How to use the filter order calculator
- Enter the largest loss in the passband in decibels (for example 1 dB) and the passband edge frequency.
- Enter the smallest loss in the stopband (for example 40 dB) and the stopband edge frequency.
- Put the stopband edge above the passband edge for a low-pass filter, and below it for a high-pass filter.
- Read the order a Butterworth filter and a Chebyshev filter need, the exact values before rounding up, and the Butterworth -3 dB frequency.
Formula
Frequency ratio = stopband edge / passband edge for low-pass (inverted for high-pass)
e = (10^(As/10) - 1) / (10^(Ap/10) - 1)
Butterworth order: n = log10(e) / (2 log10(ratio))
Chebyshev order: n = acosh(sqrt(e)) / acosh(ratio). Round either up to a whole number.
The order is the number of poles, and each pole adds 20 dB per decade of slope. A higher order separates the passband from the stopband more sharply and costs more parts.
1 dB at 1 kHz and 40 dB at 2 kHz
The ratio is 2 and e = (10^4 - 1) / (10^0.1 - 1) = 38,617. A Butterworth filter needs n = log10(38617) / (2 log10 2) = 7.6185, which rounds up to order 8. A Chebyshev filter needs only acosh(sqrt(38617)) / acosh(2) = 4.5361, so order 5.
With order 8 the Butterworth filter has a -3 dB frequency of 1088.1 Hz and a loss of 42.3 dB at 2 kHz, which clears the 40 dB requirement. A high-pass filter with 3 dB at 500 Hz and 30 dB at 100 Hz needs order 3 as a Butterworth (2.1472 before rounding) and order 2 as a Chebyshev (1.8097). A steeper case of 0.5 dB at 10 kHz and 60 dB at 15 kHz needs a Butterworth of order 20.
Butterworth or Chebyshev
A Butterworth filter has a flat passband, and a Chebyshev filter trades ripple in the passband for a steeper cutoff, so it needs a lower order for the same limits. If the ripple is acceptable, a Chebyshev filter saves parts. If a flat response matters, such as in audio, a Butterworth is the usual choice, at the cost of a higher order.
Each pair of poles is usually built as one active stage, and the gain of a stage depends on its amplifier, which the op-amp gain calculator can help with. To convert losses between decibels and ratios, use the decibel calculator.
From order to a design
The order is the first step in a design. After that you pick the cutoff frequency, the pole positions and the component values, and you check the real response with the tolerance of the parts. For the resonance of the circuits you choose, see the RLC impedance and resonance calculator.
The numbers on this page describe the ideal analog prototype. A digital filter uses other methods, and a built filter is affected by part tolerance, amplifier limits and loading.
Assumptions and limits
- Losses are positive decibels, and the stopband loss is larger than the passband loss.
- The edge frequencies differ. Their ratio is always taken above 1.
- The order of the Chebyshev filter is for a type I filter with ripple equal to the passband loss.
- The results are for ideal analog filters.
Frequently asked questions
How do I find the order of a filter?
Use the ratio of the edge frequencies and the two loss limits in the formula for the type you want, then round up to the next whole number.
Why does a Chebyshev filter need a lower order?
It allows ripple in the passband, and that freedom gives a steeper transition between the passband and the stopband.
Does this work for a high-pass filter?
Yes. Enter a stopband edge below the passband edge. The page inverts the ratio.
What does the -3 dB frequency tell me?
It is where a Butterworth filter of the rounded order would lose 3 dB. It sits above the passband edge, because the order was rounded up and the filter is better than the minimum.