Principles and methods of the calculation of transfer characteristics of disk piezoelectric transformers Текст научной статьи по специальности «Электротехника, электронная техника, информационные технологии»

PA^IOE^EKTPOHIKA TA TE^EKOMYfflKA^Ï

PA^HOMEKTPOHHKA H TЕ^ЕKOММУНHKAЦHH

RADIO ELECTRONICS AND TELECOMMUNICATIONS

UDC 621.373.826.032:534.232.082.73

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.

KEYWORDS: piezoelectric transformer, axially symmetric oscillations, physical processes, mathematical model.

NOMENCLATURE

U1 is an amplitude value of electric potential difference;

is an imaginary unit; ra is an angular frequency; t is a time;

xk are coordinates of the point, in which it is

determined the displacement of the piezoelectric material particles from the equilibrium position;

n is a set of geometrical and physical and mechanical properties of the piezoelectric transformer; Z1 is an electrical impedance of the input electrode 1;

Ii is an amplitude value of the electric current in the

conductor, which connects an input electrode 1 with a source of the electrical signals;

2dj is a width of the input ring electrode;

H, E are vectors of the conjugate magnetic and electric fields;

B, D are vectors of the magnetic and electric induction of the electromagnetic field components;

up, uz are amplitude values of the radial and axial

components of the material particles displacement vector of dynamically deformable piezoelectric disk.

INTRODUCTION Thanks to its unique properties piezoceramics has applications in various fields of engineering and

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 [2-4]. 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].

1 PROBLEM STATEMENT

The operation principle of piezoelectric transformers is generally known [8].

When applying an electrical potential difference

U1emt 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

u (xk) eirat. Fluctuations of material particles are

accompanied by dynamic deformations smn (xk )rat of

infinitely small elements of a piezoelectric volume. Due to the direct piezoelectric effect the harmonically varying

in time according to e1 rat polarization charges with a

surface density qm (xk) e1 rat arise in a deformable

piezoelectric. Some of these charges are collected by the second pair of electrodes, which like the first pair, partially covers the surface of the piezoelectric plate. The polarization charge on the second pair of electrodes

causes an electric current i (t) = I e1 rat in the conductor, which connects one of the electrodes of the second pair to the load impedance Zn. The voltage drop

transformer. Obviously, the transformation ratio K (ra , n) is equal to the ratio of the output signal to the input one, i.e.

ZnI

K (ra,n) = ^ =~ n

t j irat rj T ,

U0e = Z„I e

irat

is an output signal of the piezoelectric

U1 U

and is a mathematical model of a piezoelectric transformer [9].

The practical value of the analytical structure K (ra, n) 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 [10-13].

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 [14-19] 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.

3 MATERIALS AND METHODS

Let us consider the disk with the radius R and the thickness a (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 (p,<p, z ). Electric polarization direction defines the properties and the matrices construction of piezoceramic disk's material constants.

Z„

T

X

R 2

a

Uid6™ 1

J Zi

lu0eit];

Ri

2d.

2d i

n

R

Figure 1 - Diagram of the piezoelectric disk transformer that operates on radial vibrations

The matrix of elastic moduli of piezoceramic disk polarized across the thickness looks like

-11

-xp

c12 cE 0 0 0

cE c22 cE c23 0 0 0

cE c33 0 0 0

cE c44 0 0

C55 0

-66

(1)

E E E

where X,p =1;...;6 are Voigt indices; c11 = c22 ^ c33 ;

E = E = E . E = E . E = I E _ E \ c12 = c13 = c23 ; c44 = c55 ; c66 = ^c11 c12 j/2 •

The matrix of piezomoduli ek p (k = 1;2;3; p = 1;2;...;6) can be written as follows [10]

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 2dl (Fig.

1)) the electrical potential difference U^^ from a source of electrical signals with the output impedance Zi is applied. Obviously, on the electrode 1 we will have another value of the electrical potential U0elwt, where \U0| < Uj, that can be written as follows

^o =

Z + Z1

(4)

Electrical impedance Zl can be determined from Ohm's law for electrical circuit section

Z = Uo/A .

(5)

If on the surface of the electrode 1 we have

harmonically time varying electric charge q (t) = Q1eirat,

the electric current amplitude value is determined as follows [20]

A = - ira 61.

(6)

The amplitude value of the electric charge Q1 is determined by the axial component Dz (p, a) of the electric induction vector

R + dj

Q1 = 2n J pDz (p,a)dp .

R1 - d1

(7)

Electrical condition of any material object is determined by Maxwell's equations

ekp =

where e31 = e32 * e

0 0 0 0 e15

0 0 0 e24 0

;31 e32 e33 0 0

(2)

= (e

! )/ 2.

33 ' ^ _ c-24 _ V 33 c'31 ;

The matrix of the dielectric permittivity tensor %bmn has diagonal structure and

(3)

X11 0 0

X mn = X22 0

X33

rot H = J + rotE = -

d D ~dt '' d B

d t '

(8) (9)

where J = rE 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 r = 0. In this case, Maxwell's equation (8) for harmonically varying fields takes the following form

rot H = ira D.

(10)

where Xi1 = X22 * x33.

Calculating the divergence of the left and right side of (10), we can come to the following conclusion

divD = 0 .

(11)

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)

rotE = 0.

(12)

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

E = - grad ® .

(13)

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

Dm = emkj3kj - Xmn (grad ®)n

(14)

where emkj » emP

Dp= 2 e153pz - Xir

5®

,(1)

dp

(15)

Dz = e313pp + e323 99 + e333 zz X33 = e31 (3pp + 39q>) + e333zz - X33

5®

d z 5®(1)

d z

(16)

(P is a Voigt index, by which it is changed a couple of symmetrical tensor indices k, j ) is an element of the matrix of piezoelectric constants; 3 kj. is

a component of infinitesimal deformations tensor; %3mn is a component of the dielectric permittivity tensor; (grad ®)n is the n -th component of scalar potential

gradient vector. When writing the equation (14) in a cylindrical coordinate system we should consider the following correspondence between the symbols (p,9, z) of the coordinate axes of the cylindrical coordinate system and the numbers k = 1,2,3 of the coordinate axes xk of the Cartesian coordinate system: 1 » p ; 2 ; 3 » z.

From the general expression (14) the next follows

where D9 = 0 because of the axial symmetry of the problem under consideration;

3pz = (d upjd z + d uj dp)^2 is a shear deformation. In

(16) piezoelectric moduli of the same value e31 and e32 (see comment to the matrix (2)) are written, as is usual in solid mechanics, by the same symbol e31 . Components

3pp=5 UplQp , 39K = up/p and 3 zz =d uzld z

determine compression and expansion deformations along the coordinate lines of a cylindrical coordinate system.

