Quartz Crystal Resonator
A quartz crystal behaves as a very high-Q resonator modelled by a motional branch (L₁, C₁, R₁) in parallel with a shunt capacitance C₀. This tool computes the series and parallel resonant frequencies that bracket the crystal's usable range, the motional Q, the capacitance ratio, and how far a load capacitor pulls the frequency — the numbers behind oscillator design and trimming.
Equations & Parameters ▸
\(f_s=\dfrac{1}{2\pi\sqrt{L_1 C_1}} \qquad f_p=f_s\sqrt{1+\dfrac{C_1}{C_0}} \qquad Q=\dfrac{2\pi f_s L_1}{R_1}\)
\(r=\dfrac{C_0}{C_1} \qquad \dfrac{\Delta f}{f}\approx\dfrac{C_1}{2\,(C_0+C_L)}\ \text{(pull to load }C_L)\)
\(r=\dfrac{C_0}{C_1} \qquad \dfrac{\Delta f}{f}\approx\dfrac{C_1}{2\,(C_0+C_L)}\ \text{(pull to load }C_L)\)
| L1 | Motional inductance (H) — large for a crystal (henries). |
| C1 | Motional capacitance (fF), very small. |
| R1 | Motional (series) resistance / ESR (Ω), sets the Q. |
| C0 | Shunt (holder + electrode) capacitance (pF). |
| CL | Load capacitance (pF), optional — gives the pulled frequency. |
Reference: Butterworth–Van Dyke crystal model; W. L. Smith & standard quartz-crystal application notes.
Inputs
H
HenriesfF
FemtofaradsΩ
For QpF
Holder cappF
For pullingResults
Resonance
Series resonance fs—
Parallel resonance fp—
fp − fs spacing—
Figures
Quality factor Q—
Capacitance ratio C0/C1—
Pulled frequency (at CL)—
Diagram