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Mathematical Modelling of Rod-Type Piezo-Electric Transducers for Acoustoelectronic
Devices
Bazilo С. V.1, Bondarenko M. O.1, Khltvnyt V. V.\ Тотепко M. H2, Тотепко V. I2
1Cherkasy State Technological University 2Cherkasy Institute of Fire Safety named after the heroes Chernobyl National University of Civil Protection of Ukraine
E-mail: maxxiuni23&gniaiL com
The work is devoted to the peculiarities of the construction and study of mathematical models of rodtype piezoelectric transducers, which are widely used in various acoustoelectronic devices (hydroacoustic means of target detection, ultrasonic non-destructive testing, medical diagnostics, etc.). In contrast to the existing mathematical models of piezoelectric transducers (based on amplitude-phase dependences, resonant piezoelectric transducers, equivalent circuits, etc.), the proposed mathematical model makes it possible to establish a dependence, which is a mathematical description of the acoustic coupling that exists in a solid piezoceramic rod between wave fields on its various areas. An algorithm for calculating a mathematical model of rod-type piezoelectric transducers is presented and based on the determination of the transformation ratio in the case of the inverse piezoelectric effect. Analytical dependencies, which make it possible to determine the electrical impedance and the amplitude value of the potential in the electrical circuit of the piezoelectric transducer, are obtained. It is shown that these dependencies underlie the expression for determining the transformation ratio, which is a mathematical model of a rod piezoelectric transducer. At the same time, the principle of operation of such a transducer provides for the use of longitudinal vibrations in a prismatic rod. The results of the mathematical modelling are presented on the example of a rod transducer with a square cross-section made of piezoelectric ceramics of the PZT type (plumbum zirconat.e titanate). The performed comparisons of the calculated and experimentally obtained values of the frequency dependence of the modulus of the transformation ratio of the piezoceramic transducer showed a high convergence between them (the discrepancy between the results of mathematical modelling and the experimentally obtained data for the same value of the operating frequency does not exceed 8.5%).
Key words: piezoelectric transducer: acoustoelect.ronics: mathematical model: electrical signal generator DOI: 10.20535/RADAP. 2021.86.58-67
Introduction
Recently, acoustoelectronic devices have found more and more widespread use due to their miniaturization, nmltifnnctionality, and the possibility of integrating the manufacturing technologies of these devices with microelectronic and microcircnitry technologies. At the same time, the increase in the operational marmfactnrability of snch devices and the versatility of manufacturing technologies leads to a decrease in their cost, which increases the demand for them in various fields of science and technology. At the same time, the concept of sustainable development and the "Industry 4.0" initiatives require the nse of renewable technologies and materials that bring biological, environmental and economic safety. The most promising materials for acoustoelectronic devices from the point of view of "Industry 4.0", according to the authors of fl], are piezoelectrics.
So, today, synthetic piezoelectrics (piezocerami-cs) are nsed in location-type devices (hydroacoustic target detection, ultrasonic non-destrnctive testing, medical diagnostics), in primary transducers of electrical measurement systems for non-electrical quantities (information sensors in systems for automatic monitoring and control of objects and technological processes) [2]. This should also include polarized ferroelectrics, which are nsed to create varions mi-croelectromechanical structures (MEMS), a new link in instrumentation, which has been developing more than vigorously since the mid-eighties of the twentieth centnry [3].
However, the variety of practical applications of piezoelectrics should be accompanied by appropriate methodological (scientific) support. These are methods of calculation, design and, especially, modelling of piezoelectric transducers for varions purposes [4].
1 The relevance of the study based on the analysis of recent publications
Many publications arc devoted to the construction and stndy of mathematical models of piezoelectric transducers. However, many of them describe only the processes occurring in a piezoelectric element with a fully electroded surface. For example, in [5]. studies of forced vibrations of a piezoceramic plate are carried out taking into account the viscoelasticity of the material at the frequency of the main resonance under the action of an external mechanical harmonious load for cases of rigid fixation and hinge support of the plate contour. In [6]. studies of the vibration characteristics of a thin piezoelectric ceramic disk with different ratios of its dimensions are presented. The works [7] and [8] disenss the problem of determining the spectrum of natural frequencies and modes of vibration. In [9]. expressions for the instantaneous power are derived. In [10]. a mathematical model of resonant piezoelectric transducers with fixed and free ends is developed. A mathematical model of the state of a piezoelectric with a gradient excitation field in the plane of the crystal element is presented in [11]. In [12]. the main parameters of piezoelectric ceramics are determined by measuring the maximum and minimum conductivity. The work [13] investigates the amplitnde-phase dependences for radial displacements, the sum of principal stresses and admittance components in the vicinity of resonant and antiresonant frequencies. The principles of calculation of the piezoelectric elements with partial electrodes covering are investigated in work [14].
