Filter Q Reference
Quality factor (Q) in filters — relationship to damping, bandwidth, and peaking.
Reference
Definitions
- Q factor
- Q = f_r / Δf_{-3dB} = 1 / (2ζ) for 2nd-order
- Damping ratio ζ
- = 1 / (2Q)
- Characteristic eqn
- s² + (ω₀/Q) s + ω₀²
Second-order Q → behavior
| Q | Damping | Step response | Peak in freq |
|---|---|---|---|
| 0.5 | Critical | Slow monotonic | No peak |
| 0.577 | Bessel (flat delay) | Slight rise | No peak |
| 0.707 | Butterworth | Small overshoot (4%) | No peak (max flat) |
| 1.0 | Chebyshev-ish | Overshoot ~16% | Mild peak |
| 2.0 | Under-damped | Ringing | 6 dB peak |
| 10 | Very under-damped | Long ring | 20 dB peak |
Relation to bandwidth
- Δf (bandwidth)
- = f_r / Q
- High Q → narrow band
- Selective; rings on transients
- Low Q → wide band
- Gentle response, fast settling
Parametric EQ
- EQ Q typically 0.7 – 10 depending on purpose.
- Narrow Q (high value) for surgical notching.
- Wide Q (low value) for musical tone shaping.
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