Why do some colored traces “jump” between K samples?

Root locus branches can **cross** or nearly coincide; a greedy matcher that permutes roots between successive **K** values can pick the wrong pairing for a step. Increasing **K sweep max** or zooming mentally along **σ** usually clarifies the geometry; for publication-grade plots you would branch-track with algebraic machinery beyond this teaching sketch.

Do the drawn asymptotes replace the actual locus?

No—they are the **Evans** centroid and angle recipe for **n − m** branches going to infinity as **K → ∞**. The polylines still come from the polynomial roots; the dashed rays are a **reference** when **n > m**.

Why block new zeros when deg N would reach deg D?

The plant must stay **strictly proper** for the usual unity-gain root-locus story (finite high-frequency behavior). The UI therefore refuses zeros that would make **deg N ≥ deg D**.

Is Durand–Kerner guaranteed to converge?

No—like many simultaneous root finders it is **heuristic** but works reliably here on moderate degrees with warm starts. If you place roots pathologically close, rerun by nudging a pole or **K**; the defaults target classroom polynomials.

Root Locus: 1 + K·G(s) = 0

This page studies unity negative feedback with a scalar plant G(s) = g₀ ∏(s − z_i) / ∏(s − p_j) edited on the same s-plane click map as the Bode/Nyquist pole–zero lab. The closed-loop characteristic equation is 1 + K G(s) = 0, equivalently D(s) + K g₀ N₀(s) = 0 where D and N₀ are monic polynomials built from the listed poles and zeros. A logarithmic sweep of K traces each closed-loop root as a colored polyline (a discrete sketch of the root locus). Asymptote directions and the real-axis segment rule (odd count of real poles+zeros to the right) are drawn as teaching overlays when deg D > deg N. Roots at each K are found with Durand–Kerner iteration on the monic polynomial; successive K samples reuse the previous root vector as the initial guess and apply a greedy permutation so branches stay visually continuous—this is a visualization aid, not a proof of Evans’s construction. The current K slider recomputes roots with the nearest sweep sample as a warm start; gold rings mark those roots and a counter flags right-half-plane closed-loop poles for quick instability feedback.

Who it's for: Undergraduates in classical control who have met the root-locus sketch rules and want a fast numeric picture tied to the same pole–zero placement UX as the companion Bode/Nyquist page.

Key terms

Root locus
Characteristic equation
Unity feedback
Open-loop poles and zeros
Asymptotes
Real-axis rule
Durand–Kerner
Closed-loop stability

How it works

Unity negative feedback with forward gain K and fixed plant G(s) = g₀ ∏(s−z)/∏(s−p) gives the characteristic equation 1 + K G(s) = 0 ⟺ D(s) + K g₀ N₀(s) = 0. The root locus is the set of closed-loop poles in the s-plane as K increases from 0 (you start at the open-loop poles) toward ∞ (branches tend toward zeros and asymptotes when n > m). Click to edit poles (●) and zeros (×) like the companion bode-nyquist-pole-zero page; dashed gray rays show asymptote angles (2q+1)π/(n−m) from the centroid; thick slate segments mark real-axis loci (positive K rule). Gold rings mark the closed-loop roots at the Current K slider. Roots are found with a Durand–Kerner polish on D + K g₀ N (teaching-grade numerics, not a certified control toolbox).

Key equations

1 + K G(s) = 0 ⟺ D(s) + K g₀ N₀(s) = 0

σ_a = (Σ p_i − Σ z_i) / (n − m), θ_k = (2k+1)π / (n − m)

Frequently asked questions

Why do some colored traces “jump” between K samples?: Root locus branches can cross or nearly coincide; a greedy matcher that permutes roots between successive K values can pick the wrong pairing for a step. Increasing K sweep max or zooming mentally along σ usually clarifies the geometry; for publication-grade plots you would branch-track with algebraic machinery beyond this teaching sketch.
Do the drawn asymptotes replace the actual locus?: No—they are the Evans centroid and angle recipe for n − m branches going to infinity as K → ∞. The polylines still come from the polynomial roots; the dashed rays are a reference when n > m.
Why block new zeros when deg N would reach deg D?: The plant must stay strictly proper for the usual unity-gain root-locus story (finite high-frequency behavior). The UI therefore refuses zeros that would make deg N ≥ deg D.
Is Durand–Kerner guaranteed to converge?: No—like many simultaneous root finders it is heuristic but works reliably here on moderate degrees with warm starts. If you place roots pathologically close, rerun by nudging a pole or K; the defaults target classroom polynomials.

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Root Locus: 1 + K·G(s) = 0

How it works

Key equations

Frequently asked questions

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Root Locus: 1 + K·G(s) = 0

How it works

Key equations

Frequently asked questions