The works considered above are not united by any systematic approach, have the character of scattered episodes, on the basis of which it can be argned that at present there is a need to create an integral methodology for constructing mathematical models of piezoelectric transducers, which could be used as a theoretical basis for calculating their characteristics and parameters.
There are also a number of works based on the use of equivalent circuits [15 18]. Mathematical models of piezoelectric transducers built based on the analysis of the so-called equivalent circuits do not take into account the obvious fact that the motion of material particles of a piezoelectric disk must satisfy the second and third laws of Newton. Which is guaranteed that they are inadequate to real objects and those physical processes occurring in them [19].
Tims, the relevance of the development of physically meaningful mathematical models of rod-type piezoelectric transducers for acoustoelectronic devices remains now.
2 Statement of the problem of modelling a rod piezoelectric transducer
Consider the design of the transducer, which is shown in Fig. . A prismatic rod of length L with a rectangular cross-section a x 6 is made of piezoelectric ceramics of the PZT type (plumbum zirconate ti-tanate) polarized along the Ox3 axis. The bottom surface x3=0 of the rod is electroded and grounded. On the top surface x3 = a there are two electrodes (positions 1 and 2 in Fig. 1). which form the primary and secondary electrical circuits of the piezoelectric transducer (piezotransformer). The primary circuit consists of a source of the electrical potential difference Uielut (U\ is the amplitude value of the electrical potential difference; i = \f—1 is the imaginary unit; w is the cyclic frequency of the change in the sign of the potential difference; t is time) with the output electrical resistance Z1 and electrode 1. The secondary electrical circuit of the piezotransformer consists of electrode 2 (position 2 in Fig. ) and an electrical load Z2, on which the potential difference U2elut is released.
U.,ekot
Fig. 1. Schematic presentation of the construction of rod piezoelectric transducer
3 Construction of mathematical model of rod-type piezoelectric transducers
Obviously, the system function (mathematical model) of the piezoelectric transformer is the transformation ratio K (w, n) (the symbol n in the list of arguments of the complex-valued function K (w, n) means a set of geometric, physical-mechanical and electrical parameters of the piezoelectric transformer), which is determined in the following way
K (u, n) = ^. (1)
When applying to electrode 1 (position 1 in Fig. 1) the difference of electric potentials from an external source the forced longitudinal oscillations of material particles of the piezoelectric arise in the region l1 < x2 < l2 of piezoceramic rod through the inverse piezoelectric effect. Thanks to elastic bonds, all material particles (elementary volumes of a deformed solid) of piezoceramic rod are involved in the process of longitudinal harmonic vibrations. As a result of the direct piezoelectric effect, an electric charge Q2 (w) appears on the electrode 2 of the secondary electrical circuit, which with its electric field sets in motion free carriers of electricity in the conductor of the secondary electrical circuit. Electric current I2 (w) = —iwQ2 (w) elut, flowing through the conductor in the secondary electrical circuit, forms on the electrical load Z2 the difference of electrical potentials U2elut = —iuZ2Q2 (w) etut . Tims
U2 = —1UZ2Q2 M . (2)
The electric charge Q2 (w) is determined through the axial component of the electric induction vector D3\xk) in a piezoelectric deformed in the region l3 < x2 < l4 as
I4 b
Q2 (w) = j j d3\x\,x2,a) dx\dx2.
13 0
where
-il
-il
u12
rE c22
^13 r-E c23
u33
0 0 0
r-E
C44
0 0 0 0
r-E
■55
0 0 0 0 0
fE
c66
u66
= (cEi -
= c
°22 "i2)/2;
.E . 33
,E i2
E E E E
i3
u23) °44
Matrix of piezoelectric moduli
0000 ei5 0
lekl3 1 = 0 0 0 e24 0 0
e3i e32 633 0 0 0
where ei5 = e24 = (e33 - &3i)/2-, e31 = e32 = e33;
- Matrix of dielectric permittivities xlj-, whose elements are experimentally determined in the mode of constancy (equality to zero) of elastic deformations (symbol e) in the volume of the investigated piezoelectric
I Xij I
Xii 0 0
X22 0
X33
(6)
where x?i = X22 = X33-
Since shear deformations in the considered piezoceramic rod (Fig. 1) are absent, then the generalized Hooke's law for different sections of the rod can be written as follows
^T^fc) = cii£iT)(xfc) + ci2e2T')(xfc) +
r.E r(m)i
+ cE2e(T\xk) - e3iE3m)(xk), (7)
(m).
a2,2 \xk) = ce2£1 i \xk) + c22£2 2 \xk) +
+ cE2£3s\xk) - e3i£3T)(xk), (8)
(m).
