Butterworth Q Factor

Q = 1 / (2 × sin(π/(2N)))

Quick answer

This calculator solves Q = 1 / (2 × sin(π/(2N))) for Q. With Filter Order 500, the result is 159.2.

Calculator

Result
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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.

Symbols and units

Symbol Meaning Default unit Accepted units
N Filter Order — —

Step by step

  1. 1. What you know: Filter Order 500
  2. 2. The relation: Q = 1 / (2 × sin(π/(2N)))
  3. 3. Substitute: N = 500
  4. 4. Result: Q = 159.2

Worked examples

Minimum range

Given: Filter Order 25

Result: Q = 7.963

Low range

Given: Filter Order 250

Result: Q = 79.58

Typical

Given: Filter Order 500

Result: Q = 159.2

High range

Given: Filter Order 800

Result: Q = 254.6

Maximum range

Given: Filter Order 8

Result: Q = 2.563

How Q changes with Filter Order

Filter Order (input) Q (result)
50 15.92
125 39.79
250 79.58
375 119.4
500 159.2
750 238.7
1000 318.3
2000 636.6
3500 1114
5000 1592

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Frequently asked questions

How do you calculate Butterworth Q Factor?

Use Q = 1 / (2 × sin(π/(2N))). With Filter Order 500, Q works out to 159.2.

What is the formula for Butterworth Q Factor?

The formula is Q = 1 / (2 × sin(π/(2N))). The Butterworth filter is designed for maximally flat amplitude response in the passband, meaning no ripple.

Related Concepts

All Filters formulas →
RC Low-Pass Filter RC High-Pass Filter Bandpass Bandwidth Filter Rolloff Rate Notch Frequency Butterworth Q Factor Sallen-Key Cutoff Frequency Chebyshev Filter Order

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