Quadrature oscillator

[bonzer]

Hello everyone!  Please help to understand a curiosity.

Here we have a famous two integrator quadrature oscillator.
It doesn't allow a full control of barkhausen condition because it is in open loop always about -180° (for any frequency).
T(jw) = 1/[(jwRC)^2]= -1/(wRC)^2

So it isn't that good because my RC makes my T(jw0) = -1 only in magnitude but phase is always -180.

I just wanted to let you know that - I know this so we don't talk about this again. So with real components you couldn't get precise -1 and you'd rather have instability or decreasing to zero. Therefore some components are used like optocouplers or two zenner diodes on the inverting amplifier etc to change the gain and get stable amplitude. (you can find solutions for that online)

The reason why I posted here this circuit is to ask you about another thing.
When I have my real integrator circuit my op amp introduces a little phase shift error epsilon (see attachment) because of the influence of that differential Vd voltage which is not null and is +90° compared to the output voltage because of that pole introduced by the op amp. I know the best is to have that Vd as small as possibile but anyway let's immagine I can't deal with that and it still has an effect.

In open loop I have 2epsilon and thefore when I connect the loop in the point I marked there in the attachment I could have some consequences because of that.
My question is what exactly happens in terms of distortion? Is it really dangerous? I think it could give you a frequency that is slightly different but I don't know.
Actually there exist some ways to avoid this error but anyway let me know what you think about this and how it changes the result.