Bode- diagrams

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)... 

When T2 is greater than T equation (6.115) is an aetive lag network, whose Bode diagram is shown in Figure 6.40. The relationships between T, T2, uum and 0m are as given in equations (6.104) and (6.105), exeept, in this ease, 0m is negative. The same eomment applies to Figure 6.32 whieh shows the relationship between the spaeing of reeiproeals of T2 and T and 0m-... 

Fig. 6.41 Lag compensated and uncompensated open-loop bode diagram for Example 6.7.

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. 

The script file examp61h.m shows how it is possible to customize a Bode diagram... 

Example 6.2 Bode Diagram %Second-order system clf... 

After eleven iterations, hinfopt identifies that 7 in equation (9.176) has a best value of 0.13. The command sigma calculates the data for a singular value Bode diagram as shown in Figures 9.32, 9.34 and 9.35. Other information printed in the command window is given below... 

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. 

To use the Ziegler-Nichols rules, it is necessary to plot the open-loop Bode diagram without the controller. All other transfer functions are assumed to be unity. 

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. 

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