(3)
<m) 33
(Xk) = C^2£iî)(xk ) + C^2£22)(xk) +
+ C33£33)(xk) - e33E3m)(xk).
(9)
As follows from the equations of the physical state of a deformed piezoelectric [20], the components of the electric induction vector are determined by the components of the elastic strain tensor. The degree of influence of one or another component on the electrical state of a deformed piezoelectric is determined by the components of the matrix of piezoelectric modules = ^hij-
With the above method of electric polarization of a piezoceramic rod, the matrices of its material constants have the following design:
- Matrix of elastic moduli cEx, which are experimentally determined in the mode of constancy (equality to zero) of the electric field strength (symbol E) in the volume of the deformable piezoelectric
where ), a^^txj. ), a'^)(xk ) are the axial
components of the mechanical stresses in the rn-th region of a deformable piezoceramic rod; m =1, 2, ..., 5 is the number of the area of the rod (Fig. 1), in which the stress-strain state is determined; when writing relations (7) (9), material constants of the same value are denoted, as is customary in the mechanics of a deformable solid, by the same symbols. Compression-tension deformations in the rn-th area of the rod along the coordinate axes, Oxi5 Ox2 and Ox3 (Fig. ) are denoted by symbols eir\xk)
<m)
(m)f
(4)
u55)
(5)
4?}(xh) and 4™)(xh) respectively; E3m\xh) is an axial component of the electric field strength vector in the rn-th region of a deformable piezoceramic rod.
We will consider snch a range of frequencies of changing the sign of the potential difference at the output of the generator of electrical signals, in which the wavelength of elastic vibrations of the material particles of the rod is commensurate with its length L. So, according to the definition of a prismatic rod, strong inequalities a./L C ^d b/L C 1 must be fulfilled without fail, then it can be argned that in this frequency range the stress-strain state of the rod does not change within the area of its cross-section.
Suppose that the rod vibrates in a vacuum and does not come into contact with other material objects. In this case, duo to Newton's third law the following relations must be fulfilled on its faces:
<m) rii
(xk )
0,
= 0, b
(10)
E
E
E
E
E
E
E
X1
(m) '22
(Xk)
(m) '22
(Xk )
T<3^ \xk)
X2 = L
0.
(11)
(12)
cfieim)(xfc ) + cf2e3mT)(xfc ) = -cf2e2m^)(xfc)+e3iS3m)(xfc )
E (m)/
cf2eiT)(xfc) + c33£3s)(xk) = -cf2e22")(xfc)+e33^3m)(xfc).
(13)
The system of equations (13) determines the components of the deformation tensor e(m^)(xk) and
e33 )(xfc):
cE
en (xk) = (c33 - cu) £22 ) + A0
"12£33
„E Jm),
(m)
+ A (4^31 - Cf2e33) E^)(xk), A0
^3 ) (xk ) = - (cfl - cll) ^2 ) (xfc ) + 7o
(14)
-12 7 £22 „E \ ^(m)t
+ 7 (cfi633 - 4631) E3m)(xk), Ar
account the structure of matrices (5) and (6), to write the following calculation formulas:
^1m)(xfc) = xhE1m)(xk), D2m)(xk) = x11^2m)(^fc)
^(m)
(m),
Am)i
(17)
Conditions (10) and (12) in the considered frequency range are satisfied not only on the surface of the rod. but also at any point inside its volume. This allows us to write relations (7) and (9) in the following form
D3m)(xk ) = e31 £^"m^)(xk) + e2m)(xfc) Sm)
+ 633^^)+ x33E3m)(*k). (18)
+ (m)f
Since on the edges x1 conditions [22]
0, b and x2 = 0, L the
D(T)(xk )
0, D2m\xk )
xi = 0, b
0
must be met, then ,E<m)(xk) = 0 mid E^^)(xk) = 0 on the surface and in the volume of the deformable piezoceramic rod. Hence it follows that D<<m)(xk) = ,D<m)(xk) = 0 V xk € V, where V is the volume of the prismatic rod.
Tims, the electrical state of a deformable piezoceramic rod is determined by the axial component d3™\x2) of the electric induction vector, the value of which does not depend on the coordinate x3. For this reason, the fundamental condition for the absence of free carriers of electricity div D <m)(xk) = dDim\xk)/d x3 = 0 is fulfilled automatically. Substituting expressions (14) into relation (18), we obtain a description of the electric polarization of a piezoceramic rod for the mode of uniaxial stress state:
œ2=0, L ( m)
where Ao = cf cf — (cf2) . When writing expressions (14), it was taken into account that the physical state of a deformable piezoelectric remains unchanged within its cross-sectional area.
