Nyquist Plots
Why Nyquist Plots?
What Is A Nyquist Plot?
High Frequency Asymptotes
Systems With Poles At The Origin
What If?
Problems
You are at:  Basic Concepts - System Models - Frequency Response - Nyquist Plots

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:
• High frequency asymptote,
• Low frequency asymptote or DC gain point,
• Be able to show the direction of increasing frequency along the Nyquist plot.
• Use an analysis program like Mathcad or Matlab to get a Nyquist plot.

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 frequency response function is G(jw) = 1/(jw + 1).
• The phase angle is -45o when the angle of the denominator is +45o.
• The angle of the denominator is tan-1(w).
• Solving for the frequency, w, we get w = 1.0.
• If w = 1.0, then f =  w/2p = .159 Hz.
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.

• The low frequency portion of the plot is near +1.  That makes sense since the DC gain is 1 for G(s) = 1/(s + 1).
• The high frequency portion of the plot is near the origin in the G(jw) plane.  That makes sense because the magnitude becomes small as frequency gets large.
• The high frequency portion of  the plot approaches the origin at an angle of -90o.  That makes sense because the phase approaches -90o as the frequency gets large.
What's wrong with all of this?  Is there something else we should note?
• The frequency is a parameter of the plot, and unless we do something, there will be no indication of what frequency corresponds to a particular point on the plot.
• We can indicate direction of increase of frequency with small arrows along the plot.  Using those arrows is  more- or-less standard practice.
• We will always assume that the plot starts at zero frequency and frequency goes to infinity.
At this point, you have seen one Nyquist plot.  We need to consider a few more points about Nyquist plots.
• You need to learn what Nyquist plots look like for different systems, including second order systems, higher order systems, systems with resonant peaks and systems with poles and zeroes at the origin of the s-plane.
• You need to learn how you can generate 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.

• First, consider a more general transfer function.  Most transfer functions are a ratio of polynomials in s.  Here is a typical example - shown in factored form.
• This system has m zeroes.
• This system has n poles.
• It is almost always true that the denominator is of higher order than the numerator so,
n  > m, i.e.

#Poles > # Zeroes

• Although, on occasion we have:
n = m, i.e.

#Poles = # Zeroes

• The system has n poles and m zeroes.
• We remind you that a stable system will have all of the poles in the left half of the s-plane, so all of the p's will be negative in G(s).  Also, zeroes will usually be in the left half of the s-plane, but it's possible that is not the case.
Now, let us assume - at least for the moment - that:
• The transfer function has no poles at s = 0.  We normally say that the system has no poles at the origin.
• We are going to examine the behavior of the Nyquist plot for large frequencies.
• Let s = jw in the transfer function to obtain:
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.

• The angle is determined by the power of j.  You get -90o for every j.
• For example, if n = 4, and m = 1, then n - m = 3, and for high frequencies the Nyquist plot would have an angle of -270o.
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.
• A first order system, G(s) = 1/(s + 1)
• Click on the button to see the high frequency asymptote.
• The high frequency asymptote is at -90o which is where it should be for a system with one more pole than zero.
• A second order system, G(s) = 1/(s + 1)2
• Click on the button to see the high frequency asymptote.

• The high frequency asymptote is at -180o which is where it should be for a system with two more poles than zeroes.
• A third order system, G(s) = 1/(s + 1)3
• Click on the button to see the high frequency asymptote.

• The high frequency asymptote is at -270o which is where it should be for a system with three more poles than zeroes.
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

• The angle of this limiting form is determined by the j-term.
• The angle is determined by the power of j.  You get -90o for every j.
• For our example, we have n = 3, and m = 2, so n - m = 1, so for high frequencies the Nyquist plot would have an angle of -270o.
Here is another example.
• Here's a system with 5 poles and 2 zeroes.
• Some poles are repeated - double - poles.
• The complete Nyquist plot is below.
• While the plot is fairly complex - especially for a system without any complex poles, the high frequency asymptote is still -270o.
• Some interesting details of the plot include the following.
• With low frequency zeroes, the phase lead from the zeroes starts to overtake the phase lag of the one low frequency pole.
There are other interesting things that can happen.
• Systems with complex poles can have resonant peaks.  Here is the transfer function of a system with complex poles.
• Here is the Nyquist plot for the system.
• The resonant peak causes the magnitude of the response to be larger than the DC gain (which is 1.0).  In the plot the DC gain is 1.0, and the plot starts from 1.  As the frequency increases the magnitude gets larger as the angle becomes negative.  It's a little larger than 1.6 when the phase reaches -90o.
• It's also interesting to look at the density of points on the plot.  For these plots we have been using a logarithmic frequency spacing.  Looking at just the points, with no connecting lines between points, the plot below shows what you get.  In this plot we made the points a little larger (and messier, apparently!) than in the previous plots.  The negative frequency portion remains the same for comparison.
• Notice how the pointson the plot are more widely spaced when the magnitude is larger.
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.

• The transfer function form is shown below.  The transfer function shown has a single pole at the origin, s = 0, and another pole at s = -1.
• Substituting s = jw  we have the G(jw) shown below.
• The j term in the denominator contributes a constant -90o to the phase.
• The real problem is encountered when we consider what happens at low - zero, or almost zero - frequency.
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.
• The system transfer function is shown below again.  It has a pole at s = 0, and at s = -1.
• The high frequency asymptote is what we expect, - 180o.
• For low frequencies, the Nyquist plot goes to infinity - at an angle of -90o.
In our example system, the frequency response may be better viewed in a video.  Here's a video of that frequency response.

• Use the arrow buttons to control the frequency of the point being displayed.
• Adjust the frequency to get a phase angle of -135o.
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.
• There's a misconception - easy to pick up - that the plot should approach the axis.
• You can find that misconception drawn as though it were true in many textbooks.  Check it out.
• But this plot is correct, and we can show that.
Let's compute the real part of this frequency response.
• Start by manipulating the frequency response function.
G(jw) = 1/[jw(jw + 1)] = 1/(-w2 + jw)
• Multiply numerator and denominator by the conjugate of the denominator, to obtain the following.
= - w2/( w4 + w2) - jw/( w4 + w2)
• The interesting point is that the real part approaches -1 as the frequency gets small.
• For low frequencies, the real part stays at -1 even though the imaginary part becomes negatively infinite for frequencies approaching zero.
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:

• Systems with resonant peaks.
• Multiple poles at the origin.
• Zeroes at the origin.
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.
• We assume you have some transfer function, G(s).  Normally, that transfer function would be a ratio of polynomials, and it may be expressed in terms of zero and pole factors in the numerator and denominator.
• You need to determine a range of frequencies for which you want to plot the Nyquist plot.
• Normally, you would want to include frequencies that extend above the highest corner frequency and below the lowest corner frequencies.   (Or above and below the resonant frequencies if there are complex poles.)
• Within the range of frequencies you need to select frequencies at which you will compute G(jw).
• For the selected frequencies you need to compute G(jw).
• Finally, plot Im(G(jw)) against Re(G(jw)) for all of the selected frequencies.
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.
• You could use an evenly spaced set of frequencies - like 1,2,3 . . .1034,1035 . .
• You could use a set of frequencies that are evenly spaced logarithmically.
• For frequency response plots, logarithmic spacing often works best, and that's the best explanation for the popularity of Bode' plots.
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