Butterworth Q Factor

Q = 1 / (2 × sin(π/(2N)))

Calculator

Result
—

Formula

Q = 1 / (2 × sin(π/(2N)))

Description

The Butterworth filter is designed for maximally flat amplitude response in the passband, meaning no ripple. The Q factor of each second-order section in a Butterworth filter depends on the total filter order. This formula gives the Q of the most critical (highest Q) pole pair, which determines the component sensitivity and the peak in the frequency response near the cutoff. Higher-order Butterworth filters require higher Q sections that are more sensitive to component tolerances. The Butterworth response provides a good balance between passband flatness, transition steepness, and phase response.

Variables

  • Q — Quality factor of the highest-Q pole pair
  • N — Filter order

Practical Notes

For a 2nd-order Butterworth: Q = 0.707 (the classic value). For 4th-order: Q values are 0.541 and 1.307 for the two sections. Chebyshev filters trade passband flatness for steeper rolloff (higher Q sections). Bessel filters have lower Q for best phase response. When building active filters, the required Q determines op-amp gain-bandwidth requirements.

Related Formulas

RC Low-Pass Filter RC High-Pass Filter Bandpass Bandwidth Filter Rolloff Rate Notch Frequency Sallen-Key Cutoff Frequency

Need more features?

Save calculations, import telemetry data, simulate battery discharge, and collaborate with your team.

Try the App