®(1) is an electrical potential in the ring area (Rj -dj <p<Rj + dj; 0<9<2n; 0<z<a} under the electrode 1.

Expressions (15) and (16) substituting into condition (11) gives a second order differential equation in partial derivatives relative to the required scalar potential

®(1)(p, z) of the electric field in a deformable

piezoelectric.

In the particular case of a sufficiently thin disk when aIR < 1, 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 (p = R; 0 <9< 2n; 0 < z <a} , the shear deformation becomes zero on this surface and on surfaces z = 0 and z = a. In addition, on the surface covered by the electrode z = 0 the radial component Dp = 0. The radial component Dp = 0 on the side surface of the

piezoceramic disk [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 Dp = 0 V(p,9,z)eV, where V is a

volume of the disk. In this case, the vector of electric induction is completely determined by only one non-zero axial component Dz, and the condition (11) takes the form

d D^/d z = 0;

(17)

where D^-1 further underlines the fact that we are talking about electric induction vector at the ring area (Rj - dj <p< Rj + dj; 0 <9< 2n; 0 < z <a} under the electrode 1.

From condition (17) it implies that the axial component D^ is a function of the radial coordinate p 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 X >> a (X is an elastic wave length.

Because of

3pp + 399 = 5 Up/5p + Up/p = [5(p Up V^pI/p ,

definition (16) can be written as follows

d

Dij)(p)= ^ p dp

p u

(1)'

d u

(1)

+ e-

33

d:

X33 "

5®'

(1)

5;

(18)

a D()(p)= ^ A p dp

u«(p,a)-u(1)(p, 0)

Let

pj u() (p, z) dz

(19)

"X33

®(1) (a)-®(1) (0)

v(1)(p) =

1 a

(p)=rj v(p,z )dz,

a 0

(20)

and u(1) (p) 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

®(1) (a) - ®(1) (0) = U0, then (19) takes the form

p dp

d!»=

p upj) (p)

C33

(p,a) - u(1) (p,0)

y3 U0

X33 •

a

(21)

Substituting (21) into definition (7) of the amplitude value of electric charge, we can obtain

Q1 = 2n 1e

p upj) (p)

R1 + d1

R1 - d1

+ 12! j p u(,j) (p,a) - u(j)(p,0)

a - d1 L

(( + dj )2 - (( - dj) U0

d p -

X33

2a

(22)

We set

,(1)(z )=-L-Rj+dj p u (1)

v ' 2 dR

d1R1 Rj - dj

j p u(1)(p, z)dp , (23)

where u^1 (z) is an averaged over the area of the ring (R1 - d1 <p< Rj + dj; 0 <9< 2n} axial component of the material particles u (p, z) displacement vector in the

ring area under the electrode 1. With the definition (23) relation (22) can be written as follows

Q1 = 2n e3

(Rj + dj )u()(Rj + dj)-

where ^^ (p, z) and u(1) (p, z) are amplitude values of

the components of the material particles displacement vector in the ring area

(R1 -d1 <p<R1 + d1; 0<9<2n; 0<z<a}.

Integrating with respect to z the left and right side of (18), and taking into account the condition (17), we obtain the following result

-(Rj - dj)u«(( - dj)

+ 4nd1 R1 — a

u(j) (a) - u( (0)

(1),

- C'U

(24)

where Cj3 = 4nd1 R1X£33/a 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

u(1),

(p) = U0F«(p), u(j)(z) = U0F^(z) , (25)

/1)

(1),

where functions Fp(1) (p) and F(1 (z) differ from the

averaged components ^^ (p) and u(1 (z) of the material

particles displacement vector only by a constant factor U0, and have the meaning of displacements sensitivity in

the ring area {R1 - d1 < p < R1 + d1; 0 < 9 < 2n; 0 < z < a}

to the amplitude values of electrical potential difference

on the ring electrode 1. The dimension of Fp(j) (p) and

Fz(1)(z) is m/V. Functions Fp(1) (p) and f(! (z)

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 U0 = 1 V is applied to this electrode.

Following suggestions (25), the expression (24) for the electric charge Q1 calculation can be written as follows

Qi = U 0C3 Fj (ra,Hj) ,

(26)

where dimensionless function F1 (ra, n1) is defined follows

as

F1 (>П1 ) =

2х3з^1

1 + ^ R

1 У

Fp(1)(Rt + d )-

1 - d.

л

R

Fp(1)(Rt - dx)

1У

Fi1)(a)- F«(0)

X33

- 1 .

(27)

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 Z1 :

= -

1

iraC! F (со, П )

(28)

Zj = 1,

[/(iraCf) .

Substituting (28) into the formula (4), we obtain

Uo =

U

1 - iroCfF (со,П1 )

(29)

It should be emphasized that the potential difference U0 is determined by components averaged values of the material particles displacement vector of the ring area {R _dj <p<R1 + dj; 0<9<2n; 0<z<a}. This fact is

of fundamental importance, since there is the possibility of equations joint solutions of a deformable piezoelectric motion.

In the case when a strong inequality a/R << 1 takes place, i.e. when the disk can be considered as infinitely thin, the situation is considerably simplified, since the deformation ezz becomes linearly dependent on the sum of deformations epp and e.

From the generalized Hooke's law [20] for the elastic media with piezoelectric properties

deformations epp, ефф and ezz and can be defined by the

and ctzz correspond to compression and expansion deformations epp, sw following expressions:

(sw + 1 zz )

3pp = CnSpp + c12 (( + S- I + e3

дФ

СТфф = C12Spp + С118фф + C12S zz + e31

3zz = c12 (pp + 8фф) + 4Szz + e33

5 z дФ д z '' дФ

17 '

(30)

(31)

(32)

If the dielectric under the ring electrode 1 does not have piezoelectric properties, i.e. e31 = e33 = 0, the

function F (ra, n1) = _ 1 and the expression (28) becomes as well-known formula for capacitor reactive resistance calculation with capacitance Cf, i.e.

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 (z = 0) and top (z = a) 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:

CTzP = 3

zp lz = 0 ;a z

= 0;a

= 0.

