Q.What should be the value of loop gain for sustained oscillations?

An oscillator is built from an amplifier plus a feedback network arranged so that a fraction of the output is returned to the input in phase — this is positive (regenerative) feedback. Because the returned signal aids the input, the net input rises, and this is exactly the mechanism that lets a circuit sustain an output with no external signal.

For an amplifier of open-loop gain $A$ with a feedback fraction $\beta$, the overall (closed-loop) voltage gain with positive feedback is

The product $A\beta$ is the loop gain. When $A\beta = 1$ the denominator becomes zero and $A_f \to \infty$: the circuit delivers an output with no input, i.e. it has become an oscillator.

The first signal comes from noise — the random thermal motion of electrons in the circuit. At switch-on the loop gain is deliberately made $A\beta > 1$ so this tiny noise builds up; as the amplitude grows, amplifier non-linearity reduces the effective gain until $A\beta$ settles at exactly 1, giving a constant-amplitude output. …