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This article deals with 2-D modeling of coupled vibrations of finite piezoelectric resonators. A general solution for all the physical quantities in the Cartesian and cylindrical coordinate systems is deduced from the governing equations by expansion in series summation of trigonometric functions of thickness coordinate and trigonometric or Bessel functions of the lateral one. The essential difference between this model and the earlier ones is that instead of expressing mainly in the thickness coordinate and integration through the thickness, the solutions are expressed in the form of double Fourier series augmented by single Fourier or Fourier-Bessel series, which contributes to better satisfy the mechanical and electrical boundary conditions. The dynamic stiffness matrix of the system is developed. Electrical impedances of a typical piezoelectric parallelepiped under stress-free and symmetrical loading conditions and its frequency spectrum for different width-to-thickness ratios are calculated using our model as well as by the finite element method. A comparison shows an excellent agreement. Finally, theoretical and measured electrical impedances of a piezoelectric parallelepiped and a piezoelectric disk are compared and discussed. The 2-D theoretical model proposed here is shown to be accurate and efficient for coupled vibration analysis of piezoelectric resonators and is applicable for any set of finite dimensions and crystal symmetry.

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This paper describes a non-contact method to characterize isotropic and anisotropic planar multilayer structures using a genetic algorithm. The method is based on the determination of critical angles, where the maxima of the modulus of transmission coefficient of the structure appear, and which correspond to the generation of guided waves. The optimization process minimizes the error between the reference critical angles and associated amplitudes of the transmission coefficient, with the corresponding estimated ones. The estimation of elastic parameters is demonstrated for acrylic and oak plates as well as for a bi-layered structure composed of oak and a thin layer of gesso. It is shown that to obtain satisfactory optimization results, it is necessary for guided modes of higher order than the zero ones to be taken into account. Results also show that some elastic constants such as C33 and C55 retrieved from the transmission coefficient are very sensitive to the optimization.

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Symmetric reflector ultrasonic transducers (SRUTs) are a new type of ultrasonic transducer that take advantage of the ultrasonic wave emitted on the rear face of the active element. In this work, the electroacoustic modeling and characterization of such a structure is reported. Using the well-known Krimholtz, Leedom, et Matthaei (KLM) model, the electroacoustic response of a SRUT based on a piezoelectric ceramic with one matching layer is calculated. Simulations show that the optimal acoustic impedance of the matching layer should be lower than for conventional transducers, leading to a relative bandwidth of approximately 50%. The characterization of such a transducer based on a piezoceramic plate has been carried out. Bandwidth and sensitivity are reported. They are found to be close to the simulation results and demonstrate that these new types of transducers need to be designed according to new rules compared to conventional ones.

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The radial and thickness extensional vibration modes in piezoelectric cylinders are always inevitably coupled due to the finite dimension, Poisson's ratio, and piezoelectric effect. In this paper, an analytical model based on the superposition method is developed to obtain the coupled dynamic response of a piezoelectric cylinder under an applied voltage. The problem can be described mathematically by three partial differential equations with mixed boundary conditions in the cylindrical coordinates system. To solve this, the problem is decomposed first into two building block - vibrations in radial and thickness directions. In each building block, the expressions of displacements and electric potential are assumed and then the induced dynamic responses, such as in-plane stress and electric displacements, are calculated. Finally, the vibration responses of the two building blocks are superimposed to satisfy the mixed boundary conditions using Fourier and Fourier-Bessel series expansions. Electrical impedance of a typical piezoelectric disk and frequency spectrum of piezoelectric cylinders of different diameter-to-thickness ratios are calculated by the present analytical method as well as by finite element method. Comparison shows an excellent agreement. This analytical model can be applied to material characterization and the design and the optimization of the active elements of piezoelectric devices.

