barcode reader project in c#.net Note that, if 71 = 72, then r = rr = 72 and l = 1. The reader should verify these results. in Software

Generating Code-128 in Software Note that, if 71 = 72, then r = rr = 72 and l = 1. The reader should verify these results.

Note that, if 71 = 72, then r = rr = 72 and l = 1. The reader should verify these results.
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Terms Used to Describe an Underdamped System
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Of these three cases, the underdamped response occurs most frequently in control systems. Hence a number of terms are used to describe the underdamped response
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FIGURE 8-3 Terms used to describe an underdamped second-order response.
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quantitatively. Equations for some of these terms are listed below for future reference. In general, the terms depend on 5 and/or r. All these equations can be derived from the time response as given by Eq. (8.17); however, the mathematical derivations are left to the reader as exercises.
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1. Overshoot. Overshoot is a measure of how much the response exceeds the
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ultimate value following a step change and is expressed as the ratio A/B in Fig. 8.3. The overshoot for a unit step is related to 5 by the expression Overshoot = exp(-ml/ Jl - {*) (8.24)
This relation is plotted in Fig. 8.4. The overshoot increases for decreasing 4.
0.6 b 0.8 0
FIGURE 8-4 Characteristics of a step response of an underdamped second-order system.
HIGHER-ORDER
SYSTEMS:
SECOND-ORDER
TRANSPORTATION
2. Decay ratio. The decay ratio is defined as the ratio of the sizes of successive peaks and is given by CIA in Fig. 8.3. The decay ratio is related to 4 by the expression Decay ratio = exp(-2r[/ ,/g) = (overshoot)* (8.25) which is plotted in Fig. 8.4. Notice that larger 6 means greater damping, hence greater decay. 3. Rise time. This is the time required for the response to first reach its ultimate value and is labeled t ,. in Fig. 8.3. The reader can verify from Fig. 8.2 that t, increases with increasing 5. 4. Response time. This is the time required for the response to come within +5 percent of its ultimate value and remain there. The response time is indicated in Fig. 8.3. The limits 55 percent are arbitrary, and other limits have been used in other texts for defining a response time. 5. Period of oscillation. From Eq. (8.17), the radian frequency (radians/time) is the coefficient of t in the sine term; thus, o, radian frequency = J=F (8.26) 7 Since the radian frequency w is related to the cyclical frequency f by w = 2~f, it follows that (8.27) where T is the period of oscillation (time/cycle). In terms of Fig. 8.3, T is the time elapsed between peaks. It is also the time elapsed between alternate crossings of the line Y = 1. 6. Natural period of oscillation. If the damping is eliminated [C = 0 in Eq. (8.1). or 5 = 01, the system oscillates continuously without attenuation in amplitude. Under these natural or undamped conditions, the radian frequency is l/r, as shown by Eq. (8.26) when 5 = 0. This frequency is referred to as the natural frequency wn:
The corresponding natural cyclical frequency f,, and period T, are related by the expression (8.29) Thus, r has the significance of the undamped period. From Eqs. (8.27) and (8.29), the natural frequency is related to the actual frequency by the expression
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which is plotted in Fig. 8.4. Notice that, for 5 < 0.5, the natural frequency is nearly the same as the actual frequency. In summary, it is evident that 5 is a measure of the degree of damping, or the oscillatory character, and r is a measure of the period, or speed, of the response of a second-order system.
Impulse Response
If a unit impulse s(t) is applied to the second-order system, then from Eqs. (8.11) and (4.1) the transform of the response is 1 Y(s) = (8.30) r*s* + 2lTS + 1 As in the case of the step input, the nature of the response to a unit impulse will depend on whether the roots of the denominator of Eq. (8.30) are real or complex. The problem is again divided into the three cases shown in Table 8.1, and each is discussed below.
CASE I IMPULSE RESPONSE FOR 5 < 1. The inversion of Eq. (8.30) for
5 < 1 yields the result (8.31) which is plotted in Fig. 8.5. The slope at the origin in Fig. 8.5 is 1 .O for all values of t.
FIGURE 8-5
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