Editing Crystal oscillator (section)

==Modeling==

===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.

===Resonance modes===
{{anchor|Pull capacitor}}A quartz crystal provides both series and parallel resonance. The series resonance is a few kilohertz lower than the parallel one. Crystals below 30&nbsp;MHz are generally operated between series and parallel resonance, which means that the crystal appears as an [[inductive reactance]] in operation, this inductance forming a parallel resonant circuit with externally connected parallel capacitance. 
[[File:Xtal response.jpg|thumb|220px|right|Frequency response of a 100kHz crystal, showing series and parallel resonance]]
Any small additional capacitance in parallel with the crystal pulls the frequency lower. Moreover, the effective inductive reactance of the crystal can be reduced by adding a capacitor in series with the crystal.  This latter technique can provide a useful method of trimming the oscillatory frequency within a narrow range; in this case inserting a capacitor in series with the crystal raises the frequency of oscillation. For a crystal to operate at its specified frequency, the electronic circuit has to be exactly that specified by the crystal manufacturer.  Note that these points imply a subtlety concerning crystal oscillators in this frequency range: the crystal does not usually oscillate at precisely either of its resonant frequencies.

Crystals above 30&nbsp;MHz (up to >200&nbsp;MHz) are generally operated at series resonance where the impedance appears at its minimum and equal to the series resistance. For these crystals the series resistance is specified (<100&nbsp;Ω) instead of the parallel capacitance. To reach higher frequencies, a crystal can be made to vibrate at one of its [[overtone]] modes, which occur near multiples of the fundamental resonant frequency. Only odd numbered overtones are used. Such a crystal is referred to as a 3rd, 5th, or even 7th overtone crystal. To accomplish this, the oscillator circuit usually includes additional [[LC circuit]]s to select the desired overtone.


===Temperature effects===
A crystal's frequency characteristic depends on the shape or "cut" of the crystal. A tuning-fork crystal is usually cut such that its frequency dependence on temperature is quadratic with the maximum around 25&nbsp;°C.{{Citation needed|date=November 2016}} This means that a tuning-fork crystal oscillator resonates close to its target frequency at room temperature, but slows when the temperature either increases or decreases from room temperature. A common parabolic coefficient for a 32&nbsp;kHz tuning-fork crystal is −0.04 ppm/°C<sup>2</sup>:{{Citation needed|date=November 2016}}

: <math>f = f_0\left[1 - 0.04~\text{ppm}/^\circ\text{C}^2 \cdot (T - T_0)^2\right].</math>

In a real application, this means that a clock built using a regular 32&nbsp;kHz tuning-fork crystal keeps good time at room temperature, but loses 2 minutes per year at 10&nbsp;°C above or below room temperature and loses 8 minutes per year at 20&nbsp;°C above or below room temperature due to the quartz crystal.