стр. 52 |

If there were two roots on the imaginary axis (they come in pairs of complex

conjugates) and all other roots were on the left side of the imaginary axis, the loop

would be oscillating with a constant amplitude. The value of KC that generates this

case is KCU.

There are several ways to proceed from Eq. (6-2.4), and control textbooks [1] are

delighted to show you so. In this book we are interested only in the п¬Ѓnal answer,

that is, KCU, not in the mathematics. For this case, which we call the base case, the

KCU value and the period at which the loop oscillates, which in Chapter 2 we called

the ultimate period TU are

%CO

KCU = 23.8 and TU = 28.7 sec

%TO

Let us now learn what happens to these values of KCU and TU as terms in the loop

change.

137

CONTROL LOOP STABILITY

imaginary

Unstable

region

Stable

region

real

Stable

region

Unstable

region

Figure 6-2.1 Roots of the characteristic equation.

6-2.1 Effect of Gains

Let us assume that a new transmitter is installed with a range of 75 to 125В°F. This

means that the transmitter gain becomes

(100 - 0)%TO 100 %TO %TO

= =2

(125 - 75)в€ћF 50в€ћF в€ћF

Thus, the transfer function of the transmitter becomes

2.0

H=

10 s + 1

and the characteristic equation

900 s3 + 420 s 2 + 43s + (1 + 1.6KC ) = 0

The new ultimate gain and ultimate period are

%CO

KCU = 11.9 and TU = 28.7 sec

%TO

Thus, a change in any gain in the вЂњloopвЂќ (in this case we changed the transmitter

gain, but any other gain change would have the same effect) will affect KCU. Fur-

thermore, we can generalize by saying that if any gain in the loop is reduced, KCU

increases. The reciprocal is also true: If any gain in the loop increases, KCU reduces.

The change in gains does not affect the ultimate period.

138 BLOCK DIAGRAMS AND STABILITY

6-2.2 Effect of Time Constants

Let us now assume that a new faster transmitter (with the same original range of

50 to 150В°F) is installed. The time constant of this new transmitter is 5 sec. Thus the

transfer function becomes

1.0

H=

5s + 1

and the characteristic equation

450 s3 + 255s 2 + 38 s + (1 + 0.8 KC ) = 0

The new ultimate gain and ultimate period are

%CO

KCU = 25.7 and TU = 21.6 sec

%TO

This change in transmitter time constant has affected KCU and TU. By reducing the

transmitter time constant, KCU has increased, thus permitting a higher gain before

reaching instability, and TU has been reduced, thus resulting in a faster loop.

The effect of a change in any time constant cannot be generalized as we did with

a change in gain. Again install the original transmitter, and consider now that a

change in design results in a faster exchanger; its new transfer function is

50

G1 =

20 s + 1

and the characteristic equation

450 s3 + 255s 2 + 38 s + (1 + 0.8 KC ) = 0

The new ultimate gain and ultimate period are

%CO

KCU = 18.7 and TU = 26.8 sec

%TO

The effect of a reduction in the exchanger time constant is completely different from

that obtained when the transmitter time constant was changed. In this case, when

the time constant was reduced, KCU also reduced. It is difп¬Ѓcult to generalize;

however, we can say that by reducing the smaller (nondominant) time constants,

KCU increases, whereas reducing the larger (dominant) time constants, KCU

decreases. Usually, the smaller time constants are those of the instrumentation such

as transmitters and valves.

6-2.3 Effect of Dead Time

Back again to the original system, but assume now that the transmitter is relocated

to another location farther from the exchanger, as shown in Fig. 6-2.2. This location

139

CONTROL LOOP STABILITY

SP

TC

22

Steam

Process TT

TT

22

22

fluid

T (t ) original

Ti ( t )

стр. 52 |