Nyquist Plots
Why Nyquist Plots?
What Is A Nyquist Plot?
High Frequency Asymptotes
Systems With Poles At The Origin
What If?
Problems
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Why Nyquist Plots?

        Nyquist Plots are a way of showing frequency responses of linear systems.  There are several ways of displaying frequency response data, including Bode' plots and Nyquist plots.

  Bode' plots use frequency as the horizontal axis and use two separate plots to display amplitude and phase of the frequency response.
  Nyquist plots display both amplitude and phase angle on a single plot, using frequency as a parameter in the plot.
  Nyquist plots have properties that allow you to see whether a system is stable or unstable.  It will take some mathematical development to see that, but it's the most useful property of Nyquist plots.
        Nyquist Plots were invented by Nyquist - who worked at Bell Laboratories, the premiere technical organization in the U.S. at the time.  He was interested in designing telephone amplifiers to be placed in ocean-floor cables.  In those days, between the first and second world wars, undersea cables were the only reliable means of intercontinental communication.

        Undersea telephone cables needed to be reliable, and to have a constant gain that did not change as the amplifier aged.  In those days, electronic amplifiers were constructed with tubes, and tubes had gains that could change dramatically as they aged.

        The solution to the aging problem was to design feedback amplifiers.  However, those amplifiers could become unstable.  One morning - going to work on the Staten Island ferry, before the Verrazano Narrows bridge - Nyquist had an inspiration, and wrote his work, literally, on the back of an envelope as he rode.  Today, millions of control system students are tortured by instructors making them apply the Nyquist Stability criterion, and it is widely used in control system design.

        So, what is a Nyquist plot anyway?

  A Nyquist plot is a polar plot of the frequency response function of a linear system.

  That means a Nyquist plot is a plot of the transfer function, G(s) with s = jw.  That means you want to plot G(jw).

G(jw) is a complex number for any angular frequency, w, so the plot is a plot of complex numbers.

The complex number, G(jw), depends upon frequency, so frequency will be a parameter if you plot the imaginary part of G(jw) against the real part of G(jw).

        In this lesson, we will introduce you to Nyquist plots - what they look like for different kinds of systems.  You need to think about what you will get from this lesson.  Here are the goals.
  Given a Transfer Function:
  Be able to sketch a Nyquist plot, manually
       including the following:

What Is A Nyquist Plot?

       An example of a Nyquist plot will illustrate what a Nyquist plot is.

We will take a very simple system:  G(s) = 1/(s+1).
If we substitute s = jw, we get G(jw) = 1/(jw + 1).
Now, compute the real and imaginary parts of G(jw) by converting the denominator to a real number.
or:

Now, the real part of the frequency response function is:

Real(G(jw)) = 1/(1+w2)

And, the imaginary part is:

Imag(G(jw)) = jw/(1+w2)

- or you may prefer that we express this as:

 Imag(G(jw)) = w/(1+w2) - leaving off the j.

        Now, to generate a Nyquist plot we would need to plot the imaginary part on the vertical axis of a plot, and the real part on the horizontal axis.  Here is a video of that operation.

        The point at which the phase angle becomes -45o is important.  You can read the frequency from the clip.  Determine the frequency on the clip at which the phase is closest to -45o.

        Now, since the transfer function, G(s), is 1/(s + 1) for this example, we can determine what should have been the answer, not just the closest frame on the video.  Let's determine the frequency at which the phase angle is -45o.

        The video of the Nyquist plot isn't really a true Nyquist plot.  A true Nyquist plot shows the frequency response function for all frequencies, not just a single -albeit moving - point.  So, let's take a look at the Nyquist plot for G(s) = 1/(s + 1).  Here it is!

Now, let us look at some interesting points in this Nyquist plot.

        What's wrong with all of this?  Is there something else we should note?         At this point, you have seen one Nyquist plot.  We need to consider a few more points about Nyquist plots.         First, let us examine a few general properties of Nyquist plots.  Then, there are a number of special cases that you need to understand.


High Frequency Asymptotes

        There are other points you need to note about Nyquist plots. Let's start by considering how a Nyquist plot is affected when the system has a higher order.
 

n  > m, i.e.

#Poles > # Zeroes

n = m, i.e.

#Poles = # Zeroes

        Now, let us assume - at least for the moment - that: Then, if we let the frequency become very large.  In the limit, each jw term will "overpower" the corresponding z or p term in G(jw) and we will have:

G(jw) ~= 1/(jwn-m)

        The angle of this limiting form is what we are interested in now, and the angle is determined by the j-term.

        Here are some examples.  For each example, think about the asymptotes, then click on the hot word or the Nyquist plot to show the high frequency asymptote when you have determined what the angle should be.
        The example third order system is not easily seen.  However, you can change the scale for that system, and see things more clearly.  If you have a problem seeing the asymptote you may want to change scales when you have to do this kind of analysis.

        Now, here's a question for you.


Problems

1.  What is the high frequency asymptote of a system that has three poles and two zeroes?

  90o
 -90o
-180o



        If you had problems with the problem, remember, the high frequency expresion is:

G(jw) -> K/(jw)n-m for large w

Here is another example.
        There are other interesting things that can happen.
        There are numerous other peculiarities that you can find in these plots, but were are going to go on to some special cases that are important.

Systems With Poles At The Origin

        Systems with poles at s = 0 - otherwise referred to as poles at the origin - present interesting complications on Nyquist plots.  Let's look at the problem and examine a simple system with a pole at the origin.

        Clearly, when there is a pole at the origin, the frequency response approaches infinity as frequency approaches zero.  To get a better understanding of exactly what happens, we will look at a specific example.
        In our example system, the frequency response may be better viewed in a video.  Here's a video of that frequency response.

        There's one interesting observation about this particular frequency response.  For low frequencies, the phase angle is very close to zero.  However, looking at the plot - with an scale that shows more of the low frequency behavior - it appears that the low frequency portion of the plot is not asymptotic to the negative imaginary axis.  That is, in fact, the situation.  The real part takes on a fixed value while the imaginary part goes toward negative infinity as the frequency approaches zero on the plot.         Let's compute the real part of this frequency response.
G(jw) = 1/[jw(jw + 1)] = 1/(-w2 + jw)
= - w2/( w4 + w2) - jw/( w4 + w2)
        There are other situations you should be aware of.  Systems with more than one pole at the origin can have even more interesting behavior.  Here is a system with two poles at the origin.  The transfer function is 1/s2(s + 1).  Here the low frequency asymptote doesn't even approach a constant.  Yet, the low frequency phase is -180o.

What If?

        Clearly there are going to be a lot of little points that can produce interesting results in a Nyquist plot.  Some things to be wary of include the following:

        Your assignment now, is to investigate these possibilities.  You can do that in either Mathcad or Matlab.  In either case, try the possibilities listed above, and see what you can invent on your own.  Follow up on your curiosity and see what you can find.  You may even produce some interesting artwork since some Nyquist plots can look like rosettes and other artful things.  Have fun.
 


        Let us examine how you would actually generate a Nyquist plot.         Some of these chores may be taken care of automatically, if you use a control system or mathematical analysis package.  Still, you should understand that choices will be made for you if you don't make them yourself.  One important item is the choice of frequencies.  Consider some of your options.         Considering these two options, you will almost always find that an evenly spaced set of frequencies will really produce points that are "jammed" together at the higher frequencies.  Logarithmically spaced frequencies are perfect for Bode' plots because they produce points evenly spaced on a logarithmic frequency scale, but the same choice works pretty well for Nyquist plots.  When you use a package - Mathcad or Matlab, for example - that choice of point density will often work best.


Problems