Equations Of Linear Piezoelectricity Research Articles

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

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

FEM Analysis of Piezoelectric Resonator Polarization Process

The polarization of the piezoelectric resonator depends on the direction of the applied electric field. The direction of the applied electric field is determined by the shape of the resonator and the position of the electrodes. In case of resonators with electrodes incompletely covering their bases, an inhomogeneous electric field is generated, which results in an inhomogeneous polarization of the resonator. The resonator will be polarized in some places either in a direction other than the desired one or not polarized at all. The aim of this work is to analyze the polarization process on resonators with electrodes incompletely covering their bases. The physical description is given by the linear piezoelectric equations, the Gaussian equation for the description of the electric field and by Newton’s law of force. On this basis, a FEM model is developed and used to analyze the polarization process. The results of the calculation of the electric field vector distribution are presented. Finally, the areas are identified in which polarization in the desired direction is achieved in the resonator as well as the ones where no polarization occurs.

SC-Cut Quartz Resonators for Dynamic Liquid Viscosity Measurements.

This article proposes an innovative viscosity sensor based on the thickness-shear vibration of stress compensated (SC)-cut quartz resonator. The thickness-shear mode is first analyzed and further studied with fluid-structure interaction between the resonator and the viscous fluid loading. The characteristic equation is derived based on the 3-D linear piezoelectric equations and solved for sensitivity analysis. Then laboratory experiment is carried out to validate the theory. To conduct the viscosity measurement, the SC-cut quartz resonator is integrated with a U-tube test fixture, which is designed and fabricated for sensor housing to avoid the influence of the mass of the fluid. The resonator is tested with various viscosities by tuning the ratio of glycerol/water mixture. Experiment results show consistency with the analytical solution, which together present an improved sensitivity of viscosity measurement by using SC-cut quartz resonator comparing to other resonator-based viscosity sensors. The proposed viscosity sensor is sensitive, accurate, and portable, and therefore can be applied to real-time, on-site measurement or sampling of fluidic samples.

Sensitivity analysis of multilayered C-axis inclined zigzag zinc oxide thin-film bulk acoustic resonators for CO2 concentration monitoring

This paper proposes a theoretical analysis of an innovative C-axis inclined zigzag multilayered ZnO thin-film bulk acoustic resonator (FBAR) for CO2 concentration monitoring. Analytical study is carried out with 3-D linear piezoelectric equations with appropriate boundary conditions. Numerical results show that the thickness of FBAR has a constant influence on the sensitivity while C-axis inclined angle has a significant influence on the sensitivity for the longitudinal mode of multilayered (double-layered in this paper) FBAR CO2 sensor. The selectivity and sensitivity of the proposed sensor can be further improved by the dopant in ZnO FBAR for real environment monitoring.

An analysis of PN junctions in piezoelectric semiconductors

We present a theoretical study on the equilibrium state of a PN junction, which is created by two piezoelectric semiconductor half spaces doped oppositely, based on the equations of linear piezoelectricity and the conservation of charge for holes and electrons. The nonlinearity associated with the drift currents of electrons and holes, which appears as products of the unknown carrier concentrations and the unknown electric field, is linearized for small carrier concentration perturbations. An analytical solution is rigorously derived, which is able to show the shaping of the PN junction near the interface. The electromechanical fields and concentrations of electrons and holes near the interface of the PN junction, and the forward-bias current-voltage characteristics of the PN junction under different applied stresses, are calculated and discussed. The effects of a few physical parameters on the properties of the PN junction are investigated as well.

Power delivery to a piezoelectric transducer embedded in an elastic body by acoustic waves

ABSTRACTWe studied the transmission of power to a piezoelectric transducer embedded in an elastic body by acoustic waves. A four-layer mechanics model was constructed. A theoretical analysis was performed using the equations of linear piezoelectricity and elasticity. The input admittance, the output voltage and the efficiency of the power transmission structure were calculated. It was shown that significant energy transmission can be produced at a few resonant frequencies. The effects of various physical and geometric parameters were examined. The performance of the structure is sensitive to these parameters and design optimization is necessary for effective power transmission. It was found that the location of the piezoelectric transducer inside the elastic body does not matter much. As long as the strain in the transducer is large and does not change its sign along the transducer thickness, a significant output can be received.

Shear-horizontal piezoelectric waves in an aluminum nitride film on a silicon substrate

ABSTRACTThe propagation of shear-horizontal waves in a piezoelectric film of aluminum nitride on a silicon substrate is studied. Three different electrode configurations are considered for thin film acoustic wave resonator application. A theoretical analysis is performed. The equations of linear piezoelectricity and anisotropic elasticity are used for the film and the substrate, respectively. Real and imaginary dispersion curves as well as electromechanical mode shapes are obtained. The effects of electrode configuration on the distribution of the electromechanical fields and the dispersion curves of long thickness-twist waves as well as energy trapping are examined.

