Nichols Plots

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. 

For a given system and closed-loop gain displayed in the root locus plot, we can generate its corresponding time response (step and impulse) and frequency response (Bode, Nyquist, and Nichols) plots. 

We do not use Nichols plot (log magnitude versus phase) much anymore, but it is nice to know that we can do it just as easily ... 

In Chap. 12 we will learn this new Chinese language (including several dialects Bode, Nyquist, and Nichols plots). In Chap. 13 we will u.se frequency-domain methods to design closedloop feedback control systems. Finally, in Chap. 14, we will briefly discuss some frequency-domain and other methods for experimentally identifying a process. 

There are three different kinds of plots that are commonly used to show how magnitude ratio (absolute magnitude) and phase angle (argument) vary with frequency CO. They are called Nyquist, Bode (pronounced "Bow-dee ), and Nichols plots. After defining what each of them is, we will show what some common transfer functions look like in the three different plots. 

The final plot that we need to learn how to make is called a Nichols plot. It is a single curve in a coordinate system with phase angle as the abscissa and log modulus as the ordinate. Frequency is a parameter along the curve. Figure 12.24 gives Nichols plots of some simple transfer functions. 

Nichols plots, fj) First-Older tag (h) first-order Iead (c) deadtime (d) deadtime and lag (e) integrator (/) integrator and lag (g) second-order underdamped lag... 

At this point you may be asking why we need another type of plot. After all, both the Nyquist and the Bode plots are simply different ways to plot complex numbers. Well, as we will see in Chap. 13, all of these plots (Nichols, Bode, and Nyquist) are very useful for designing control systems. Each has its own individual application, so we have to learn all three of them. Keep in mind, however, that the main workhorse of our Chinese language is the Bode plot. We usually make it first since it is easy to construct from its individual simple elements. Then we use the Bode plot to sketch the Nyquist and Nichols plots. 

The Pjj s are points on Bode, Nyquisl, or Nichols plots of the distillation-column transfer functions. 

Sketch Nyquist, Bode, and Nichols plots for the following transfer functions 1... 

As we have seen in the three examples above, the contour usually is the only one that we need to map into the G B plane. Therefore from now on we will make only polar (or Bode or Nichols) plots of... 

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. 

Naturally we also can show closedloop stability or instability on Bode and Nichols plots. The ( — 1, 0) point has a phase angle of — IS0° and a magnitude of unity or a log modulus of 0 decibels. The stability limit on Bode and Nichols plots is, therefore, the (0 dB, —180°) point. At the limit of closedloop stability... 

Phase margins of around 45° are often used. Figure 13.7 shows how phase margin is found on Bode and Nichols plots. 

All the Nyquist, Bode, and Nichols plots discussed in previous sections have been for openloop system transfer functions B(j ). Frequency-response plots can be made for any type of system, openloop or closedloop. The two closedloop transfer functions that we derived in Chap. 10 show how the output is affected in a closedloop system by a setpoint input and by a load. Equation (13.28) gives the closedloop servo transfer function. Equation (13.29) gives the closedloop load transfer function. 

To use a Nichols chart, we first construct the openloop G, B Bode plots. Then we drawn an openloop Nichols plot of Finally we sketch this... 

J. (a) Make Bode, Nyquist, and Nichols plots of the system with Xj = 1 ... 

Nichols plots, (a) First-order lag. (b) First-order lead, (c) Deadtime. id) Deadtime and lag. ie) Integrator. (/) Integrator and lag. ig) Second-order underdamped lag. 

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