Modeling of piezoelectric transducers with combined pseudospectral and finite-difference methods

A new hybrid finite-difference (FD) and pseudospectral (PS) method adapted to the modeling of piezoelectric transducers (PZTs) is presented. The time-dependent equations of propagation are solved using the PS method while the electric field induced in the piezoelectric material is determined through a FD representation. The purpose of this combination is to keep the advantages of both methods in one model: the adaptability of FD representation to model piezoelectric elements with various geometries and materials, and the low number of nodes per wavelength required by the PS method. This approach is implemented to obtain an accurate algorithm to simulate the propagation of acoustic waves over large distances, directly coupled to the calculation of the electric field created inside the piezoelectric material, which is difficult with classical algorithms. These operations are computed using variables located on spatially and temporally staggered grids, which attenuate Gibbs phenomenon and increase the algorithm's accuracy. The two-dimensional modeling of a PZT plate excited by a 50 MHz sinusoidal electrical signal is performed. The results are successfully compared to those obtained using the finite-element (FE) algorithm of ATILA software with configurations spatially and temporally adapted to the FE requirements. The cost efficiency of the FD-PS time-domain method is quantified and verified.     

压电陶瓷复合换能器固有频率的有限差分法*

为丰富换能器固有频率的研究理论，以及提供一种新的计算方法供工程人员选择，提出了计算其固有频率的有限差分法。以由径向极化的压电陶瓷圆管与金属预应力管沿径向复合而成的二元压电陶瓷复合换能器为例，建立并推导了其向振动的数学模型及其有限差分形式，给出了换能器径向振动的特征方程。利用MTALAB对计算实例的径向振动的固有频率进行编程计算，理论计算结果与已有实验结果符合很好，验证了有限差分法计算压电陶瓷复合换能器固有频率的可行性及准确性。通过仿真计算，给出了换能器径向振动固有频率与其结构尺寸的影响关系：换能器径向振动的固有频率随压电陶瓷圆管内径的增大而降低，随换能器壁厚比的增大而降低。该文所建立的换能器径向振动固有频率的有限差分法同样适用于结构形式相近的换能器及其他元器件。     

Bonding and impedance matching of acoustic transducers using silver epoxy

Silver epoxy was selected to bond transducer plates on glass substrates. The properties and thickness of the bonding medium affect the electrical input impedance of the transducer. Thus, the thickness of the silver epoxy bonding layer was used as a design parameter to optimize the structure for the transducer input impedance to match the 50 Ω output impedance of most radio frequency (RF) generators. Simulation and experimental results show that nearly perfect matching is achieved without using any matching circuit. At the matching condition, the transducer operates at a frequency band a little bit below the half-wavelength resonant frequency of the piezoelectric plate. In experiments, lead titanate (PT) piezoelectric plates were employed. Both full-size, 11.5 mm × 2 mm × 0.4 mm, and half-size, 5.75 mm × 2 mm × 0.4 mm, can be well matched using optimal silver epoxy thickness. The transducer assemblies demonstrate high efficiency. The conversion loss from electrical power to acoustic power in soda-lime glass is 4.3 dB. This loss is low considering the fact that the transducers operate at off-resonance by 12%. With proper choice of silver epoxy thickness, the transducer can be matched at the fundamental, the 3rd and 5th harmonic frequencies. This leads to the possible realization of triple-band transducers. Reliability was assessed with thermal cycling test according to Telcordia GR-468-Core recommendation. Of the 30 transducer assemblies tested, none broke until 2900 cycles and 27 have sustained beyond 4050 cycles.     

Modeling of functionally graded piezoelectric ultrasonic transducers

The application of functionally graded material (FGM) concept to piezoelectric transducers allows the design of composite transducers without interfaces, due to the continuous change of property values. Thus, large improvements can be achieved, as reduction of stress concentration, increasing of bonding strength, and bandwidth. This work proposes to design and to model FGM piezoelectric transducers and to compare their performance with non-FGM ones. Analytical and finite element (FE) modeling of FGM piezoelectric transducers radiating a plane pressure wave in fluid medium are developed and their results are compared. The ANSYS software is used for the FE modeling. The analytical model is based on FGM-equivalent acoustic transmission-line model, which is implemented using MATLAB software. Two cases are considered: (i) the transducer emits a pressure wave in water and it is composed of a graded piezoceramic disk, and backing and matching layers made of homogeneous materials; (ii) the transducer has no backing and matching layer; in this case, no external load is simulated. Time and frequency pressure responses are obtained through a transient analysis. The material properties are graded along thickness direction. Linear and exponential gradation functions are implemented to illustrate the influence of gradation on the transducer pressure response, electrical impedance, and resonance frequencies.     

