barcode reader using c#.net INTRODUCTION TO FREQUENCY RESPONSE in Software

Painting ANSI/AIM Code 128 in Software INTRODUCTION TO FREQUENCY RESPONSE

INTRODUCTION TO FREQUENCY RESPONSE
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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 right-hand ordinate of Fig. 16.4.
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Rrst-Order
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Systems in Series
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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 first-order 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)
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(16.20)
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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.
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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 first-order systems in series, it is rewritten as
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(s + l)( 45s + 1)
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(16.21)
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The time constants are q = 1 and 72 = 4. The factor k in the numerator corresponds to the steady-state 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 first-order 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 steady-state 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 16-5 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.
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