What transfer function form does this tool accept?

The tool works with the open-loop gain function G(s)H(s) expressed as its poles and zeros. Enter each pole or zero as a real number (e.g. 0, -2, -5). Complex poles and zeros are not yet supported — use the real part for an approximation, or refer to the angle-of-departure rule output for guidance.

How many branches does a root locus have?

The number of branches equals the order of the denominator polynomial (number of poles). Each branch starts at an open-loop pole (K=0) and ends at an open-loop zero or goes to infinity along an asymptote (K→∞).

What is the centroid of the asymptotes?

The centroid σ_a = (Σpoles − Σzeros) / (n − m), where n is the number of poles and m is the number of zeros. It is the point on the real axis from which all asymptotes radiate.

How is a break-away point found?

Break-away (and break-in) points occur where dK/ds = 0 on the real axis segments of the locus. The tool evaluates this numerically across the real axis and flags local maxima of −1/G(s)H(s) as candidates.

Does this tool plot the actual locus?

This tool computes all the construction rules and annotates the real-axis candidate points. A full complex-plane plot requires a graphing library; use the numerical results here alongside MATLAB rlocus() or Python-control for a visual.

Root Locus Helper

Name: Root Locus Helper
Availability: InStock
Author: Nham Vu

Enter open-loop poles and zeros to compute root locus rules: asymptotes, centroid, break-away points, departure angles, and gain margin.

Open-Loop Transfer Function

Poles — values of s where denominator = 0

Comma-separated real numbers

Zeros — values of s where numerator = 0 (optional)

Leave blank for no finite zeros

Gain range for break-away search (K min / max)

Transfer Function G(s)H(s)

Enter poles and zeros above

Click Analyze Root Locus to see construction rules

Stability (unity-feedback, K > 0)

Rule 1 — Branches

Rule 2 — Symmetry

Locus is symmetric about the real axis.

Rule 3 — Real-Axis Segments

Rule 4 — Asymptotes

Rule 5 — Centroid σ_a

Rule 6 — Break-Away / Break-In

Gain K at Break-Away Candidates

s (real axis)	K = −1/G(s)H(s)	Type

Rule 7 — Angle of Departure / Arrival

Summary

Enter open-loop poles and zeros to compute root locus rules: asymptotes, centroid, break-away points, departure angles, and gain margin.

How it works

Enter real-valued open-loop poles (comma-separated) in the Poles field.
Enter real-valued open-loop zeros (comma-separated) in the Zeros field, or leave blank.
Click Analyze to compute all 7 root-locus construction rules instantly.
Review the results: number of branches, real-axis segments, asymptote data, centroid, break-away candidates, and stability assessment.
Use the gain table to find approximate gain K at candidate break-away points.
Adjust pole/zero values and re-analyze to explore how plant changes affect locus shape.

Use cases

Verify hand-computed root locus rules for a homework or exam problem.
Quickly determine if a system is stable under unity feedback before simulation.
Find the gain at a break-away point to set an initial PID proportional gain.
Compare how adding a zero (derivative action) reshapes the locus toward stability.
Teach root locus construction rules with an interactive step-by-step breakdown.
Cross-check Bode or Nyquist analysis with root locus stability conclusions.

Frequently Asked Questions

Last updated: 2026-07-24 · Reviewed by Nham Vu