Fabrication and comparison of PMN-PT single crystal, PZT and PZT-based 1-3 composite ultrasonic transducers for NDE applications

This paper describes fabrication and comparison of PMN-PT single crystal, PZT, and PZT-based 1-3 composite ultrasonic transducers for NDE applications. As a front matching layer between test material (Austenite stainless steel, SUS316) and piezoelectric materials, alumina ceramics was selected. The appropriate acoustic impedance of the backing materials for each transducer was determined based on the results of KLM model simulation. Prototype ultrasonic transducers with the center frequencies of approximately 2.25 and 5 MHz for contact measurement were fabricated and compared to each other. The PMN-PT single crystal ultrasonic transducer shows considerably improved performance in sensitivity over the PZT and PZT-based 1-3 composite ultrasonic transducers.     

On the optimization of the effective electromechanical coupling coefficients of a piezoelectric body

Effective electromechanical coupling coefficients are defined based on the expression for the internal energy of a piezoelectric body [B. S. Aronov, "Energy analysis of a piezoelectric body under nonuniform deformation," J. Acoust. Soc. Am. 113, 2638-2646 (2003)]. The condition is considered under which the effective coupling coefficient can be maximized up to the value of the material coupling coefficient for any given distribution of deformation. A simple practical way to optimize the effective coupling coefficient of a transducer by changing its electrode shape is illustrated with examples of transducers vibrating in longitudinal and flexural modes.     

Determination of acoustic impedances of multi matching layers for narrowband ultrasonic airborne transducers at frequencies <2.5 MHz - Application of a genetic algorithm

The effective ultrasonic energy radiation into the air of piezoelectric transducers requires using multilayer matching systems with accurately selected acoustic impedances and the thickness of particular layers. One major problem of ultrasonic transducers, radiating acoustic energy into air, is to find the proper acoustic impedances of one or more matching layers. This work aims at developing an original solution to the acoustic impedance mismatch between transducer and air. If the acoustic impedance defences between transducer and air be more, then finding best matching layer(s) is harder. Therefore we consider PZT (lead zirconate titanate piezo electric) transducer and air that has huge acoustic impedance deference. The vibration source energy (PZT), which is used to generate the incident wave, consumes a part of the mechanical energy and converts it to an electrical one in theoretical calculation. After calculating matching layers, we consider the energy source as layer to design a transducer. However, this part of the mechanical energy will be neglected during the mathematical work. This approximation is correct only if the transducer is open-circuit. Since the possibilities of choosing material with required acoustic impedance are limited (the counted values cannot always be realized and applied in practice) it is necessary to correct the differences between theoretical values and the possibilities of practical application of given acoustic impedances. Such a correction can be done by manipulating other parameters of matching layers (e.g. by changing their thickness). The efficiency of the energy transmission from the piezoceramic transducer through different layers with different thickness and different attenuation enabling a compensation of non-ideal real values by changing their thickness was computer analyzed (base on genetic algorithm). Firstly, three theoretical solutions were investigated. Namely, Chebyshev, Desilets and Souquet theories. However, the obtained acoustic impedances do not necessarily correspond to a nowadays available material. Consequently, the values of the acoustic impedances are switched to the nearest values in a large material database. The switched values of the acoustic impedances do not generally give efficient transmission coefficients. Therefore, we proposed, in a second step, the use of a genetic algorithm (GA) to select the best acoustic impedances for matching layers from the material database for a narrow band ultrasonic transducer that work at frequency below the 2.5 MHz by considering attenuation. However this bank is rich, the results get better. So the accuracy of the propose method increase by using a lot of materials with exact data for acoustic impedance and their attenuation, especially in high frequency. This yields highly more efficient transmission coefficient. In fact by using increasing number of layer we can increase our chance to find the best sets of materials with valuable both in acoustic impedance and low attenuation. Precisely, the transmission coefficient is almost equal to unity for the all studied cases. Finally the effect of thickness on transmission coefficient is investigated for different layers. The results showed that the transmission coefficient for air media is a function of thickness and sensitive to it even for small variation in thickness. In fact, the sensitivity increases when the differences of acoustic impedances to be high (difference between PZT and air).     

