Poles in Motion
In a feedback control system, the closed-loop poles determine everything: stability, oscillation frequency, damping, and settling time. As the designer adjusts the loop gain K, these poles move continuously through the complex plane. The root locus traces their complete trajectory, giving the engineer a map of all possible system behaviors achievable through gain adjustment alone.
Construction Rules
Evans developed elegant rules for sketching the root locus by hand: branches start at open-loop poles (K=0) and end at open-loop zeros (K→∞). Segments of the real axis to the left of an odd number of real poles+zeros lie on the locus. Asymptotes guide branches departing to infinity, with angles and centroid determined by the pole-zero excess. These rules remain invaluable for building design intuition.
Breakaway and Stability
When two branches meet on the real axis, they break away into the complex plane, indicating the onset of oscillatory behavior. The breakaway gain is found by solving dK/ds = 0. As gain increases further, the complex branches may cross the imaginary axis — the critical stability boundary. This crossing gain, determinable via the Routh-Hurwitz criterion, sets the maximum allowable gain for stable operation.
Design Applications
Root locus is not just an analysis tool — it guides compensator design. Adding a lead compensator (zero-pole pair) reshapes the locus to pass through a desired pole location, achieving target damping and bandwidth. Lag compensation shifts the locus to improve steady-state accuracy without destabilizing. This simulator lets you place poles and zeros interactively and watch the locus reshape in real time.