FEM Analysis of the Effect of Polarization on the Electromechanical Coupling Factor of Resonators with a Wrap-Around Electrode

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The published results show a difference in the size of the electromechanical coupling coefficient in piezoceramic resonators with wrap-around electrodes, compared to resonators with fully electroded bases. The results can be applied in the design of new types of piezoceramic resonators.

Abstract

The efficiency of electromechanical conversion in piezoelectric elements is assessed according to the magnitude of the electromechanical coupling coefficient. For power applications of piezoelectric elements, it is desirable that the efficiency of electrical to mechanical energy conversion be as high as possible. In case of resonators with electrodes incompletely covering their bases, an inhomogeneous electric field is generated, which results in inhomogeneous polarization of the resonator. The resonator will be polarized in some places either in a direction other than the desired one or not polarized at all. The degree of polarization of the resonator is one of the factors affecting the electromechanical coupling coefficient. The aim of this work is to analyze the magnitude of the electromechanical coupling coefficient of resonators with electrodes incompletely covering the bases and to compare the results with the electromechanical coupling coefficient of resonators with fully electroded bases. The physical description is given by the linear piezoelectric equations, the Gaussian equation for the electric field, and by Newton’s law of force. On this basis, a FEM model is developed and used to analyze the electromechanical coupling coefficient. The result of the calculation of the electromechanical coupling cofficient is that for resonators with a wrap-around electrode in studied shape and dimensions there will be a decrease in the electromechanical coupling coefficient compared to a fully electroded resonator of identical dimensions. The presented conclusions are compared with analytically and experimentaly achieved results.

1. Introduction

The electromechanical coupling coefficient indicates the degree to which the piezoelectric element is able to convert the input energy $W_{in}$ into the output energy $W_{out}$. When the element is in actuator mode, the input energy is electrical energy, while the output energy is mechanical energy, and vice versa in sensor mode. The magnitude of the electromechanical coupling coefficient can be determined either theoretically based on knowledge of the material constants (static value), or calculated from the measured series and parallel resonance frequencies (dynamic value) [1].

In a simple way, the properties of a resonator can be described by an equivalent electrical circuit with lumped parameters. It describes the resistor impedance in the vicinity of resonance and antiresonance frequencies [2]. From the equivalent electrical circuit of a piezoelectric resonator, the effective electromechanical coupling coefficient $k_{eff}$, [2], can be determined. This coefficient may be expressed independently of the mode of oscillation as

$$k_{eff}^2=\frac{f_P^2-f_S^2}{f_P^2}\approx \frac{f_a^2-f_r^2}{f_a^2},$$

(1)

where $f_S$, $f_P$ are the series and parallel resonance frequencies expressed from the electrical equivalent circuit, and $f_r$ and $f_a$ are the resonance and antiresonance frequencies meeting the requirement of zero imaginary impedance component.

It is possible to show the relationship between the effective electromechanical coupling coefficient and the electromechanical coupling coefficients of certain types of oscillations of resonators of certain shapes [2], an example being the thickness mode of oscillation of circular fully electroded resonators. In a general case, however, the dependence cannot be simply defined and one must make do with the general relation (1). This is also the case for circular resonators with wrap-around electrodes. Because of the axially asymmetrical electrodes, the magnitude of the electromechanical coupling coefficient cannot be expressed analytically. As shown in [3], the wrap-around electrodes cause inhomogeneous polarization in the resonator; moreover, in the area where the electrodes are wrapped-around, polarization is not achieved at all. It can be therefore expected that the electromechanical coupling coefficient will be lower for resonators with wrap-around electrodes than for resonators of the same dimensions with fully electroded bases. In the presented work, we will verify this hypothesis computationally and validate it by comparison with experimental results.

In order for the obtained results for resonators with fully electroded bases to be comparable with the results calculated for resonators with wrap-around electrodes, we will use the relation (1) to calculate the coefficient of electromechanical coupling. It will therefore be necessary to determine the resonance and antiresonance frequencies of the oscillation for which we determine $k_{eff}$. The vibration to be studied will be the fundamental thickness oscillation.

The study of the resonance characteristics of piezoelectric resonators with incomplete or asymmetric electrodes is already the subject of [4,5], or more, recently [6,7]. In the publications [8,9], the analytical solution is compared with theexperimental results. However, in all these publications, the authors focus on resonators where a certain form of symmetry can be seen in the electrode surfaces, e.g., circular electrodes with different diameters but a common axis of circles, or rectangular electrodes on a resonator of rectangular plan, where two planes of symmetry can be found.

