atoms, molecules, or ions are packed in a regularly ordered

February 25, 2023

Operation[edit]

A crystal is a solid in which the constituent atoms, molecules, or ions are packed in a regularly ordered, repeating pattern extending in all three spatial dimensions.

Almost any object made of an elastic material could be used like a crystal, with appropriate transducers, since all objects have natural resonant frequencies of vibration. For example, steel is very elastic and has a high speed of sound. It was often used in mechanical filters before quartz. The resonant frequency depends on size, shape, elasticity, and the speed of sound in the material. High-frequency crystals are typically cut in the shape of a simple rectangle or circular disk. Low-frequency crystals, such as those used in digital watches, are typically cut in the shape of a tuning fork. For applications not needing very precise timing, a low-cost ceramic resonator is often used in place of a quartz crystal.

When a crystal of quartz is properly cut and mounted, it can be made to distort in an electric field by applying a voltage to an electrode near or on the crystal. This property is known as inverse piezoelectricity. When the field is removed, the quartz generates an electric field as it returns to its previous shape, and this can generate a voltage. The result is that a quartz crystal behaves like an RLC circuit, composed of an inductor, capacitor and resistor, with a precise resonant frequency.

Quartz has the further advantage that its elastic constants and its size change in such a way that the frequency dependence on temperature can be very low. The specific characteristics depend on the mode of vibration and the angle at which the quartz is cut (relative to its crystallographic axes).^[13] Therefore, the resonant frequency of the plate, which depends on its size, does not change much. This means that a quartz clock, filter or oscillator remains accurate. For critical applications the quartz oscillator is mounted in a temperature-controlled container, called a crystal oven, and can also be mounted on shock absorbers to prevent perturbation by external mechanical vibrations.

Modeling[edit]

Electrical model[edit]

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

Schematic symbol and equivalent circuit for a quartz crystal in an oscillator

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.,

{\begin{aligned}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}}{2C_{0}}}\right)\quad \left(C_{0}\gg C_{1}\right)\end{aligned}}

where $s$ is the complex frequency ( $s=j\omega$ ), $\omega _{\mathrm {s} }$ is the series resonant angular frequency, and $\omega _{\mathrm {p} }$ is the parallel resonant angular frequency.

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 pF load has its specified parallel resonant frequency when a 6.0 pF capacitor is placed across it. Without the load capacitance, the resonant frequency is higher.

--
You received this message because you are subscribed to the Google Groups "2top-manitotasy-1" group.
To unsubscribe from this group and stop receiving emails from it, send an email to 2top-manitotasy-1+unsubscribe@googlegroups.com.
To view this discussion on the web visit https://groups.google.com/d/msgid/2top-manitotasy-1/CALML-R2eoQwL8rOje0f7iXfC%3DA3Maec6Gb2M5TjyAyoo6jhXJg%40mail.gmail.com.

Search This Blog

Alexander City