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In this paper, new frequency equations of a shear mode c-axis tilted FBAR (Film Bulk Acoustic Wave Resonator) sensor operating in viscous liquid media are derived by using the transfer matrix method to predict the frequency shift (FS) induced by fluid viscosity. Top and bottom electrodes of FBAR are considered as a superposition of a pure elastic model and a perfectly conducting model. Employing the perturbation method and developed frequency equations, an approximate FS expression is obtained to study the effects of viscous liquid properties on frequency shifts for the sensor operating in water-glycerol solutions. Numerical results by the developed frequency equations are compared with the published experimental results and those predicted by the traditional Stockbridge-Kanazawa equation, which was originally derived for quartz crystal microbalances (QCMs). Meanwhile, the influence of the electrode material and thickness on FS is also discussed to demonstrate the outperformance of the developed equations over the Stockbridge-Kanazawa equation. Results show that the analytical equations for frequency shift analysis have more potential than the well-known Stockbridge-Kanazawa equation to predict frequency shift characteristics of FBAR sensors in terms of accuracy and efficiency.

Mode-coupled vibrations in a thickness-shear (TSh) mode and laterally finite film bulk acoustic resonator (FBAR) with one face in contact with Newtonian (linearly viscous and compressional) liquid are investigated. With boundary conditions and interface continuity conditions, exact dispersion curves in FBAR sensors contacting with two kinds of liquids are obtained, and they are compared with the dispersion curves in a bare sensor without liquid contact. Frequency spectra, describing mode couplings between the main TSh modal branch and undesirable modal branches, are calculated by employing weak boundary conditions at lateral free edges constructed based on the variational principle. Mode shapes of mechanical displacements in both the sensor and liquid layer are presented, and mode transformations are observed due to the liquid contact and lateral edge effect. The effect of liquid thickness on frequency spectra is also studied. Numerical results reveal that the generation of shear wave in the liquid layer results in the shifts of spectrum curves along the frequency axis and hence it is the main factor of frequency shifts of FBAR sensors. The compressional wave causes the shifts of spectrum curves along the lateral aspect ratio axis. Then for a given FBAR sensor, the liquid thickness changes could also cause frequency shifts. Therefore, desirable vibration modes should be chosen based on the frequency spectra to avoid strong mode couplings and to eliminate frequency shifts induced by the liquid thickness changes in real applications.

The second (H2) and third (H3) harmonic generations in surface acoustic wave (SAW) resonators based on the YZ-cut LiNbO 3 operating in both longitudinal leaky SAW (LLSAW) mode and Rayleigh SAW mode were investigated and compared. For the Rayleigh SAW mode, the patterns and trends of nonlinearly excited harmonics in the YZ-cut LiNbO 3 were the same as those of 128° YX LiNbO 3 except that the peaks of nonlinear harmonic responses in YZ-cut LiNbO 3 were denser than the latter. However, for the YZ-cut LiNbO 3 LLSAW resonators, the peaks of the nonlinear harmonic responses would be 20 to 30 dBm lower than those in the Rayleigh SAW resonators. For the YZ-cut LiNbO 3 LLSAW resonators, we found that the quasi-periodic peaks in the nonlinear harmonics response would disappear at a substrate thickness of 500 μm. Results showed that the nonlinear harmonic generation concerns in SAW filters and duplexers may be substantially reduced by the use of longitudinal leaky SAW based on YZ-cut LiNbO 3 .

The heterostructures made of magnetostrictive and piezoelectric layers have been recently investigated for the application of magnetic sensors and antenna. In this paper we investigate the possibility of FeGaB thin film on quartz crystal substrate with different orientations of quartz layer. Response of magnetic FeGaB/AlN nanoplate resonator under RF magnetic excitation was simulated by finite element program COMSOL. The results were compared with available experiment and confirmed the validity of available material properties of FeGaB thin film and current simulation method. Quartz/FeGaB composite resonator structures based on different modes of vibration were studied. Quartz/FeGaB resonator based on in-plane extensional modes was found to have a large magnetoelectric coupling effect. These findings can be used to design RF magnetic sensors based on Quartz/FeGaB bilayer plate.

