PRINCIPLES AND METHODS OF THE CALCULATION OF TRANSFER CHARACTERISTICS OF DISK PIEZOELECTRIC TRANSFORMERS

Bazilo C. V. – PhD, Associate Professor, Associate Professor of Department of Instrument Making, Mechatronics and Computerized Technologies, Cherkasy State Technological University, Cherkasy, Ukraine. ABSTRACT Context. Thanks to its unique properties piezoceramics has applications in various fields of engineering and technology. Disk piezoelectric devices are widely used in the elements of information systems. Research has shown that piezoelectric transformers can compete with traditional electromagnetic transformers on both efficiency and power density. The final goal of mathematical modeling of the vibrating piezoelectric elements physical condition is a qualitative and quantitative description of characteristics and parameters of existing electrical and elastic fields. Objective. The purpose of this paper is to set out the principles of mathematical models construction that are sufficiently adequate to real devices and occurring physical processes using the simplest example of axially symmetric radial oscillations of the piezoelectric disk. Method. Mathematical models of piezoelectric transformers working with axially symmetric radial oscillations of piezoceramic disks are constructed with a minimal number of assumptions simplifying the real situation. This allows us to state that the proposed construction scheme delivers mathematical models that are sufficiently adequate to the real objects and physical processes that exist in them. Results. Main results of this work can be formulated as follows: mathematical model of piezoelectric transformer with ring electrode in the primary electrical circuit is constructed; high sensitivity of frequency characteristic of piezoelectric transformer to the values of the output impedance of the electrical signal source in the primary electrical circuit is demonstrated. Conclusions. As a result of research of real device’s mathematical model a set of geometrical, physical and mechanical and electrical parameters of a real object can be determined which provides realization of technical parameters of piezoelectric functional element specified in technical specifications. The cost of the saved resources is the commercial price of the mathematical model. Prospects for further research can be to build a mathematical model of a piezoelectric transformer with sector electrodes.


NOMENCLATURE
U is an amplitude value of electric potential difference; 1 i = − is an imaginary unit; ω is an angular frequency; t is a time; technology. The relevance of the use of various functional elements of piezoelectronics in radio electronics, information and power systems is explained by their high reliability and small dimensions, which solves the problem of miniaturization of such systems. Piezoelectric disks with surfaces partially covered electrodes are often used to create various functional piezoelectronic devices. Disk piezoelectric devices are widely used in the elements of information systems. In disk piezoelectric elements with surfaces partially covered by electrodes we can simultaneously excite oscillations of compression-tension and transverse bending vibrations. Manipulating the geometric parameters of electrodes and their location relative to each other, you can have a significant effect on the energy of oscillatory motion particular type of material particles of piezoelectric disk volume. It should be especially noted that this piezoelectric element has compatibility with microsystem technology, so it can be made as microelectromechanical structures (MEMS) [1]. One of the main elements of functional piezoelectronics is piezoelectric transformer (PT). Research has shown that PTs can compete with traditional electromagnetic transformers on both efficiency and power density [2][3][4]. PTs are therefore an interesting field of research [5]. The favorable attributes of the PT are low weight and size and potentially low cost. One additional important characteristic is the high voltage isolation of the ceramic materials used to build PTs [6]. In addition, a piezoelectric transformer is more suitable for mass production than traditional, coil-based transformers [7].

PROBLEM STATEMENT
The operation principle of piezoelectric transformers is generally known [8].
When applying an electrical potential difference to pair of electrodes that are partially cover the front and bottom surfaces of the piezoelectric plate, harmonic oscillations of material particles are excited in a volume of the plate, which, in general, can be described by the displacement vector of material particles ( ) ω and is a mathematical model of a piezoelectric transformer [9]. The practical value of the analytical structure ( ) ω, Π K that adequately describes the physical processes in the real object is evident.
