Basic Physical Principles
Materials that are used for crystal oscillator applications exhibit the piezoelectric effect. This phenomenon produces an electric field within a material when a mechanical force is applied.

Conversely, if an electrical drive signal is applied to the crystal, a mechanical vibration results. If the driving signal is periodic in nature, then the resulting vibration will also be periodic. Very accurate electric signal frequencies can be produced by certain materialswith the appropriate shape, electrode geometry, and ambient conditions.

For this reason, crystal resonators are used in applications such as television, cellular radio communications, and electronic test equipment where accurate synthesis of signal frequencies is required throughout.

The relationship between the electric field, electric displacement, electric polarization, and mechanical stress and strain is given by what are called constituitive relations. The electrical quantities are related in the following manner:

D = e0 E + P

where D is the electrical displacement, E is the electric field and P is the electric polarization. The term e 0 is the free-space permittivity. The polarization depends on the applied electric field. In a piezoelectric material an electric polarization can also result from an applied stress or strain. 

This is called the direct piezoelectric effect. Therefore the polarization is proportional to the stress (T) or strain (S) that is P = dT or P = eS

In the converse piezoelectric effect, the stress or strain forces are generated by the electric field applied to the crystal. Thus, we have S = dE or T = eE

The constants of proportionality d and e are called piezoelectric stress and strain coefficients. The stress and strain forces are represented by matrix quantities, and the coefficients are tensor quantities. A tensor mathematically represents the fact that the polarization can depend on the stress or strain in more than one direction.

This is also true for the relationship between the stress or strain and the electric field. Many other physical properties in crystals also exhibit this nature, which is called anisotropy. Thus when a property is anisotropic, its value depends on the direction of orientation in the crystal.

For the direct piezoelectric effect, the total polarization effect is the sum of these two contributions, an applied electric field and applied mechanical force.

Based on the relationship between the electric displacement and the electric polarization it is then possible to write equations that relate the displacement D to the applied stress or strain. Electric displacement is the quantity that is preferred in experiment and engineering.

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