Piezo element

Structurally, the piezoelectric element is a piezo-ceramic with coated electrodes. Piezo elements can be of various shapes: in the form of disks, rings, tubes, plates, spheres, etc. For vibrators and generators, piezo elements are combined into piezo stacks to achieve the best performance.

Piezoelectric effect

Piezoelectric substances (piezoelectrics), in particular, piezoelectric ceramics, have the property that during deformation under the action of external mechanical pressure, electric charges appear on their surface. This effect is called the direct piezoelectric effect and was discovered in 1880 by the Curie brothers.

Shortly thereafter (in 1881), the inverse piezoelectric effect was confirmed, namely, that such a substance located between two electrodes reacts to an applied voltage across it by changing its shape. The first effect is currently used for measurements, and the second for the excitation of mechanical pressures, deformations, and oscillations.

More detailed studies of the piezoelectric effect showed that it is explained by the property of the unit cell of the material structure. In this case, the unit cell is the smallest symmetric unit of material, from which a microscopic crystal can be obtained by its repeated repetition. It was shown that a necessary pre-condition for the appearance of the piezoelectric effect is the absence of a center of symmetry in the unit cell.

Here we can briefly explain the piezoelectric effect by the example of barium titanate, often used piezoelectric ceramics with a relatively simple unit cell design. The barium titanate BaTiO₃, like many other piezoceramic substances, is similar in structure to perovskite (СаТiО₃), by which this class of materials is named. The unit cell at temperatures above the critical, which is also called the Curie point, is cubic. If the temperature is below this critical one, then the unit cell is distorted tetragonally towards one of the edges. As a result, the distances between positively and negatively charged ions also change (Figure 1, for BaTiO₃ instead of Pb - Ba). The displacement of ions from their initial position is very small: it makes up a few percents of the unit cell parameter. However, such a displacement leads to a separation of the centers of gravity of the charges inside the cell, so that an electric dipole moment appears. According to the energy conditions, the dipoles of neighboring unit cells of a crystal are ordered by regions in the same direction, forming the so-called domains.

The polarization directions of domains are distributed in a polycrystalline structure according to a static law. Thus, unordered accumulations of individual microcrystals in the structure of matter, which are formed only in sintered ceramics, in the macroscopic sense, cannot produce any piezoelectric effect. The use of piezoelectric properties of unit cells is possible only after the so-called polarization process, in which a larger number of domains are aligned parallel to each other under the action of a strong electric field imposed on ceramics. Polarization is usually carried out at a temperature slightly below the Curie temperature in order to facilitate the orientation of the domains. After cooling, this ordered state remains stable.

Mechanical compression or tension, acting on a piezoelectric plate parallelly to the direction of polarization, leads to the deformation of all unit cells. In this case, the centers of gravity of the charges are mutually displaced inside the unit cells, which are now predominantly parallel, and the result is a charge on the surface [2].

Piezoelectric constitutive equations

Polarized piezoelectric materials are characterized by several coefficients and relationships. The four possible forms for piezoelectric constitutive equations are shown below [13]:

• Stress - Charge form
• Stress - Voltage form
• Strain - Charge form
• Strain - Voltage form

• where {T} is 6x1 mechanical stress vector, Pa,
• {S} is 6x1 mechanical strain (elastic deformation) vector, m/m,
• {D} is 3x1 the electric charge density displacement (electric displacement field, electric flux density, dielectric displacement) vector, C/m²,
• {E} is 3x1 electric field vector, V/m,
• [c^E/D] is 6x6 elastic stiffness coefficient (for E/D = constant) matrix, N/m²
• [s^E/D] is 6x6 elastic compliance coefficient (for E/D = constant) matrix, m²/N,
• [ε^S/T] is 3x3 dielectric permittivity (for S = constant/T = 0) matrix, F/m,
• [e] is 3x6 piezoelectric stress coefficient matrix, C/m² or N/Vm,
• [h] is 3x6 piezoelectric deformation coefficient matrix, N/C or V/m,
• [d] is 3x6 piezoelectric strain (charge) coefficient matrix, C/N or m/V,
• [g] is 3x6 piezoelectric voltage or pressure constant matrix, m²/C or Vm/N,
• t (superscript) is transposed matrix

Properties of piezoceramics

The relationship between the applied force and the resulting response of the piezoelectric element depends on: the piezoelectric properties of piezoceramic, the size and shape of the sample, the direction of electrical and mechanical excitation.