The nonlinear intermodulation effect in piezoelectric devices was studied By using the three-dimensional equations of nonlinear piezoelectricity and the method of harmonic balance, we have formulated and derived coupled sets of nonlinear piezoelectric equations in the frequency domain for intermodulation effects. The coupled sets of equations could be embedded in COMSOL FEM models. In order to check the validity and accuracy of the FEM models we studied the third order intermodulation effects in AT-cut quartz resonators. Our three-dimensional FEM model results compared well with the analytical one-dimensional results by Tiersten and with the experimental results by Smythe. The FEM models could be used for simulating intermodulation effects in BAW and SAW piezoelectric devices provided that the nonlinear material constants are available and are accurate.

Recent research work on the theoretical and experimental possibilities of acoustically actuated MEMS RF magnetoelectric antennas using piezoelectric-magnetostrictive multilayers have been published, for example in ref. [1], [2]. A mechanism of electromagnetic-acoustic coupling was achieved in the composite ferromagnetic/piezoelectric heterostructure. The magnetostrictive thin film provided the function of converting the RF magnetic signal to acoustic vibration of the composite resonator while the piezoelectric body realized the transduction of acoustic wave to electric signal. In this paper, we proposed a novel MEMS RF FeGaB/Quartz receiver sensor where FeGaB had a relatively stronger piezomagnetic coefficient compared to other ferromagnetic materials such as nickel. The quartz resonator had the advantages of high Q, low resistance, and high frequency-temperature stability. Experiments were set up to measure the remnant magnetization state of FeGaB thin film deposited on AT- and SC-cut quartz wafers. Consistent comparison results were obtained between the simulated and measured magnetization curves of FeGaB thin films. A 10 MHz AT-cut quartz resonator coated with 200-nm-thick FeGaB layer was fabricated and tested. The frequency response of a similar uncoated quartz resonator was also tested so that the difference between the responses of coated and uncoated resonator respectively could yield a qualitative view of the magnetic field response. A special setup of experiments were taken to reduce the electric field effect of the RF signal. The simulated frequency responses of the coated and uncoated resonators showed trends that were consistent with the experimental results.

The characteristics of nonlinear harmonic generation of bulk acoustic wave (BAW) modes in surface acoustic wave (SAW) devices were studied and presented. Specifically the BAW modes were nonlinearly generated ($f$-2$f$-3$f$) by the fundamental SAW mode in a 128° YX LiNO 3 temperature compensated SAW (TC-SAW) resonator. These BAW modes interfered with the desired SAW mode of the TC-SAW resonator. The method and formulation of nonlinear frequency response of piezoelectric BAW and SAW was presented, and FEM was used to solve three sets of coupled nonlinear equations in the frequency domain. For SAW devices, nonlinearly excited BAW waves in 128° YX LiNO 3 TC-SAW devices had been observed experimentally and reported [1]. We demonstrated our formulation/method for this nonlinear harmonic generation problem, and we obtained good, consistent simulation results. The 2nd harmonic (H2) generation was due to a nonlinearly excited bulk wave mode polarized and travelling along the thickness direction of substrate. The mode was almost normal to the top and bottom surfaces of substrate, forming a standing wave between top and bottom surfaces. Therefore, the substrate thickness, and bottom surface roughness had effects on the H2 response. The H3 response was due to a similar type of nonlinearly excited BAW wave.

Forced mode coupling vibration of ZnO thin film resonator with its c-axis along thickness direction operating at thickness-extensional mode is studied in this paper. The zone dispersion curve is obtained since the mechanical damping of ZnO thin film is considered in this case. The displacement solutions are constructed based on the zone dispersion curve and the Mason model solution, which are then substituted into weak boundary conditions at the left and right edges determining the undetermined constants. The effect of lateral electrode size on suppressing spurious modes is particularly studied. The results show that with lateral electrode size increasing, more spurious modes occurs in the band between fs and fp but their amplitude strength become weaker. Therefore, in realistic resonators a larger lateral boundary is more desirable and then the band between fs and fp would become smoother.

