This tutorial provides an introduction to the basics of piezoelectrithành phố. This includes an introduction to lớn the nature of piezoelectrithành phố, và a description of the two main families of piezoceramic materials (hard doped & soft doped). In this tutorial, you will also be introduced lớn the constitutive sầu equations as well as the properties of piezoceramic material at high field. You will also find a description of the thermal properties of piezoceramic material, as well as an overview helping you select a ceramic material.

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Nature of Piezoelectricity

The piezoelectric effect was discovered by Jacques và Pierre Curie in 1880. The initial observation was the appearance of dielectric charge on a crystal proportional lớn an applied mechanical bức xúc. Soon thereafter, the converse effect i.e. the geometrical strain of a crystal proportional khổng lồ an applied electrical field, was discovered.

Basics on piezoelectric material

Piezoelectriđô thị is the property of some materials to develop electric charge on their surface when mechanical găng tay is exerted on them. An applied electrical field produces a linearly proportional strain in these materials. The electrical response to lớn mechanical stimulation is called the direct piezoelectric effect, & the mechanical response to electrical simulation is called the converse piezoelectric effect.

Different piezoelectric materials

Piezoelectric effect is exhibited by most of the materials that possess a non-centrosymmetric crystal structure. Some naturally occurring crystalline materials possessing these properties are quartz và tourmaline. Some artificially produced piezoelectric crystals are Rochelle salt, ammonium dihydroren phosphate and lithium sulphate. Another class of materials possessing these properties is piezoelectric ceramics.

In contrast lớn the naturally occurring piezoelectric crystals, piezoelectric ceramics are of a “polycrystalline” structure. The most commonly produced piezoelectric ceramics are lead zirconate titanate (PZT), barium titanate and lead titanate. Polycrystalline ceramic materials have sầu several advantages over single crystal piezoelectric materials, including the ease of fabrication and forming of various shapes & sizes. In contrast, single crystals must be cut along certain crystallographic directions, limiting the possible geometric shapes, but offer superior piezoelectric properties, except Curie and phase transition temperatures.

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PZT crystal structure


PZT crystal structure

PZT has crystal structures belonging khổng lồ the perovskite family with the general formula AB03. In the following figure, the ikhuyễn mãi giảm giá, cubic perovskite structure is shown. PZT crystallites are centro-symmetric cubic (isotropic) above sầu the Curie temperature and exhibit tetragonal symmetry (anisotropic structure) below the Curie temperature.

Poling process

Piezoelectric ceramics consist of grains (crystallites), each of these grains contains domains that are randomly oriented before poling, as shown in the left figure below. As a result, the net polarization of the material is zero và therefore ceramic does not exhibit piezoelectric properties. During poling process, adequate DC electrical field is applied and this applied electric field orients the domains in the electric field direction (as seen in the middle figure below) and lead khổng lồ a remanent polarization of the material (as seen in the right figure below).


Poling process of PZT

Hard và Soft Piezoceramic Material

Although there are several types of piezoelectric ceramic materials available today, most can be placed into one of two general categories: “Hard” or “Soft” PZT materials. The perovskite structure is very tolerant khổng lồ element substitution (doping) – therefore the terms “hard” & “soft” are used. Even small amounts of a dopant (~1%) may cause substantial changes in the properties of a material.

Characteristics of hard piezoceramic material

Hard piezoelectric ceramics have higher mechanical unique factor and are suitable for dynamic/on-resonance applications. Since higher mechanical unique factor provides more efficient energy conversion (from electrical to lớn work), hard materials can withstand high cấp độ of electrical excitation and mechanical stress, generate less heat during this process & are not easy poled or depoled except at elevated temperature. Compared to lớn soft piezoelectric materials, hard piezoelectric materials lack the strain because of the lower d coefficients.

Characteristics of soft piezoceramic material

Soft piezoelectric ceramics have higher piezoelectric coefficients compared to hard piezoelectric ceramics, at the expense of unique factor. Soft piezoelectric ceramics also provide higher sensitivity and permittivity & are well suited for static or sengươi static applications, where large strain is required. Soft piezoelectric ceramics, when operated in dynamic mode at high field suffer from high dielectric losses & low unique factors, which may lead khổng lồ overheating over an extended period of operation.

