The Nyquist plot is a graph of the magnitude and phase of a transfer
function evaluated along the jw axis, with the graph displayed as real part
vs. imaginary part or magnitude vs. phase. The Nyquist plot contains the
same magnitude and phase information as the Bode plot. In the Nyquist plot,
however, there is only a single graph, and frequency is not explicitly shown
in the plot; it is a parameter along the graph.
Perhaps the main use of the Nyquist plot in control system analysis is the application of the Nyquist stability criterion, discussed in later examples. The Nyquist plot and stability criterion can be applied to transfer functions with poles and/or zeros in the right-half of the s-plane. The usual interpretation of a Bode plot limits its application to transfer functions having no poles or zeros in the right-half plane. In that respect, the Nyquist plot is more general than the Bode plot.
A simple system is represented by the transfer function
where the gain K = 1. The MATLAB function "nyquist" can be used to compute the real and imaginary parts of the transfer function. The user can specify a set of frequency values or have MATLAB choose the frequencies at which the function is to be evaluated. With no output arguments, "nyquist" will make a plot of the data; with output arguments, no plot is made.
The plot linked below is the Nyquist plot for this transfer function. The
solid (red) line is for positive frequencies, and the dashed (green) line
is for negative frequencies. Three specific frequencies are shown on the
graph: w = 0.1 r/s, w = 1 r/s, and w = 10 r/s. The line drawn from the
origin to the point on the graph at w = 1 r/s represents the transfer
function G(jw) evaluated at w = 1 r/s. The length of the line is the
magnitude and the angle that the line makes with respect to the positive
real axis is the phase angle. The real and imaginary parts of G(j1) can
be read off the axes at that point.
Nyquist plot for G1(s)
From the plot, notice that the graph starts at a magnitude of 1 and a phase angle of 0 degrees, which is G(j0). As frequency goes to infinity, the magnitude goes to 0 (more poles than zeros in G(s)), and the phase goes to -90 degrees (1 more pole than zero in G(s)). Also note that for negative frequencies, the graph is the mirror image about the real axis of the graph for positive frequencies. The Nyquist plot is always symmetric about the real axis.
If the gain in the transfer function is changed, the Nyquist plot is changed
in a very simple manner. If the gain is kept positive, then changing the
gain changes the magnitude of the transfer function at each frequency,
without changing the phase angle. Therefore, the graph just gets larger or
smaller, corresponding to increases or decreases in gain. If the gain is
made negative, 180 degrees is added to the phase at each frequency, so the
entire curve is rotated by that amount. The next figure shows the Nyquist
plot for the previous transfer function for gains K of
{1, 2, 4, -1, -2, -4}. Only positive frequencies are shown; negative
frequencies would give the mirror images of the graphs shown.
Effects of gain changes on G1(s)
A second system is described by the transfer function
With three poles and no zeros, the phase (for w > 0) decreases monotonically
from 0 degrees to -270 degrees, and the magnitude decreases monotonically
from 1 to 0. For negative frequencies, the mirror image is obtained. Three
frequencies are again indicated on the graph to provide a sense of scale to
the plot. Notice that for frequencies greater than 10 r/s, the graph is
essentially at the orgin. Without making a second plot with an expanded
scale, no information is available in that part of the plot. That is one
advantage of Bode plots, where both very large and very small magnitudes
are visible due to the logarithmic scale used.
Nyquist plot for G2(s)
Another system is described by the transfer function
This transfer function shows the effect of zeros on the Nyquist plot. Each
zero (in the left-half plane) has a phase which goes from 0 degrees to +90
degrees as frequency varies from 0 to infinity. The smallest pole or zero
in this transfer function is the pole at s=-0.4. Therefore, the Nyquist
graph starts having negative phase due to that pole. The three zeros at
s=-1 then start providing positive phase, with the result that the phase
goes positive and the magnitude increases. The three poles at s=-5
provide negative phase and decreasing magnitude. As frequency goes to
infinity, the net result is a magnitude of 0 (more poles than zeros) and a
phase of -90 degrees (1 more pole than zero). The frequency of w=2.089 r/s
is the frequency at which the maximum positive phase shift occurs, and was
obtained from the data.
Nyquist plot for G3(s)
The last system is described by the transfer function
This system also has a collection of poles and zeros. The phase goes
negative first since the smallest term is the pole at s=-0.1. The phase
then goes positive and the magnitude increases due to the zeros at s=-0.6
and s=-4. Finally, the phase goes negative and the magnitude decreases
due to the poles at s = -20, s=-130, and s=-1000. The net result is a
magnitude of 0 (more poles than zeros) and a phase of -180 degrees (2 more
poles than zeros).
Nyquist plot for G4(s)
Assuming that the gain is positive and that there are no poles or zeros in
right-half of the s-plane, then starting (w = 0) and ending (w =
+infinity) points of the graph in a Nyquist plot are determined by two
things:
Magnitude Curve |
Phase Curve |
|
M1 if N = 0 0 in N < 0 |
||
M2 if n = m Inf if n < m |
where M1 and M2 are the magnitudes of the transfer function when it is evaluated at s=0 and s=infinity, respectively, given by
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Latest revision on 8/27/97
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