Draw Nyquist Plot By Hand: Learn How to Determine Stability

• Bypass
In summary, the conversation discusses how to draw a Nyquist plot for a transfer function by substituting s with jω. The conversation also mentions choosing different values for ω and calculating the complex value of the denominator to plot the result. The person also asks for key points to plot in order to determine the stability of a closed loop system.
Bypass
I am trying to understand how to draw nyquist plot.
Lets say the transfer function is

Subbing in jω for s,

***Note: should be (jw)^4 , (jw)^3, (jw)^2

Then separating the Real and Imaginary part,

So when
w = 0, in the nyquist plot it is infinity
w = infinity, it is 0
Imaginary intercept is 1.25
For real intercept I am not sure. Since imaginary part is 0 only when w is infinity, i plug in infinity for w in real part. Would the real intercept be infinity or 0? It would be infinity/infinity but numerator is lower order than higher order so would it be 0 instead?

In any case, how am i supposed to plot the rough nyquist plot or at least be able to determine the stability using these 4 points?

In Matlab the Nyquist plot comes out like this

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Last edited:
Sorry, I cannot see what you are doing. You have:

Substituting s with jω, you should get:

50 / ( ( jω )4 + 5( jω )3 + 4( jω )2 ) =

50 / ( ω4 - j5ω3 - 4ω2 )

Now, choose a ω and calculate the complex value of the denominator. Do the division and plot the result.

Example: ω=1 → point = ( -4.412 + j7.353 )

Last edited:
Hesch said:
Sorry, I cannot see what you are doing. You have:

Substituting s with jω, you should get:

50 / ( ( jω )4 + 5( jω )3 + 4( jω )2 ) =

50 / ( ω4 - j5ω3 - 4ω2 )

Now, choose a ω and calculate the complex value of the denominator. Do the division and plot the result.

Example: ω=1 → point = ( -4.412 + j7.353 )

Sorry about that, just a typo. But it is not possible to plug in every single point that encircles the RHP of S-Plane to see how many times the nyquist plot encircles -1 to determine the stability. So my question is what are some key points that I need to plot so I can determine stability of a closed loop system.

Try ω = 1, 2, 3, 4 . . . .

If distances are to small then continue: . . . . 8, 16, 32

If distance is to large between ω=2 and ω=3, then try ω=2.5. It's a "cut and try" process.

1. How do I draw a Nyquist plot by hand?

To draw a Nyquist plot by hand, you will need to plot the real and imaginary parts of the transfer function on a polar coordinate system. Start by determining the frequency range and plotting the corresponding values for the real and imaginary parts. Then, connect the points to create a curve that represents the Nyquist plot.

2. What is the purpose of drawing a Nyquist plot?

The Nyquist plot is used to determine the stability of a system by analyzing the frequency response. It helps to identify if the system is stable, marginally stable, or unstable.

3. How do I determine stability from a Nyquist plot?

To determine stability from a Nyquist plot, you need to look at the number of encirclements of the -1 point on the real axis. If the curve encircles the point in a clockwise direction, the system is unstable. If it does not encircle the point or encircles it in a counterclockwise direction, the system is stable.

4. What are the key characteristics of a stable Nyquist plot?

A stable Nyquist plot will have no encirclements of the -1 point on the real axis, indicating that the system is stable for all frequencies. It will also have a curve that approaches the origin at higher frequencies, indicating a stable and well-damped system.

5. Can I use a computer program to draw a Nyquist plot?

Yes, there are many computer programs and software that can automatically generate a Nyquist plot from a transfer function. However, it is important to understand how to draw it by hand to fully grasp the concept and interpretation of the plot.

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