Root Locus Calculator
Free root locus calculator for control systems. Enter numerator and denominator polynomials, vary the gain K, and visualize closed-loop poles, stability and damping directly in the s-plane.
Full original guide (expanded)
Root Locus Calculator
Enter the numerator and denominator of your open-loop transfer function and explore how the closed-loop poles move in the complex plane as the gain K varies.
For unity-feedback SISO systems with transfer function G(s) = N(s) / D(s).
Interactive root locus plotter
Comma or space separated, in descending powers of s.
Comma or space separated, in descending powers of s.
Root locus plot (s-plane)
Real axis (horizontal), imaginary axis (vertical).
Closed-loop pole summary for selected K
| Pole # | Re(s) | Im(s) | |s| | Damping ratio ζ |
|---|---|---|---|---|
| Run the calculator to see the closed-loop poles. | ||||
Root locus basics
The root locus is a classical control-design tool that shows how the closed-loop poles of a feedback system move in the complex plane as a system parameter (typically the loop gain K) varies. It is particularly useful to:
- Assess stability for a range of gains.
- Inspect damping ratio and natural frequency of dominant poles.
- Choose a suitable gain K that meets transient-response requirements.
Closed-loop characteristic equation
Consider a unity-feedback SISO system with open-loop transfer function
G(s) = N(s) / D(s)
The closed-loop transfer function is
T(s) = \(\dfrac{K G(s)}{1 + K G(s)}\)
The characteristic equation of the closed-loop system is therefore
1 + K G(s) = 0
Substituting G(s) = N(s) / D(s) gives
D(s) + K N(s) = 0
For each value of K, the roots of this polynomial are the closed-loop poles. The root locus is the collection of these poles as K moves from 0 to a large positive value.
Interpreting the root locus
- Stability: For continuous-time systems, the system is stable when all closed-loop poles lie in the left half-plane (negative real part).
- Damping ratio: For a complex pole \(s = \sigma \pm j\omega\), the damping ratio is \(\zeta = -\sigma / \sqrt{\sigma^2 + \omega^2}\).
- Transient response: Poles further to the left (more negative real part) generally yield faster responses; lightly damped complex pairs cause overshoot and oscillations.
How to use this calculator effectively
- Model your system and derive the open-loop transfer function \(G(s)\) in factored or polynomial form.
- Expand \(N(s)\) and \(D(s)\) and enter their coefficients in descending powers of \(s\).
- Choose a plausible range for \(K\) (for example from 0 to 100 or 0 to 1,000 depending on your scaling).
- Click Plot root locus and inspect how the poles move as you adjust the gain slider.
- Select a range of \(K\) values that keeps poles in the left half-plane with acceptable damping ratios.
Limitations and numerical notes
- The tool is intended for moderate-order polynomials typical of teaching and practical design.
- Very high order systems may be numerically ill-conditioned; consider model reduction if necessary.
- The calculator assumes standard unity feedback. For non-unity feedback, rewrite your loop as an equivalent unity-feedback system first.
Root locus FAQ
What is a root locus in control systems?
The root locus is a plot of the closed-loop poles of a feedback system in the complex s-plane as a parameter, usually the loop gain K, varies from 0 to a large value. It lets you see how stability and transient response change with gain.
Which transfer functions can I enter here?
You can enter any rational SISO transfer function of the form G(s) = N(s) / D(s), where N(s) and D(s) are polynomials with real coefficients. Enter the coefficients of N(s) and D(s) in descending powers of s.
How is the closed-loop characteristic polynomial built?
Assuming unity feedback, the characteristic equation is 1 + K G(s) = 0. With G(s) = N(s) / D(s), this becomes D(s) + K N(s) = 0. For every gain K in the chosen range, the tool forms this polynomial and finds its roots.
What does the damping ratio column tell me?
For a complex pole s = σ + jω, the damping ratio is ζ = −σ / √(σ² + ω²). Values 0 < ζ < 1 correspond to underdamped behavior with overshoot, ζ = 1 is critically damped, and ζ > 1 corresponds to overdamped real poles.
Can I use this tool for discrete-time systems?
The visualization and formulas are designed for continuous-time systems in the s-plane. For discrete-time systems in the z-plane, you should map your design to an equivalent continuous-time representation or use a dedicated z-plane root locus tool.
Formula (LaTeX) + variables + units
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T(s) = \(\dfrac{K G(s)}{1 + K G(s)}\)
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Last code update: 2026-01-19
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