A FEM-based method to determine the complex material properties of piezoelectric disks

Numerical simulations allow modeling piezoelectric devices and ultrasonic transducers. However, the accuracy in the results is limited by the precise knowledge of the elastic, dielectric and piezoelectric properties of the piezoelectric material. To introduce the energy losses, these properties can be represented by complex numbers, where the real part of the model essentially determines the resonance frequencies and the imaginary part determines the amplitude of each resonant mode. In this work, a method based on the Finite Element Method (FEM) is modified to obtain the imaginary material properties of piezoelectric disks. The material properties are determined from the electrical impedance curve of the disk, which is measured by an impedance analyzer. The method consists in obtaining the material properties that minimize the error between experimental and numerical impedance curves over a wide range of frequencies. The proposed methodology starts with a sensitivity analysis of each parameter, determining the influence of each parameter over a set of resonant modes. Sensitivity results are used to implement a preliminary algorithm approaching the solution in order to avoid the search to be trapped into a local minimum. The method is applied to determine the material properties of a Pz27 disk sample from Ferroperm. The obtained properties are used to calculate the electrical impedance curve of the disk with a Finite Element algorithm, which is compared with the experimental electrical impedance curve. Additionally, the results were validated by comparing the numerical displacement profile with the displacements measured by a laser Doppler vibrometer. The comparison between the numerical and experimental results shows excellent agreement for both electrical impedance curve and for the displacement profile over the disk surface. The agreement between numerical and experimental displacement profiles shows that, although only the electrical impedance curve is considered in the adjustment procedure, the obtained material properties allow simulating the displacement amplitude accurately.     

Piezoelectric resonators with mechanical damping and resistance in current conduction

A novel design method for high Q piezoelectric resonators was presented and proposed using the 3-D equations of linear piezoelectricity with quasi-electrostatic approximation which include losses attributed to mechanical damping in solid and resistance in current conduction. There is currently no finite element sofware for estimating the Q of a resonator without apriori assumptions of the resonator impedance or damping. There is a necessity for better and more realistic modeling of resonators and filters due to miniaturization and the rapid advances in frequency ranges in telecommunication. We presented new three-dimensional finite element models of quartz and barium titanate resonators with mechanical damping and resistance in current conduction. Lee, Liu and Ballato’s 3-D equations of linear piezoelectricity with quasi-electrostatic approximation which include losses attributed to mechanical damping in solid and resistance in current conduction were formulated in a weak form and implemented in COMSOL. The resulting finite element model could predict the Q and other electrical parameters for any piezoelectric resonator without apriori assumptions of damping or resistance. Forced and free vibration analyses were performed and the results for the Q and other electrical parameters were obtained. Comparisons of the Q and other electrical parameters obtained from the free vibration analysis with their corresponding values from the forced vibration analysis were found to be in excellent agreement. Hence, the frequency spectra obtained from the free vibration analysis could be used for designing high Q resonators. Results for quartz thickness shear AT-cut and SC-cut resonators and thickness stretch poled barium titanate resonators were presented. An unexpected benefit of the model was the prediction of resonator Q with energy losses via the mounting supports.     

Measurements of attenuation and electromechanic coupling constant of piezoelectric films in microwave resonators

It was found that for arbitrary high overtone and thin film microwave resonators the results of the measurements of the difference between frequencies of resonance and antiresonance on any harmonic of the resonator together with the measurement of the frequency difference between the peculiarities on the frequency dependence of imagine part of the electric impedance of the resonator give a simple way of the evaluation of the losses in the materials composing resonator structures and of the evaluation of the electromechanical constant of the piezoelectric film exciting acoustic waves.     

Electric impedance and amplitude-frequency response of a one-dimensional ultrasonic liquid-filled resonator with flat piezoelectric plates

The impedance method is used to determine the electric impedance of a resonator. The amplitude-frequency response of a one-dimensional liquid-filled ultrasonic resonator is calculated by directly solving the wave equations and piezoelectric effect equations under the corresponding boundary conditions. An analysis of the amplitude-frequency response shows that the simple analytical expression obtained from the aforementioned solution is in good agreement with experimental data. An anomalous variation of the electric current in the radiating piezoelectric plate versus the excitation frequency is theoretically revealed near the high-Q resonance peaks. This effect is confirmed experimentally. It gives rise to errors in the measured absorption coefficient and multiply broadens the resonance peaks when the measurements are performed near the resonance frequencies of the piezoelectric plates.     