In the available literature, the author has not found a relevant study concerning circular, partially electroded resonators and effects of this feature on the resonance frequency of the thickness oscillation. Neither for resonators electroded symmetrically according to the axis of symmetry of the disc, nor for asymmetrical electroding similar to the wrap-around electrodes. Neither experimental nor theoretical study has been found. However, mention should be made of the work of [10], where incomplete or axially asymmetric electroding is studied, but not for thickness oscillation and not in the electrode configurations usable for comparison with wrap-around electrodes. Similar results are reported in [11], where the application of symmetrical electroding on a circular resonator in order to determine the Poisson number of piezoelectric ceramics is presented. However, the electrode variants and types of excited oscillations presented in his work are not relevant to the case studied here.

In publications [12,13,14,15], the finite element method is used as a tool to determine the resonance frequency, antiresonance frequency, and electromechanical coupling coefficient. The change in the electromechanical coupling coefficient with the size of the area covered by the electrode in a circular resonator is also examined, but only configurations with electrodes symmetric to the resonator symmetry and almost exclusively for radial oscillation are studied. None of the works mentioned above deals with the issue of changing the electromechanical coupling coefficient of resonators with electrodes not symmetrical with respect to the axis of symmetry of the resonator; resonators with a wrap-around electrode can be included among them. At the same time, the thickness mode of the resonator oscillation is considered in none of these works. The presented work is therefore unique in its analysis of the influence of asymmetric electrodes on the electromechanical coupling coefficient of the resonator thickness oscillation.

2. Materials and Methods

Consider the piezoelectric continuum with density $\varrho$, characterized by material tensors $c,e,\epsilon$. We denote the volume of the continuum as $\Omega$ and its boundary as $\Gamma$. Behavior of the piezoelectric continuum is in time range $\langle 0,\tau \rangle$ governed by two differential equations [16]. Gauss’s law (2) and Newton’s law of motion (3),

$$\nabla ·\mathbf{D}=\frac{\partial }{\partial x_k}(e_{kij}·{\mathbf{S}}_{\mathrm{ij}}+{\epsilon }_{kj}^S·{\mathbf{E}}_{\mathrm{j}})=0,$$

(2)

$$\nabla ·\mathbf{T}=\varrho \frac{{\partial }^2{\mathbf{u}}_i}{\partial t^2}=\frac{\partial }{\partial x_j}(c_{ijkl}^E·{\mathbf{S}}_{\mathrm{kl}}-e_{kij}·{\mathbf{E}}_{\mathrm{k}})i=1,2,3,x\in \Omega ,t\in \langle 0,\tau \rangle ,$$

(3)

where T, D, S, E, u are stress, the vector of electric displacement, strain, electric field, mechanical displacement. Initial and boundary conditions are added:

$$\begin{array}{ccc}\hfill \phi \left(0\right)& =& {\phi }_0\hfill \end{array}$$

(4)

$$\begin{array}{ccc}\hfill u\left(0\right)& =& u_0\hfill \end{array}$$

(5)

$$\begin{array}{ccc}\hfill \phi & =& {\phi }_D\mathrm{on}{\Gamma }_1,\hfill \end{array}$$

(6)

$$\begin{array}{ccc}\hfill {\mathrm{D}}_{\mathrm{k}}{n}_k& =& {\mathrm{D}}_{\mathrm{N}}\mathrm{on}{\Gamma }_2,\hfill \end{array}$$

(7)

$$\begin{array}{ccc}\hfill u_i& =& u_{iD}i=1,2,3,\mathrm{on}{\Gamma }_1,\hfill \end{array}$$

(8)

$$\begin{array}{ccc}\hfill {\mathrm{T}}_{\mathrm{ij}}{n}_j& =& t_{iN}i=1,2,3,\mathrm{on}{\Gamma }_2.\hfill \end{array}$$

(9)

The right-hand sides ${\phi }_D$, ${\mathrm{D}}_{\mathrm{N}}$, $u_{iD}$, and $t_{iN}$ represent excitation of the continuum. If we discretize the Equations (2) and (3) by the finite element method, this set of equations will have the form [16,17],

$$\left(\begin{array}{cc}\mathbf{M}& 0\\ 0& 0\end{array}\right)\left(\begin{array}{c}\ddot{\mathrm{U}}\\ 0\end{array}\right)+\left(\begin{array}{cc}\mathbf{K}& \mathbf{P}\\ {\mathbf{P}}^{\mathrm{T}}& \mathbf{E}\end{array}\right)\left(\begin{array}{c}\mathrm{U}\\ \Phi \end{array}\right)=\left(\begin{array}{c}\mathrm{F}\\ \mathrm{Q}\end{array}\right),$$

