Material Constants: Definitions
The piezoelectric effect involves the interaction between the electrical and mechanical behavior of a material: the material constants below describe that interaction. Such information is essential in the application of these ceramics.
Density, r, of a material is expressed as the ratio of the mass of a body to its volume. Density can be measured using a balance and a micrometer. A buoyancy technique provides better accuracy and has the advantage that measurements can be made on any shape.
The crystal structure of a material changes at the Curie temperature from piezoelectric (non-symmetrical) to a non-piezoelectric (symmetrical) form, and the material loses its piezoelectric properties. This phase change is accompanied by a peak in the dielectric constant at that temperature. The Curie temperature, Tc, is expressed in degrees Celsius.
The dielectric constant is defined as the ratio of the permittivity of the material to the permittivity of free space. This is generally measured well below the mechanical resonance. The dielectric constant is derived from the static capacitance measurements at 1 kHz using a standard impedance bridge. The relations are expressed as shown in Figure 2 by the associated equation.
KT: relative dielectric constant of the material
eo: relative permitivity of free space
(8.854 x 10-12 F/m)
h: distance between electrodes (m)
A: area of the electrodes (m2)
Co: measured capacitance at 1kHz (F)
Fig.2: Notation for Relative Permittivity
Dielectric Loss Factor
The dielectric loss factor, also known as dissipation factor, is defined as the tangent of the loss angle (Tan d). The loss factor represents the ratio of resistance to reactance of a parallel equivalent circuit of the ceramic element, Figure 3-1. The dielectric constant (K) and loss factor (Tan d) can be measured using a standard impedance bridge or an impedance analyzer, both of which provide a direct reading. Measurements are generally carried out at 1 kHz to measure static (away from mechanical resonance) parameters at low applied electric field.
Mechanical Quality Factor
The mechanical Q (also referred to as QM) is the ratio of the reactance to the resistance in the series equivalent circuit representing the piezoelectric resonator (Figure 3.2). The QM is also related to the sharpness of the resonance frequency.
Fr resonance frequency (Hz)
Fa anti-resonance frequency (Hz)
Zm resistance at Fr (ohm)
C0 static capacitance (Farad)
Alternatively QM can also be determined using the equation:
F1, F2 are -3dB points on the frequency/impedance curve from the resonance frequency Fr.
Fig.3: Equivalent Circuit of a Ceramic Element
The frequency constant, N, is the product of the resonance frequency and the linear dimension governing the resonance. The various modes of resonance are shown schematically in Figure 4.
N1=Fr D (Hz.m) Radial Mode Disc
N2=Fr l (Hz.m) Length Mode Plate
N3=Fr l (Hz.m) Length Mode Cylinder
N4=Fr h (Hz.m) Thickness Mode Disc, Plate
N5=Fr h (Hz.m) Shear Mode Plate
Piezoelectric Coupling Coefficient
The coupling coefficient (sometimes referred to as the electromechanical coupling coefficient) is defined as the ratio of the mechanical energy accumulated in response to an electrical input or vice versa. The piezoelectric coupling coefficient can be expressed in the following equation:
The coupling coefficients can be calculated for the various modes of vibration from the following equations:
Jo: Bessel function of the first kind and
J1: Bessel function of the first kind and
sE: Poisson' Ratio.
h1: Lowest positive root of
( 1+sE) J1 (h) = hJ0 (h)
The above relationship is plotted in Figure 5.1. It has been assumed that sE = 0.31 for lead zirconate titanate and h1 = 2.05.
where DF = Fa-Fr (Hz).
Figures 5.2 and 5.3 show the relationships for k33 and k31 coupling coefficient; kt and k15 can be calculated similar to k33 but using the appropriate resonance frequencies.
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:
The values of Keff are also shown in Figure 5.4.
Piezoelectric Charge Coefficients
The piezoelectric charge coefficient is the ratio between the electric charge generated per unit area and an applied force and is expressed in Coulomb/Newton (C/N).
The d constants are calculated from the equation:
k: electro-mechanical coupling coefficient
KT: relative dielectric constant
sE: compliance (10-12m2/N)
Piezoelectric Voltage Coefficient
The piezoelectric voltage coefficient is the ratio of the electric field produced to the mechanical stress applied and is expressed as Volt.meter/Newton (Vm/N).
The g-constants are calculated from the equation:
d: charge density
e: relative permittivity
Young's modulus describes the mechanical stiffness properties and is expressed as the ratio of 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 stress and strain and the electrical conditions. The inverse of Young's modulus, Y, is the elastic compliance, s.
r: density (kg/m3)
u: sonic velocity ( = 2pFl ) (m/sec)
l: length of the element (m)
The ageing rate of a piezoelectric ceramic is a measure of how certain material parameters vary as a function of time (age) The most important parameters that age with time are the dielectric constant, frequency constants and the resonant frequency. The ageing of ceramics has a logarithmic function with time.
t1, t2: number of days after polarization
P1, P2: measured parameters such as
capacitance, resonance frequency, etc.