Analytical analysis of a beam flexural-mode piezoelectric actuator for deformable mirrors

A beam flexural-mode piezoelectric bimorph actuator is analyzed based on linear piezoelectricity, and the performance of the actuator is studied. The beam bimorph piezoelectric actuator (BBPA), which is a sand- wich compound consisting of a lower and an upper piezoelectric ceramic surface layer and a middle layer made of metal, is driven to flexural deformation. The statistical analytical solution and dynamical solutions from the three-dimensional equations of linear piezoelectricity are derived, and the dependence of the performance upon the physical parameters of the BBPA is evaluated. Numerical results illustrate the strengthened performance achieved by adjusting the geometrical and material parameters of the BBPA.©2015SocietyofPhoto-OpticalInstrumentation

Theoretical and experimental investigations of thickness- stretch modes in 1-3 piezoelectric composites

Bulk piezoelectric ceramics operating in thickness-stretch (TSt) modes have been widely used in acoustic-related devices. However, the fundamental TSt waves are always coupled with other modes, and the occurrence of these spurious modes in bulk piezoelectric ceramics affects its performance. To suppress the spurious modes, 1-3 piezoelectric composites are promising candidates. However, theoretical modeling of multiphase ceramic composite objects is very complex. In this study, a 1-3 piezoelectric composite sample and a bulk piezoelectric sample are fabricated. The electrical impedance of these two samples are compared. A simple analytical TSt vibration mode from the three dimensional equations of linear piezoelectricity is used to model the performance of 1-3 piezoelectric composites. The theoretical results agree well with the experimental results.

Propagation of shear-horizontal waves in piezoelectric plates of cubic crystals

Shear-horizontal waves in a piezoelectric plate of cubic crystals are studied theoretically. Both electroded and unelectroded plates are considered. Exact solutions from the equations of linear piezoelectricity are obtained. The dispersion curves and the distributions of the mechanical displacement and electric potential are presented. There exists a nondispersive face-shear wave. The rest of the waves are dispersive thickness-twist waves. The waves separate into symmetric and antisymmetric groups about the plate middle plane. The dispersion curves for electroded and unelectroded plates are very similar to each other for real wave numbers but have qualitatively different behaviors for pure imaginary wave numbers. For unelectroded plates, neglecting the electric field in the surround free space does not have significant effects.

Modeling of 1-3 piezoelectric composites operating in thickness-stretch vibration mode

For bulk piezoelectric ceramics plates, the fundamental thickness-stretch (TSt) waves are always coupled to the in-plane extension waves and the symmetric thickness-shear waves. The occurrence of these spurious modes in bulk piezoelectric ceramics plates is undesirable as it may interfere with the operation of transducers. 1–3 piezoelectric composites are promising candidates to suppress the spurious modes mentioned above. However, theoretical modelling of multiphase ceramic composite objects is very complex. In this study, a simple analytical TSt vibration model is constructed from three-dimensional equations of linear piezoelectricity. The mechanical damping is considered in the model by introducing a complex elastic constant. The performance of 1–3 piezoelectric composites is analysed and the electrical impedance results from theoretical and experimental analysis are compared. The results show that there is excellent agreement between the experimental electrical impedance and that obtained by the theoretical TSt vibration analysis. This indicates that 1–3 piezoelectric composites can be operated in a nearly pure TSt vibration mode near the fundamental resonance. The analytical model we present is valid for analysing 1–3 piezoelectric composites plates with large aspect ratios quickly and efficiently.

Efficient Waveform Preservation of Discrete Spectrum Signals With Multilayered Piezoelectric Transformer

Transformation of square pulse signals by the multilayer Rosen type piezoelectric transformer (PT) is investigated with the goals to preserve the signal waveform and achieve maximum efficiency for power electronics and LED lighting applications. The voltage gain of the transformer is calculated from equations of linear piezoelectricity and motion for a broad frequency range that includes multiple resonant frequencies of PT. The transformation of the square pulse is simulated numerically by applying the voltage gain function to the Fourier spectrum of the input signal, and then, restoring the time-domain output signal from its spectrum. A half-bridge MOSFET amplifier is built to drive the PT in the experimental setup. The measurements of the input and output waveforms and their Fourier spectra verify the calculations. It is shown that the components of the spectrum and the waveform of the square pulse signal can be preserved in the output signal by using multiple resonant frequencies of the transformer. With the accurate match between the signal harmonics and the resonant frequencies of the transformer voltage gain, the square pulse signal provides higher output power than a sine or a square wave, provided that the dominant harmonics of all signals have the same amplitude.

Thickness-twist and face-shear waves in piezoelectric plates of monoclinic crystals

We study the propagation of thickness-twist and face-shear waves in piezoelectric plates of monoclinic crystals, which include rotated Y-cut quartz and langasite as special cases. The equations of linear piezoelectricity are used. Exact solutions are obtained for both fully-electroded and unelectroded plates. Dispersion relations are plotted for both AT-cut quartz and Y-cut langasite plates. The difference in frequencies between the exact piezoelectric solutions obtained in the present paper and the approximate elastic solutions in the literature is found to be of the order of 1%, which is significant in acoustic wave resonator and sensor applications.

