Q.A Hartley oscillator circuit is to generate a frequency of $24\,\text{kHz}$. The inductors used have inductances $L_1 = 1\,\text{mH}$ and $L_2 = 2\,\text{mH}$. Find the value of capacitance of the capacitor used.

The Hartley oscillator is an LC oscillator that takes its feedback from a tapped (two-section) inductor — hence it is said to use inductive feedback. Its tank circuit is a single capacitor $C$ in parallel with two inductors $L_1$ and $L_2$ connected in series, the junction between them being the tapping point. The output is developed across $L_1$ and the feedback voltage across $L_2$.

A common-emitter transistor amplifier supplies the amplification. Resistors $R_1$, $R_2$ and $R_E$ set the DC bias, $C_E$ is the emitter bypass capacitor, $C_C$ is the coupling capacitor, and a radio-frequency choke (RFC) — a short circuit for DC but an open circuit for AC — isolates the DC supply from the collector oscillations.

Working: at switch-on the collector current charges $C$, which then discharges through $L_1$ and $L_2$, setting up tank oscillations. The voltages across the two inductor sections are in opposite phase, so the tank contributes $180^\circ$ and the transistor another $180^\circ$, giving the $360^\circ$ loop shift needed for oscillation.

The frequency of oscillation is

(if the coils are magnetically coupled, $L = L_1 + L_2 \pm 2M$, with $+2M$ for series-aiding and $-2M$ for series-opposition windings). The feedback factor is $\beta = \dfrac{L_2}{L_1}$, and to satisfy $A\beta = 1$ the required amplifier gain is $A = \dfrac{L_1}{L_2}$. Hartley oscillators are widely used as local oscillators in radio receivers and as RF signal generators.