Substituting expressions (14) into relation (8), we obtain a record of the generalized Hooke's law for a uniaxial stress state of a piezoceramic rod in the low frequency region in the following form
4m)(x2)= e*31s2m)(x2)+ X%3E3m)(*2)
( m)
( m) 22
(X2)= YE^2) - e31^3m)(x2) ,
( m)
(19)
where X33 = X33 + (e3icf3 + e33cfi — ^^cft)/Ao is the dielectric permittivity for the mode of constancy (equality to zero) of mechanical stresses a(™)(xk) and
( m) 33
(xk ).
(15)
is
where y3 = cf — (cf2)2 (eft + c3f — 2^)^ the Young's modulus of piezoceramics polarized along the axis Ox3 for the mode of uniaxial compression-tension of the rod along the axis Ox2; e31 = e31 + cf2 [—e3icf3 — e33cft + cf2 (e3i + 633)]/Ao is the piezoelectric modulus for uniaxial stress state mode of a piezoelectric rod.
The electrical state of a deformable piezoceramic rod is determined by the electric induction vector D<m)(xk), the fc-th component of which is generally determined as follows [21]:
D<T)(^k) = ek^m)(xk) + xk,E(m\xk). (16)
Since shear deformations in the considered piezoceramic rod are absent by definition, from the general formulation (16) it is possible, taking into
In 1, 3 and 5, where there is no electroding
on the surface x3 = a, the electric induction Dc^)(x2) (n =1, 3, 5) is equal to zero on the surface x3 = a and, as a consequence, at any point in the volume of the n-th area. Equating to zero the left side of expression (19), we come to the conclusion that in these areas £^n)(x2) = — e^e^(x2)/x33 ■ In this case, the generalized Hooke's law for the n-th region is written in the following form
a2,)(x2) = YDs2'2)(x2) ,
(20)
where YD = Yf + (e31)2/x33 is the Young's modulus for the mode of constancy (equality to zero) of electric induction in a deformable piezoelectric.
In order to determine the axial component E^k)(x2) (k = 2, 4) of the electric field strength vector under the electrodes of the primary and secondary electrical circuits of piezoelectric transformer, we write E^k)(x2) = —d$< k)(x2,x3)/dx3 , where $< k)(x2,x3) is the scalar potential of the electric field in the fc-th region of the deformable piezoceramic rod. Substituting this definition into relation (19) and integrating the result obtained
0
0
0
0
«
with respect to a variable x3 in the range from zero to a, we come to the conclusion that
D(sk\x2)_ e*!42^2) - X%3
( k) (
U(k)
(21)
where U(h) is the amplitude valne of the electric potential on the surface x3 = a in the fc-th region. It is obvious (Fig. ) that U(2) = U0, and u(4) = U2. Comparing two physically equivalent definitions of electrical induction d3\x2 ), that is, equating each other the right-hand sides of expressions (19) and (21), we obtain formulas for calculating the amplitude value of the axial component of the electric field strength vector in areas 2 and 4:
E(i\x2) = —Uo/a, E)4)(X2) = —U2/a. (22)
Tims, the amplitude values of mechanical stresses in the of the rod under the electrodes of the primary and secondary electrical circuits of the transformer are calculated by the following formulas:
r(4)
_(2) ) _ VE (2) ) + * U0
^22 (x2) _ I £22 (x2) + e31-,
a
o^(x2)_ YES242)(X2)+ eh^.
(23)
da22)(x2) . 2 (k), s
+ po^ U2 (X2)
8X2
0,
(24)
coating of the surface x3 = a and on electroded areas of the rod, respectively.
For any method of fixing a piezoceramic rod on the conditional boundaries identified in the process of calculating the regions, the conditions for dynamic and kinematic conjugation of solutions (25) must be satisfied. These conditions are written as follows
1) _
42V3)
T(4),
) , ^(l2)_ 43V2) ,
4!)(
(3)
3) _
22 42)
.(4),
22 r(4)
1)
22
(2) 2
(4)
22 (5) í
(12)
), u^ (I4) _ «20)(14).
22
(3) 2
(5)
(12),
(26)
(27)
Compression-tensile deformations along the axis Ox2, i.e., the values £22)(x2) (k _ 1, 2, 3, 4, 5) are determined through the amplitude value of the longitudinal component u2k\x2) of the displacement vector of material particles of the fc-th section of the piezoceramic rod. In this case e2'2(x2) _ du2^(x2)/dx2 . Elastic displacements u2f)(x2) from the equilibrium position of the material particles of the rod satisfy the equation of steady-state harmonic vibrations
where p0 is the density of piezoceramics.
