Nyquist: transfer function of G(s) = 1 / (2s+1) (s^2+s+1)

[SarahJennings]

Good Morning,

Im currently very stuck with Nyquist work.

I have the transfer function of

G(s) = 1 / (2s+1) (s^2+s+1)

I have to determine the expression for the magnitude |g(jw)|

I have come up with |g(jw)| = 1 + jw +jw^2 / 1+2jw

However i don't think its correct. My next part is to determine the expression for the phase then find the start and end of the Nyquist plot where w = infinity and w =0. (example phase = 0 degrees or 180 degrees and phase = + or - 90 degrees respectively.

Then i am to sketch the Nyquist diagram.

Can anyone help me with this as i have spent over a week struggling now!

Thanks,

Sarah

 

[Ishuda]

Good Morning,

Im currently very stuck with Nyquist work.

I have the transfer function of

G(s) = 1 / (2s+1) (s^2+s+1)

I have to determine the expression for the magnitude |g(jw)|

I have come up with |g(jw)| = 1 + jw +jw^2 / 1+2jw

However i don't think its correct. My next part is to determine the expression for the phase then find the start and end of the Nyquist plot where w = infinity and w =0. (example phase = 0 degrees or 180 degrees and phase = + or - 90 degrees respectively.

Then i am to sketch the Nyquist diagram.

Can anyone help me with this as i have spent over a week struggling now!

Thanks,

Sarah

First of all, the magnitude of something (function, number, phasor, ...) is a real non-negative number so you definitely have the magnitude of g(jw) incorrect [I assume g and G are the same here].

Next, you seem to have a misunderstanding of the Nyquist plot. As you have indicated, you need to look at g(jw) but the plot is in (x,y) plane where x represents the real part of g(jw) and y represents the imaginary part of g(jw). For a refresher on Nyquist plots, you might look at
http://www.facstaff.bucknell.edu/mastascu/eControlHTML/Freq/Freq6.html

Finally, just what is G(s) [g(s)?]? is it
(1) G(s) = [ 1 / (2s+1) ] (s^2+s+1) = \(\displaystyle \dfrac{s^2+s+1}{2s+1}\)
or
(2) G(s) = [ 1 / (2s+1) (s^2+s+1) ] = \(\displaystyle \dfrac{1}{(2s+1) (s^2+s+1)}\)
or ...?

 

[SarahJennings]

First of all, the magnitude of something (function, number, phasor, ...) is a real non-negative number so you definitely have the magnitude of g(jw) incorrect [I assume g and G are the same here].

Next, you seem to have a misunderstanding of the Nyquist plot. As you have indicated, you need to look at g(jw) but the plot is in (x,y) plane where x represents the real part of g(jw) and y represents the imaginary part of g(jw). For a refresher on Nyquist plots, you might look at
http://www.facstaff.bucknell.edu/mastascu/eControlHTML/Freq/Freq6.html

Finally, just what is G(s) [g(s)?]? is it
(1) G(s) = [ 1 / (2s+1) ] (s^2+s+1) = \(\displaystyle \dfrac{s^2+s+1}{2s+1}\)
or
(2) G(s) = [ 1 / (2s+1) (s^2+s+1) ] = \(\displaystyle \dfrac{1}{(2s+1) (s^2+s+1)}\)
or ...?

Thank you for messaging me back! I have been on the website before I came on here. It's cleared some up but if I'm honest I'm really struggling! Haha. Must be a blonde moment.

What I have to work with is what you have pointed out for number 2. Any help you could provide pushing me in the right direct will be greatly received.

thank you so so much!!

 

[Ishuda]

Thank you for messaging me back! I have been on the website before I came on here. It's cleared some up but if I'm honest I'm really struggling! Haha. Must be a blonde moment.

What I have to work with is what you have pointed out for number 2. Any help you could provide pushing me in the right direct will be greatly received.

thank you so so much!!

Given that, we would like have the real and imaginary part of the complex fraction
G(s) = 1 / [ (2s+1) (s^2+s+1) ] = \(\displaystyle \dfrac{1}{(2s+1) (s^2+s+1)}\)
evaluated at s=jw. First, I would compute the real [call that u(w)] and imaginary [call that v(w)] parts of
h(jw) = \(\displaystyle \dfrac{1}{(2jw+1) ((jw)^2+jw+1)}\, =\, u(w)\, -\, j\, v(w)\)
remembering that j² = -1. Thus you would have
G(jw) = \(\displaystyle \dfrac{1}{h(jw)}\, =\, \dfrac{1}{u(w)\, -\, j\, v(w)}\, =\, \dfrac{u(w)\, +\, j\, v(w)}{u^2(w)\, +\, v^2(w)}\)

Thus separating G(jw) into its real and imaginary part gives
G(jw) = x(w) + j y(w)
where
x(w) = \(\displaystyle \dfrac{u(w)}{u^2(w)\, +\, v^2(w)}\)
and
y(w) = \(\displaystyle \dfrac{v(w)}{u^2(w)\, +\, v^2(w)}\)

So, for the Nyquist plot, choose 'several values of the frequency w [including w=0 and w=\(\displaystyle \infty\), compute x and y, and plot them in the (x,y) plane.