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Ultrasonic transducers performance can be seriously deteriorated by loss of adhesion between some constitutive elements such as the active element, the backing, or the matching layer. In the present work, the influence of bonding delaminations on the performance of a single-element ultrasonic transducer, which is composed of a piezoelectric disk, a backing, and a matching layer, is studied numerically and experimentally. Based on the positions between layers, two cases, i.e., delaminations between ceramic and backing or between ceramic and matching layer, are considered. Each case involves three different types of delaminations, which are marked as delamination type (DT)-I, II, and III. DT-I, a circular shape delamination, starts from the center and expands towards the peripheric zone; DT-II, an annular shape delamination, starts from the peripheric zone and expands towards the center; DT-III is a sector shape delamination with a given angle. The numerical simulations are performed by the finite element method and the influence of delaminations on the electromechanical admittance (EMA) of the transducer is investigated. 3D printed backings and matching layers are mounted on a PZT sample to assemble delaminated single-element transducers. An impedance analyzer is used for experimental measurements. Comparison between numerical and experimental results shows a reasonable agreement making changes in EMA an interesting indicator to inform about the occurrence and severity of delaminations in a single-element ultrasonic transducer.

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The transducer is an essential part of all ultrasonic systems used for applications such as medical diagnostics, therapy, nondestructive evaluation, and cleaning because its health condition is vital to their proper operation. Defects within the active element, backing or other constitutive elements, and loss of adhesion between layers can significantly weaken the performance of a transducer. The objective of this work is to determine procedures to monitor the behavior of a single-element probe during its lifetime and detect degradations before they significantly affect the performance of the system. To achieve this, electromechanical admittance (EMA)-based method is envisaged numerically and experimentally. A simplified single-element transducer consisting of a piezoceramic disk, a bonding layer, and a backing is studied and the influence of bonding delamination on EMA is investigated. This study considers three different types of delaminations, which are named, respectively, "center" (circular delamination from the center of the disk toward the peripheric zone), "peripheric" (annular delamination from the peripheric zone toward the center), and "wedge" (wedge-shaped delamination with a given angle). For each case, a numerical model based on the finite-element (FE) method is developed: a 2-D FE analysis is implemented for the first two types of delaminations, taking advantage of their axisymmetric structure, and "wedge" delamination is modeled in 3-D. Then, transducers with different shapes of 3-D printed backings are mounted and experiments are conducted using an impedance analyzer. Finally, experimental results are found to be in good agreement with numerical solutions and it shows that changes in EMA can particularly reveal the occurrence and extent of delamination in an ultrasound probe.

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For transducer design, it is essential to know the acoustic properties of the materials in their operating conditions. At frequencies over 15 MHz, standard methods are not well adapted because layers are very thin and backings have very high attenuation. In this article, we report on an original method for measuring the acoustic properties in the 15-25 MHz frequency range, corresponding to typical skin-imaging applications, using a backing/piezoelectric multilayer structure. Onto a porous Pb(Zr0.53Ti0.47O3 (PZT) substrate, a piezoelectric PZT-based layer with a thickness of [Formula: see text] was deposited and directly used to excite an acoustic signal into water. Herein, the measured signal corresponds to the wave that is first reflected on a target in water, then propagates back to the multilayer structure, and is transmitted through the thick film and further to the rear face of the porous backing, where it is again reflected and returns to the piezoelectric thick film, thus avoiding overlap with the electrical excitation signal. Two types of PZT backings with similar porosity of ~20% and spherical pores with size of 1.5 and 10 [Formula: see text] were processed. The ultrasound group velocities were measured at ~3500 m/s for both samples. The acoustic attenuation of the backings with pore size of 1.5 and 10 [Formula: see text] were 12 and 33 dB/mm, respectively, measured at 19 MHz. This advanced measuring technique demonstrated potential for the simple measurements of acoustic properties of backing at high frequencies in operating conditions. Importantly, this method also enables rapid determination of the minimum required thickness of the backing to act as a semi-infinite medium, for high-frequency transducer applications.

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Barium titanate (BaTiO3) is increasingly studied to replace lead-based piezoelectric materials, such as those which belong to the lead zirconate titanate (PZT) family, due to lead toxicity. In many applications, such as Tonpilz transducers, piezoelectric materials undergo mechanical stress simulation of which is important to control and predict electroacoustic effects. Thus, this article deals with a fully tensorial model that allows to simulate the behaviors of electrical displacements and elastic strains under mechanical stress. Simulated curves are compared with experimental ones obtained for BaTiO3 samples. It can be verified that the hysteretic curves of strains are well predicted for unpoled samples as well as for poled ones. The order of values and global behavior of the theoretical electrical displacement are also verified, even if a less precise agreement is observed. The optimized values of the physical parameters, such as d33 , are discussed, and improvements both of the model and the optimization procedure are finally proposed in order to better predict the mechanical behavior of BaTiO3.

