Butterworth Filters

Imagine a filter whose frequency response in the passband is as flat as a perfectly calm lake — not a single ripple anywhere. That is the defining property of the Butterworth filter, described by British engineer Stephen Butterworth in his 1930 paper "On the Theory of Filter Amplifiers." Where other filter families sacrifice flatness to gain sharper rolloff, the Butterworth refuses to compromise: it achieves the most perfectly flat magnitude response of any realizable filter family. This makes it the natural choice for any application where signal amplitude must be preserved accurately across the entire passband — audio and sensor signal conditioning, measurement front ends, and harmonic filters where the wanted signal must emerge undistorted while the harmonics are rejected.

The Butterworth filter is also the simplest of the classical filter families to design. Its component values can be derived from straightforward formulas or read directly from normalized prototype tables, then scaled to any cutoff frequency and impedance level. You do not need to be a mathematician to build one; with this lesson, a clear design procedure, and a few components from the parts drawer, you can build a properly specified filter from first principles.

The Butterworth Magnitude Response

The Butterworth filter is defined by its magnitude-squared frequency response. For a low-pass filter of order n with cutoff frequency f_c:

This formula tells you everything about how the filter behaves. At f = 0 (DC), (f/f_c)²ⁿ = 0, so |H|² = 1 — full output, no attenuation. At f = f_c, (f/f_c)²ⁿ = 1, so |H|² = 1/2 — output power is half input power, which is −3 dB. At frequencies far above f_c, (f/f_c)²ⁿ grows very large and |H|² approaches zero — strong attenuation.

The critical insight is the exponent 2n. A first-order filter (n = 1) gives |H|² = 1/(1 + (f/f_c)²). A seventh-order filter (n = 7) gives |H|² = 1/(1 + (f/f_c)¹⁴). At twice the cutoff frequency (f = 2f_c), the attenuation in dB is:

These numbers are critically important for filter design. If you need, say, 40 dB of attenuation at twice the cutoff frequency, you need at least a 7th-order Butterworth. In practice, you choose f_c so that the frequencies you must reject are as far above it as possible while the wanted signal stays comfortably inside the passband — every extra bit of frequency ratio lets a lower-order filter meet the requirement.

Butterworth low-pass filter magnitude response for orders 1 through 7. All curves are exactly −3 dB at the cutoff frequency. Higher orders give steeper rolloff but the passband remains perfectly flat for all orders.

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The "Maximally Flat" Property

The term "maximally flat" has a precise mathematical meaning: the Butterworth response has the maximum number of derivatives equal to zero at f = 0. Practically, this means the response is flat to the highest possible degree in the passband — it does not begin to roll off until it is close to the cutoff frequency. All other classical filter families (Chebyshev, elliptic, Bessel) sacrifice some degree of passband flatness to gain some other property. The Butterworth makes no such compromise.

The consequence is that a Butterworth low-pass filter introduces essentially zero amplitude variation well inside its passband. A filter with f_c = 9 MHz has a response less than 0.01 dB below the DC response at 7.0 MHz, and less than 0.05 dB down at 7.3 MHz. A signal anywhere in that range passes through with negligible amplitude change, while frequencies above 14 MHz are progressively attenuated.

The Normalized Low-Pass Prototype

The genius of filter design tables is the concept of the normalized prototype. Instead of computing component values for every combination of cutoff frequency, source impedance, and load impedance, mathematicians solved the problem once for a specific "normalized" case:

• Cutoff frequency: 1 radian per second (ω_c = 1 rad/s, or f_c = 1/(2π) Hz)
• Source and load impedance: 1 ohm

For these normalized conditions, the component values — all in units of ohms, henrys, or farads that happen to equal small numbers — are tabulated for each order n. The normalized prototype values are called g₁, g₂, g₃, …, g_n, where each g represents either a series inductor or a shunt capacitor in the ladder network.

To design a filter for any real cutoff frequency and impedance, you simply scale the normalized values. This two-step process (look up normalized values, then scale) makes filter design accessible without needing to solve differential equations from scratch each time.

The Ladder Topology

A Butterworth low-pass filter is built as a ladder network: alternating series and shunt elements. For a filter connected between equal source and load impedances (50 ohms in and 50 ohms out), the standard topology for odd-order filters is:

For even-order filters, or when source and load impedances are not equal, the topology may start with a shunt element instead. For filters with equal 50-ohm source and load, the standard odd-order topology starting with a series inductor is the most common choice — it provides a low-resistance path for DC (the series inductors present almost zero DC resistance), which is convenient when the filter also has to pass a DC bias or supply current.

Normalized Component Value Tables

The following table gives the normalized component values g_k for Butterworth low-pass prototype filters of orders 1 through 7. For a prototype with source resistance R_S = 1 Ω and cutoff frequency ω_c = 1 rad/s, even-numbered g values are shunt capacitors (in farads) and odd-numbered g values are series inductors (in henrys). The last value g_n+1 is the normalized load resistance (always 1 for Butterworth with equal terminations).

