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barcode reader using c#.net (K,/600)e0~0396S 0.202s + 1 in Software
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USS Code 128 Printer In Visual Studio .NET Using Barcode generation for ASP.NET Control to generate, create Code 128 Code Set A image in ASP.NET applications. Generating Code 128A In .NET Framework Using Barcode creator for .NET framework Control to generate, create Code128 image in .NET applications. BT; e QO386# I I
ANSI/AIM Code 128 Creation In VB.NET Using Barcode encoder for VS .NET Control to generate, create Code 128 image in Visual Studio .NET applications. Create USS Code 39 In None Using Barcode maker for Software Control to generate, create ANSI/AIM Code 39 image in Software applications. FIGURE 171 Control system for stirredtank heater of Example 16.3.
EAN 128 Encoder In None Using Barcode generation for Software Control to generate, create EAN128 image in Software applications. Draw Code 128B In None Using Barcode encoder for Software Control to generate, create Code128 image in Software applications. The Bode diagram for G(s) is plotted in Fig. 17.2. As usual, the constant factor K,/600 is included in the definition of the ordinate for AR. At the frequency of 43 rad/min, the phase lag is exactly 180 and AR  = 0.12 K,/600 UPC  13 Creation In None Using Barcode maker for Software Control to generate, create EAN13 Supplement 5 image in Software applications. UPCA Supplement 2 Generation In None Using Barcode printer for Software Control to generate, create UPCA Supplement 5 image in Software applications. 90 g 8135 d c 180 225 270 1 2 FIGURE 172 Bode diagram for openloop transfer function of control system for stirredtank heater: (KJwC)~ ~~ [l/(qs + l)]. (Block diagram shown in Fig. 17.1.) Printing MSI Plessey In None Using Barcode creation for Software Control to generate, create MSI Plessey image in Software applications. Read Code39 In VB.NET Using Barcode decoder for .NET framework Control to read, scan read, scan image in Visual Studio .NET applications. 10 20 w Code 128 Code Set C Decoder In Visual Basic .NET Using Barcode reader for .NET framework Control to read, scan read, scan image in VS .NET applications. Create ECC200 In Java Using Barcode encoder for BIRT Control to generate, create DataMatrix image in BIRT reports applications. F REQUJ3iCY
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Data Matrix 2d Barcode Maker In ObjectiveC Using Barcode generation for iPad Control to generate, create ECC200 image in iPad applications. UCC.EAN  128 Reader In C# Using Barcode recognizer for .NET Control to read, scan read, scan image in .NET framework applications. Therefore, if a proportional gain of 5000 Btu/(hr)( F) is used, 5ooo AR = 0.12600 = 1 This is the AR between the signals E and B. Note that it is dimensionless as it must be, since E and B both have the units of temperature. The control system is redrawn for K, = 5000 in Fig. 17.3~2, with the loop opened. That is, the feedback signal B is disconnected from the comparator. It is imagined that a set point disturbance R = sin 43t is applied to the opened loop. Then, since the openloop AR and phase lag am unity and 180 B = sin(43t  180 ) =  sin43t Now imagine that, at some instant of time, R is set to zero and simultaneously the loop is closed. Figure 17.3b indicates that the closed loop continues to oscillate indefinitely. This oscillation is theoretically sustained even though both R and U are zero. Now suppose K, is set to slightly higher value and the same experiment repeated. This time, the signal E is amplified slightly each time it passes around the loop. Thus, if K, is set to 5001, after the first time around the loop the signal E becomes (5001/5000) sin 43t. After the second time, it is (5001/5000)2sin43t, etc. The phaseangle relations are not affected by changing K,. We thus conclude Bsin 4 3 t e (4  0.0396s
Before closing loop
After closing loop (b) FIGURE Sustained
173 closedloop oscillation.
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that, for K, > 5000, the response is unbounded, since it oscillates with increasing amplitude. Using the definition of stability presented in Chap. 14, it is concluded that the control system is unstable for K, > 5000 because it exhibits an unbounded response to the bounded input described above. (The bounded input is zero in this case, for U = R = 0.) The condition K, > 5000 corresponds to AR>1 for the openloop transfer function, at the frequency 43 tad/mm, where the openloop phase lag is 180 . This argument is not rigorous. We know the response B only if E remains consrant in amplitude because of the definition of frequency response. If, however, the change in K, is very small, so that E is amplified infinitesimally, then B will closely approximate the fmquency response. While this does not prove anything, it shows that we am justified in suspecting instability and that closer investigation is warranted. A rigorous proof of stability requires application of the Nyquist stability criterion [See Coughanowr and Koppel (1965) or Kuo (1987)], which uses the theory of complex variables. For our purposes, it is sufficient to proceed with heuristic arguments. The Bode Stability Criterion
It is tempting to generalize the results of the analysis of the tanktemperature control system to the following rule. A control system is unstable if the openloop frequency response exhibits an AR exceeding unity at the frequency for which the phase lag is 180 . This frequency is called the crossover frequency. The rule is called the Bode stability criterion. Actually, since the discussion of the previous section was based on heuristic arguments, this rule is not quite general. It applies readily to systems for which the gain and phase curves decrease continuously with frequency. However, if the phase curve appears as in Fig. 17.4, the more general Nyquist criterion must usually be used to determine stability. Other exceptions may occur. Fortunately, most process control systems can be analyzed with the simple Bode criterion, and it therefore finds wide application. 8 f 180 f Frquency 

