Q.Write the expression for frequency of oscillations of a Colpitts oscillator.

The Colpitts oscillator is an LC oscillator that takes its feedback from a capacitive voltage divider — hence it uses capacitive feedback. Its tank circuit is a single inductor $L$ in parallel with two capacitors $C_1$ and $C_2$ connected in series, their common junction being grounded. The feedback voltage is developed across $C_2$.

The amplifier and biasing are the same as in other transistor LC oscillators: $R_1$, $R_2$ and $R_E$ set the DC bias, $C_E$ is the bypass capacitor, $C_C$ the coupling capacitor, and the RFC (radio-frequency choke) isolates the AC oscillations from the DC supply.

Working: at switch-on the rising collector current charges $C_1$ and $C_2$. Because their common terminal is earthed, the voltages across the two capacitors are in opposite phase; the voltage across $C_2$ is fed back to the input. When the capacitors are charged they discharge through $L$, sustaining the oscillation. The tank provides $180^\circ$ and the amplifier $180^\circ$, giving the required $360^\circ$. …