# how to draw nyquist plot for [exp(-s) /s]: function with delay

Discussion in 'Homework Help' started by shameem, Jun 18, 2015.

1. ### shameemNew Member

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I want to know how can we draw exp(-s) /s or any other function which have delay in it.
I have tried with MATLAB the result is there in attachment, but I cannot make it by using normal steps of nyquist plotting.

thanks

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3. ### shameemNew Member

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Here is the plot for via MATLAB

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5. ### steveBWell-Known MemberMost Helpful Member

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I'm not sure if I'm understanding your question or your confusion correctly, but here is my comment.

you can create the system transfer function with the following commands.

s = zpk('s')
sys = exp(-10*s/(s+1))

The first command establishes "s" as the variable, and the second command makes the proper transfer function.

Nyquist plotting can be done with nyquist command, if you wish, by typing nyquist(sys); or, you can do a Bode plot with the command bode(sys).

Did I understand you correctly?

6. ### shameemNew Member

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As I mention I already know how to make it in MATLAB, but I need to show to my professor by solving it on paper with normal method of nyquist plotting.

Regards

7. ### steveBWell-Known MemberMost Helpful Member

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Ah ok, I misunderstood the question. So doing by hand is easy in principle. The basic idea of the Nyquist plot and the Bode plot, is identical. I know you only asks about the Nyquist plot, but the two are so similar, that it is good to consider both at the same time.

Remember that the transfer function is complex in value, and the input to the transfer function is the variable "s" , which is also complex. If you think about it, it is impossible to draw a plot of a complex function versus a complex input variable, because you would need 4 dimensions to plot it. However, if we restrict "s" to be imaginary only, with s=jw, where j=sqrt(-1), then we can make three dimensional plots of T(w) versus w. But, three dimensional plots are a little unwieldy, so typically we use two dimensional plots to keep it simple.

For the Bode plot, we make two separate two dimensional plots. One plot is the magnitude of the transfer function versus w, and the second is the phase of the transfer function versus w.

For the Nyquist plot, we make a 2d parametric plot of the Imaginary component of the transfer function versus the real part of the transfer function, and we let w be a parametric variable to describe the plot.

Mathematically, we can derive the Nyquist plot for your transfer function as follows.

T(s)=exp(-10 s)/(s+1)

T(jw)=exp( -10 j w)/(1+j w)=(cos(10 w) - j sin(10 w))/(1+jw)

T(w)=(cos(10 w) - j sin(10 w))*(1-jw)/(1+w^2)

Re(T(w))= (cos(10 w) - w sin(10 w))/(1+w^2)

Im(T(w))=(-sin(10 w)-w cos(10 w))/(1+w^2)

With the above two formulas, let w go from minus infinity to plus infinity and plot Im(T) versus Re(T) to get the parametric plot.

Please check my math, as I did this quickly on the fly to give you the general idea of how to do it.

Last edited: Jun 18, 2015
8. ### shameemNew Member

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thank you
it helped me a lot