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MESB374 System Modeling and Analysis Chapter 11 Frequency Domain Design - Bode
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MESB374 System Modeling and Analysis
Chapter 11Frequency Domain Design - Bode
Bode Magnitude Plot: plots the magnitude of G(j) in decibels w.r.t. logarithmic frequency, i.e.,
Bode Phase Plot: plots the phase angle of G(j) w.r.t. logarithmic frequency, i.e.,
A better way to graphically display the frequency response!
Bode Plots
10 10dB( ) 20log ( ) vs log G j G j
10( ) vs log G j Benefits:
– Display the dependence of magnitude of the frequency response on the input frequency better, especially for magnitude approaching zero– Log axis converts the multiplications and divisions into additions and subtractions, which are easier to handle graphically– Allow straight-line approximations for quick sketch
Bode PlotsEx:
2
2
1( )100
G j
2 2
1 ( ) 1( ) ( )10 ( ) 10( )
s jG s G js s j j
G j( ) 20 10log ( )G j G j( ) 0.1 0.2 0.5 1 2 5
10 20 50 100
2
1 2
( ) ( 1) 10
tan atan2 10 ,
G j j j
1.00490.5098
0.2233
0.1407
0.1096
0.09120.0711
0.04480.0196
0.0100
0.0428-5.8520
-13.0211
-17.0329-19.2012
-20.7988
-22.9671-26.9789
-34.1480
-40.0428
-83.8623-79.8358
-66.2974
-50.7016-37.8750
-37.8750
-50.7106
-66.2974
-79.8358
-84.8623
Bode Plots of LTI SystemsTransfer Function
Frequency Response
Bode Magnitude Plot
Bode Phase Plot
11 1 0 1 2
11 1 0 1 2
( )( ) ( )( )( )( ) ( )
m mm m m m
n nn n
b s b s b s b b s z s z s zG ss a s a s a s p s p s p
10 10 10 101
10 1 10 1
1 120log ( ( ) ) 20log 20log 20log( ) ( )
20log ( ) 20log ( )
mn
G j bj p j p
j z j z
11 2
1
1 2
( ) ( )( ) ( ) ( ) ( )( ) ( )
( ) ( ) ( )
m mm m
n
n
b j z j zG j b j z j z j zj p j p
j p j p j p
11
1 1
( ) ( ) 1 1( ) ( ) ( )( ) ( ) ( ) ( )m m
m mn n
b j z j zG j b j z j zj p j p j p j p
ExampleEx: Find the magnitude and the phase of the following transfer function:
9 1 180 3( )
1 1 11 1 12 4 5
j jG j
j j j
3 2
3 2
3 3 13 12 9( )2 22 76 80 2 2 4 5
3 3 1 11 1 1 192 2 4 5 3 3
1 1 1 1 1 1801 1 1 1 1 12 4 5 2 4 5
s s ss s sG ss s s s s s
s s s s s s
s s s s s s
10 10 10 10
10 10 10
9 120log ( ) 20log 20log 20log 180 3
1 1 120log 1 20log 1 20log 12 4 5
G j j j
j j j
9 1( ) 1 180 3
1 1 11 1 12 4 5
G j j j j
j j j
Bode Plots of 1st Order PolesStandard Form of Transfer Function:
Frequency Response:
Q: By just looking at the Bode diagram, can you determine the time constant and the steady state gain of the system ?
