Q.Draw the circuit diagram of Colpitts oscillator and explain its action. Write the expression for its frequency and feedback factor.

The Colpitts oscillator is an LC oscillator whose tank is a single inductor L with two series capacitors $C_1$ and $C_2$, taking capacitive feedback across $C_2$.

[!ANSWER]

It oscillates at $f = \dfrac{1}{2\pi\sqrt{LC_{eq}}}$ with $C_{eq} = \dfrac{C_1C_2}{C_1+C_2}$, and its feedback factor is $\beta = \dfrac{C_1}{C_2}$.

[!TLDR]

The Colpitts oscillator's tank is an inductor L with a series capacitive divider $C_1$–$C_2$; the transistor and tank each add $180^\circ$; it oscillates at $f = \dfrac{1}{2\pi\sqrt{LC_{eq}}}$, $C_{eq}=\dfrac{C_1C_2}{C_1+C_2}$, with $\beta = \dfrac{C_1}{C_2}$.

Circuit: Figure 6.2.4 shows the Colpitts oscillator. $R_1$, $R_2$ and $R_E$ provide the DC bias to the transistor, $C_E$ is the bypass capacitor and $C_C$ the coupling capacitor, and RFC (radio-frequency choke) achieves isolation between AC and DC operation. The tank circuit consists of an inductor L with two capacitors $C_1$ and $C_2$ connected in series across it, their common terminal being grounded.

Action: When the DC power supply is switched on, the collector current starts rising and charges the capacitors $C_1$ and $C_2$. Since their common terminal is grounded, the potentials across the two capacitors are always in opposite phase. The voltage across $C_2$ is fed back to the amplifier input. When the capacitors are fully charged they discharge through the inductor L, and the interchange of energy in the tank sustains the oscillations. The tank circuit produces a $180^\circ$ phase shift and the amplifier produces another $180^\circ$; thus the total phase shift of $360^\circ$ is produced, which is essential for oscillation.

Feedback factor: The voltage feedback fraction is

The values of $C_1$ and $C_2$ are chosen so that $A\beta = 1$, satisfying the Barkhausen criterion.

Frequency: Since the two tank capacitors are in series, the frequency of oscillation is

[!ANSWER]

Frequency $f = \dfrac{1}{2\pi\sqrt{LC_{eq}}}$ with $C_{eq} = \dfrac{C_1C_2}{C_1+C_2}$; feedback factor $\beta = \dfrac{C_1}{C_2}$.