CC BY
8
Visnyk N'l'UU KP1 Seriia tiadiotekhnika tiadioaparatobuduuannia. 2023, Iss. 91, pp. 37—45
UDC 534.232.082.73
Mathematical Modelling of Disk Piezoelectric Transducers for Acoustoelectronic Devices
Bazilo С. V., Bondarenko M. O., Usyk L. M., Faure E. V., Kovalenko Yu. I.
Chorkasv State Technological University, Cherkasy, Ukraine
E-mail: ma-.r-.rium23&gmail. com
This study has presented an algorithm for assembling, solving, and analyzing the results obtained by mathematical modeling of the disc piezoelectric transducers, which are widely used in hydroacoustics, microelectronics, microcircuit engineering (for example, as components of receiving antennas of hydroacoustic communication devices). The models developed in this study enable us to establish dependencies, which represent a mathematical description of the electroacoustic connection between the wave fields in different sections of the disc piezoelectric transducers. Analytical dependences obtained by mathematical modelling make it possible to establish the electrical impedance and quality factor together with the amplitude values of the electric charge and current on the electroded surfaces of the piezoelectric disk, subject to the inverse piezoelectric effect conditions. A complete calculation of the problem of harmonic radial oscillations of disc piezoelectric transducers allowed the authors to significantly expand the list of physical and mechanical parameters of the piezoelectric material, which had been previously determined experimentally. The research has revealed the dependence of the change in electrical impedance on the values of the electromechanical coupling coefficient, the wave number of elastic oscillations, and the Voigt indices. The study has also determined a high agreement between the electric impedance modules of discs made of lead zirconat.e titanate PZT piezoelectric ceramics with and without the piezoelectric effect (the difference between the impedance values in these cases did not exceed 18%).
Keywords: piezoelectric transducer: acoustoelectronics: mathematical mode: impedance: disk element DOI: 10.20535/RADAP. 2023.91.37-45
Introduction
Piezoelectric components and equipment designed with piezoelectric components have found multiple applications in modern science and technology [1]. Indeed, literally every research and technology field benefits from applying piezoelectric transducers nowadays. This fact is due to their high accuracy, reliability, manufacturability, multifunctionality, as well as their ability to miniaturize and integrate with various microelectronic and micromochanical components.
Energy independence from external sources may-become an absolute advantage of piezoelectric components in the near future. Moreover, in some cases, piezoelectric components, when being an element of a device, can generate electric current for other electronic components of the same device [2].
At the same time, having analyzed the market of modern piezoelectric transducers (according to marketing research of the companies taking the lead in manufacturing piezoelectric components intended for various purposes, such as: Zhejiang Jiakang Electronics Co., Changzhou Keliking Electronics Co., PI Ceramic GmbH and others [3,4]) we have established that a considerable share (about 56%) of the entire variety of
form factors of piezoelectric transducers is accounted for disc piezoelectric transducers. This share amounts to nearly 615 million US dollars as of the third quarter of 2022.'
However, further developments in the technological base of piezoelectric transducers with introducing expanded functional capabilities would not be possible without strengthening the scientific and theoretical foundations of piezoelectric technology by improving methodical, technological, and mathematical support.
Improvements in the mathematical support that underlies the piezoelectric transducers development theory, followed by their applied implementation as ready-made components, offer new methods to calculate, design and model disk-shaped piezoelectric transducers [5].
1 Relevance of the research based on the publication analysis results
Disk piezoelectric transducers serve as soundreceiving and sound-radiating devices, various sensors, and as components of receiving antennas
38
Bazilo С. V., Bondarenko M. О., Usyk L. M„ Faure E. V., Kovalenko Yu. 1.
