barcode reader using c#.net B=T; e-5 FIGURE 16-3 Proportional control of heated, stirred tank. in Software

Creation Code 128B in Software B=T; e-5 FIGURE 16-3 Proportional control of heated, stirred tank.

B=T; e-5 FIGURE 16-3 Proportional control of heated, stirred tank.
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INTRODUCTION
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TO FREQUENCY RESPONSE
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were closed, the control system would have a tendency to add more heat when the inlet temperature Ti is at its high peak, because B is then negative and -K,B becomes positive. (Recall that the set point R is held constant at zero.) Conversely, when the inlet temperature is at a low point, the tendency will be for the control system to add less heat because B is positive. This is precisely opposite to the way the heat input should be controlled. Therefore, the possibility of an unstable control system exists for this particular sinusoidal variation in frequency. Indeed, we shall demonstrate in Chap. 17 that, if K, is taken too large, the tank temperature will oscillate with increasing amplitude for all variations in U and hence we have an unstable control system. The fact that such information may be obtained by study of the frequency response (i.e., the particular solution for a sinusoidal forcing function) justifies further study of this subject.
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BODE DIAGRAMS
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Thus far, it has been necessary to calculate AR and phase lag by direct substitution of s = jw into the. transfer function for the particular frequency of interest. It can be seen from Eqs. (16.3), (16.13), and (16.15) that the AR and phase lag are functions of frequency. There is a convenient graphical representation of their dependence on the frequency that largely eliminates direct calculation. This
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is called a Bode diagram and consists of two graphs: logarithm of AR versus logarithm of frequency, and phase angle versus logarithm of frequency. The Bode
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diagram will be shown in Chap. 17 to be a convenient tool for analyzing control problems such as the one discussed in the preceding section. The remainder of the present chapter is devoted to developing this tool and presenting Bode diagrams for the basic components of control loops.
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First-Order
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(16.16) m=J7k
The AR and phase angle for the sinusoidal response of a first-order system are
Phase angle = tan- (--or) (16.17) It is convenient to regard these as functions of or for the purpose of generality. From Eq. (16.16) log AR = -; log [(w# + l] (16.18) The first part of the Bode diagram is a plot of Eq. (16.18). The true curve is shown as the solid line on the upper part of Fig. 16.4. Some asymptotic considerations can simplify this plot. As (or) + 0, Eq. (16.16) shows that AR + 1. This is indicated by the low-frequency asymptote on Fig. 16.4. As (07) + cc), Eq. ( 16.18) becomes asymptotic to log AR = - log(wr)
FREQUENCY
RESPONSE
1 d =o g .s PE E
0 s -2og 8 -40
0.01 -
FIGURE 16-4 Bode diagram for first-order system.
which is a line of slope - 1, passing through the point wr = 1 AR=1
This line is indicated as the high-frequency asymptote in Fig. 16.4. The frequency WC = VT, where the two asymptotes intersect, is known as the cornerfrequency; it may be shown that the deviation of the true AR curve from the asymptotes is a maximum at the corner frequency. Using wC = l/r in Eq. (16.16) gives AR = 1 = 0.707 Jz as the true value, whereas the intersection of the asymptotes occurs at AR = 1. Since this is the maximum deviation and is an error of less than 30 percent, for engineering purposes it is often sufficient to represent the curve entirely by the asymptotes. Alternately, the asymptotes and the value of 0.707 may be used to sketch the curve if mote accuracy is required. In the lower half of Fig. 16.4, we have shown the phase curve as given by Eq. (16.17). Since cj = tan- (-au) = - taf (WT)
it is evident that 4 approaches 0 at low frequencies and -!W at high frequencies. This verifies the low- and high-frequency portions of the phase curve. At the comer frequency, 0, = l/r, c#Q = -tan- (o,7) = -tan- (l) = -45O
There am asymptotic approximations available for the phase curve, but they are not so accurate or so widely used as those for the AR. Instead, it is convenient to note that the curve is symmetric about -45 .
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