(10)

where $\mathbf{K}$ is the stiffness matrix, $\mathbf{M}$ is the mass matrix, $\mathbf{P}$ is the piezoelectric matrix, $\mathbf{E}$ is the electric matrix, $\mathrm{U}$ is the displacement vector, $\Phi$ is the vector of electric potentials, $\mathrm{F}$ is the vector of external forces, and $\mathrm{Q}$ is the electric charge vector. When a piezoelectric resonator is excited by an electric voltage applied to its electrodes, the determining quantity is the difference in electric potentials between the electrodes. According to [16,17], we shall consider one of the electrodes grounded; the vector of electric potentials corresponding to the nodes of the finite element mesh at the location of the grounded electrode is

$${\Phi }_b=0,$$

(11)

the vector of electric potentials corresponding to the nodes of the mesh at the places of the ungrounded electrode is denoted by ${\Phi }_n$. The system (10) can be written as

$$\left(\begin{array}{ccc}\mathbf{M}& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)\left(\begin{array}{c}\ddot{\mathrm{U}}\\ 0\\ 0\end{array}\right)+\left(\begin{array}{ccc}\mathbf{K}& {\mathbf{P}}_{\mathrm{e}}& {\mathbf{P}}_{\mathrm{n}}\\ {\mathbf{P}}_{\mathrm{e}}^{\mathrm{T}}& {\mathbf{E}}_{\mathrm{e}}& {\mathbf{P}}_{\mathrm{en}}\\ {\mathbf{P}}_{\mathrm{n}}^{\mathrm{T}}& {\mathbf{P}}_{\mathrm{en}}^{\mathrm{T}}& {\mathbf{E}}_{\mathrm{n}}\end{array}\right)\left(\begin{array}{c}\mathrm{U}\\ {\Phi }_e\\ {\Phi }_n\end{array}\right)=\left(\begin{array}{c}\mathrm{F}\\ 0\\ {\mathrm{Q}}_{\mathrm{n}}\end{array}\right),$$

(12)

where ${\mathrm{Q}}_{\mathrm{n}}$ is the vector of electric charges in the nodes belonging to the ungrounded electrode, and ${\Phi }_e$ is the vector of electric potentials in the nodes outside the electrode. By eliminating ${\Phi }_e$ from the second equation of the system (12) we obtain

$$\left(\begin{array}{cc}\mathbf{M}& 0\\ 0& 0\end{array}\right)\left(\begin{array}{c}\ddot{\mathrm{U}}\\ 0\end{array}\right)+\left(\begin{array}{cc}\overline{\mathbf{K}}& \overline{\mathbf{P}}\\ {\overline{\mathbf{P}}}^{\mathrm{T}}& \overline{\mathbf{E}}\end{array}\right)\left(\begin{array}{c}\mathrm{U}\\ {\Phi }_n\end{array}\right)=\left(\begin{array}{c}\mathrm{F}\\ {\mathrm{Q}}_{\mathrm{n}}\end{array}\right),$$

(13)

where

$$\begin{array}{ccc}\hfill \overline{\mathbf{K}}& =& \mathbf{K}-{\mathbf{P}}_{\mathrm{e}}{\mathbf{E}}_{\mathrm{e}}^{-1}{\mathbf{P}}_{\mathrm{e}}^{\mathrm{T}},\hfill \end{array}$$

(14)

$$\begin{array}{ccc}\hfill \overline{\mathbf{E}}& =& {\mathbf{E}}_{\mathrm{n}}-{\mathbf{P}}_{\mathrm{en}}^{\mathrm{T}}{\mathbf{E}}_{\mathrm{e}}^{-1}{\mathbf{P}}_{\mathrm{en}},\hfill \end{array}$$

(15)

$$\begin{array}{ccc}\hfill \overline{\mathbf{P}}& =& {\mathbf{P}}_{\mathrm{n}}-{\mathbf{P}}_{\mathrm{e}}{\mathbf{E}}_{\mathrm{e}}^{-1}{\mathbf{P}}_{\mathrm{en}}.\hfill \end{array}$$

(16)

At resonance, we assume [16,17], that the electrical impedance takes the minimum value and the electrodes are short circuited (that is, both electrodes have the same electrical potential), i.e., ${\Phi }_n=0$. In resonance we can regard $\mathrm{F}=0$, the system (13) can be modified to the form

$$\left(\begin{array}{cc}\mathbf{M}& 0\\ 0& 0\end{array}\right)\left(\begin{array}{c}\ddot{\mathrm{U}}\\ 0\end{array}\right)+\left(\begin{array}{cc}\overline{\mathbf{K}}& \overline{\mathbf{P}}\\ {\overline{\mathbf{P}}}^{\mathrm{T}}& \overline{\mathbf{E}}\end{array}\right)\left(\begin{array}{c}\mathrm{U}\\ 0\end{array}\right)=\left(\begin{array}{c}0\\ {\mathrm{Q}}_{\mathrm{n}}\end{array}\right).$$

(17)

The resonance frequencies are then obtained as solutions of a generalized eigenvalue problem in the form

$$\left(\overline{\mathbf{K}}-\lambda \mathbf{M}\right)\mathrm{U}=0.$$

(18)

