Modeling and analysis of dual-output piezoelectric transformer operating at the thickness-shear vibration mode.

In our previous study, the multioutput piezoelectric transformer operating at the thickness-shear vibration mode was proposed and experimentally investigated. By designing a new construction of support and lead wire connection, a power density of 52.7 W/cm3 and a total output power of 169.8 W were achieved at a temperature rise less than 20 degrees C. In this work, a theoretical model was developed for the dual-output piezoelectric transformer operating at the thickness-shear vibration mode. The equivalent circuit parameters of the piezoelectric transformer were derived. Based on this, the impedance characteristics, equivalent inductance, capacitance ratio, voltage gain, and efficiency of the piezoelectric transformer were calculated. The theoretical results were verified by experimental data. Furthermore, the effect of the transformer size on the voltage gain, efficiency, output power and power density, and the effect of the load of one output on the voltage gain of another output were analyzed. Some useful guidelines were achieved by these analyses.     

An improved equivalent circuit model of radial mode piezoelectric transformer

In this paper, both the equivalent circuit models of the radial mode and the coupled thickness vibration mode of the radial mode piezoelectric transformer are deduced, and then with the Y-parameter matrix method and the dual-port network theory, an improved equivalent circuit model for the multilayer radial mode piezoelectric transformer is established. A radial mode transformer sample is tested to verify the equivalent circuit model. The experimental results show that the model proposed in this paper is more precise than the typical model.     

Piezoelectric beams and vibrating angular rate sensors

A tuning fork angular rate sensor made out of a single piece of quartz has been studied. The piezoelectric effect is used both to excite a reference vibration in the plane of the tuning fork and to detect a vibration normal to this plane. The amplitude of the second vibration is directly proportional to the applied angular velocity. The structure is made rigid in order for it to survive in a harsh environment. This implies that the only vibrationally active areas are the tines of the tuning fork. The performance of the sensor is predicted with the help of a phenomenological piezoelectric beam theory. This theory shows that it suffices to study the two-dimensional (2-D) dielectric field in the cross-sections of the beams in order to obtain the values of the piezoelectric equivalent components. Estimates of these values can be obtained without the use of special computer programs. The predictions are shown to be in agreement with measurements.     

Modeling and analysis of dual-output piezoelectric transformer operating at thickness-shear vibration mode

In our previous study, the multioutput piezoelectric transformer operating at the thickness-shear vibration mode was proposed and experimentally investigated. By designing a new construction of support and lead wire connection, a power density of 52.7 W/cm/sup 3/ and a total output power of 169.8 W were achieved at a temperature rise less than 20 /spl deg/C. In this work, a theoretical model was developed for the dual-output piezoelectric transformer operating at the thickness-shear vibration mode. The equivalent circuit parameters of the piezoelectric transformer were derived. Based on this, the impedance characteristics, equivalent inductance, capacitance ratio, voltage gain, and efficiency of the piezoelectric transformer were calculated. The theoretical results were verified by experimental data. Furthermore, the effect of the transformer size on the voltage gain, efficiency, output power and power density, and the effect of the load of one output on the voltage gain of another output were analyzed. Some useful guidelines were achieved by these analyses.     

Advances in high-Q piezoelectric resonator materials and devices

In order to compare piezoelectric materials and devices, an intrinsic parameter, the motional time constant tau(1)/sup (m/)=(omega/sub m/Q/sub m/)(-1) for a particular mode m is employed. The use of tau(1)/sup (m/) follows from the accommodation of acoustic loss in the elastic compliance/stiffness and the establishment of material coefficients that are elements of viscosity matrices. Alternative and fully equivalent definitions of tau(1) are given based on the RC time constant derived from the equivalent circuit representation of a crystal resonator, acoustic attenuation, logarithmic decrement, and viscosity or damping. For quartz devices, the variation of tau(1): for any simple thickness mode, for the Y'X shear mode for rotated Y-cuts, and with diameter-thickness ratio for AT-cuts is discussed. Other factors such as mounting loss and loss caused by crystal inhomogeneities (dislocations, defect positions in the resonator, and impurity migration under vibrational stress) are briefly considered with quartz devices as the model. Some new piezoelectric materials/material constants/devices are reviewed and their motional time constants are compared. A physical parameter, composed of acoustic velocity, piezoelectric coupling, and tau(1) is identified which aids in understanding the maximum frequency limitations of plate resonators.     

