What determines the stability of a closed-loop system using root locus?

• A. The location of poles in the left half-plane
• B. The total number of zeros
• C. The gain margin of the system
• D. The phase margin of the system

Answer: (A)

The stability of a closed-loop system using root locus is determined by the location of the poles in relation to the complex plane. Specifically, for a system to be stable, all poles must lie in the left half-plane. If any poles are located in the right half-plane or on the imaginary axis, the system will exhibit instability, potentially leading to unbounded output or oscillatory behavior.

Root locus is a graphical method used to analyze how the roots (or poles) of a system change with varying feedback gain. By observing the trajectories of these roots as the gain is varied, one can ascertain the conditions for stability. When the poles are displaced to the left half-plane, the system becomes stable; conversely, if they shift to the right half-plane, the system becomes unstable.

The other options focus on elements that, while critical in analyzing system performance and gain settings, do not directly determine the fundamental stability provided by pole locations in the left half-plane. The total number of zeros affects the behavior and performance but does not indicate stability directly. Gain margin and phase margin are important for assessing system robustness and sensitivity to perturbations, but they arise from the characteristics of the feedback system, not the inherent stability rules defined by pole locations.