2 REVIEW OF THE LITERATURE Many publications have been devoted to the construction and research of mathematical models of piezoelectric transformers. Starting with the monograph [8], the basics of the calculation of piezoelectric transformers' transfer characteristics were considered, for example, in [10][11][12][13].
However, in many papers only processes occurring in a piezoelectric disk with a surface, fully covered by electrodes, are described. There are also a number of works of a disparate character devoted to the solution of the problem of electromechanical oscillations of piezoelectric elements with separated electrodes (transformer type). The constructions of piezoelectric transformer of a planar transverse-longitudinal and rod type are considered in [10] and [11], respectively. In [12] the analysis of the dependence of transformation coefficient of disk piezoelectric transformer on the location of secondary electrode, on the width of secondary electrode, and on the value of electrical load applied to secondary electrode was made. In [13] the radial axisymmetric oscillations of thin piezoceramic disk with a surface, partially covered by electrodes, are considered.
In many papers [14][15][16][17][18][19] the described methods of piezoelectric transformers models constructing are mostly based on the use of equivalent electrical circuits and it does not allow analyzing of stress-strain state of solids with the piezoelectric effects.
Based on the above, it can be argued that currently there are no reliable and valid methods of constructing of mathematical models of piezoelectric transformers, which could be used as a theoretical basis for characteristics and parameters calculating of this class of functional elements of modern piezoelectronics.
The purpose of this paper is to set out the principles of mathematical models construction that are sufficiently adequate to real devices and occurring physical processes using the simplest example of axially symmetric radial oscillations of the piezoelectric disk.

MATERIALS AND METHODS
Let us consider the disk with the radius R and the thickness α (Fig. 1) made of piezoelectric ceramics PZT with thickness polarization during its manufacture i.e. along the coordinate axis z of the cylindrical coordinate system ( ρ , ϕ, z ). Electric polarization direction defines the properties and the matrices construction of piezoceramic disk's material constants.
where 31 ( ) 15 24 33 The matrix of the dielectric permittivity tensor ε χ mn has diagonal structure and where 11 22 33 Let us assume that the thickness of the electrodes is negligible in comparison with the disk thickness.
On the ring electrode 1 (its width is equal to 1 2d ( Fig.   1)) the electrical potential difference 1 from a source of electrical signals with the output impedance i Z is applied. Obviously, on the electrode 1 we will have another value of the electrical potential 0 U , that can be written as follows Electrical impedance 1 Z can be determined from Ohm's law for electrical circuit section The amplitude value of the electric charge 1 Q is determined by the axial component Electrical condition of any material object is determined by Maxwell's equations where = J r E is a surface density of the conduction current; r is a specific electric conductivity of the material. Since the piezoelectric ceramic is a fairly good isolator it can be considered that 0 ≅ r . In this case, Maxwell's equation (8) for harmonically varying fields takes the following form Calculating the divergence of the left and right side of (10), we can come to the following conclusion Equation (11) has the meaning of the condition of absence of free carriers of electricity in a volume of the ideal dielectric.
In [21] it is shown that at a frequency range up to 10 MHz, the magnetic component of the electromagnetic field in a deformable piezoelectric ceramics by several orders less than electrical component. It gives the basis for (9) rot E 0 ≅ .
Equation (12) suggests that the electric field in a volume of the deformed piezoceramics is irrotational, i.e. potential and it can be described by a scalar electric potential, and With the definition (13), known [20,21] expression for calculating of the m -th electric induction vector component in a volume of a deformable piezoelectric can be written as follows   Expressions (15) and (16) substituting into condition (11) gives a second order differential equation in partial derivatives relative to the required scalar potential ρ , z of the electric field in a deformable piezoelectric.
In the particular case of a sufficiently thin disk when 1 α < R , it can be argued that in the frequency range in which the length of the elastic wave is larger than the thickness of the piezoelectric disk, electrical and elastic fields in its volume is almost independent of the axial coordinate values z , i.e., practically do not change their values according to thickness of the disk.
