Useful Relationships
Useful Relationships
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 = sET + dE
D = dT + .gif){kind=small}
TE
E = -gT + (.gif){kind=small}
T)-1D
S = sDT + gD
E = field (Vm-1)
T = Stress (Nm-2)
S = Strain (dimensionless)
D = Dielectric displacement (Cm-2)
and the superscripted permittivity .gif){kind=small}
and compliance s denotes the quantity kept constant under boundary conditions (e.g .gif){kind=small}
T is the permittivity under constant stress).
"d" and "g" are piezoelectric constants, related by the general expression:
d = .gif){kind=small}
r.gif){kind=small}
og
where:.gif){kind=small}
r = relative permittivity (or dielectric constant).gif){kind=small}
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 kD.
| Dielectric Loss tan Ceramics with a low tan |  |
| Mechanical/ Quality Factor Qm The frequency difference fz - f, is the frequency bandwidth at about 3dB where the amplitude is 1 /SQR(2) of its maximum value. |
QM = fr f2 - f1 (only where Q>3) |
| Direction Dependence 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. | .gif) |
| Permittivity: E.g. | |
| Compliance: sii E.g. s55D is the shear strain to shear stress ratio at constant electric displacement, for shear about an axis perpendicular to the poling direction. |
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= series resistance series reactance .gif){kind=small}
should be employed for transducers which are to be run continuously at high power levels.
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ij.gif){kind=small}
11T is the permittivity for a material whose dielectric displacement and field are in the 1 direction under conditions of constant stress.
