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open type. The external power (120 V) needed for each component is also shown in Fig. 10.2. Electricity is needed for the transducer, controller, and converter. A source of 20 psig air is needed for the converter. To see how the components interact with each other, consider the process to be operating at steady state with the outlet temperature equal to the set point. If the temperature of the hot process stream increases, the following events occur: After some delay the thermocouple detects an increase in the outlet temperature and produces a proportional change in the signal to the controller. As soon as the controller detects the rise in temperature, relative to the set point, the controller output increases according to proportional action. The increase in signal to the converter causes the output pressure from the converter to increase and open the valve wider in order to admit a greater flow of cooling water. The increased flow of cooling water will eventually reduce the output temperature and move it toward the set point. From this qualitative description, we see that the flow of signals from one component to the next is such that the temperature of the heat exchanger should return toward the set point. In a well-tuned control system, the response of the temperature will oscillate around the set point before coming to steady state. We shall give considerable attention to the transient response of a control system in the remainder of this book. Further discussion will also be given on control valves in Chap. 20 and on controllers in Chap. 35. For convenience in describing various control laws (or algorithms) in the next part of this chapter, the transducer, controller, and converter will be lumped into one block as shown in Fig. 10.3. This concludes our brief introduction to valves and controllers. We now present transfer functions for such devices. These transfer functions, especially for controllers, are based on ideal devices that can be only approximated in practice. The degree of approximation is sufficiently good to warrant use of these transfer functions to describe the dynamic behavior of controller mechanisms for ordinary design purposes.
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FIGURE 10-3 Equivalent block for transducer, controller, and converter.
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ELEMENTS
IDEAL TRANSFER FUNCTIONS Control Vale
A pneumatic valve always has some dynamic lag, which means that the stem motion does not respond instantaneously to a change in the applied pressure from the controller. From experiments conducted on pneumatic valves, it has been found that the relationship between flow and valve-top pressure for a linear valve can often be represented by a first-order transfer function; thus
Q(s) Kv -=7,s + 1 P(s)
where K, is the steady-state gain, i.e., the constant of proportionality between steady-state flow rate and valve-top pressure, and T is the time constant of the valve. In many practical systems, the time constant of the valve is very small when compared with the time constants of other components of the control system, and the transfer function of the valve can be approximated by a constant
Q(s) - K P(s) "
Under these conditions, the valve is said to contribute negligible dynamic lag. To justify the approximation of a fast valve by a transfer function, which is simply K,, consider a first-order valve and a first-order process connected in series, as shown in Fig. 10.4. According to the discussion of Chap. 7, if we assume no interaction, the transfer function from P(s) to Y(s) is Y(s)
P(s) K,KP
(7,s + l)(rps + 1)
The assumption of no interaction is generally valid for this case. For a unit-step change in P,
K,KP
s (7 s + l)(rps + 1)
the inverse of which is
Y(t) = (K,Kp+ - -$$/ - ;e- Tp)]
Valva +Z--~~~ ;:lM Block diagram for a first-order valve and a first-order
LINEAR CLOSED-LOOP SYSTEMS
If r,, 4~ rp, this equation is approximately Y(t) = K,Kp(l - e- +) The last expression is the unit-step response of the transfer function
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