Dear all,

I have a conceptual question about offset in RC integrators.

By an RC integrator I mean a classic opamp with its positive input grounded, R attached to its negative input and C positioned between its negative input and output.

Assume you can apply a sine wave, ideally centered around the ground/reference level of the integrator: no problem here, the input gets integrated and becomes nicely a cosine.

Now, if you would apply a signal which has a different DC level than the ground/reference level of the integrator, the output would diverge (and get stuck on the supply voltage in a real implementation). This is also nothing abnormal: an integrator is supposed to have a infinite gain at DC.

My question is now: how can any RC integrator every be made to work (e.g. for sine wave type of signals) taking into account the finite offset of the opamp?
The offset of the opamp will always cause a finite shift in DC voltage between the input and the ground/reference level of the RC integrator, hence its output will diverge (towards the supply voltage).

What am I missing here?

Hi Visjnoe,

I think, you correctly have described the situation.
However, it's even worse:
Since the "integrator" has
(a) a finite dc gain (even without R in parallel to C), and  
(b) the opamp gain decreases with rising frequencies (and even has a second pole),
the phase shift of the "integrator" reaches the ideal value of 90 deg. for ONE frequency wo only! Below wo the phase shift will be less and above wo larger than 90 deg.
That means - strictly spoken, a real integration is possible only for one single frequency wo.
By the way, because of some other reasons, the situation is about the same if you use an OTA instead of a voltage opamp.

In practical circuits an integrator is never designed
to run forever.

Never!

However, the circuit you describe is often used
(with a few switches) to integrate over a defined
period of time and use the result,
then be reset to zero, and then integrate again.

If you want to do things like create a cosine from a sine wave you are better off to use a 90 degree phase shift method of some form.

There are ways to get rid of the DC offset, but free running integrators are not generally used in function circuits.