Q.To obtain sustained oscillation in an oscillator,

An oscillator needs positive feedback — feeding part of the in-phase output back to the input — so the signal keeps reinforcing itself and sustains oscillations instead of dying out.

An oscillator is essentially an amplifier with a feedback network that returns part of the output signal back to the input. For sustained oscillations without any external input, the feedback signal must:

• be in phase with the original input signal (positive/regenerative feedback), so it adds to and reinforces the input rather than opposing it, and …

The Barkhausen criterion for sustained oscillations requires unity loop gain, zero (or 2π) net phase shift, and positive feedback -- all three conditions simultaneously.

Working

For an oscillator circuit to produce sustained (undamped, constant-amplitude) oscillations, the feedback signal fed back to the input must reinforce the original signal exactly, cycle after cycle. This requires the Barkhausen criterion:

1. The magnitude of the loop gain (open-loop amplifier gain $\times$ feedback fraction $\beta$) must equal exactly 1 -- if greater than 1, oscillations grow (until amplitude-limited by non-linearity); if less than 1, they die out.
2. The total phase shift around the feedback loop must be $0°$ or a multiple of $360°$ ($2\pi$) -- so the fed-back signal arrives exactly in phase with (adds constructively to) the original input. …

The Colpitts oscillator's tank circuit uses a capacitive voltage-divider (two series capacitors with a single inductor) to derive its feedback signal, which is why it is characterised as using capacitive feedback.

An LC oscillator sustains continuous oscillation by feeding a fraction of the tank circuit's output signal back to the amplifying device (transistor or op-amp) in the correct phase to satisfy the Barkhausen criterion (loop gain $\ge 1$, loop phase shift $=0°$ or a multiple of $360°$). Different LC-oscillator topologies differ mainly in how this feedback fraction is tapped from the tank circuit.

In a Colpitts oscillator, the tank circuit consists of a single inductor $L$ in parallel with two capacitors $C_1$ and $C_2$ connected in series with each other. The junction between $C_1$ and $C_2$ forms a capacitive voltage divider, and the feedback voltage fed back to the amplifier's input is tapped from this junction — hence 'capacitive feedback'. The oscillation frequency is set by $L$ and the series combination of $C_1,C_2$: $f=\dfrac{1}{2\pi\sqrt{LC_{eq}}}$ with $C_{eq}=\dfrac{C_1 C_2}{C_1+C_2}$.

…