Library / Circuits, electronics and noise / ELN-38
Circuits, electronics and noise
Butterworth low-pass magnitude response
Magnitude response of an nth-order Butterworth low-pass filter with unity DC gain and a −3 dB frequency fc.
- Ref
- ELN-38
- Rev
- v1
- Field
- Circuits, electronics and noise
- Result
- |H| [V/V]
A Butterworth low-pass filter of order n has |H| = 1/√(1 + (f/fc)^(2n)). It is maximally flat in the passband and exactly −3 dB (0.707) at fc for every order. Above fc the response falls at 20n dB per decade, so a higher order gives a sharper cut. At f = fc/2 the gain is still very close to 1 for a high order.
Magnitude response of an nth-order Butterworth low-pass filter with unity DC gain and a −3 dB frequency fc. The response is 1/√2 at fc for every order, and falls faster for higher orders.
Calculator
- Gain in decibels−12.3 dB
- Roll-off rate far above fc, 20n40 dB/decade
- Gain at f = 2 fc0.2425 V/V
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.
- At 2,000 Hz an order-2 filter with fc = 1,000 Hz has a gain of 0.2425, which is −12.3 dB.
- At f = fc the gain is always 0.7071 (−3.01 dB) for every order; only the steepness around that point changes.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (V/V) | Local step | Change in result |
|---|---|---|---|---|---|---|
| fc Cutoff (−3 dB) frequency | 10 to 100,000 Hz | 2.5e-5 | 1 | 1 | +10% | +19.4 % |
| f Frequency | 1 to 100,000 Hz | 1e-4 | 1 | 0.9999 | +10% | −16.6 % |
| n Filter order | 1 to 10 | 9.766e-4 | 0.4472 | 0.4462 | +1 | −48.9 % |
Sweep of Filter order
| Filter order | Gain magnitude (V/V) |
|---|---|
| 1 | 0.4472 |
| 2.125 | 0.2235 |
| 3.25 | 0.1045 |
| 4.375 | 0.04814 |
| 5.5 | 0.02209 |
| 6.625 | 0.01013 |
| 7.75 | 0.004645 |
| 8.875 | 0.00213 |
| 10 | 9.766e-4 |
Sweep one variable
Variables and how each one contributes
Gain magnitude
What the equation gives you. Everything else feeds it.
Frequency
The frequency at which the gain is wanted, in Hz. A higher frequency gives a lower gain. Raising it by 10 % (to 2,200) moves the result by −16.6 %, from 0.2425 to 0.2023 V/V.
Cutoff (−3 dB) frequency
The −3 dB cut-off frequency, in Hz. A higher cut-off raises the gain at a fixed frequency. Raising it by 10 % (to 1,100) moves the result by +19.4 %, from 0.2425 to 0.2895 V/V.
Filter order
The filter order, a whole number from 1 upward. A higher order sharpens the cut, so the gain above fc falls and the gain below fc rises. Raising it by 10 % (to 2.2) moves the result by −12.3 %, from 0.2425 to 0.2127 V/V.
Worked example with the values above
- Frequency ratio: f/fc = 2,000 / 1,000 = 2.
- Raise to 2n = 4: 16.
- |H| = 1/√(1 + 16) = 0.24254.
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
- Treating f₀ as the −3 dB frequency when Q is not the Butterworth value.
- Using angular frequency where frequency in hertz is needed, a factor of 2π.
- Rounding the filter order down: the ceiling is needed.
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