Barkhausen criteria and oscillation principle

Barkhausen criteria: An amplifier with feedback produces sustained sinusoidal oscillation at the frequency for which the loop gain Aβ satisfies both:

1. Magnitude condition: |Aβ| = 1 (loop gain is exactly unity)
2. Phase condition: the total phase shift around the loop is 0° or 360° (positive feedback), that is, ∠Aβ = 0°, 360°, ...

Here A is the amplifier gain and β is the feedback factor.

Principle of oscillation

An oscillator converts dc power into ac of a fixed frequency without any input signal. It consists of an amplifier, a positive feedback network, and a frequency-determining network (LC, RC or crystal).

         +---------+
 Vs=0 -->|    A    |----+----> Vo
         +---------+    |
              ^         |
              |   +-----+
              +---|  β  |<-+
                  +-----+

1. At switch-on, the noise and the turn-on transient present in the circuit have components of many frequencies.
2. The amplifier amplifies these. The feedback network passes back a fraction β in phase (0° or 360° for only one frequency fo). So only that component is reinforced.
3. At first, the loop gain is slightly above 1 (|Aβ| > 1), so the amplitude grows exponentially.
4. As the amplitude increases, the amplifier saturates or a limiter reduces the gain, until |Aβ| = 1. The amplitude then becomes constant: a sustained oscillation of frequency fo.

With closed-loop gain Af = A/(1 − Aβ): when Aβ = 1 the denominator is zero, so the gain is infinite, which means an output exists with no input. If |Aβ| < 1 the oscillation dies out; if |Aβ| > 1 (and not limited) the output is distorted.