PSPICE controlled-source models of analogous circuit for Langevin type piezoelectric transducer

The design and construction of wide-band and high efficiency acoustical projector has long been considered an art beyond the capabilities of many smaller groups. Langevin type piezoelectric transducers have been the most candidate of sonar array system applied in underwater communication. The transducers are fabricated, by bolting head mass and tail mass on both ends of stacked piezoelectric ceramic, to satisfy the multiple, conflicting design for high power transmitting capability. The aim of this research is to study the characteristics of Langevin type piezoelectric transducer that depend on different metal loading. First, the Mason equivalent circuit is used to model the segmented piezoelectric ceramic, then, the impedance network of tail and head masses is deduced by the Newton's theory. To obtain the optimal solution to a specific design formulation, PSPICE controlled-source programming techniques can be applied. A valid example of the application of PSPICE models for Langevin type transducer analysis is presented and the simulation results are in good agreement with the experimental measurements.     

Analysis of unidirectional broadband piezoelectric spherical shell transducers for underwater acoustics

This paper presents an analytical investigation and experimental verification of the properties of unidirectional broadband piezoelectric acoustic transducers utilizing axisymmetric vibrations of both complete and incomplete spherical piezoelectric shells and is a continuation of a previous paper that presented the electromechanical modal analysis part of the problem [J. Acoust. Soc. Am. 130(2), 753-763 (2011)]. The analysis covers the treatment of the acoustic radiation and reception problem by including analysis of the acoustic impedances and diffraction coefficients as a function of geometry and modal excitation as well as providing specific design examples including multimode spherical acoustic transducers with conformal baffles, and transducers made of incomplete shells (e.g., hemispheres and caps) with free circumferential boundary conditions. The energy method is used to obtain equivalent parameters for a multi-contour electromechanical circuit representation of the transducer and to calculate the transducer performance characteristic as sound projectors and as receivers. Experimental results are obtained on representative piezoceramic transducer prototypes and are in good agreement with the analytical results.     

Design and modeling of inversion layer ultrasonic transducers using LiNbO3 single crystal

Using inversion domain engineering controlled by heating temperature, the LiNbO(3) (LNO) piezoelectric plate with both odd and even-order thickness-extensional modes can be excited simultaneously. Therefore, the inversion layer ultrasound transducer is expected to be capable of operating over a wider frequency range. In this paper, the electrical impedance and the acoustic characteristics of LiNbO(3) (LNO) inversion layer transducer have been studied by finite element modeling (FEM). The transducer designed for this study uses a 36 degrees rotated Y-cut LiNbO(3) thin plate with an active element thickness of approximately 100 microm. First the electrical and elastic properties of the 36 degrees rotated Y-cut LNO were obtained by transforming a basic piezoelectric matrix for Z-cut LNO. In order to validate the FEM using the transformed properties several pieces of pure and 50% inversion layer LNO were tested on the electrical impedance analyzer. The modeled impedance characteristics were consistent with the measured data. Next the model was used to design 50-60 MHz transducers using pure and 30% inversion LNO. Two lambda/4 matching layers and a Tungsten loaded epoxy backing were used in these designs. The modeled results show that an over 90% bandwidth transducer can be made with proper matching and 30% inversion layer.     

Interaction of light with acoustic microwave waves excited by nonperiodic multielement transducers. Part I

The feasibility of effective wide-band acoustooptic interaction in the short-wave part of the microwave range by means of nonperiodic multielement piezoelectric transducers is analyzed. The variation of the pitch or the period of the transducer sections is discussed. The impedance of a piezoelectric element with an arbitrary number of transformable layers and the acoustic power radiated from each of the piezoelements of the transducer are calculated. A 13-section transducer designed for a center frequency of 9 GHz is considered as an example.     

Phase transitions of the energy-relative enstrophy theory for a coupled barotropic fluid–rotating sphere system

The statistical equilibrium of a coupled barotropic fluid–rotating solid sphere system is simulated using a energy-relative enstrophy spherical model in a wide range of parameter space by Monte Carlo (MC) methods [J.M. Hammersley, D.C. Handscomb, Monte Carlo Methods, Methuen & Co, London, Wiley, New York City, 1964; C.C. Lim, J. Nebus, Vorticity, Statistical Mechanics and Simulations, Springer, Berlin, 2006]. The energy-relative enstrophy model does not have the low temperature defect of the classical energy–enstrophy theory [R.H. Kraichnan, Statistical dynamics of two-dimensional flows, J. Fluid Mech. 67 (1975) 155–175] because of its microcanonical constraint on relative enstrophy. This model also differs from previous work in not fixing the angular momentum. A family of spin–lattice models are derived as convergent finite dimensional approximations to the total kinetic energy. MC simulations are used to calculate the mean nearest neighbor parity as order parameter or indicator of phase transitions in the system.     

