Routh-Hurwitz Stability Calculator

• •Use the Routh-Hurwitz Stability Calculator to check your hand calculations on homework or exam problems — comparing its result with your own working helps identify arithmetic errors before they propagate.
• •Use it during the early design phase to rapidly iterate on parameters and narrow down feasible configurations before committing time to detailed finite element simulations or full design packages.
• •Use it when reviewing a colleague's calculation or checking a vendor's data sheet for plausibility — a quick sanity check can prevent costly downstream errors.
• •Use it to generate reference data for a technical report or presentation without manual computation, ensuring consistent, reproducible numbers throughout the document.
• •Use it in the field when a quick estimate is needed and a full engineering software package is not available.

The Routh-Hurwitz Stability Calculator is a precision engineering calculation tool designed for students, engineers, and technical professionals. Build the full Routh array for any-order polynomial, determine stability, and count right-half-plane poles from the characteristic equation All calculations are performed using established engineering formulas from the relevant scientific literature and standards. Follow the units printed beside each field. Where a unit selector is provided, the selected unit is converted internally. Results are computed in your browser without sending data to a server, which keeps your inputs private. This calculator is intended as a supplementary tool for learning and design exploration; always verify results against authoritative references for safety-critical applications.

The Routh-Hurwitz stability criterion determines whether all roots of a polynomial have negative real parts (corresponding to a stable system) without explicitly computing them. For a polynomial P(s) = a_n·s^n + a_{n-1}·s^(n-1) + ... + a_0, the method builds an array (Routh array) with the polynomial coefficients. The number of sign changes in the first column equals the number of roots with positive real parts (unstable roots). If the first column has no sign changes, no root lies in the right half plane; the system is stable unless a row of zeros or a zero constant term a₀ reveals roots on the jω axis, which make it marginally stable if each is simple and unstable if one repeats. The first two rows of the array come from the polynomial coefficients; subsequent rows are computed using specific formulas involving determinants. Special cases include: (1) a row of zeros indicates roots symmetric about the origin (such as a pair ±jω on the imaginary axis), requiring an auxiliary polynomial substitution; (2) a first-column zero with non-zero other entries requires a small ε substitution or the modified method. The Routh-Hurwitz criterion is useful for checking stability without factoring the characteristic polynomial and for determining stability boundaries as functions of system parameters (gain K, controller settings, etc.). For high-order systems, the determinant computations are tedious but straightforward. The calculator builds the full Routh array and identifies the number of right-half-plane poles, providing a clear stability conclusion.

• •Closed-loop stability verification: check whether a feedback control system is stable for a given controller gain without explicit pole calculation.
• •Stability range determination: find the range of controller gains that maintain closed-loop stability, often expressed as a limit on proportional gain.
• •Higher-order polynomial analysis: for polynomials of degree > 4 where root finding is numerically difficult, Routh-Hurwitz provides stability information analytically.
• •Characteristic polynomial examination: verify that a system's characteristic polynomial has all left-half-plane roots.
• •Educational tool: Routh-Hurwitz is taught in introductory control courses as a fundamental stability analysis technique.