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Im
Unit circle
1
GM
(–1, j0) |
Re |
PM
$ |
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$ |
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$ $ |
Controller |
Plant |
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R(s) |
C(s) |
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+ |
4 |
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K1 |
s(s2 + 2s + 4) |
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– |
$ # #
$
$ # #
$
# #
$ #
$
#
$
#
$
$
$ |
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$
$
$ $
$
$ $
$
$
$
# |
$ |
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# # |
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# # # $ |
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Unit circle on new scale
New scale |
–1 |
–1 |
ω = 2.0 |
|
–0.6 |
–0.5 |
–0.4 |
–0.3 |
–0.2 |
Original scale
ω = 1.5
ω = 1.0
ω = 0.5
0.1 |
Im |
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ω = 2.5 |
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–0.1 |
0 |
0 |
Re 0.1 |
|
° |
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PM=50 |
–0.1 |
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–0.2
–0.3
–0.4
–0.5
–0.6
$ |
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$ |
# |
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$ |
# |
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$
G(s)H(s) = K
s(1+Ts) Im
Im
G(s)H(s) = K
(1+Ts)
Re
Re
(a) First-order type zero system |
(b) Second-order type one system |
Im
K |
Re |
G(s)H(s) = s2(1+Ts) |
(c)Third-order type two system
$ |
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# |
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$ |
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$ |
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$ |
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$ |
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#
$
$ |
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# # |
$ |
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# # |
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$
$ |
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# # |
30 |
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20 |
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System just unstable |
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10 |
(ii) |
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0 |
GM = 6 dB |
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|
(dB) |
(i) |
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–10 |
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Gain |
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–20 |
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–30 |
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–40 |
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–50 |
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10–1 |
100 |
101 |
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Frequency (rad/s) |
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(a) Bode Gain
–50
Phase (degrees)
–100
–150 |
PM = 50° |
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–180 |
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–200 |
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–250 |
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–300 |
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100 |
101 |
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10–1 |
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Frequency (rad/s)
(b) Bode Phase