(33)

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. д aij Jд z s 0 . It follows that the condition (33) must be

satisfied at any point of the volume F 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:

\ e33 дФ c- (+£фф)- #^• (34)

c33 c33 u z

2 (

Substituting expression (34) into (30), (31) and (16), it produces the following results:

CTpp = С11ерр + С12ефф + e31 д z

СТфф = C12Spp + С118фф + e31

Dz = e31 (pp + !фф) - X33

дФ

17 '

дФ

17

дФ д z '

(35)

(36)

(37)

3ij = cijkl!kl + ekij

дФ д xk

where а .. is a component of the resulting mechanical stresses tensor, follows that in a polarized across the thickness piezoceramic disk normal stresses арр, а,

фф

where cu = cr, -

cj

( 4 Г/'

33

42

c12 (1 c1 2/ c3 3 );

e31 = e31 - e33cj2/cj3 are material constants for planar

stress-strain state of the polarized across the thickness piezoceramic element; x 33 = x33 +

e3^/ c3j is a

dielectric permittivity of the polarized across the thickness piezoceramic disk for constancy mode (equality

to zero) of the normal mechanical stresses ctzz . Equations (35) and (36) in combination with ctzp = ctzz = 0V(p,z)eV suggest that uz = 0 in the

entire oscillating disk.

The expression (29) takes the form

U = ■

U,

(38)

1 - ifflCf F1(0)(co, nj )

where Cj3 = 4nd1 RJx^3/a is a static electrical

capacitance of the ring area of infinitely thin disk under the electrode 1;

2X33^1

1 +

1 -

33u

d ^(1)

R

Fp(1)(R1 +d )-

1

R

Fp(1)(Rt - d )

1

- 1.

(39)

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, in the ring area 2

{R2 -d2 <p<R2 + d2 ; 0<9<2n; 0<z<a} the

amplitude values ^(p, z) and «(2)(p, z) of the

harmonically time varying components of the material particles displacement vector of the oscillating piezoelectric disk can be represented as follows:

«p'(p, z)= U0 Fp(2) (p, z) '(p,z) =U0 F<2)(p,z),

(40)

where U0 is an electric potential difference on the exciting ring electrode 1 (Fig. 1); Fp(2)(p, z) and F^ (p, z) are

displacements sensitivities in the ring area 2.

The amplitude value U2 of the voltage drop on electrical load Zn, i.e. on the input impedance of the electronic circuit which is directly connected to the ring

electrode 2, is defined as follows

U2 = ZnI 2,

(41)

q2 = c)u0f2(m,n2)- c)u.

where C) = 4nd2 R2 %33/a is a capacitance of the ring area 2;

(42)

static electrical

F2 (ro,n2) =

2%33d2

d2 1 + -2-R.

Fp(2) (2 + d2 )-

2

1 -Hi

R2

Fp(2)(R2 - d2 )

X33

Fz(2) (a)- F))

(i

Fp()) (p) and F^2-1 (z) are averaged sensitivities.

Substituting (42) into current definition I2, and obtained result into (41), we can come to the conclusion that

U2 = fn (ffl)U0F) ( H) )

(43)

where fn (ra) = - /'roC|ZnJ(1 - zraC|Zn) is a switching

on function or load characteristic of the output ring electrode of the piezoelectric transformer.

In the short-circuit mode (Zn = 0) function

fn (ra) = 0 and U2 = 0. This fact is very clear and does not require any mathematical calculations to prove its validity. In idle mode, when Zn ^ &, switching on function fn (ra) if ra = 0 is equal to zero, and at an arbitrarily small ra > 0 fn (ra) = 1, i.e. in this mode

switching on function is a function of Heaviside. It follows that the piezoelectric receiver of elastic vibrations is not capable to register the static pressures and deformations. This statement is not so obvious to practitioners actually cancels a large group of devices of piezoelectronics, which are presented in [22].

The rate of change of the switching on function fn (ra) is determined by the time constant in = C |Zn of the circuit that connects the receiver electrode to an electrical load. The function module values fn (ra),

depending on the value of the dimensionless quantity Qn = raxn are shown in Fig. 2.

where 12 = - /'raQ2 is an amplitude of the electric current in the conductor, which connects the electrode 2 and the electrical load Zn ; Q 2 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 Z1 , we can obtain the following definition of the charge Q2 :

1.0

0-8 0.6 0.4 0,2 0

K, («>)i

V

0 2 4 6 8 10 Figure 2 - Changing of switching on function module of the acoustic waves piezoelectric receiver

After substituting (29) into (43), we can write the following definition of the transformation K (ra,n) of the piezoelectric transformer

* (со, П)= U2 = (ra)F2 ^ U 1 - iroCf F1 (со,П1 )Zi

(44)

In the case of very thin piezoceramic disk, when a strong inequality a/R << 1 takes place an expression (44) can be written as follows

К (0)(co, П) =

= U 2 = f(0) (C0)F2(0)(CQ, П 2 )

U1 1 - iraCf F1(0,(co,П1 )Zi

(45)

where /(0) (со) = - iraC23Zn/(1 - iraC23Zn) ; C23 = 4nd2R2хЗз/a ;

F2(0)(ra , П2 ) =

2Хз3 d2

1 +

R

■2 У

Fp(2)( + d2 )-

1 -~L

R

2У

Fp(2)(R2 - d2 )

; f/0) (со,П1 ) is defined by

Figure 3 - Calculation scheme of disk piezoelectric transformer

It is obvious that the work of function piezoelectronic element, which is schematically shown in Fig. 3, is fully described by transformation ratio K (ra,n) = U 2/ U1 ,

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

д ^(p)

(39).

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.

4 EXPERIMENTS

Let us consider a disk piezoelectric transformer (Fig. 3), primary electrical circuit of which consists of electric potential difference generator U1e"ot (where U1 is an amplitude value of electric potential difference) with output electrical impedance Zg 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 Zn , on which an electric potential difference

U2emt 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).

pp

(p)= cu

p w + c _p_ ^ 12

(p) , . U2

дp

+ e^, (46)

(2)(p)= cD + ^

PP

дp

(47)

д «p3)(p)

'(p) , . U0

+ e^ ——, (48)

m (p)= cd д up4) (p) + d up4) (p)

p a

up

where

pp

11 Я + c12

дp p

(49)

= E - i E \2 / E • = E 1л - E / e \ •

11 = Vc12/ / c33 ; c12 = c12 y1 CJ2/C33);

11 = + (e31 ) /X33 ; c12 = c12 + (e31 ) /X33 are

moduli of elasticity for the mode of axially symmetric radial oscillations of the piezoceramic disk material particles in the areas under the electrodes (area No.1, where p e [0,R1 ] , and area No.3, where p e [R2, R3 ]) and in the areas where there are no electrodes (area No.2, where pe[ R1, R2 ], and area No.4, where

Pe[ R3 , R]).

