# Phase shift oscillator: confused about phase shift per stage?

I am just learning about the RC phase shift oscillator with 3 RC stages and an opamp. From what I've learned - each stage will introduce a phase shift of $$\60^\circ\$$ so the total phase shift will be $$\180^\circ\$$ which added to the $$\180^\circ\$$ introduced by the inverting opamp should make the feedback signal in phase with the output.

Now from the derivation of the transfer function, the value of $$\\dfrac{1}{\omega RC}\$$ comes out to be $$\\sqrt{6}\$$ to produce a $$\\beta = \dfrac{1}{29}\$$ and gain $$\A = -29\$$ (barkhausen condition). But won't this mean that the phase shift per stage is $$\\tan^{-1}\left( \dfrac{1}{\omega RC} \right)\ = \tan^{-1}(\sqrt{6}) = 67.8^\circ\$$ which is not $$\60^\circ\$$ ? Am I incorrect in my use of the phase shift formula ? Any help is appreciated!

• Look back at your amplifier -- at the frequency such that the shift per stage is 60 degrees, what is the gain all the way around the loop? As long as the overall phase is correct, the loop gain can be greater than one; the phase shift that you get equating the gain to one (assuming this is what you did) isn't as important. Apr 24, 2021 at 18:44
• – jonk
Apr 24, 2021 at 18:50

You're thinking of a phase shift amplifier with three cascaded RC filters, with all the R's and C's the same value. The issue is that the stages load one another -- this is why the gain is $$\\frac{1}{29}\$$ instead of 0.65 or so, and why the shift seems wrong.