By their nature, piezoelectric materials are anisotropic crystals. Figure 3 shows the different directions and orientation axes of the piezoelectric material. Axes 1, 2 and 3 are the corresponding analogs of the X, Y, Z axes of the classical orthogonal coordinate system, while the axes 4, 5, and 6 define the axes of rotation. The direction of axis 3 is the direction of polarization [1]. This direction is established during production by means of a high constant voltage that is created between the electrodes.

a) Relative dielectric constant

The relative dielectric constant is the ratio of the dielectric permittivity of the material (in this case and ) to the dielectric permittivity of a vacuum (ε₀).

and ,

where ε₀ = 8,85· 10^-12, F/m

The superscript shows the boundary conditions acting on the material in the process of determining the value of the relative dielectric constant. In particular, the index T (in this case) says that the dielectric constant is measured on a free (not clamped) sample [3]. And the S index shows that measurements occur when the piezoelectric ceramics are constantly deformed (in the clamped state). The first subscript indicates the direction of the dielectric displacement, and the second - the electric field [1]. The formula for calculating the relative dielectric constant is as follows:

,

• where - dielectric permittivity (one of two or ), F/m
• t – distance between electrodes, m,
• A – area of ​​the electrode, m²,
• C – capacitance, F

b) Resonance frequency

The natural frequency f₀ of the plate by thickness calculated by the following formula

,

where с – speed of sound in the material, m/s [2]

c) Electromechanical coupling constant

The electromechanical coupling coefficients k_p, k₃₃, k₁₅, k_t and k₃₁ describe the ability of the piezoelectric element to convert energy from electrical to mechanical and vice versa.The square of the electromechanical coupling coefficient is defined as the ratio of the accumulated converted energy of one type (mechanical or electrical) to the input energy of the second type (electrical or mechanical). The index shows the relative directions of electrical and mechanical quantities and the type of oscillations. They can be associated with the mode of oscillations of a simple transducer of a certain shape. Planar coupling coefficient k_p means the interconnection of electrical and mechanical energy in a thin circular disk polarized in thickness and oscillating in the radial direction — a planar mode (Figure 5a). Transverse coupling coefficient k₃₁ refers to a long thin bar with electrodes on a long surface. The type of oscillations - tensile and compressive strain in length (Figure 5b). Thickness coupling coefficient k_t is associated with a thin disk or plate and determines the tensile and compression in thickness (Figure 5c). Longitudinal coupling coefficient k₃₃ corresponds to a long thin bar with electrodes at its ends and polarized in length. The type of oscillations is tensile and compression lengthwise (Figure 5d). Shear coupling coefficient k₁₅ describes the energy converted to shear oscillations in thickness (Figure 5e) [4].

This coefficient can be calculated through the resonant and antiresonant frequency by the formula.

,

• where f_r – resonant frequency, Hz,
• f_a – antiresonant frequency, Hz [5]

To measure these frequencies, an impedance analyzer is usually used, with which it is possible to obtain the dependence of impedance on the frequency of piezoelectric ceramics (Figure 6).

By nature, the resonant frequency occurs when the system has very little resistance, while antiresonance occurs when the system has very high resistance. In Figure 6, a frequency that has a minimum impedance is considered resonant (f_r), and the frequency with maximum impedance - antiresonant (f_a).

Figure 5 – Types of oscillation samples of piezoceramics of different shapes

d) Elastic constant

The elastic properties of piezoelectric materials are characterized by elastic compliance () or elastic stiffness (). Elastic compliance determines the amount of strain due to an applied mechanical stress. Due to the fact that under the influence of mechanical stress, the ceramic generates an electrical response, which counteracts the resultant strain, the effective Young's modulus for a short circuit of electrodes is less than at open circuited. In addition, the stiffness is different in different directions, so electrical and mechanical conditions are indicated for accurate determination of the magnitude. The superscript E indicates that measurements occur at a constant electric field (short circuit). While the superscript D indicates the boundary condition of the constant dielectric displacement, i.e. measurements occur at open circuit. The first digit in subscript shows the direction of strain, the second subscript indicates the direction of mechanical stress [4].

e) Piezoelectric constant

There are four piezoelectric constants [10]:
• d [C/N] or [m/V] - piezoelectric strain or charge coefficient;
• e [C/m²] or [N/Vm] - piezoelectric stress coefficient;
• g [m²/C] or [Vm/N] - piezoelectric voltage or pressure coefficient;
• h [N/C] or [V/m] - piezoelectric deformation coefficient.