The amplitude values of the radial components of the material particles displacements vectors in the areas No.1, ..., No.4, are defined as follows:

u«(p)= A J (yp), (50)

up2) (p)= A2 J (y1p) + A3Л (y1p), (51)

к

«(3)(p)= a-a (yp) + a5n (yp), «(4) (p)= A6- M + A7N (Ylp),

(52)

(53)

where A1, ..., A7 are frequency-dependent constants of the radial displacements of material particles in various areas; J1 (z), N (z) (z = yp ; z = y1p) are Bessel and Neumann functions [25] of the first order;

are wave numbers

Y = <°/\lcn/po and Y1 = ro/№1!PO of the radial oscillations in the areas under the electrodes, and in the areas where there are no electrodes; p0 is a

piezoceramics density.

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:

-pW'

u (3),

(«1 )-<#(R ) = 0, (r )-«p% ) = 0,

(R )-<$(R ) = o,

(R )-«P3)(R2 ) = 0, (R )-<«(R ) = 0,

pp

(R3 )-«(4)(R ) = 0.

(54)

(55)

(56)

(57)

(58)

(59)

If boundary p = R of the piezoceramic disk is free from mechanical contacts with other material objects, then on the contour p = R next condition should be satisfied

pp

(R )= 0.

(60)

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 A1 ,

A7 . 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:

D .

11 '

ml2 = J0 (YjRj)- (1 - kj) Jj (yjRj V(yjRj )];

5=v1 + K321; K321=( e3j )7(x33cu); kj = cD2/^

m13 = S [W (YjRj) - (1 - kj) N (YjRj V(yjRj )];

m14 = mj5 = mj6 = mj7 = 0 ; Pj = - qU2jQ; q = e3\R/(cna) ; Q = YR ; m2j = Jj (yRj );

m22 = Jj (yJrI ); m31 = 0; m23 = ni (yJrI ); m24 = m25 = m26 = m27 = 0; P2 = 0;

m32 =5[ J0 (Y1R2 ) - (1 - k1 ) J1 (Y1R2 )/(Y1R2 )] ; m33 = S[N0 (Y1R ) -(1 - kj) Nj (Y1R2 )/(yjR ) ;

m34 = J0 (YR2 )-(! - k)J1 (yR2 )(yR2 ); m35 = N0 (yR2 )-(l - k) N1 (yR2 V(yR2 );

m36 = m37 =0; P3 = qU0/Q; m41 =0; m42 = ji (yIr2 );

m43 = ni (yIr2 ); m44 = Jj (yr2 ); m45 = Nj (yr2 ); m46 = m47 = 0; P4 = 0 ; m51 = m52 = m53 = 0 ;

m54 = J0 (yR3 )-(l - k )Jj (yR3 )(YR3);

m55 = N0 (yR3 )-(j - k) NI (yR3 V(YR3) ;

m56 = S[J0 (Y1R3) - (1 - kj) Jj (Y1R3)/(yIR)]; m57 = S [N0 (Y1R3) -(1 - kj) Nj (Y1R3 )/(yiR ) ;

P5 =-qU0lQ; m61 = m62 = m63 =0; m64 = Jj (yr3 );

m65 = ni ( yr2 ); m66 = Jj (yIr3 ); m67 = nI (yIr3 ); P6 = 0 ; m71 = m72 = m73 = m74 = m75 = 0;

m76 = J0 (YjR)-(j - k1 )Jj (Yjr)(yIr); m77 = N0 (YjR) - (1 - kj )Nj (YjR)/(YjR) ; P7 = 0.

Solutions for constants A1, A4 and A5, that define the radial displacements of disk material particles under the electrodes of primary and secondary electrical circuits of piezoelectric transformer are as follows:

A = -Q(U2An + U0A12); An = A12 =

Q

Do

BA

Do B.

(62)

A4 =Q(A41 + U0A42) ; A41 ; A42 ; (63)

Do B

Z mA = Pj, ( j »k = 1>2 »•••»7). (61) A5 =-Q(U2A51 + Uo A52 ); A51 = A52 =

Do B

52 .

k = 1

DO

Do

(64)

The coefficients

V

and right-hand parts Pj of where D0 is a determinant of the system of equations

equations (61) have the following form:

»11 = -o (yR1 ) - (1 - k) (yR1 V(YR1 ); k = C12Ien ;

(61), and B11, matrices:

B52 are determinants of the following

=

Bi =

=

Bu =

-m22 -m23 0 0 0 0

m32 m33 -m34 -m35 0 0

m42 m43 -m44 -m45 0 0

0 0 m54 m55 -m56 -m57

0 0 m64 m65 -m66 -m67

0 0 0 0 m76 m77

-m12 -m13 0 0 0 0

-m22 -m23 0 0 0 0

m32 m33 -m34 + m54 -m35 + m55 -m56 -m57

m42 m43 -m44 -m45 0 0

0 0 m64 m65 -m66 -m67

0 0 0 0 m76 m77

m21 -m22 -m23 0 0 0

0 m32 m33 —m35 0 0

B41 = 0 0 m42 0 m43 0 —m45 m55 0 —m56 0 —m57 ;

0 0 0 m65 —m66 —m67

0 0 0 0 m76 m77

m11 "m12 —m13 0 0 0

m21 "m22 -m23 0 0 0

0 m32 m33 —m35 + m55 —m56 —m57

0 m42 m43 —m45 0 0

0 0 0 m65 —m66 —m67

0 0 0 0 m76 m77

m21 -m22 —m23 0 0 0

0 m32 m33 —m34 0 0

B51 = 0 0 m42 0 m43 0 —m44 m54 0 —m56 0 —m57 ;

0 0 0 m64 —m66 —m67

0 0 0 0 m76 m77

m11 —m12 —m13 0 0 0

m21 —m22 —m23 0 0 0

0 m32 m33 —m34 + m54 —m56 —m57

0 m42 m43 —m44 0 0

0 0 0 m64 —m66 —m67

0 0 0 0 m76 m77

Substituting definition (62) of the constant A1 into the equation (50), and obtained result into the formula for potential calculating U2 we can come to the conclusion that

*

U2 = 2fe (U2A11 + U0A12 )j ( ),

X33 R^

which implies that

K2 (Q,n) =

U2 = U0K2 (Q,n) ; (65)

2fe (rc) K2 A,2 [j (PRjR)/(^/R)] ;

1 - 2fe (co)K^ An [j (QRj/R)/(rç/R)] ;

where K2 = (e31) j(cnX33) is a squared electromechanical coupling coefficient for the mode of

radial oscillations of thickness polarized piezoceramic disk material particles.

