Polar plot system stability

In this statement, we have used "polar plot of G0l" to replace a mouthful of words. We have added G0L-plane in the wording to emphasize that we are using an analysis based on Eq. (7-2a). The real question lies in what safety margin we should impose on a given system. This question leads to the definitions of gain and phase margins, which constitute the basis of the general relative stability criteria for closed-loop systems. 

Example 13.4. Figure 13.Sa shows the polar plot of an interesting system that has conditional stability. The system openloop transfer function has the form... 

In Chap. 12 we presented three different kinds of graphs that were used to represent the frequency response of a system Nyquist, Bode, and Nichols plots. The Nyquist stability criterion was developed in the previous section for Nyquist or polar plots. The critical point for closedloop stability was shown to be the 1,0) point on the Nyquist plot. 

The lead compensator contributes phase advance to the system and thus increases the overall system stability (Section 7.10.4). The degree of phase advance provided is a function of frequency. At the same time this type of compensator increases the overall system amplitude ratio, which has the effect of reducing the the stability of the system. However, the major contribution of phase advance occurs at those frequencies where the open-loop polar plot is adjacent to the (-1,0) point on the complex plane. The increase in amplitude ratio takes place at lower frequencies and, consequently, the effect of this is much less significant. As the ratio of r,/r2 is increased, the maximum phase advance supplied by the lead compensator also increases, i.e. the greater is the stabilising effect of the compensating element011. 

There are two basic types of specifications commonly used in the frequency domain. The first type, phase margin and gain margin, specifies how near the openloop GM iu)Gc ia)) polar plot is to the critical (- 1,0) point. The second type, maximum closedloop log modulus, specifies the height of the resonant peak on the log modulus Bode plot of the closedloop servo transfer function. So keep the apples and the oranges straight. We make openloop transfer function plots and look at the (- 1, 0) point. We make closedloop servo transfer function plots and look at the peak in the log modulus curve (indicating an underdamped system). But in both cases we are concerned with closedloop stability. 

When a freshly polished metal surface is exposed to a slurry solution, the initial OCP of the system often exhibits a time-dependent behavior. These OCP transients indicate the degree of surface reconstmctions, which can arise from a metal s surface oxidation, passivation, dissolution, and/or from chemisorption of various solution species (Lagudu et al., 2013). The time needed for OCP stabilization indicates the extent of surface reorganization, and this is a useful feature of the transients. On the other hand, LSV-based measurements of Ecoir generate the polarization plots and icoir data these results are useful to evaluate the active/passive characteristics of a metal surface, as well as to examine the relative strengths of anodic and cathodic processes... 

The Nyquist stability criterion is similar to the Bode criterion in that it determines closed-loop stability from the open-loop frequency response characteristics. Both criteria provide convenient measures of relative stability, the gain and phase margins, which will be introduced in Section J.4. As the name implies, the Nyquist stability criterion is based on the Nyquist plot for GqiXs), a polar plot of its frequency response characteristics (see Chapter 14). The Nyquist stability criterion does not have the same restrictions as the Bode stability criterion, because it is applicable to open-loop unstable systems and to systems with multiple values of co or cOg. The Nyquist stability criterion is the most powerful stability test that is available for linear systems described by transfer function models. 

This approach (comprising, however, only the LW and EL forces) was developed independently by Derjaguin and Landau (1941) and Verwey and Overbeek (Verwey and Overbeek, 1948, 1999), and became known, after these authors, as the DLVO theory, and the corresponding energy vs. distance plots as DLVO plots. In the absence, or virtual absence, of polar (AB) interactions (i.e., in mainly apolar media), the DLVO theory correlates admirably with the stability of particle suspensions. However, in the cases of particle suspensions in polar and especially aqueous media, disregarding the influence of polar interactions by using simple DLVO plots usually leads to severely unrealistic models (van Oss et ai, 1990a). In other words, in all polar systems one should take into account AG j, as a function of , as ... 

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