84 Chapter 3 / Mathematical Modeling of Mechanical Systems and Electrical Systemswe have
(3–37)
Notice that the second operational-amplifier circuit acts as a sign inverter as well as a
gain adjuster.
When a PID controller is expressed as
Kpis called the proportional gain, is called the integral time, and is called the
derivative time. From Equation (3–37) we obtain the proportional gain Kp,integral time
and derivative time to be
When a PID controller is expressed as
Kpis called the proportional gain,Kiis called the integral gain, and Kdis called the
derivative gain. For this controller
Table 3–1 shows a list of operational-amplifier circuits that may be used as con-
trollers or compensators.
Kd=
R 4 R 2 C 1
R 3
Ki=
R 4
R 3 R 1 C 2
Kp=
R 4 AR 1 C 1 +R 2 C 2 B
R 3 R 1 C 2
Eo(s)
Ei(s)
=Kp+
Ki
s
+Kd s
Td=
R 1 C 1 R 2 C 2
R 1 C 1 +R 2 C 2
Ti=
1
R 1 C 1 +R 2 C 2
Kp=
R 4 AR 1 C 1 +R 2 C 2 B
R 3 R 1 C 2
Ti , Td
Ti Td
Eo(s)
Ei(s)
=Kpa 1 +
Ti
s
+Td sb
=
R 4 AR 1 C 1 + R 2 C 2 B
R 3 R 1 C 2
c 1 +
1
AR 1 C 1 + R 2 C 2 Bs
+
R 1 C 1 R 2 C 2
R 1 C 1 + R 2 C 2
sd
=
R 4 R 2
R 3 R 1
aR 1 C 1 +R 2 C 2
R 2 C 2
+
1
R 2 C 2 s
+R 1 C 1 sb
Eo(s)
Ei(s)
=
Eo(s)
E(s)
E(s)
Ei(s)
=
R 4 R 2
R 3 R 1
AR 1 C 1 s+ 1 BAR 2 C 2 s+ 1 B
R 2 C 2 s
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