How an octave-up pedal works: rectifiers

What you're hearing

Two copies, one upside down, and the higher one wins. The op-amp is an inverting stage: its gain is −Rf/Ri = −10 kΩ / 10 kΩ = −1 (chapter 3 explains how: the op-amp's two rules). So its output is the input turned upside down, exactly: when the input is at +1 V, the copy is at −1 V. Classic octave-up pedals make that copy with a centre-tapped transformer, or with one transistor whose collector and emitter give two opposite copies (a phase splitter). D1 comes from the input, D2 from the copy, and both meet at r. Whichever side is above r conducts, and the other is reverse-biased and passes almost nothing. On the top half of the wave, D1 passes the input. On the bottom half, D2 passes the flipped copy. The bottom half comes out on top: that's full-wave rectification, |input|, less a diode's drop.

Flipping the bottom half doubles the frequency. A sine's bottom half is the same shape as its top half. Flip it up and each cycle becomes two identical bumps, so the wave repeats twice as often: a 110 Hz note repeats every 4.5 ms (1/220 Hz), and you hear 220 Hz, an octave up. Because the copy is an exact mirror, nothing at 110 Hz is left. What's added are the new note's own harmonics, 440 Hz, 660 Hz and so on: all even multiples of the original. For a perfect rectifier, the 440 Hz harmonic is one fifth the size of the 220 Hz one (14 dB under it), and the 660 Hz one 3/35 of it (21 dB under).

Half-wave keeps the old note. Take D2 out and the bottom half is simply cut off: r sits at 0 V for half of each cycle. The wave still repeats only once per cycle, so the note stays at 110 Hz, with even harmonics added. That's a buzzier, hollow version of the same pitch, not an octave.

Each diode costs half a volt. A silicon diode passes almost nothing until about 0.3 V, so small signals barely get through:

At 2 V in, 150 µA flows through Rl, the 10 kΩ to ground, and the diode holds 0.50 V of the input. A fading note suffers most: from 1 V down to 0.3 V, 10.5 dB quieter, the octave falls 32 dB and sputters out. That's why a bare diode rectifier needs a loud signal in front of it. A precision rectifier fixes this by putting the diode inside an op-amp's feedback, so the op-amp makes up the drop.

Rectifying makes an offset; the capacitor takes it away. A rectified wave never goes below 0 V, so it has an average, a DC offset: 0.84 V at 2 V in (half-wave: 0.42 V). A perfect full-wave rectifier's average is 2/π, 64% of its peak; the diode's drop flattens the bottoms and lowers it. C1 and R2 make a high-pass filter: f = 1/(2πRC) = 1/(2π × 1 MΩ × 10 nF) ≈ 15.9 Hz. It blocks the offset and passes the 220 Hz note almost untouched. At 2 V in, the output then swings from −0.84 V to +0.66 V around zero. The offset builds with each note and settles in about 50 ms (five times RC, 10 ms). The third button removes C1, and the Scope shows the whole wave sitting above zero.

Why it likes single, pure notes. The trick needs a wave whose two halves mirror each other and which crosses zero only twice a cycle. Single notes high on the neck, played on the neck pickup with the tone down, are close to sine waves, so they double cleanly. Bright tones with strong harmonics don't: a saw folds into a triangle at the same pitch, with no octave at all, and a square comes out almost flat. Chords are worse. Two notes a fifth apart, 110 Hz and 165 Hz, come out with their octaves (220 Hz and 330 Hz), but also their difference, 55 Hz, and their sum, 275 Hz, which are louder than the octaves here. Rectifying acts partly like multiplying the signal by itself, so every pair of notes makes sum and difference tones.

Try this

• Turn input from 2 V down to 0.3 V and watch the Scope. The bumps shrink much faster than the input, and below about 0.5 V there's little left: the diodes' drop.
• Switch between "Full-wave" and "Half-wave" on a held note. Full-wave jumps an octave; half-wave stays at the note's pitch and sounds hollow and buzzy.
• Press "Full-wave, no capacitor". The wave sits above zero: that's the offset C1 removes.
• Open it in SynthMaker to change the parts. Try 1N34A germanium diodes: they drop about half as much, so quieter notes get through.

The parts list

circuit Octave  setting input: 2 V, from 0.2 V to 10 V  setting output: 0.87 V, from 0.1 V to 5 V  U1 gnd m b opamp  Ri in m 10k  Rf b m 10k  D1 in r 1N4148  D2 b r 1N4148  Rl r gnd 10k  C1 r out 10n  R2 out gnd 1Mend

This is the full-wave version. Half-wave puts a # at the start of D2's line; the third button replaces C1 with a 1 Ω resistor.

How we checked it

Checked against ngspice, a standard circuit simulator: the whole circuit, in each of its three versions, driven by a 1 kHz sine, is within 0.002% of ngspice across the audible band at the 2× rate it runs at, also down on the diodes' knee at 0.5 V in. Measured on a 120 Hz note, full-wave leaves no trace of the original note at all: its octave is followed by the 4× harmonic 17 dB lower and the 6× 31 dB lower.

Build it step by step

• The lesson: A distortion pedal from six parts (opens in a new tab). Ned builds a diode circuit part by part and swaps between silicon and germanium diodes.
• The video: This distortion pedal is six lines of text: a real circuit, simulated. It loads only when tapped.
• Half-wave, held: Random voltages from noise (opens in a new tab) charges a capacitor through one diode, so it keeps a pulse's peak after the pulse has gone.

This distortion pedal is six lines of text: a real circuit, simulated

Next and previous

• Start of the chapter: How a diode clipper works

All circuits

Further reading

• Half-wave and full-wave rectifiers, their average voltage and their output's harmonics: Wikipedia: Rectifier.
• How an op-amp removes the diode's drop: Wikipedia: Precision rectifier.
• The one-transistor way to make two opposite copies: Wikipedia: Phase splitter.
• Why two notes through a nonlinear circuit make sum and difference tones: Wikipedia: Intermodulation.
• The diode model (Shockley's equation, as SPICE uses it): ngspice documentation.