The Bode diagram

The teehnique uses asymptotes to quiekly eonstruet frequeney response diagrams by hand. The eonstruetion of diagrams for high-order systems is aehieved by simple graphieal addition of the individual diagrams of the separate elements in the system. The modulus is plotted on a linear y-axis seale in deeiBels, where 

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The Bode diagram is a logarithmie version of the frequeney response diagrams illustrated in Figures 6.4(b) and (e), and also Figure 6.6, and eonsists of... 

The Bode diagram, given in Figure 6.10, is the mirror image, about the frequeney axis, of the first-order lag system. Note that the transfer funetion given in equation (6.35) is also that of a PD eontroller. 

The Bode diagram for a pure integrator is shown in Figure 6.12. 

Asymptote interseetion (break frequeney) oeeurs at XjT, i.e. 2rad/s. The Bode diagram is shown in Figure 6.13. 

Example 6.2 (See also Appendix 1, examp62.m) Draw the Bode diagram for... 

Flenee the absolute log modulus at a = is 20 dB). The Bode diagram is given by Figure 6.14. Note in Figure 6.14 that the phase eurve was eonstrueted by reading the phase from Figure 6.11(b), an oetave either side of ain. 

Construet, on log-linear graph paper, using asymptotes, and validate using MATLAB or a similar tool, the Bode diagrams for... 

Phase Margin (PM) The phase margin is the ehange in open-loop phase, required when the open-loop modulus is unity, (or OdB on the Bode diagram) to make the elosed-loop system just unstable. 

The Nichols chart shown in Figure 6.26 is a rectangular plot of open-loop phase on the x-axis against open-loop modulus (dB) on the jr-axis. M and N contours are superimposed so that open-loop and closed-loop frequency response characteristics can be evaluated simultaneously. Like the Bode diagram, the effect of increasing the open-loop gain constant K is to move the open-loop frequency response locus in the y-direction. The Nichols chart is one of the most useful tools in frequency domain analysis. 

Figure 6.27 (see also Appendix 1, fig627.m) shows the Nichols chart for K = 4 (controller gain K = 1). These are the settings shown in the Bode diagram in Figure 6.23(a), curve (i), and (b), where... 

Thus from equation (6.99) it can be seen that the system designer has complete flexibility since, K, T and T2 are not linked. For a lead network, T must be greater than T2. The Bode diagram for an active lead network is shown in Figure 6.31. From equation (6.99)... 

Plot the Bode diagram on log-linear paper and determine... 

Construct, using asymptotes, the Bode diagram and read off values of open-loop modulus and phase for the following frequencies... 

The frequency of maximum phase advance is to occur at the frequency that corresponds to —180° on the Bode diagram constructed in section (a). The lower break frequency XjTx is to be half this value and the upper break frequency l/r2 is to be twice this value. Evaluate T and T2 and calculate values of 0 for the frequencies specified in section (a). Construct the Bode diagram for the compensation element for the condition K = X, and read off values of modulus at the same frequencies as the calculated phase values. 

Vary the magnitude of the frequency in the forcing disturbance and study its effect on the phase angle and amplitude ratio of the system. Use the information to construct the Bode diagram for the system, i.e., plots of phase angle and amplitude ratio versus frequency. 

Draw the Bode diagrams of the following transfer functions ... 

The Bode diagram for a first order system is given in Figure 7.45. 

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. 

This heuristic argument forms the basis of the Bode stability criterion(22,24) which states that a control system is unstable if its open-loop frequency response exhibits an AR greater than unity at the frequency for which the phase shift is —180°. This frequency is termed the cross-over frequency (coco) for reasons which become evident when using the Bode diagram (see Example 7.7). Thus if the open-loop AR is unity when i/r = —180°, then the closed-loop control system will oscillate with constant amplitude, i.e. it will be on the verge of instability. The greater the difference between the open-loop AR (< I) at coc and AR = 1, the more stable the closed-loop... 

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°... 

The Bode diagram of G (.r) is obtained by breaking down the transfer function into its constituent parts, plotting each separately and performing a graphical summation. 

The polar plot is an alternative to the Bode diagram for representing frequency response data and is the locus of all points occupied by the tip of a vector in the complex plane whose magnitude and direction are determined by the amplitude ratio and phase shift, respectively, as the frequency of the forcing function applied to the system is varied from zero to infinity. 

The Bode diagrams (in honour of H. W. Bode) constitute a convenient way to represent the frequency response characteristics of a system. As we can see from Eqs. (17.14a) and (17.14b), the amplitude ratio and the phase shift of the ultimate response of a system are functions of the frequency to. The Bode diagrams consist of a pair of plots showing ... 

Let us now examine the Bode diagrams of some simple dynamic systems that we have encountered in previous chapters. 

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