CATALOG

EQUATIONS

Electromechanical Relations

The electromechanical characteristics of a piezoelectrical ceramic element can he represented in the simplest form by the equivalent circuit:

The series branch L, C, and R represents the converted mechanical properties - effective mass, compliance and mechanical loss. Co is the clamped electrical capacitance. This basic circuit is applicable at frequencies only near the first fundamental resonance, well-removed from any other resonant modes.

If the electrical impedance of an appropriately shaped piezoelectric ceramic element is measured as a function of frequency, a characteristic plot is obtained.

From this impedance information and a knowledge of the dimensions, mass, capacitance and dissipation factor, nearly all of the material properties can be determined.

Channel Industries uses a modern automated system to measure and evaluate the piezoelectric resonator. This system is comprised of a Hewlett-Packard computer and impedance analyzer with provision for printing out tabular data and impedance plots.

At frequencies well below the lowest frequency fundamental resonance of a ceramic element, the piezoelectric, dielectric, and elastic properties are interrelated.

For a definitive treatment of the piezoelectric relations, refer to "IRE Standards on Piezoelectric Crystals, 61 IRE 14.S1."

Static and Low Frequencies

The free displacement of a piezoelectric element at static or low frequencies is illustrated by the following linear equations:

Parallel mode   Dt = d₃₃ V

Transverse mode Dl  = d₃₁ (l/ t) · V

At high electric fields, the strain/field relationship (effective d) becomes non-linear in a positive direction. This gives larger free displacernents (100% in some materials) than the displacement calculated from low field d coefficients.  Non-linearity leads to hysteresis, mechanical creep, and high power dissipation.

It is generally recommended that static fields be applied only in the direction of the original poling field (+E). A negative field (-E) may be applied, but the magnitude must be controlled, or partial to full depoling will result.  Similarly, a low frequency dynamic field can cause depolarization due to the negative half cycles.

It is not possible to provide absolute values on permissible E, but Table V below lists some estimated maximum guidelines for low frequencies.

At the full free displacement, the developed force goes to zero. To determine the actual displacement when working against a mechanical force, a linear interpolation can be made between the zero displacement force (F_b) and the full displacement zero force.

Displacement actual = Disp_F (F_B - F_act) / F_B

The blocked force:

in the parallel mode  F_B = d₃₃ Y^E₃₃ (A / t) V

in the transverse mode  F_B = d₃₁ Y^E₁₁ w V