Design analysis of miniature quartz resonator using two-dimensional finite element model

This study focused on 2-D miniature quartz plates. By assigning appropriate boundary condition using finite element modeling (FEM), the vibration of a quartz plate was analyzed for converse piezoelectric effect. The quality and stability of the resonance of a quartz plate was determined by examining changes on the response curve of resonant frequency when the length of plate was decreased or increased. A graphical user interface (GUI) was adopted to assist the finite element software to calculate the frequency responses with different length of a large number of quartz plates, and to conclude a detailed curve of resonant frequency versus size. With this diagram, changes of the resonant mode for quartz plates caused by length variation can be easily observed. An optimum size of the quartz plate is obtained from the curve. Moreover, analyses were also conducted on the electrode coverage of a quartz plate and the mass-loading effect of metallic electrodes for this study, to discuss the influence on the resonant frequencies of quartz plates.     

Determination of the electromechanical coupling factor of gallium orthophosphate (GaPO4) and its influence on resonance-frequency temperature dependencies.

The quartz homeotype gallium orthophosphate (GaPO4) is a representative of piezoelectric single crystals of large electromechanical coupling factor. It is known that its coupling factor kappa26 associated with the resonators vibrating in the thickness-shear mode is approximately two times greater than that of quartz. This property increases the spacing between the series and parallel resonance frequencies of resonators, as well as the difference between the resonance frequency temperature dependencies of the fundamental and harmonic resonance frequencies of resonators vibrating in the thickness-shear mode. In this paper, the methods for determination of the coupling factor kappa26 are presented, and the computed values are compared with the measured ones. The influence of the coupling factor to the resonance-frequency temperature dependencies of the fundamental and third harmonics of selected rotated Y-cut GaPO4 resonators vibrating in the thickness-shear mode is presented. The purely elastic case for a laterally unbounded plate, which corresponds closely to the limiting case of high harmonic resonance frequency-temperature behavior was assumed for the calculations. The computed temperature coefficients for the Y-cut orientation and calculated turnover point temperatures TTP for different (YX1) orientations are presented.     

Two-dimensional analysis using one-dimensional FEM forstraight-crested waves in arbitrary anisotropic crystal plates andaxisymmetric piezoelectric vibrations in ceramic disks

We have developed a hybrid method applicable to straight-crested waves in arbitrary anisotropic crystal plates and to axisymmetric piezoelectric vibrations in ceramic disks. The solutions to two-dimensional (2-D) equations of motion are described with a linear combination of eigenmodes guided by a pair of parallel edges. The guided eigenmodes and their amplitudes are determined by using one-dimensional (1-D) finite element method (FEM). The method developed here provides rapid convergence with small matrix size compared with 2-D FEM. Computer programs have been developed for three examples, SC- and AT-cut quartz plates and barium titanate (BaTiO₃) disks, for which the frequency spectra and the corresponding mode shapes were calculated. The frequency spectra of AT-cut quartz plates are compared with those obtained from Mindlin's plate equations, with the aim of examining the accuracy of the straight-crested wave solutions for Mindlin's plate equations. A convergence study is also presented for BaTiO₃ disks     

用二维等效电路模型研究大截面圆柱变幅杆的振动

由振动方程的近似解得到了具有大尺寸比的圆柱变幅杆的二维等效电路,用于带外载或不带外载的圆柱轴对称振动分析,以及尺寸比变化引起的频率变化评价。与有限元及一维模型进行了对比。发现该模型计算频率比一维模型更准确,与有限元模态分析相一致。进而由此确定了这些频率所对应的振动模态。计算发现轴向和周向水负载对应的圆柱振动频率接近,可用于超声变幅杆的优化。     

Determination of the electromechanical coupling factor of gallium orthophosphate (GaPO/sub 4/) and its influence on resonance-frequency temperature dependencies

The quartz homeotype gallium orthophosphate (GaPO/sub 4/) is a representative of piezoelectric single crystals of large electromechanical coupling factor. It is known that its coupling factor k/sub 26/ associated with the resonators vibrating in the thickness-shear mode is approximately two times greater than that of quartz. This property increases the spacing between the series and parallel resonance frequencies of resonators, as well as the difference between the resonance frequency temperature dependencies of the fundamental and harmonic resonance frequencies of resonators vibrating in the thickness-shear mode. In this paper, the methods for determination of the coupling factor k/sub 26/ are presented, and the computed values are compared with the measured ones. The influence of the coupling factor to the resonance-frequency temperature dependencies of the fundamental and third harmonics of selected rotated Y-cut GaPO/sub 4/ resonators vibrating in the thickness-shear mode is presented. The purely elastic case for a laterally unbounded plate, which corresponds closely to the limiting case of high harmonic resonance frequency-temperature behavior was assumed for the calculations. The computed temperature coefficients for the Y-cut orientation and calculated turnover point temperatures T/sub TP/ for different (YX1) orientations are presented.     