• 
• 
• 
• 

• where D is the electric charge density displacement (electric displacement field, electric flux density, dielectric displacement), C/m²,
• E is electric field, V/m,
• T is mechanical stress, Pa,
• S is mechanical strain (elastic deformation),
• i (subscript) mean electrodes are perpendicular to "i" axis,
• j (subscript) mean that applied stress, or piezoelectrically induced strain, is in "j" direction.

The piezo modulus d_ij (piezoelectric strain or charge coefficient) – is the ratio of the mechanical strain to the applied electric field or of induced electric charge to mechanical stress (C/N) [2]

,

• where Δx_s is the change of plate thickness, m,
• U_s is the applied voltage, V

It is useful to remember that large d_ij values ​​lead to large mechanical strain, which is usually achieved when designing ultrasonic transducers. d₃₃ is used when the force is directed in the direction of the polarization axis (figure 5d). d₃₁ is used when a force is applied at a right angle to the polarization axis, herewith a charge appears at the electrodes, just as in the previous case (figure 5b). d₁₅ indicates that the charge is accumulated on the electrodes that are at right angles to the original polarizing electrodes and that the resulting mechanical oscillations are shear (figure 5e).

The piezo modulus e_ij (piezoelectric stress coefficient) is the ratio of the developed stress to the applied electric field.

, [10]

• where c_ii is the material's elastic stiffness constant under conditions of constant electric field, Pa

The piezo modulus g_ij (piezoelectric pressure constant) is the ratio of the received voltage to the applied pressure.

,

• where U_e – received voltage, V,
• t - the thikness, m,
• p_x – applied pressure, Pa.

Index “33” shows that the electric field and mechanical stress are directed along the polarization axis. The “31” index means that the pressure is applied at a right angle to the polarization axis, and the voltage is received from the same electrodes as in the case of “33”. The index “15” implies that the applied voltage is shear and the resulting electric field is perpendicular to the polarization axis. A high value of g_ij leads to high output voltages, which is desirable for sensors.

The piezo modulus h_ij (piezoelectric deformation coefficient) is the ratio of the received voltage to the external change in thickness (developed strain).

, [11]

• where U_e is the resultant voltage across the piezoelectric element, V,
• Δx_e is an external change in thickness, m.

f) Poisson’s ratio

Poisson's ratio is the ratio of the relative contraction strain (transverse, lateral or radial strain) normal to the applied load to the relative extension strain (or axial strain) in the direction of the applied load.

,

• where µ is Poisson’s ratio,
• Δa is the absolute increment of thickness, m,
• a is thickness after deformation, m,
• Δl is the absolute increment of length, m,
• l is length after deformation, m

g) Temperature coefficients

Temperature coefficient shows the change in various material properties (resonant frequency, capacitance, dimensions) with a change in temperature [6]

,

• where α_f is the temperature coefficient of resonant frequency (TCF), ppm/˚С,
• f(t₁) is the resonant frequency at t₁, Hz,
• f(t₂) is the resonant frequency at temperature t₂, Hz,
• f₂₀ is the resonant frequency at temperature of 20˚С, Hz,
• Δt is the temperature difference Δt = t₂ - t₁, ˚С

,

• where α_C is the temperature coefficient of capacitance (TCC), ppm/˚С,
• C(t₁) is capacity at temperature t₁, F,
• C(t₂) is capacity at temperature t₂, F,
• C₂₀ capacity at a temperature of 20˚С, F

,

• where α_l is the coefficient of linear thermal expansion (CLTE), ppm/˚С,
• l(t₁) is length at temperature t₁, m,
• l(t₂) is length at temperature t₂, m,
• l₂₀ is length at temperature of 20˚С, m

h) Aging rate

The aging rate is a measure of the change in the resonant frequency and static capacitance with time. To calculate this rate, after polarization, the transducer electrodes are connected together, and the sample is heated for a certain period of time. The resonance frequency and capacitance are measured every 2ⁿ (1,2,4, and 8) days. The aging rate is calculated by the following formula [1]:

,

• where AR is the aging rate for resonant frequency or static capacitance,
• t₁, t₂ is number of days aged after polarization,
• , is resonant frequency or static capacitance through t₁ and t₂ days after polarization

i) Mechanical quality factor

Q-factor is a quantitative characteristic of the resonant properties of oscillatory systems, indicating how many times the amplitude of forced oscillations at resonance exceeds the amplitude of forced oscillations at a frequency much lower than the resonant one with the same amplitude of the exciting force [8]. The quality factor is equal to the ratio of the natural frequency ω of the resonant system to the width Δω of the frequency band, at the boundaries of which the energy of the system at forced oscillation is half the energy at the resonant frequencies.