A multiple degree of freedom electromechanical Helmholtz resonator

The development of a tunable, multiple degree of freedom (MDOF) electromechanical Helmholtz resonator (EMHR) is presented. An EMHR consists of an orifice, backing cavity, and a compliant piezoelectric composite diaphragm. Electromechanical tuning of the acoustic impedance is achieved via passive electrical networks shunted across the piezoceramic. For resistive and capacitive loads, the EMHR is a 2DOF system possessing one acoustic and one mechanical DOF. When inductive ladder networks are employed, multiple electrical DOF are added. The dynamics of the multi-energy domain system are modeled using lumped elements and are represented in an equivalent electrical circuit, which is used to analyze the tunable acoustic input impedance of the EMHR. The two-microphone method is used to measure the acoustic impedance of two EMHR designs with a variety of resistive, capacitive, and inductive shunts. For the first design, the data demonstrate that the tuning range of the second resonant frequency for an EMHR with non-inductive shunts is limited by short- and open-circuit conditions, while an inductive shunt results in a 3DOF system possessing an enhanced tuning range. The second design achieves stronger coupling between the Helmholtz resonator and the piezoelectric backplate, and both resonant frequencies can be tuned with different non-inductive loads.     

水下圆柱形Helmholtz共振器的声学特性分析

理论分析了水下圆柱形Helmholtz共振器的声学特性. 综合考虑壁面弹性和辐射阻抗的影响,基于电-声类比的基本原理,建立了较为完善的水下圆柱形Helmholtz共振器的低频集中参量模型. 利用电路分析的基本方法,得到了系统的输入阻抗和声压传递函数表达式. 仿真分析了主要结构参数对共振器声学特性的影响,得出了一些有意义的结论. 在充水驻波罐中对自制的Helmholtz共振器进行了测量,并对实验结果进行了详细地误差分析. 去除压电水听器对测量结果的影响后,实验与仿真结果基本一致,从而验证了理论分析的正确性. 关键词： Helmholtz共振器 共振频率 传递函数 辐射阻抗     

Theoretical modeling and experimental realization of dynamically magnified thermoacoustic-piezoelectric energy harvesters

Conventional thermoacoustic-piezoelectric (TAP) harvesters convert thermal energy, such as solar or waste heat energy, directly into electrical energy without the need for any moving components. The input thermal energy generates a steep temperature gradient along a porous medium. At a critical threshold of the temperature gradient, self-sustained acoustic waves are developed inside an acoustic resonator. The associated pressure fluctuations impinge on a piezoelectric diaphragm, placed at the end of the resonator. In this study, the TAP harvester is coupled with an auxiliary elastic structure in the form of a simple spring–mass system to amplify the strain experienced by the piezoelectric element. The auxiliary structure is referred to as a dynamic magnifier and has been shown in different areas to significantly amplify the deflection of vibrating structures. A comprehensive model of the dynamically magnified thermoacoustic-piezoelectric (DMTAP) harvester has been developed that includes equations of motions of the system?s mechanical components, the harvested voltage, the mechanical impedance of the coupled structure at the resonator end and the equations necessary to compute the self-excited frequencies of oscillations inside the acoustic resonator. Theoretical results confirmed that significant amplification of the harvested power is feasible if the magnifier?s parameters are properly chosen. The performance characteristics of experimental prototypes of a thermoacoustic-piezoelectric resonator with and without the magnifier are examined. The obtained experimental findings are validated against the theoretical results. Dynamic magnifiers serve as a novel approach to enhance the effectiveness of thermoacoustic energy harvested from waste heat by increasing the efficiency of their harvesting components.     

The determination of electrical parameters of quartz crystal resonators with the consideration of dissipation

Recently, as the dissipation of quartz crystal through material viscosity is being considered in vibrations of piezoelectric plates, we have the opportunity to obtain electrical parameters from vibration solutions of a crystal plate representing an ideal resonator. Since the solutions are readily available with complex elastic constants from Mindlin plate equations for thickness-shear vibrations, the calculation of resistance and other parameters related to both mechanical deformation and electrical potential is straightforward. We start with the first-order Mindlin plate equations of a piezoelectric plate for the thickness-shear vibration analysis of a simple resonator model. The electrical parameters are derived with emphasis on the resistance that is related to the imaginary part of complex elastic constants, or the viscosity. All the electrical parameters are frequency dependent, enabling the study of the frequency behavior of crystal resonators with a direct formulation. Through the full consideration of complications like partial electrodes and supporting structures, we should be able obtain electrical parameters for practical applications in resonator design.     

Analysis of the electrically forced vibrations of piezoelectric mesa resonators

We study the electrically forced thickness-shear and thickness-twist vibrations of stepped thickness piezoelectric plate mesa resonators made of polarized ceramics or 6-ram class crystals. A theoretical analysis based on the theory of piezoelectricity is performed, and an analytical solution is obtained using the trigonometric series. The electrical admittance, resonant frequencies, and mode shapes are calculated, and strong energy trapping of the modes is observed. Their dependence on the geometric parameters of the resonator is also examined.     