b
11( ) , 0
1pG ss
1
1 2 2
11
1( ) , 01
1( )1
( ) atan2( ,1)tan
p
p
p
G jj
G j
G j
Phas
e (d
eg);
Mag
nitu
de (d
B)
-40
-20
-30
Frequency (rad/sec)0.01/ 0.1/ 1/ 10/ 100/
-90
-45
0
20dB/decade
break frequency
10 1
2 210
10
20log ( )
10log 1
10dB, 1 or
13dB, 1 or
1-20log , 1 or
p
b
b
bb
G j
Example• 1st Order Real Poles
Transfer Function:
Plot the straight line approximation of G(s)’s Bode diagram:
50 1( ) 10 15 15
G ss s
G ss
( )50
5
Frequency (rad/sec)
Phas
e (d
eg);
Mag
nitu
de (d
B)
0
5
10
15
20
-90
-45
0
10-1
100
101
102
10 10120log ( ) 20 20log 1 1
5
G jj
1( ) 15
G j j
1 5b
Bode Plots of 1st Order ZerosStandard Form of Transfer Function
Frequency Response1( ) 1 , 0zG s s
0G j j
G j
G j
z
z
z
1
12 2
11
1
1
1
( )
( )
( ) ( , )tan
,
atan2
Frequency (rad/sec)0.01/ 0.1/ 1/ 10/ 100/
40
20
30
0
45
90
Phas
e (d
eg);
Mag
nitu
de (d
B)
10 1
2 210
10
20log ( )
10log 1
10dB, 1 or
13dB, 1 or
120log , 1 or
p
b
b
bb
G j
Example• 1st Order Real Zeros
Transfer Function:
Plot the straight line approximation of G(s)’s Bode diagram:
( ) 0.7 1G j j
( ) 0.7 0.7G s s
Phas
e (d
eg);
Mag
nitu
de (d
B)
Frequency (rad/sec)
0
5
10
15
20
0
45
90
10-1
100
101
102
10 10 1020log ( ) 20log 0.7 20log 1G j j
( ) 1G j j
-3
Example• Lead Compensator
Transfer Function:
Plot the straight line approximation of G(s)’s Bode diagram:
1( ) 1 1 15
G j jj
G s ss
( )
35 355
Phas
e (d
eg);
Mag
nitu
de (d
B)
-10
0
10
20
30
2
Frequency (rad/sec)
-90
0
90
10-1
100
101
10
35 1( ) 1 15 15
G j jj
10
10 10 1020log 7
120log ( ) 16.9 20log 1 20log 1 15
G j jj
17
1st Order Bode Plots Summary• 1st Order Poles
– Break Frequency
– Mag. Plot Approximation0 dB from DC to b and a straight line with 20 dB/decade slope after b
– Phase Plot Approximation0 deg from DC to . Between and 10b , a straight line from 0 deg to 90 deg (passing 45 deg at b). For frequency higher than 10b, straight line on 90 deg.
• 1st Order Zeros
– Break Frequency
– Mag. Plot Approximation0 dB from DC to b and a straight line with 20 dB/decade slope after b
– Phase Plot Approximation0 deg from DC to . Between and 10b , a straight line from 0 deg to 90 deg (passing 45 deg at b). For frequency higher than 10b , straight line on 90 deg.
11( ) , 0
1pG ss
1 rad/sb
110b
1( ) 1 , 0zG s s
1 rad/sb
110b
110b
Note: By looking at Bode plots you should be able to determine the relative order of the system, its break frequency, and DC (steady-state) gain. This process should also be reversible, i.e., given a transfer function, be able to plot a straight line approximation of Bode plots.
110 b
Bode Plots of IntegratorsStandard Form of Transfer Function
Frequency Response
10 0 10 10
10 10
1 120log ( ) 20log 20log
120log 20log
pG jj
G ssp01( )
1G jj
G j
G j
p
p
p
0
0
0
1
90
( )
( )
( )
Phas
e (d
eg);
Mag
nitu
de (d
B)
Frequency (rad/sec)
-60
-40
-20
0
20
0.1 1 10 100 1000-135
-90
-45
0
20dB/decade
Bode Plots of DifferentiatorsStandard Form of Transfer Function
Frequency Response
10 0 10
10
20log ( ) 20log
20logzG j j
G s sz0( )
G j j
G j
G j
z
z
z
0
0
0 90
( )
( )
( )
Frequency (rad/sec)
Phas
e (d
eg);
Mag
nitu
de (d
B)
0.1 1 10 100 1000
-20
0
20
40
60
0
45
90
135
20dB/decade
Example• Combination of Systems
Transfer Function:
Plot the straight line approximation of G(s)’s Bode plots:
35 1 11 15 151 1( ) 7 1 1 1
5
G s ss s
G j jj j
G s ss s
( )( )
35 355
Phas
e (d
eg);
Mag
nitu
de (d
B)
-10
0
10
20
30
2
Frequency (rad/sec)
-90
0
90
10-1
100
101
10
10
10 1020log 7
10 10
20log ( ) 16.9 20log 1
1 120log 20log 1 15
G j j
j j
1( ) 1 90 1 15
o
G j jj
SUM THEM
Example• Combination of Systems
Transfer Function:
Plot the straight line approximation of G(s)’s Bode plots:
2500 2500 1 1 1
1 150 5 50 5 1 150 5
1 1 1( ) 10 1 11 150 5
G ss s s ss s
G jjj j
G ss s s
( )( )
250055 2502
Phas
e (d
eg);
Mag
nitu
de (d
B)
-120
-80
-40
0
40
Frequency (rad/sec)
10-1 100
101
102
103
-270
-180
-90
0
10
10 1020log 10
10 10
120log ( ) 20 20log 1 150
1 120log 20log 1 15
G jj
j j
20dB/decade
40dB/decade
60dB/decade
Bode Plots of Complex PolesStandard Form of Transfer Function
Frequency Response
22
11 2 and ( )2 1
r n p rG j
2
2 22( ) , 1 0
2n
pn n
G ss s
2 2
2
121
p
nn
G jj
2 22 2 2
2 2
1
41
p
n n
G j
2
2 2
2
2
21
2= atan2 , 1
pnn
n n
G j j
n n n n n
Phas
e (d
eg);
Mag
nitu
de (d
B)
Frequency (rad/sec)
-80
-60
-40
-20
0
20
-180
-90
0
Peak (Resonant) Frequency and Magnitude for 1 0.707
2
40dB/decade
115 n
2 5 n
2nd Order System Frequency Response
decreases decreases
decreases
Pha
se (d
eg);
Mag
nitu
de (d
B)
Frequency (rad/sec)
-60
-40
-20
0
20
40
-180
-135
-90
-45
0
0.1n n 10n
increases
decreases
2nd Order System Frequency ResponseA Few Observations:• Three different characteristic frequencies:
– Natural Frequency (n)
– Damped Natural Frequency (d):
– Resonant (Peak) Frequency (r):
• When the damping ratio , there is no peak in the Bode magnitude plot. DO NOT confuse this with the condition for over-damped and under-damped systems: when the system is under-damped (has overshoot) and when the system is over-damped (no overshoot).
