Root Locus
The root locus traces how closed-loop poles move through the s-plane as a single gain is varied from zero to infinity.
Poles on the Move
The root locus is a plot of the closed-loop pole locations as the loop gain K varies from zero to infinity. Because closed-loop poles determine stability and transient behavior, watching them move as gain changes gives a direct, visual design method for choosing K.
The governing equation
Closed-loop poles satisfy 1 + K*L(s) = 0, where L(s) is the open-loop transfer function with its gain factored out. This splits into a magnitude condition and an angle condition. The angle condition, that the total angle from open-loop poles and zeros equals an odd multiple of 180 degrees, defines which points lie on the locus.
Construction rules
- Branches start (K = 0) at the open-loop poles and end (K -> infinity) at the open-loop zeros or at infinity.
- The number of branches equals the number of open-loop poles.
- Segments of the real axis to the left of an odd count of real poles and zeros belong to the locus.
- Branches heading to infinity follow asymptotes whose angles and centroid are set by simple formulas.
What it reveals
The locus shows at a glance how gain affects the system. Where branches cross into the right half-plane, the loop becomes unstable, and the gain at that crossing is the stability limit, found via Routh-Hurwitz or directly. Where branches bend toward the imaginary axis, damping falls and overshoot rises. The designer picks a gain that places the dominant poles at a desired damping and speed.
Design with compensators
Adding a compensator introduces new open-loop poles and zeros that reshape the entire locus. A lead compensator's zero pulls the branches leftward, improving damping and speed; a lag compensator raises low-frequency gain with little effect on the dominant pole positions. The root locus makes the effect of each addition visible before any simulation.
Though computers now compute pole locations instantly, the root locus remains valued for the geometric intuition it builds about how feedback gain reshapes a system's dynamics.