Quartz Crystal Resonators
A quartz crystal is a piezoelectric plate whose mechanical resonance is extraordinarily stable and high-Q — the reason it sets the frequency of nearly every clock, radio, and microprocessor. Electrically it behaves as a very sharp resonant circuit, and to the surrounding oscillator it looks like the lumped model below. Understanding that model is the key to choosing a crystal, hitting a target frequency, and trimming it with a load capacitor.
The Butterworth–Van Dyke Model
The crystal is modelled by a motional branch — the electrical image of the mechanical resonance — in parallel with the static shunt capacitance \(C_0\) of the electrodes and holder:
The motional inductance \(L_1\) is huge (henries) and the motional capacitance \(C_1\) tiny (femtofarads) — impossible with real coils and capacitors, which is exactly why the mechanical resonance is so much sharper than an LC tank.
Series and Parallel Resonance
The motional branch alone resonates at the series frequency \(f_s\). Adding \(C_0\) creates a second, slightly higher parallel (anti-resonant) frequency \(f_p\); the crystal is inductive only in the narrow window between them, where oscillators operate:
The fractional split \(f_p-f_s\) is only a few hundred ppm because \(C_1\ll C_0\). The capacitance ratio \(r=C_0/C_1\) (typically 200–1000) sets how far the crystal can be pulled.
Quality Factor
The motional resistance \(R_1\) (the mechanical loss) sets a Q that dwarfs any LC circuit:
Fundamental-mode crystals reach Q of 10⁴–10⁶; precision SC-cut resonators exceed 10⁶. This enormous Q is what gives quartz oscillators their part-per-million (or better) stability.
Pulling with a Load Capacitor
A "parallel-resonant" crystal is specified with a nominal load capacitance \(C_L\); the oscillator's external capacitance shifts the operating frequency by:
Increasing \(C_L\) lowers the frequency toward \(f_s\); decreasing it raises the frequency toward \(f_p\). This is how a trimmer or varactor tunes a crystal oscillator (a VCXO), and why using the wrong load capacitance leaves a crystal off-frequency.
Typical Parameters
| Type | Frequency | C₀ | R₁ (ESR) | Q |
|---|---|---|---|---|
| Watch tuning fork | 32.768 kHz | 1–2 pF | 30–50 kΩ | 10⁴–10⁵ |
| AT-cut fundamental | 1–30 MHz | 3–7 pF | 10–100 Ω | 10⁴–10⁵ |
| AT-cut overtone | 30–200 MHz | 3–6 pF | 20–80 Ω | 10⁴–10⁵ |
| SC-cut (OCXO) | 5–10 MHz | 2–5 pF | 50–150 Ω | >10⁶ |
Beyond about 30–40 MHz, fundamental plates become too thin, so crystals run on the 3rd, 5th, or 7th overtone. Frequency also drifts slightly with temperature (set by the crystal cut) and ages over time — the reasons TCXOs and OCXOs add temperature compensation or an oven.