Phase shift oscillator: confused about phase shift per stage?

Question

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!

Answer 1

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.

If you were to put a unity-gain buffer between each stage, then your math would work out -- but you'd be using up op-amps like mad.

Answer 2

Yes - you are incorrect in using the mentioned formula. This formula does apply for one single RC stage only. However, in your circuit there are three RC stages which are coupled together. Hence, it is not allowed to split up the total phase shift into three equal parts. You must treat the thre stages as one single system.