in hydroacoustic communication devices [6. 7]. As indicated in the studies [8. 9]. the standard design of piezoelectric stack transducers, as a rule, involves compiling disk elements into a single design, which expands the functional properties of such type of multiarray transducers, hence improving the operational properties of devices where these transducers are used. The study [8] presents findings related to the functional features of a new three-layer circular piezoelectric transducer, and first introduces an analytical model containing closed-form equations, which are important tools for predicting and optimizing the transducer’s operation. Employing hence both the plate theory and the constitutive equations of piezoelectric materials, an analytical formula was found to describe the deflection of the transducer as a function of electrical loads (electromechanical characteristics of the transducer). The authors have conducted verification tests to demonstrate that the results obtained with the developed solution correspond satisfactorily to both literature and numerical data. Based on the obtained analytical model, the influence of selected dimensionless variables on the transducer’s performance has been investigated. It has been demonstrated that parameters including the dimensions and mechanical properties of the piezoelectric disc significantly affect the transducer’s performance. In addition, the authors of the study [9] discuss the results of investigating piezoelectric stack transducers connected in varied sequences. Designing and developing piezoelectric matrix options were implemented for series, parallel, series-parallel, and parallel-series connections of piezoelectric transducers. Modeling these options and analyzing the obtained results were conducted with the electronic circuit simulation software package PSIM. Verification of the simulation results has confirmed the feasibility to determine such operating modes of the piezoelectric matrix that provide the optimal output power, which can satisfy the minimum energy requirement for powering the device with the considered variant of the configuration of piezoelectric components with low load.
A number of surveys, namely those conducted by C. Bazilo. O. Petrishchev. I. Yanchevskiy. A. Zagorskis and others, focus on theoretical and applied research of piezoelectric technology and investigate the physical processes that occur in disk piezoelectric transducers with differently shaped electrodes [10 12]. For example, in the study [10]. piezoelectric sensors for environmental monitoring are modeled and investigated. This analysis proves the absence of reliable and well-founded methods to build mathematical models of disc piezoelectric transducers, which could be used as a theoretical basis for calculating the characteristics and parameters of this class of functional components used in modern piezoelectric electronics. Analysis and further COMSOL Multiphysics software platform modeling of the physical processes that occur in
piezoelectric transducers have established a significant expansion of the range of their mechanical characteristics and an increase in sensitivity when using these transducers in environmental monitoring tasks. Surveys such as [11. 12] undertake research of electromechanical and energy characteristics of piezoceramic elements during radial movements, which occur when their operating frequencies are close to resonance/anti-resonance values.
At the same time, the studies [13.14] that focus on the possibility of modeling variously shaped piezoelectric transducers using equivalent circuits have demonstrated that it is impossible to consider mechanical processes and phenomena in piezoceramics. while establishing a connection between these processes and the electromechanical characteristics of piezoceramic material is complicated.
Thus, having analyzed the above-mentioned scientific publications, along with a number of publications [15 18] dedicated to the mathematical description of physical processes in piezoelectric disc transducers, the authors of this article established the absence of unified mathematical models of piezoelectric disc actuators. Any clear sequence of calculating their technical and operational characteristics is also absent.
Therefore, we consider development of a mathematical model of piezoelectric disc transducers and experimental confirmation of the results obtained with the help of the developed models as a topical problem, the solution of which is the main goal of this research.
2 Problem statement for modelling of a thin piezo-
ceramic disk in the mode of thickness oscillations
Figure 1 demonstrates a disc with PZT-type piezoceramics polarized in direction Ox3. Its surfaces x3 = 0 x3 = — a are metallized f ]. The surface x3 = —a has zero potential, and the potential Ф (t) = U0elut is supplied to the surface x3 = 0 from an external generator, where U0 is the voltage amplitude; i = \f—T; ш is the circular frequency of the charge sign reversal in the electric potential; t is time.
We consider the disk sufficiently thin, i.e., the strong inequality a/R С 1 takes place. Apart from that, we consider that the disk is isolated from any mechanical contacts with other material objects. For definiteness, we assume that the disk is in vacuum.