Let us define the amplitude value U0 of electric

potential difference on the electrode of the piezoelectric transformer's primary electric circuit. It is obvious that

U o =

UZ 3

^ + Z3

(66)

3 i \

03 = 2n j pD^3' (p)dp =

(67)

= C ?U

2ae,

R3X33 (1 -p2 )u 0

,(3)

(R )-«(3)(R )

-1}

where C? =

nX33

(( - R22 ))

is a static electrical

After calculating the values u( ) (R2) and u( ) (R3) according to the formula (68) it can be written that

Q3 = C3aU0K3 (Q, n), where

K3 (Q'n)= IK^ {[ K2 (Q,n)^41 + A42 ] J (Q) +

+ [K2 (Q,n) A5I + A52] N(Q)j - 1; J(q) = [J1 (QR3/R) - pj1 (PQR3/R)]/(QR3/R); N(Q)= [N1 (QR3/R) - PN1 (PQR3/R)]/((R3/R) .

© Bazilo C. V., 2018

DOI 10.15588/1607-3274-2018-4-1

After charge determining Q3 the electrical impedance

Z3

is

determined

by the

expression

Z3 = - 1 [iro C3eK3 (Q, n)], from which the definition of potential difference on the ring electrode follows

U0 =-

U,

1 - /roC?ZgK3 (Q,n)

(69)

where Z3 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 Z3 = U0/I3 , where I3 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 I3 = - iroQ3, where Q3 is an amplitude value of

polarization charge under the ring electrode, which is defined as follows:

Substituting (69) into (65) we can come to the conclusion that

u 2=U

K2 (Q, n)

1 - iroC?ZgK3 (Q,n)

from which the formula for the transfer ratio calculation follows

U

K ( n)= 7T=

K2 (Q, n)

U1 1 - iroC?ZgK3 (Q,n)

(70)

capacitance of the ring electrode; P = R2/R3 is a

geometrical parameter of the ring.

Substituting (63) and (64) for the calculation of constants A4 and A5 into definition (52), and taking into account the expression (65), we obtain the following formula for the calculation of displacements u(3) (p):

u(3) (p)= UQ^1 {[K2 (Q,n) A41 + A42] Jj (Qp/R) +

q ^Cn a

+ [K2 (Q,n)Ail + A52] N (Qp/R)j . (68)

Analytical structure (70) is a mathematical model of piezoelectric ring-dot transformer with ring electrode in the primary circuit.

5 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 Zg . If

the value of Zg = 0 the feedback disappears and transfer ratio is completely determined by a frequency dependent function K2 (Q,n).

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 Zg and,

accordingly, by a decrease of potential difference U0 (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 Zg is

accompanied by significant changes in the frequency characteristic of piezoceramic disk transformer.

2

Fig. 5 illustrates an influence of mechanical Qi -factor of disk material on a change of transformation ratio in a narrow band near the first electro-mechanical resonance of the radial oscillations of free (not fixed) piezoceramic disk. The numerical values of quality factor are indicated near the corresponding curves.

All calculations were performed for piezoceramic disk

with radius

R = 33•10-

and thickness

a = 3 • 10 3 m , made of thickness polarized PZT type piezoceramics with following parameters:

p0 = 7400 kg/m3 ; c^ = 112 GPa ; cf2 = 62 GPa ;

,2 .

= - 9 C/m

2

4 = 100 GPa ; e33 = 20 c/n X33 =1800 x0; x0 = 8,85 • 10-12 F/m is a dielectric constant; Qi = 100 is a quality factor of piezoceramics; Zn = 10 kOhms is an electrical load value; Q = rax0 is a

dimensionless quantity, where т0 = RP0 is a piezoceramic disk time constant. The frequency f = 15206 Hz corresponds to the value Q = 1. The

value of the electrical impedance module of the electrical signal source is shown in the figures field.

jKlfi.tlj

Zq= 5 Ohms

V A -я

]КЮЖ|

4 . 6

a

|к{п,п(

—Г — I I 2g= 20 Ohms

JJ К / МГ2

I ! 1 M Zg= 50 Ohms

I

1 I — /

s

ю

4

8

0 2 4,6

c d

Figure 4 - Influence of the signal source output impedance Zg

on a frequency-dependent change of the transfer ratio module, when RjR = 12/25 , R2/R = 15/25 and R3/R = 0.999 :

a - Zg = 5 Ohms; b - Zg = 10 Ohms; c - Zg = 20 Ohms; d - Zg = 50 Ohms

400 350 300 350 200 150 100 50 0

[Kian

Z * 5 Ohms

250 1

200 N

1 <0

100

/

60

50

40

50

20

10

[кю,п|

2.4

2.45 a

2.5 °2.4

250 V 1 1 10 Ohms

200 50

J

10 0 ,

О

2.45 b

2.5

|к(п,п]

200 25 0

1 >0

10 ) J}

-1 » Q'

2.4

2.45

c

2.45 d

Figure 5 - Influence of the signal source output impedance Zg on

a frequency-dependent change of the transfer ratio module, when Rj/R = 12/25 , R2/R = 15/25 and R3/R = 0.999 : a - Zg = 5 Ohms; b - Zg = 10 Ohms;

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 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 ring-dot disk transformer's transformation coefficient. The calculation is based on the same parameters as in the

calculation of the curves \K (Q ,n) 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 R = 33 • 10-3 m, the thickness a = 3 • 10-3 m and RjR = 12/25, R2/R = 15/25, R3/R = 0.999. The values

of the modulus of transformation coefficient of the piezoceramic disk transformer are plotted along the ordinate axis, and the frequency f (dimensionless value Q ) - on the abscissa axis. The frequency f = 15206 Hz corresponds to the value Q = 1.

3

m

c - Zj, = 20 Ohms; d - = 50 Ohms

K(£2,n)

1

1 1

1 1

\ \

\ ll

ll 1

1 1 i

J \ ---

0 20 40 60 80 100 120 140 f, kHz

0 2 4 6 8 10n

Figure 6 - Calculated (solid line) and experimentally obtained (dashed line) curves of the frequency dependence of the modulus of piezoceramic ring-dot disk transformer's transformation coefficient

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 a. In other words, the thickness of the electrodes, which, as a rule, does not exceed 15 ^m, was not taken into account for constructing a mathematical model of piezoelectric transformer based on piezoceramic thin disk (a/R << 1). It should also be noted that mathematical

model (70) was built for ring-dot piezoelectric transformer (see Fig. 3) with surfaces partially covered by electrodes (area 1, p e [0, Rj ], and area 3, where

p e [ R2, R3 ]) and in the areas where there are no

electrodes (area 2, where p e [Rx, R2 ] , and area 4,

where p e [R3, R]).