Substituting the definition of mechanical stresses a22h2)(x2) on different parts of the piezoceramic rod into equation (24), we write down its general solution in the following form
u21)(x2 ) = A1 sin 7x2 + B cos 7x2 , (2)
u2 )(x2) = A2 sin Xx2 + A3 cos Xx2, 43)(x2) = A4 sin 7x2 + A5 cos 7x2, (25) u(24)(x2) = A6 sin Ax2 + A7 cos Xx2, u2\x2) = sin 7X2 + Ag cos 7X2,
where A1 ..., Ag and B are the constants to be determined; 7 = p0/YD , X = p0/YE are wave rmmbers of longitudinal vibrations of material particles of piezoelectric ceramics in areas without electrode
Conditions at the boundaries x2 = ^d x2 = L are determined by the method of fixing the ends of the rod. Suppose that the ends of the rod are free from contacts with other material objects. It follows from this that, as required by Newton's third law, on the surfaces x2 = ^d x2 = L the following conditions must be fulfilled
a(2)(0) = 0, a(52)(L)=0. (28)
It is easy to calculate that the total number of conditions (26) (28) exactly corresponds to the number of sought constants.
To complete the construction of a mathematical model of a rod transformer, it is necessary to explicitly define the constants A1,..., Ag mid B. From the first boundary condition (28) it obviously follows that the constant B = 0. The remaining nine constants A1,...,Ag are determined from eight conditions ( ), (27) of conjugation of solutions (25) of equation (24) and the second boundary condition (28). Substituting the definitions of stresses o22(x2) a11^ longitudinal displacements u2h)(x2) into these conditions, we obtain the following system of linear algebraic equations
mijAj = Pi, i,j = 1, 2 ,..., 9, (29)
where mij are frequency dependent coefficients, the
numerical values of which are determined as follows:
m11 = —Yd7sin7I4; m12 = —YeXcosX£1;
TO13 = YeXsinX£1; (TO14 ^ TO1g) = 0 m21 = 0;
m22 = YeXcosXl2; to23 = —YEXsinXl2;
TO24 = —YD 7 cos 7I2; TO25 = YD7 sin 7I2;
(TO26 ^ TO29) = 0 (TO31 ^ TO33) = 0;
TO34 = Yd7cos7I3; TO35 = —Yd7sin7I3;
to36 = —YeX cos X£3; to37 = YEX sin X£3;
(TO38 ^ TO3g) = 0 (TO41 ^ TO45) = 0;
to46 = YeXcos X£4 ; to47 = —YEXsinX£4; to48 = —Yd7cos7I4; TO4g = Yd7sin7I4; TO51 = cos 1; TO52 = — sin X£1; TO53 = — cos AI1; (TO54 ^ TO5g) = 0 TO61 = 0 TO62 = sin AI2; TO63 = cos AI2; TO64 = sin 7I2; TO65 = — cos 7I2; (TO66 ^ TO6g) = 0 (TO71 ^ TO73) = 0; TO74 = sin 7I3; TO75 = cos 3; TO76 = — sin AI3; TO77 = — cos AI3; (TO78 ^ TO7g) = 0 (TO81 ^ TO85) = 0 TO86 = sin AI4; TO87 = cos X£4; TO88 = — sin 7I4; TO89 = — cos 7I4; (TOg1 ^ TOg7 ) = 0 TOg8 = YD7 cos 7L;
a
1
1
3
2
2
to99 = —Yd 7 sin 7L;
Pi = el-JJo/a-, P2 = -ehU0/a; P3 = e*31U2/a; Pa = -eiiU2/a- (P5 - P9HO.
From the design of the system of equations (29) it clearly follows that the stress-strain state of the rod in all its sections without exception depends simultaneously on the amplitude values of the potentials and U2, that is, on the potentials on the electrodes of the primary and secondary electrical circuits of the piezoelectric transformer. The latter means that the potentials ^^d U2 are related to each other by a linear dependence. This dependence is a formal (mathematical) description of the acoustic (mechanical) connection that exists in a solid piezoceramic rod between wave fields (material particles) in its various sections.
In order to determine the coupling coefficient between the amplitude values of the potentials U0 and U2, we construct an expression for calculating the potential U2.
In accordance with the design diagram shown in Fig. 1, it is necessary to write down that
9 x 9 of the coefficients at the constants Aj, i.e., the main determinant of the system of equations (29).
