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FIG 1.18. The gain of a first-order lag is governed by the asymptotes.
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This observation is of particular significance for two reasons: 1. Load variations are normally introduced by turning the valve in the outflow line, thus changing k. 2. In most processes, including this one, k would not be a constant even if the load were fixed, because the relationship between input and output is not linear. In a real liquid-level process, fo=Cdh where C is the flow coefficient of the valve opening. k=dh 1 f. 7=Ic: Consequently k = C/d& so even if C remains fixed, k still varies with level. Again, fortunately, this does not affect the dynamic gain. The time constant TV, of such a process is not a constant, but varies with ik. But this is of little consequence, because the dynamic gain is constant. The ratio V/F must be recognized as the determining factor. It will appear again and again in different processes, with different forms of variables, but it is the fundamental time constant of any flowing system. Its units are those of time. For example, gal/(gal/min) = minutes. The phase angle between input and output of a first-order lag is the negative of +D in the vector diagram of Fig. 1.17. As r0 approaches zero, 4~ approaches +90 , and therefore the true phase lag approaches 90 . In the steady state, however, the vertical vector is zero, hence the phase angle is zero. The phase of a first-order lag is mathematically described as +1 = -~~n~lZ$!$! 0 Substituting for V/Fk, 41 = -tan-l 2777--1 To (1.24) Then,
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Since the phase la,g can never exceed 90 , the first-order lag cannot oscillate under proportional control. This was also true of the integrating process. Therefore we can make a general statement that a singlecapacity process can be controlled without oscillation at zero proportional band. This means that the valve will be driven fully open or fully closed on an infinitesimal error, so that the loop is operating at top speed all the time. Since the proportional band is zero, no offset can develop. A single-capacity process must therefore be categorized as the easiest to control.
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FIG 1.19. This is how a singlecapacity process would react to zero proportional band.
Time
Figure 1.19 illustrates the set-point response of a single-capacity process to zero proportional band. As soon as the set point is changed, the valve will open wide, delivering maximum inflow. The level will rise as rapidly as possible, which is a function of both k and the present value of level. If no control were provided, the measurement would follow the projected path. But when the new set point is reached, the inflow will be reduced instantaneously to a value equal to the outflow. This assumes that all elements in the loop, excepting the tank, are capable of instantaneous response. If this is not so, the process is not single-capacity. Examples of pure single-capacity processes are rare. The most common one is a tank being filled through a valve which is rigidly coupled to a float. The level is prevented from overshooting the set point because the rigid coupling eliminates any delay in feedback action. Whereas the non-self-regulating process cycled uniformly with an integrating controller, the self-regulating will not. The phase shift of the self-regulating process only reaches -90 at a period of zero. As a result, the loop could only oscillate at zero period, where the gain of both process and controller are zero. The loop cannot, therefore, sustain oscillations.
A Two-capacity Process
Having established the ease with which a single-capacity process may be controlled, the complications involved in adding a second capacity may be evaluated. Since each capacity contributes a phase lag approaching 90 , the total phase lag in the loop can only approach 180 . As a result, the loop can oscillate only at zero period. This is exactly like a first-order lag with an integrating controller. Adding another lag anywhere in the Ioop will change the previous level process to two-capacity, as shown in Fig. 1.20. A chamber is attached to the tank; although we wish to control tank level, chamber level is measured, which lags behind tank level. The time const ant of the chamber is its volume divided by the maximum rate at which liquid can enter. This time constant will be designated TV. Control of a two-capacity process is easiest to illustrate if one of the capacities is non-self-regulating.
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