A systems feedback representation of piezoelectric transducer operational impedance

An analysis of thickness-mode piezoelectric transducer impedance under a variety of mechanical load conditions is presented. A wideband, systems model is used to describe this impedance, which is representative of the electrical input impedance in the transmitting mode and the electrical output impedance for the receiving mode.For a fixed input voltage, the total current flowing through the transducer is considered as the vector sum of two current quantities. Firstly, an input current arising directly from the applied voltage source and secondly, a feedback current, which is generated by secondary piezoelectric action. By treating the problem in this manner, an extremely valuable insight into the nature of the electrical characteristics is provided. Complex variations in the impedance behaviour are readily explained and the physical processes that contribute to such deviations are clearly isolated.     

Optical ac coupling to overcome limitations in the detection of optical power fluctuations

A high-sensitivity detection method for optical power fluctuations is demonstrated based on photodetection in reflection of an optical resonator with a specific impedance matching. That resonator is used to reduce the carrier power reflected by the resonator while preserving the power fluctuation sidebands for frequencies above the resonator bandwidth. A sensitivity of 7 x 10(-10) Hz(-1/2) for relative power fluctuations was achieved with only 3 mA of detected photocurrent and 99.6% of the power remained for downstream experiments. As in the widely used ac coupling of electrical signals, this technique overcomes dynamic-range limits and reduces detector noise associated with large carrier amplitudes of the optical field.     

Generalized electrical analysis of low-pass and high-pass birdcage resonators

The radio-frequency 'birdcage resonator' has found wide use in MRI/MRS for its field homogeneity and signal-noise characteristics. This paper presents a general analysis, derived from lumped element transmission line theory, of the electrical behavior of unloaded, N-column birdcage resonators applicable to several versions of the basic design including low-pass and high-pass coils. Analytic expressions and computer results are presented for both types of coil describing resonant frequencies, input and characteristic impedances, dispersion relations, pass-bands, resonant peak bandwidth and Q. Theoretical expressions for normalized resonant frequency difference ratios independent of element values and resonator geometry have been developed for generic low- and high-pass coils. Experimental measurements of resonant frequencies were made for six coils, and the average agreement with theoretical predictions was approximately 4%.     

Nonlinear mechanical behavior of piezocomposites for ultrasonic transducers

A comparative study is carried out between the nonlinear behavior of a composite and the piezoceramic used to obtain it. This characterization is necessary for using the composite in power transducer applications. A study of the losses and the resonator stiffness variations has also been done. Both these effects, as well as the possibility of the frequency hysteresis, show different behavior in the composites, since the increases in the ceramics are different from those in the composites. In this study two measurement methods are used: principally the motional impedance increase with the motional current measurements. The results obtained are normalized in order to make them independent of the resonator size, and thus make the comparison between the composite and the ceramic easier. The figure of the mechanical loss tangent tan delta m versus the mean strain shows that the losses can be greater in the ceramic than in the composite for soft ceramics. The dependence behavior of the losses and stiffness variations versus the mean strain is studied for both resonators.     

体声波力传感器灵敏度的微分-综合分析法

为了分析基于应力/应变效应的体声波(BAW)力传感器的敏感机理、准确计算其灵敏度,提出了一种用于BAW力传感器灵敏度分析的微分-综合分析法。该方法借鉴了微积分的原理,在Mason等效电路模型中将一个完整的BAW谐振器替换为多个谐振器微元的并联,从而将谐振器有源区面积A上应力/应变场的有限元计算结果与压电薄膜材料的力学特性、谐振器微元的电声学特性关联起来;最后,在射频电路仿真软件中进行等效电路的综合,得到整个BAW谐振器在应力/应变场作用下的阻抗特性曲线及其串/并联谐振频率。当BAW谐振器微元的划分足够细密时,获得的灵敏度分析结果将足够精确。为了论证该方法的原理,给出了一个直观的校核案例。以一个嵌入式FBAR结构的四梁BAW加速度计表头为例,介绍了该方法用于BAW力传感器灵敏度分析的详细过程。虽然案例中只讨论了一种应力/应变型BAW力传感器的单一力敏机理,但该方法具有普适性。并且,当谐振器微元小到接近其压电材料晶格的尺度时,就能与压电薄膜的力-声-电特性的第一性原理计算结果关联起来,实现从微观材料特性到介观器件物理的多尺度计算。     