Previous papers and experiments have well documented the drive level dependency in a single mode. The parameters governing drive level dependency in quartz resonators however are still not well understood. We show in this paper the nonlinear interactions of the 3rdovertone of thickness shear (f3) mode and the fundamental thickness shear mode (f1) on the drive level dependency of a quartz resonator. We have developed a finite element program/model for nonlinear frequency responses of f-2f-3f modal interactions and drive level dependency in quartz resonators. Our model results compared well with the measured results in quartz 33-MHz fundamental mode AT-cut and SC-cut resonators for which the fundamental mode and third overtone mode frequencies were closely matched by design. The fundamental mode interacted nonlinearly with the third overtone mode. An interesting effect was observed that the frequency shift (drive level dependency) was greater for when the 3rd harmonic 3f frequency was higher than the 3rd overtone f3 frequency, especially for the AT-cut resonators.

The thickness-extensional mode in layered piezoelectric plates can be inevitably interfered by some undesirable eigen-modes due to the lateral edge effect. A reported theoretical method to predict high-frequency coupled vibrations is the 2D high-order plate theory derived based on approximate dispersion relations. In this paper, based on accurate dispersion relations a novel approach called Frequency Spectrum Quantitative Prediction (FSQP) is developed to quantitatively investigate coupled vibration behaviors in piezoelectric multilayered plates operating at ultra-high frequency. Two significant sub-goals need to be achieved: one is the accurate dispersion relations of the layered structure; the other is the variational formulation for the layered plates with piezoelectric and/or elastic phases. Lastly, the objective equation, i.e., frequency spectra, describing coupling strengths between the thickness-extensional mode and unwanted eigen-modes, is derived. A reported numerical example of a piezoelectric thin-film resonator is considered to demonstrate the correctness and superiority of the proposed methodology. Mode shapes of mechanical displacements are investigated in detail to illustrate the application of frequency spectra to suppress the undesirable eigen-modes. Numerical results show that the proposed approach FSQP is efficient and more accurate than the existing 2D high-order plate theory in quantitative predictions of high-frequency coupled vibrations in layered piezoelectric plates.

Incremental piezoelectric equations for small vibrations superposed on initial deformations are presented. The equations are implemented in COMSOL finite element models (FEA). Equations are validated by comparing the results for the force sensitivity coefficient Kf of a circular quartz plate subjected to a pair of diametrical forces with measured data. The model results show a consistent trend with the experimental results, and the relative difference between our FEA results and Ballato's measured result is about 13%. A detailed study of the acceleration sensitivity of a rectangular AT-cut quartz plate is presented. The plate resonator is fixed along one edge as a cantilever. For AT-cut quartz resonators with the crystal digonal X-axis perpendicular to plate X-axis, the in-plane acceleration sensitivity is found to be negligible compared with the out-of-plane (Y-axis) acceleration sensitivity. For AT-cut quartz resonators with the crystal digonal X-axis parallel to plate X-axis, the Y-axis acceleration sensitivity is found to be rectified, that is the fractional change in frequency is positive with respect to both positive and negative Y-axis accelerations. The Y-axis acceleration sensitivity is small in comparison with the in-plane acceleration sensitivity for small body forces. However, for large body forces, the Y-axis acceleration sensitivity dominates because it increases nonlinearly with the Y-axis acceleration. The resonator rectified acceleration sensitivity is confirmed by phase noise measurements. For reduced acceleration sensitivity, two pairs of electrodes along the plate edges reduce the bending of the plate resonator and subsequently reduce acceleration sensitivity. We present a new method using these edge electrodes in which a dc bias field is employed to control the resonant frequency of resonator subjected to g body forces. A dc bias field with an appropriate dc bias voltage could potentially yield a reduction of acceleration sensitivity in Y-axis direction of about two orders of magnitude.

