Frequency Response
Q
UESTION
5
For the closed–loop system shown in Figure 5.1 with
(
29
(
29
(
29
2
0.4
4
8
0.4
s
s
G s
s
s

+
=
+
+
, complete the
following:
a.
Sketch the Bode magnitude and phase plots by hand. Sketch the plots for each term and
then sketch the total plots. Graph the Bode magnitude and phase plots using Matlab.
b.
Compute the gain margin and the frequency at which it occurs. Determine the equations
required to solve for the phase margin and the frequency at which it occurs.
c.
Using the Bode plots sketched in part (a), estimate the gain and phase margins and the
frequencies at which these occur.
G(s)
K
U(s)
R(s)
+

Y(s)
E(s)
Figure 5.1
2
0.4
4
s
s

+
term
The magnitude is
M
= 20log
10
(4) = 12.04 dB for 0 <
ω
≤ 2 rad/s.
The magnitude is
M
= 12.04 + 40 dB/dec for
ω
> 2 rad/s.
The phase is
φ
= 360° for
ω
< 0.2 rad/s.
The phase is 360° >
φ
> 180° at a slope of –90°/dec for 0.2 ≤
ω
≤ 20 rad/s.
The phase is
φ
= 180° for
ω
> 20 rad/s.
(
29
1
8
s
+
term
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Frequency Response
The magnitude is
M
= –20log
10
(8) = –18.06 dB for 0 <
ω
≤ 8 rad/s.
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 Spring '11
 LANDERS
 Signal Processing, Decibel, Electronics terms, rad/s

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