Oscillation criteria for a phase shift oscillator?

From:
http://en.wikipedia.org/wiki/Phase_shift_oscillator

According to Wikipedia, there is a long equation, relating all the component values in the system, which must be satisfied before the phase-shift oscillator will oscillate (I suppose it's a different equation for each variety of phase shift oscillator). I don't understand why this is. It's a filter with a phase shift of 180° at some frequency f, with a feedback amplifier. I thought it was guaranteed to oscillate as long as the total gain of the filter and amplifier was above unity at f.

So, apparently I misunderstand something something. Can someone please tell me what the discrepancy was in my knowledge?

I found that article after spending maybe 60-90 minutes coming up with a differential equation to describe a different phase shift oscillator, which I designed in my head. It turned out not to oscillate because the phase shift was only 180° at an infinite frequency.

It's not the gain of the amplifier that has to be more than 1, it is the overall gain, including the feedback network.

In other words the feedback network acts as an attenuator and the amplifier has to make this up at 180° phase shift.

Wiki is correct in that the attenuation of the 3 stage phase shift network is 29 so the amplifier gain needs to exceed this.

This is stated in a very short equation near the beginning of the article.

The long equation you refer to is for the most general case where each of the three stages has different component values.

I will leave it as an exercise to reduce this to the much simpler version for the more normal case where all are the same.

Incidentally different feedback networks introduce different attenuations so for instance the Wein bridge has an attenuation of 3.

And variable gain amps fix a multitude of problems.

As far as I understand, each RC network shift the output 60 degrees to the left when compared with input.

You have three RC networks, so the output of the first RC network is 60 degrees to the left of the input, output of the second RC network is 120 degrees to the left of the input, output of the third RC network is 180 degrees tot he left of the input.

The output of the third RC network is then fed to the inverting input of the amplifier. This shift the output of the third RC network 180 degrees to the right and now the input to the first RC network (initial input into the circuit) and output of the amplifier (the overall output of the circuit) is in phase.

At least that is how I understand it. If I am off, let me know.

The problem is most designs have a variable shift per segment, it isn't a smooth 60°. The important part is all the shifts added together is 180°, this is the important part.