Calculator Inputs
Example Data Table
This example shows a common engineering setup for testing the calculator.
| Field | Example value | Meaning |
|---|---|---|
| Numerator coefficients | 1 | Constant numerator for G(s) = K / (s³ + 6s² + 8s) |
| Denominator coefficients | 1, 6, 8, 0 | Open-loop poles at 0, -2, and -4 |
| Minimum gain | 0 | Starts the locus at open-loop poles |
| Maximum gain | 100 | Shows branch movement over a broad gain interval |
| Gain samples | 81 | Creates a smooth engineering plot |
| Reference damping ratio | 0.5 | Useful for approximate transient design goals |
| Reference natural frequency | 2.0 | Adds a circular guide for pole placement |
Formula Used
For a unity negative-feedback system, the closed-loop characteristic equation is:
1 + K G(s)H(s) = 0
If the open-loop transfer function is written as:
G(s)H(s) = N(s) / D(s)
then the root locus comes from:
D(s) + K N(s) = 0
Key engineering relations
- Closed-loop poles: roots of
D(s) + K N(s) - Asymptote count:
n - m, wherenis pole count andmis zero count - Asymptote centroid:
(sum of poles - sum of zeros) / (n - m) - Asymptote angles:
(2q + 1) × 180° / (n - m) - Damping ratio:
ζ = -σ / √(σ² + ω²)for a poles = σ ± jω - Natural frequency:
ωn = √(σ² + ω²)
This calculator samples the gain range numerically, solves the characteristic polynomial at each gain, and connects the pole locations into locus branches.
How to Use This Calculator
- Enter numerator coefficients in descending powers of
s. - Enter denominator coefficients in descending powers of
s. - Choose minimum and maximum gain values for the root locus sweep.
- Set the number of gain samples for smoother or faster plotting.
- Optionally enter a reference damping ratio and natural frequency guide.
- Press Generate Root Locus to calculate poles and draw the plot.
- Review centroid, asymptotes, stable ranges, and dominant pole trends.
- Export the results table using the CSV or PDF buttons.
Frequently Asked Questions
1) What does the root locus plot show?
It shows how closed-loop pole locations move in the complex plane as gain changes. This helps engineers study stability, damping, oscillation tendency, and possible controller tuning ranges.
2) Why are numerator and denominator coefficients entered separately?
Root locus starts from the transfer function form N(s)/D(s). Separate inputs let the calculator build the characteristic equation, find poles and zeros, and track how poles move with gain.
3) What is the meaning of the centroid?
The centroid is the real-axis location where root-locus asymptotes intersect. It helps estimate branch behavior when the system has more poles than zeros.
4) Why is the imaginary-axis crossing gain approximate?
This calculator samples the gain range numerically. A crossing is detected when stability changes between two nearby gains, so the reported value is an approximation within the chosen sample spacing.
5) What does the damping ratio tell me?
Damping ratio estimates how oscillatory the dominant mode is. Higher values generally mean less overshoot, while lower values usually indicate stronger oscillation and slower settling.
6) Can I use this for higher-order systems?
Yes. The calculator numerically solves the characteristic polynomial for each sampled gain. Very high orders can still be analyzed, though interpretation becomes more complex and numerical sensitivity can increase.
7) What happens when poles and zeros nearly cancel?
The plot may show branches passing very close to each other. In physical systems, near cancellation can still leave sensitive dynamics, so design decisions should not rely only on visual cancellation.
8) Why might my system show no stable gain range?
That can happen if the pole paths stay on or cross into the right-half plane for the sampled gain interval. Expanding the gain range or redesigning the compensator may be necessary.