# How to draw a Nyquist plot by hand using standard plotting strategies?

I want to draw by hand the Nyquist plot of an RC filter's transfer function. I do not want to use the "tricks" that help with building it faster (evaluating just some limits points) - I want to draw it using the usual plotting strategies. Specifically, I have x values (real part) and compute y values (imaginary part). Ultimately I want to use Desmos graphing calculator.

The imaginary part is $$-\frac{\omega RC}{(1+(\omega RC)^2}$$ and the real part is $$\frac{1}{1+(\omega RC)^2}$$ with $$\\omega =2\pi f\$$. I believe Nyquist plots plot the imaginary part of the transfer function with the real part as a variable. Hence, should I replace $$\\omega\$$ in the imaginary part with the real part, i.e., $$\mathrm{Im}(\omega) = -\mathrm{Re}(\omega) \times \frac{RC}{1+(\mathrm{Re}(\omega)RC)^2}$$?

I tried that, but the result in Desmos isn't similar to the Nyquist plot generated using Matlab or Maple.

This is what I get with Desmos:

and this is what it should look like:

How do I obtain the correct plot using "standard" function plotting techniques?

• check this example desmos.com/calculator/sprwnkggss, you can plot the nyquist plot by adding a graph (r_e(t), i_m(t)), and you would have to give t the range of frequencies you want to plot, If you want the full nyquist plot that would be -Inf+Inf; as a recommendation however I would suggest you to learn matlab/python/octave or a language of choice, it is much more useful. In matlab you can simply do plot(re,im), you even have toolboxes which you can pass the transfer function and will do the nyquist plot Commented Jan 2, 2022 at 16:59
• Thank you. I did not want to use a tool because I have not understood how to plot this by myself. Now I know. I use either Maple, Matlab or Desmos for validation Commented Jan 2, 2022 at 17:30
• You're not really drawing it by hand if you're using Desmos, are you? Commented Jan 6, 2022 at 4:45

Your RC lowpass filter has the transfer function $$\H(s) = \frac{1}{RCs+1} \$$ and the frequency characteristic $$\H(j\omega) = \frac{1}{j\omega RC+1} \$$.

For the sake of understanding, let's say $$\R=10 \; \text{k}\Omega \$$, $$\C=27 \; \mu\text{F}\$$. So the frequency characteristic becomes $$H(j\omega) = \frac{1}{j\omega \cdot10 \;\text{k}\Omega \cdot 27 \;\mu\text{F}+1}=\frac{1}{\frac{j\omega}{3.70}+1}$$ From this, you can draw a straight line approximation of a Bode Plot, because we can read from $$\H(j\omega) \$$ that the cut-off frequency is $$\3.70 \: \text{rad/s}\$$.

Observations:

1. At DC, the Bode Plot shows a gain of $$\1 \;\frac{\text{V}}{\text{V}}\$$ and $$\0^\circ\$$ phase shift $$\\rightarrow\$$ The Nyquist Plot starts at $$\(1,0)\$$.
2. At $$\0.37 \; \text{rad/s}\$$ a linear negative phase shift starts but the gain is still $$\ 1 \; \frac{\text{V}}{\text{V}}\$$ $$\\rightarrow\$$ The Nyquist Plot goes clock-wise and maintains the same distance to the origin.
3. At $$\3.70 \; \text{rad/s}\$$ the gain starts decreasing by 20dB/decade and the phase is still decreasing at the same rate $$\\rightarrow\$$ The distance to the origin in the Nyquist Plot is getting shorter and the plot still moves in the clock-wise direction.
4. At $$\37 \; \text{rad/s}\$$ the phase is $$\-90^\circ\$$ and the gain is forever rolling off $$\\rightarrow\$$ The Nyquist Plot approaches the origin fast.

Below is a drawing is an approximate drawing I made in paint.net (I apologize for the bad penmanship), next to a plot of the Nyquist Plot made with MATLAB: -

I managed to generate the plot with Excel and desmos. Excel :

• in one column generetate omega data = 2pif, f = between 0 and 10000 (or whatever you want)
• in 2nd column compute values for Re(f), for each f you have choosen
• in 3rd column compute values for Im(f)
• select 2nd and 3rd column, insert chart > XY chart.

Excel will plot each pair Re(f) - Im(f)

• you can plot one expresion against another with the syntax (exp1(x), exp2(x)), min<x<max