Effect of initial stress on Love waves in a piezoelectric structure carrying a functionally graded material layer

The effect of initial stress on the propagation behavior of Love waves in a piezoelectric half-space of polarized ceramics carrying a functionally graded material (FGM) layer is analytically investigated in this paper from the three-dimensional equations of linear piezoelectricity. The analytical solutions are obtained for the dispersion relations of Love wave propagating in this kind of structure with initial stress for both electrical open case and electrical short case, respectively. One numerical example is given to graphically illustrate the effect of initial stress on dispersive curve, phase velocity and electromechanical coupling factor of the Love wave propagation. The results reported here are meaningful for the design of surface acoustic wave (SAW) devices with high performance.

Thickness-shear vibration of a circular cylindrical ceramic cylinder with unattached electrodes and air gaps

We analyze thickness-shear vibration of an axially poled circular cylindrical tube with unattached electrodes and air gaps. Both free and electrically forced vibrations are studied. Exact solutions are obtained from the equations of linear piezoelectricity. Resonant frequencies and the impedance of the transducer are calculated from the solution. Results show that the resonant frequencies are sensitive to the dimensions of the air gaps when the gaps are thin. The impedance depends strongly on the air gaps.

A circular cylindrical, radially polarized ceramic shell piezoelectric transformer

We propose a circular cylindrical shell piezoelectric transformer of radially polarized ceramics operating in thickness-stretch modes. Two theoretical analyses involving different approximations are performed on the proposed transformer using the 3-D equations of linear piezoelectricity. The transforming ratio, input admittance, and efficiency of the transformer are calculated and compared with those of a previously analyzed circular cylindrical shell transformer of axially polarized ceramics operating in shear modes.

Thickness vibration of piezoelectric plates of 6 mm crystals with tilted six-fold axis and two-layered thick electrodes

We perform a theoretical analysis of thickness vibrations in piezoelectric plates of crystals with 6 mm symmetry. The six-fold axis is tilted with respect to the plate surfaces. The major surfaces of the plate are covered with two layers of electrodes of different metals. The equations of linear piezoelectricity are used for the crystal plate. The electrodes are modeled by the equations of elasticity. Thickness vibrations frequencies and modes as well as impedance are calculated and examined.

Transmitting electric energy through a closed elastic wall by acoustic waves and piezoelectric transducers

Transmission of electric energy through a closed elastic wall by piezoelectric transducers and acoustic waves is studied based on the linear theory of piezoelectricity and elasticity. A theoretical analysis is performed. For the structure and motion considered, the 3-D equations of linear piezoelectricity reduce to a 1-D mathematical problem. An exact solution is obtained. Transmitted voltage, current, power, efficiency and stress distribution are obtained. Their dependence on various parameters is examined. The model and results of this paper are closer to real situations compared with those in a previous analysis.

Thickness resonances dispersion characteristics of a lossy piezoceramic plate with electrodes of arbitrary conductivity

A theoretical description of the dissipative phenomena in the wave dispersion related to the "energytrap" effect in a thickness-vibrating, infinite thicknesspolarized piezoceramic plate with resistive electrodes is presented. The three-dimensional (3-D) equations of linear piezoelectricity were used to obtain symmetric and antisymmetric solutions of plane harmonic waves and investigate the eigen-modes of thickness longitudinal (TL) up to third harmonic and shear (TSh) up to ninth harmonic vibrations of odd- and even-orders. The effects of internal and electrode energy dissipation parameters on the wave propagation under regimes ranging from a short-circuit (sc) condition through RC-type relaxation dispersion to an opencircuit (oc) condition are examined in detail for PZT piezoceramics with three characteristic T -mode energy-trap figure-of-merit c-(D)(33)/c-(E)(44) values - less, near equal and higher 4 - when the second harmonic spurious TSh resonance lies below, inside, and above the fundamental TL resonanceantiresonance frequency interval. Calculated complex lateral wave number dispersion dependences on frequency and electrode resistance are found to follow the universal scaling formula similar to those for dielectrics characterization. Formally represented as a Cole-Cole diagram, the dispersion branches basically exhibit Debye-like and modified Davidson Cole dependences. Varying the dissipation parameters of internal loss and electrode conductivity, the interaction of different branches was demonstrated by analytical and numerical analysis. For the purposes of dispersion characterization of at least any thickness resonance, the following theorem was stated: the ratio of two characteristic determinants, specifically constructed from the oc and sc boundary conditions, in the limit of zero lateral wave number, is equal to the basic elementary-mode normalized admittance. As was found based on the theorem, the dispersion near the basic and nonbasic TL and TSh resonances reveal some simple representations related to the respective elementary admittance and showing the connection between the propagation and excitation problems in a continuous piezoactive medium.

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 software for estimating the Q of a resonator without apriori assumptions of the resonator im- pedance 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-electro- static 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 as- sumptions of damping or resistance. Forced and free vibration analyses were per- formed and the results for the Q and other electrical parameters were obtained. Comparisons of the Q and other electrical parameters obtained from the free vibra- tion 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.