ECE 460: Homework 8
Instructor: Stan Baek
Due: Wednesday, March 30, at 2:10pm
Note 1.
Your solutions should be neat and legible; always show your work.
Problem 1.
( 40 points) Sketch the
asymptotes
of the Bode plot magnitude and phase for
each of the following transfer functions using the semilog graph attached at the end of this
homework assignment. Find the gain margin, the phase margin, the gain crossover
frequency, and the phase crossover frequency of the systems.
(a)
G
(
s
) =
(
s
+ 6)
(
s
+ 0
.
5)(
s
+ 1)(
s
+ 20)
Solution:
G
(
s
)
=
6(
s/
6 + 1)
0
.
5(
s/
0
.
5 + 1)(
s
+ 1)20(
s/
20 + 1)
=
3
5
(
s/
6 + 1)
(
s/
0
.
5 + 1)(
s
+ 1)(
s/
20 + 1)
20 log(3
/
5) =

4
.
4 dB. There is no gain crossover frequency. Although the asymptotes of
the phase plot reaches 180
◦
, the actual plot never reaches 180
◦
. Therefore, there is no
phase crossover frequency. Since both gain and phase crossover frequencies do not exist,
the gain margin and phase margin are both the infinity. Compared with the Bode plot
generated by MATLAB shown in Figure 1a, the asymptotes are good approximations.
(b)
G
(
s
) =
(
s
+ 20)(
s
+ 100)
(
s
+ 1)(
s
2
+ 6
s
+ 25)
Solution:
G
(
s
)
=
20(
s/
20 + 1)100(
s/
100 + 1)
(
s
+ 1)25((
s/
5)
2
+ 6
/
5(
s/
5) + 1)
=
80(
s/
20 + 1)(
s/
100 + 1)
(
s
+ 1)((
s/
5)
2
+ 6
/
5(
s/
5) + 1)
20 log(80) = 38 dB,
ζ
= 3
/
5 = 0
.
6. The gain crossover frequency is 12 rad/s and the phase
crossover frequency is 10 rad/s. The gain margin is approximately 8 dB and the phase
margin is 0
◦
according to the asymptotes of the Bode plot. Compared with the Bode plot
generated by MATLAB shown in Figure 1b, the phase plot is quite different. The main
reason is that we have not incorporated the effect of
ζ
shown in Figure 10.17 on page 547
of our textbook.
1
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(c)
G
(
s
) =
(
s
+ 4)
s
(
s
+ 2
s
2
+ 10)
Solution:
G
(
s
)
=
(
s
+ 4)
s
(2
s
2
+
s
+ 10)
=
4(
s/
4 + 1)
2
s
(
s
2
+
s/
2 + 5)
=
2(
s/
4 + 1)
5
s
((
s/
√
5)
2
+ 1
/
(2
√
5)(
s/
√
5) + 1)
=
2
5
(
s/
4 + 1)
s
((
s/
√
5)
2
+ 1
/
(2
√
5)(
s/
√
5) + 1)
20 log(2
/
5) =

8 dB,
ζ
= 1
/
(4
√
5) = 0
.
11. The gain crossover frequency is 0.4 rad/s and
the phase crossover frequency is 7 rad/s. The gain margin is approximately 36 dB and the
phase margin is 55
◦
according to the asymptotes of the Bode plot. Compared with the
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 Winter '14

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