Back to basics:

Hello to all,

I have made an observation and I wonder if a desription of this   effect can be found in the literature:

If the classical Miller integrator stage is fed with a sinusoidal amplitude which drives the output into saturation, the output voltage is - of course - a "clipped" wave. However, the phase shift is LESS than +90 deg. - depending on the degree of clipping. Perhaps it´s something which is similar to the effect of "wind-up" in control systems.
But this works only if the saturation effect takes place inside the integration opamp. It does not happen if signal clipping occurs after integration.

I think, I am able to explain the effect, however, I ask myself how to predict/calculate this effect and - more than that - why nobody has reported about that up to now.  Or am I wrong ?
Regards
LvW

Thank you, VIVEK, for your kind reply.

I know, that clipping is a non-linear effect - nevertheless, I think one can compare the clipped output signal with an ideal one and define a phase difference between both. Perhaps its more accurate not to speak about a phase shift but rather a time shift.

But pay attention: I have made a (nearly) catastrophic typing error regarding the sign of the additional phase shift.
It is not negative - that means the total phase is not LESS than +90 deg, rather it is positive and the total phase is e.g. +90.5 or 91 deg.
That means the time shift is larger than 1/4 of the period.

Some remarks regarding the importance of this observation:
The very well known two-integrator oscillator (without any additional amplitude regulation) is able to oscillate at a frequency which leads to a phase shift of each (real) inverting opamp integrator of less than +90 deg. That is a fact ! Does this mean the loop gain of the oscillator is less than 360 deg ? The only explanation is that this neg. phase error is compensated by the positive "error" caused by signal clipping.
I have checked this by simulation with the result that clipping gets larger for higher frequencies (larger neg. phase errors).

Question: Do you have any other explanation for the fact that stable oscillations of this circuit are possible without any additional circuitry ?  It´s interesting, all authors describing this oscillator (without an external regulation mechanism) suppress any comments regarding the amplitude and the quality of the output signal.
More than that - nobody even explains WHY the circuit oscillates at one frequency only ! Remember: The loop gain for frequencies below 1/RC also is real and above unity! Why does the circuit exhibit no oscillations at lower freqiencies and goes into saturation ?
I think, only the above mentioned observation gives the answer.

I am very interested to get your comments.
Regards
Lutz