A general purpose finite element model that yielded directly the nonlinear frequency response of the second harmonic generation in SAW IDT resonators was developed. The model enabled us to study the effects of the nonlinear material constants, voltage drive, duty factor, and IDT geometry on f-2f nonlinear interactions. Specifically we have applied our model to the study of nonlinear frequency response of second harmonic generation in lithium niobate YX-127o SAW IDT resonators. Lithium niobate was chosen because a complete set of its nonlinear material constants was known and published in the literature. A complete set of nonlinear piezoelectric equations along with their material constants for a lithium niobate YX-127o SAW IDT 836 MHz resonator was implemented in a finite element program. Two sets of finite element matrix equations were derived for the fundamental SAW mode and its 2nd harmonic mode, respectively. We demonstrated how two frequency domain interfaces could be coupled together to simulate the f-2f nonlinear interactions. The fundamental Rayleigh SAW mode was excited by a drive voltage at the IDT that in turn generated the 2nd harmonic mode at twice the frequency. A higher voltage drive lead to higher 2f interactions. Effects of the 2nd harmonic generation on the fundamental SAW mode and its Q were observed when the voltage drive was greater than 0.5V. For example, when the drive voltage was 0.1V the fundamental mode was clean. But when the drive voltage was increased to 0.5V the fundamental mode exhibited Q reduction and stronger f-2f interactions were observed, namely, there were stronger interactions of the 2f BAW mode with the f SAW mode. There was an optimal duty factor of 0.4 to 0.5 for the least f-2f interactions. Thinner electrode thickness resulted in stronger f-2f interactions.

The nonlinear characteristics of SAW resonators have been of great interest recently. We present a new nonlinear solution algorithm for the f-2f nonlinear coupling phenomenon in SAW IDT resonators. Specifically we have applied our algorithm to obtain the f-2f frequency response of the lithium niobate YX-127o SAW IDT resonators. Lithium niobate was chosen because a complete set of its nonlinear material constants was known and published in the literature. The algorithm consisted of solving the nonlinear finite amplitude model in the time domain to obtain the time dependent response, and then performing a fast Fourier transform of the time dependent response. In order to minimize the transient response, the initial conditions for the time domain solution were obtained from the frequency response of the linear model. Results were presented for a one-electrode IDT SAW resonator at increasing drive voltages. The charge on the electrode IDT for the nonlinearly generated 2f mode increased with the square of the magnitude of f mode drive level voltage.

The experimental studies of temperature coefficient of frequency (TCF) of SAW resonators were usually modeled by TCF models first proposed by Bechmann, Ballato and Lukaszeck[1]. These models unfortunately were not frame invariant with regards to the material property tensors of the crystal substrate and metal interdigital transducers (IDTs). Furthermore the models did not have the 3rd order nonlinear elastic constants that were needed to account for the effects of thermal strains from mountings, boundary constraints, and material interfaces. We employed a Lagrangian model of TCF of SAW resonators wherein governing equations for incremental vibrations superposed on thermal strains were implemented in a finite element program. The model used material coordinates referred to a fixed reference frame at a reference temperature, therefore the material property tensors were frame invariant. In addition it had a term that was a product of the 3rd order nonlinear elastic constants and thermoelastic strains. This product could account for the thermoelastic strains from interfacial stresses and boundary conditions. The objective of our study was to provide an accurate and detailed finite element model of the effects of the crystal substrate, metal IDTs, and stack layers on the TCF of SAW resonators. Good comparisons of model results with the experimental results were found for the 1 GHz lithium tantalate YXl 42° shear horizontal SAW resonator. Characterization of the electrode film properties was found to be important, especially when the film thickness was increased from 2000 A to 4000 A.