When calculating the antiresonance frequency, we use the system of Equation (13) again. In accordance with [18,19], we consider disconnected electrodes, thus the electric charge on the ungrounded electrode meets the condition

$${\mathrm{Q}}_{\mathrm{n}}=0.$$

(19)

However, the part of the system of equations related to the grounded electrode was eliminated due to the condition (11), see the transition from (10) to (12). At the same time, $\mathrm{F}=0$ will apply; the system (13) can be written as

$$\left(\begin{array}{cc}\mathbf{M}& 0\\ 0& 0\end{array}\right)\left(\begin{array}{c}\ddot{\mathrm{U}}\\ 0\end{array}\right)+\left(\begin{array}{cc}\overline{\mathbf{K}}& \overline{\mathbf{P}}\\ {\overline{\mathbf{P}}}^{\mathrm{T}}& \overline{\mathbf{E}}\end{array}\right)\left(\begin{array}{c}\mathrm{U}\\ {\Phi }_n\end{array}\right)=\left(\begin{array}{c}0\\ 0\end{array}\right).$$

(20)

From the second equation we express

$${\Phi }_n={\overline{\mathbf{E}}}^{-1}{\overline{\mathbf{P}}}^{\mathrm{T}}\mathrm{U}.$$

(21)

We then obtain the antiresonance frequency as a solution to the generalized problem of eigenvalues in the form

$$\left(\widehat{\mathbf{K}}-\lambda \mathbf{M}\right)\mathrm{U}=0,$$

(22)

where

$$\widehat{\mathbf{K}}=\overline{\mathbf{K}}-\overline{\mathbf{P}}{\overline{\mathbf{E}}}^{-1}{\overline{\mathbf{P}}}^{\mathrm{T}}.$$

(23)

Knowing the values of the resonance frequencies, calculated according to the relation (22), and the values of the antiresonance frequencies according to the relation (22), the values of the effective electromechanical coupling coefficients can be obtained by substituting in the relation (1).

2.1. Task Assignment

We calculate $k_{eff}$ of the fundamental thickness oscillation for resonators with wrap-around electrodes, where we expect inhomogeneous polarization due to electrodes being wrapped around. We compare the result with the calculation of $k_{eff}$ thickness oscillation of a resonator of identical dimensions but fully electroded on both circular surfaces, where homogeneous polarization can be expected throughout the resonator volume. By comparing the calculated $k_{eff}$ of the resonators with wrap-around electrodes with the $k_{eff}$ of the fully electroded resonators, we conclude the effect of polarization on the electromechanical coupling coefficient.

This approach is applied when computing $k_{eff}$ of two types of piezoelectric transducers, labeled (a) and (b), shown in Figure 1. The transducers differ in the shape of the electrode wrap-around. In case of (a), the overlay is in the shape of a circular segment, in case of (b), it is in the shape of intersecting circles.

To make the results comparable, we will consider both piezoelectric transducers of the same size. The criterion of equivalence for the wrap-around electrodes needs to be established. We will consider equivalent the electrodes that will have the same area of wrap-around electrode in both cases.

Let $\alpha =d/t$, where d is the diameter of the disc and t its thickness. We will consider a disc with a dimensional ratio of $\alpha \gg 1$, where the basic thickness mode of the oscillation is significantly distinct. In the monograph [1] (chapter 5.3), the frequency spectrum of a resonator with entire electrodes on both circular surfaces with ratio $\alpha =23.7$, made of NCE51 piezoelectric ceramics, is analyzed. We will thus work with a resonator that maintains the dimensions of resonator from chapter 5.3 of the monograph [1], except that it will not be fully electroded on both circular surfaces. A diagram of the dimensions of the transducers is shown in Figure 2. The dimensions of the transducers are given in

Table 1. Dimensions of the transducers.

d (mm)	47.9	t (mm)	2.02
$s_1$ (mm)	4.66	$r_1$ (mm)	9.50
$s_2$ (mm)	2.00	$r_2$ (mm)	7.50

.

The dimensions of the wrap-around are chosen so that the area of the wrapped around electrodes is identical in both cases. The ratio of the wrap-around to the total circular area is approximately $1/25$.

We consider the transducers made of NCE51 piezoelectric ceramics, again in accordance with Chapter 5.3 in [1]. The values of the elements of the above-mentioned tensors are given in

Table 2. Material properties of NCE51 ceramics.

