What three conditions in a Barkhausen criterion must be met for oscillation to occur?

Answer 1

The Barkhausen stability criterion is just an obvious observation. It is a necessary but NOT sufficient criterion for oscillations.

The Barkhausen stability criterion states that an oscillator will oscillate when the total phase shift from input to output and back to input is an integral multiple of 360 degrees and the system gain is equal to 1.

The proper Barkhausen topology is defined as a loop of an amplitude determining inverting nonlinear amplifier (four terminal two-port ) and a linear passive frequency determining feed-back four terminal two-port circuit. It is obvious that the loop gain is 1 and the phase shift is a multiple of 360 degrees when the loop is closed. The Barkhausen criterion is necessary but not sufficient.

Question: Who was the first to open the loop and claim that the criterion is sufficient ?

Linear oscillators are mathematical fiction. Statements like "the non-linear characteristic will bring the poles to the imaginary axis" are of course nonsense. The bias-point of the amplifier will vary with time so the small-signal model of the circuit will vary with time i.e. the eigenvalues of the linearized Jacobian of the differential equations will move around with time in the complex frequency domain. The mechanism behind the steady state behavior is a kind of energy balance. Energy is received from the power source when the poles are in RHP. Energy is dissipated in the loss elements when the poles are in LHP. The frequency is a kind of time average of the imaginary part of the complex pole pair. The Barkhausen criteria is just the obvious observation that the gain around the closed loop of an amplifier and a feed-back circuit is 1 (one) and the phase-shift is 0 (zero or a multiple of 2pi). The Barkhausen criteria is NOT sufficient for start-up of oscillations. Phase-noise is an expression for the variation of the imaginary part of the complex pole-pair during the period of the oscillations.