The root locus shows how the closed-loop poles of a feedback system move in the s‑plane as a single gain K varies from 0 to ∞.

Why it matters: Closed-loop pole locations determine stability (must be in LHP), speed (further left = faster), and oscillatory behavior (further from real axis = more oscillation).

How to read the plot:

• Branches start at open-loop poles (✕) when K=0
• Branches end at open-loop zeros (○) when K→∞
• Extra branches (n‑m of them) go to infinity along asymptotes
• The dashed magenta line is the stability boundary (Im axis)

Key rules (Evans, 1948):
• n branches, where n = number of open-loop poles
• Branches lie on the real axis to the left of an odd number of real poles+zeros
• Branches are symmetric about the real axis
• Asymptote angles: θ_k = (2k+1)π/(n‑m), centroid: σ_a = (Σp_i ‑ Σz_j)/(n‑m)

Try this: Place two poles at ‑1±j2, then slide K. Watch the branches stay in the LHP — the system is always stable. Now place a third pole at ‑5 and watch branches cross into the RHP at some K.