Root Locus
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Root locus · CC BY-SA 4.0
The root locus traces how the closed-loop poles move in the complex plane as the gain K increases from 0 to infinity. Poles start at the open-loop pole locations and migrate toward the zeros. When a pole crosses the imaginary axis, the system goes unstable.
How poles move
At K=0, the closed-loop poles are the open-loop poles. As K increases, poles move along branches. Some branches head toward the open-loop zeros. If there are more poles than zeros, the remaining branches head to infinity along asymptotes. The angles and intersection point of the asymptotes follow simple formulas.
Breakaway points
When two poles on the real axis approach each other, they meet at a breakaway point, then split off the real axis as a complex conjugate pair. This is where the system transitions from overdamped to underdamped behavior.
Stability margins
The root locus tells you how much gain you can add before the system goes unstable. The gain margin is the factor by which you can multiply K before a pole hits the imaginary axis. Read it directly from the plot: find the K value where the locus crosses the imaginary axis.