1.3 Electrical Model

The equivalent electric circuit of a crystal is shown in Figure 1-4. The series RLC network is called the motional arm and gives an electrical description of the mechanical behavior of the crystal, where C₁ represents the elasticity of the quartz, L₁ represents the vibrating mass, and R₁ represents losses due to damping. C₀ is called the shunt or static capacitance and is the sum of the electrical parasitic capacitance due to the crystal housing and electrodes. If a capacitance meter is used to measure the crystal capacitance, only C₀ will be measured (C₁ will have no effect).

By using the Laplace transform, two resonant frequencies can be found in this network. The series resonant frequency, f_s, depends only on C₁ and L₁. The parallel or anti-resonant frequency, f_p, also includes C₀. See Figure 1-5 for the reactance vs. frequency characteristics.

$f_s=\frac{1}{2\pi \sqrt{L_1{C}_1}}$

$f_s=\frac{1}{2\pi \sqrt{L_1{C}_1}}\sqrt{1+\frac{C_1}{C_0}}$

Crystals below 30 MHz can operate at any frequency between the series and parallel resonant frequencies, which means that they are inductive in operation. High-frequency crystals above 30 MHz are usually operated at the series resonant frequency or overtone frequencies, which occur at multiples of the fundamental frequency. Adding a capacitive load, CL, to the crystal will cause a shift in frequency given by Equation 1-3. The crystal frequency can be tuned by varying the load capacitance, and this is called frequency pulling.

$\mathrm{\Delta }f=f_s\left(\frac{C_1}{2(C_0+C_L)}\right)$