barcode reader using c#.net The Second-Order System in Software

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As shown in Example 16.2, the frequency response of a system with a secondorder transfer function
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G(s) =
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is given by Eq. (16.13), repeated here for convenience, AR = 1 (1 - 6JV)2 + (25wr)2 -2507 1 - (OX)2 (16.13)
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Phase angle = tan-
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0 -45 8 -90 -135 ,-180 0.1 0.2 0.5 1.0 UT--t 2.0 5.0 10.0
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FlGURJI 16-6 Bode diagram for second-order system 1/($s2 + 2$7s + 1).
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If OT is used as the abscissa for the general Bode diagram, it is clear that 5 will be a parameter. That is, there is a different curve for each value of 5. These curves appear as in Fig. 16.6. The calculation of phase angle as a function of o from Eq. (16.13) requires careful attention. The calculation can be done most clearly with the aid of a plot of tan-lx (or arctan x) as shown in Fig. 16.7. As or goes from zero to unity, we see from EQ. (16.13) that the argument of the arctan function goes from 0 to --CQ and the phase angle goes from 0 to -90 as shown by the branch from A to B in Fig. 16.7. As WT crosses unity from a value less than unity to a value greater than unity, the sign of the argument of the arctan function in Eq. (16.13) shifts from negative to positive. To preserve continuity in angle as or crosses unity, the phase angle must go from -90 to - 180 as OT goes from unity to +a and the branch of the arctan function goes from C to D (in Fig. 16.7). The arctan function available in calculators and digital computers normally covers the principal branches of the arctan function, shown as BAE in Fig. 16.7. For this reason, one must be very careful in calculating the phase angle with Eq. (16.13). If a calculator programmed for the principal branches of the arctan
INlXODUCI ION
TO FREQUENCY RJ SPONSE
__-----
---w--
-270"
FIGURE 16-7 Use of plot of tan-lx for computing phase angle of second-order system.
function is used and the argument is positive, one obtains the correct phase angle by subtracting 180 from the answer given by the calculator. Notice that for WT = 1, the phase angle is -!JO , independently of 3. This verifies that all phase curves intersect at -90 as shown in Fig. 16.6. We may now examine the amplitude curves obtained from Eq. (16.13). For WT C 1, the AR, or gain, approaches unity. For OT > 1, the AR becomes asymptotic to the line 1 AR = (OT)2 This asymptote has slope -2 and intersects the line AR = 1 at or = 1. The asymptotic lines are indicated on Fig. 16.6. For 5 2 1, we have shown that the second-order system is equivalent to two first-order systems in series. The fact that the AR for c L 1 (as well as for & < 1) attains a slope of -2 and phase of - 180 is, therefore, consistent. Figure 16.6 also shows that, for 5 < 0.707, the AR curves attain maxima in the vicinity of WT = 1. This can be checked by differentiating the expression for the AR with respect to or and setting the derivative to zero. The result is
(UT) mM = Jl 352
l c 0.707
(16.24)
for the value of or at which the maximum AR occurs. The value of the maximum AR, obtained by substituting (~7)~~~ into Eq. (16.13) is
(AR),,
= x&-F
6 < 0.707
0.8 0.6
FREQUENCY
RESF'ONSE
Maximum A. R.
FIGURE 16-8 Maximum AR versus damping for second-order system.
A plot of the maximum AR against f is given in Fig. 16.8. The frequency at which the maximum AR is attained is called the resonant frequency and is obtained from Eq. (16.24),
The phenomenon of resonance is frequently observed in our everyday experience. A vase may vibrate when the stereo is playing a particular note. As a car decelerates, perceptible vibrations may occur at particular speeds. A suspension bridge oscillates violently when scouts march across stepping at a certain cadence. It may be seen that AR values exceeding unity are attained by systems for which 5 < 0.707. This is in sharp contrast to the first-order system, for which the AR is always less than unity. The curves of Fig. 16.6 for l < 1 am not simple to construct, particularly in the vicinity of the resonant frequency. Fortunately, almost all second-order control system components for which we shall want to construct Bode diagrams have 5 > 1. That is, they are composed of two first-order systems in series. Actually, the curves of Fig. 16.6 are presented primarily because they are useful in analyzing the closed-loop frequency response of many control systems.
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