Root Locus of Second-Order Feedback

A unity negative feedback loop with a plant transfer function L(s) = K / [s(s+2)] has a closed-loop characteristic equation of s² + 2s + K = 0. The roots of this equation dictate the stability and transient behavior of the system as the gain K varies.

Experiment: Tuning the Gain

Use the interactive module to observe how the poles move and the step response changes.

1. Set Gain K to 0.5 and observe that the roots are real and distinct (-0.293 and -1.707). The response is overdamped.
2. Set Gain K to 1. The roots converge to a repeated pole at -1. The system is critically damped.
3. Set Gain K to 2. The roots split vertically into a conjugate pair (-1 ± i). The response becomes underdamped and exhibits overshoot.
4. Adjust Time to see how the transient response evolves over time.