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

Question

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?

Answer 1

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: -

Answer 2

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)

Desmos: https://www.desmos.com/calculator/iptojz0tpm

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