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Circuits, electronics and noise
Minimum Butterworth low-pass filter order
Smallest whole filter order whose Butterworth response meets a passband limit Ap (dB) at the passband edge fp and a stopband attenuation As (dB) at the stopband edge fs.
- Ref
- ELN-39
- Rev
- v1
- Field
- Circuits, electronics and noise
- Result
- n [poles]
A Butterworth filter must pass a signal with no more than Ap decibels of loss at the passband edge fp and give at least As decibels of loss at the stopband edge fs. Setting |H|² at those two frequencies gives the order n ≥ ln[(10^(As/10) − 1)/(10^(Ap/10) − 1)] / [2 ln(fs/fp)]. The order must be a whole number, so the result is rounded up to the next integer, which gives a margin on both limits.
Smallest whole filter order whose Butterworth response meets a passband limit Ap (dB) at the passband edge fp and a stopband attenuation As (dB) at the stopband edge fs. It needs fs > fp > 0 and As > Ap > 0.
Calculator
- Unrounded order4.289
- Transition ratio, fs/fp2
- Roll-off needed, As − Ap19 dB
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- To pass 1,000 Hz within 1 dB and stop 2,000 Hz by 20 dB needs 4.289, so a 5-pole filter.
- Each extra pole adds 20 dB per decade of roll-off, but also another amplifier stage and more component tolerance to manage.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (poles) | Local step | Change in result |
|---|---|---|---|---|---|---|
| fp Passband edge frequency | 100 to 1,900 Hz | 1 | 58 | 57 | +10% | 0 % |
| fs Stopband edge frequency | 1,100 to 10,000 Hz | 2 | 32 | 30 | +10% | −20 % |
| As Minimum stopband attenuation | 5 to 80 dB | 2 | 15 | 13 | +10% | 0 % |
| Ap Maximum passband attenuation | 0.1 to 3 dB | 4 | 7 | 3 | +10% | 0 % |
Sweep of Minimum stopband attenuation
| Minimum stopband attenuation (dB) | Minimum order (poles) |
|---|---|
| 5 | 2 |
| 14.38 | 4 |
| 23.75 | 5 |
| 33.13 | 7 |
| 42.5 | 9 |
| 51.88 | 10 |
| 61.25 | 12 |
| 70.63 | 13 |
| 80 | 15 |
Sweep one variable
Variables and how each one contributes
Minimum order
What the equation gives you. Everything else feeds it.
Passband edge frequency
The passband edge frequency, in Hz. A higher passband edge, closer to the stopband, needs a higher order. Raising it by 10 % (to 1,100) moves the result by +0 %, from 5 to 5 poles.
Stopband edge frequency
The stopband edge frequency, in Hz. A higher stopband edge relaxes the transition, so the order falls. Raising it by 10 % (to 2,200) moves the result by −20 %, from 5 to 4 poles.
Maximum passband attenuation
The largest allowed loss in the passband, in dB. A looser passband limit lets the cut-off sit lower, so the order falls. Raising it by 10 % (to 1.1) moves the result by +0 %, from 5 to 5 poles.
Minimum stopband attenuation
The smallest required loss in the stopband, in dB. A higher requirement needs a sharper filter, so the order rises. Raising it by 10 % (to 22) moves the result by +0 %, from 5 to 5 poles.
Worked example with the values above
- Numerator: ln[(10^(20/10) − 1)/(10^(1/10) − 1)] = 5.9463.
- Denominator: 2 ln(fs/fp) = 2 ln(2) = 1.3863; ratio = 4.2894.
- Round up: n = 5 poles.
Assumptions and limits
- Ideal amplifiers with enough bandwidth, and ideal resistors and capacitors.
- A source with low output impedance and a load that is much larger than the filter’s output impedance.
Common mistakes
- Rounding the order down, which fails the specification.
- Mixing up the passband and stopband frequencies.
- Using decibel values with the wrong sign, as attenuations should be positive.
Statistics and uncertainty
Component tolerance moves the cut-off frequency and the quality factor: 1 % parts give about 1 % in f₀ but can move Q a lot when the design sits near instability. Real amplifier gain-bandwidth limits the filter at high frequency.
Where it is used
- Anti-alias and noise-reduction filters before an ADC.
- Choosing order, cut-off and quality factor of analogue filters from a specification.
Maths you need
- Transfer function: output over input as a function of frequency
- Decibel: 20 log₁₀ of a voltage ratio
- Quality factor: Q, how sharply a resonance peaks