The acceleration sensitivity and its related problem of force-frequency in quartz resonators were studied. We studied the effects of initial nonlinear strains versus the effects of nonlinear elastic constants on the force-frequency problem of quartz resonators subjected to bending forces. We found that for quartz thickness shear resonators subjected to bending, the force frequency effect could be described primarily by the effects of nonlinear elastic constants with negligible effects from initial nonlinear strains. Since the acceleration sensitivity of quartz resonators is due mainly to bending forces, we proposed a set of simplified equations for these problems which were easier to implement in commercial finite element software.

We present a new method in which the electrode stress can be used to improve the frequency-temperature (f-T) behavior of ultra-high frequency (UHF) quartz resonators. The electrode films are treated as composite layers of different metals in a plate resonator. A FEM model for the effects of non-homogeneous thermal stress on the f-T behavior of quartz resonators was developed. For our 1 GHz AT-cut resonator, the use of chromium-aluminum electrodes yields improved frequency-temperature behavior compared to the case where aluminum electrodes are used alone. The UHF quartz resonators must be treated as composite plates of quartz and electrode film since the ratio of electrode thickness to quartz plate thickness is significant. The quartz-aluminum composite plate rotates the f-T curve clockwise while the quartz-chromium composite plate rotates the f-T curve counter-clockwise. The titanium was found to be a good electrode metal due to its small effect on the f-T curve.

Theoretical analyses and designs of high-Q, quartz thin film resonators are presented. The resonators operate at an ultra-high frequency of 3.4 GHz for application to high-frequency timing devices such as cesium chip-scale atomic clocks. The frequency spectra for the 3.4-GHz thin film quartz resonators, which serve as design aids in selecting the resonator dimensions/configurations for simple electrodes, and ring electrode mesa designs are presented here for the first time. The thin film aluminum electrodes are found to play a major role in the resonators because the electrodes are only one third the thickness and mass of the active areas of the plate resonator. Hence, in addition to the material properties of quartz, the elastic, viscoelastic, and thermal properties of the electrodes are included in the models. The frequency-temperature behavior is obtained for the best resonator designs. To improve the frequency-temperature behavior of the resonators, new quartz cuts are proposed to compensate for the thermal stresses caused by the aluminum electrodes and the mounting supports. Frequency response analyses are performed to determine the Q-factor, motional resistance, capacitance ratio, and other figures of merit. The resonators have Q's of about 3800, resistance of about 1300 to 1400 ohms, and capacitance ratios of 1100 to 2800.

Common piezoelectric resonators such as quartz resonators have a very high Q and ultra stable resonant frequency. However, due to small material nonlinearities in the quartz crystal, the resonator is drive level dependent, that is, the resonator level of activity and its frequency are dependent on the driving, or excitation, voltage. The size of these resonators will be reduced to one fourth of their current sizes in the next few years, but the electrical power which is applied will not be reduced as much. Hence, the applied power to resonator size ratio will be larger, and the drive level dependency may play a role in the resonator designs. We study this phenomenon using the Lagrangian nonlinear stress equations of motion and Piola–Kirchhoff stress tensor of the second kind. Solutions are obtained using COMSOL for the AT-cut, BT-cut, SC-cut and other doubly rotated cut quartz resonators and the results compared well with experimental data. The phenomenon of the drive level dependence is discussed in terms of the voltage drive, electric field, power density and current density. It is found that the drive level dependency is best described in terms of the power density. Experimental results for the AT-, BT- and SC-cut resonators in comparison with our model results are presented. Results for new doubly rotated cuts are presented. The effects of spurious modes, quality factor and air damping on DLD are presented.

