Root locus: follow the poles
Root locus shows where closed-loop poles move as a gain changes.
First, picture it
Instead of testing a thousand gain values in time simulations, trace every possible closed-loop pole position as K sweeps from zero upward. Crossing into the right half-plane warns of instability.
What the model says
For L(s)=K G₀(s) in unity negative feedback, poles satisfy 1+K G₀(s)=0. The branches begin at open-loop poles and end at open-loop zeros or infinity. Points on the locus satisfy both an angle condition and a magnitude condition. The method assumes a chosen loop and linear model; delays and saturations need more care.
1+K G₀(s)=0; ∠G₀(s)=(2n+1)180°For positive K, the negative real-axis direction follows from K G₀(s)=−1.
As K rises from 0, poles at 0 and −2 meet at −1, then become −1 ± j√(K−1). This example never crosses into the right half-plane for K>0.
Make it concrete
In the motor lab, raising Kp can first speed response, then make control effort and oscillation worse. Root locus explains the linear-model pole movement behind that trend.
A root-locus plot is not the time response itself; infer a trend from poles, then verify with the actual model and limits.
Check your understanding+
What happens when a closed-loop pole crosses into the right half-plane?
Its corresponding mode grows exponentially; the linear closed loop becomes unstable.