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RESPONSE OF FIRST-ORDER SYSTEMS
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Lag = GE = 0.0555 min
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In general, the lag in units of time is given by Lag = 360f when 4 is expressed in degrees.
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The response of the thermometer reading and the variation in bath temperature are shown in Fig. 5.8. It should be noted that the response shown in this figure holds only after sufficient time has elapsed for the nonperiodic term of Eq. (5.24) to become negligible. For all practical purposes this term becomes negligible after a time equal to about 37. If the response were desired beginning from the time the bath temperature begins to oscillate, it would be necessary to plot the complete response as given by Eq. (5.24).
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In this chapter several basic concepts and definitions of control theory have been introduced. These include input variable, output variable, deviation variable, transfer function, response, time constant, first-order system, block diagram, attenuation, and phase lag. Each of these ideas arose naturally in the study of the dynamics of the first-order system, which was the basic subject matter of the chapter. As might be expected, the concepts will find frequent use in succeeding chapters. In addition to introducing new concepts, we have listed the response of the first-order system to forcing functions of major interest. This information on
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temperature Ultimate periodic response *
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t, min FIGURE 5-8 Response of thermometer in Example 5.2.
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LINEAR OPEN-LOOP SYSTEMS
the dynamic behavior of the first-order system will be of significant value in the remainder of our studies.
PROBLEMS
5.1. A thermometer having a time constant of 0.2 min is placed in a temperature bath,
and after the thermometer comes to equilibrium with the bath, the temperature of the bath is increased linearly with time at a rate of l /min. What is the difference between the indicated temperature and the bath temperature (a) 0.1 min, (b) 1.0 min after the change in temperature begins (c) What is the maximum deviation between indicated temperature and bath temperature, and when does it occur (d) Plot the forcing function and response on the same graph. After a long enough time, by how many minutes does the response lag the input 5.2. A mercury thermometer bulb is 4 in. long by b in. diameter. The glass envelope is very thin. Calculate the time constant in water flowing at 10 ft/sec at a temperature of 100 F. In your solution, give a summary which includes (a) Assumptions used (b) Source of data (c) Results 5.3. Given a system with the transfer function Y(s)/X(s) = (Tls + l)/(Tzs + 1). Find Y(r) if X(t) is a unit-step function. If Tl/T2 = 5, sketch Y(r) versus tlT2. Show the numerical values of minimum, maximum, and ultimate values that may occur during the transient. Check these using the initial-value and final-value theorems of Chap. 4. 5.4. A thermometer having first-order dynamics with a time constant of 1 min is placed in a temperature bath at 100 F. After the thermometer reaches steady state, it is suddenly placed in a bath at 110 F at t = 0 and left there for 1 min, after which it is immediately returned to the bath at 100 F. (a) Draw a sketch showing the variation of the thermometer reading with time. (b) Calculate the thermometer reading at t = 0.5 min and at t = 2.0 min. 5.5. Repeat Prob. 5.4 if the thermometer is in the 110 F bath for only 10 sec. 5.6. A mercury thermometer, which has been on a table for some time, is registering the room temperature, 75 F. Suddenly, it is placed in a 400 F oil bath. The following data are obtained for the response of the thermometer.
llme, set 0
1 2.5 5 8 10
Thermometer reading, F 75
140 205 244 282 328 385
Give two independent estimates of the thermometer time constant.
RESPONSE OF FIRST-ORDER SYSTEMS
5.7. Rewrite the sinusoidal response of a first-order system [Eq. (5.24)] in terms of a cosine wave. Reexpress the forcing function [Eq. (5.19)] as a cosine wave, and compute the phase difference between input and output cosine waves. 5.8. The mercury thermometer of Prob. 5.6 is again allowed to come to equilibrium in the room air at 75 F. Then it is placed in the 400 F oil bath for a length of time less than 1 set, and quickly removed from the bath and reexposed to the 75 F ambient conditions. It may be estimated that the heat-transfer coefficient to the thermometer in air is one-fifth that in the oil bath. If 10 set after the thermometer is removed from the bath it reads 98 F, estimate the length of time that the thermometer was in the bath. 5.9. A thermometer having a time constant of 1 min is initially at 50 C. It is immersed in a bath maintained at 100 C at t = 0. Determine the temperature reading at t = 1.2 min. 5.10. In problem 5.9, if at t = 1.5 min, the thermometer is removed from the bath and put in a bath at 75 C, determine the maximum temperature indicated by the thermometer. What will be the indicated temperature at t = 20 min 5.11. A process of unknown transfer function is subjected to a unit-impulse input. The output of the process is measured accurately and is found to be represented by the function y(t) = te- . Determine the unit-step response of this process.
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