• As , r n and G(j)increases; also the phase transition from 0 deg to 180 deg becomes sharper.
21 2r n
21d n
r d n
Example• Second-Order System
Transfer Function:
Plot the straight line approximation of G(s)’s Bode diagram:
2
2 22( ) , 1 0
2n
pn n
G ss s
G ss s
( )
250010 25002
Phas
e (d
eg);
Mag
nitu
de (d
B)
-120
-80
-40
0
40
Frequency (rad/sec)
10 0 101 102 10310
4
-180
-90
02 2500, 50n n 102 10, 0.12n
n
21 2r n n 0.1
11 42.65 n 0.1
2 5 58.7n
Bode Plots of Complex ZerosStandard Form of Transfer Function
Frequency Response
115 n
22
2 2
2( ) , 1 0n nz
n
s sG s
2
2 2
21znn
G j j
2 2
22 2 2
2 2
1
41
z p
n n
G j G j
2
2
2 2
2atan2 , 1n n
z pG j G j
Frequency (rad/sec)
n n n n n
Phas
e (d
eg);
Mag
nitu
de (d
B)
-20
0
20
40
60
80
0
90
180
2 5 n
Bode Plots of Poles and Zeros Bode plots of zeros are the mirror images of the Bode plots of the identical poles w.r.t. the 0 dB line and the 0 deg line, respectively:
Let G sG s
G jG j
G j G j
G j G j
G j G j
pz
pz
p z
p z
p z
( )( )
( )( )
( ) ( )
log ( ) log ( )
( ) ( )
|
||
1
1
20 2010 10
-40
-20
0
20
40
-180
0
180
Frequency (rad/sec)
Pha
se (d
eg)
M
agni
tude
(dB
)
n nn
2nd Order Bode Diagram Summary• 2nd Order Complex Poles
– Break Frequency
– Mag. Plot Approximation0 dB from DC to n and a straight line with 40 dB/decade slope after n. Peak value occurs at:
– Phase Plot Approximation0 deg from DC to . Between and n , a straight line from 0 deg to 180 deg (passing 90 deg at n). For frequency higher than n , straight line on 180 deg.
2
2 22( ) , 1 0
2n
pn n
G ss s
rad/sb n
15 n 1
5 n
• 2nd Order Complex Zeros
– Break Frequency
– Mag. Plot Approximation0 dB from DC to n and a straight line with 40 dB/decade slope after n.
– Phase Plot Approximation0 deg from DC to . Between and n , a straight line from 0 deg to 180 deg (passing 90 deg at n). For frequency higher than n , straight line on 180 deg.
22
2 2
2( ) , 1 0n nz
n
s sG s
rad/sb n
15 n 1
5 n
2
2
1 2 1( )
2 1
r n
p r MAXG j
• Combination of SystemsTransfer Function:
Plot the straight line approximation of G(s)’s Bode diagram:
Example
2
2
2000 25 25 1 1 2500( ) 1200 2500 25 10 25001200
s sG ss s ss
G s s ss s s s
( ) ( )( )( )
2000 25200 10 2500
2
2
10
2
10 10 1020log 0.1
10 10 2
25 120log ( ) 20 20log 20log25
1 250020log 20log1 2500 101200
jG j
j
jj
2
2
2590
25
1 25001 25001
200
oj
G j
jj
-80
-60
-40
-20
0
20
40
Mag
nitu
de (d
B)
10 -1 100 101102 10 3
-270
-180
-90
0
90
180
Frequency (rad/sec)
Pha
se (d
eg)
-80
-60
-40
-20
0
20
40
Mag
nitu
de (d
B)
10 -1 100 101102 10 3
-270
-180
-90
0
90
180
Frequency (rad/sec)
Pha
se (d
eg)