Mathematical Modelling of Disk Piezoelectric Transducers for Aeoustoeleetronie Devices
39
Fig. 1. Calculation scheme of a piezoceramic disk
The action of an externally applied electric potential difference induces an external electric field in the disk, which acts on the piezoceramic ions through the Coulomb forces, which means that an inverse piezoelectric effect appears in the volume of the disk. Here we assume that the frequency ш is such that the spatial inhomogeneity scale of the stress-strain state in the disk is commensurate with its thickness. Since the inequality a/R C 1 is satisfied, one can safely neglect the influence of the lateral disk boundary on the characteristics and parameters of its stress-strain state and consider that the displacement vector u (xk) of the material particles in the piezoceramics is determined on average by the axial component u3. If we take into consideration that the component u3 solely depends on the coordinate x3, then the real stress-strain state of the disk is reduced to a uniaxial deformed state. A uniaxial deformed state is a stress-strain state of an elastic body in which the strain tensor has only one nonzero component. In the model situation formulated above, such a component is the quantity £33 = du3/dx3 . The two components that define compression and tension along the axes Oxi and Ox2, that is, along the radial and circumferential axes of the cylindrical coordinate system, are equal to zero.
Under the above conditions, the electrical impedance Zei(u) of a disk oscillating through its thickness should be determined.
3 Construction of a mathematical model of thin piezoceramic disk in the mode of thickness oscillations
As a rule [20], qualitative and quantitative parameters of the stress-strain state of an oscillating piezoceramic element are determined before calculating the electric current in the conductor that connects the disk’s electroded surface to the electric potential generator (Fig. 1). To perform these estimates, we note the generalized Hooke’s law in the inverse formulation [211
&ij ^ijkl^kl + ^kij Ek , (1)
which we obtain from the Taylor’s expansions of the functions £jj, Dm mid S
(гы, Ek and T. The symbol sftkl in expression (1) denotes the component of the piezoceramics’ elastic compliance tensor. The symbol dkij denotes the piezoelectric charge modulus. There exist clearly defined relations between the sfjkl rnid dkij values, as well as between the elastic moduli с^к1 and the piezoelectric moduli ekij
c
E E
afi * fi-y
d
dkf3
ekas
E
(2)
where a, ft, 7 are Voigt indices; 5ai is the Kronecker symbol.
From relation (1) it follows that
£i = sfi^i + 4^2 + 4a3 + d3iE3 = 0, (3)
£2 = sfi ai + 4^2 + 43°з + ^32 E 3 = 0, (4)
£3 = 4 ai + s32a2 + 4a3 + d33E 3 = 0. (5)
In relations (3) (5), the notation with Voigt indices is used for the components of strain and stress tensors. Shear deformations £p (ft = 4, 5, 6) due to the physical content of the problem identification are equal to zero. Since matrices sEp and dka are similar to matrices and ek in their structure and the relations between
d3i = d32. In this
case, it follows from equations (3) and (4) that
CTl = 02
d3i
qE + qE *11 + *12
E3
S
E
13
4 + s
E
12
^3.
(6)
Substituting the stresses o1 mid o2 determined byexpression (6) into relation (5), we obtain the notation
_ V3 .
£3 = +
d33 - 2
■4^31
qE + ~E 611 + *12
Eз,
(7)
where ME is the elastic modulus for the uniaxial deformed state mode (compression tension through the thickness) of the disk. Wherein
1
~МЁ
kE -ь 33
2(4)2
4 + 4*'
(8)
Generally,
follows
we can note the Hooke’s law (7) as
^3
where
3U)
M £3 - e(e)E3,
s13^31
d,33 - 2
qE + qE *11 + *12
ML
(9)
(10)
We assign the value e(^) to represent the piezoelectric modulus for the mode of uniaxial compression tension of the disk.
ME mid e(v)
through the elastic moduli cEp modules eka.
40
Bazilo С. V., Bondarenko M. О., Usyk L. M„ Faure E. V., Kovalenko Yu. 1.
The elastic moduli matrix for a polarized through the thickness PZT-piozocoramic plate is rioted as follows
rE rE C11 c12 rE c13 0 0 0
cE c22 cE c23 0 0 0
r-E 1 = rE c33 0 0 0
^afi \ rE C44 0 0
rE c55 0
„Е
c66
(И)
cfi = c|2; cf3 = cf3 cf4 = c||.