体声波力传感器灵敏度的微分-综合分析法

为了分析基于应力/应变效应的体声波(BAW)力传感器的敏感机理、准确计算其灵敏度,提出了一种用于BAW力传感器灵敏度分析的微分-综合分析法。该方法借鉴了微积分的原理,在Mason等效电路模型中将一个完整的BAW谐振器替换为多个谐振器微元的并联,从而将谐振器有源区面积A上应力/应变场的有限元计算结果与压电薄膜材料的力学特性、谐振器微元的电声学特性关联起来;最后,在射频电路仿真软件中进行等效电路的综合,得到整个BAW谐振器在应力/应变场作用下的阻抗特性曲线及其串/并联谐振频率。当BAW谐振器微元的划分足够细密时,获得的灵敏度分析结果将足够精确。为了论证该方法的原理,给出了一个直观的校核案例。以一个嵌入式FBAR结构的四梁BAW加速度计表头为例,介绍了该方法用于BAW力传感器灵敏度分析的详细过程。虽然案例中只讨论了一种应力/应变型BAW力传感器的单一力敏机理,但该方法具有普适性。并且,当谐振器微元小到接近其压电材料晶格的尺度时,就能与压电薄膜的力-声-电特性的第一性原理计算结果关联起来,实现从微观材料特性到介观器件物理的多尺度计算。     

利用遗传算法由电阻抗计算四阶带通箱的低频响应

提出了一种基于电阻抗得到四阶带通扬声器系统低频频率响应的方法。该方法无需消声室,首先建立扬声器系统的电阻抗集中参数电路模型并测量得到电阻抗曲线,然后运用遗传算法优化模型中的元件参数值,使得由模型计算得到的阻抗曲线与测得的阻抗曲线相吻合,再根据模型计算得到低频响应曲线。测量结果表明理论曲线与实测曲线相吻合,说明基于电阻抗得到低频响应的方法是可行的和有效的。     

京胡和二胡的共鸣器

本文主要是对京胡和二胡共鸣器的理论分析。从膜与琴筒内声波的耦合振动出发,给出共振频率和共振响应的近似解,并与实验结果进行比较。对于二胡的共鸣器,由实验的声压分布推算琴筒开口格板的声阻抗率作为格板的影响。文中还给出共鸣器辐射场声压频率特性的例子。     

Free vibration analysis of piezoelectric coupled circular plate with open circuit

A vibration analysis of a circular steel substrate surface bonded by a piezoelectric layer with open circuit is presented. A solution for the electrical potential along the thickness direction of the piezoelectric layer satisfying the open circuit electric boundary condition is developed for the first time. The mechanical model and solutions for the vibration analysis of the piezoelectric coupled circular plate are then established based on the developed electrical potential, the Kirchhoff plate model, and the Maxwell equation. The first four mode shapes and the corresponding resonant frequencies of the plate with two standard boundary conditions are presented in numerical simulations and compared with those of a piezoelectric coupled plate with the closed circuit condition. The simulations show that the resonant frequencies of the open circuit piezoelectric coupled plate are higher than those of the closed circuit piezoelectric coupled plate. Corresponding discussions are thus provided for the higher piezoelectric effect from the open circuit piezoelectric layer.     

The calculation of electrical parameters of AT-cut quartz crystal resonators with the consideration of material viscosity

Electrical parameters like resistance and quality factor of a quartz crystal resonator cannot be determined through vibration analysis without considering the presence of material dissipation. In this study, we use the first-order Mindlin plate equations of piezoelectric plates for thickness-shear vibrations of a simple resonator model with partial electrodes. We derive the expressions of electrical parameters with emphasis on the resistance that is related to the imaginary part of complex elastic constants, or the viscosity, of quartz crystal. Since all electrical parameters are frequency dependent, this procedure provides the chance to study the frequency behavior of crystal resonators with a direct formulation. We understand that the electrical parameters are strongly affected by the manufacturing process, with the plating techniques in particular, but the theoretical approach we presented here will be the first step for the precise estimation of such parameters and their further applications in the analysis of nonlinear behavior of resonators. We calculated the parameters from our simple resonator model of AT-cut quartz crystal with the first-order Mindlin plate theory to demonstrate the procedure and show that the numerical results are consistent with earlier measurements.     

Piezoacoustic wave spectra using improved surface impedance matrix: application to high impedance-contrast layered plates

Starting from the general modal solutions for a homogeneous layer of arbitrary material and crystalline symmetry, a matrix formalism is developed to establish the semianalytical expressions of the surface impedance matrices (SIM) for a single piezoelectric layer. By applying the electrical boundary conditions, the layer impedance matrix is reduced to a unified elastic form whether the material is piezoelectric or not. The characteristic equation for the dispersion curves is derived in both forms of a three-dimensional acoustic SIM and of an electrical scalar function. The same approach is extended to multilayered structures such as a piezoelectric layer sandwiched in between two metallic electrodes, a Bragg coupler, and a semi-infinite substrate as well. The effectiveness of the approach is numerically demonstrated by its ability to determine the full spectra of guided modes, even at extremely high frequencies, in layered plates comprising up to four layers and three materials. Negative slope in f-k curve for some modes, asymptotic behavior at short wavelength regime, as well as wave confinement phenomena made evident by the numerical results are analyzed and interpreted in terms of the surface acoustic waves and of the interfacial waves in connection with the bulk waves in massive materials.