1. Frequency Response Method
Frequency response analysis and design methods
consider response to sinusoids methods rather than steps
and ramps.
Frequency response is readily determined experimentally
in sinusoidal testing.
Frequency response is readily obtained from the system
transfer function (s = jω) , where ω is the input frequency).
Link between frequency and time domains is indirect.
Design criteria help obtain good transient time response.
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2. The frequency response of a system is steady-state
response of the system to a sinusoidal input signal.
For linear dynamic systems, the steady state output of
the system is a sinusoid with the same frequency as the
input, but differing in amplitude and phase angle (there is a
phase shift in the output).
The frequency response can be computed for a single
frequency, and can be plotted for a single frequency, and
can be plotted for a range of frequencies.
Frequency Response Method
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6. Learning Outcomes
At the end of this lecture, students should be
able to:
Sketch the bode diagram for a given system
Identify the system’s stability based on the
determined Gain Margin & Phase Margin
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7. There are two types of Bode plots:
The Bode straight-line approximation to the log-
magnitude (LM) plot, LM versus w (with w on
a log scale)
The Bode straight-line approximation to the
phase plot, (w) versus w (with w on a log
scale)
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9. Five types of terms in H(jw)
1) K (a constant)
2) (a zero) or (a pole)
3) jw (a zero) or 1/jw (a pole)
4) Any of the terms raised to a positive integer power.
5) Complex zero/poles
1
w
1 j
w
2
2 2 2
0 0
2 2
0 0
2 w w 1
1 j - (a complex zero) or (a complex pole)
2 w w
w w
1 j -
w w
1
1
w
1 j
w
2
1
w
For example, 1 j (a double zero)
w
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10. 1. Constant term in H(jw)
If H(jw) = K = K/0
Then LM = 20log(K) and (w) = 0 , so the LM and phase responses are
LM (dB)
w
0
w
0o
(w)
1
20log(K)
10 100
Summary: A constant in H(jw):
• Adds a constant value to the LM graph (shifts the entire graph up or down)
• Has no effect on the phase
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12. 2. A) 1 + jw/w1 (a zero): The straight-line
approximations are:
2
-1
1 1 1
2
-1
1 1
w w w
If H(jw) 1 j 1 tan
w w w
w w
Then LM 20log 1 and (w) tan
w w
To determine the LM and phase responses, consider 3 ranges
for w:
1) w << w1
2) w >> w1
3) w = w1 12
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13. So the Bode approximations (LM and phase) for
1 + jw/w1 are shown below.
Summary: A 1 + jw/w1 (zero) term in H(jw):
• Causes an upward break at w = w1 in the LM plot. There is a 0dB
effect before the break and a slope of +20dB/dec or +6dB/oct after
the break.
• Adds 90 to the phase plot over a 2 decade range beginning a
decade before w1 and ending a decade after w1 .
LM
w
0dB
= +20dB/dec
w
90o
(w)
= + 6dB/oct
20dB
w1
slope
0o
10w1
45o
w1 10w1
0.1w1
= +45 deg/dec
slope
(for 2 decades)
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16. 2) B (a pole): The straight-line approximations
are:
To determine the LM and phase responses, consider 3 ranges for w:
1) w << w1
2) w >> w1
3) w = w1
-1
2 2
1
-1
1
1 1 1
-1
2
1
1
1 1 0 1 w
If H(jw) tan
w w
1 j w w w
1 tan 1
w
w w w
1 w
Then LM 20log and (w) -tan
w
w
1
w
1
1
w
1 j
w
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17. So the Bode approximations (LM and phase) for
are shown below. 1
1
w
1 j
w
LM
w
0dB
= -20dB/dec
w
-90o
(w)
= - 6dB/oct
-20dB
w1
slope 0o
10w1
-45o
w1 10w1
0.1w1
= -45 deg/dec
slope
(for 2 decades)
Summary: A 1 + jw/w1 (zero) term in H(jw):
• Causes an downward break at w = w1 in the LM plot. There
is a 0dB effect before the break and a slope of -20dB/dec or -
6dB/oct after the break.
• Adds -90 to the phase plot over a 2 decade range beginning
a decade before w1 and ending a decade after w1 .
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24. This transfer function has 4 forms:-
i. Factor Constant, K=10
ii. Factor
iii. Factor
iv. Factor
j
1
2
1
1
j
10
1
1
j
Sketch for magnitude and phase!
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26. Stability
)
(
)
(
log
20 M
GM
G
M LM
j
G
G
M
j
G
o
M
)
(
180
Gain Margin
Phase Margin
The system is stable if BOTH
0
0
M
M
G
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27. Example : Determine the system’s stability
GM
ΦM
dB
GM 5
.
29
)
5
.
29
(
o
o
o
M 63
)
117
(
180
Since
0
0
M
M
G
STABLE
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28. Example : Determine the system’s stability
GM
ΦM
dB
GM 8
.
20
)
8
.
20
(
o
o
o
M 73
)
253
(
180
Since
0
0
M
M
G
UN-STABLE
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29. Gain Cross Over frequency
A gain cross over frequency is any frequency
at which the amplitude ratio for GH = 0.
The phase margin is the additional negative
phase shift necessary to make the phase
shift of GH equals to +,- 180. at a gain cross
over frequency.
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30. Phase Cross Over Frequency
A Phase Cross Over Frequency is any
frequency at which the phase shift of GH is +-
180.
The gain margin of the feed back system is
the additional dB amplitude necessary to
make the amplitude of GH unity at a phase
cross over frequency.
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31. Possibilities
If the phase shift crosses +- 180, at more
than one frequency, the gain margin is the
smallest value of the two possibilities.
If there is no phase cross over frequency, the
gain margin can said to be infinite.
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