# Ideal OpAmp differentiator stability

So, for OpAmp stability, we don't want $$\A_{OL}\beta\$$ to be -1, which we can derive from the Bode plot by looking at: $$\20\log(A_{OL})-20\log(1/\beta)\$$ and looking at the rate of closure to indicate the phase shift I have at that point.

I understand, that $$\1/\beta\$$ is the noise gain, which is something different from the signal gain, well at least in the case of the inverting configuration. The default differentiator is an inverting configuration with the input resistor replaced by a capacitor.

It follows that $$\\beta = \frac{X_C}{X_C+R}\$$, hence my $$\1/\beta(s) = 1 + sRC\$$ if I inspect this in my $$\s\$$-domain.

This gives me a zero with a frequency of $$\f = \frac{1}{2\pi RC}\$$ and hence my aplitude Bode plot should be a straight line and then rise with 20 dB/dec - this will intersect my $$\A_{OL}\$$ which has a -20 dB/dec rolloff by the dominant pole (assuming a single pole OpAmp) and give me an unstable ROC as I have a 40 dB/dec ROC leading to a zero phase margin.

However, when I look at some application notes (e.g. TI's Handbook of Operation Amplifier Application, see screenhot) I typically see that they simply argue that the overall transfer function is a straight line with a +20 dB/dec. This is obviously correct, as: $$\ \frac{u_{out}(s)}{u_{in}(s)}=-\frac{R}{X_C} = -sRC\$$. And then they derive from that an unstable ROC. However, for any ROC analysis, don't I need to look at the noise gain, not my signal gain. Is this right?