The elements of the elastic compliance matrix sf^ are calculated from the elements of the elastic moduli matrix in the following notation
s
E Afi
(—1)л+м
мЛм
Aq ’
(12)
M\p is the algebraic cofactor at the element that is located at the crosshairs of the A-th raw
and the Aq = cfx cf!c|3 — 2(cf3)2
c12 сз3 — 2(c^3)2
The elastic compliances included into the definitions of ME and are determined in accordance with expression (12) as follows
rE
c12
S
s
s
s
E
11
E
12
E
13
E
33
с?1сз3— (ci3 /Aq;
«12 «33 — (cf3)2 /AQ; [c12 «13 — «13 «И ] / AQ;
(«fT — (cf3)2l/Ao.
(13)
Substituting relations (13) into the definition (8) of the elastic modulus ME results in the following relation
M1
c33AQ
(cf1 — cf2 C33 (Cf1 + cf2 ) — 2 ( cf3 )2
(14)
According to the expression (2). the piezoelectric moduli d3^id d33 are defined by the following relations
jp jp jp jp
«31 = e3 a sa1 = e31 su + s21 +e33«31;
E EE E
«33 = e3a sa3 = e31 + s23 +e33«33.
(15)
Thus, under uniaxial compression tension of a polarized through thickness piezoceramic plate (disk) with the elastic properties of a transversally isotropic solid (cf2 = cf3), the following notation of the generalized Hooke’s law can be used
«3 = «33^3 — 033E3. (17)
In order to determine the resulting electric field strength E3 that we have in the relation ( ), we consider the condition when free electricity carriers are absent in the volume of the deformable piezoceramic. For the case under consideration, we obtain dD3/дх3 = 0 from the general formulation of this condition div D = 0, where D3 = e33e3 + x33E-Since e3 = du3/dx3 and E3 = — дФ/дх3 , where Ф is the electric potential of the resulting field within the volume of the deformable piezoelectric, we note the expression for the vertical component of the electric induction vector
D3
du3
633
0X3
X33
д Ф
8x3
(18)
where D3, u3 and Ф are amplitude values of harmonically time-varying quantities.
The vertical component D3 of the electric induction vector does not depend on the values of the coordinate x3. Therefore, by integrating the left and the right parts of relation ( ) with respect to the variable x3
within the range from —a to 0, we obtain
D3 = — [«3 (0) —«3 (—«)] — ^ [Ф (0) — Ф(— a)]. (19)
a a
Obviously, the closing square bracket in the formula (19) is equal to the amplitude of the electric potential Uq (Fig. ) supplied to the electroded surface x3 = 0 from an electrical signal generator. Therefore
D3 = — [«3(0)— «3 (—«)] — Uq. (20)
a a
By substituting the right side of the expression (20) into the left side of the above definition of the quantity D3 we obtain
e33p~ +x33^3 = — [«3(0) —«3 (—a)] — ^33Uq, 0x3 a a
When noting the relations (15), we considered the notation of the piezoelectric moduli matrix е^а
|cfca|
0 0 0 0 615 0
= 0 0 0 e24 0 0
631 632 633 0 0 0
e32 and e15 = e24-
(16)
where e31
Substituting expressions (13), (14) and (15) into the definition ( ) of the piezoelectric modulus e^) we
e(v) = e33
that cf2 = cf3
um), from expression ( ) it follows that ME = c33.
whence it follows that
E3
e33 du3 + 633 X33 dx3 ax33
[«3(0) — U3 (—a)]
a
(21)
In the formula (21) for calculating the vertical component of the resulting electric field strength vector, the first two terms determine the internal electric field that arises in the volume of the deformable piezoceramic due to the ions being displaced from the equilibrium position. The third term determines the strength of the electric field created by an external source, that is, the electrical signal generator.
Mathematical Modelling of Disk Piezoelectric Transducers for Aeoustoeleetronie Devices
41
Substituting the relation (21) into the generalized Hooke’s law (17) produces
way. The final result of solving this system of equations is the notation
a3 = eg p3 - [«a (0) -«3 (-«)] + — U0, (22)
oxз ^х3з a
where cg3 = eg (1 + Kg) is the elasticity modulus in к iduction constancy (equal to
zero); K3 = eg/(x33cg3) is squared electromechanical coupling coefficient in the mode of thickness oscillations in a polarized through thickness piezoceramic
K3 < 0, 5.
