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I have never analyzed a Wien Bridge oscillator until now and I have never read one of the thousands of explanations about it on the internet, but here is what I see:
1) R1, R2, C1 and C2 form a bandpass filter that peaks at one frequency and applies positive feedback to the opamp to cause oscillation.
2) R3, R4 and the Jfet provide negative feedback so that the gain is about 3 times. If the gain is less than 3 times then it does not oscillate because I guess the bandpass filter has a loss of 3 times. If the gain is more than 3 times then the output signal level keeps increasing until the opamp has clipping.
3) The diode rectifies the output signal and C5 filters it into a negative DC voltage that is almost the peak voltage of the output signal.
4) The rectified signal's DC voltage decreases or increases the conduction of the Jfet to set the output level.
Look at one of the thousands of explanations in Google and reply about any difference from what I said.
In addition, it should be clear that the Diode-C5 path - together with the FET - establish an automatic amplitude control.
That means: To avoid hard limiting (clipping) of the output signal (because the closed loop gain of the opamp with negative feedback must be somewhat larger than "3") the FET circuitry reacts upon rising amplitudes and provides a kind of soft limiting. As a result, the quality of the output signal (THD) remains fairly good.
In "the good old days" we used a little incandescent light bulb instead of the Jfet. When starting, the filament is cold then it has a low resistance causing the circuit's voltage gain to be high. When the output level became too high then the bulb glowed and since its filament is at a high temperature its resistance is increased enough to reduce the circuit's voltage gain. It took time for the bulb to react causing the output level to bounce up and down for a duration of one second if anything disturbed the circuit like changing its frequency with a switch or a dual pot.
To be a bit more precise: This method (usage of a light bulb) for stabilizing the amplitude of a WIEN-type oscillator (of course with a tube as active element) was first described already in 1939 (Stanford Univ., master degree thesis from W.R. Hewlett).
The first product produced by Hewlett Packard was a sinewave gernerator using a Wien bridge oscillator and a light bulb. The manufacturing was done in Hewlett's garage.
If you want an explanation, this is the best you can find over the wide internet. It is made by James K. Roberge who has written the book "Operational Amplifiers: Theory and Practice". I have just found out that MIT (Massachusetts Institute of Technology) has a lot of well known authors and video lectures from them ! What a great source !
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