Defintions and Equations
The relative dielectric constant is defined as the ratio of the permittivity
of the material to the permittivity of free space. This generally measured
well below the mechanical resonance. The dielectric constant is derived
from the static capacitance measurements at 1kHz using a standard impedance
Dielectric Loss Factor
The dielectric loss factor is defined as the tangent of the loss angle
(tan δ). The loss factor represents the ratio of resistance to reactance
of a parallel equivalent circuit of the ceramic element. The loss factor
cam be measured directly using an impedance bridge.
Mechanical QM Factor
The mechanical QM factor is 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 frequency.
Alternatively the QM factor can also be determined using the
The frequency constant, N, is the product of the resonance frequency and
the linear dimension governing the resonance. N is also equal to have
the sound velocity in the same directions. The various modes of resonance
are shown schematically in the Modes of Vibration
N1 = frL Transverse mode, thin bar
Np = frD Radial mode, disc
Nt = frT Thickness mode, disc
N3 = frL Length mode, cylinder
Ns = frT Shear mode, plate
Piezoelectric Coupling Coefficient
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 similar to kt by using the appropriate resonance
Another parameter, keff, 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:
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:
Piezoelectric Voltage Coefficient
The piezoelectric voltage coefficient is the ratio of the electric field
produced to the mechanical stress applied (Vm/N).
The g constants are calculated from the equation:
Young’s modulus describes the mechanical stiffness properties and
is expressed as the ratio fo stress to strain. In a piezoelectric material,
mechanical stress produces an electrical response, which opposes the resultant
strain. The value of the Young’s modulus depends on the direction
of the stress and strain and the electrical conditions. The inverse of
Young’s modulus, Y, is the elastic compliance, s.
The aging rate of a piezoelectric ceramic is an index of the change of
certain material parameters with time:
T1, t2 are number of days after polarization
P1, P2 are measured parameter.