As expected, the absolute values of the frequencies of resonances in calculation and experiment differ from each other. So, following the calculation, the frequencies of the first second and third electromechanical resonances are respectively equal to frX = 37193 Hz, fr2 = 88194 Hz

and fr3 = 135330 Hz; the frequency ratio

The experimental values of the same quantities are, respectively, fr1 = 34491 Hz , fr2 = 83728 Hz ,

fr3 = 132325 Hz and Z = J 2/f = 2.428 . If the

/ Jrl

experimental data are assumed to be true, the error in determining the frequency ratio is AZ = 2.3%. 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.

ACKNOWLEDGMENTS

This work was made within the framework of a state budgetary research topic "Development of highly efficient intellectual complex for creation and research of piezoelectric components for instrumentation, medicine and robotics" of Cherkasy State Technological University (No. 0117U000936).

REFERENCES

1. Varadan V., Vinoy K. and Dzhoze K. VCh MEMS i ikh primenenie. Moscow, Tekhnosfera, 2004, 528 p.

2. Bove T., Wolny W., Ringgaard E., and Breboel K. New type of piezoelectric transformer with very high power density. Applications of Ferroelectrics, 2000, Vol. 1, pp. 321-324. DOI: 10.1109/ISAF.2000.941564

3. Flynn A. M. and Sanders S. R. Fundamental limits on energy transfer and circuit considerations for piezoelectric transformers, IEEE Transactions on Power Electronics, 2002, Vol. 17, pp. 8-14. DOI: 10.1109/63.988662

4. Shao W., Chen L., Pan C., Liu Y., and Feng Z. Power density of piezoelectric transformers improved using a contact heat transfer structure, IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, 2012, Vol. 59, pp. 73-81. DOI: 10.1109/TUFFC.2012.2157

5. Andersen T., Andersen M. A. E., Thomsen O. C. Simulation of Piezoelectric Transformers with COMSOL. Proceedings of the 2012 COMSOL Conference in Milan. URL: https://www.comsol.dk/paper/download/151765/andersen_p aper.pdf

6. Ivensky G., Shvartsas M., and Ben-Yaakov S. Analysis and Modeling of a Piezoelectric Transformer in High Output Voltage Applications. URL: http://cbucc.com/glecture_file/2002060132032.pdf

DOI: 10.1109/APEC.2000.822822

7. Huang Y.-T., Wu W.-J., Wang Y.-C., Lee C.-K. Multilayer Modal Actuator-Based Piezoelectric Transformers, IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 2007, Vol. 54, No. 2. DOI: 10.1109/TUFFC.2007.249

8. Lavrinenko V. V. P'ezoe'lektricheskie transformatory. Moscow, E'nergiya, 1975, 112 p.

9. Matematicheskaya e'nciklopediya. Tom 2: D - Koo. -Moscow, Sovetskaya e'nciklopediya, 1979, pp. 951-952.

10. Karlash V. L. Longitudinal and lateral vibrations of a plate piezoceramic transformer. Ukr. J. Phys., 2006, Vol. 51, No. 10, pp. 986-992.

11. Karlash V. L. Vynuzhdennye e'lektromexanicheskie kolebaniya pryamougol'nyx p'ezokeramicheskix sterzhnej s razdelennymi e'lektrodami, Prikladnaya mexanika, 2013, Tom 49, No. 3, pp. 125-134.

12. Bogdan A. V., Petrishhev O. N., Yakimenko Yu. I., Yanovskaya Yu. Yu. Issledovanie xarakteristik p'ezoe'lektricheskogo transformatora na osnove radial'nyx kolebanij v tonkix p'ezokeramicheskix diskax, E'lektronika i svyaz ', Tematicheskij vypusk «E 'lektronika i nanotexnologii», 2009, P. 1, pp. 269-274.

13. Bogdan A. V., Petrishhev O. N., Yakimenko Yu. I., Yanovskaya Yu. Yu. Matematicheskoe modelirovanie kolebanij tonkix p'ezokeramicheskix diskov dlya sozdaniya funkcional'nyx e'lementov p'ezoe'lektroniki, E'lektronika i svyaz ', Tematicheskij vypusk «E 'lektronika i nanotexnologii», 2009, part. 2, pp. 35-42.

14. Buchacz A., Placzek M., Wrobel A. Modelling of passive vibration damping using piezoelectric transducers - the

УДК 621.373.826.032:534.232.082.73

mathematical model, Maintenance and reliability, 2014, Vol. 16, No. 2, pp. 301-306.

15. Livingston D., Kiran Kumar P., Venugopal. N. Modeling and simulation of multiple piezoelectric transformer converters, International Journal of Emerging Technology and Advanced Engineering, 2013, Vol. 3, No. 8, pp. 237245.

16. Paerand Yu., Kryvoshei D. Secondary power supplies with piezoelectric transformers. Electronics and Nanotechnology. Kyiv, 2011, pp. 55.

17. Buchacz A., Wrobel A. Piezoelectric layer modelling by equivalent circuit and graph method, Journal of Achievements in Materials and Manufacturing Engineering, 2007, Vol. 20, No. 1-2, pp. 299-302.

18. Cheng K. W. E., Ho Y. L., Ho S. L., Kwok K. W., Wang X. X., Chan H., Xue X. D. Modeling and analysis of piezoelectric transformer using multi-mesh loop matrix circuit under square-wave excitation conditions. IAS, 2005, pp. 2207-2211.

19. Huang Y., Ling Z. Progress in research on piezoelectric transformers, Electronic Components & Materials, 2004, Vol. 23, No. 1, pp. 7-10.

20. Petrishhev O. N. Garmonicheskie kolebaniya p'ezokeramicheskix e'lementov. Chast' 1. Garmonicheskie kolebaniya p'ezokeramicheskix e'lementov v vakuume i metod rezonansa - antirezonansa. Kiev, Avers, 2012, 300 p.

21. Grinchenko V.T., Ulitko A.F., Shul'ga N.A. Mexanika svyazannyx polej v e'lementax konstrukcij, Vol. 5. E'lektrouprugost'. Kiev, Naukova dumka, 1989, 280 p.

22. Dzhagupov R. G., Erofeev A. A. P'ezoe'lektronnye ustrojstva vychislitel'noj texniki, sistem kontrolya i upravleniya: Spravochnik. SPb, Politexnika, 1994, 608 p.

