Because of the anisotropic nature of Piezo ceramics, effects are dependent on direction. To identify directions the axes, termed 1, 2, and 3, are introduced (analogous to X, Y, Z of the classical right hand orthogonal axial set). The axes 4, 5 and 6 identify rotations (shear).
The direction of polarization (3 axis) is established during the poling process by a strong electrical field applied between two electrodes. For actuator applications the piezo properties along the poling axis are most essential (largest deflection).
It should be clearly understood that the piezoelectric coefficients described here are not independent constants. They vary with temperature, pressure, electric field, form factor, mechanical and electrical boundary conditions etc. The coefficients only describe material properties under small signal conditions. Compound components such as Piezo stack actuators, let alone preloaded actuators or lever amplified systems cannot be described sufficiently by these material parameters. This is why each component or system manufactured by PI is characterized by specific data such as stiffness, load capacity, displacement, resonant frequency, etc., acquired by individual measurements.
Piezoelectric materials are characterized by several coefficients:
- dij: Strain coefficients [m/V]:
strain developed (m/m) per electric field applied (V/m)
or (due to the sensor / actuator properties of Piezo material).
Charge output coefficients [C/N]:
charge density developed (C/m²) per given stress (N/m²).
- gij: Voltage coefficients or field output coefficients [Vm/N]:
open circuit electric field developed (V/m) per applied mechanical stress (N/m²)
or (due to the sensor / actuator properties of Piezo material)
strain developed (m/m) per applied charge density (C/m²).
- kij: Coupling coefficients [no Dimensionss].
The coefficients are energy ratios describing the conversion from mechanical to electrical energy or vice versa.
k² is the ratio of energy stored (mechanical or electrical) to energy (mechanical or electrical) applied.
Other important parameters are the Young's modulus Y (describing the elastic properties of the material) and the relative dielectric coefficients (permittivity) e (describing the capacitance of the material).
To link electrical and mechanical quantities double subscripts (e.g. dij) are introduced. The first subscript gives the direction of the excitation, the second describes the direction of the system response.
d33 applies when the electric field is along the polarization axis (direction 3) and the strain (deflection) is along the same axis. d31 applies if the electric field is in the same direction as before, but the strain is in the 1 axis (orthogonal to the polarization axis)
In addition the superscripts "S, T, E, D" are introduced. They describe an electrical or mechanical boundary condition.
S = strain = constant (mechanically clamped)
T = stress = constant (not clamped)
E = field = constant (short circuit)
D = electrical displacement = constant (open circuit)
The individual piezoelectric parameters are related by several equations that are not explained here because they are not important for the user of piezo actuators.