Filter Q to Damping Ratio

Formula

Description

The quality factor Q and damping ratio ζ are reciprocally related parameters that describe the same physical behavior of a second-order system. Low damping (ζ < 1, underdamped) corresponds to high Q and produces ringing and overshoot in the step response. Critical damping (ζ = 1, Q = 0.5) gives the fastest response without overshoot. A Butterworth second-order filter has Q = 0.707 (ζ = 0.707), providing a good compromise between speed and overshoot. Control systems and filter design use whichever parameter is more convenient for the analysis.

Variables

• Q — Quality factor (dimensionless)
• ζ — Damping ratio (dimensionless)

Practical Notes

Key values: ζ = 1 (Q = 0.5) critically damped, ζ = 0.707 (Q = 0.707) Butterworth, ζ = 0.5 (Q = 1.0) moderate peaking, ζ = 0.1 (Q = 5) significant ringing. In control systems, ζ = 0.7 is often targeted for fast settling with <5% overshoot. For filters, the Q determines the shape of the magnitude response near the cutoff frequency.