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barcode reader using c#.net INTRODUCTION TO FREQUENCY RESPONSE in Software
INTRODUCTION TO FREQUENCY RESPONSE Read ANSI/AIM Code 128 In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. Generate USS Code 128 In None Using Barcode creation for Software Control to generate, create Code 128 Code Set C image in Software applications. It should be stated that, in a great deal of the literature on control theory, amplitude ratios (or gains) are reported in decibels. The decibel is defined by Decibels = 20 log ia Thus, an AR of unity corresponds to zero decibels and an amplitude ratio of 0.1 corresponds to 20 decibels. The abbreviation for the decibel is db. The value of the AR in decibels is given on the righthand ordinate of Fig. 16.4. Read Code 128B In None Using Barcode decoder for Software Control to read, scan read, scan image in Software applications. ANSI/AIM Code 128 Generation In Visual C# Using Barcode generator for .NET Control to generate, create Code 128 image in .NET applications. RrstOrder
Drawing Code 128 Code Set A In VS .NET Using Barcode generation for ASP.NET Control to generate, create Code128 image in ASP.NET applications. Code 128 Code Set B Generator In .NET Framework Using Barcode printer for Visual Studio .NET Control to generate, create Code128 image in .NET applications. Systems in Series
Code 128 Code Set B Printer In VB.NET Using Barcode printer for VS .NET Control to generate, create Code 128B image in Visual Studio .NET applications. UCC128 Encoder In None Using Barcode printer for Software Control to generate, create EAN 128 image in Software applications. The advantages of the Bode plot become evident when we wish to plot the frequency response of systems in series. As shown in Example 16.3, the rules for multiplication of complex numbers indicate that the AR for two firstorder systems in series is the product of the individual ARs: A R = Jz&JqTT Similarly, the phase angle is the sum of the individual phase angles r#~ = tan (wrt) + tan (7.072) (16.19) Making ANSI/AIM Code 128 In None Using Barcode maker for Software Control to generate, create Code 128A image in Software applications. Code 39 Extended Drawer In None Using Barcode maker for Software Control to generate, create Code 3/9 image in Software applications. (16.20) Drawing Barcode In None Using Barcode printer for Software Control to generate, create barcode image in Software applications. UCC  12 Printer In None Using Barcode generator for Software Control to generate, create GTIN  12 image in Software applications. Since the AR is plotted on a logarithmic basis, multiplication of the ARs is accomplished by addition of logarithms on the Bode diagram. The phase angles are added directly. The procedure is best illustrated by an example. 4State Customer Barcode Generation In None Using Barcode encoder for Software Control to generate, create 4State Customer Barcode image in Software applications. EAN128 Maker In .NET Using Barcode creation for Reporting Service Control to generate, create EAN 128 image in Reporting Service applications. Example 16.4. Plot the Bode diagram for the system whose overall transfer function is 1 (s + l)(S + 5) To put this in the form of two firstorder systems in series, it is rewritten as 1D Barcode Generator In VS .NET Using Barcode printer for .NET Control to generate, create 1D Barcode image in VS .NET applications. Creating Bar Code In Java Using Barcode creation for BIRT Control to generate, create bar code image in Eclipse BIRT applications. (s + l)( 45s + 1) UPC Code Creator In VS .NET Using Barcode creation for ASP.NET Control to generate, create UPC Code image in ASP.NET applications. Decode Barcode In VB.NET Using Barcode scanner for VS .NET Control to read, scan read, scan image in Visual Studio .NET applications. (16.21) Make Barcode In None Using Barcode generation for Font Control to generate, create barcode image in Font applications. Generating Code 128B In Visual C#.NET Using Barcode creation for .NET Control to generate, create Code 128 Code Set A image in .NET framework applications. The time constants are q = 1 and 72 = 4. The factor k in the numerator corresponds to the steadystate gain. From Eqs. (16.21) and (16.19) Hence, logAR=logf;log(w2+1);log or
co2 1 [(I
s +l (16.22) log AR = log /s + log (AR)I + log (AR), FREQUENCY
RESPONSE
where (AR)t (AR)2 are the ARs of the individual firstorder systems, each with unity gain. Equation (16.22) shows that the overall AR is obtained, on logarithmic coordinates, by adding the individual ARs and a constant corresponding to the steadystate gain. The individual ARs must be plotted as functions of log w rather than log (wr) because of the different time constants. This is easily done by shifting the curves of Fig. 16.4 to the right or left so that the comer frequency falls at w = l/r. Thus, the individual curves of Fig. 16.5 are placed so that the comer frequencies fall at WC1 = 1 and w,s = 5. These curves are added to obtain the overall curve shown. Note that in this case the logarithms are negative and the addition is downward. To complete the AR curve, the factor log f should be added to the overall curve. This would have the effect of shifting the entire curve down by a constant amount. Instead of doing this, the factor f is incorporated by plotting the overall curve as ARti instead of AR. This procedure is usually more convenient. Asymptotes have also been indicated on Fig. 16.5. The sum of the individual asymptotes gives the overall asymptote, which is seen to be a good approximation to the overall curve. The overall asymptote has a slope of zero below o = 1,  1 for w between 1 and 5, and 2 above w = 5. Its slope is obtained by simply adding the slopes of the individual asymptotes. To obtain the phase angle, the individual phase angles are plotted and added according to Eq. (16.20). The factor i has no effect on the phase angle, which approaches  180 at high frequency. FIGURE 165 Bode diagram for 0.2/[(s + 1)(0.2s +
l)]. INTRODUCTION
TO FREQUENCY RESPONSE
Graphical Rules for Bode Diagrams
Before proceeding to a development of the Bode diagram for other systems, it is desirable to summarize the graphical rules that were utilized in Example 16.4. Consider a number of systems in series. As shown in Example 16.3, the overall AR is the product of the individual ARs, and the overall phase angle is the sum of the individual phase angles. Therefore, log (AR) = log (AR), + log (AR), + . . . + log (AR), and 4 = 41 + 42 + . . . + 4% (16.23) where n is the total number of systems. Therefore, the following rules apply to the true curves or to the asymptotes on the Bode diagram: 1. The overall AR is obtained by adding the individual ARs. For this graphical addition, an individual AR that is above unity on the frequency response diagram is taken as positive; an AR that is below unity is taken as negative. To understand this, recall that the logarithm of a number greater than one is positive and the logarithm of a number less than one is negative. A convenient way to combine two or more individual AR curves is to use a pair of dividers to transfer distances at a selected value of w. 2. The overall phase angle is obtained by addition of the individual phase angles. 3. The presence of a constant in the overall transfer function shifts the entire AR curve vertically by a constant amount and has no effect on the phase angle. It is usually more convenient to include a constant factor in the definition of the ordinate. These rules will be of considerable value in later examples. Let us now proceed to develop Bode diagrams for other control system components.