${\epsilon }_{11}^S\left(1\right)$	906	$c_{12}^E$$\left(\mathrm{Pa}\right)$	2.04 $\times {10}^{10}$
${\epsilon }_{33}^S\left(1\right)$	823	$c_{13}^E$$\left(\mathrm{Pa}\right)$	8.85 $\times {10}^{10}$
$e_{31}$$(\mathrm{C}/{\mathrm{m}}^2)$	−4.80	$c_{33}^E$$\left(\mathrm{Pa}\right)$	17.2 $\times {10}^{10}$
$e_{33}$$(\mathrm{C}/{\mathrm{m}}^2)$	17.2	$c_{44}^E$$\left(\mathrm{Pa}\right)$	2.04 $\times {10}^{10}$
$e_{15}$$(\mathrm{C}/{\mathrm{m}}^2)$	13.7	$c_{66}^E$$\left(\mathrm{Pa}\right)$	2.30 $\times {10}^{10}$
$c_{11}^E$$\left(\mathrm{Pa}\right)$	12.9 × 1010	$\varrho$$(\mathrm{kg}/{\mathrm{m}}^3)$	7850

and are taken from [1,20]. In the table, permittivity is given as relative permittivity.

The polarization of the above-mentioned resonators with a wrap-around electrode is analyzed in the work by [3]. We will consider the polarization directions as calculated in [3]. Thus, we know the polarization directions of piezoelectric ceramics and consequently the macroscopic piezoelectric properties that are determined by polarization.

2.2. Geometry and Finite Element Mesh

The electromechanical coupling coefficient is calculated using the relation (1), i.e., based on the knowledge of the resonance and antiresonance frequency of the respective oscillation. The resonance and antiresonance frequencies are determined by the procedure in the introduction of Section 2. The thickness oscillation that is symmetric to the resonator symmetry axis will be examined. The resonator has the cylindrical shape, so the axis of symmetry is the axis of the cylinder representing the resonator. It is therefore appropriate that the chosen discretization of the resonator area reflects the symmetry of the oscillation and is therefore homogeneous. The use of inhomogeneous grids may lead to:

• An axially asymmetrical shape of the oscillation that does not reflect reality;
• Inaccuracies in the calculation of the resonance frequency.

We can say that this is a form of discretization error. By using a finite element mesh, the edges of which will have a comparable size in the whole discretized area, we eliminate the mentioned errors in the obtained solution. At the same time, identical finite element meshes will be used to calculate the resonance and antiresonance frequencies of resonators with wrap-around electrodes and to calculate the resonance and antiresonance frequencies of a fully electroded resonator. The use of identical meshes has the following benefits:

• The same discretization error can be expected for the calculation of the resonance frequency of a resonator with fully electroded circular surfaces, as well as for resonators with wrap-around electrodes;
• A very similar numerical error in the calculation of resonance frequencies can be expected for all analyzed cases.

Under these circumstances, the results of the calculation of the resonance frequencies obtained for each case can be compared. The differences between the results will be negligible because:

• Resonators fully electroded on circular surfaces and resonators with wrap-around electrodes have the same diameter and thickness;
• Both resonator types are made of the same material and its density determining the mass matrix $\mathbf{M}$ is the same;
• With respect to the condition (11) the boundary conditions are also the same.

The only difference between the analysed variants is due to the anisotropy of the elastic coefficients tensor. In the case of a resonator fully electroded on circular surfaces, the homogeneous polarizing electric field results in an almost perfect orientation of the individual ceramic grains in the direction of the applied electric field, i.e., in the direction perpendicular to the thickness of the resonator.

In the case of resonators with wrap-around electrodes, we will consider the orientation of the ceramic grains in the directions calculated in the work [3]. Thus, the difference of the individual analyzed variants will be apparent only in those elements of the stiffness matrix $\mathbf{K}$, which will correspond to the finite elements in which the material properties differ.

The thickness oscillation we are analyzing is symmetric along the resonator axis. Therefore, the geometry corresponding to the symmetric half of the resonator can be used for the calculation, taking into account the boundary conditions

$$\begin{array}{ccc}\hfill {\mathrm{D}}_{\mathrm{k}}{n}_k& =& 0,\hfill \end{array}$$

(24)

$$\begin{array}{ccc}\hfill {\mathrm{T}}_{\mathrm{ij}}{n}_j& =& 0,\hfill \end{array}$$

(25)

in the plane of symmetry. The finite element meshes used are shown in Figure 3. They have a sufficient number of nodes for the calculation of the fundamental thickness oscillation over the base diameter, both variants (a) and (b) contain a total of about 14,000 nodes.

3. Results

3.1. Calculation of Resonance and Antiresonance Frequencies for Fully Electroded Resonators, Model Verification

The resonance frequency of the thickness oscillation will be first determined for a resonator whose thickness and diameter are given in Table 1, with fully electroded circular surfaces. The result will be compared with the analytical calculation and the experimentally determined value. The calculation will be performed on the finite element meshes shown in Figure 3. We will compare the results obtained on both types of finite element meshes. The material properties are given in Table 2. We consider the polarization direction to be homogeneous throughout the resonator volume, in the direction of the resonator thickness, i.e., along the z axis, as shown in Figure 3. The calculated frequency spectrum contains the oscillation mode shown in Figure 4.

