
n 2 n
f
w
f
w 0.1 n
n
w 10 n
-
w 100 n w
0
1.5
z = 2
-20
-40
0.707
-60
z = 2
-80
0.4
-100
0.2
-120
-140
0.05
z = 2 z = 2 z = 2
-160
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19
-180
Frequency Response Plots (14)
2
j
j
=
+
T
20log 1
dB
10
2
z w 2 2 w
w + w
n
n
j
j
w
= +
+
fi
T (
) 1
z w 2 2 w
w w
n
n
/
j
1
tan
-
z w w 2 2 2 w w /
1
n 2 n
f =
-
dBT
1.5
z = 2
40
30
0.707
z = 2
20
0.4
z = 2
10
0
0.2
0.05
z = 2
w
z = 2
-10
w
w 0.1 n
n
w 10 n
w 100 n
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20
-20
Frequency Response Plots (15)
2
j
j
=
+
20log 1
T
10
dB
2
z w 2 2 w
w + w
n
n
j
j
w
= +
+
fi
T (
) 1
z w 2 2 w
w w
n
n
/
j
1
tan
-
z w w 2 2 2 w w /
1
n 2 n
f =
-
0.05
f
z = 2
180
0.2
z = 2
160
0.4
z = 2
140
120
100
0.707
80
z =
z = 2 2 1.5
60
40
w
20
w
w 0.1 n
n
w 10 n
w 100 n
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21
0
Frequency Response Plots (16)
N
K
jw
(
) N
1 ) N jw
(
+ 1
w j z
20
N
dB/decade
20 log K
1
20
dB/decade
N
w
10
20
N
dB/decade
1
w
z
w
w
o
o
90N
90N
w
o0
o0
o
w
w
-
z
w
10z
90N
z 10
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22
-
Frequency Response Plots (17)
1
N
N
N
j
j
j
j
+
+
z w 2 2 w
zw 12 w
w + w
2
w + w
2
k
k
n
n
1 w j p
+ 1
1
1
w
k
p
N
40
dB/decade
w
w
20
N
dB/decade
40
N
dB/decade
w
w
n
o
180N
w
w
k 10
k
w 10 k
p
10 p
p 10
w
o0
w
o0
w
o0 w
o
w
- -
n
w 10 n
180N
o
90N
n 10
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23
- -
Frequency Response Plots (18)
Ex. 2
=
=
w T (
)
+
+
w +
500 + w 5)(
w j j
w j
(
10)
(1
w j
j
/10)
w j 10 / 5)(1
=
w T (
)
+
+ w
(1
w j
j
/10)
o
=
90
1 w tan (
/ 5)
1 w tan (
/10)
+
w j 10 / 5)(1 w j 10 + w / 5 1
1
w j
j
/10
=
+
+
+
20 log 10 20 log
w j
20 log
20log
10
10
10
10
dBT
+
+ w
1 w j
1
/ 5
1 j
1
/10
- - - -
fi
o
1
+
+
90
tan
tan
w
1 / 5
1 /10
1 w
= f
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24
- -
Frequency Response Plots (19)
Ex. 2
=
=
w T (
)
+
+
w +
500 + w 5)(
w j j
w j
(
10)
(1
w j
j
/10)
w j 10 / 5)(1
=
+
+
+
20log 10 20log
w j
20log
20log
+
+ w
1 w j
1
/ 5
1 j
1
/10
dBT 10 10 10 10
dBT
20 log 10
40
10
K
30
20
20log K
10
w
10
w
0
-10
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-20 0.1 1 5 10 20 100 500 200 25
Frequency Response Plots (20)
Ex. 2
=
=
w T (
)
+
+
w +
500 + w 5)(
w j j
w j
(
10)
(1
w j
j
/10)
w j 10 / 5)(1
=
+
+
+
20log 10 20log
w j
20log
20log
+
+ w
1 w j
1
/ 5
1 j
1
/10
dBT 10 10 10 10
dBT
20 log 10
40
10
30
jw
(
)N
20
20 log
jw
20
N
dB/decade
10
w
10
1
w
0
-10
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-20 0.1 1 5 10 20 100 500 200 26
Frequency Response Plots (21)
Ex. 2
=
=
w T (
)
+
+
w +
500 + w 5)(
w j j
w j
(
10)
(1
w j
j
/10)
w j 10 / 5)(1
=
+
+
+
20log 10 20log
w j
20log
20log
+
+ w
1 w j
1
/ 5
1 j
1
/10
dBT 10 10 10 10
dBT
20 log 10
40
10
1
30
N
20
w j p
+ 1
20 log
jw
10
w
p
10
w
0
20
N
dB/decade
20 log
10
1 jw+
1
/ 5
- -10
