Sketching Nyquist Diagrams for Transfer Functions: A Helpful Guide"

[Neofit]

Homework Statement

I need to be able to sketch Nyquist diagrams for transfer functions. I spent a lot of time but I cannot wrap my head around the idea of mapping the GH(jw) in the complex plane. Let's consider the following example for this question:
$$GH(s) = \frac{4} {s(s+2)^2}$$

2. The attempt at a solution
The transfer function is factored as $$\frac{4}{jw}\times\frac{1}{jw+2}\times\frac{1}{jw+2}$$
$$MAG \frac{4}{jw}=\frac{4}{w}; ARG=\frac{-pi}{2}$$
$$MAG \frac{1}{jw+2}=\frac{1}{ \sqrt{w^2\times2^2} }; ARG=-tan^-1{w}$$
Then I make a table for all three elements of the transfer function and calculate their magnitude and angle for some values of frequency, including zero and infinity. After that, I convert the resultant polar coordinates to rectangular and plot them. This is what I understand I need to do, and it is not working.

I have had only one lecture on this topic and did not had a chance to ask the professor anything. Please offer some help - I really need it! An example on how to sketch the above TF's Nyquist diagram will be very appreciated. Also, how can I decide what frequency values to use in the calculation?

Thanks

 

[Hesch]

I don't know Nyquist diagrams, but anyway I will suggest:

H(s) = 4/(s(s+2)²) =>
H(s) = 4/(s(s²+4s+4)) =>
H(s) = 4/(s³+4s²+4s)

Now substitute s by jω and do some calculations

H(jω) = 4/(-4ω²+j(4ω-ω³))

Say ω=1, you will get

H(jω) = 4/(-4+j3) = (-0.64 - j0.48) No ARG, no MAG, just plot it as is.

how can I decide what frequency values to use in the calculation?

Well, try ω=0, 1, 2, 3 . . .
If a too big hole appears between 1 and 2 then try a value in the middle.

 

[Neofit]

Thank you very much! Because of your explanation I finally understand how to do the diagrams.