ECE 421

Nyquist Example #1

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 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 below is the Nyquist plot for this transfer function. The solid line is for positive frequencies, and the dashed 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 origin. 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 = -0.2. 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 the 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:

the number of poles at the origin
the difference between the total number of poles and the total number of zeros in G(s).
Specifically, let N = the number of poles of G(s) at s = 0, n = the total number of poles in G(s) (including those at s=0), and m = the total number of zeros in G(s). The following table gives the relationship between the value of the magnitude curve and those quantities and between the value of the phase curve and those quantities in the limits as frequency goes to 0 and goes to infinity -- under the assumptions of positive gain and no poles or zeros in the right-half of the s-plane. If the assumptions are false, the phase relationships are modified slightly, as described in the next set of examples (no changes in the magnitude relationships). Note that at low frequencies only N is a factor, and at high frequencies (n-m) is the deciding factor.

Value of
Magnitude Curve
Value of
Phase Curve
Inf if N > 0
M1 if N = 0
0 in N < 0
- 90*N degrees
0 if n > m
M2 if n = m
Inf if n < m
- 90*(n-m) degrees

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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Lastest revision on Wednesday, June 7, 2006 12:38 PM