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-20 0.1 1 5 10 20 100 500 200 27
Frequency Response Plots (22)
Ex. 2
=
=
w T (
)
+
+
w +
500 + w 5)(
w j j
w j
(
10)
(1
w j
j
/10)
w j 10 / 5)(1
=
+
+
+
20log 10 20log
w j
20log
20log
+
+ w
1 w j
1
/ 5
1 j
1
/10
dBT 10 10 10 10
dBT
20 log 10
40
10
1
30
N
20
w j p
+ 1
20 log
jw
10
w
p
10
w
20log
0
10
20
N
dB/decade
20 log
10
1 jw+
1
/10
1 jw+
1
/ 5
- -10
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-20 0.1 1 5 10 20 100 500 200 28
Frequency Response Plots (23)
Ex. 2
=
=
w T (
)
+
+
w +
500 + w 5)(
w j j
w j
(
10)
(1
w j
j
/10)
w j 10 / 5)(1
=
+
+
+
20log 10 20log
w j
20log
20log
+
+ w
1 w j
1
/ 5
1 j
1
/10
dBT 10 10 10 10
Sl ope = 0 + 20 – 20
dBT
20 log 10
40
10
30
20
20 log
jw
10
w
10
20log
10
0
20 log
1 jw+
1
/10
10
1 jw+
1
/ 5
-10
29
-20 1 10 5 100 200 500 0.1 20 s i tes.google.com/site/ncpdhbkhn
Frequency Response Plots (24)
Ex. 2
=
=
w T (
)
+
+
w +
500 + w 5)(
w j j
w j
(
10)
(1
w j
j
/10)
w j 10 / 5)(1
o
1
1
=
+
+
90
tan
tan
w
1 / 5
1 /10
w
o90
f f 90
jw
(
)N
w
o
90N
- -
w
60 40 20 0 -20 -40 -60
0.1
0.5
1
5
10
20
50
100
200
500
30
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-90
Frequency Response Plots (25)
Ex. 2
=
=
w T (
)
+
+
w +
500 + w 5)(
w j j
w j
(
10)
(1
w j
j
/10)
w j 10 / 5)(1
o
1
1
=
+
+
90
tan
tan
w
1 / 5
1 /10
w
o90
f f 90
N
1 w j p
+ 1
w
p
10 p
tan
w
p 10 o0
- 1
1 /5w
- -
o
60 40 20 0 -20 -40 -60
90N
0.1
0.5
1
5
10
20
50
100
200
500
31
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- -90
Frequency Response Plots (26)
Ex. 2
=
=
w T (
)
+
+
w +
500 + w 5)(
w j j
w j
(
10)
(1
w j
j
/10)
w j 10 / 5)(1
o
1
1
=
+
+
90
tan
tan
w
1 / 5
1 /10
w
o90
f f 90
N
1 w j p
+ 1
w
p
10 p
tan
w
- 1
tan
1 /10
w
p 10 o0
- 1
1 /5w
- -
o
60 40 20 0 -20 -40 -60
90N
0.1
0.5
1
5
10
20
50
100
200
500
32
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- -90
Frequency Response Plots (27)
Ex. 2
=
=
w T (
)
+
+
w +
500 + w 5)(
w j j
w j
(
10)
(1
w j
j
/10)
w j 10 / 5)(1
o
1
1
f
=
+
+
90
tan
tan
w
1 / 5
1 /10
w
o90
f 90
w
tan
w
- 1
tan
1 /10
- 1
1 /5w
- -
60 40 20 0 -20 -40 -60
0.1
0.5
1
10
20
5
50
100
200
500
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33
-90
Frequency Response Plots (28)
Ex. 3
+
+
w j
w j
=
=
T
w (
)
2
2
+ w
20) w +
+
+
w j
(1
j
j
/ 20) + w 4 /10 (
j
/10)
(
j
5)
40
j
)
w w j (
j
100
8(1 + w 2 / 5) 1
1000( 2 w
40
dBT
20 log 8
20
10
w
0
-20
K
-40
20 log K
10
-60
w
-80
-100
0.1
1
5
10
20
50
100
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34
Frequency Response Plots (29)
Ex. 3
+
+
w j
w j
=
=
T
w (
)
2
2
+ w
20) w +
+
+
w j
(1
j
j
/ 20) + w 4 /10 (
j
/10)
(
j
5)
40
j
)
w w j (
j
100
8(1 + w 2 / 5) 1
1000( 2 w
40
+
20 log 1
dBT
10
jw 20
20 log 8
20
10
w
0
-20
N
w j z
+ 1
-40
20 dB/decade
N
-60
z
w
-80
-100
0.1
1
5
10
20
50
100
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35
