text: A typical rule of thumb is to make the small signal loop gain at the oscillation frequency 2 or 3.  When the power is turned on, oscillation is started by the power turn-on transient or random electronic noise present in the circuit.  Noise guarantees that the circuit will not remain "balanced" precisely at its unstable DC equilibrium point (Q point) indefinitely. Due to the narrow passband of the filter, the response of the circuit to a noise pulse will be sinusoidal, it will excite a small sine wave of voltage in the loop. Since for small signals the loop gain is greater than one, the amplitude of the sine wave increases exponentially.
  During startup, while the amplitude of the oscillation is small, the circuit is approximately linear, so the analysis used in the Barkhausen criterion is applicable.  When the amplitude becomes large enough that the amplifier becomes nonlinear, technically the frequency domain analysis used in normal amplifier circuits is no longer applicable, so the "gain" of the circuit is undefined.  However the filter attenuates the harmonic components produced by the nonlinearity of the amplifier, so the fundamental frequency component  mainly determines the loop gain (this is the "harmonic balance" analysis technique for nonlinear circuits).
  The sine wave cannot grow indefinitely; in all real oscillators some nonlinear process in the circuit limits its amplitude, reducing the gain as the amplitude increases, resulting in stable operation at some constant amplitude.   In most oscillators this nonlinearity is simply the saturation  (limiting or clipping) of the amplifying device, the transistor, vacuum tube or op-amp.  The maximum voltage swing of the amplifier's output is limited by the DC voltage provided by its power supply.  Another possibility is that the output may be limited by the amplifier slew rate.