How it works
A quartz crystal exploits the piezoelectric effect: mechanical deformation produces an electric charge, and an applied electric field produces mechanical deformation. The equivalent circuit has a series RLC branch (L_m, C_m, R_m) representing the mechanical resonance, all in parallel with a packaging capacitance C_0. This gives two resonance frequencies: series resonance f_s = 1/(2π√(L_m·C_m)) where impedance is minimum (≈R_m, a few ohms), and parallel resonance f_p = f_s√(1 + C_m/C_0) a few kHz above f_s where impedance is maximum. Oscillator circuits operate either at f_s (Pierce oscillator, common in microcontrollers) or between f_s and f_p (parallel resonant mode).
Key points to remember
Crystal Q factor is typically 10,000–100,000 — compared to 50–200 for LC oscillators — which is why crystals provide 10–100× better frequency stability. Frequency stability of a standard AT-cut crystal is ±20–50 ppm over 0–70°C; a TCXO achieves ±0.5 ppm; an OCXO achieves ±0.01 ppm. The AT-cut crystal has near-zero temperature coefficient at room temperature due to the specific 35°10' cut angle relative to the crystal axis. A crystal can be pulled (shifted in frequency) by a small amount — typically ±100 ppm — by adding a small variable capacitor in series or parallel, forming a VCXO (Voltage-Controlled Crystal Oscillator) used in PLLs.
Exam tip
The examiner always asks you to draw the electrical equivalent circuit of a crystal and identify the two resonant frequencies — label L_m, C_m, R_m, and C_0, write the formulas for f_s and f_p, and explain why the impedance is minimum at f_s and maximum at f_p.