The elasticity modulus C33 counts in the consistent (coherent) action of elastic forces and Coulomb forces in the volume of the deformable piezoelectric. Therefore, cg3 > eg.
The amplitudes of both the stress tensor a3 and the displacement vector u3 components that vary harmonically in time must satisfy Newton’s second law in differential form, which, as applied to the situation under consideration, is a notation
A = В =
e33^0 (1 -cos 7a)
where A = sin 7a
С33Л ja 7
633^0 sin 7a
С33Л ja 7
Kl tg(7«/2 )
(27)
= sin ja ■ Л0
1+Kf (70/2)
It then follows that the amplitudes of the harmonically time-varying displacements u3 of the piezoceramic disk’s material particles, which satisfy the fundamental laws of mechanics, i.e., Newton’s second and third laws, are to be determined by the following expression
«3 = —■
e33Uo
сз3Л
(1—cos 70) sin 7 a
cos 7^3 + sin 7^3
7a
70
(28)
Substituting expression (28) into definition (20) of the vertical componentwe obtain
do-----+ Pou2«3 = 0 Ухк € V,
ox 3
(23)
D3
X33U0
0Л0 ’
where p0 is density; V denotes the volume of the piezoceramic disc.
By substituting the definition (22) of the resulting voltage a3 into the equation ( ) we compose
-2«3
+ 7 2«3
0 Ухк € V,
(24)
where Л0
1 -
К2
tg(-ya/2 ) (7o./2 ) ‘
I0
1+K2
The amplitude value harmonically changing in time within the conductors is determined through the value D3
I0 = -iuSD3 = шСЪ , Л0
(29)
where 7 = (xg/cgfpg is the wave number of elastic vibrations in the volume of the piezoceramic disk. Indeed, the solution of the equation (24) is
«3 = A cos 7ж3 + В sin 7^3, (25)
where C0 = S\33/a is the dynamic electrical capacitance of the electroded piezoceramic disc.
The formula below defines the electrical impedance Zei (ш) of an oscillating piezoceramic disk by Ohm’s law for a section of an electrical circuit
where A and В are the constants to be defined from the boundary conditions, which in their physical essence represent Newton’s third law in differential form.
With regard to the problem being solved, it then follows that
а3|жз = (0,-a) =0 Ухк € S, (26)
where S denotes the metallized surfaces x3 = (0; —a) (Fig. 1) of the oscillating disk.
Let expression (25) for calculating the amplitude of the displacement vector’s vertical component «3 be substituted into the definition (22) of the stress tensor component a3 amplitude harmonically varying in time. In the result obtained, we equate the coordinate value to zero first, and then assign x3 = —a, we equate the expressions obtained to zero, as required by condition (26). In this case we obtain a system of two algebraic equations that contain the two required coefficients, A and B.
The indicated system of equations is solved with respect to the required coefficients A mid В in a unique
Ze‘<“> = T0 = Л"' <30)
Next., we consider the physical content of the results obtained.
If the dielectric placed between the electroded surfaces does not possess piezoelectric properties, then K32 = 0 mid Л0 = 1. The expression (30) acquires the meaning of a formula widely used in electrical engineering to calculate the electric capacitance reactance of a flat capacitor. When K3 = 0, the frequency-dependent change in the electrical impedance Zei(u) becomes much more complicated due to alterations introduced to the function Л0.
Figure 2 contains a graph reflecting the dynamics of the Л0 function calcula • K3 = 0, 5. With
^ 0, we have Л0 = 1/( 1 + Kg). As the argument
ja
definite frequency, which is marked with symbol шг in Figure , the function becomes Л0 = 0. It then follows at once that Zei (шг) = 0. At a frequency of шг, when Л0 =0, the displacement amplitudes of the
42
Bazilo С. V., Bondarenko M. О., Usyk L. M„ Faure E. V., Kovalenko Yu. 1.
