16.06 Principles of Automatic Control
Lecture 18
Bode Plot Construction (continued)
Note that phase of sα term is
=pjωqα “=j α “ α=j
“α ¨ 90˝
To plot 1 ` s{a term, note that
|1 ` jω{a| “p1 ` ω 2 {a2 q1{2
$
’1,
ω!a
&
“ ω{a, ω " a
’?
%
2, ω “ a
Example:
Kpsq “ 1 ` s{20
1
dB
10
2
40
|K|
10
1
20
2 ~ 3dB
+1, 20 dB/dec
0
10 0
10
10
1
10
2
ω 10
3
What about the phase?
=1 ` jω{a “ tan ´1 ω{a
K
11 °
90°
45°
0 0
10
ω
4
a/5
10
1
20
a
10
2
10
3
5a
That is, the phase varies by 90˝ over the frequency range p a5 , 5aq.
Some people find it easier to draw the construction lines with breakpoints at a{10, 10a.
• Easier to draw
• Less phase error
2
• Middle segment is not technically an asymptote anyway
K
6°
90°
45°
0 0
10
For K “
10
1
,
1`s{a
ω
10
2
1
20
10
2
10
200
3
the above magnitude and phase plots are flipped about |K| “ 1 or =K “ 0˝ .
a
0
0
K
|K|
-1
10
−45
−1
−90
10
ω
−2
10
0
10
1
10
2
10
ω
3
10
0
10
1
10
2
Bode Rules:
Rule 1: Manipulate the transfer function into Bode form.
Rule 2: Determine α for K0 sα term. Plot the low-frequency asymptote with slope α(or 20α
dB/dec) through the point ω “ 1, 1 ¨ 1 “ K0 .
3
10
3
Rule 3: Complete the composite magnitude asymptotes. At each break point, change the
slope by ˘1, or ˘ 2, as appropriate.
Rule 4: Sketch in approximate magnitude curve. (see FPE for more details).
Rule 5: Plot the low frequency asymptote of the phase curve pφ “ α ¨ 90˝ q.
Rule 6: The approximate phase is found by changing the phase by ˘90˝ or ˘180˝ at each
breakpoint.
Rule 7: Locate the asymptotes for each phase curve, at break points 1{5 and 5 times (or
1{10 and 10 times) the frequency of the magnitude break point.
Rule 8: Graphically add the asymptotes, and draw the approximate phase curve.
Example:
2000ps ` 0.5q
sps ` 10qps ` 50q
2p1 ` s{0.5q
“
sp1 ` s{10qp1 ` s{50q
KGpsq “
The magnitude break points are
0 0.5
x 10
x 50
The phase break points are
0 0.05, 5
x 1, 100
x 5, 500
4
103
Choose appropriate scales on log-log paper
Magnitude
102
101
1
0.1
0.01
0.01
0.1
1
10
100
1000
100
1000
Frequency, ω (rad/sec)
103
Plot low-frequency asymptote
Magnitude
102
101
K0=2, α=-1
1
0.1
0.01
0.01
0.1
1
10
Frequency, ω (rad/sec)
5
103
Complete the composite magnitude asymptotes
Magnitude
102
10
1
slope=-1
K0=2, α=-1
slope=0
slope=-1
50
10
1
0.5
10
slope=-2
0.1
0.01
0.1
1
10
100
1000
Frequency, ω (rad/sec)
103
Sketch the appropriate magnitude curve
Magnitude
102
101
slope=-1
K0=2, α=-1
slope=0
slope=-1
50
1
0.5
10
slope=-2
0.1
0.01
0.1
1
10
Frequency, ω (rad/sec)
6
100
1000
0˚
approx.
-30˚
Phase
-60˚
smoothed
version
-90˚
-120˚
-150˚
-180˚
0.01
0.1
1
10
100
1000
Frequency, ω (rad/sec)
0˚
5
-30˚
1
Phase
-60˚
- 45˚/dec
-90˚
- 90˚/dec
0.05
-120˚
100
-150˚
500
500
500
-180˚
0.01
- 45˚/dec
0.1
0.5
1
10
Frequency, ω (rad/sec)
7
50 100
1000
0˚
5
-30˚
Phase
-60˚
-90˚
45˚/dec
-90˚/dec
0.05
-120˚
100
-150˚
- 45˚/dec
500
-180˚
0.01
0.1
0.5 1
10
Frequency, ω (rad/sec)
8
50 100
1000
MIT OpenCourseWare
http://ocw.mit.edu
16.06 Principles of Automatic Control
Fall 2012
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.