Q.With a circuit diagram, explain the working of a Hartley oscillator. Write the expression for its frequency and feedback factor.

The Hartley oscillator is an LC oscillator that uses a tapped inductor ($L_1, L_2$) with a capacitor C as its tank and takes inductive feedback across $L_2$.

[!ANSWER]

It oscillates at $f = \dfrac{1}{2\pi\sqrt{LC}}$ with $L = L_1 + L_2$, and its feedback factor is $\beta = \dfrac{L_2}{L_1}$.

[!TLDR]

When the supply is switched on the tank ($L_1,L_2,C$) sets up oscillations; the transistor and tank each add $180^\circ$ (total $360^\circ$); it oscillates at $f = \dfrac{1}{2\pi\sqrt{LC}}$, $L = L_1+L_2$, with feedback factor $\beta = \dfrac{L_2}{L_1}$.

Circuit: Figure 6.2.3 shows the Hartley oscillator. $R_1$, $R_2$ and $R_E$ provide the necessary DC bias to the transistor, $C_E$ is the emitter bypass capacitor and $C_C$ the coupling capacitor. RFC is the radio-frequency choke, which acts as a short circuit for DC and an open circuit for AC, so it isolates the DC power supply from the collector oscillations. The two inductors $L_1$ and $L_2$ with the capacitor C form the tank circuit; the output is taken across $L_1$ and the feedback voltage across $L_2$.

Working: When the DC power supply is switched on, the collector current increases and charges the capacitor C. When the capacitor is fully charged it discharges through the inductors $L_1$ and $L_2$, setting up oscillations in the tank. The potentials developed across the two inductors are always in opposite phase. The voltage across $L_2$ is fed back into the amplifier input and the voltage across $L_1$ is taken as the output. The tank circuit provides a $180^\circ$ phase shift and the transistor amplifier provides another $180^\circ$, so the total phase shift around the loop is $360^\circ$, which is essential for sustained oscillation.

Feedback factor: The voltage feedback ratio is

The values of $L_1$ and $L_2$ are chosen so that $A\beta = 1$ (i.e. $A = L_1/L_2$), satisfying the Barkhausen criterion.

Frequency: The frequency of oscillation is decided by the tank elements,

(If the coils are magnetically coupled, $L = L_1 + L_2 \pm 2M$.)

[!ANSWER]

Frequency $f = \dfrac{1}{2\pi\sqrt{LC}}$ with $L = L_1 + L_2$; feedback factor $\beta = \dfrac{L_2}{L_1}$.