Modern Control Engineering

144 Chapter 4 / Mathematical Modeling of Fluid Systems and Thermal Systems

A–4–5. Draw a block diagram of the pneumatic controller shown in Figure 4–30. Then derive the transfer

function of this controller. Assume that Assume also that the two bellows are identical.

If the resistance Rdis removed (replaced by the line-sized tubing), what control action do we get?

If the resistance Riis removed (replaced by the line-sized tubing), what control action do we get?

Solution.Let us assume that when e=0the nozzle–flapper distance is equal to and the con-

trol pressure is equal to In the present analysis, we shall assume small deviations from the

respective reference values as follows:

small error signal

small change in the nozzle–flapper distance

small change in the control pressure

small pressure change in bellows I due to small change in the control pressure

small pressure change in bellows II due to small change in the control pressure

small displacement at the lower end of the flapper

In this controller, is transmitted to bellows I through the resistance Rd. Similarly, is trans-

mitted to bellows II through the series of resistances RdandRi. The relationship between and is

where derivative time. Similarly,pIIandpIare related by the transfer function

where integral time. The force-balance equation for the two bellows is

whereksis the stiffness of the two connected bellows and Ais the cross-sectional area of the

bellows. The relationship among the variables e, x, and yis

The relationship between and xis

pc=Kx (K>0)

pc

x=

b

a+b

e-

a

a+b

y

ApI-pIIBA=ks y

Ti=RiC=

PII(s)

PI(s)

=

1

Ri Cs+ 1

=

1

Ti s+ 1

Td=RdC=

PI(s)

Pc(s)

=

1

Rd Cs+ 1

=

1

Td s+ 1

pI pc

pc pc

y=

pII=

pI=

pc=

x=

e =

P

–

c^.

X

–

RdRi.

e

a

b

CC

X+x

Pc+pI

Pc+pII

Ps

III

Ri

Rd

Pc+pc

y

Figure 4–30

Schematic diagram

of a pneumatic

controller.

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