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Barium titanate (BaTiO3) is being studied extensively to replace lead-based piezoelectric materials, such as the lead zirconate titanate (PZT) family, due to lead toxicity. As a result, researchers are turning to materials such as BaTiO3 and seek to improve their properties with the use of dopants. In many applications such as Tonpilz transducers, piezoelectric materials undergo mechanical stress which is important to control and predict their electro-acoustic performance. Thus, this study deals with a fully tensorial model that allows us to simulate the behaviors of electrical displacements and elastic strains under mechanical stress. The simulated curves are compared with the experimental ones obtained for a doped BaTiO3 composition and the hysteretic curves of strains are in good agreement both for the unpoled and poled samples. The values and global behavior of the theoretical electrical displacement are also found to be in fair agreement, though some discrepancies are observed. The optimized values of the physical parameters, such as d33 , are discussed and improvements both of the model and the optimization procedure are finally proposed to better predict the mechanical behavior of the doped BaTiO3 piezoceramics.

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The standard fabrication method for 1-3 piezocomposites for ultrasound transducers is the "dice and fill" method (DFM) in which lateral periodicity is introduced. This contributes to the appearance of spurious modes that can drastically affect the performance of the device if they appear near its thickness mode frequency, thus limiting the effective frequency range. A new 1-3 piezocomposite fabricated with a super-cell structure [1-3 super cell (13SC)] was designed in order to overcome these limitations. It consists of the merging of several periodic cells with 47% PZT volume fraction and epoxy resin as the matrix. Two lateral periodicities in one direction are defined as well as two different kerfs. The chosen cell shape is composed of five nonaligned square section rods ( 1 ×1 mm 2 ). For comparison of performance, two regular 1-3 piezocomposites (the same materials and equivalent periodicities) were fabricated by DFM. Electroacoustic responses in water were measured for the three composites being considered as transducers. Successive regular thinnings (from 2.8 to 1.1 mm) were carried out for each sample to increase the operating frequency (from around 0.4 to 1.3 MHz) and study the evolution of the characteristics (bandwidth and sensitivity). The experimental results confirmed the behavior of those obtained with numerical simulations, showing that the 13SC composite can be used in this entire frequency range, unlike regular composites.

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An electrophoretic deposition (EPD) process with high deposition rate was used to fabricate a curved piezoelectric thick film devoted to high-frequency transducers for medical imaging. Niobium-doped lead zirconate titanate (PZTNb) powder was stabilized in ethanol to prepare a suspension with high zeta potential and low conductivity. A gold layer, pad-printed and fired on a curved porous PZT substrate, was used as the working electrode for the deposition of the PZTNb thick film. This substrate was chosen because it has the required properties (acoustic impedance and attenuation) to be used directly as a backing for the high-frequency transducer, leading to a simplified process for transducer assembly with this integrated structure. PZT-Nb thick films were also deposited by EPD on flat gold-coated alumina substrates as a reference. The thickness of the films was between 20 and 35 µm, and their electromechanical performance was comparable to standard PZT bulk ceramics with a thickness coupling factor of 48%. For the curved thick film, the thickness coupling factor was slightly lower. The corresponding integrated structure was used to fabricate a transducer with a center frequency of 40 MHz and an f-number of 2.8. It was integrated into a realtime ultrasound scanner and used to image human forearm skin; the resulting images showed, for the first time, the efficacy of the EPD process for these imaging applications.

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Recently, a second-order formalism of piezoelectric structures under an external mechanical stress was developed. Because the yield strength of lithium niobate was unknown, this study was not able to describe and evaluate realistic benefits of a prestress load on electromechanical properties. Therefore, in this study, experimental determination of the yield strength of lithium niobate sample is performed and shows that this limit is close to 110 MPa. Then, the nonlinearities and evolutions of electroacoustic parameters of this piezoelectric material under mechanical stress are numerically studied. By varying the initial prestress, as well as azimuthal and elevation angles, the cut planes in which a prestress induces significant benefits on velocities and coupling coefficient are identified. Finally, approximate relations describing changes between electroacoustic parameters defined in the two coordinate systems of the study are determined.

