Definitions

Dielectric Constant Dielectric Loss Factor Mechanical Quality Factor Frequency Constants Piezoelectric Coupling Coefficients Piezoelectric Charge Coefficients Piezoelectric Voltage Coefficients Elastic Compliance Ageing Rate

Dielectric Constant

The dielectric constant K (relative permittivity) is defined as the ratio of the permittivity of the material to the permittivity of free space (i.e., e/e0). K is derived from the capacitance as measured with a capacitance bridge. This measurement is generally performed at a frequency well below the mechanical resonance (the standard is 1 kHz), thus giving the "free" dielectric constant KT (in contradistinction to the "clamped" dielectric constant KS).

Go to the top

Dielectric Loss Factor

The dielectric loss factor is defined as the tangent of the loss angle, tan d , i.e. the ratio of the imaginary part e" to the real part e' of the permittivity. In a parallel equivalent circuit of the ceramic element, the loss factor represents the ratio of conductance Gp to susceptance B. tan d  is also the ratio of resistance Rs to reactance X of a series equivalent circuit. The loss factor can be measured directly using a capacitance bridge.

Go to the top

Mechanical Quality Factor

The mechanical quality factor Qm is defined as the ratio of the strain in phase to the strain out of phase with stress. This can be expressed using complex mechanical coefficients (cf. elastic compliance below), e.g.:

Qm can be found as the ratio of the reactance to the resistance in the series equivalent circuit representing the piezoelectric resonator.

The Qm factor is also related to the sharpness of the resonance and a better value can be obtained by using the resonance width directly:

Go to the top

Frequency Constants

The frequency constant, N, is the product of the resonance frequency and the linear dimension governing the resonance. N is also equal to half the sound velocity of a longitudinal wave in the same direction.  Length mode, cylinder    Thickness mode, thin disc or plate    Transverse mode, thin bar    Radial mode, disc    Shear mode, plate    Circumferential mode, tube

Go to the top

Piezoelectric Coupling Coefficients

The coupling coefficient (electromechanical coupling coefficient) is defined as the ratio of the mechanical energy accumulated in response to an electrical input or vice versa.

The coupling coefficient can be calculated for the various modes of vibration:

k33 and k15 can be calculated similarly to kt by using the appropriate resonance frequencies. Another parameter, k eff, is frequently used to express the effective coupling coefficient of an arbitrary resonator, either at fundamental resonance or at any overtone and is expressed as follows:

Go to the top

Piezoelectric Charge Coefficients

The piezoelectric charge coefficient is the ratio of electric charge generated per unit area to an applied force (C/N).

The d constants are calculated from the equation:

Go to the top

Piezoelectric Voltage Coefficients

The piezoelectric voltage coefficient is the ratio of the electric field produced to the mechanical stress applied (V m/N).

The g constants are calculated from the equation:

Go to the top

Elastic Compliance

In an isotropic material (e.g., a typical metal) the mechanical stiffness properties can be described by a single parameter and are conventionally expressed as the ratio of stress to strain, Young's modulus Y . In a piezoelectric material, the mechanical properties are direction-dependent and thus correctly described by a tensor. Two equivalent descriptions are used, the elastic compliance matrix s (ratio of strain to stress) and the mechanical stiffness matrix c (ratio of stress to strain). s and c can defined as complex in order to include mechanical losses (cf. Qm above).     Furthermore, the mechanical properties depend on the electrical conditions, since a mechanical stress produces an electrical response which opposes the resultant strain. The s and c matrices are therefore provided with superscript E for constant electric field ("shorted" conditions) or D for constant dielectric displacement ("open" conditions). As seen below in the examples for s, the elastic properties for open and shorted conditions are related by piezoelectric coupling coefficients.     For a poled piezoceramic, the Young's moduli parallel and perpendicular to the poling direction are found as 1/s33 and 1/s11, respectively (for applicable electrical conditions).

Go to the top

Ageing Rate

Certain materials parameters of a piezoelectric ceramic change with time, and usually this time dependence is logarithmic as expressed by the ageing rate:

Where: t1, t2 are two time values (e.g., number of days after polarisation) P1, P2 are corresponding values of the measured parameter

Go to the top

Ferroperm Piezoceramics A/S Hejreskovvej 18A DK-3490 Kvistgård Tel.: +45 49 12 71 00; Fax: +45 49 13 81 88 e-mail: pz@ferroperm-piezo.com