# Drawing nyquist diagram by hand for a system with time delay

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.

• youtube.com/watch?v=sof3meN96MA watch these Commented Oct 3, 2021 at 12:31
• it does not enter time delay into open-loop equation. Commented Oct 5, 2021 at 12:37

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.