The Piezoelectric Effect
When you squeeze a piezoelectric crystal, positive and negative ions shift asymmetrically within the unit cell, generating a surface charge proportional to the applied stress. This direct piezoelectric effect was discovered by Jacques and Pierre Curie in 1880. The converse effect — applying a voltage to produce mechanical strain — was predicted thermodynamically and confirmed shortly after. Both effects are described by the same coefficient d33.
The d33 Coefficient
The subscript '33' indicates that both the electric field (or polarization) and the mechanical stress (or strain) are along the same axis — the poling direction. This longitudinal mode gives the largest response in most ferroelectric ceramics. Other coefficients like d31 (transverse mode) and d15 (shear mode) describe coupling along perpendicular directions and are critical for bending actuators and shear sensors respectively.
Coupling Factor and Energy Conversion
The electromechanical coupling factor k33 quantifies what fraction of input energy (mechanical or electrical) is converted to the other form. It depends on d33, the permittivity, and the elastic compliance. A k33 of 0.7 means 49% energy conversion efficiency (k-squared). Single crystals of PMN-PT achieve k33 above 0.9, making them the most efficient electromechanical transducers known.
Material Selection for Applications
Sensor applications prioritize high d33 and low permittivity (high g33 = d33/epsilon) for voltage sensitivity. Actuators need high d33 and high strain. Energy harvesters optimize the d33-squared-over-epsilon figure of merit. This simulator lets you explore how changing material constants shifts the performance envelope across these application spaces.