Q.With a circuit diagram, explain the working of crystal oscillator. Write the expression for its frequency of oscillation.

A crystal oscillator uses a piezoelectric quartz crystal as the frequency-controlling element in the feedback path of a transistor amplifier, giving very high frequency stability. …

[!TLDR]

A quartz crystal (working on the piezoelectric effect) replaces the tank in a transistor oscillator; excited at series resonance it fixes the frequency at $f_s = \dfrac{1}{2\pi\sqrt{LC_s}}$ with very high stability.

Principle: A crystal oscillator works on the piezoelectric effect — when an AC voltage is applied across a suitably cut crystal (usually quartz) it vibrates at the frequency of the applied voltage, and conversely a mechanical vibration of the crystal generates an AC voltage across it. Electrically the crystal behaves as a very high-Q tuned circuit: a series R–L–$C_s$ branch in parallel with the mounting capacitance $C_m$.

Circuit: Figure 6.4.4 shows the BJT crystal oscillator. $R_1$, $R_2$ and $R_E$ provide the DC bias, $C_E$ is the bypass capacitor and $C_C$ the coupling capacitor, and RFC is the radio-frequency choke that isolates the DC supply from the collector oscillations. A capacitive voltage divider $C_1$–$C_2$ is connected at the collector, and the crystal is connected as a series element in the feedback path from the collector to the base. …