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

A crystal oscillator replaces the LC or RC tank with a vibrating quartz crystal to achieve far higher frequency stability. It works on the piezoelectric effect: when an AC voltage is applied across a suitably cut crystal it vibrates mechanically at that frequency, and conversely, mechanically vibrating the crystal generates an AC voltage across it. Quartz, Rochelle salt and Tourmaline are piezoelectric; quartz is the common choice because it is inexpensive, readily available and mechanically robust.

Electrically, a vibrating crystal behaves like a very high-$Q$ tuned circuit. Its AC equivalent circuit is a series branch of $R$, $L$ and $C_s$ in parallel with a mounting capacitance $C_m$, where $L$ represents the crystal's mass, $C_s$ its elasticity, $R$ its mechanical friction, and $C_m$ the capacitance of the two metal electrodes with the slab as dielectric. This gives the crystal two close resonant frequencies: a series resonant frequency

$f_s = \dfrac{1}{2\pi\sqrt{L C_s}}$ …