How it works
The characteristic equation is s² + 2ζω_n·s + ω_n² = 0. Closed-loop poles are s = −ζω_n ± jω_n√(1−ζ²) for 0 < ζ < 1. The real part σ = ζω_n determines the exponential decay rate; the imaginary part ω_d = ω_n√(1−ζ²) is the damped natural frequency. When ζ = 0, poles are purely imaginary at ±jω_n and the output oscillates indefinitely at ω_n. When ζ = 1, both poles coincide at s = −ω_n (critically damped, no overshoot). When ζ > 1, poles are both real and negative (overdamped, exponential settling).
Key points to remember
The four response types based on ζ: underdamped (0 < ζ < 1) — oscillatory with overshoot; critically damped (ζ = 1) — fastest non-oscillatory; overdamped (ζ > 1) — slow exponential; undamped (ζ = 0) — sustained oscillation. For ζ = 0.5, %Mp ≈ 16.3%. Bandwidth ω_BW ≈ ω_n√(1 − 2ζ² + √(4ζ⁴ − 4ζ² + 2)), which simplifies to approximately ω_n for small ζ. The angle of departure from complex poles and the distance from the origin (= ω_n) are key pole-zero map facts. Increasing ω_n while keeping ζ constant moves poles radially outward, speeding up response proportionally.
Exam tip
The examiner always asks you to sketch the pole locations in the s-plane and identify the response type for a given second-order G(s) — mark ω_n as the distance from origin, ζω_n as the real part, and ω_d as the imaginary part on the diagram.