An exact analysis of a rectangular plate piezoelectric generator

We study thickness-twist vibration of a finite, piezoelectric plate of polarized ceramics or 6-mm crystals driven by surface mechanical loads. An exact solution from the three-dimensional equations of piezoelectricity is obtained. The plate is properly electroded and connected to a circuit such that an electric output is generated. The structure analyzed represents a piezoelectric generator for converting mechanical energy to electrical energy. Analytical expressions for the output voltage, current, power, efficiency, and power density are given. The basic behaviors of the generator are shown by numerical results.     

Electric equivalent circuit for flexural vibrations in piezoelectric materials

General expressions for the values of the components of an electric equivalent circuit are derived for flexural vibrations. These expressions are applied to some interesting electrode configurations. The obtained values are in good agreement with experimental values and values specified by manufacturers of watch crystals. It is noted that solving Laplace's equation for the dielectric field is sufficient in order to obtain the values of the vibration amplitude, the piezoelectric current, and the equivalent components. The piezoelectric part of the electric field need only be considered if very accurate values of the resonance frequencies are desired. It is shown how accurate estimates can be obtained without the need of advanced calculation tools.     

Radially composite piezoelectric ceramic tubular transducer in radial vibration

The radially composite piezoelectric tubular transducer is studied. It is composed of radially poled piezoelectric and a long metal tube. The electro-mechanical equivalent circuit of the radially poled piezoelectric and metal tube in radial vibration is obtained. Based on the force and velocity boundary conditions, the six-port electro-mechanical equivalent circuit for the composite tubular transducer is given and the resonance/anti-resonance frequency equations are obtained. The relationship between the resonance frequency and the dimensions is analyzed. Numerically simulated results obtained by the finite element method are compared with those from the analytical method. Composite piezoelectric tubular transducers are designed and manufactured. The resonance/anti-resonance frequencies are measured, and it is shown that the theoretical results are in good agreement with the simulated and experimental results. It is expected that radially composite piezoelectric tubular transducers can be used as high-power ultrasonic radiators in ultrasonic applications, such as ultrasonic liquid processing.     

Evaluation on mass sensitivity of SAW sensors for different piezoelectric materials using finite-element analysis

The mass sensitivity of the piezoelectric surface acoustic wave (SAW) sensors is an important factor in the selection of the best gravimetric sensors for different applications. To determine this value without facing the practical problems and the long theoretical calculation time, we have shown that the mass sensitivity of SAW sensors can be calculated by a simple three-dimensional (3-D) finite-element analysis (FEA) using a commercial finite-element platform. The FEA data are used to calculate the wave propagation speed, surface particle displacements, and wave energy distribution on different cuts of various piezoelectric materials. The results are used to provide a simple method for evaluation of their mass sensitivities. Meanwhile, to calculate more accurate results from FEA data, surface and bulk wave reflection problems are considered in the analyses. In this research, different cuts of lithium niobate, quartz, lithium tantalate, and langasite piezoelectric materials are applied to investigate their acoustic wave properties. Our analyses results for these materials have a good agreement with other researchers' results. Also, the mass sensitivity value for the novel cut of langasite was calculated through these analyses. It was found that its mass sensitivity is higher than that of the conventional Rayleigh mode quartz sensor.     

State vector approach of free-vibration analysis of magneto–electro-elastic hybrid laminated plates

In this paper, state variable formulation for free vibration of the laminated structures bonded and embedded actuators and sensors (piezoelectric and/or piezomagnetic) is established. The present mixed type formulation has the great advantage that the size of the system to be solved is independent of the number of layers and is of order three for free vibration of the laminate with bonded and embedded piezoelectric and/or piezomagnetic layers. Analytical solutions of 3D free vibration of simply supported piezoelectric and piezomagnetic composite plates have been presented. The transfer matrices for either closed circuit or open circuit piezoelectric and piezomagnetic layers are derived. The special case of elastic layer is also treated. The assembly procedure is described for the different electric and magnetic surface conditions. Numerical examples are analyzed to study the vibration characteristics of smart laminate plates with different stacking sequence and different span to thickness ratio.     