23. Tixonov A. N. Matematicheskaya model'. V kn. Matematicheskaya e'nciklopediya. Moscow, Sovetskaya e'nciklopediya, 1982, Vol. 3, Koo - Od., pp. 574-575.

24. Petrishhev O. N., Bazilo K. V. Principy i metody rascheta peredatochnyx xarakteristik diskovyx p'ezoe'lektricheskix transformatorov. Chast' 2. Metodika rascheta parametrov i xarakteristik prostejshego diskovogo p'ezoe'lektricheskogo transformatora. Visnik Cherkas'kogo derzhavnogo texnologichnogo universitetu, 2015, No. 4, pp. 10-23.

25. Spravochnik po special'nym funkciyam s formulami, grafikami i matematicheskimi tablicami. Ed. M. Abramovic and I. Stigan. Moscow, Nauka, 1979, 832 p.

Received 24.04.2018.

Accepted 10.09.2018.

ПРИНЦИПИ ТА МЕТОДИ РОЗРАХУНКУ ПЕРЕДАТОЧНИХ ХАРАКТЕРИСТИК ДИСКОВИХ П'еЗОЕЛЕКТРИЧНИХ ТРАНСФОРМАТОР1В

Базшо К. В. - канд. техн. наук, доцент, доцент кафедри приладобудування, мехатронжи та комп'ютеризованих технологш Черкаського державного технолопчного уншерситету, Черкаси, Украша.

АНОТАЦ1Я

Актуальшсть. Завдяки сво!м ушкальним властивостям п'езокерамжа знаходить застосування в рiзних областях технiки i технологи. Дисюж п'езоелектричщ пристро! широко використовуються в елементах шформацшних систем. Дослiдження показали, що п'езоелектричн трансформатори можуть конкурувати з традицiйними електромагнiтними трансформаторами як за ефектившстю, так i за щшьшстю потужностi. Кiнцевою метою математичного моделювання фiзичного стану коливальних п'езокерамiчних елементiв е якiсний i кшьюсний опис характеристик i параметрiв юнуючих в них електричних та еластичних полiв.

Мета роботи - запропонувати принципи побудови математичних моделей, яю в достатнiй мiрi адекватнi реальним пристроям i фiзичним процесам, що ввдбуваеться в них.

Метод. Математичш моделi п'езоелектричних трансформаторiв, що працюють з використанням вiсесиметричних радiальних коливань п'езокерамичних дискiв, побудованi з тшмальним числом припущень, що спрощують реальну © Ба2Йо С. V., 2018 БО! 10.15588/1607-3274-2018-4-1

ситуацта. Це дозволяе стверджувати, що запропонована схема побудови доставляв математичш модел1, яю в достатнш Mipi адекватно вiдповiдають реальним об'ектам i фiзичним процесам, яю в них iснують.

Результати. Основш результати ще! роботи можна сформулювати наступним чином: побудована математична модель п'езоелектричного трансформатора з кшьцевим електродом в первинному електричному колц показана висока чутливгсть частотно1 характеристики п'езоелектричного трансформатора до змш значень вимдного опору джерела електричного сигналу в первинному електричному кол^

Висновки. В результата дослвдження математично1 моделi реального пристрою можна визначити той набiр геометричних, фiзико-механiчних та електричних параметр1в реального об'екта, який забезпечуе реалiзацiю технiчних показник1в функцiонального елемента п'езоелектрошки, обумовлених в технiчному завданнi. Варлсть збережених ресурсiв становить комерцiйну цшу математично1 моделi. Перспективи подальших дослщжень можуть полягати в побудовi математично! моделi п'езоелектричного трансформатора з секторними електродами.

КЛЮЧОВ1 СЛОВА: п'езоелектричний трансформатор, вюесиметричш коливання, фiзичнi процеси, математична модель.

УДК 621.373.826.032:534.232.082.73

ПРИНЦИПЫ И МЕТОДЫ РАСЧЕТА ПЕРЕДАТОЧНЫХ ХАРАКТЕРИСТИК ДИСКОВЫХ ПЬЕЗОЭЛЕКТРИЧЕСКИХ ТРАНСФОРМАТОРОВ

Базило К. В. - канд. техн. наук, доцент, доцент кафедры приборостроения, мехатроники и компьютеризованных технологий Черкасского государственного технологического университета, Черкассы, Украина.

АННОТАЦИЯ

Актуальность. Благодаря своим уникальным свойствам пьезокерамика находит применение в различных областях техники и технологии. Дисковые пьезоэлектрические устройства широко используются в элементах информационных систем. Исследования показали, что пьезоэлектрические трансформаторы могут конкурировать с традиционными электромагнитными трансформаторами как по эффективности, так и по плотности мощности. Конечной целью математического моделирования физического состояния колеблющихся пьезокерамических элементов является качественное и количественное описание характеристик и параметров существующих в них электрических и упругих полей.

Цель работы - предложить принципы построения математических моделей, которые в достаточной мере адекватны реальным устройствам и происходящим в них физическим процессам.

Метод. Математические модели пьезоэлектрических трансформаторов, работающих с использованием осесимметричных радиальных колебаний пьезокерамических дисков, построены с минимальным числом упрощающих реальную ситуацию предположений. Это позволяет утверждать, что предложенная схема построения доставляет математические модели, которые в достаточной мере адекватны реальным объектам и физическим процессам, которые в них существуют.

Результаты. Основные результаты настоящей работы можно сформулировать следующим образом: построена математическая модель пьезоэлектрического трансформатора с кольцевым электродом в первичной электрической цепи; показана высокая чувствительность частотной характеристики пьезоэлектрического трансформатора к изменениям значений выходного сопротивления источника электрического сигнала в первичной электрической цепи.

Выводы. В результате исследования математической модели реального устройства можно определить тот набор геометрических, физико-механических и электрических параметров реального объекта, который обеспечивает реализацию технических показателей функционального элемента пьезоелектроники, оговоренных в техническом задании. Стоимость сохраненных ресурсов составляет коммерческую цену математической модели. Перспективы дальнейших исследований могут заключаться в построении математической модели пьезоэлектрического трансформатора с секторными электродами.

КЛЮЧЕВЫЕ СЛОВА: пьезоэлектрический трансформатор, осесимметричные колебания, физические процессы, математическая модель.

