Phase lag compensation

Using passive components, a phase lag compensator may be constructed, whose transfer function is of the form 

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

Identify what modulus attenuation is required to provide an aeeeptable phase margin and lienee determine the spaeing between 1/72 and l/T i (i.e. 6dB attenuation requires a one oetave spaeing, 12 dB attenuation needs a two oetave spaeing, ete.). 

Position l/T i one deeade below the eompensated modulus erossover frequeney, and lienee ealeulate ujm using equation (6.104). 

A process plant has an open-loop transfer function 

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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 ... 

The shape of the magnitude plot resembles that of a PI controller, but with an upper limit on the low frequency asymptote. We can infer that the phase-lag compensator could be more stabilizing than a PI controller with very slow systems.1 The notch-shaped phase angle plot of the phase-lag compensator is quite different from that of a PI controller. The phase lag starts at 0° versus -90°... 

From the perspective of a root locus plot, a phase-lag compensator adds a large open-loop zero... 

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... 

This compensation method now exhibits a -180 degree phase lag at low frequencies, then beginning at one-tenth the error amplifier s Alter pole (/ep) the phase lag increases to its high frequency limit of -270 degrees. 

This eompensation is intended for voltage-mode eontrolled forward eonverters whieh exliibit a seeond order output filter pole eharaeteristie. This would also inelude a quasi-resonant forward-mode eonverter that uses variable frequency, voltage-mode control. The T-C filter has a severe 180 degree phase lag and a -40dB/decade gain rolloff. To get any sort of wide bandwidth from the supply at all, this type of compensation must be used. 

The method performed above with the plaeement of the poles and zeroes will yield a minimum value for the exeess phase of 45 degrees, whieh is satisfaetory. If other pole and zero loeations are attempted, then loeate the maximum phase lag point of the L-C filter at the geometrie mean frequency between/ez2 and/epi. This will guarantee the best phase performance. The amount of phase boost of the compensation design will be... 

There are several other methods of achieving stability in potentiostatic circuits. A capacitor may be added between the counter and reference electrodes to reduce phase shift in the critical frequency region. Some caution must be exercised since a low-resistance reference electrode then becomes the counterelectrode at high frequencies. A particularly interesting method is known as input lead-lag compensation a series RC is connected between the input terminals of the control amplifier, and a second resistor is connected between the noninverting input and common. This form of compensation has minimum effect on the slew rate of the control amplifier. Further details can be found in the book by Stout and Kaufman listed in the bibliography. 

The presence of significant amounts of dead time in a control loop can cause severe degradation of the control action due to the additional phase lag that it contributes (see Example 7.7). One method for compensating for the effects of dead time in the control loop has been suggested by SMITH<30>. This consists of the insertion of an additional element which is often termed the Smith predictor as it attempts to predict the delayed effect that the manipulated variable will have upon the process output. 

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. 

Since the true value of the process dead time is not 0.8 but 1.0, the compensation is not perfect. There is a residual dead time equal to 1.0 - 0.8 = 0.2, which has not been compensated by the dead-time compensator. Thus uncompensated dead time gives rise to additional phase lag and leads eventually to a crossover frequency. If the ultimate gain is smaller than 100, the system with Kc = 100 is unstable. Indeed, for the... 

The stability of the laser wavelength can, of course, never exceed that of the reference wavelength. Generally it is worse because the control system is not ideal. Deviations AX(t) = -A.l(0 r cannot be compensated immediately because the system has a finite frequency response and the inherent time constants always cause a phase lag between deviation and response. 

It has been pointed out that dynamic compensation generally takes the form gq/gm- ft niay be recalled, however, that the ratio of two vector quantities like these resolves into the ratio of their magnitudes and the difference between their phase angles. Since dead-time elements have unity gain, their ratio is also unity their only contribution is phase lag. This is why the ratio gg/gm appears as the difference t, — r between the dead times. 

This equation expresses the phase lag between x-axis polarization and y-axis polarization when the light passes through normal to the sample. The polarizer and analyzer are placed normal to each other. They are placed at 45° with respect to the x andy axes. Botii components deviate by b only when they pass through the sample. Figure 10 illustrates the principle of the measiuement. Babinet s compensator is positioned between the sample... 

In this section we present an advanced control technique, time-delay compensation, which deals with a problematic area in process control—namely, the occurrence of significant time delays. Time delays commonly occur in the process industries because of the presence of distance velocity lags, recycle loops, and the analysis time associated with composition measurement. As discussed in Chapters 12 and 14, the presence of time delays in a process hmits the performance of a conventional feedback control system. From a frequency response perspective, a time delay adds phase lag to the feedback loop, which adversely affects closed-loop stabihty. Consequently, the controller gain must be reduced below the value that could be used if no time delay were present, and the response of the closed-loop system will be sluggish compared to that of the control loop with no time delay. 

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

Exploitation of this approach is straightforward in flow analysis the sample zone is handled, stopped inside the detector for a pre-set time interval and then directed towards waste. The initial and final measurements of the analytical signal are considered. The blank value is directly compensated by this approach and lag-phase effects become of minor concern, as demonstrated in the spectrophotometric determination of ethanol in whole blood [359]. The samples were injected without prior treatment other than dilution with a buffer solution and the stop period was 50—70 s. Sensitivity could be varied by selecting other stop periods. 

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