piozocoramic disk material particles increase indefinitely. This phenomenon is caused by the material particles being displaced within the volume of the deformable disk that generate an internal electric field. the direction of which coincides with the external electric field. In other words, the internal electric field complements the external electric field, hence the strength of the resulting electric field increases. This leads to an increase in the displacement amplitudes of the piozocoramic material particles. In its turn, the strength of the resulting electric field increases again, and so do the displacement amplitudes of the material particles. In the case when an ideal generator of a harmonically time-varying electric potential difference is present, the displacement amplitudes of material particles in an ideal (without loss of elastic vibrations energy) piezoelectric increase indefinitely at шр frequency. In this case, infinitely large polarization charges are produced and electric currents of infinitely large amplitude flow through the conductors. The described situation corresponds to the short circuit mode, i.e., a Zei(ur) = 0 condition in the circuit of an ideal electric potential difference generator. Naturally, the electrical load, that is, the oscillating piozocoramic disk, consumes the maximum possible amount of energy from the electrical signal generator in this case. The physical state of any system that operates on the energy produced by an external source, in which the system consumes the maximum possible amount of energy from the source, is called resonance. For this reason, <xr frequency is called the resonance frequency, and the physical state of the oscillating disk is called the electromechanical resonance [22].
Fig. 2. Frequency graph for the Л0 function
In a real-life situation, currents of finite amplitude flow along the conductors of the electrical circuit, which is shown in Figure 1. We can define two factors causing this effect. First, an electrical signal generator has a finite internal resistance Rr, while an ideal electrical voltage generator has zero resistance. Secondly, the piozocoramic material absorbs the energy of elastic vibrations and converts it into heat.
The integral estimate of energy losses in the volume of a dynamically deformed solid is a dimensionless quantity Q0, which is called the quality factor. The numerical value of the Q0 parameter is inversely proportional to the amount of energy loss in the material during the sign inversion of the stress-strain state. In this case, the wave number 7 becomes a complexvalued function of the frequency and is determined as 7 = 70 (1-i/(2Qo)) 70 = ш/ °c^3/po ;
0 C33 is the static modulus of elasticity. The complex number 7 transforms the function Л into a category of a complex variable function, which at шг frequency-lias a non-zero finite value. In this case, the electrical impedance of the oscillating disk acquires Z0 value, which is an active resistance by its content. This statement has been fully confirmed by the experimentally observed facts.
If the symbol e denotes the value 1/(2Q0), which is obviously a small parameter, then the expansion of the function Л0
small parameter e with an accuracy of e2 and higher orders of magnitude produces the following expression for the electrical impedance at the frequency of the first electromechanical resonance
У ( ) у Кз sin Xp cos Xp\
el (шр)= 0 =2(1 + K22) Q0txpCe0 xpcos2Xp ’
where xp = шра j ^2y/0c33/p0 j .
The latter relation defines the numerical value of the piezoceramics’ quality factor at the frequency of the first electromechanical resonance
^ -^3 [^p sin cos ^p\
0 2(1 + K3) Z0UpCq xpcos2Xp
For PZT-type piezoceramics, the quality factor Q0 has a value of 100.. .200 relative units at frequencies (1...2) MHz.
At the frequency ша, when 70/2 = -n/2, the function Л0 tends to infinity, accordingly, |Ze;(wa)| ^ to. In this case, the amplitude of the electric current in the conductors of the electrical circuit (Fig. 1) tends to zero. The piezoelectric disk ceases to consume energy frorn the source, i.e., from the generator of the electrical potential difference. Hence ша is called the frequency of electromechanical antiresonance. The physical essence of electromechanical antiresonance lies in the fact that the polarization charge in the volume of the oscillating disk completely compensates for the electric charge induced by the electric potential difference generator on the electroded surfaces of the piezoceramic disk. Electromechanical antiresonance is the result of the algebraic addition of the polarization charge and the electric charge induced by the generator. If there is no external generator of electric potential difference, the electromechanical antiresonance is not observed. This is a typical situation in the case of using piezoelectric elements as an elastic vibrations receiver. In a real-life
Mathematical Modelling of Disk Piezoelectric Transducers for Aeoustoeleetronie Devices
43
situation, the electrical impedance of the piezoceramic disk at ша frequency exceeds the electrical impedance Z0 at шг frequency by almost three orders of magnitude.
