Formula used
A transfer function can be written as: G(s) = N(s) / D(s).
- Zeros are roots of N(·)=0.
- Poles are roots of D(·)=0.
- For continuous-time, stability usually requires Re(poles)<0.
- For each pole p = σ + jω, \(ω_n=\sqrt{σ^2+ω^2}\) and \(ζ=-σ/ω_n\) when \(ω_n>0\).
How to use this calculator
- Enter numerator coefficients from highest power to constant.
- Enter denominator coefficients the same way.
- Pick s or z as your variable label.
- Submit to compute roots and render the complex-plane plot.
- Use auto-scale for large-magnitude roots, then export.
Example data table
This example corresponds to a second-order denominator and a constant numerator.
| Case | N(s) | D(s) | Zeros | Poles | Interpretation |
|---|---|---|---|---|---|
| Sample | 1 | s² + 2s + 2 | — | -1 ± 1j | Stable, underdamped response. |
FAQs
1) What coefficients should I enter?
Enter real coefficients ordered from the highest power to the constant term. Use spaces, commas, or semicolons. Fractions like 1/2 are accepted.
2) What do poles and zeros mean?
Zeros are input frequencies where the transfer function output goes to zero. Poles are natural modes where the system tends to resonate or grow. Their locations strongly shape transient and frequency response.
3) How is stability determined here?
For continuous-time models, the tool checks whether every pole has a negative real part. Any pole with positive real part indicates instability. Poles near the imaginary axis often imply slow decay or oscillation.
4) Why are there repeated or close roots?
Repeated roots exist for some polynomials, and numerical methods can return values that are extremely close. Slight differences are usually rounding effects. Auto-scale and higher precision in exports can help inspection.
5) Does gain change poles or zeros?
A nonzero scalar gain multiplies the numerator but does not move roots. Poles depend only on the denominator, and zeros depend only on the numerator. Gain affects magnitude, not pole-zero locations.
6) Can I use this for discrete-time systems?
Yes. Set the variable label to z for clarity. Interpretation differs: discrete-time stability typically requires poles to lie inside the unit circle. This plot still helps visually, but stability logic shown is continuous-time oriented.
7) Why might the plot look off-scale?
Very large roots can compress the view. Use Auto-scale to expand the axes to fit all points. Unit scale is useful when you want to compare against the unit circle for z-plane inspection.