If the disk is gently fixed along the surface  [21], on the surface of ring electrode 1 and on the disc symmetry axis, i.e. on the axis Oz . The combination of these facts suggests that in thin piezoceramic disk, in a first approximation, it can be considered that volume of the disk. In this case, the vector of electric induction is completely determined by only one non-zero axial component z D , and the condition (11) takes the form where ( ) D is a function of the radial coordinate ρ and is independent of the axial coordinate values z , which is in full agreement with the above mentioned adopted suggestion about a weak dependence of the physical characteristics of the fields on the axial coordinate values in the frequency range in which the following inequality holds λ >> α ( λ is an elastic wave length. Because of ( ) definition (16) can be written as follows where ( ) ( ) Integrating with respect to z the left and right side of (18), and taking into account the condition (17), we obtain the following result and is an averaged over the thickness of the disk radial component of the material particles displacement vector in the ring area under the electrode 1. Since Substituting (21) into definition (7) of the amplitude value of electric charge, we can obtain We set where ( ) ( ) is a static electric capacity of the piezoceramic volume under the ring electrode No. 1.
Since by definition the piezoelectric transformer is a linear physical system, the averaged components of the material particles displacement vector can always be represented as follows where functions ( ) ( ) to the amplitude values of electrical potential difference on the ring electrode 1. The dimension of ( ) ( ) numerically equal to the material particles averaged displacements of the ring area under the electrode 1 when the electric potential difference with the amplitude value of 0 1 = U V is applied to this electrode. Following suggestions (25), the expression (24) for the electric charge 1 Q calculation can be written as follows ( ) where dimensionless function Substituting (26) into the definition (6) of the electric current amplitude, and the obtained result into Ohm's law (5) for the circuit section, we can get the estimated ratio for the electrical impedance 1 Z : ( ) ( ) Substituting (28) into the formula (4), we obtain ( ) It should be emphasized that the potential difference In the case when a strong inequality 1 α << R takes place, i.e. when the disk can be considered as infinitely thin, the situation is considerably simplified, since the deformation ε zz becomes linearly dependent on the sum of deformations ρρ ε and ϕϕ ε .
From the generalized Hooke's law [20] for the elastic media with piezoelectric properties where σ ij is a component of the resulting mechanical stresses tensor, follows that in a polarized across the thickness piezoceramic disk normal stresses ρρ σ , ϕϕ σ and σ zz correspond to compression and expansion deformations ρρ ε , ϕϕ ε and ε zz and can be defined by the following expressions: ( ) 11 12 31 ρρ ρρ ϕϕ In expressions (30)-(32) material constants of the same value (the elements of matrices (1) and (2)) are written by the same symbols.
On the bottom ( 0 = z ) and top ( = α z ) surfaces of the piezoceramic disk free from mechanical contacts with other material objects in accordance with Newton's third law the following conditions should take place: Since the disk is very thin, it can be argued that the quantitative characteristics of its stress-strain state does not depend on the axial coordinate values z , i.e.
. It follows that the condition (33) must be satisfied at any point of the volume V of a thin piezoceramic disk. Substituting into the left side of (32) a zero, we obtain the following definition for the compression and expansion deformations in the axial direction: Substituting expression (34) into (30), (31) and (16), it produces the following results: 11 12 31 * ρρ ρρ ϕϕ in the entire oscillating disk. The expression (29) takes the form Now let us consider the processes that occur in an area of the ring electrode 2, i.e. output electrode of the piezoelectric transformer. Obviously where 0 U is an electric potential difference on the exciting ring electrode 1 (Fig. 1 where 2 2 = − ω I i Q is an amplitude of the electric current in the conductor, which connects the electrode 2 and the electrical load n Z ; 2 Q is an amplitude value of the electric charge on the ring electrode 2.
Acting in the same manner as in the determination of the electrical impedance 1 Z , we can obtain the following definition of the charge 2 Q : ( ) z are averaged sensitivities.
