the phase-shift oscillator
The phase-shift oscillator is another simple recipe for a sine wave, built from one amplifier and a ladder of three identical RC sections. The idea is delightfully direct: an inverting amplifier flips its signal by 180 degrees, and we just need a network that adds another 180 degrees so the signal comes back around in step. A single RC section can shift phase by up to (but never quite reaching) 90 degrees, so you cannot get 180 from one. Stack three sections, each contributing 60 degrees, and together they hit 180 at one particular frequency.
Walk it through. Each RC section is a small high-pass that advances the phase. At the frequency where the three together total exactly 180 degrees, the loop's total phase becomes 180 (amplifier) plus 180 (network) equals 360, which is zero net, satisfying Barkhausen. For three equal sections this happens at f equals 1 over (2 times pi times R times C times the square root of 6). At that same frequency the three-stage ladder attenuates the signal to one twenty-ninth of its input. So the amplifier must supply a gain of at least 29 to bring the loop gain up to 1.
It is a great teaching circuit because you can see the Barkhausen conditions being met section by section. In practice it is less popular than the Wien bridge for precision work: its frequency is harder to tune (you would have to vary three components together), and its output has more distortion because it sits closer to the edge of stable operation. It shines as a cheap fixed-frequency tone source and as the clearest hands-on illustration of how phase around a loop creates oscillation.
With each section using R equal to 10 kilohm and C equal to 10 nanofarad, the frequency is 1 over (2 times pi times 10000 times 10 nanofarad times the square root of 6), about 650 Hz. The op-amp is set to a gain of around 29 to overcome the network's one-twenty-ninth loss.
Three RC sections add up to 180 degrees; the inverting amp adds the other 180, closing the loop in phase.
You cannot get 180 degrees from one or two RC sections, because a single section approaches but never reaches 90 degrees; three is the practical minimum. The exact gain-29 figure assumes ideal, identical sections that do not load each other, which real components only approximate.