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+w^{2})
And, the imaginary part is:
Imag(G(jw))
= jw/(1+w^{2})
- or you may prefer that we express this
as:
Imag(G(jw))
= w/(1+w^{2})
- 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 -45^{o} is important.
You can read the frequency from the clip. Determine the frequency
on the clip at which the phase is closest to -45^{o}.
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 -45^{o}.
The frequency response
function is G(jw)
= 1/(jw
+ 1).
The phase angle is -45^{o}
when the angle of the denominator is +45^{o}.
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 -90^{o}.
That makes sense because the phase approaches -90^{o} 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/(jw^{n-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 -90^{o} 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 -270^{o}.
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 -90^{o} 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 -180^{o} 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 -270^{o} 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?
90^{o}
-90^{o}
-180^{o}
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 -90^{o} 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 -270^{o}.
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 -270^{o}.
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 -90^{o}.
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 -90^{o} 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, - 180^{o}.
For low frequencies, the
Nyquist plot goes to infinity - at an angle of -90^{o}.
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 -135^{o}.
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/(-w^{2}
+ jw)
Multiply numerator and
denominator by the conjugate of the denominator, to obtain the following.
= - w^{2}/(
w^{4}
+ w^{2})
- jw/(
w^{4}
+ w^{2})
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 -180^{o}.
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.