Below you can see a comparison of the characteristics of the hard and soft doped piezoceramic material.

Type of ceramicSoft piezoceramic materialHard piezoceramic material
Piezo constants (strain in static)HighLow
Dielectric constants (capacitance)HighLow
Dielectric losses (self-heating)HighLow
Coercive sầu field (depolarization)LowHigh
Quality factors (strain at resonance)LowHigh

Constitutive Equations

Because of the anisotropic nature of piezoelectric ceramics, properties vary depending on direction. To identify directions in a piezoelectric ceramic element, a specific coordinate system is used. Three axes are defined, termed 1, 2, and 3, analogous to lớn X, Y, and Z of the classical three-dimensional orthogonal phối of axes.

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Piezoelectric coefficients and directions


The polar, or 3 axis, is determined by the direction of the poling. Unless the component needs khổng lồ be utilized in shear mode, electric field is applied in direction 3. Directions 1 và 2 are physically equivalent so they can be defined arbitrarily, perpendicular to direction 3 and to each other. The directions termed 4, 5 & 6 correspond to lớn tilting (shear) motions around axes 1, 2 and 3 respectively.

In shear mode, after poling, electrodes are stripped & redeposited perpendicular to axis 1. In this case, once electric field is applied, the component shears in one dimension without any change in other dimensions.

Piezoelectric materials can be characterized by several coefficients. Piezoelectric coefficients with double subscripts liên kết electrical & mechanical quantities. The first subscript provides the direction of the electric field, or the dielectric charge produced. The second subscript provides the direction of the mechanical bức xúc or strain.

The piezoelectric constants relating the mechanical strain produced by an applied electric field are termed the piezoelectric deformation constants, or the “d” coefficients. They are expressed in meters per volt . Conversely, these coefficients which are also called piezoelectric charge constants may be viewed as relating the charge collected on the electrodes to lớn the applied mechanical bức xúc. The units can therefore also be expressed in Coulombs per Newton .

In addition, several piezoelectric material constants may be written with a “superscript” which specifies either a mechanical or an electrical boundary condition. The superscripts are T, E, D, and S, signifying:

T=constant stress=mechanically freeE=constant field=short circuitD=constant electrical displacement=open circuitS=constant strain=mechanically clamped

Here are three examples of parameters used in the piezoelectric equations together with an explanation of their notation:


Fundamental piezoelectric equations

There are different ways of writing the fundamental equations of the piezoelectric materials, depending on which variables are of interest. The two most comtháng forms are (the superscript t stands for matrix-transpose):


These matrix relationships are widely used for finite element modelling. For analytical approaches, in general only some of the relationships are useful so the problem can be further simplified. For example this relationship, extracted from line 3 of the first matrix equation, describes strain in direction 3 as a function of bức xúc and field.


Just like any other elastic material, strain is proportional lớn the applied găng. But in addition for piezoelectric materials, an additional piezoelectric term is present, relating strain to electric field also.

Limitations of the linear constitutive sầu equations

There are a number of limitations of the linear constitutive equations. The piezoelectric effect is actually non-linear in nature due to lớn hysteresis and creep.

Furthermore, the dynamics of the material are not described by the linear constitutive equations. Piezoelectric coefficients are temperature dependent. Piezoelectric coefficients show a strong electric field dependency.

Properties of Piezoceramic Material at High Field

Piezoelectric materials exhibit non-linearity, hysteresis & creep. This section provides typical material data to lớn understand and compensate these effects.

Linearity: Actuators (individual & stacked multilayer) và benders

The stroke versus applied voltage relationship for piezo electric actuators is not perfectly linear as predicted by the piezoelectric equations. Typical performances are shown in the following figures. As it can be seen, the extension vs voltage curve is actually slightly S-shaped. At low voltage, the curve for increasing voltage is concave sầu upward & the shape is cthảm bại to quadratic.

The example below shows the displacement during charging of an actuator using the piezoelectric material NCE57. Higher resolution curves can be found in the “hysteresis” section. Non-linearity implies that stroke at 1kV/mm is less than expected from the linear extrapolation using stroke at the maximum recommended field (which corresponds khổng lồ 3kV/mm).