4 Discussion and experimental confirmation of simulation results
Figure 3 shows the electrical impedance modulus of a PZT 19 piezoceramic disc. The disc’s niateri-c,®=106 GPa; p0 3;
e33=18C/m2; X33 = 1000Xo; Xo=8.85T0-12 F/m; the ceramics quality factor is Q0=100. The radius R of the disk exceeds the thickness a of the disk ten times (Д/а=10). The electrical impedance modulus values in ohms are plotted on the ordinate. The dimensionless quantity ja/(2-к) values are plotted on the abscissa. The inset in the figure field shows a change in the Zei(u) modulus in the vicinity of the electromechanical resonance frequency.
Fig. 3. Graphs for the electrical impedance modulus of the PZT-19 piezoceramic disc, calculated with (solid curve) and without (dashed curve) piezoelectric effect
It follows from the above report that the numerical values of the resonance and antiresonance frequencies are determined by the values of the piezoceramic material constants. From this obvious fact follows the possibility of solving the inverse problem, that is. determining the measured values of the electromechanical resonance frequencies and antiresonance by recalculating the values of the piezoceramic material constants. This possibility determines the relevance and practical significance of mathematical modeling and the subsequent experimental study of oscillating piezoceramic elements and their electrical impedance.
Suppose that for the disk under study, an experimental determination of the frequency-dependent change in the electrical impedance modulus Zei(u) in a wide frequency range is performed and a graph similar to that shown in Figure 3 is performed.
We consider the following values to be experimentally determined:
— disc dimensions a, R, in meters;
— disc mass m, in kilograms;
— frequencies of the first resonance fr and the first antiresonance fa in hertz, measured to the nearest hertz;
— electrical impedance modulus Z0 at the frequency of the first electromechanical resonance, in ohms;
— electrical impedance modulus Zei (шп)
in ohms at low frequency fn, where fn C f(p\ where frP^ is the frequency of the first radial resonance, whereas = fr/20 .
The first step will be to del ie the density
of the piezoceramics p0 = m/ (naR2). The known value Zei(un)
capacitance of the disc C3 = 1/[2nfnZe; (шп)\. In this case we naturally assume that Л0 « 1. The dielectric constant is d mined from the known capacitance X33 = aCy(KR2).
The fa
resonance satisfies the condition nfaa/vD = Д2, where vD = y/0cD3/p0 is the propagation velocity of elastic perturbations of compression tension in the disk material. The condition noted above determi-v D
modulus 0сД = (2fao)2p0
jra/2 = nfra/vD = nfr/(2fa) the condition results in the following notation for the electromechanical resonance
whence
Kj tg [nfr/(2fa) ]
1+Д1 ^ [lTfr/(2 fa )]
0,
k2
[*fr / (2fa)\
tg [-Kfr/(2fa)\ — [ъfr/(2fa)]
(32)
The known values of the squared electromechanical coupling coefficient Д3 md dielectric constant \з3 define the dielectric constant x33 = Хз3/( 1 + K3)- The known values 0сД and Д| define the elastic modulus 0C33 = 0сз3/(1 + K3^^^ace K3 = e\oJ{0),
e33 =
K3V°^3xh- From the formula ( ) it follows that
Qo= 3
к3{[ъ/г/(2 fa )]—sm[nfr/(2fa)]cos[nfr / (2fa)]}
4(1+K2) Zo (nfr)2/(2fa) C^cos2[-Kfr/(2fa)]
Thus, the results of measuring the electromechanical resonance and antiresonance frequencies in the mode of thickness oscillations allow us to establish the numerical values of the elastic modulus 0c^3, piezoelectric modulus e33, and permittivity x33- Quite apart from that, the measured value of the electrical impedance at the resonance frequency can determine the quality factor of the piezoceramics at this frequency.
44
Bazilo С. V., Bondarenko M. О., Usvk L. M„ Faure E. V., Kovalenko Yu. 1.
Measuring the electrical impedance of a piezoceramic disk in a low-frequency range or in the mode of radial oscillations significantly expands the range of experimentally determined piezoceramic physical and mechanical parameters.
