the Wien-bridge oscillator

The Wien-bridge oscillator is the friendliest way to make a clean, low-distortion sine wave at audio frequencies, from tens of hertz up to a few hundred kilohertz. It is the classic circuit you reach for to build a test-tone generator. It uses just an op-amp, four resistors, and two capacitors, and it produces a pure tone you could feed straight into an amplifier and hear as a smooth, hum-free whistle.

Its secret is a frequency-selective network made of an RC series section and an RC parallel section, arranged so that at exactly one frequency the signal passes through with zero phase shift and is attenuated to one third. That frequency is f equals 1 over (2 times pi times R times C). The op-amp is wired as a non-invering amplifier whose gain must be exactly 3 to cancel that one-third loss and make the loop gain 1 with zero phase, satisfying Barkhausen. For example, with R equal to 16 kilohm and C equal to 10 nanofarad, f is about 1 kHz. The other two resistors set the gain: a feedback resistor twice the size of the grounded one gives exactly 3.

Here is where the real art lives. If the gain is set to a fixed 3 it is impossible to hold: a hair under 3 and the oscillation dies, a hair over and it grows until the op-amp clips into an ugly distorted near-square wave. The classic fix, used since Hewlett-Packard's very first product, is amplitude stabilization: a small lamp or a JFET or a diode network gently lowers the gain as the amplitude rises, automatically parking the loop gain at exactly 1. That feedback loop is what turns a fussy textbook equation into a rock-steady, low-distortion tone.