Root Locus: Watching Every Closed-Loop Pole Move at Once

One equation, traced as a knob turns

Close a feedback loop around a plant G(s) with a simple proportional gain K and the closed-loop poles are the roots of the characteristic equation, 1 + K·G(s)H(s) = 0. Fix G and H, sweep K continuously from 0 to ∞, and each root traces a continuous path in the complex s-plane. The root locus is the picture of every one of those paths at once — a map of exactly how every closed-loop pole moves as you turn one dial.

At K = 0 the equation reduces to G(s)H(s) having a pole, so every locus branch starts exactly at an open-loop pole. As K → ∞ the equation instead forces G(s)H(s) toward 0, so branches end at an open-loop zero — or, if there are more poles than zeros, race off to infinity along asymptotic straight lines. Every branch is anchored at both ends by the plant you already know; only the path in between is new information.

The angle condition, and why the sketching rules aren't arbitrary

A point s₀ lies on the locus exactly when it satisfies the angle condition — the sum of angles from every open-loop zero to s₀, minus the sum of angles from every open-loop pole to s₀, equals 180° (mod 360°). Everything you're taught as a sketching rule for root locus — real-axis segments, asymptotes, breakaway points — is just a shortcut derived from that one condition, so it's worth keeping in view:

real axis          on the locus iff an ODD number of real poles+zeros lie to its right
asymptotes         (2k+1)·180° / (#poles - #zeros), centred at
                    σ = (Σ poles - Σ zeros) / (#poles - #zeros)
breakaway point    where dK/ds = 0 along a real-axis segment
                    (two real poles approach, collide, and split off the axis)
imaginary crossing  where the locus crosses jω — read off the gain that
                    puts the system exactly on the stability boundary

Once s₀ is confirmed on the locus by the angle condition, the exact gain that places a closed-loop pole there falls out of the companion magnitude condition: K = 1 / |G(s₀)H(s₀)|, computed as the product of distances from every pole to s₀ divided by the product of distances from every zero to s₀. Angle finds where; magnitude tells you how much gain puts a pole exactly there.

Reading stability, and reading performance

Stability is the coarse read: any branch that crosses into the right half-plane means there is a gain, past that crossing, that makes the closed loop unstable — the imaginary-axis crossing point is literally the maximum gain you can use. But root locus gives you something Bode analysis does not hand you directly: the exact location of every closed-loop pole for a chosen K, which means you can read transient performance straight off the plot. A line at constant angle θ = cos⁻¹(ζ) from the origin is a locus of constant damping ratio ζ; a circle of radius ω_n centred at the origin is a locus of constant natural (undamped) frequency. Overlay those two families of curves on the root locus and picking a gain becomes picking the intersection point that gives you the overshoot and settling time you actually want.

settling time  ts ≈ 4 / (ζ·ωn)     — set by how far LEFT the pole sits (real part)
overshoot      Mp  = e^(-ζπ/√(1-ζ²))  — set by the ANGLE of the pole from the real axis
speed of response                    — set by how far the pole sits from the origin

Beyond a bare gain: adding poles and zeros to reshape the map

Root locus isn't limited to tuning a single proportional gain — it is exactly as useful for deciding where to put a compensator's poles and zeros before you tune its gain. Add a zero (as a PD or lead compensator does) and the branches bend left, toward more negative real parts, because the angle condition now needs less contribution from the poles to reach 180° — that's the mechanism behind a lead compensator's classic promise of faster settling and a shorter transient. Add a pole at the origin (a PI or lag compensator, for zero steady-state error) and the branches tend to bend right, toward the imaginary axis, trading away some stability margin for the improved steady-state accuracy integral action buys you — which is exactly why a lag compensator's added pole is conventionally placed very close to the origin, so its rightward pull on the locus is felt mainly at very low gain.

This is the real value of the tool: it turns "will this compensator design work" from an algebra problem into a geometry problem you can sketch by hand and verify by eye before ever touching MATLAB or running a single simulation.

Frequently asked questions

Try it live

Everything above runs in your browser — open Root Locus and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.