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Pad-printed thick-film transducers have been shown to be an interesting alternative to lapped bulk piezoceramics, because the film is deposited with the required thickness, size, and geometry, thus avoiding any subsequent machining to achieve geometrical focusing. Their electromechanical properties are close to those of bulk ceramics with similar composition despite having a higher porosity. In this paper, padprinted high-frequency transducers based on a low-loss piezoceramic composition are designed and fabricated. High-porosity ceramic cylinders with a spherical top surface are used as the backing substrate. The transducers are characterized in view of imaging applications and their imaging capabilities are evaluated with phantoms containing spherical inclusions and in different biological tissues. In addition, the transducers are evaluated for their capability to produce high-acoustic intensities at frequencies around 20 MHz. High-intensity measurements, obtained with a calibrated hydrophone, show that transducer performance is promising for applications that would require the same device to be used for imaging and for therapy. Nevertheless, the transducer design can be improved, and simulation studies are performed to find a better compromise between low-power and high-power performance. The size, geometry, and constitutive materials of optimized configurations are proposed and their feasibility is discussed.

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In many industrial processes where online control is necessary such as in the food industry, the real time monitoring of visco-elastic properties is essential to ensure the quantity of production. Acoustic methods have shown that reliable properties could be obtained from measurements of velocity and attenuation. This paper proposes a simple, real time ultrasound method for monitoring linear medium properties (phase velocity and attenuation) that vary in time. The method is based on a pulse echo measurement and is self-calibrated. Results on a silica gel are reported and the importance of taking into account the changes of the mechanical loading on the front face of the transducer will be shown. This is done through a modification of the emission and reception transfer parameters. The simultaneous measurement of the input and output currents and voltages enables these parameters to be calculated during the reaction. The variations of the transfer parameters are in the order of 6% and predominate other effects. The evolution of the ultrasonic longitudinal wave phase velocity and attenuation as a function of time allows the characteristic times of the chemical reaction to be determined. The results are well correlated with the gelation time measured by rheological method at low frequency.

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The transient analysis of piezoelectric transducers is often performed using finite-element or finite-difference time-domain methods, which efficiently calculate the vibration of the structure but whose numerical dispersion prevents the modeling of waves propagating over large distances. A second analytical or numerical simulation is therefore often required to calculate the pressure field in the propagating medium (typically water) to deduce many important characteristics of the transducer, such as spatial resolutions and side lobe levels. This is why a hybrid algorithm was developed, combining finite- difference and pseudo-spectral methods in the case of 2-D configurations to simulate accurately both the generation of acoustic waves by the piezoelectric transducer and their propagation in the surrounding media using a single model. The algorithm was redefined in this study to take all three dimensions into account and to model single-element transducers, which usually present axisymmetrical geometry. This method was validated through comparison of its results with those of finite-element software, and was used to simulate the behavior of planar and lens-focused transducers. A high-frequency (30 MHz) transducer based on a screen-printed piezoelectric thick film was fabricated and characterized. The numerical results of the hybrid algorithm were found to be in good agreement with the experimental measurements of displacements at the surface of the transducer and of pressure radiated in water in front of the transducer.

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In a recent publication [E. Filoux, S. Callé, D. Certon, M. Lethiecq, F. Levassort, Modeling of piezoelectric transducers with combined pseudospectral and finite-difference methods, J. Acoust. Soc. Am. 123 (6) (2008) 4165-4173], a new finite-difference/pseudospectral time-domain (FD-PSTD) algorithm was presented and used to model the generation of acoustic waves by a piezoelectric resonator and their propagation in the structure and the surrounding water. In this paper, the model has been extended to simulate the two-dimensional behaviour of a complete single-element transducer, composed of the resonator, a backing and a front matching layer. This further version of the model takes into account the mechanical loss in materials, and enables the calculation of electrical impedance, which is a characteristic of high interest to optimize the performance of ultrasonic transducers. The impedance curves of a PZT [URL: http://www.ferroperm-piezo.com (last viewed 04/2008); B. Jaffe, R.S. Roth, S. Marzullo, Piezoelectric properties of lead zirconate-lead titanate solid-solution ceramics, J. Appl. Phys. 25 (1954) 809-810] plate-based high-frequency transducer, with a 50 MHz thickness resonant frequency, were compared to those of a KLM model [R. Krimholtz, D.A. Leedom, G.L. Matthei, New equivalent circuit for elementary piezoelectric transducers, Electron. Lett. 6 (1970) 398-399] in the one-dimensional case. The acoustical properties were also found to be in good agreement with those obtained using the finite element (FE) method of ATILA software in two-dimensional configuration.

