Phase lead compensation

A passive lead network (using two resistors and one eapaeitor) has a transfer fune-tion of the form 

To find cutn, differentiate equation (6.103) with respeet to lo, and equate to zero. This gives 

The value of 0m depends upon the spaeing of XjTx and l/T 2 on the logtu axis, see 

Set Kto a. suitable value so that any steady-state error eriteria are met. 

Plot the open-loop frequeney response and obtain the phase margin and the modulus erossover frequeney. (i.e. the frequeney at whieh the modulus passes through 0 dB) 

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The nice feature of the phase-lead compensator, and for that matter a real PD controller, is that it limits the high frequency magnitude. In contrast, an ideal PD controller has no upper limit and would amplify high frequency input noises much more significantly. 

Design a cascade lead compensator that will ensure stability and provide a phase margin of at least 30°, a bandwidth greater than 5rad/s and a peak closed-loop modulus Mp of less than 6dB. 

The open-loop transfer function is third-order type 2, and is unstable for all values of open-loop gain K, as can be seen from the Nichols chart in Figure 6.33. From Figure 6.33 it can be seen that the zero modulus crossover occurs at a frequency of 1.9 rad/s, with a phase margin of —21°. A lead compensator should therefore have its maximum phase advance 0m at this frequency. Flowever, inserting the lead compensator in the loop will change (increase) the modulus crossover frequency. 

Fig. 6.35 Bode gain and phase for lead compensator, design one.

Fig. 6.37 Open-loop bode gain and phase for design two lead compensator.

X Example 8.13. Derive the magnitude and phase lag of the transfer functions of phase-lead and phase-lag compensators. In many electromechanical control systems, the controller Gc is built with relatively simple R-C circuits and takes the form of a lead-lag element ... 

Here, z0 and p0 are just two positive numbers. There are obviously two possibilities case (a) z0 > po, and case (b) z0 < p0. Sketch the magnitude and phase lag plots of Gc for both cases. Identify which case is the phase-lead and which case is the phase-lag compensation. What types of classical controllers may phase-lead and phase-lag compensations resemble ... 

Example 8.14. Designing phase-lead and phase-lag compensators. Consider a simple unity feedback loop with characteristic equation 1 + GCGP = 0 and with a first order process... 

Can sense as the basis of phase-lead and phase-lag compensator design... 

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. 

Curve a in Fig. 7.62 shows the open-loop polar plot for the heat exchanger system described in Example 7.6. with Kc = 1.8 and t,= 2.5 (see also Example 7.8 and Fig. 7.55). Clearly this indicates an unstable system (Section 7.10.5). If a lead compensator with r, = 1 min and r2 = 0.1 min (r,/r2 = 10) is inserted into the loop, as shown in Fig. 7.63, then the system becomes stable (curve b in Fig. 7.62) due to the additional phase lead supplied by the compensator. (Using these values of r, and r2, Kc can now be increased by almost a factor of ten before the system becomes unstable). 

The properties of the lead compensator must be considered in order to select suitable values for rt and tj. The AR and phase shift of such a compensator are (from equations 7.104, 7.105 and 7.149) ... 

The output of the element represented by equation 7.155 lags the input. However, the destabilising effect of this additional lag is more than offset by an associated decrease in amplitude ratio. This decrease is more pronounced as the difference between r, and tj is increased. Lag compensators can be designed to produce different total open-loop stability specifications (e.g. in terms of allowable gain margin, phase margin, etc.) in a manner similar to that for lead compensators. 

V comp Phase shift of lead, lag or lag-lead compensator Vpm Angle representing phase margin on Nyquist diagram... 

The introduction of the gas phase leads to the formation of cavities behind the impeller blades. As a result, the power number and the impeller pumping capacity are reduced. Hence, the impeller speed has to be increased to compensate for the loss of pumping capacity. Consequently, the critical impeller speed for solid suspension was always higher in the presence of the gas phase (N g)... 

The photoisomerization of chiral azobenzene dopants often leads to the dramatic change in their HTPs. When it is used in combination with another non-photoresponsive chiral dopant with opposite chirality to form a compensated system, the change in HTP can be utilized to reversibly switch the LC phase between compensated nematic and cholesteric [49]. 

For the most part, n p and p Hp. The densities of electronic carriers fall into the relatively narrow range of 0.3 to 3 x 10 m . Unless otherwise noted, the values listed in Table 7.3 were calculated assuming w Pp = a. The advantages and limitations of using this approximation are discussed in Ref [44]. In previous reports, the assumption n wp was made instead, which leads to slightly different values for the electronic parameters. Whichever assumption is made, however, does not change the fundamental notion that many of the MAX phases are compensated conductors with n p and p pp, and a two-band model is needed to explain their electronic transport parameters. 

Elementary instrument servo with phase-lead (derivative) compensator stable... 

Even though certain CCC features, such as the relatively low efficiency when compared to HPLC, are considered drawbacks for this technique, they can be compensated by the characteristic selectivity and high loading capacity. Thus, the volume ratio of active stationary phase/mobile phase in the CCC column, usually around 80% in contrast to the less than 10% of HPLC columns, and the accessibility of this liquid stationary phase, lead to a much higher loading capacity with lower solvent consumption for a given amount of product processed in CCC. Moreover, problems related to adsorption of analytes on the support are avoided and the whole amount of sample injected can be recovered. These characteristics make CCC specially suited for preparative purposes whose scalability as process technique has already been demonstrated [18, 19]. 

In all of these oxide phases it is possible that departures from the simple stoichiometric composition occur dirough variation of the charges of some of the cationic species. Furthermore, if a cation is raised to a higher oxidation state, by the addition of oxygen to tire lattice, a conesponding number of vacant cation sites must be formed to compensate tire structure. Thus in nickel oxide NiO, which at stoichiomen ic composition has only Ni + cations, oxidation leads to Ni + ion formation to counterbalance the addition of extra oxide ions. At the same time vacant sites must be added to the cation lattice to retain dre NaCl sUmcture. This balanced process can be described by a normal chemical equation thus... 

The main consequences are twice. First, it results in contrast degradations as a function of the differential dispersion. This feature can be calibrated in order to correct this bias. The only limit concerns the degradation of the signal to noise ratio associated with the fringe modulation decay. The second drawback is an error on the phase closure acquisition. It results from the superposition of the phasor corresponding to the spectral channels. The wrapping and the nonlinearity of this process lead to a phase shift that is not compensated in the phase closure process. This effect depends on the three differential dispersions and on the spectral distribution. These effects have been demonstrated for the first time in the ISTROG experiment (Huss et al., 2001) at IRCOM as shown in Fig. 14. 

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