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The temperature-measuring element, which senses the bath temperature T and transmits a signal T,,, to the controller, may exhibit some dynamic lag. From the discussion of the mercury thermometer in Chap. 5, we observed this lag to be first-order. In this example, we shall assume that the temperature-measuring element is a first-order system, for which the transfer function is (9.18) T (s) where the input-output variables T and TA are deviation variables, defined as
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T = T - T , T; = T,,, - T,,,,
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T,i,(s)
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1 =7,s + 1
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Note that, when the control system is at steady state, T, = Tm, , which means that the temperature-measuring element reads the true bath temperature. The transfer function for the measuring element may be represented by the block diagram shown in Fig. 9.6.
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THE CONTROL SYSTEM
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Controller and Fhal Control Element
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For convenience, the blocks representing the controller and the final control element are combined into one block. In this way, we need be concerned only with the overall response between the error and the heat input to the tank. Also, it is assumed that the controller is a proportional controller. (In the next chapter, the response of other controllers, which are commonly used in control systems, will be described.) The relationship for a proportional controller is q = K,E+A where E TR K, A = TR - T,,, = set-point temperature = proportional sensitivity or controller gain = heat input when E = 0 (9.19)
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At steady state, it is assumed* that the set point, the process temperature, and the measured temperature are all equal to each other; thus TR, = T, = T,, Let E be the deviation variable for error; thus E = E-E, where E, = TR, - Tm, Since TR, = T,,,,, E, = 0 and Eq. (9.21 becomes E =E-(-J=E This result shows that E is itself a deviation variable. Since E, = 0, Eq. (9.19) becomes at steady state qs = Kcc, + A = 0 + A = A Equation (9.19) may now be written in terms of qs; thus q = Kc6 + qs
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(9.20)
(9.21)
(9.22)
Q = K,E where Q = q - qs The transform of Eq. (9.23) is simply
(9.23)
Q(s) = K,E(s)
(9.24)
*In a practical situation, the equality among the three variables, T, T,,,, and TR, at steady state as given by Eq. (9.20) can always be established by adjustment of the instruments. The equality between T and T, can be achieved by calibration of the measuring element. The equality between T,,, and TR can be achieved by adjustment of the proportional controller.
LINEAR CLOSED-LOOP
SYSTEMS
T;(s)
FIGURE 9-7 Block diagram of proportional controller.
Note that E, which is also equal to E , may be expressed as E = TR - TR, - (T, - T,,) or E = T; - T; (9.26) Equation (9.25) follows from the definition of E and the fact that TR, = T,,,,. Taking the transform of Eq. (9.26) gives
E( S)
(9.25)
= T;(s) - T;(s)
(9.27)
The transfer function for the proportional controller given by Eq. (9.24) and the generation of error given by EQ. (9.27) may be expressed by the block diagram shown in Fig. 9.7. We have now completed the development of the separate blocks. If these are combined according to Fig. 9.2, there is obtained the block diagram for the complete control system shown in Fig. 9.8. The reader should verify this figure.
SUMMARY
It has been shown that a control system can be translated into a block diagram that includes the transfer functions of the various components. It should be emphasized that a block diagram is simply a systematic way of writing the simultaneous differential and algebraic equations that describe the dynamic behavior of the components. In the present case, these were Eqs. (9.10), (9.18), and (9.24) and the definition of E. The block diagram clarifies the relationships among the variables of these simultaneous equations. Another advantage of the block-diagram representation is that it clearly shows the feedback relationship between measured variable and desired variable and how the difference in these two signals (the
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