Characteristics of Real Processes in Visual Studio .NET

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Characteristics of Real Processes
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Each of the three elements above changes the dimension of what is passing through. In order t,o arrive at a dimensionless loop gain, the dimensional gain of all three elements must be included in the product.
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Transmitter Gain
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In the liquid-level process presented in Chap. 1, the measurement h was defined as representing the fractional contents of the tank. This trick enabled us to find the time constant of the vessel in terms of its capacity V and its nominal throughput F. When instrumenting a plant, however, it is not necessary that every liquid-level t,ransmitter be scaled to measure the entire volume of the vessel. If, instead, the transmitter span represents only a small percentage of the vessel volume, the vessel will have effectively shrunk to the span of the transmitter. To state it another way: for control purposes, those part,s of the vessel beyond the range of the transmitter do not exist. Reducing the span of a transmitter is equivalent to reducing the proportional band of the controller. If a particular damping, hence a particular loop gain is to be achieved, the proportional band estimate must, take into account the span of the transmitter. In order to facilitate the evaluation of systems more complex than the liquid-level process, the transmitter gain will be explicitly defined:
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GT = 100 % ~
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Gain is the ratio of output to input. The numerator in Eq. (2.5) is the output that will be produced for a full-span change in input. Obviously G, is not a pure number-it has the dimensions of the measurement. Suppose a level transmitter were caalibrated to a range of 20 to 100 in. of water. Its gain would t,hen be loo%/80 in., or 1.25%/in. The fact that GT has dimension indicates that it is an incomplete t erm. The ot,her gains around the loop must be multiplied by G, in order to make the loop gain dimensionless. It is entirely possible that (;I17 is not constant. This would be the case if the transmit,tcr were nonlinear. E ew transmitters are sufficiently nonlinear to show any marked effect on rontrol-loop stability. A change in gain of at least; 1.5/l would be necessary to cause difficulty. Some temperature measurements are nonlinear, b u t seldom to this extent. The most, notable (UC of a nonlinenr transmitter is the differential flowmeter, whose output varies with t,hc square of flow through the primary element. Each flow transmitter has its own particular span. But in addition, the differential flow transmitter has the nonlinear relat,ionship. & can he detcrmincd on the basis of t,ransmitter span, with the nonlin6arity applied as a cocffic4cnt. Let h = dimensionless differential (fraction of
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1 Understanding
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h 0.5
FIG 2.5. The gain of a dilferential flow transmitter is directly proportional to ROUL
full scale) and f = dimensionless flow. h = f and
Then, (2.6)
The derivative dh/dj is the dimensionless gain of the transmitter. dimensional gain is then
As an example, look at a differential flowmeter whose scale is 0 to gptn. c&f = 2,~(0.2%/gpm). At full-scale flow, GT = O.A ;;Jgpnt; at 50 percent flow, G7< = 0.27F ,/gpnl; at zero flow, GYY = 0. The result of the nonlinearity is that the control loop will not perform c onsistcntly at, different, rates of flow. If the proportional band is adjusted for acceptable damping at 50 percent flow, the loop n-ill be undamped at 100 percent flow and sluggish near zero flow. The problem can be readily resolved, however, by inserting a square-root extractor, whose output would be linear with flow.
5 0 0
Referring again t o the liquid-level procaess of Chap. 1, the time constant of the vessel was bused upon the rated flow F which the control valve was capable of delivering. The time caottstattt thus depends on valve size; consequently, the proportional band is :L function of valve size. 1,ooliitig
Characteristics of Real Processes
at it another way, an oversize valve would only be operated over part of its travel-the span of stem travel would be less than 100 percent. Therefore the proportional band must be wider to compensate. The gain of a valve can be defined as the change in delivered flow vs. percent change in stem position. The gain of a linear valve is simply the rated flow under nominal process conditions at full stroke: (2.9) If a linear valve were able to deliver 500 grim fully open at stipulat ed process condit)ions, G,. would be 5 gpnl/Lz. Sotice that valve gain has dimension, as did transmitter gain, but now the percent sign is in the denominator. The valve is at the output of the controller, whereas the transmit,ter is at the input. Controller gain is therefore in terms of ojc/ yO, hence dimensionless. V&es cannot be manufactured to the same tolerance as transmitters. So there is no such thing as a truly linear valve. But perfect linearity is not essential, because a control loop does not demand it. Some valves are deliberately characterized t,o particular nonlinear functions, in order t o better carry out certain specific duties. The most commonly used characterized valve is the equal-percent age type. The name equal-percentage is very subject to misinterpretation. It means that a given increment in stem position will change the flow by a certain percentage of the present flow, regardless of the value of the present flow. To state it mathematically,
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