Frequency Response Plots (30)
Ex. 3
+
+
w j
w j
=
=
T
w (
)
2
2
+ w
20) w +
+
+
w j
(1
j
j
/ 20) + w 4 /10 (
j
/10)
(
j
5)
40
j
)
w w j (
j
100
8(1 + w 2 / 5) 1
1000( 2 w
40
+
20 log 1
dBT
10
jw 20
20 log 8
20
10
w
0
20 log
10
1 jw
-20
1 jw )N
(
-40
-60
1
w
20 dB/decade
N
-80
-100
0.1
1
5
10
20
50
100
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36
-
Frequency Response Plots (31)
Ex. 3
+
+
w j
w j
=
=
T
w (
)
2
2
+ w
20) w +
+
+
w j
(1
j
j
/ 20) + w 4 /10 (
j
/10)
(
j
5)
40
j
)
w w j (
j
100
8(1 + w 2 / 5) 1
1000( 2 w
40
+
20 log 1
dBT
10
20 log
10
2
jw 20
(1
/ 5)
1 jw+
20 log 8
10
20
w
0
20 log
10
1 jw
-20
N
-40
1 w j p
+ 1
p
-60
w
N- 20 dB/decade
-80
-100
0.1
1
5
10
20
50
100
s i tes.google.com/site/ncpdhbkhn
37
Frequency Response Plots (32)
Ex. 3
+
+
w j
w j
=
=
T
w (
)
2
2
+ w
20) w +
+
+
w j
(1
j
j
/ 20) + w 4 /10 (
j
/10)
(
j
5)
40
j
)
w w j (
j
100
8(1 + w 2 / 5) 1
1000( 2 w
40
+
20 log 1
dBT
10
20 log
10
2
jw 20
(1
/ 5)
1 jw+
20 log 8
10
20
w
0
20 log
10
1
1 jw
-20
N
2
j
j
+
z w 2 2 w
w + w
k
k
1
-40
kw
w
-60
40
N
dB/decade
-80
20 log
10
2
+
1 +
1
w 4 /10 (
/10)
w j
j
-100
0.1
1
5
10
20
50
100
s i tes.google.com/site/ncpdhbkhn
38
-
Frequency Response Plots (33)
Ex. 3
+
+
w j
w j
=
=
T
w (
)
2
2
+ w
20) w +
+
+
w j
(1
j
j
/ 20) + w 4 /10 (
j
/10)
(
j
5)
40
j
)
w w j (
j
100
8(1 + w 2 / 5) 1
1000( 2 w
40
dBT
20
w
0
-20
-40
-60
-80
-100
0.1
1
5
10
20
50
100
39
Frequency Response - s i tes.google.com/site/ncpdhbkhn
Frequency Response Plots (34)
Ex. 3
+
+
w j
w j
=
=
T
w (
)
2
2
+ w
20) w +
+
+
w j
(1
j
j
/ 20) + w 4 /10 (
j
/10)
(
j
5)
40
j
)
w w j (
j
100
8(1 + w 2 / 5) 1
1000( 2 w
90
f
50
w
0
-50
N
-90
w j z
+ 1
o
90N
-180
-200
w
z
10z
o0 z 10
-270
-300
-360
0.1
0.2
0.5
1
10
50
100
200
500
2 s i tes.google.com/site/ncpdhbkhn
40
Frequency Response Plots (35)
Ex. 3
+
+
w j
w j
=
=
T
w (
)
2
2
+ w
20) w +
+
+
w j
(1
j
j
/ 20) + w 4 /10 (
j
/10)
(
j
5)
40
j
)
w w j (
j
100
8(1 + w 2 / 5) 1
1000( 2 w
90
f
50
w
0
-50
-90
1 )N jw
(
-180
-200
w
o
90N
-270
-300
-360
0.1
0.2
0.5
1
10
50
100
200
500
2 s i tes.google.com/site/ncpdhbkhn
41
-
Frequency Response Plots (36)
Ex. 3
+
+
w j
w j
=
=
T
w (
)
2
2
+ w
20) w +
+
+
w j
(1
j
j
/ 20) + w 4 /10 (
j
/10)
(
j
5)
40
j
)
w w j (
j
100
8(1 + w 2 / 5) 1
1000( 2 w
90
f
50
w
0
-50
1
-90
N
w
j p
+ 1
-180
p
10p
-200
w
p 10 o0
o
90N
-270
-300
-360
0.1
0.2
0.5
1
10
50
100
200
500
2 s i tes.google.com/site/ncpdhbkhn
42
-
Frequency Response Plots (37)
Ex. 3
+
+
w j
w j
=
=
T
w (
)
2
2
+ w
20) w +
+
+
w j
(1
j
j
/ 20) + w 4 /10 (
j
/10)
(
j
5)
40
j
)
w w j (
j
100
8(1 + w 2 / 5) 1
1000( 2 w
90
f
50
w
0
-50
1
N
-90
2
j
j
+
z w 2 2 w
w + w
k
k
w
-180
k
w 10 k
-200
w
1 kw 10 o0
o
180N
-270
-300
-360
0.1
0.2
0.5
1
10
50
100
200
500
2 s i tes.google.com/site/ncpdhbkhn
43
-
Frequency Response Plots (38)
Ex. 4 Find the transfer function from the Bode plot?
