the Barkhausen criterion
/ BARK-how-zen /
The Barkhausen criterion is the two-part rule that tells you whether a feedback loop will sing a steady note or stay silent. Imagine pushing a child on a swing. To keep the swing going at a constant height, two things must be true: you must push at the right moment in the cycle (timing, or phase), and each push must replace exactly the energy lost to friction (strength, or gain). Push at the wrong moment and you fight the swing; push too softly and it dies; push too hard and it climbs higher each time. An oscillator is the same balancing act in electrical form.
Stated precisely, around the closed loop the signal must come back to its starting point unchanged. First condition: the loop gain (the amplifier's gain times the fraction fed back) must equal exactly 1 in magnitude. Second condition: the total phase shift all the way around the loop must be zero, or equivalently a whole multiple of 360 degrees, so the returning signal arrives perfectly in step to reinforce. Write the loop gain as a complex number A times B; Barkhausen says A times B equals 1 with zero net phase. The frequency at which the phase condition is met is the frequency the circuit oscillates at.
This single idea designs every sine-wave oscillator: build a frequency-selective network that gives zero phase shift at just one frequency, then set the amplifier gain to overcome that network's loss exactly. The honest subtlety: Barkhausen is a necessary condition, not a complete guarantee, and at exactly loop gain 1 the amplitude would be set by the initial noise, which is fragile. Real designs deliberately make the small-signal loop gain a little above 1 so oscillation starts reliably, then rely on amplitude stabilization to pull it back down to 1 once running.
In a phase-shift oscillator, an inverting amplifier already supplies 180 degrees. The RC network must add another 180 degrees at the oscillation frequency to make the total zero (mod 360). At that frequency the network attenuates the signal to one twenty-ninth, so the amplifier gain must be at least 29 to make the loop gain reach 1.
Both halves must hold at once: unity loop gain AND zero net phase, at the same frequency.
Barkhausen is necessary but not sufficient: it predicts the frequency and the gain needed, but real start-up requires loop gain a bit over 1, and a passing simulation can still fail to oscillate without a noise kick. It says nothing about amplitude, which a separate mechanism must set.