Half of the resonator cross section is shown, and the center of the resonator is on the left. The shape of the oscillation corresponds to the shape of the thickness oscillation in Section 5.3 in [1]. The calculated resonance frequencies are

$$\begin{array}{c}\hfill f_{ra}=988,961\mathrm{Hz} \mathrm{for} \mathrm{mesh} \mathrm{variant} (a),\\ \hfill f_{rb}=988,915\mathrm{Hz} \mathrm{for} \mathrm{mesh} \mathrm{variant} (b).\end{array}$$

The calculated antiresonance frequencies are

$$\begin{array}{c}\hfill f_{aa}=1,115,829\mathrm{Hz} \mathrm{for} \mathrm{mesh} \mathrm{variant} (a),\\ \hfill f_{ab}=1,115,712\mathrm{Hz} \mathrm{for} \mathrm{mesh} \mathrm{variant} (b).\end{array}$$

According to [1], the resonance frequency of the fundamental thickness oscillation can be expressed as

$$f_{rt}=\frac{\eta }{2\pi }\sqrt{\frac{c_{33}^D}{\varrho }},$$

(26)

where $\eta$ is the wavenumber, $c_{33}^D$ is the element of the elastic modulus tensor at constant electric induction, and $\varrho$ is the density. The wavenumber $\eta$ must satisfy the relevant boundary conditions on the circular surfaces of the resonator and the transcendental equation applies to it:

$$\mathrm{tg}\left(\eta b\right)=\frac{\eta t}{{2k_t}^2}.$$

(27)

The electromechanical coupling coefficient for the thickness mode of the oscillation can be expressed as

$$k_t=\frac{e_{33}}{\sqrt{c_{33}^D{\epsilon }_{33}^S}}.$$

(28)

Let us reiterate that the polarization is assumed to be in the direction of the resonator thickness, i.e., in the direction perpendicular to the circular surfaces. Therefore, the components of material tensors with coefficients 33 appear in the relation (28). The resonance frequency of the fundamental mode of the thickness oscillation calculated for said resonator is

$$f_{rt}=994,270\mathrm{Hz}.$$

The difference between the value of the resonance frequency $f_{rt}$ calculated analytically and the numerically obtained values of $f_{ra}$ and $f_{rb}$ is 0.6%, which can be considered a very good agreement. We can conclude that the numerical model is thus verified.

Using a similar procedure as with the resonance frequency, the antiresonance frequency can be analytically calculated, see [1]. The value is

$$f_{at}=1,121,000\mathrm{Hz}.$$

It can be noted that the values calculated by the finite element method (FEM) model are 0.6% lower in the case of both resonance and antiresonance frequencies. Since the calculation of both values is identical in principle and the same values of material parameters are used, a similar difference can be expected between the values obtained analytically and numerically.

The experimentally determined value of the resonance frequency reported in [1] is

$$f_{re}=1,001,600\mathrm{Hz}$$

Compared to $f_{ra}$, $f_{rb}$, it differs by approximately 1.3%. Let us now compare the difference between the frequencies $f_{ra}$, $f_{rb}$ that were calculated on different meshes, as documented in Figure 3. The difference is in the order of 0.001%. The calculation performed on both variants of the meshes yields comparable results. It can be thus concluded that we have a verified model through which we can capture the trends in the resonance frequency changes of the thickness oscillation caused by the electrodes being wrapped around, respectively by the inhomogeneous polarization of the resonator. At the same time, it can be stated that the different discretization of variants (a) and (b) has only a negligible effect on the calculated resonance frequency of the thickness oscillation.

The experimentally determined value of the antiresonance frequency reported in [1] is

$$f_{ae}=1,112,500\mathrm{Hz}.$$

Compared to $f_{aa}$, $f_{ab}$ differs by approximately 0.3%.

3.2. Calculation of Resonance and Antiresonance Frequencies of Resonators with Wrap-Around Electrodes

The calculation of the resonance frequency of resonators with a wrap-around electrode will be based on the assumptions made in Section 2.2. In particular, identical finite element meshes will be used for the calculation so that the same discretization error can be achieved and the results from Section 3.1 can be compared with the results obtained for resonators with wrap-around electrodes. The meshes used will be the finite element meshes shown in Figure 3, according to the electrode structure (a) or (b) as shown in Figure 2.

The material properties listed in Table 2 will be used. The polarization and its direction, i.e., its influence on the orientation of the material tensors, are taken from [3]. In this way, the influence of different polarization in the parts with the wrap-around electrode compared to the homogeneous polarization in the whole transducer in the case of a fully electroded resonator is taken into account.