Notice the symmetry: each row reads the same left-to-right as right-to-left. This is because the Butterworth filter has equal source and load impedances, making the filter symmetric — you can turn it around and it performs identically. Also notice that the middle value of each odd-order filter equals 2.0000; this is a mathematical property of the Butterworth normalization.

Frequency and Impedance Scaling

The normalized prototype values (all referenced to 1 ohm and 1 rad/s) must be scaled to match your actual design frequency and impedance. The scaling formulas are straightforward:

These formulas come from two facts: impedance scaling multiplies inductors by R and divides capacitors by R; frequency scaling divides all component values by 2π·f_c. The two scalings are applied simultaneously in the formulas above.

Calculating Stopband Attenuation

Before committing to a specific filter order, it is worth calculating exactly how much attenuation your filter will provide at the harmonic frequencies. The formula is:

For filter design, work backwards: you know the required attenuation (say, 43 dB) and the frequency ratio (f_harmonic / f_c), and you want to find the minimum order n:

Always round n up to the next integer (since you cannot build a non-integer order filter). A slightly higher order gives more margin.

Worked Design Example: Square Wave to Sine Wave

A very common bench requirement is a clean sine wave from a square-wave clock — for testing an amplifier, for a reference signal, or to drive a circuit that needs a sinusoidal input. A square wave contains its fundamental plus odd harmonics at 3×, 5×, 7× and so on, with amplitudes of 1/3, 1/5, 1/7 of the fundamental. A low-pass filter that passes the fundamental and removes the harmonics turns it into a sine wave. Let's design one first for a case where Butterworth struggles, then for one where it shines.

The worked example shows an important real-world lesson: Butterworth filters provide clean, flat passbands but need a higher order than a Chebyshev to meet the same specification when the frequencies to be rejected are close to the cutoff. When the stopband frequencies are well above the cutoff — roughly 2.5 times or more — or when passband flatness is the primary concern, the Butterworth is the simplest and best choice.

Converting a Low-Pass Butterworth to a High-Pass Filter

The normalized low-pass prototype can be converted to a high-pass filter by a simple element substitution: every series inductor becomes a series capacitor with value 1/g_k, and every shunt capacitor becomes a shunt inductor with value 1/g_k. After substitution, apply the same frequency and impedance scaling formulas. The result is a high-pass filter with −3 dB cutoff at the same f_c and with the same rolloff rate, but now passing high frequencies and blocking low ones.

A 3rd-order Butterworth high-pass filter with f_c = 100 MHz and R = 50 Ω would protect a 433 MHz receiver module from strong broadcast interference: it would attenuate signals below 100 MHz (where the powerful AM, shortwave and FM broadcast transmitters live) while passing 433 MHz with virtually no loss.

Practical Construction Tips for LC Butterworth Filters

Building a working Butterworth filter requires attention to the practical aspects of component selection, winding technique, and layout.

Inductor Construction

For LC low-pass filters below about 30 MHz, inductors are typically wound on powdered-iron toroid cores. T68-2 (red/clear) cores are popular for 1–30 MHz; T68-6 (yellow/clear) cores are used above 10 MHz. The number of turns for a given inductance can be calculated from the core's A_L value (inductance per 100 turns, in µH). For a T68-2 core, A_L ≈ 57 µH per 100 turns. Use the formula: N = 100 × √(L / A_L) where L is in µH and A_L is in µH/100 turns. For the 1.33 µH inductor above: N = 100 × √(1.33/57) = 15 turns. Above about 50 MHz, small air-wound coils or surface-mount chip inductors are used instead.

Wind inductors with the tightest practical winding, covering 60–80% of the core circumference, and secure the windings with a drop of coil dope or RTV silicone. Use enameled copper wire sized for the current: at signal levels, almost any wire works; for 100 W into 50 Ω, peak current is approximately √(2 × 100/50) = 2 A, so 24 AWG is adequate.

Capacitor Selection

LC filters need high-Q, stable capacitors. C0G (NP0) ceramic capacitors are the standard choice: they have low series resistance, capacitance that barely changes with temperature or voltage, and good self-resonance characteristics. Silver-mica capacitors are a traditional alternative for values under 1,000 pF. Avoid X7R and Z5U ceramics, which have high losses at high frequencies and a capacitance that changes with temperature and applied voltage, shifting the filter response. For filters carrying significant power, check that the capacitor's voltage rating exceeds the peak signal voltage.

The capacitor's voltage rating must exceed the peak RF voltage: for 100 W into 50 Ω, peak voltage = √(2 × P × R) = √(2 × 100 × 50) = 100 V. Use capacitors rated at 200 V minimum (500 V is safer).

Layout and Shielding

A poor physical layout can ruin an otherwise well-designed filter. Keep the filter in a shielded metal enclosure. Orient the inductors so their magnetic fields are perpendicular to each other (alternating horizontal and vertical, or with 90° rotation between adjacent toroids) to minimize mutual coupling. Minimize lead lengths and keep ground connections short and direct. Run the signal path in a straight line if possible, with input and output connectors on opposite ends of the enclosure.