How a wah pedal works: coils and resonance

What you're hearing

A coil and a capacitor make a resonance. A coil (an inductor) passes slow changes and fights fast ones. A capacitor does the opposite. Put the two side by side and they swap energy back and forth at one frequency: f = 1/(2π√(LC)). Fed through a resistor, the pair passes that frequency loudest and the rest more quietly: a band-pass filter. At the peak here, the output stands 30 to 34 dB above what it is at 100 Hz.

This coil and capacitor alone would ring at 3.0 kHz. With L = 500 mH and C = 5.6 nF: f = 1/(2π√(0.5 H × 5.6 nF)) ≈ 3.0 kHz. A wah needs to go much lower than that, and a pedal can't easily turn a coil or a capacitor. So the circuit makes the capacitor look bigger instead.

The pedal changes how big the capacitor looks. The second transistor amplifies the output node about 30 times, upside down. Its output comes back to the capacitor's far end through the pedal's pot, a divider of Ra (1 kΩ) and Rpedal (the treadle). When one end of a capacitor is driven the opposite way, more current flows through it, so the coil sees a larger capacitor. At the heel (Rpedal 100 kΩ), nearly all of the amplified signal comes back. The coil sees about 178 nF, 32 times the real 5.6 nF, and the peak falls to about 530 Hz. At the toe (100 Ω), little comes back. The coil sees about 7.4 nF, and the peak rises to about 2.6 kHz. That's all the treadle does: it turns a pot.

Q is how narrow the peak is. Q is the peak's frequency divided by its width, measured where it's 3 dB down. A high Q is a narrow peak that rings after each note; a low Q is a broad, gentle lift. Here the Q is about 6.0 at the heel (a peak 88 Hz wide at 533 Hz) and falls to about 2.0 at the toe (1.3 kHz wide at 2.6 kHz). So the heel is the ringiest, most vocal end. As built, with the pedal at 3.3 kΩ, the peak is at 1.0 kHz with a Q of 4.7.

R3 sets how much the peak can ring. The sound reaches the coil through R3, 47 kΩ. A larger resistor disturbs the ringing pair less, so the peak gets narrower, though quieter. At the pedal's 3.3 kΩ setting, R3 at 15 kΩ gives a Q of about 2.9, and 150 kΩ gives about 6.0. The buttons swap R3 and set the output to match the peak's level, so you hear the width change, not the volume.

Why it sounds like a voice. Our vowels are made by resonances in the mouth and throat called formants. In an average male voice, the second formant runs from about 600 Hz ("oo") to 2.4 kHz ("ee"). This wah's single peak sweeps through the same band, about 530 Hz to 2.6 kHz. Moving it from heel to toe is heard as a vowel opening, which is why the pedal is named after the sound "wah".

The first transistor keeps the coil from being loaded. The first transistor is a buffer: its emitter follows the input, so whatever comes before can't damp the coil. The coil's own wire, 60 Ω, is written as a resistor. In SynthMaker a coil is otherwise ideal, so its resistance is part of the parts list, not hidden in the part.

Try this

• Turn pedal slowly from heel (100 kΩ) to toe (100 Ω). Hear the peak rise from about 530 Hz to 2.6 kHz, ringing most at the heel. Turning it back and forth in time with a note is the wah.
• Press the three R3 buttons with the pedal in the middle. At 15 kΩ the wah is broad and gentle. At 150 kΩ it's narrow and almost whistles.
• Open the example sounds. In "Coil Wah", an LFO rocks the pedal once a beat. In "Envelope Wah", an Envelope opens it on every note of a bass, the quack of an envelope filter. To change the parts yourself, open it in SynthMaker.

The parts list

circuit CoilWah  setting input: 0.1 V, from 0.01 V to 1 V  setting output: 0.02 V, from 0.005 V to 0.4 V  setting pedal: 3.3k, from 100k to 100R  V1 vcc gnd 9V  C1 in b1 47n  R1 vcc b1 470k  R2 b1 gnd 470k  Q1 vcc b1 e1 2N3904  Re1 e1 gnd 10k  C2 e1 a 1u  R3 a out 47k  C3 out b2 220n  R4 vcc b2 470k  R5 b2 gnd 82k  Q2 c2 b2 e2 2N3904  Rc vcc c2 10k  Re2 e2 f 220  Re3 f gnd 1k  Ce f gnd 10u  C4 c2 d 1u  Ra d w 1k  Rpedal w gnd pedal  Ctune out w 5.6n  L1 out l 500mH  RL l gnd 60end

The pedal setting is written from the bigger value (100k down to 100R), so turning the knob up moves towards the toe.

How we checked it

Checked against ngspice, a standard circuit simulator: across the pedal's travel, its peak frequency is within 0.2% and its Q within 1% of ngspice's. With the pedal at 3.3 kΩ, a 1 kHz sine of 0.1 V through it is within 0.056 mV rms of ngspice at the 2× rate it runs at, against a peak of 31.5 mV.

Build it step by step

• The manual: Coils and the wah builds this circuit part by part, with the coil on its own first.
• The example sounds: "Coil Wah" (a clavinet-like line, the pedal rocked by an LFO) and "Envelope Wah" (a bass, the pedal opened by an Envelope on every note).

Next and previous

All circuits

Further reading

• Resonance, Q and the band-pass: Wikipedia: RLC circuit.
• Vowel formants and their frequencies: Wikipedia: Formant (its table of average vowel formants).
• The wah pedal's history and use: Wikipedia: Wah-wah pedal.
• M. Holters and U. Zölzer, "A Generalized Method for the Derivation of Non-Linear State-Space Models from Circuit Schematics", EUSIPCO 2015, the method SynthMaker uses. PDF