Editing Crystal oscillator (section)

===Electrical model===
A quartz crystal can be modeled as an electrical network with low-[[Electrical impedance|impedance]] (series) and high-[[Electrical impedance|impedance]] (parallel) resonance points spaced closely together. Mathematically, using the [[Laplace transform]], the impedance of this network can be written as:

[[File:Crystal oscillator.svg|thumb|220px|right|Schematic symbol and equivalent circuit for a quartz crystal in an oscillator]]

: <math>Z(s) = \left( {\frac{1}{s\cdot C_1}+s\cdot L_1+R_1} \right) \left\| \left( {\frac{1}{s\cdot C_0}} \right) \right. ,</math>

or

: <math>\begin{align}
                           Z(s) &= \frac{s^2 + s\frac{R_1}{L_1} + {\omega_\mathrm{s}}^2}{\left(s \cdot C_0\right)\left[s^2 + s\frac{R_1}{L_1} + {\omega_\mathrm{p}}^2\right]} \\[2pt]
  \Rightarrow \omega_\mathrm{s} &= \frac{1}{\sqrt{L_1 \cdot C_1}}, \quad \omega_\mathrm{p}
                                 = \sqrt{\frac{C_1 + C_0}{L_1 \cdot C_1 \cdot C_0}}
                                 = \omega_\mathrm{s} \sqrt{1 + \frac{C_1}{C_0}} \approx \omega_\mathrm{s} \left(1 + \frac{C_1}{2 C_0}\right) \quad \left(C_0 \gg C_1\right)
\end{align}</math>

where <math>s</math> is the complex frequency (<math>s=j\omega</math>), <math>\omega_\mathrm{s}</math> is the series resonant [[angular frequency]], and <math>\omega_\mathrm{p}</math> is the parallel resonant angular frequency.

{{anchor|Load capacitance|Load capacitor|Padding capacitor}}Adding [[capacitance]] across a crystal causes the (parallel) resonant frequency to decrease. Adding [[inductance]] across a crystal causes the (parallel) resonant frequency to increase. These effects can be used to adjust the frequency at which a crystal oscillates. Crystal manufacturers normally cut and trim their crystals to have a specified resonant frequency with a known "load" capacitance added to the crystal. For example, a crystal intended for a 6&nbsp;pF load has its specified parallel resonant frequency when a 6.0&nbsp;pF capacitor is placed across it.  Without the load capacitance, the resonant frequency is higher.