Let us first pay attention to the calculated shape of the oscillation. In the case of a fully electroded resonator, the thickness oscillation is rotationally symmetric along the axis of symmetry of the resonator. For resonators with wrap-around electrodes, the thickness oscillation is symmetric only along the plane of symmetry of the electrodes. The damping of the oscillation can be seen in the area under the wrap-around electrode; see Figure 5, where the oscillation for wrapping-around according to variant (a) is shown. The damping of the oscillation for variant (b) is similar.

The calculated resonance frequencies are

$$\begin{array}{c}\hfill f_{ra}=1,006,413\mathrm{Hz} \mathrm{for} \mathrm{the} \mathrm{electrode} \mathrm{structure} (a),\\ \hfill f_{rb}=1,004,792\mathrm{Hz} \mathrm{for} \mathrm{the} \mathrm{electrode} \mathrm{structure} (b).\end{array}$$

Compared to the fully electroded resonator, there is an increase in the value of the resonance frequency in both variants of the mesh, namely

• By 1.8% in the case of the electrode structure (a);
• By 1.6% in the case of the electrode structure (b).

The calculated antiresonance frequencies are

$$\begin{array}{c}\hfill f_{aa}=1,126,303\mathrm{Hz} \mathrm{for} \mathrm{the} \mathrm{electrode} \mathrm{structure} (a),\\ \hfill f_{ab}=1,125,329\mathrm{Hz} \mathrm{for} \mathrm{the} \mathrm{electrode} \mathrm{structure} (b).\end{array}$$

and conclusions similar to those for resonance frequencies can be drawn for them.

3.3. Calculation of the Effective Coefficient of Electromechanical Coupling

As mentioned in Section 2.2, the reference element is a circular fully electroded resonator with dimensions according to Table 1.

3.3.1. Calculation of $k_{eff}$ for Fully Electroded Resonators

We calculate $k_{eff}$ for fully electroded resonators from the resonance and antiresonance frequencies obtained

• Computationally analytically;
• Experimentally;
• Through the FEM model for mesh variants (a) and (b) as shown in Figure 3.

Analytical calculation of the resonance and antiresonance frequencies is performed according to the relations in [1]. The calculated values are

$$\begin{array}{ccc}\hfill f_{rt}& =& 994,270\mathrm{Hz},\hfill \\ \hfill f_{at}& =& 1,121,000\mathrm{Hz}.\hfill \end{array}$$

By substituting these values into the relation (1) we obtain the value of the effective coefficient of the electromechanical coupling

$$k_{efft}=0.4619.$$

The experimentally determined values are taken from [1]

• Resonance frequency $f_{re}=1,001,600\mathrm{Hz}$;
• Antiresonance frequency $f_{ae}=1,112,500\mathrm{Hz}$.

By substituting these values into the relation (1) we obtain the effective coefficient of the electromechanical coupling

$$k_{effe}=0.4352.$$

The values of the resonance and antiresonance frequencies for the finite element mesh variants (a) and (b) according to Figure 3 were determined in Section 3.1. By substituting into the (1) relation, we obtain the value of the effective coefficient of the electromechanical coupling

$$k_{effa}=0.4631$$

for the mesh variant (a) and

$$k_{effb}=0.4630$$

for the mesh variant (b).

The obtained result is consistent with the fact that the values of resonance and antiresonance frequencies are both shifted by 0.6% compared to the analytical solution. The calculation of $k_{eff}$ given by the relation (1) can then be expected to be almost identical in value for both the analytical and numerical solutions.

3.3.2. Calculation of $k_{eff}$ for Resonators with Wrap-Around Electrodes

The values of the resonance and antiresonance frequencies for both variants of the mesh according to Figure 3 were calculated in Section 3.2. By substituting into the relation (1) we obtain the value of the effective coefficient of the electromechanical coupling

$$k_{effa}=0.4490$$

for the wrap-around electrode according to the variant (a) and

$$k_{effb}=0.4503$$

for the wrap-around electrode according to the variant (b). The electromechanical coupling coefficient calculated for the fully electroded resonator in Section 3.3.1 is equal to 0.46 for both electrode structures. It can be thus concluded that the calculated value of the electromechanical coupling coefficient is lower in the case of resonators with wrap-around electrodes.

4. Discussion

It can be hypothesized that for the resonators with the wrap-around electrode in the shape and dimensions according to the scheme in Figure 2 there will be an increase in the resonance frequency compared to the fully electroded resonators of the same dimensions. The differences between a fully electroded resonator and a resonator with the wrap-around electrode, which are also very plausible causes of the increase in resonance frequency, are:

• Inhomogeneous polarization, especially in the area of electrode wrap-around, resulting in inhomogeneous material properties;
• Inhomogeneous distribution of the excitation electric field in the resonator.

