Piezo element
Structurally, the piezoelectric element is a piezoceramic 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.
Diameter: 10 mm
Thickness: 1 mm
Material: CTS26
Voltage: 5V
Excitation frequency: 1MHz
Oscillation scale:: 30000:1
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 precondition 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 BaTiO_{3}, like many other piezoceramic substances, is similar in structure to perovskite (СаТiО_{3}), 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 BaTiO_{3} 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 socalled 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 socalled 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.
Modern design tools allow you to calculate / model separately the piezoelectric element or the entire piezoelectric transducer. In coordination with "Engineering solutions", you can order parameters calculation of the piezoelectric transducer
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/m^{2},
 {E} is 3x1 electric field vector, V/m,
 [c^{E/D}] is 6x6 elastic stiffness coefficient (for E/D = constant) matrix, N/m^{2}
 [s^{E/D}] is 6x6 elastic compliance coefficient (for E/D = constant) matrix, m^{2}/N,
 [ε^{S/T}] is 3x3 dielectric permittivity (for S = constant/T = 0) matrix, F/m,
 [e] is 3x6 piezoelectric stress coefficient matrix, C/m^{2} 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, m^{2}/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 (ε_{0}).and ,
where ε_{0} = 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, m^{2},
 C – capacitance, F
,
where с – speed of sound in the material, m/s [2]
Oscillation scale 200000:1
Oscillation scale 10000:1
Oscillation scale 10000:1
c) Electromechanical coupling constant
The electromechanical coupling coefficients k_{p}, k_{33}, k_{15}, k_{t} and k_{31} 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 k_{31} 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 k_{33} 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 k_{15} 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
Figure 6  The dependence of impedance on the frequency of piezoceramics [6]
 There are four piezoelectric constants [10]:
 d [C/N] or [m/V]  piezoelectric strain or charge coefficient;
 e [C/m^{2}] or [N/Vm]  piezoelectric stress coefficient;
 g [m^{2}/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/m^{2},
 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. d_{33} is used when the force is directed in the direction of the polarization axis (figure 5d). d_{31} 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). d_{15} 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.
,
 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
,
 where α_{f} is the temperature coefficient of resonant frequency (TCF), ppm/˚С,
 f(t_{1}) is the resonant frequency at t_{1}, Hz,
 f(t_{2}) is the resonant frequency at temperature t_{2}, Hz,
 f_{20} is the resonant frequency at temperature of 20˚С, Hz,
 Δt is the temperature difference Δt = t_{2}  t_{1}, ˚С
,
 where α_{C} is the temperature coefficient of capacitance (TCC), ppm/˚С,
 C(t_{1}) is capacity at temperature t_{1}, F,
 C(t_{2}) is capacity at temperature t_{2}, F,
 C_{20} capacity at a temperature of 20˚С, F
,
 where α_{l} is the coefficient of linear thermal expansion (CLTE), ppm/˚С,
 l(t_{1}) is length at temperature t_{1}, m,
 l(t_{2}) is length at temperature t_{2}, m,
 l_{20} is length at temperature of 20˚С, m
,
 where AR is the aging rate for resonant frequency or static capacitance,
 t_{1}, t_{2} is number of days aged after polarization,
 , is resonant frequency or static capacitance through t_{1} and t_{2} days after polarization
Qfactor 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 Qfactor.
Curie temperature is the temperature above which the piezoelectric material loses its properties [2].,
 where ρ – density, kg/m^{3},
 m – mass, kg,
 V – volume, m^{3}.
Manufacture of piezo elements
Most of the compositions of piezoelectric ceramics based on chemical compounds with formula ABO_{3} (for example, BaTiO_{3}, PbTiO_{3}) with a perovskitetype crystal structure and different solid solutions on their basis (e.g., systems BaTiO_{3} — CaTiO_{3}, BaTiO_{3} — CaTiO_{3} — CoCO_{3}, NaNbO_{3} — KNbO_{3}). Particularly widely used as piezoelectric materials are compounds of the РbТiO_{3} — PbZrO_{3} system (the socalled PZT system). Of practical interest is also a number of compounds with the formula AB_{2}O_{6}, for example PbNb_{2}O_{6}, 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 socalled 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].
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 David H. Johnson. Simulation of an ultrasonic piezoelectric transducer for NASA/JPL Mars rover. PA, USA.: Cybersonic, Inc. of Erie, 2003.
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