What is the closed-loop characteristic equation in Root Locus?

• A. G(s) = 0
• B. H(s) = 1
• C. 1 + K G(s) H(s) = 0
• D. K = G(s)H(s)

Answer: (C)

The closed-loop characteristic equation in Root Locus is derived from the principles of control theory, specifically the feedback system analysis. In control systems, the goal is often to analyze the stability and behavior of the system as the gain (K) varies.

The expression '1 + K G(s) H(s) = 0' is fundamental in control theory because it represents the condition for the closed-loop system to have poles in the s-plane, which are crucial for determining system stability. Here, G(s) is the open-loop transfer function, and H(s) is the feedback transfer function. When this equation is set to zero, it signifies points where the characteristic equation of the closed-loop system intersects with the s-axis on the Root Locus plot, which can indicate potential poles of the system.

This characteristic equation helps in analyzing how variations in gain (K) impact the location of these poles and, consequently, the stability and transient response of the system. Understanding this relationship allows engineers to design more effective control systems by adjusting gain values and ensuring desired performance characteristics.

The other options do not represent the closed-loop characteristic equation correctly; one indicates open-loop conditions, while others suggest relationships not tied directly to closed-loop stability analysis. Therefore, focusing on the