Substituting (42) into current definition 2 I , and obtained result into (41), we can come to the conclusion that In the case of very thin piezoceramic disk, when a strong inequality 1 R α << takes place an expression (44) can be written as follows where ( ) ( ) Expressions (44) and (45), which have a sense of mathematical models of piezoelectric transformers operating on axially symmetric radial oscillations of piezoceramic disks, are built with a minimal number of simplifying assumptions.
To fill the definition (44) or (45) by a specific physical meaning, it is necessary to determine the components of the material particles displacement vector of the oscillating piezoceramic disk. This procedure is the subject of a separate investigation.

EXPERIMENTS
Let us consider a disk piezoelectric transformer (Fig. 3), primary electrical circuit of which consists of electric potential difference generator 1 (where 1 U is an amplitude value of electric potential difference) with output electrical impedance g Z and ring electrode (position 1 in Fig. 3). The secondary electrical circuit consists of an electrode in the form of a circle (position 2) with connected electronic circuit to it with input electrical impedance n Z , on which an electric potential difference 2 ω i t U e is formed. The primary and secondary circuits of piezoelectric transformer do not have a galvanic connection. The energy exchange between the primary and secondary circuits is carried out by means of axisymmetric radial vibrations of the piezoceramics material particles in the volume of thickness polarized disk (position 3 in Fig. 3). It is obvious that the work of function piezoelectronic element, which is schematically shown in Fig. 3, is fully described by transformation ratio which is a mathematical model of the device under consideration. Scheme of construction of piezoelectric transformer's mathematical model is outlined in [23]. The elastic stresses and displacements of material particles of piezoelectric ceramics in the areas under the electrodes, and in the areas where there are no electrodes are determined in [24]. Following the method which is described in [24] we can write that  In the conditional separation boundaries the amplitudes of displacements and stresses should satisfy the conditions of dynamic and kinematic coupling, which can be written as follows: If boundary ρ = R of the piezoceramic disk is free from mechanical contacts with other material objects, then on the contour ρ = R next condition should be satisfied Substituting expressions (46)-(53) into conditions (54)-(60), we obtain an inhomogeneous system of linear algebraic equations, which consists of seven equations, that contain seven sought constants 1 A , …, 7 A . It is obvious that this system of equations is solved in one way. In general terms, mentioned system of equations can be written as follows: Solutions for constants 1 A , 4 A and 5 A , that define the radial displacements of disk material particles under the electrodes of primary and secondary electrical circuits of piezoelectric transformer are as follows: where ( ) ( ) Let us define the amplitude value 0 U of electric potential difference on the electrode of the piezoelectric transformer's primary electric circuit.
It is obvious that where 3 Z is an electric impedance of the area No.3 under the ring electrode of the piezoelectric transformer's primary electric circuit. In accordance with Ohm's law for the electrical circuit section 3 3 I is an amplitude of the alternating current in the conductor, which connects the generator of electrical potential difference with the ring electrode. As before, we assume that 3 3 Q is an amplitude value of polarization charge under the ring electrode, which is defined as follows: is a static electrical capacitance of the ring electrode; 2 3 β = R R is a geometrical parameter of the ring. Substituting (63) and (64) for the calculation of constants 4 A and 5 A into definition (52), and taking into account the expression (65), we obtain the following formula for the calculation of displacements ( ) ( ) After calculating the values ( ) ( ) according to the formula (68) it can be written that ( ) After charge determining 3 Q the electrical impedance 3 Z is determined by the expression ( ) , from which the definition of potential difference on the ring electrode follows ( ) Substituting (69) into (65) we can come to the conclusion that from which the formula for the transfer ratio calculation follows Analytical structure (70) is a mathematical model of piezoelectric ring-dot transformer with ring electrode in the primary circuit.

RESULTS
Expression (70), which determines the transfer ratio of piezoelectric device, has a structure which is typical for electronic devices with negative feedback. It is clearly seen that the depth of feedback is directly proportional to the value of the signal source output impedance g Z . If the value of 0 g Z = the feedback disappears and transfer ratio is completely determined by a frequency dependent function ( ) 2 K Ω , Π .
