Example Problems and Solutions 145
From the equations just derived, a block diagram of the controller can be drawn, as shown in
Figure 4–31(a). Simplification of this block diagram results in Figure 4–31(b).
The transfer function between Pc(s)andE(s)is
For a practical controller, under normal operation is
very much greater than unity and Therefore, the transfer function can be simplified as
follows:
where
Thus the controller shown in Figure 4–30 is a proportional-plus-integral-plus-derivative one.
If the resistance Rdis removed, or Rd=0, the action becomes that of a proportional-plus-
integral controller.
Kp=
bks
aA
Kpa 1 +
1
Ti s
+Td sb
=
bks
aA
a
Ti+Td
Ti
+
1
Ti s
+Td sb
Pc(s)
E(s)
bksATi s+ 1 BATd s+ 1 B
aATi s
TiTd.
@KaATi sC(a+b)ksATi s+ 1 BATd s+ 1 BD@
Pc(s)
E(s)
=
b
a+b
K
1 +K
a
a+b
A
ks
a
Ti s
Ti s+ 1
ba
1
Td s+ 1
b + – + – + –
E(s) X(s) Pc(s)
K
a
a+b
b
a+b
A
ks
PI(s)
PII(s)
1
Td s + 1
1
Ti s + 1
(a)
(b)
ab+b K
E(s) X(s) Pc(s)
aATis
(a+b)ks(Tis+ 1) (Tds+ 1)
Figure 4–31
(a) Block diagram of
the pneumatic
controller shown in
Figure 4–30;
(b) simplified block
diagram.