Phase margin

Phase margin. This is the value of the phase of a closed-loop system at the gain cross-over frequency (G(s) = OdB). 

Next find the closed-loop gain cross-over frequency by deciding how much phase margin you desire in your system. A good value is 45 degrees. Ignoring any effect of the Q of the T-C filter, the gain cross-over point is found from... 

Fig. 6.19 Gain margin (GM) and phase margin (PM) on the Nyguist diagram.

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. 

Time domain performanee speeifieations are deseribed in seetion 3.7 and Figure 3.21. Frequeney domain performanee speeifieations are given in terms of gain and phase margins to provide adequate stability together with information relating to the elosed-loop frequeney response. Figure 6.24 shows the elosed-loop frequeney response of a eontrol system. The elosed-loop modulus is usually defined as... 

Gain margin = 8.15dB Phase margin = 60 Bandwidth = 2.02rad/s... 

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)... 

Set dJm to the modulus erossover frequeney and estimate the phase advanee 0m required to provide a suitable phase margin. From equations (6.104) and (6.105), determine T and T2. 

Plot the eompensated system open-loop frequeney response. Note that the modulus erossover frequeney has now inereased. Reduee the eompensator gain K so that the modulus erossover frequeney returns to its original value, and the desired phase margin is met. 

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. 

From Figure 6.34 it ean be seen that to aehieve the desired phase margin of at least 30° then the eompensator must provide in the order of an additional 20° of phase advanee, i.e. 57° in total, at the modulus erossover frequeney. 

Closed-loop peak Mp = 5.5 dB Gain margin = 13.75dB Bandwidth = 5.09 rad/s Phase margin = 30.6 ... 

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.). 

The compensated and uncompensated open-loop frequency response is shown in Figure 6.41. From this Figure the compensated gain margin is 12.5 dB, and the phase margin is 48°. In equation (6.117), K does not need to be adjusted, and can be set to unity. When responding to a step input, the steady-state error is now 4.6%. 

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. 

Find the frequency where the magnitude Gc(jto)Gp(jco) is 1. This particular frequency is the phase crossover frequency, to = tocp. We then find the angle between GcGp and -180°. This is the phase margin, PM. The formal definition is... 

The Nyquist stability criterion can be applied to Bode plots. In fact, the calculation using the Bode plot is much easier. To obtain the gain margin, we find the value of GCGP which corresponds to a phase lag of-180°. To find the phase margin, we look up the phase lag corresponding to when GCGP is 1. 

The gain and phase margins are used in the next section for controller design. Before that, let s plot different controller transfer functions and infer their properties in frequency response analysis. Generally speaking, any function that introduces additional phase lag or magnitude tends to be destabilizing, and the effect is frequency dependent. 

The concept of gain and phase margins derived from the Nyquist criterion provides a general relative stability criterion. Frequency response graphical tools such as Bode, Nyquist and Nichols plots can all be used in ensuring that a control system is stable. As in root locus plots, we can only vary one parameter at a time, and the common practice is to vary the proportional gain. 

With the straight textbook definition and explanation, the gain margin and phase margin of the... 

We can also use the Bode plot of G to do phase margin calculations. From the textbook definition, we are supposed to find the phase angle ( > = ZG0L where G0L = 1. If the phase margin is 45°, should be -135°. It appears that we need to know Kc beforehand to calculate GOL, but we do not. 

We use the fact that ZG is identical to ZGOL, and cocp is the same in both plots, so we can go backward. On the G Bode plot, we can find the value of G (jcocp) which corresponds to a phase lag of, say, -135°. Now G0L = KC G and since G0L = 1 at the phase margin, we can find K,. = l/ G (joocp) that will provide a phase margin of 45° without trial-and-error. 

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