# Is the Nyquist plot a plot of Re(G(s)) v.s. Im(G(s) or Re(G(jω)) v.s. Im(G(jω))

It seems that people in control theory are divided over whether the Nyquist Plot/Diagram (that is the plot, not the contour!) is a plot of $$\text{Re}(G(s)) \text { v.s. } \text{Im}(G(s))$$ or $$\text{Re}(G(j\omega)) \text{ v.s. } \text{Im}(G(j\omega))$$ After a quick internet search, I have found that

The popular web resources that uses the former convention includes:

People who uses the latter convention includes:

The rest does not seem to care about this distinction and does not label their plots properly.

In any case, from a purely academic perspective, which is correct? I need as many electrical engineers to weigh in as possible.

Side note: if this is any worth I think the former is correct since we are not only mapping the Nyquist contour on the imaginary axis but also over whichever semicircle happen to be on the contour, and those are not on the imaginary axis!

## 2 Answers

From a practical point of view the Nyquist plot is the mapping of the imaginary axis from $-j\infty$ to $j\infty$ through the function 1 + G(s).

From an academic point of view it is the mapping of the Nyquist contour through 1+G(s).

The Nyquist contour is again the imaginary axis from $-j\infty$ to $j\infty$ and then an arc that starts at $0+j\omega$ and goes clockwise to $0-j\omega$ with $\omega$ approaching infinity. The idea is to get a contour that encompasses the whole right half plane. Then Cauchy's principle of argument can be used to determine the difference (!) between the number of zeros and poles in the enclosed region.

If G(s) has low-pass characteristic then points at infinity will map into 1 since G(s) approaches zero. All practical systems have low-pass characteristic so the semicircle can be omitted.

For my opinion, one should only ask: Which steps are necessary to create a Nyquist plot in the complex frequency plane? Answer: Replacing the variable s bei jw and computing the real and the imaginary part, respectively, of the function G(s=jw). Then, the results are plottet as Im[G(jw)] vs. Re[G(jw)].

These are the same steps which are also necessary to create a BODE plot - however, in this case we do not use Im and Re but the magnitude SQRT(Re²+Im²) and the phase arctan (Im/Re).