Conclusions
This project was undertaken to design an algorithm for assembling and solving a mathematical model of piezoelectric disc transducers, as a result of which the dependences of a piezoceramic disc’s electrical impedance on the cyclic frequency have been obtained. Geometric, mechanical, and electrical parameters of these transducers have also been established.
Analytical dependences were obtained, according to which the electrical impedance, the quality factor and amplitude values of the electric charge and electric current on the electroded surfaces of a piezoceramic disc can be determined, provided that the reverse piezoelectric effect is observed. This enabled us to conduct a complete calculation of the problem of a piezoelectric disk’s harmonic radial oscillations, by which we have significantly expanded the set of physical and mechanical parameters of piezoelectric ceramics, that, as a rule, are determined experimentally.
The study has revealed the dependence of the change in electrical impedance which significantly depends on the change in the function Л0, which, in its turn, depends on the value of the electromechanical coupling coefficient, the wave number of elastic oscillations, and the Voigt indices. The analysis has demonstrated that an increase in the value of 7а./к from 0 to 1 results in a sharp decrease of Л0 from 0,66 down to -2,0. The authors have also determined that at the resonance frequency шг (Л0 = 0,5), the oscillating piezoelectric disc consumes the maximum possible amount of energy from the electric signal generator.
The comparative analysis of the electric impedance modules obtained with and without consideration of the piezoelectric effect in the disc made of PZT-19 piezoelectric ceramics demonstrated a high coinciding in these data (the difference between the impedance values in these cases did not exceed 18%).
The proposed paper presents the results obtained during the experimental scientific and technical project ‘'Developing a highly efficient mobile ultrasonic system to intensify the extraction process while manufacturing concentrated functional beverages for combatants”, that is being implemented by the authors (state registration entry number: 0121U109660, entry date: 12.03.2021).
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Математичие моделюваипя дискових шезоелектричиих nopoTuopioua'iiu для акустоелектрошшх иристро'ш
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Математичне моделювання дискових п’езоелектричних перетворювач!в для акустоелектронних пристроУв
Базгло К. В., Бондаренко М. О., Усик JI. М.,
Фауре Е. В., Коваленко Ю. I.
В матер!алах статт! представлено алгоритм побу-дови та досл!джегшя математичпих моделей дискових п’езоелектричиих перетворювач!в. що зпаходять широко застосуваппя в пдроакустицй м!кроелектроп!гц. м!-кросхемотехшц! (паприклад, як компопепти приймаль-пих аптеп прилад!в г1дроакустичиого зв’язку). Переваги розроблегшх в статт! моделей полягають у можлгшост! встаповлшшя за ix допомогою залежпостей. як! е мате-матичшгм описом електроакустичиого зв’язку м!ж хви-..льовими полями па р1зпих д1.ляпках п’езоелектричпого перетворювача дисково! форми.
Отримаш шляхом математичиого моделювання апа-л1тич1Й залежпост! дозволяють розрахувати зпачешы електричиого 1мпедапсу та добротпост! разом з амп.л1-тудпими значениями електричиого заряду та струму па електродовапих поверхпях п’езоелектричиого диску за умов зворотпого п’езоелектричиого ефекту. Проведений повпий розрахупок задач! щодо гармошйиих рад!алышх колгшаиь дискових п’езоелектричшгх перетворювач!в дозволрш суттево розширити перел!к ф!зико-мехашчпих параметр!в п!езоматер!алу. як! рашше визпачалися екс-перимепта..лыю.
Показана залежп!сть змши електричиого !мпедап-су в!д зпачепь коеф!ц!епту електромехашчиого зв’язку. хвильового числа пружпих колршаиь та !пдекс!в Фойг-та. Також встаповлепа висока зб!жп!сть м!ж модулями електричиого !мпедапсу диск!в з п’езоелектричио'! ке-рам!ки сорту ЦТС (циркопат-титапат свшщю), як з урахуваппям. так ! без урахуваппя п’езоелектричпого ефекту (розб!жп!сть м!ж значениями !мпедапсу в цргх вршадках по переврпцувала 18%).
Ключоог слова: п’езоелектрргтгшй перетворювач: акустоелектрогика: математр1чпа модель: !мпедапс: дас-коврш елемепт