EDIT: It should have been
h(jw) = \(\displaystyle (2jw+1) ((jw)^2+jw+1)\, =\, u(w)\, -\, j\, v(w)\)
remembering that j² = -1

 

[SarahJennings]

Given that, we would like have the real and imaginary part of the complex fraction
G(s) = 1 / [ (2s+1) (s^2+s+1) ] = \(\displaystyle \dfrac{1}{(2s+1) (s^2+s+1)}\)
evaluated at s=jw. First, I would compute the real [call that u(w)] and imaginary [call that v(w)] parts of
h(jw) = \(\displaystyle \dfrac{1}{(2jw+1) ((jw)^2+jw+1)}\, =\, u(w)\, -\, j\, v(w)\)
remembering that j² = -1. Thus you would have
G(jw) = \(\displaystyle \dfrac{1}{h(jw)}\, =\, \dfrac{1}{u(w)\, -\, j\, v(w)}\, =\, \dfrac{u(w)\, +\, j\, v(w)}{u^2(w)\, +\, v^2(w)}\)

Thus separating G(jw) into its real and imaginary part gives
G(jw) = x(w) + j y(w)
where
x(w) = \(\displaystyle \dfrac{u(w)}{u^2(w)\, +\, v^2(w)}\)
and
y(w) = \(\displaystyle \dfrac{v(w)}{u^2(w)\, +\, v^2(w)}\)

So, for the Nyquist plot, choose 'several values of the frequency w [including w=0 and w=\(\displaystyle \infty\), compute x and y, and plot them in the (x,y) plane.

Thank you again for your help. This Nyquist plots is driving me nuts.

I have to find ins an expression for the phase to plug the values of the frequency in and them compute then for the plot. I have found a website that says to use excel. However it has driven me mad. I'm sure it's not coming out right.

Im sorry for continually asking for help on this but it appears to be a mega weakness for me!! ???

thank you you for taking the time to help me!

Sarah

 

[Ishuda]

Thank you again for your help. This Nyquist plots is driving me nuts.

I have to find ins an expression for the phase to plug the values of the frequency in and them compute then for the plot. I have found a website that says to use excel. However it has driven me mad. I'm sure it's not coming out right.

Im sorry for continually asking for help on this but it appears to be a mega weakness for me!! ???

thank you you for taking the time to help me!

Sarah

To start, please show your work in trying to find u(w) and v(w) where
\(\displaystyle (2jw+1) ((jw)^2+jw+1)\, =\, u(w)\, -\, j\, v(w)\)

That is u(w) is the real part of the expression on the left and v(w) is the imaginary part of that expression. Both u and v should be real valued functions of w [the radial frequency].

 

[Daz]

To start, please show your work in trying to find u(w) and v(w) where
\(\displaystyle (2jw+1) ((jw)^2+jw+1)\, =\, u(w)\, -\, j\, v(w)\)

That is u(w) is the real part of the expression on the left and v(w) is the imaginary part of that expression. Both u and v should be real valued functions of w [the radial frequency].

Thanks again for replying. I have tried to source help from a tutor however it has not helped as they also are unsure of how to do this. Brilliant!!! Its so frustrating i feel like giving up.

(2jw+1) ((jw)^2+jw+1)\, =\, u(w)\, -\, j\, v(w)

to

(2jw+1) ((jw)^2+jw+1)\, +\, j\, v(w)=\, u(w)\

and

(2jw+1) ((jw)^2+jw+1)\, - \, u(w)\, = -\, j\, v(w)

To be honest im really struglling, this is a complete guess!!!

Thanks you again for helping.

Sarah

 

[Ishuda]

To start, please show your work in trying to find u(w) and v(w) where
\(\displaystyle (2jw+1) ((jw)^2+jw+1)\, =\, u(w)\, -\, j\, v(w)\)

That is u(w) is the real part of the expression on the left and v(w) is the imaginary part of that expression. Both u and v should be real valued functions of w [the radial frequency].

