Draw Nyquist Plot By Hand
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.
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.
Change it to:
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:
For example the transfer function given above can be represented in the following form:
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:
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.
4. Find The Positions of “0+” & “+∞” On The Plot, And Connect Them
- 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″.
- 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.
- Mirror the nyquist path plotted in part 2 across the real axis.
- 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.