Modelling nonlinearity in piezoceramic transducers: From equations to nonlinear equivalent circuits

Quadratic nonlinear equations of a piezoelectric element under the assumptions of 1D vibration and weak nonlinearity are derived by the perturbation theory. It is shown that the nonlinear response can be represented by controlled sources that are added to the classical hexapole used to model piezoelectric ultrasonic transducers. As a consequence, equivalent electrical circuits can be used to predict the nonlinear response of a transducer taking into account the acoustic loads on the rear and front faces. A generalisation of nonlinear equivalent electrical circuits to cases including passive layers and propagation media is then proposed. Experimental results, in terms of second harmonic generation, on a coupled resonator are compared to theoretical calculations from the proposed model.     

Ultrasonic imaging using air-coupled P(VDF/TrFE) transducers at 2 MHz

A reflection non-contact ultrasonic microscope system working both in amplitude and phase difference modes at 2 MHz has been developed using an air-coupled concave transducer made of piezoelectric polymer films of poly(vinylidene fluoride/trifluoroethylene) [P(VDF/TrFE)]. The transducer is composed of three 95 μm-thick P(VDF/TrFE) films stacked together, each of which is activated electrically in parallel by a driving source. The transducer has a wide aperture angle of 140° and a focal length of 10 mm. The measured two-way transducer insertion loss is 80 dB at 1.83 MHz. Despite 20 dB higher insertion loss than that estimated from Mason’s equivalent circuit, we have obtained clear amplitude acoustic images of a coin with transverse resolution of 150 μm, and clear phase difference acoustic images of the rough surface of a paper currency bill with depth resolution of sub-micrometer. Using two planar transducers of P(VDF/TrFE), we have also successfully measured in through-transmission mode the sound velocity and absorption of a 3 mm-thick silicone-rubber plate. The present study proves that, owing to its low acoustic impedance and flexibility, P(VDF/TrFE) piezoelectric film is very useful for high frequency acoustic imaging in air in the MHz range.     

Properties and characteristics of P(VDF/TrFE) transducers manufactured by a solution casting method for use in the MHz-range ultrasound in air

Highly effective piezoelectric polymer transducers operating in air at high frequencies have been successfully made by casting a solution of ferroelectric poly(vinylidene fluoride-co-trifluoroethylene) P(VDF/TrFE) directly on a backing metal plate, and their performance has been evaluated. By utilizing this method, it has been possible to develop the three kinds of transducers that operate respectively at 4, 6 and 10 MHz in air. For precise evaluation of the performance of the P(VDF/TrFE) transducers, the absorption loss in air was measured up to 10 MHz. It was confirmed that the empirical formula obtained from the measured absorption values in air at high frequencies was in alignment with its theoretical value. In addition, a high lateral resolution acoustic image of a ROM-Chip (amplitude-image) at 6 MHz in air was successfully displayed using an air coupled concave type P(VDF/TrFE) transducer by bonding an epoxy adhesive.     

Modeling NDT piezoelectric ultrasonic transmitters

Ultrasonic NDT applications are frequently based on the spike excitation of piezoelectric transducers by means of efficient pulsers which usually include a power switching device (e.g. SCR or MOS-FET) and some rectifier components. In this paper we present an approximate frequency domain electro-acoustic model for pulsed piezoelectric ultrasonic transmitters which, by integrating partial models of the different stages (driving electronics, tuning/matching networks and broadband piezoelectric transducer), allows the computation of the emission transfer function and output force temporal waveform. An approximate frequency domain model is used for the evaluation of the electrical driving pulse from the spike generator. Tuning circuits, interconnecting cable and mechanical impedance matching layers are modeled by means of transmission lines and the classical quadripole approach. The KLM model is used for the piezoelectric transducer. In addition, a PSPICE scheme is used for an alternative simulation of the broadband driving spike, including the accurate evaluation of non-linear driving effects. Several examples illustrate the capabilities of the specifically developed software.     

