Draw Nyquist Plot By Hand

nyquist-example1

What is a Nyquist Plot?

Nyquist plot is a plot used mostly in control and signal processing and can be used to predict the stability and performance of a closed-loop system.

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Use the following instructions to draw Nyquist plot by hand from a transfer function.

1. Change Transfer Function From “s” Domain To “jw” Domain

First, If the transfer function G(s) is given in “S” domain, transfer it to “jw” domain.

Example:

eq11

Change it to:

eq21

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2. Find The Magnitude & Phase Angle Equations

Write an equation explaining the Magnitude and Phase Angle of the transfer function (now in “jw” form) that would look like:

eq3

For example the transfer function given above can be represented in the following form:

eq4

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3. Evaluate At Point “0+” and “+∞” points

Evaluate the magnitude and phase angle equations found above, at ω (omega) values of “0+” and “+∞” points.

In the above example:

eq51

Note 1: The ω (Omega) value of “0+” means an angle very close to zero but slightly larger. The ε (epsilon) in the phase angle (in example above) is due to ω being slightly larger than zero. This will be later used in drawing the nyquist plot.

Note 2: In above example, evaluating the phase angle (ω), at “0+” yeilds a phase angle of “-180 – ε”. The reason is that a slightly greater angle than zero would produce slightly greater “tangent” than zero.

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4. Find The Positions of “0+” & “+∞” On The Plot, And Connect Them

  1. Using the values found from the above section, find the positions of “0+” and “+∞” on the Real and Imaginary axis: In the above example, the point at “0+” is located at “-180 - ε” degrees which is slightly more negative than “-180″.
  2. Connect the points together. The second point is at “0″ on real axis with “-90″ degrees. Therefore the nyquist path coming from the “ω=0+” should approach the “ω=+∞” at a “-90″ degrees. The curvy path is not exact as we are only drawing the plot by hand.
  3. Mirror the nyquist path plotted in part 2 across the real axis.
  4. Connect the “ω=0-” to “ω=0+”. This should be done clock-wise. While in this example’s case the clock-wise path is the closest, that is not the case all the time.

Nyquist Plot

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Here is a second Nyquist Plot Example to help you better understand the topic.