Sl ope = 0
dBH
jw
(
)N
40
30
20 dB/decade
N
20
1
w
jw
10
w
0
-10
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44
-20 0.1 1 5 10 20 100 200 500
Frequency Response Plots (39)
Ex. 4 Find the transfer function from the Bode plot?
Sl ope = 0
dBH
K
20 log 10
40
10
30
20 log K
10
20
w
jw
10
w
0
-10
45
s i tes.google.com/site/ncpdhbkhn
-20 0.1 1 5 10 20 100 200 500
Frequency Response Plots (40)
Ex. 4 Find the transfer function from the Bode plot?
Sl ope = 0
dBH
1
20 log 10
40
10
w j p
N
+ 1
30
p
w
20
N- 20 dB/decade
jw
10
w
0
20log
10
1 jw+
1
/ 5
-10
46
s i tes.google.com/site/ncpdhbkhn
-20 0.1 1 5 10 20 100 200 500
Frequency Response Plots (41)
=
Ex. 4 Find the transfer function from the Bode plot?
w T (
)
+
+ w
w j 10 / 5)(1
(1
w j
j
/10)
Sl ope = 0
dBH
1
N
20 log 10
40
10
w j p
+ 1
30
p
w
20
N- 20 dB/decade
20 log
10
jw
1 jw+
1
/10
10
w
0
20log
10
1 jw+
1
/ 5
-10
s i tes.google.com/site/ncpdhbkhn
47
-20 0.1 1 5 10 20 100 200 500
Frequency Response Methods
1. Introduction 2. Frequency Response Plots 3. Performance Specifications in the
Frequency Domain
4. Frequency Response Methods Using Control
System Software
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48
Performance Specifications in the Frequency Domain (1)
2
w
w =
z
<
z 1 2
,
0.707
resonant
n
-
2
=
=
<
z
-
M
w G j (
)
z 1 2
0.707
z 2
,
-
(
) 1
w p
r
0.05
z = 2
dBT
20
0.2
z = 2
0.4
z = 2
10
1
=
G
w (
)
2
0
z =
j
j
2 0.707
+
1
z w 2 2 w
1.5
z = 2
w + w
n
n
-10
-20
w
-30
w
w 0.1 n
n
w 10 n
s i tes.google.com/site/ncpdhbkhn
w 100 n 49
-40
2
=
=
<
z
-
Performance Specifications in the Frequency Domain (2) z 2 M 0.707
w G j (
z 1 2
)
,
) 1
(
w p
r
5.5
5
4.5
4
3.5
pw
M
3
2.5
2
1.5
1 0.1
0.2
0.3
0.4
0.5
0.6
0.7
-
zw
z
nt
= +
q +
y t ( ) 1
Be
w cos(
)
t 1
Resonant magnitudes should be relatively small: Mpω < 1.5, for example
s i tes.google.com/site/ncpdhbkhn
50
-
Frequency Response Methods
1. Introduction 2. Frequency Response Plots 3. Performance Specifications in the Frequency
Domain
4. Frequency Response Methods Using
Control System Software
s i tes.google.com/site/ncpdhbkhn
51
Ex.
=
T s ( )
+
Frequency Response Methods Using Control System Software s 500 + s 5)(
10)
(
s
s i tes.google.com/site/ncpdhbkhn
52
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