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how can I draw the Nyquist diagram of Kexp(-Ts)/(s*(s^2+s+a) by hand for different a and K values and examine the stability of the system depending on T value? I drew for the case with no time delay but when the time delay is added, drawing the Nyquist diagram will be difficult. I would be glad if anyone could help me in this.

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The nyquist plot is nothing more than just a polar representation of the bode plot (assuming we neglect negative frequencies). So if you know how $e^{-Ts}$ affects the bode plot, you can essentially just pick a certain amount of frequencies, get their phase and magnitudes from the bode plot, and compute its corresponding position in the Nyquist plot. Just know that the magnitude in the Nyquist plot is in a linear scale and not in dB. Then compute the limit cases and connect the dots.

I am expecting the plot will now spiral to the origin for frequencies far beyond the nyquist frequency, and the limit $j\omega \rightarrow 0$ will approach the same limit value for the system without the sampling delay.

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    $\begingroup$ The OP is looking for methods on how to actually draw these plots by hand, not just a statement of "draw it". $\endgroup$ Commented Oct 3, 2021 at 13:10
  • $\begingroup$ adding the e^(-jωT) is simply a rotation in the complex plane . . . $\endgroup$
    – Pete W
    Commented Oct 3, 2021 at 14:30

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