Biomedical Engineering Reference
In-Depth Information
The parameters of the second-order system can be derived from the Bode plot. Knowing
the general form of a second-order system:
kω
n
G(s)
=
(11.5)
s
2
+
2
ζω
n
s
+
ω
n
where:
k
is the system gain
ω
n
is the system's natural frequency
ζ
is the system's damping ratio.
The DC gain (
k
) of a system can be calculated from the magnitude of the Bode plot
when
s
=0 as
k
=
10
M(
0
)
20
. The natural frequency of a second-order system occurs when
the phase of the response is
−
90
◦
relative to the phase of the input (
ω
n
=
ω
−
90
◦
). Where
(
ω
−
90
◦
) is the frequency at which the phase plot is at
−
90
◦
.
The damping ratio of a system can be found with the DC gain and the magnitude of
the Bode plot when the phase plot is
−
90
◦
:
k/
2
10
M
(
−
90
◦
)
ζ
=
×
20
DC gain
:
k
=
10
M(
0
)
⇒
k
=
1
20
(11.6)
M
(
0
)
=
0
ω
n
=
ω
−
90
◦
ω
−
90
◦
=
⇒
ω
n
=
9.3
(11.7)
9.3
⎨
K/
2
×
10
M
−
90
◦
20
ζ
=
1
⇒
ζ
=
=
1.26
(11.8)
⎩
10
−
0.4
×
2
M
−
90
◦
=
20 log
(
0.4
)
=−
7.96 dB
Therefore the transfer function of the system is:
86
86
G(s)
=
86
=
(11.9)
s
2
+
2
(
1.26
)(
9.3
) s
+
s
2
+
23.4
s
+
86
Please note that
G
(
s
) is the transfer function of the closed-loop system. To find the open
loop transfer function,
H
(
s
), consider Figure 11.12.
a
(
s
)
b
(
s
)
+
−
G
(
s
)
Kp
H
(
s
) =
(a)
(b)
Figure 11.12
(a,b) Closed-loop system and its equivalent transfer function
