amplitude stabilization
Amplitude stabilization is the trick that keeps an oscillator's swing from either fading away or growing until it clips. Barkhausen says oscillation is sustained only when the loop gain is exactly 1. But exactly is impossible to hold with fixed parts: components drift, temperature shifts, and the gain is never precisely right. So a real oscillator is designed with loop gain a little above 1 to start reliably, and then needs an automatic referee that gently pulls the gain back down to exactly 1 once the amplitude reaches the level you want. Without it, the output relentlessly grows until it slams into the supply rails and turns the lovely sine into a clipped, distorted square.
The mechanism is a slow, gentle negative-feedback loop that watches the amplitude and adjusts the gain. The classic Wien-bridge trick is a small incandescent lamp in the gain-setting leg: as the amplitude rises, more current warms the filament, its resistance rises, and that lowers the amplifier's gain, automatically settling the loop gain to 1. Modern versions rectify the output to a DC level and use it to control a JFET acting as a voltage-variable resistor, or use back-to-back diodes that start to conduct at high swing and softly reduce gain. The key is that this correction must act slowly compared to one cycle, so it sets the amplitude without distorting the waveform.
Amplitude stabilization is where a sine-wave oscillator's purity is won or lost. Stabilize too aggressively and the gain wobbles within each cycle, adding distortion; stabilize too softly and the amplitude bounces or takes ages to settle after start-up. The honest point is that a high-quality, low-distortion sine generator is really a careful amplitude-control design wrapped around a simple oscillator core. Relaxation oscillators sidestep this entirely by deliberately driving the output to the rails, which is why they make square waves, not clean sines.
In a Wien bridge needing a gain of 3, a lamp in the lower gain leg starts cold with low resistance, giving a gain slightly above 3 so the oscillation grows. As the swing rises the lamp warms, its resistance climbs, and the gain drops to exactly 3, parking the amplitude wherever the lamp's self-heating balances.
A slow gain-reducing feedback loop parks the loop gain at exactly 1 once the desired amplitude is reached.
It must act slowly relative to one cycle. A fast-acting limiter (like simply letting the op-amp clip) does control amplitude, but at the cost of heavy harmonic distortion, which is exactly what you are trying to avoid in a clean sine source.