Piezo Ceramics Tutorial 12 of 15

Useful Relationships

Piezoelectric Equations and Constants
To a good approximation, the interaction between the electrical and mechanical behaviour of the piezoelectric medium can be described by the following relationships:
S = s^ET + dE
D = dT + ^TE
E = -gT + (^T)^-1D
S = s^DT + gD

E = field (Vm^-1)
T = Stress (Nm^-2)
S = Strain (dimensionless)
D = Dielectric displacement (Cm^-2)

and the superscripted permittivity and compliance s denotes the quantity kept constant under boundary conditions (e.g ^T is the permittivity under constant stress).

"d" and "g" are piezoelectric constants, related by the general expression:

d = _r_og

where:
_r = relative permittivity (or dielectric constant)
_o = permittivity of free space ( 8.85x10^-12Fm^-1)

The piezoelectric constants are defined as follows:

	direct effect			reverse effect
d=	charge density developed	CN^-1	d =	strain developed	mV^-1
	applied mechanical stress			applied field

g=	electric field developed	VmN^-1	g=	strain developed	m2C^-1
	applied mechanical stress			applied charge density

As well as the above there are other parameters to be considered which characterise a piezoelectric material; of prime importance are the coupling coefficient, loss factor and the mechanical quality factor.

The Coupling Coefficient
This parameter determines the efficiency of energy conversion in the component (but not the overall efficiency of the ceramic as a transducer) and is defined as follows:

(i) For an electrically stressed component

k² = stored mechanical energy

             total stored energy

(ii) For a mechanically stressed component

k² = stored electrical energy

            total stored energy

The derivation of k from critical frequencies is complex and graphical solutions are generally used to facilitate calculations of k from (fn - fm)/fm. (see IRE Standards on Piezoelectric Crystals: Measurements of Piezoelectric Ceramics, 1961.)

An approximate solution which depends on the shape of the piece, the mode of vibration as well as the material and is useful in design is given by:

This expression is often used for thick (1Ot > d) discs and is then called k_D.

Dielectric Loss
The efficiency of a transducer depends on the mechanical and dielectric loss as well as the coupling coefficient. The dielectric loss is usually the most significant factor and is the ratio of the effective series resistance to the effective reactance, or as in the diagram to the right. It is the tangent of the loss angle

 tan = series resistance
            series reactance

Ceramics with a low tan should be employed for transducers which are to be run continuously at high power levels.

Mechanical/ Quality Factor Qm
is defined as the ratio of the energy supplied per cycle to the energy dissipated per cycle and can be calculated from:
Morgan Electro Ceramics Tutorial
where C is the low frequency (1 kHz) capacitance and Zm the minimum impedance. QM can also be determined approximately from the frequency response curve as right:

The frequency difference fz - f, is the frequency bandwidth at about 3dB where the amplitude is 1 /SQR(2) of its maximum value.

Morgan Electro Ceramics Tutorial

QM =     fr

f2 - f1
(only where Q>3)

Direction Dependence
Because poled piezoelectric ceramics are anisotropic and the direction of polarising may be freely chosen, a method of identifying the axes of a component is necessary in order to specify its parameters.

The direction of polarisation is conventionally taken as the 3 axis, with axes 1 and 2 perpendicular to this. The terms 4, 5 and 6 refer to shear stains associated with the 1, 2 and 3 directions.

This axis notation is used when specifying mast of the piezoelectric parameters discussed above.

Permittivity:_ij
i - direction of dielectric displacement.
j - direction of electric field.

E.g. ₁₁^T is the permittivity for a material whose dielectric displacement and field are in the 1 direction under conditions of constant stress.

Compliance: s_ii
i - direction of strain.
j - direction of stress.

E.g. s₅₅^D is the shear strain to shear stress ratio at constant electric displacement, for shear about an axis perpendicular to the poling direction.

Indepth technical information can also be found in the Technical Publication Archives as Acrobat® pdf files.