Dynamic gain

The stiffness characteristic of the positioner/actuator varies with frequency. Figure 8-75Z indicates the stiffness of the positioner/actu-ator is high at low frequencies and is directly related to the locked-stem pressure gain provided by the positioner. As frequency increases, a dip in the stiffness curve results Trom dynamic gain attenuation in the pneumatic amplifiers in the positioner. The value at the bottom of the dip is the sum of the mechanical stiffness of the spring in the actu-... 

A variety of devices have been designed by using transparent electro-optical ceramic materials, including variable optical attenuators (VOA), polarization controllers (PC), sinusoidal filters, dynamic gain flattening filters, tunable optical filters, and (2-switches, which have been described in detail in Ref. [129]. A brief description is given for some devices as follows. 

Notice that this dynamic-gain asymptote does not contain k. In fa it is identical to the gain of the non-self-regulatlng process. Althou the steady-state gain can be changed simply by turning the valve at tl bottom of the tank, this does not affect the dynamic gain. 

The time constant ti, of such a process is not a constant, but varies with h. 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. 

State gain of l/k. But it was pointed out that this steady-state gain had no influenee whatsoever on the dynamic gain of the process. The... 

The problem has been identified as variable dynamic gain. It is a common problem, not often recognized, still less often anticipated. It occurs in processes where the values of the secondary dynamic elements, principally dead time, vary with flow. These variations cause proportionate changes in the period of the loop, which affects the dynamic gain of the principal capacity. 

Consider the heat exchanger as a single-capacity plus dead-time process where the dynamic gain of the capacity is expressed as... 

Do not test for steady-state gain. In Chap. 1 it was pointed out that the steady-state gain of a single-capacity liquid-level process is not constant. It varies with both flow and level. Yet the dynamic gain is... 

Do not test for time constants. There arc several methods available for finding the time constants in a linear system. But, as in the single-capacity level process, the time constant may vary with flow without affecting dynamic gain. The likelihood of a nonlinear element in a... 

If the time constant of this capacity is known, its dynamic gain ( i at can be calculated. Combining this with known values of transmitter and valve gain, together with the controller proportional band, yields... 

The composition of a ])roduct leaving a 50-tray distillation colunin e.xhihits a dead time of lO min following a change in reflux flow, ruder propor-tional-plus-resi t control, estimate (a) the i)eriod of oscillation, (h) the reset time, ( ) the dynamic gain of the process. 

The purpose of the analysis is not to show how an analysis should be made, but rather to explain why a flow loop behaves the way it does. Because many dynamic elements are present, all of the same order of magnitude, dynamic gain is high. The proportional band of a flow controller is rarely less than 100 percent, making reset mandatory. Where the valve and transmitter are in the same line, the period of oscillation will invariably fall within 1 to 10 sec. The presence of noise precludes the use of derivative. As long as these factors are appreciated, there is little reason to spend time analyzing flow loops. 

The dynamic gain of the process is principally that of the primary time constant ... 

Since the nominal flow f has already been identified as a constant, process gain is also constant. (This is another illustration of the case where process steady-state gain varies with flow, but the time constant does too, so dynamic gain is invariant. Steady-state gain, as calculated above, is only meaningful at the rated flow F.)... 

A linear control loop is identified by Its constant dynamic gain, which applies the same damping to disturbances of all magnitudes. This statement, holds true whether the loop consists entirely of linear elements or includes a nonlinear function intentionally Introduced to compensate another function naturally occurring in the process. 

Exact compensation may be obtained by programming the settings of the controller as functions of flow. Because the period of the loop varies directly with dead time, derivative and reset time ou t to vary inversely with flow. And since process dynamic gain varies inversely with flow, the proportional band should too. Knowing this, it is possible to write a flow-adapted control algorithm ... 

Since the second method does not test the process, the current value of loop gain is unknown until a disturbance Identifies It. Identification must then be carried out, and parameter adjustment made carefully to prevent overcorrection. Identification consists principally of factoring the response curve Into high- and low-frequency components whose ratio represents the dynamic gain of the closed loop. The load-response curve shown in Fig. 6.21 is so separated. 

Where the dynamic adaptive system controlled the dynamic gain of a loop, its counterpart seeks a constant steady-state process gain. This implies, of course, that the steady-state process gain is variable and that one particular value is most desirable. 

The terms K , g , Kq and g, represent the steady-state and dynamic gain terms for the manipulated variable and load. The feedforward control system is to be designed to solve for m, substituting r fix c ... 

Because this dead time varies inversely with flow, hence with thermal power, the process exhibits variable dynamic gain. But the feedforward signal Q is a multiplier in both power and temperature loops, providing gain adaptation. 

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