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🔌 Butterworth Filter Calculator

Calculate Butterworth filter component values (inductors and capacitors) for Tee and Pi, low-pass and high-pass configurations. Enter cutoff frequency and source/load impedance to get exact values.

Filter Topology

Cutoff Frequency (fc)

Frequency Unit

Impedance (R) in Ohms

What is this tool?

A Butterworth filter is a type of electronic filter designed to provide the flattest possible passband response — hence its nickname, the maximally flat filter. British engineer Stephen Butterworth first described it in his 1930 paper “On the Theory of Filter Amplifiers.” Unlike Chebyshev or elliptic designs that accept passband ripple in exchange for steeper roll-off, the Butterworth design prioritises a smooth, ripple-free signal in the passband, making it ideal for audio, instrumentation and communication applications where signal fidelity matters more than extreme frequency selectivity. The two most common passive LC topologies for a 3rd-order Butterworth filter are the **Tee** and **Pi** networks, each of which can be configured as low-pass (LP) or high-pass (HP). In a Tee network the series arms are first, flanking a central shunt element. In a Pi network the shunt arms are at the ends, flanking a central series element. The choice between Tee and Pi often comes down to practical considerations: which is easier to fabricate, which has more convenient standard component values, or which performs better at the impedances you are working with. This calculator handles all four combinations — Tee LP, Tee HP, Pi LP and Pi HP — in a single tool. You select the topology and type, enter the cutoff frequency and the source/load impedance (R), and the calculator returns the three component values needed. The Butterworth normalised g-values for a 3rd-order filter are g1 = 1.0, g2 = 2.0, g3 = 1.0, which scale to actual L and C through simple frequency-impedance denormalisation. For audio crossover design, RF front-end filtering, or removing unwanted harmonics from a signal path, the Butterworth response offers the cleanest trade-off between passband flatness and transition-band steepness. If you need sharper roll-off and can tolerate some ripple, consider using our Chebyshev filter calculator instead. For active RC designs using op-amps, the Sallen-Key filter calculator is more appropriate.

How it works

The Butterworth filter component values are derived from normalised prototype low-pass filter tables. For a 3rd-order Butterworth (n = 3), the normalised element values (g-values) are g1 = 1.0, g2 = 2.0, g3 = 1.0, with a normalised cutoff of ωc = 1 rad/s and impedance R = 1 Ω. **Frequency-impedance scaling** converts these normalised values to real component values at your target cutoff frequency (fc) and impedance (R): **Low-Pass Tee / Pi (inductors in series arms, capacitor in shunt):** - L = g × R / (2π × fc) - C = g / (2π × fc × R) **High-Pass Tee / Pi (capacitors in series arms, inductor in shunt):** - C = 1 / (g × 2π × fc × R) - L = R / (g × 2π × fc) The table below shows how the g-values map to components for each topology: | Topology | Series Element 1 | Shunt Element | Series Element 2 | |---|---|---|---| | Tee LP | L1 = 1.0 × R/(2πfc) | C = 2.0 / (2πfcR) | L2 = 1.0 × R/(2πfc) | | Pi LP | C1 = 1.0 / (2πfcR) | L = 2.0 × R/(2πfc) | C2 = 1.0 / (2πfcR) | | Tee HP | C1 = 1.0 / (2πfcR) | L = 2.0 × R/(2πfc) | C2 = 1.0 / (2πfcR) | | Pi HP | L1 = 1.0 × R/(2πfc) | C = 2.0 / (2πfcR) | L2 = 1.0 × R/(2πfc) | As a derived metric, this calculator also shows the **3 dB cutoff frequency** (which equals fc by design) and the **attenuation at 2×fc** (approximately 18 dB for a 3rd-order Butterworth, since the roll-off rate is n × 20 dB/decade = 60 dB/decade). For quick reference, the following table shows pre-computed capacitor values (in pF) for a 50 Ω system at various cutoff frequencies (Tee LP configuration, C = 2.0/(2πfcR)): | fc (MHz) | C (pF) | L1=L2 (μH) | |---|---|---| | 1 | 6,366 | 7.96 | | 10 | 636.6 | 0.796 | | 100 | 63.66 | 0.0796 | | 1,000 | 6.366 | 0.00796 |

How to use

Select the filter topology (Tee or Pi) using the dropdown.
Select the filter type (Low-Pass or High-Pass).
Enter the cutoff frequency (fc) in Hz, kHz or MHz.
Enter the source/load impedance (R) in ohms.
Click Calculate to get the three component values (L1, L2/C2, C or C1, L, C2).

Frequently Asked Questions

What is the difference between Tee and Pi Butterworth filter topologies?

A Tee network has two series elements flanking one shunt element (shaped like the letter T), while a Pi network has two shunt elements flanking one series element (shaped like \u03C0). Both achieve the same Butterworth response. The choice depends on practical factors such as which standard component values are available, ease of fabrication, and physical layout constraints. In some impedance ranges, one topology may yield more convenient or realisable component values than the other.

How steep is the roll-off of a 3rd-order Butterworth filter?

A 3rd-order Butterworth filter rolls off at 60 dB per decade (18 dB per octave) above the cutoff frequency. At 2×fc the attenuation is approximately 18 dB. If you need sharper selectivity, you can cascade filter sections for higher orders — each additional order adds 20 dB/decade. Alternatively, consider a Chebyshev design which achieves steeper roll-off at the cost of passband ripple.

Why does the Butterworth filter have a maximally flat response?

The Butterworth design positions all transmission zeros at infinity and distributes the poles of the transfer function on a circle in the complex plane, which mathematically produces the flattest possible magnitude response in the passband. No other all-pole filter of the same order can achieve a flatter passband without introducing ripple.

Can I use this calculator for audio crossover networks?

Yes. The 3rd-order Butterworth is a popular choice for speaker crossovers because of its flat passband and 18 dB/octave slope. For a crossover at 2 kHz with 8 \u03A9 speakers, enter fc = 2000 Hz and R = 8 \u03A9. The calculator will return the inductor and capacitor values for both Tweeter and Woofer paths. Note that real speaker impedances vary with frequency, so the actual crossover point may shift slightly.

What happens if my actual component values differ from the calculated ones?

Small deviations (5–10%) from the ideal values will slightly shift the cutoff frequency and may introduce minor passband ripple, but the filter will still function. If you must use standard E12/E24 value series, pick the nearest available value and verify the resulting cutoff frequency. For critical applications, use precision components (1% or better) and measure the actual filter response with a network analyser.

Does this calculator support unequal source and load impedances?

This calculator assumes equal source and load impedances (R), which is the standard assumption for symmetric Butterworth designs. For unequal impedances, the g-values and component scaling formulas change, and you would need an impedance-transforming network. That scenario is beyond the scope of this tool.

Tips & Advice

When using this calculator, keep in mind that real-world components have parasitic effects that become significant at high frequencies. Inductors have series resistance and inter-winding capacitance; capacitors have equivalent series resistance (ESR) and lead inductance. For RF work above 100 MHz, consider using surface-mount components with tight tolerances and minimal parasitic effects. The Butterworth design assumes ideal, lossless L and C elements; real attenuation and passband flatness will deviate from theory. Also remember to check the self-resonant frequency (SRF) of your inductors — operation above the SRF will cause the inductor to behave capacitively. For audio applications (20 Hz–20 kHz), large inductor values may be impractical; consider active filter topologies instead. If you need a steeper roll-off than 60 dB/decade, cascade two 3rd-order sections or use a higher-order design.

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