Feedback physical content which exists in piezoelectric transformers is practically obvious. Displacements levels of piezoelectric disk material particles increases significantly at a frequency of electromechanical resonance of radial oscillations. This is accompanied by an increase of deformations and as a consequence, by an increase of levels of polarization charges on the electrodes of the primary electrical circuit. Because of this the amplitude of the electric current in the primary circuit increases, which is accompanied by an increase of voltage drop on the resistance g Z and, accordingly, by a decrease of potential difference 0 U (see. Fig. 3).
The transfer ratio modeling of piezoelectric transformer according to (70) have been conducted, the results of which are shown in Fig. 4. As follows from the results shown in Fig. 4, the parameter change g Z is accompanied by significant changes in the frequency characteristic of piezoceramic disk transformer. From the results shown in Fig. 4, 5 it can be concluded that each set of physical and mechanical piezoelectric parameters, each primary and secondary circuit electrodes configuration and fixed electrical load of piezoelectric transformer is corresponded to a fixed value of electrical signal source output impedance g Z , with which the maximum transfer ratio is realized in a specified frequency range.
In Fig. 6 it is shown the calculated (solid line) and the experimentally obtained (dashed line) curves of the frequency dependence of the modulus of piezoceramic ringdot disk transformer's transformation coefficient. The calculation is based on the same parameters as in the calculation of the curves ( ) K Ω , Π shown in Fig. 4.
Naturally, the dimensions of the disk transformer in the calculation and experiment are chosen to be the same, i.e., the radius   6 DISCUSSION When building the model, it was assumed that the thickness of the electrodes located on the surfaces of the disk is very small in comparison with the thickness of the disk α . In other words, the thickness of the electrodes, which, as a rule, does not exceed 15 μm, was not taken into account for constructing a mathematical model of piezoelectric transformer based on piezoceramic thin disk ( 1 α << R ). It should also be noted that mathematical model (70) was built for ring-dot piezoelectric transformer (see Fig. 3 If the experimental data are assumed to be true, the error in determining the frequency ratio is 2 3 . % Δζ = . The obtained results are explained very simply. The numerical values of the frequencies of resonances s are determined by the dimensions and physicomechanical parameters of the material of disk element. The ratio of the resonances frequencies of the same disk is determined practically only by its dimensions. For this reason, a very satisfactory match between the theoretically and experimentally determined resonance frequency ratios is observed. The discrepancy between the absolute values of the resonance frequencies is explained by the discrepancy between the physicomechanical parameters of the piezoceramics, which were incorporated into the calculation and which are inherent in the experimentally investigated object. Comparing the curves, we can conclude that the quality factor of the material of the experimentally investigated sample is at least 1.2 times larger than included in the quality factor calculation. Thus, it can be asserted that the character of the variation of both curves, shown in Fig. 6, in a fairly wide frequency range coincides with accuracy to details. This means that the qualitative content of the expression (70) is adequate to the processes that occur in real object. In other words, expression (70) is a mathematical model of piezoelectric ring-dot transformer with ring electrode in primary electrical circuit and sufficiently adequate to the real object and the processes occurring in it. The latter allows us to assume that the mathematical description of the stress-strain state of the disk transformer also corresponds quite well to the real state of things.

CONCLUSIONS
Physical processes in piezoelectric transformers, which operate using axially symmetric radial oscillations of the piezoceramic disk, are considered. The scheme of mathematical models constructing of the ring-dot piezoelectric transformer that is sufficiently adequate to real objects and occurring physical processes is proposed.
Main results of this work can be formulated as follows: -mathematical model of piezoelectric transformer with ring electrode in the primary electrical circuit is constructed; -high sensitivity of frequency characteristic of piezoelectric transformer to the values of the output impedance of the electrical signal source in the primary electrical circuit is demonstrated. Received