Dear Lutz, I was interested in your paper and the DIO circuit... and maybe because it is related to my intuitive explanation of the op-amp integrator.

In particular, two moments impressed me. The first was your introduction where you show that two integrators - inverting and non-inverting, in a loop can oscillate. I felt the urge to imagine this in an intuitive way over time (not in the frequency domain) because I know that is the only way to achieve it.

I began to see some resemblance to the relaxation circuit of a ramp oscillator where an integrator and a non-inverting comparator with hysteresis are connected in a loop… as though the second integrator is replaced with the comparator.

In my opinion, this type of oscillator does not obey the Barkhausen criteria because here the term "gain" does not make sense. There are two integrators in a loop here and not amplifiers. As if this is more of a kind of relaxation generator...

In contrary - the working of this oscillator can be used as a proof of Barkhausen`s criterion because the circuit automatically adjusts itself to Barkhausens oscillation condition.

Explanation: For zero deg within the loop the gain is larger than unity. Hence, the circuit starts to oscillate - and as soon as the rising amplitudes reach the upper limit and tend to be clipped there will be an automatic phase correction which shifts the frIn contequency to a value with unity loop gain. Therefore, only a very small amount of amplitude clipping can be observed. This is an automatic amplitude limiting without any extra non-linear path (as required for all other oscillator circuits)

Lutz,

I try with my purely intuitive thinking in time domain to make sense of what you say...

According to my ideas, these are two circuits in a loop, each with local negative feedback (implemented by a capacitor). In this situation, how can we talk about a total loop gain greater than one? The integrators do not amplify the fast AC changes, so each of them has a zero gain and the total loop gain is zero too...

The second moment in your paper impressed me was your "Detailed explanation".

I completely agree with it and I would supplement it with another intuitive explanation.

The active circuit of the op-amp inverting amplifier can be considered as a simple RC integrating circuit to which a variable voltage source is added in series. It copies the voltage drop across the capacitor and adds it to the input voltage thus compensating it. In this way, two voltage sources are connected in series - main (input) and supplemental (compensating).

However, when the op-amp output voltage reaches the supply voltage, the supplemental voltage stops changing. The magic of this "voltage compensation" disappears and the circuit becomes passive again.

Only to clarify that by "intuitive explanation in (over, through) time" I mean exactly the way you have explained what happens when the supply voltage is reached (the section Detailed explanation).

Except for the introduction, the other things are not interesting to me... not to mention that they are even unpleasant to me…

I understand and explain circuit operation by following it, step by step, over time…

In this regard, I must admit that in fact I have not yet been able to explain in this way how the oscillations arise in all these RC active circuits. Barkhausen criterion does not explain anything, it only gives the conditions.

The main question here is what causes the direction of charge of the capacitor to be reversed. I just have some guesses that I shared in the discussions about the Wien bridge oscillator… but I am not sure if they are correct...

Here is my explanation...

These are active RC circuits with positive feedback; so while the op-amp is in linear mode (there is some loop gain), they are not stable and try to reach equilibrium.

They reach a kind of temporary equilibrium when the output voltage approaches the supply (positive or negative) voltage. Then the gain drops to almost zero and the "self reinforcing positive feedback" mechanism ceases to operate. So the circuit finds peace through a significant drop in gain...

But this is for a short time and they "go" again but in the opposite direction. What makes them do it?

As far as I remember, my explanation is that some, albeit very small noise makes the op-amp slightly out of saturation and go in the opposite direction (in the previous one it can't because it has "touched" in the supply rail).

The following conclusion comes to my mind:

While the output voltage is "moving", the circuit is in a stable state but if it "stops" next to the rail, the circuit is in an unstable state and immediately "goes" in the opposite direction...

As you would say, I have never seen such an explanation before... and that was the explanation I needed...

Quote: In this situation, how can we talk about a total loop gain greater than one? The integrators do not amplify the fast AC changes, so each of them has a zero gain.

Each active Integrator block has a gain above unity in the frequency region below the cross-over frequency (0 dB) at wo=1/RC.