A new method of determining the equivalent circuit parameters of piezoelectric resonators and analysis of the piezoelectric loading effect

An inductive method of piezoelectric resonance detection is applied to the determination of equivalent circuit parameters of piezoelectric resonators. Using this method one can measure the resonance frequency and mechanical Q-factor of a resonator directly as well as their dependences on the electrical impedance which is connected to the resonator. From the equivalent circuit analysis the changes in resonance frequency and Q-factor due to the piezoelectric loading effects are determined. Measurements on two typical commercial piezoelectric resonators, an AT-cut quartz crystal and a PZT ceramic resonator, are in good agreement with the analysis.     

The band gaps of plate-mode waves in one-dimensional piezoelectric composite plates: polarizations and boundary conditions

Theoretical studies are presented for the band structures of plate-mode waves in a one-dimensional (1-D) phononic crystal plate consisting of piezoelectric ceramics placed periodically in an epoxy substrate. The dependences of the widths and starting frequencies of first band gaps (FBG) on the filling fraction and the thickness to lattice pitch ratio are calculated for different polarizations of piezoelectric ceramics under different electric boundary conditions, i.e., short circuit (SC) and open circuit (OC). We found that the FBG always is broadened by polarizing piezoelectric ceramics, and the FBG widths with SC always are larger than that with OC for the same polarization. Our research shows that there are three critical parameters which determine the FBG: the polarized directions, the filling fraction, and the ratio of the plate thickness to the lattice pitch, respectively. Therefore, we can control the width and starting frequency of the FBG in the engineering according to need by choosing these parameters of the system.     

Thickness-shear mode shapes and mass-frequency influence surface of a circular and electroded AT-cut quartz resonator

Finite-element solutions for the fundamental thickness shear mode and the second-anharmonic overtone of a circular, 1.87-MHz AT-cut quartz plate with no electrodes are presented and compared with previously obtained results for a rectangular plate of similar properties. The edge flexural mode in circular plates, a vibration mode not seen in the rectangular plate is also presented. A 5-MHz circular and electroded AT-cut quartz plate is studied. A portion of the frequency spectrum is constructed in the neighborhood of the fundamental thickness-shear mode. A convergence study is also presented for the electroded 5-MHz plate. A new two-dimensional (2-D) technique for visualizing the vibration mode solutions is presented. This method departs substantially from the three-dimensional (3-D) ;wire-frame' plots presented in the previous analysis. The 2-D images can be manipulated to produce nodal line diagrams and can be color coded to illustrate mode shapes and energy trapping phenomenon. A contour plot of the mass-frequency influence surface for the plated 5-MHz resonator is presented. The mass-frequency influence surface is defined as a surface giving the frequency change due to a small localized mass applied to the resonator surface.     

Finite element simulation of piezoelectric transformers

Piezoelectric transformers are nothing but ultrasonic resonators with two pairs of electrodes provided on the surface of a piezoelectric substrate in which electrical energy is carried in the mechanical form. The input and output electrodes are arranged to provide the impedance transformation, which results in the voltage transformation. As they are operated at a resonance, the electrical equivalent circuit approach has traditionally been developed in a rather empirical way and has been used for analysis and design. The present paper deals with the analysis of the piezoelectric transformers based on the three-dimensional finite element modelling. The PIEZO3D code that we have developed is modified to include the external loading conditions. The finite element approach is now available for a wide variety of the electrical boundary conditions. The equivalent circuit of lumped parameters can also be derived from the finite element method (FEM) solution if required. The simulation of the present transformers is made for the low intensity operation and compared with the experimental results. Demonstration is made for basic Rosen-type transformers in which the longitudinal mode of a plate plays an important role; in which the equivalent circuit of lumped constants has been used. However, there are many modes of vibration associated with the plate, the effect of which cannot always be ignored. In the experiment, the double resonances are sometimes observed in the vicinity of the operating frequency. The simulation demonstrates that this is due to the coupling of the longitudinal mode with the flexural mode. Thus, the simulation provides an invaluable guideline to the transformer design     

A new method for continuous monitoring of series resonance frequency and simple determination of motional impedance parameters for loaded quartz-crystal resonators

A new electronic design for continuous motional series resonant frequency monitoring of loaded quartz crystal resonators is presented. Using this circuit, a low-cost method for a simple determination of equivalent circuit parameters of quartz crystal resonators is described. Measurements made with the proposed system on typical AT cut quartz crystals are in good agreement with those of an Impedance Analyzer.