,

• where Q_m – mechanical quality factor,
• f_r is the resonant frequency, Hz,
• f_a is the antiresonant frequency, Hz,
• Z_r is the resonant resistance, Ohm,
• С is static capacitance, F

Products based on piezoelectric resonance require high mechanical Q-factor.

j) Curie temperature

Curie temperature is the temperature above which the piezoelectric material loses its properties [2].

k) Density

,

• where ρ – density, kg/m³,
• m – mass, kg,
• V – volume, m³.

Manufacture of piezo elements

Most of the compositions of piezoelectric ceramics based on chemical compounds with formula ABO₃ (for example, BaTiO₃, PbTiO₃) with a perovskite-type crystal structure and different solid solutions on their basis (e.g., systems BaTiO₃ — CaTiO₃, BaTiO₃ — CaTiO₃ — CoCO₃, NaNbO₃ — KNbO₃). Particularly widely used as piezoelectric materials are compounds of the РbТiO₃ — PbZrO₃ system (the so-called PZT system). Of practical interest is also a number of compounds with the formula AB₂O₆, for example PbNb₂O₆, having a very high Curie point (~570 °С), which allows you to create piezoelectric elements for operation at high temperatures.

The process of manufacturing piezoceramics is divided into several stages. When synthesizing a given ferroelectric compound, the raw materials (oxides or salts, for example, titanium dioxide and barium oxide) are milled and mixed in amounts corresponding to the stoichiometric composition of the compound, and then subjected to heat treatment at temperatures of 900 - 1300 °C, during which is a chemical synthesis. A so-called method of precipitation from aqueous solutions is also used, at which the temperature of synthesis decreases to 750 - 1000 °C due to ideal mixing of the components. Blanks for future piezo elements of the required configuration and dimensions are obtained from the powdery synthesized material by pressing (as well as by injection molding), which are then sintered at a strictly defined temperature regime, which largely determines the properties of piezoceramics. The machining (grinding, lapping, polishing, and sawing) of the part after sintering provides it with precisely defined shape and dimensions. Electrical connections are usually made by applying silver, nickel, platinum electrodes in a screen printing process with subsequent sintering at approximate 600°C. For the polarization of ceramics, an electrical voltage is applied to the electrodes (field strength E is from 0.5 to 3 kV/mm depending on the chemical composition and polarization method). In order to reduce the field strength E, during polarization, the sample is heated to temperatures close to the Curie point (because the domains are more mobile), and then slowly cooled in the presence of a field. Aging, i.e. change in parameters (dielectric constant, piezo modules) with time, is characteristic of piezoceramics, especially noticeable in the first few days after fabrication and polarization of samples, which is caused by a change in both the mechanical stresses at the grains boundaries and the residual polarization [8].

Piezoceramics application

Piezoelectric materials have found application in a wide range of areas, such as medical instruments, control of industrial processes, semiconductor manufacturing systems, household electrical appliances, communication control systems, various measuring instruments, and other areas. Commercial systems that use piezoelectric materials - pumps, sewing machines, sensors (pressure, icing, angular velocities, etc.), optical instruments, laser printers, motors for autofocus cameras, and many others. In this case, the scope of these materials is constantly growing. Using a piezoelectric element is usually reduced to four categories: sensors, generators, power actuators, and transducers.

In generators, piezoelectric materials can generate a voltage sufficient to cause a spark between the electrodes, and thus can be used as electrodes to ignite the fuel, for gas stoves and for welding equipment. Alternatively, the electrical energy generated by the piezoelectric elements may accumulate. Such generators are excellent solid batteries for electronic circuits.

In sensors, piezoelectric materials convert physical parameters such as acceleration, pressure, and vibrations into an electrical signal.

In actuators, piezoelectric materials convert an electrical signal into a precisely controlled physical displacement, clearly setting the accuracy of mechanical tools, lenses, and mirrors.

In transducers, piezoelectric transducers can both generate an ultrasonic signal from electrical energy and convert incoming mechanical vibrations into electrical ones. Piezoelectric instruments are designed to measure distance, flow rate, and fluid level. Transducers are also used to generate ultrasonic vibrations for cleaning, drilling, welding, grinding ceramics and for medical diagnostics [1].