Root Locus Plotter

• •Use the Root Locus Plotter 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 Root Locus Plotter is a precision engineering calculation tool designed for students, engineers, and technical professionals. Plot how closed-loop poles move as gain K varies; interactive K slider shows closed-loop poles, open-loop poles, and zeros 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.

A root locus plot shows how the poles of a closed-loop system move in the complex plane as a parameter (typically the loop gain K) varies from 0 to infinity. The root locus is constructed for a characteristic equation 1 + K·G(s)·H(s) = 0. At K = 0, the closed-loop poles equal the open-loop poles. As K increases, the poles migrate along paths toward either open-loop zeros or infinity. Key construction rules: (1) the locus starts at open-loop poles (K = 0) and ends at open-loop zeros (K = ∞); (2) the number of branches equals the system order; (3) branches on the real axis exist where the number of poles+zeros to the right is odd; (4) asymptotes at ±180°/m for branches going to infinity, where m = n_poles − n_zeros; (5) breakaway points occur where d(G(s)·H(s))/ds = 0. The root locus shows the closed-loop pole locations at any gain value, enabling visual design of feedback controllers. A designer can see the maximum gain for stability, the effect of adding poles or zeros (compensator design), and the relationship between gain and damping. The calculator plots root locus interactively as gain K varies and marks dominant closed-loop pole locations.

• •Proportional gain tuning: determine the maximum gain K_max that keeps the closed-loop system stable, and choose operating gain well below this for robustness.
• •Lead-lag compensator design: evaluate the effect of adding zeros (lead) or poles (lag) on the root locus shape, and position them to achieve desired closed-loop performance.
• •Robustness analysis: see how closely the operating point is to the stability boundary (jω axis) to assess robustness to parameter variations.
• •PID controller design: interpret PID gains (P, I, D) in terms of their effect on the root locus and closed-loop pole locations.
• •Educational visualization: root locus is a powerful visual tool for understanding feedback control concepts and is taught in all undergraduate control courses.

• !Forgetting that n − m branches go to infinity (along the asymptotes); novices often draw only the bounded portion and miss the K → ∞ behavior.
• !Computing the centroid as (Σ poles)/(n − m) without subtracting Σ zeros. The correct formula is σ_a = (Σ poles − Σ zeros)/(n − m), and zero contributions are critical for compensator design.
• !Confusing breakaway points (locus leaves the real axis as K increases) with break-in points (locus returns to the real axis). Both satisfy d K/d s = 0 but have opposite local trends in K.
• !Treating open-loop poles as if they were closed-loop poles. The OL poles are only the K = 0 starting points — closed-loop pole locations move as K varies and are what determines stability.
• !Misreading asymptote angles when n − m is even. For n − m = 2, the angles are ±90° (purely vertical), which means the asymptotes never close back; the centroid is the only finite reference point.
• !Skipping the magnitude condition K = 1/|G(s)| when designing for a specific pole location. The phase condition tells you whether a point is on the locus; the magnitude condition tells you what K puts it there.