ECE 421, Spring 2006
Solutions to HW Assignment #11
Problem #B85
The Bode magnitude and phase plots are shown in Fig. 1. The transfer function has one poles at the
origin
(
N
=1)
,
so at low frequencies the magnitude plot has a slope of
−
20
db/decade and the phase plot
has a value of
−
90
◦
.
There are
n
=3
poles and
m
=2
zeros, so
n
−
m
=1
making the magnitude plot have
as
lopeo
f
−
20
db/decade and the phase plot have a value of
−
90
◦
at high frequencies.
The peaks in the magnitude curve and rapid changes in the phase curve are due to the small damping
ratios of the complex conjugate poles and zeros. The damping ratio and natural frequency of the zeros are
0.2 and 1 rad/sec, and for the poles they are 0.1333 and 3 rad/sec, respectively. The small damping ratio
of the zeros causes the negative peak in the magnitude and the sharp rise in the phase curve. The e
f
ects of
the poles are in the opposite directions.
10
2
10
1
10
0
10
1
10
2
10
3
100
80
60
40
20
0
20
40
60
Frequency (r/s)
M
a
g
n
i
t
u
d
e
(
b
)
&
P
h
s
Bode Plots for Prob. B85
Figure 1: Bode plots for Problem B85.
1
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View Full DocumentProblem #B87
The Bode magnitude and phase plots are shown in Fig. 2. The transfer function has no poles at the
origin
(
N
=0)
,
so at low frequencies the magnitude plot has a slope of
0
db/decade and the phase plot has
a value of
0
◦
.
There are
n
=3
poles and
m
=1
zeros, so
n
−
m
=2
mak
ingthemagn
itudep
lothaveas
lope
of
−
40
db/decade at high frequencies. If all the poles and zeros were in the lefthalf plane, then the phase
plot would have a value of
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 Spring '08
 Cook,G
 Nyquist plot, encirclements

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