Substituting relations (33) into the definition (32) of the amplitude value of the electric charge Q2, we can write the following
where
M
q2 = u2^2 M + Uo*o M :
(41)2
(14 -13) X33
x [£4 (cos M4 — cos Al3) — £3 (sin M4 — sin Al3)j — 1;
^0 M =
(e3i)2
(14 — I3) X33
U2 = Z2I2,
(30)
where I2 is the amplitude value of the current in the conductor of the secondary electrical circuit of the piezoelectric transformer. Earlier it was shown [21] that I2 = — iwQ2, where Q2 is the amplitude value of the electric charge on the electrode of the secondary-electrical circuit. It is obvious that
x [£2 (cos AI4 — cos Al3) — (sin AI4 — sin Al3)j.
Since I2 = —iuC%Z2U2V2 (w) — iuClZ2U0V0 (w), then after substituting the amplitude value of the current into the definition (30), we obtain the following equality
U2 [1 + luClZ2^2 M] = — luClZ2U0^0 M , whence it follows that
U2
—So (w) Uo,
(34)
where S0 (w) is an acoustic feedback coefficient, the numerical values of which are calculated by the formula
14
Q2 = bj D(4) (x2,a) dx2.
10 M =
iuC%Z2^o M
(31)
1 + luC3A Z2^2 M' (35)
It is obvious (Fig. 1) that the amplitude value of the potential is determined as follows
Substituting the calculated formula (21) into definition (31), we obtain the following result
U2
Uo
UizjP H Si + z^ M
(36)
Q2 = b<e*3i «2 (l4) —ui (l3) —(l4 —l3) X33 = where Z^ (w) is an electrical impedance of the pi-
= 6e3i [A6 (sin Al4 —sinX£3)+A7 (cos Al4 —cos Al3)] —
— U2, (32)
where = b (l4 — l3) x33/a is the dynamic electrical capacitance of the section of the piezoceramic rod under the electrode of the secondary electrical circuit.
The constants A7 are determined from the
system of equations (29) as follows:
ezoceramic rod section under the electrode of the primary electrical circuit of the piezoelectric transformer.
Let us determine the electrical impedance Z^ (w). This operation actually completes the construction of a mathematical model of a rod piezoelectric transformer.
From the well-known Ohm's law for a section of an electrical circuit, it follows that
z? M = 7°
Uo
AR = -
e*3iUo
a—■
e
3iU2
a
£2, A7
e*3iUo
£3 +
e3i^2
(37)
£4, (33)
where £- = (A61 + A62VA0 ; £2 = (A63 + A64VA0 ; £3 = (An + A72VA0 ; £i = (A73 + A7a)/A^ Ai:j are algebraic additions (determinants of matrices of 8 x 8 size), which are obtained clS ct result of deleting the ith column and j-th row from the matrix of 9 x 9 size, composed of coefficients at constants Aj in the system A0
where Ii is the amplitude value of the electric current
in the conductor of the primary electrical circuit. Since
Ii = — iuQu and the electric charge Qi is determined
(2)
through electric induction D3 ) (x2,a), then
Ii = — iwbe^
3i
,(2)
(I2) —
(2)
i) + iuC% Uo, (38)
where C2 = b
— li) X33/a is the dynamic electrical capacitance of the section of the piezoceramic rod
2
2
64
Ba:¡i.;io k. tí., boiiaapoiiko M. o., Xjiiuiiufi tí. tí., Tomoiiko M. T., Tomoiiko tí. 1.
under the electrode of the primary electrical circuit of the piezoelectric transformer.
The coefficients A2 and A3, which determine the longitudinal displacements of the material particles of the rod in the sections x2 = f^d x2 = l2, are calculated by the following formulas
A2 = -
4iuo
Ci -
-31
U2
C2,
A3 =
e3i^Q
C3 +
3i
U2
(39)
C4,
where C1 = (A21 + A22VA0 ; C2 = (A23 + A24)/Ao ; C3 = (A31 + A32VA0 ; C4 = (A33 + A34)/Ao .
Substituting the definitions (39) of the constants A2 mid A3 into the formula for calculating the displacement 42)(x2) of the material particles of the rod, and the obtained results into relation (38), we obtain the following result
h = -i(w),
(40)
where
a
a
a
a
(w)=
(e*i)2
- Il) X33
{- [Ci - Sq (w) C2] (sin X¿2 - sin Ali) + [C3 - Sq (w) C4] (cos AI2 - cos Ali)} - 1.
Taking into account the definition (40) of the amplitude valne of the current in the conductor of the primary electrical circuit, from the relation (37) we find the electrical impedance
(w)
1
-iwC¡2 (w)'
after which we obtain a calculation formula for the amplitude value of the potential U0 from the relation (36)
U1
Uq =
1 - iwC23Zi^i (w)'
(41)
k (w, n) = ^ =__^_.
