Complex poles lead to which damping type?

Complex poles lead to which damping type?

• A. Underdamped
• B. Unstable
• C. Overdamped
• D. Critically damped

Answer: (A)

Complex poles indicate an oscillatory transient that dies away over time. In a typical second-order system, when the damping ratio ζ is between 0 and 1, the poles are complex conjugates with negative real parts: s = -ζω_n ± jω_n√(1−ζ^2). This yields a time response that oscillates at the damped frequency ω_d = ω_n√(1−ζ^2) while the amplitude decays as e^(−ζω_n t). That combination—oscillation plus decay—is the hallmark of an underdamped system. If the poles were real (ζ ≥ 1), you’d get overdamped or critically damped behavior without oscillations; if a pole had a positive real part, the system would be unstable.

Complex poles indicate an oscillatory transient that dies away over time. In a typical second-order system, when the damping ratio ζ is between 0 and 1, the poles are complex conjugates with negative real parts: s = -ζω_n ± jω_n√(1−ζ^2). This yields a time response that oscillates at the damped frequency ω_d = ω_n√(1−ζ^2) while the amplitude decays as e^(−ζω_n t). That combination—oscillation plus decay—is the hallmark of an underdamped system. If the poles were real (ζ ≥ 1), you’d get overdamped or critically damped behavior without oscillations; if a pole had a positive real part, the system would be unstable.