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The design of transducers requires a clear understanding of their electromechanical behavior. This involves precise linear modeling as well as characterization. With the development of novel techniques such as harmonic imaging as well as high-power applications, nonlinear aspects must also be taken into account. In this study, harmonic generation in the mechanical displacement of a piezoceramic rod under high sinusoidal electric fields was measured. Theoretically, the nonlinearity can come from various sources: dielectric, mechanical, and electromechanical. The nonlinearity coming from external sources being eliminated or taken into account, it is shown here that the analysis, over a wide frequency range, of 2 parameters related to the harmonic distortion enables the respective identification of these sources and, at the same time, the evaluation of third-order constants of the material.

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In this paper, a new formulation of the electrical input impedance of a single element transducer is presented. The resistive part of the electrical impedance that takes into account acoustic radiation in the front medium and losses in the transducer is split into a radiation resistance on one hand and into dissipation resistances related to each transducer component on the other hand. To confirm these theoretical results, characterization methods based on temperature measurements and pulse-echo response are presented. Measurements have been conducted on 1 MHz transducers, which consist of a piezoelectric ceramic glued on a backing. The results show a good agreement between experience and theory for dissipation resistance and radiation resistance values, which confirms the theoretical approach.

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The goal of this work was to develop an extended ultrasound transducer model that would optimize the trade-off between accuracy of the calculation and computational time. The derivations are presented for a generalized transducer model, that is center frequency, pulse duration and physical dimensions are all normalized. The paper presents a computationally efficient model for lens-focused, circular (axisymmetric) single element piezoelectric ultrasound transducer. Specifically, the goal of the model is to determine the lens effect on the electro-acoustic response, both on focusing and on matching acoustic properties. The effective focal distance depends on the lens geometry and refraction index, but also on the near field limit, i.e. wavelength and source radius, and on the spectrum bandwidth of the ultrasound source. The broadband (80%) source generated by the transducer was therefore considered in this work. A new model based on a longitudinal-wave assumption is presented and the error introduced by this assumption is discussed in terms of its maximum value (16%) and mean value (5.9%). The simplified model was based on an extension of the classical KLM model for transducer structures and on the related assumptions. The validity of the implemented extended KLM model was evaluated by comparison with finite element modeling, itself previously validated analytically for the one-dimensional planar geometry considered. The pressure field was then propagated using the adequate formulation of the Rayleigh integral for both the extended KLM and finite element results. The simplified approach based on the KLM model delivered the focused response with good accuracy, and hundred-fold lower calculation time in comparison with a mode comprehensive FEM method. The trade-off between precision and time thus becomes compatible with an iterative procedure, used here for the optimization of the acoustic impedance of the lens for the chosen configuration. An experimental comparison was performed and found to be in good agreement with such an extension of the KLM model. The experiments confirm the accuracy of such a model in a validity domain up to -12 dB on the pulse-echo voltage within a relative error of 9% between experiment and modeling. This extended KLM model can advantageously be used for other transducer geometries satisfying the assumption of a predominantly longitudinal vibration or in an optimization procedure involving an adequate criteria for a particular application.

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A screen-printed PZT thick film with a final thickness of about 40 microm was deposited on a porous PZT substrate to obtain an integrated structure for ultrasonic transducer applications. This process makes it possible to decrease the number of steps in the fabrication of high-frequency, single-element transducers. The porous PZT substrates allow high acoustic impedance and attenuation to be obtained, satisfying transducer backing requirements for medical imaging. The piezoelectric thick films deliver high electromechanical performance, comparable to that of standard bulk ceramics (thickness coupling factor over 45%). Based on these structures, high-frequency transducers with a center frequency of about 25 MHz were produced and characterized. As a result, good sensitivity and axial resolution were obtained in comparison with similar transducers integrating a lead titanate (PT) disk as active material. The two transducers were integrated into a high-frequency imaging system, and comparative skin images are shown.

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