( ' ) Ui 1 - iwC22Zi'fyi (w)
(42)
K(ro ,n) 5
X
3
0 10 20 30 40 50 60 70 f, kHz
Substituting expression (41) into relation (34), we find the calculation formula for the potential U2, after which we can write expression (1) to calculate the transformation ratio K (w, n) in the following form
0
1
2
3
4
5 Q
The analytical structure (42) IS 3. mathematical model of a rod piezoelectric transformer, the principle of operation of which involves the uso of longitudinal vibrations in a prismatic rod, the design diagram of which is shown in Fig. 1.
4 Discussion and experimental confirmation of simulation results
For practical confirmation of the obtained simulation results, a practical experiment was implemented [23]. It was carried out at the research stand [24] in the frequency range 0-80 kHz and was used to confirm the results of mathematical modelling with the above sequence. The experiment was carried out under the condition of observing the total measurement error, which did not exceed 8.5%, Fig. 2 (dashed line).
Fig. 2. Calculated (solid line) and experimentally obtained (dashed line) curves of the frequency dependence of the modulus of the transformation ratio of a piezoceramic transducer
The results of mathematical modelling (the transformation ratio K (iv, n)) of rod transducer with a square cross-section made of PZT type piezoelectric ceramics were acquired according to the expression (42) and are presented in the form of dependences (solid curves) shown in Fig. 2. The following parameter values were taken for calculations: cE
ii
°i2
33
= 62 GPa
18 c /
33 ; e3i
-7 c/
= 112 GPa; 106 GPa; pQ = 7400 ka/ms; ci „. ^ _ 8, 85 • 10-9 F '
X33
Q0 = 80; a = 3 = 9 toto; L = 90 toto. The calculations were performed at the frequencies of the first two electromechanical resonances.
When comparing the results of mathematical modeling and the results of experimental measurements (Fig. 2), one can see that the calculated results agree with the actual values of the amplitnde-freqnency characteristic. At the same time, the discrepancy between the results of mathematical modeling and the experimentally obtained data at the same values of the operating frequency in the range up to 50kHz is 7.8-8.5%, and in the range of operating frequencies of 50-80 kHz it does not exceed 5.1%).
The obtained calculated results are extremely important from a practical point of view, since for the overwhelming majority of real situations the use of piezoelectric transducers becomes impossible, due to the impossibility of obtaining a posteriori information on the actual values of the transformation ratio modulus in the process of experimental use of such transducers. Therefore, the development of a mathematical model of rod-type piezoelectric transducers, the implementation of which can be automated with the involvement of modern microprocessor tools and software, will make it possible to quickly and with sufficient accuracy-build the dependence of the transformation ratio on the cyclic frequency and operating parameters of the piezoelectric transducer, which will improve the development process of such transducers, and will also reduce the time and cost of the technological process of their manufacture.
Thus, as a result of the comparison of the calculated and experimental curves of the frequency dependence of the modulus of the transformation ratio, the possibility of using the mathematical model developed by the authors for calculating the main operational characteristics of piezoceramic rod-type transducers in the process of designing acoustoelectronic devices has been proved.
Conclusions
The sequence of constructing a mathematical model of piezoelectric transducers of a rod type (rod piezoelectric transformer) is presented, the result of which is finding a complex-valued function K (w, n) on the cyclic frequency and a complex of parameters (geometric, physical-mechanical, electrical) of a pi-ezoelect ric t ransformer.
Analytical dependences was obtained that made it possible to determine the electrical impedance and the amplitude value of the potential that arise in the electrical circuit of a piezoelectric rod-type transducer with a square cross-section from piezoelectric ceramics of the PZT type.
A comparison of the calculated and experimental values of the frequency dependence of the module of the transformation ratio of the piezoceramic transducer was carried out. which showed a high convergence between these data (the discrepancy between the results of mathematical modelling and the experimentally obtained data at the same values of the operating frequency in the range up to 80 kHz did not exceed 8.5%).
The above results were obtained as part of the scientific and technical (experimental) project "Development of a highly efficient mobile ultrasonic system for intensifying the extraction process in the manufacture of concentrated functional drinks", which is carried out at the expense of the general fund of the
state budget (state registration number: 0121U109GG0. registration date: 12.03.2021).
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Математичне моделювання п'езо-
електричних перетворювач1в стриж-
ньового типу для пристро*1в акусто-електрошки
Базию К. В., Боидарсико М. О., Хлгвиий В. В., Томеико М. Г., Томенко В. I.