.HITEPATyPA / .nHTEPATYPA

1. BapagaH B. B^ МЭМС h hx npuMeHeHue / B. BapagaH, K. Bhhoh, K. flxo3e. - M. : TexHoc^epa, 2004. - 528 c.

2. . New type of piezoelectric transformer with very high power density / [Bove T., Wolny W., Ringgaard E., and Breboel K.] // Applications of Ferroelectrics. - 2000. -Vol. 1. - P. 321-324. DOI: 10.1109/ISAF.2000.941564

3. Flynn A. M. and Sanders S. R. Fundamental limits on energy transfer and circuit considerations for piezoelectric transformers // A. M. Flynn and S. R. Sanders // IEEE Transactions on Power Electronics. - 2002. - Vol. 17. - P. 8-14. DOI: 10.1109/63.988662

4. Shao W., Chen L., Pan C., Liu Y., and Feng Z. Power density of piezoelectric transformers improved using a contact heat transfer structure / [W.Shao, L.Chen, C. Pan et al] // IEEE Transactions on Ultrasonics, Ferroelectrics and

Frequency Control. - 2012. - Vol. 59. - P. 73-81. DOI: 10.1109/TUFFC.2012.2157

5. Andersen T. Simulation of Piezoelectric Transformers with COMSOL / Andersen T., Andersen M. A. E., Thomsen O. C. // Proceedings of the 2012 COMSOL Conference in Milan. URL: https://www.comsol.dk/paper/download/151765/andersen_p aper.pdf

6. Ivensky G., Shvartsas M., and Ben-Yaakov S. Analysis and Modeling of a Piezoelectric Transformer in High Output Voltage Applications. URL: http://cbucc.com/glecture_file/2002060132032.pdf

DOI: 10.1109/APEC.2000.822822

7. Multilayer Modal Actuator-Based Piezoelectric Transformers / [Y.-T. Huang, W.-J. Wu, Y.-C. Wang, C.-K. Lee] // IEEE Transactions on Ultrasonics, Ferroelectrics,

and Frequency Control. - 2007. - Vol. 54, No. 2. DOI: 10.1109/TUFFC.2007.249

8. Лавриненко В. В. Пьезоэлектрические трансформаторы / В.В. Лавриненко. - М. : Энергия, 1975. - 112 с.

9. Математическая энциклопедия. Том 2: Д - Коо. - М. : Советская энциклопедия, 1979. - С. 951-952.

10. Karlash V. L. Longitudinal and lateral vibrations of a plate piezoceramic transformer / V. L. Karlash // Ukr. J. Phys. -2006. - Vol. 51, No. 10. - P. 986-992.

11. Карлаш В. Л. Вынужденные электромеханические колебания прямоугольных пьезокерамических стержней с разделенными электродами / В. Л. Карлаш // Прикладная механика. - 2013. - Том 49, № 3. - С. 125134.

12. Богдан А. В. Исследование характеристик пьезоэлектрического трансформатора на основе радиальных колебаний в тонких пьезокерамических дисках / А. В. Богдан, О. Н. Петрищев, Ю. И. Якименко, Ю. Ю. Яновская // Электроника и связь. Тематический выпуск «Электроника и нанотехнологии». - 2009. -Ч. 1. - С. 269-274.

13. Богдан А. В. Математическое моделирование колебаний тонких пьезокерамических дисков для создания функциональных элементов пьезоэлектроники / А. В. Богдан, О. Н. Петрищев, Ю. И. Якименко, Ю. Ю. Яновская // Электроника и связь. Тематический выпуск «Электроника и нанотехнологии». - 2009. -Ч. 2. - С. 35-42.

14. Buchacz A. Modelling of passive vibration damping using piezoelectric transducers - the mathematical model / A. Buchacz, M. Placzek, A.Wrobel // Maintenance and reliability. - 2014. - Vol. 16, No. 2. - P. 301-306.

15. Livingston D., Kiran Kumar P., Venugopal. N. Modeling and simulation of multiple piezoelectric transformer converters / D. Livingston, P. Kiran Kumar, N. Venugopal // International Journal of Emerging Technology and Advanced Engineering. - 2013. - Vol. 3, No. 8. - P. 237245.

16. Paerand Yu., Kryvoshei D. Secondary power supplies with piezoelectric transformers / Yu. Paerand, D. Kryvoshei // Electronics and Nanotechnology. - Kyiv, 2011. - P. 55.

17. Buchacz A., Wrobel A. Piezoelectric layer modelling by equivalent circuit and graph method / A. Buchacz, A. Wrobel // Journal of Achievements in Materials and Manufacturing Engineering. - 2007. - Vol. 20, No. 1-2. -P. 299-302.

18. Cheng K. W. E., Ho Y.L., Ho S. L., Kwok K.W., Wang X.X., Chan H., Xue X.D. Modeling and analysis of piezoelectric transformer using multi-mesh loop matrix circuit under square-wave excitation conditions / [K. W. E. Cheng, Y. L. Ho, S. L. Ho et al]. - IAS, 2005. -P. 2207-2211.

19. Huang Y., Ling Z. Progress in research on piezoelectric transformers / Y. Huang, Z. Ling // Electronic Components & Materials. - 2004. - Vol. 23, No. 1. - P. 7-10.

20. Петрищев О. Н. Гармонические колебания пьезокерамических элементов. Часть 1. Гармонические колебания пьезокерамических элементов в вакууме и метод резонанса - антирезонанса / О. Н. Петрищев. - К. : Аверс, 2012. - 300 с.

21. Гринченко В. Т. Механика связанных полей в элементах конструкций. Т. 5. Электроупругость / В. Т. Гринченко, А. Ф. Улитко, Н. А. Шульга. - К. : Наукова думка, 1989. - 280 с.

22. Джагупов Р. Г. Пьезоэлектронные устройства вычислительной техники, систем контроля и управления: Справочник / Р. Г. Джагупов, А. А. Ерофеев. - СПб. : Политехника, 1994. - 608 с.

23. Тихонов А. Н. Математическая модель. - в кн. Математическая энциклопедия. - М. : Советская энциклопедия, 1982. -Т. 3, Коо - Од., С. 574-575.

24. Петрищев О. Н. Принципы и методы расчета передаточных характеристик дисковых пьезоэлектрических трансформаторов. Часть 2. Методика расчета параметров и характеристик простейшего дискового пьезоэлектрического трансформатора / О. Н. Петрищев, К. В. Базило // Вюник Черкаського державного технолопчного ушверситету. -2015. - № 4. - С. 10-23.

25. Справочник по специальным функциям с формулами, графиками и математическими таблицами / Под ред. М. Абрамовица и И. Стиган. - М.: Наука, 1979. - 832 с.