What is the purpose of the anti-parallel diodes in this Wien Bridge oscilator circuit?
The purpose of anti-parallel diodes in a Wien bridge oscillator is to provide non-linear feedback. For stable oscillator operation, the closed-loop gain must be 1. If it is greater than 1, the oscillations will increase with time. If it is less than 1, the oscillations will decay.
By using a non-linear feedback element, as the signal increases in amplitude, the gain decreases. If the signal decreases in amplitude, the gain increases. This permits there to be a level for the signal where the gain is exactly 1, and the oscillator is stable.
Diodes have an exponential/logarithmic voltage/current relationship. Although their use to automatically adjust gain in a Wien bridge oscillator leads to some distortion of the output so that it is not exactly sinusoidal, in practice, the distortion is often small enough for the purposes at hand. If not, another method of gain regulation is needed, or a different oscillator topology.