Asymptote Bode plot

Sinee 1 / 2 is —3 dB, the exaet modulus passes 3 dB below the asymptote interseetion at /T rad/s. The asymptotie eonstruetion of the log modulus Bode plot for a first-order system is shown in Figure 6.8. 

Even with MATLAB, we should still know the expected shape of the curves and its telltale features. This understanding is crucial in developing our problem solving skills. Thus doing a few simple hand constructions is very instructive. When we sketch the Bode plot, we must identify the comer (break) frequencies, slopes of the magnitude asymptotes and the contributions of phase lags at small and large frequencies. We ll pick up the details in the examples. 

On the Bode plot, the comer frequencies are, in increasing order, l/xp, Zq, and p0. The frequency asymptotes meeting at co = l/xp and p0 are those of a first-order lag. The frequency asymptotes meeting at co = z0 are those of a first-order lead. The largest phase lag of the system is -90° at very high frequencies. The system is always stable as displayed by the root locus plot. 

What are the low and high frequency asymptotes of the minimum phase function (s + z)/(s + p) versus the simplest nonminimum phase function (s - z)/(s + p) in a Bode plot ... 

The Bode plots are shown in Fig. 12.15. One of the most convenient features of Bode plots is that the L curves can be easily sketched by considering the low-and high-frequency asymptotes. As a> goes to zero, L goes to zero. As to becomes very large, Eq. (12.39) reduces to... 

If a transfer-function model is desired, approximate transfer functions can be fitted to the experimental curves. First the log modulus Bode plot is used. The low-frequency asymptote gives the steadystate gain. The time constants can be found from the breakpoint frequency and the slope of the high-frequency asymptote. The damping coefficient can be found from the resonant peak. 

Figure 10.19 shows the Bode plots for several values of damping coefficient The breakpoint frequency is the reciprocal of the time constant. The high-frequency asymptote has a slope of -40 dB/decade. 

A Bode plot of dF2i/dRm can be easily constructed by means of the low-frequency and the high-frequency asymptotes of its factors observing that A is a second-order polynomial in y. The amplitude drops for [Pg.152]

Having determined the difference between the number of zeros and poles of the transmittance, on the basis of the Bode plot (Fig. 2.18) we can find the point (lograj, 0) at which the first slope asymptote intercepts the logo) axis, and thus estimate the time constant V with he largest value, according to the relationship... 

The Bode diagram (Figure 7j) shows plots for GW, G2(s) and G(s) as amplitude ratio against frequency. Only the asymptotes (Section 7.10.4, Volume 3) are plotted. 

The Bode diagrams are plotted for these in Figure 7o. The asymptotes on the AR plots are summed and the sums on the ft plots are obtained by linear measurement. 

The Bode diagram in this case (Fig. 7.46) is distinguished by the fact that f is a parameter which affects both the AR and the yr plots. However, the asymptotes may be determined in the same manner as for the first-order system. It is found that, for all , the AR high frequency asymptote is a straight line of slope -2 passing through the point (1,1) and the LFA is represented by the line AR = 1. The yr plots all tend to zero degrees as tor- 0 and to -180° as When toc= lr, y/--90°... 

For a system like the one in item 8, we claim that the slope of the overall asymptotes, in the log AR versus log to plot of its Bode diagram, can be given from the algebraic sum of the slopes of the asymptotes for the individual subsystems,... 

As was derived in chapter 9, the amplitude ratio for a dead-time process is 1.0 and the phase shift -0)6. The amplitude ratio for the process becomes then AR (second-order process) x AR(dead-time process). The phase shift of the process becomes then (second-order process) + dead-time process). Figure 32.3 shows the Bode diagram in which the logarithm of the amplitude ratio and the phase shift are plotted against the frequency O). For the amplitude ratio two asymptotes emerge, one for low frequencies a>- ) (static behaviour) en one for high frequencies 0)- °° (high-frequency behaviour). The values can easily be calculated from ... 

Figure 32.4 shows the Bode diagram of a PI eontroller with integral action = 10. Rather than plotting the amplitude ratio, the ratio between amphtude ratio and controller gain is plotted on the vertieal axis. As can be seen from Eqn. (32.25), plotting log(AR) versus log(comer frequency, where the asymptotes interseet is at 1/t . The phase shift ranges from -90° to 0°. 

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