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the two chart records at different frequencies is shown in the table. From these data, show that it is reasonable to consider element B as a first-order process and calculate the time constant. Describe your method clearly.
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16.3. Plot the asymptotic Bode diagram for the PID controller:
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where K, = 10,rr = 1, rD = 100. Label comer frequencies and give slopes of asymptotes. 16.4. One way of experimentally measuring frequency response is to plot the output sine wave versus the input sine wave. The results of such a plot look like the figure shown in Fig. P16.4. This is the sinusoidal deviation in output versus sinusoidal deviation in input and appears as an ellipse centered at the origin. Show how to obtain the AR and phase lag from this plot. output
FIGURE P16-4
FREQUENCY
RESPONSE
16.5. For the transfer function shown below, sketch carefully the gain versus frequency portion of the asymptotic plot of the Bode diagram. Determine the actual (exact) value of gain and phase angle at w = 1. Determine the phase angle as w + m. G(s) =
2(0.1s + 1)
s2(1Os + 1)
Indicate very clearly the slopes of the asymptotic bode diagram of G(s). 16.6. (a) Plot accurately and neatly the Bode diagram for the process shown in Fig. P16.6 using log-log paper for gain vs. frequency and semi-log paper for phase vs. frequency. Plot the frequency as rad/min. (b) Find the amplitude ratio and phase angle for Y/X at w = 1 rad/min and w = 4 rad/min.
7 5s + 1
1 2s + 1
FIGURE P16-6
16.7. For the system shown in Fig. P16.7, determine accurately the phase angle in degrees between Y(t) and X(t) for w = 0.5. Determine the lag between the input wave and the output wave.
16.8. (a) For the transfer function given below, sketch carefully the asymptotic approximation of gain vs. frequency. Show detail such as slopes of asymptotes.
G@) = (1 s
(b) Find the actual (exact) value of gain and phase angle for o = 1 and for
w = 2. 16.9. Derive expressions for amplitude ratio and phase angle as functions of o for the
transfer function G(s) = l/(s* - 1).
16.10. The data given in the following table represent experimental, frequency response
data for a process consisting of a first-order process and a transportation lag. Determine the time constant and the transportation lag parameter. Write the transfer function for the process, giving numerical values of the parameters.
INTRODUCI M)N
TO FREQUENCY RESPONSE
Requency,
0.01 0.02 0.04 0.06 0.08 0.10 0.15 0.20 0.30 0.40 0.60 0.80 1.00 1.50 2.00 4.00 6.00 8.00 10.00 20.00 40.00
Gain
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.98 0.94 0.88 0.83 0.71 0.61 0.37 0.26 0.20 0.16 0.080 0.041
Phase angle, deg
-2.0 -6.0 -7.0 -8.5
-11.0 - 17.0 -23.0 - 36.0 -48.0 -73.0 -96.0 - 122.0 - 180.0 - 239.0
CHAPTER
CONTROL SYSTEM DESIGN BYFREQUENCY RESPONSE
The purpose of this chapter is twofold. First, it will be indicated that the stability of a control system can usually be determined from the Bode diagram of its openloop transfer function. Then methods will be presented for rational selection of controller parameters based on this Bode diagram. The material to be presented here is one of the more useful design aspects of the subject of frequency response.
Tank-Temperature Control System
It was indicated in the discussion following Example 16.3 that the control system of Fig. 17.1 might offer stability problems because of excessive phase lag. To review, this system represents proportional control of tank temperature with a delay in the feedback loop. The factor & is the process sensitivity ~/WC, which gives the ultimate change in tank temperature per unit change in heat input Q. The proportional sensitivity K,, in Btu per hour per degree of temperature error, is to be specified by the designer. The open-loop transfer function for this system is G(s) =
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