Where must the roots of the characteristic equation be located on the S-plane for a system to be stable?

• A. In the Unstable Region (Right Half Plane)
• B. On the +jω axis only
• C. In the Stable Region (Left Half Plane)
• D. At the origin (0,0) only

Answer: (C)

For a system to be considered stable, the roots of its characteristic equation must reside in the stable region of the S-plane, which is defined as the left half-plane. When the roots, or poles of the system, are located in this area, it indicates that any perturbation or input to the system will eventually decay over time, leading to stable system behavior. This is because poles in the left half-plane correspond to negative real parts, which result in exponentially decaying responses.

If the roots were found in the unstable region, which is the right half-plane, or on the imaginary axis, the system would display behavior that could lead to increasing oscillations or sustained responses, indicating instability. Thus, having the roots in the stable region ensures not only that the transient dynamics will settle down but also that the system will not experience any growth in its response that could lead to instability.