Understanding Piezoelectric Coefficients: A Guide to d33, d31, and d15

As essential functional ceramics, piezoelectric materials convert mechanical energy into electrical energy and vice versa. This direct piezoelectric effect—translating mechanical stress into electrical charge or voltage—forms the foundation of sensor applications. Key coefficients include the longitudinal coefficient (d33), transverse coefficient (d31), and shear coefficient (d15). Their sign and magnitude are material- and orientation-dependent inputs to device design; interpret them with the specimen geometry, poling direction, and measurement mode.

The Physical Nature of the Three Major Piezoelectric Coefficients

Piezoelectric coefficients are core components of the third-order tensor dᵢⱼ, where 'i' represents the electric field direction and 'j' represents the stress direction. Measured in pC/N (picocoulombs per newton), they reflect the charge output capability per unit stress. Following high-voltage polarization, the macroscopic polarization aligns with the Z-axis, defining the application scenarios for each coefficient.

Longitudinal Piezoelectric Coefficient (d33)

The d33 coefficient describes charge developed in the poling direction when normal stress is applied in that same direction. It is commonly used to compare longitudinal response, but a measured value depends on composition, poling state, geometry, and measurement method. It should not alone be treated as a complete predictor of sensor output.

Transverse Length Vibration Coefficient (d31)

The d31 coefficient links stress applied perpendicular to the poling direction with charge developed along the poling direction. It is relevant to transverse and bending configurations, such as cantilevers, when the coordinate convention and mechanical boundary conditions are defined. Its sign must be reported using the stated convention rather than assumed for every material.

Thickness Shear Vibration Coefficient (d15)

The d15 coefficient is a shear-mode coefficient that links the specified shear stress and transverse electric displacement under the adopted piezoelectric-coordinate convention. It is relevant to shear-mode devices, but its use requires a fixture and analysis method that actually excite and identify the intended shear mode. Do not infer its relative magnitude or a particular application from d15 alone.

Selecting a Piezoelectric Measurement System

Accurate measurement of these coefficients is important when moving piezoelectric ceramics from R&D to device integration. The MatMeas PCA1000 Piezoelectric Ceramic Parameter Analyzer lists d33, d31, and d15 among its measured piezoelectric parameters, with a 20 Hz–10 MHz frequency range and ±0.05% measurement accuracy. The. MatMeas PEMS1000 In-situ d33 Piezoelectric Measurement System is a separate option for dynamic d33 work; its documented force-frequency range is 30 Hz–300 Hz, with a 110 Hz default frequency.

FAQ

Q: Which system lists d33, d31, and d15 rather than only dynamic d33?

A: PCA1000 lists d33, d31, and d15 among its piezoelectric parameters, with 20 Hz – 10 MHz frequency range and ±0.05% measurement accuracy.

Q: When is PEMS1000 the narrower, dynamic-d33 choice?

A: PEMS1000 lists 30–300 Hz force frequency and a 110 Hz default. It should not be presented as a replacement for PCA1000’s multi-coefficient resonance characterization.