Робота присвячепа особливостям побудови та до-сл1джеппя математичпих моделей п'езоелектричпих пе-ретворювач!в стрижпьового типу, як! зпаходять широко застосуваппя в р1зпомаштпих пристроях акусто-електрошкп (пдроакустичпих засоб!в впявлеппя ц!лей. ультразвукового перуйшвпого контролю, медпчпо! д1-агпостпкп. тощо). На в!дмшу в!д 1спуючпх математичпих моделей п'езоелектричпих перетворювач!в (па основ! амшнтудпо-фазових залежпостей. резопапепих п'езоелектричпих перетворювач1в. екв1валептпих схем та шших). запропоповапа математичпа модель дозволяв встаповити залежп1сть. що е математичпим опи-сом акустичпого зв'язку. який 1спуе в суц!лыюму п'езокерам1чпому стрижш м!ж хвильовими полями па його р!зпих д!ляпках.
Наведено алгоритм розрахупку математичио! моде-л! п'езоелектричпих перетворювач!в стрижпьового тгшу. що базуеться па визпачеш коефщ!епта траисформац!!, який випикае при зворотпому п'езоелектричпому ефе-кт1. Отримаш апал1тичп1 залежпост!. що дозволяють визпачати електричшш 1мпедапс та амшнтудпе зпаче-ппя потепц1алу в електричпому кол! п'езоелектричпого перетворювача. Показано, що гц залежпост! лежать в основ! виразу для впзпачешш коефщ!епта трапсформа-ид1 К(ш, П), який е математичиою моделлю стрижиевого п'езоелектричпого трансформатора. При цьому. пргш-цип до такого п'езотрапсформатора передбачае викори-стаппя поздовжшх колгшапь в призматичпому стрижш.
Наведеш результати проведепого математичпого мо-делюваш1я па приклад! стрижиевого перетворювача з квадратпим поперечпим перер!зом !з п'езоелектричпо! керам!к! тгшу ЦТС. Проведен! пор!впяппя розрахо-вапих та експеримепталыго отриманих зиачень частотно! залежпост! модуля коеф!ц!епта трапсформац!! п'езокерам!чпого перетворювача показали високу зб!-жшеть м!ж пими (розб!жп!сть м!ж результатами математичпого моделюваппя та експеримепталыю отрима-пими дапими для одпакового зпачмшя робочо! частоти по перевищуе 8,5%).
Клюноог слова: п'езоелектричпий перетворювач: акустоелектрошка: математичпа модель: генератор еле-ктричних сигпал!в
Математическое моделирование пьезоэлектрических преобразователей стержневого типа для устройств аку-стоэлектроники
Базило К. В., Бондаренко М. А., Хливной В. В., Томенко М. Г., Томенко В. И.
Работа посвящена особенностям построения и исследования математических моделей пьезоэлектрических преобразователей стержневого типа, которые находят широкое применение в различных устройствах акусто-электроники (гидроакустических средств обнаружения целей, ультразвукового неразрушающего контроля, медицинской диагностики и т.д.). В отличие от существующих математических моделей пьезоэлектрических преобразователей (основанных на амплитудно-фазовых зависимостях, резонансных пьезоэлектрических преобразователях, эквивалентных схемах и т.п.), предложенная математическая модель позволяет установить зависимость, являющуюся математическим описанием акустической связи, которая существует в твердом пьезоке-рамическом стержне между волновыми полями на его различных участках.
Приведен алгоритм расчета математической модели пьезоэлектрических преобразователей стержневого типа, основанный на определении коэффициента транс-
формации, который возникает при обратном пьезоэлектрическом эффекте. Получены аналитические зависимости, позволяющие определять электрический импеданс и амплитудное значение потенциала в электрической цепи пьезоэлектрического преобразователя. Показано, что эти зависимости лежат в основе выражения для определения коэффициента трансформации К(ш, П), который является математической моделью стержневого пьезоэлектрического трансформатора. При этом, принцип действия такого пьезотрансформа-тора предусматривает использование продольных колебаний в призматическом стержне.
Приведены результаты проведенного математического моделирования на примере стержневого преобразователя с квадратным поперечным сечением из пьезоэлектрической керамики типа ЦТС. Проведены сравнения рассчитанных и экспериментально полученных значений частотной зависимости модуля коэффициента трансформации пьезокерамического преобразователя показали высокую сходимость между ними (расхождение между результатами математического моделирования и экспериментально полученными данными для одинакового значения рабочей частоты не превышает 8,5%).
Ключевые слова: пьезоэлектрический преобразователь; акустоэлектроника; математическая модель; генератор электрических сигналов