Remembering that j²=-1
\(\displaystyle (2jw+1) ((jw)^2+jw+1)\, =\, (2jw+1) (j^2 w^2 + j w + 1)\, =\, (2jw+1) (-w^2+jw +1)\)
=\(\displaystyle (2jw) (-w^2 + jw + 1) + (1) (-w^2 + jw + 1) = -2jw^3 + 2j^2 w^2 +2jw - w^2 + jw + 1\)
=\(\displaystyle -2jw^3 + 2(-1)w^2 +2jw - w^2 + jw + 1 = -3w^2 + 1 + j (-2w^3+3w)\)

Thus
u(w) = \(\displaystyle -3w^2 + 1 \)
v(w) = \(\displaystyle -(-2w^3+3w) = 2w^3-3w\)

Can you continue from there

 

[SarahJennings]

Remembering that j²=-1
\(\displaystyle (2jw+1) ((jw)^2+jw+1)\, =\, (2jw+1) (j^2 w^2 + j w + 1)\, =\, (2jw+1) (-w^2+jw +1)\)
=\(\displaystyle (2jw) (-w^2 + jw + 1) + (1) (-w^2 + jw + 1) = -2jw^3 + 2j^2 w^2 +2jw - w^2 + jw + 1\)
=\(\displaystyle -2jw^3 + 2(-1)w^2 +2jw - w^2 + jw + 1 = -3w^2 + 1 + j (-2w^3+3w)\)

Thus
u(w) = \(\displaystyle -3w^2 + 1 \)
v(w) = \(\displaystyle -(-2w^3+3w) = 2w^3-3w\)

Can you continue from there

Hi Ishuda. Again thank you so much for the help. I currently have so much scribbles on paperwork. Now that i have the Real and Imaginary parts form yourself.

Real u(w) = \(\displaystyle -3w^2 + 1 \)
Imaginary v(w) = \(\displaystyle -(-2w^3+3w) = 2w^3-3w\)

I have to find the expression for the Phase. However i believe i have gone wrong as when i work out the phase = 180 or 90. I get strange answers.

Is this correct for the Phase?

Phase = Tan^-1 (imaginary / Real )

Phase = j / 1-3w^2

Thanks for your help,

Sarah

 

[Ishuda]

Hi Ishuda. Again thank you so much for the help. I currently have so much scribbles on paperwork. Now that i have the Real and Imaginary parts form yourself.

Real u(w) = \(\displaystyle -3w^2 + 1 \)
Imaginary v(w) = \(\displaystyle -(-2w^3+3w) = 2w^3-3w\)

I have to find the expression for the Phase. However i believe i have gone wrong as when i work out the phase = 180 or 90. I get strange answers.

Is this correct for the Phase?

Phase = Tan^-1 (imaginary / Real )

Phase = j / 1-3w^2

Thanks for your help,

Sarah

It appears that you have missed a good part of you class room work. You should go back and review that. Also, reading
http://rotorlab.tamu.edu/me617/intro_freq_resp_functions.pdf
may help or look around at
https://duckduckgo.com/?q=+the+frequency+response+function&t=lm&ia=w
and/or
https://duckduckgo.com/?q=nyquist+phase&t=lm&ia=web

 

[SarahJennings]

It appears that you have missed a good part of you class room work. You should go back and review that. Also, reading
http://rotorlab.tamu.edu/me617/intro_freq_resp_functions.pdf
may help or look around at
https://duckduckgo.com/?q=+the+frequency+response+function&t=lm&ia=w
and/or
https://duckduckgo.com/?q=nyquist+phase&t=lm&ia=web

Hi, Ishuda.

Thanks for getting back to me. I currently am doing this through open university and do not attend class room lectures. This subject is something I am teaching myself.

If possible could you help me work through this? I have found I learn better when presented with the full working and then myself working back. This allows me to see what steps were taken. I'm sorry if I keep pestering you. However your help is greatly appreciated.

Up to now your help has been astounding. It has enabled me to complete a large section of my work.

Again thanks!!

Sarah

 

[stapel]

I currently am doing this through open university and do not attend class room lectures. This subject is something I am teaching myself.

If possible could you help me work through this?

Someone can probably help, but not likely within this framework. (Besides, you have loads of fully-worked examples in your open-university curriculum, plus all the various websites you've studied. So one more worked example won't make much difference, especially when you don't have background on the various intermediate steps.)

From what you've posted, it sounds as though your best bet would be to find a qualified local tutor (not the unqualified one you've already tried). That tutor, by working with you face-to-face, will be best placed to figure out what are all the holes in your background which are preventing you from understanding the current material. Then the tutor can provide the few weeks of personalized private instruction you seek.

Good luck!