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TpS + 1
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so that the combination of process and valve is essentially first-order. This clearly demonstrates that, when the time constant of the valve is much smaller than that of the process, the valve transfer function can be taken as K,. A typical pneumatic valve has a time constant of the order of 1 sec. Many industrial processes behave as first-order systems or as a series of first-order systems having time constants that may range from a minute to an hour. For these systems we have shown that the lag of the valve is negligible, and we shall make frequent use of this approximation. Controllers In this section, we shall present the transfer functions for the controllers frequently used in industrial processes. Because the transducer and the converter will be lumped together with the controller for simplicity, the result is that the input will be the measured variable x (e.g. temperature, level, etc.) and the output will be a pneumatic signal p. (See Fig. 10.3) Actually this form (X as input and p as output) applies to a pneumatic controller. For convenience, we shall refer to the lumped components as the controller in the following discussion, even though the actual electronic controller is but one of the components.
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CONTROL. The proportional controller produces an output signal (pressure in the case of a pneumatic controller, current or voltage for an electronic controller) that is proportional to the error E. This action may be expressed as
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PROPORTIONAL
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= Kc6 + ps
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(10.1)
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where
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p = output signal from controller, psig or ma
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K, = gain, or sensitivity
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E = error = set point - measured variable
ps = a constant
The error E, which is the difference between the set point and the signal from the measuring element, may be in any suitable units. However, the units of set point and measured variable must be the same, since the error is the difference between these quantities. In a controller having adjustable gain, the value of the gain K, can be varied by moving a knob in the controller. The value of ps is the value of the output signal when E is zero, and in most controllers ps can be adjusted to obtain the required output signal when the control system is at steady state and E = 0.
CONTROLLERS AND FINAL CONTROL ELEMENTS
To obtain the transfer function of Eq. (10. l), we first introduce the deviation variable
p =p-Ps
into Eq. (10.1). At time t = 0, we assume the error E, to be zero. Then E is already a deviation variable. Equation (10.1) Becomes
P(t) = K&(t)
(10.2)
Taking the transform of Eq. (10.2) gives the transfer function of an ideal proportional controller
K P(s) 4s) c
(10.3)
The term proportional band is commonly used among process control engineers in place of the term gain. Proportional band (pb) is defined as the error (expressed as a percentage of the range of measured variable) required to move the valve from fully closed to fully open. A frequently used synonym is bundwidth. These terms will be most easily understood by considering the following example.
Example 10.1. A pneumatic proportional controller is used to control temperature within the range of 60 to 1OO E The controller is adjusted so that the output pressure goes from 3 psi (valve fully open) to 15 psi (valve fully closed) as the measured temperature goes for 71 to 75 F with the set point held constant. Find the gain and the proportional band. (75 F - 71 F) Proportional band = (lOOOF _ 600F) X 100 = 10% Gain - Ap _ cl5 Psi - 3 psi) = 3 psi/~F de (7S F - 71 F) Now assume that the proportional band of the controller is changed to 75 percent. Find the gain and the temperature change necessary to cause a valve to go from fully open to fully closed. AT = (proportional band) (range) = 0.75(40 F) = 30 F 12 psi Gain = - = 0.4 p&F 30 F
From this example, we see that proportional gain corresponds inversely with proportional band; thus Proportional gain a l/proportional band The gain K, has the units of psi/unit of measured variable (e.g. psi/OFin Example 10. I). If the actual controller of Fig. 10.3~ is considered, both the input and the
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