Microfabrication of stacks of acoustic matching layers for 15 MHz ultrasonic transducers

This paper presents a novel method used to manufacture stacks of multiple matching layers for 15 MHz piezoelectric ultrasonic transducers, using fabrication technology derived from the MEMS industry. The acoustic matching layers were made on a silicon wafer substrate using micromachining techniques, i.e., lithography and etch, to design silicon and polymer layers with the desired acoustic properties. Two matching layer configurations were tested: a double layer structure consisting of a silicon–polymer composite and polymer and a triple layer structure consisting of silicon, composite, and polymer. The composite is a biphase material of silicon and polymer in 2-2 connectivity. The matching layers were manufactured by anisotropic wet etch of a (1 1 0)-oriented Silicon-on-Insulator wafer. The wafer was etched by KOH 40 wt%, to form 83 μm deep and 4.5 mm long trenches that were subsequently filled with Spurr’s epoxy, which has acoustic impedance 2.4 MRayl. This resulted in a stack of three layers: The silicon substrate, a silicon–polymer composite intermediate layer, and a polymer layer on the top. The stacks were bonded to PZT disks to form acoustic transducers and the acoustic performance of the fabricated transducers was tested in a pulse-echo setup, where center frequency, −6 dB relative bandwidth and insertion loss were measured. The transducer with two matching layers was measured to have a relative bandwidth of 70%, two-way insertion loss 18.4 dB and pulse length 196 ns. The transducers with three matching layers had fractional bandwidths from 90% to 93%, two-way insertion loss ranging from 18.3 to 25.4 dB, and pulse lengths 326 and 446 ns. The long pulse lengths of the transducers with three matching layers were attributed to ripple in the passband.     

Electrical characterization of plate piezoelectric transducers bonded to a finite substrate

This article presents a new technique for characterizing piezoelectric transducers attached to a finite substrate. It consists of determining the impedance of the transducer cleared of the effects caused by finite dimensions of the substrate. This technique is validated by comparison with measurements on a transducer mounted on an effectively half-infinite substrate. It is applied for the electrical matching of a lithium niobate plate transducer bonded to a fused quartz rod.     

具有阻抗匹配层的宽带纵向振动压电换能器设计

本文研究纵向振动压电换能器的频带展宽问题。在复合棒纵向换能器的辐射端加工适当材料的阻抗匹配层,可以使其工作在非单谐振状态下,在单层阻抗匹配层的情况下,合理地选择匹配层的厚度可以调整其谐振点之间的位置,从而改善换能器的辐射特性。本研究结果表明,对于机械品质因素Qm=6,发射响应带宽△f=4kHz的纵向振子,采用四分之一波长厚度的匹配层,在不降低发射响应的条件下,可展宽频带一倍以上。     

Assessing the relationship between the inter-rod coupling and the efficiency of piezocomposite high-intensity focused ultrasound transducers

The electroacoustic conversion efficiency of the ultrasonic transducer is a critical performance index for high-power applications. The material properties, volume fraction (VF) and aspect ratio (AR) are typically regarded as the design parameters of the piezocomposite transducer. We hypothesized that the spacing between piezoelectric rods was also a dominant factor. Therefore, the inter-rod coupling effects on the efficiency of 1–3 piezocomposite ultrasonic transducers were investigated in this study. The efficiencies of six flat and three curved 1.0 MHz PZT4 epoxy composite transducers with different geometric parameters were measured. Finite element transient analyses of the inter-rod electrical-mechanical coupling in the composites were carried out to explain the measured results. The experimental results showed that for 0.47 AR, the 79% VF transducers had lower efficiency than the 64% VF and 53% VF transducers. For 0.19 AR, the efficiency of the 59% VF transducer was not greater than the efficiency of the 39% VF transducer. Numerical analyses demonstrated that the positive peak voltage induced by the coupling of the side rods was more than twice the level induced by the coupling of the diagonal rods for any spacing. The diagonal coupling voltage peak did not change for spacings larger than 0.2 mm. Moreover, for spacings of 0.05 and 0.1 mm, the inter-rod coupling caused 24% and 20% waveform shifts of the driving voltage, respectively, while the 0.2 mm spacing coupling caused a 14% reduction in the amplitude of the driving voltage. As a result, the asymmetry of the driving voltage degraded the efficiency of the composite transducers and became more severe when the spacing was decreased. We concluded that the efficiency loss induced by inter-rod coupling as a function of spacing should be considered when designing piezocomposite transducers.