One may also ask why in the case of the mesh, or—more precisely—of the wrap-around shape according to variant (a), the growth of the resonance frequency is about 0.2% higher than in the case of the wrap-around according to variant (b). A discretization error cannot be ruled out even though the difference in the calculated resonance frequency between variants (a) and (b) is less than an order of magnitude lower than a tenth of a percent in the case of the calculated resonance frequency.

However, we can hypothesize that the difference in the calculated frequency is caused by the different amount of polarized volume of the transducer in the direction of its thickness. This, in accordance with the results given in [3], is

• A total of 88% of resonator volume in configuration (a);
• A total of 90% of resonator volume in configuration (b).

This reasoning is consistent with the conclusions made, for example, in [12]. The effect of electrode size on the resonance frequency of symmetrically electroded discs is studied there. The increase in resonance frequency with decreasing electrode area is shown. Compared with the problem of the resonance frequency of a thickness oscillation resonator with a wrap-around electrode studied here, there are two major differences:

• In [12], axially symmetrically electroded discs are studied, in contrast to resonators with a wrap-around electrode, which are symmetric only in one plane;
• In [12] the radial oscillations of the disc are studied, in contrast to the thickness oscillations studied in resonators with a wrap-around electrode.

For longitudinal oscillations of piezoelectric rods and thickness oscillations of plates, a relation for the electromechanical coupling coefficient valid for this type of oscillations [1,2], can be derived. The condition is full electroding of the opposite circular bases. For the thickness oscillations of the plate with full electrodes [2], the following applies

$$k_t^2=\frac{\pi }{2}\frac{f_r}{f_a}\mathrm{cotg}\left(\frac{\pi }{2}\frac{f_r}{f_a}\right).$$

(29)

As mentioned in the introduction to this chapter, the relationship between $k_{eff}$ and the electromechanical coupling coefficient of a given type of oscillation can be expressed. In the case of the thickness oscillation and its electromechanical coupling coefficient $k_t$, the following is true [2],

$$k_t^2=\frac{\pi }{2}\sqrt{1-k_{eff}}\mathrm{cotg}\left(\frac{\pi }{2}\sqrt{1-k_{eff}}\right).$$

(30)

For comparison, let us calculate the coefficient $k_t$ for the fully electroded resonator as well as for resonators with a wrap-around electrode. The calculation will be performed, although we know that for resonators with a wrap-around electrode it will only be approximately valid. The results of the calculation are summarized in

Table 3. Calculated values $k_t$.

	Fully Electroded	Wrap-Around (a)	Wrap-Around (b)
$k_t$	0.5013	0.4867	0.4858

.

The calculated values of $k_t$ are higher than the corresponding values of $k_{eff}$ in all of the considered cases, but the trend is the same for both quantities. In the case of resonators with wrap-around electrodes, $k_t$ is lower than in the case of a fully electroded resonator.

5. Conclusions

Based on the above, it can be hypothesized that for resonators with a wrap-around electrode in the shape and dimensions according to the scheme in Figure 2 there will be a decrease in the electromechanical coupling coefficient compared to a fully electroded resonator of identical dimensions. The hypothesis is based on the comparison of the results obtained by the same method, which is the FEM model calculation for both the resonators with a wrap-around electrode and the fully electroded resonator. The reasons that lead to the decrease in the electromechanical coupling coefficient are likely to lie in the differences between the fully electroded resonator and the resonators with a wrap-around electrode; in particular, the uneven polarization of the resonator due to the shape and placement of the electrodes in resonators with wrap-around electrodes.

In the available literature, the author has not found a study of the effect of partial electroding, asymmetrical along the resonator axis, on the electromechanical coupling coefficient. Mention may be made of the works of [8,12,14,15,21]. Here, however, the cases of partial electroding symmetrical along the axis of symmetry of the disc and for the case of radial oscillations are studied.

It can be concluded that the claim of a decrease in the electromechanical coupling coefficient during partial electrode position is consistent with the results published in [12]. In this work, the graph in Figure 10 shows the dependence of the electromechanical coupling coefficient on the size of the electrodes. However, these findings apply to a symmetrically electroded disc and radial type of oscillation. Thus, the hypothesis made in the present paper cannot be based on the results published in [12].

The results presented here can be verified experimentally; for the reasons given in Section 1 it is not possible to determine an analytical solution. The experimental determination of the frequency spectra of thickness oscillating resonators with a wrap-around electrode is the subject of the technical report of [22]. The report contains the values of the electromechanical coupling coefficient calculated from the measured resonance and antiresonance frequencies of resonators with double-sided electroding and resonators with a wrap-around electrode. The electromechanical coupling coefficient of the thickness oscillation is determined here to be lower in most cases for resonators with a wrap-around electrode than for the fully electroded resonators. Although the measurements were carried out on a small number of samples, this result supports the proposed hypothesis of a decrease in the electromechanical coupling coefficient of resonators with a wrap-around electrode.