The quartz resonator Q with aluminum electrodes was studied with respect to its fundamental thickness shear mode frequency and its viscoelastic, viscopiezoelectric, and viscopiezoelectromagnetic behaviors. The governing equations for viscoelasticity, viscopiezoelectricity, and viscopiezoelectromagnetism were implemented for an AT-cut quartz resonator. To simulate the radiation conditions at infinity for the viscopiezoelectromagnetic model, perfectly matched layers over a surface enclosing the resonator were implemented to absorb all incident electromagnetic radiation. The shape of the radiation spectrum of a 5.6 MHz AT-cut quartz resonator was found to compare relatively well the measured results by Campbell and Weber. The mesa-plate resonator was studied for a frequency range of 1.4 GHz to 3.4 GHz. The resonator Q was determined to be influenced predominantly by the quartz viscoelasticity; however at frequencies greater than 2.3 GHz, the quartz electromagnetic radiation had an increasingly significant effect on the resonator Q. At 3.4 GHz, the electromagnetic radiation accounted for about 14% of the loss in resonator Q. At frequencies less than 2 GHz, the calculated resonator Q compared well with the intrinsic Q(x) provided by the formula Q(x) = 16 x 10(6)/f where f was in MHz. At frequencies higher than 2.3 GHz, the aluminum electrodes had significant effects on the resonator Q. At 3.4 GHz, the electromagnetic radiation loss in the electrodes was an order of magnitude greater than their viscoelastic loss; hence, the vibrating aluminum electrodes became an efficient emitter of electromagnetic waves. The effects of electrical resistance in both the electrodes and quartz were determined to be negligible.

The frequency–temperature characteristics of quartz crystal resonators, particularly the frequency stability in a specific temperature range in which the vibration modes are strongly coupled, has been an important requirement in most applications. The analytical work on frequency–temperature relations has been done over the last decades in many aspects, ranging from the fundamental theory for the thermal effect on vibrations of elastic solids to simplified plate equations of a few strongly coupled vibration modes. However, it has been clearly observed that due to complications of the resonator structures such as the presence of a mounting structure and electrodes, simple and analytical solutions will not be able to consider all the factors which will have inevitable and noticeable effects. In this paper, we incorporate the frequency–temperature theory for crystal plates based on the incremental thermal field formulation by Lee and Yong into our finite element analysis implementation, which is then used to analyze the free vibrations of crystal plates with the higher-order Mindlin plate theory. The effect of electrodes on the frequency–temperature relation is also considered. The computational results are compared with experimental ones from actual products. The satisfactory agreement demonstrates the precise prediction of the frequency–temperature behavior and practical applications of the finite element analysis in product modeling and development.

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.

The frequency-temperature behavior of a piezoelectric crystal resonator can be predicted quite accurately if the resonator is under a stress-free and steady-state uniform temperature condition. The condition is however seldom achieved practically. Most practical resonators are subjected to thermal stresses. Conventional finite element analytical tools such as ANSYS cannot provide a sufficiently accurate model for the frequency-temperature behavior of piezoelectric quartz resonators. A new dynamic frequency-temperature model which accurately predicted the frequency-temperature behavior of quartz resonators affected by transient and steady state temperature changes was presented. Lagrangean equations for small vibrational (incremental) displacements superposed on initial thermal stresses and strains were employed. The initial thermal stresses and strains were obtained from the uncoupled heat and thermoelastic equations. The constitutive equations for the incremental displacements incorporated the temperatur...

The Mindlin plate equations with the consideration of thickness-shear deformation as an independent variable have been used for the analysis of vibrations of quartz crystal resonators of both rectangular and circular types. The Mindlin or Lee plate theories that treat thickness-shear deformation as an independent higher-order vibration mode in a coupled system of two-dimensional variables are the choice of theory for analysis. For circular plates, we derived the Mindlin plate equations in a systematic manner as demonstrated by Mindlin and others and obtained the truncated two-dimensional equations of closely coupled modes in polar coordinates. We simplified the equations for vibration modes in the vicinity of fundamental thickness-shear frequency and validated the equations and method. To explore newer structures of quartz crystal resonators, we utilized the Mindlin plate equations for the analysis of annular plates with fixed inner and free outer edges for frequency spectra. The detailed analysis of vibrations of circular plates for the normalized frequency versus dimensional parameters provide references for optimal selection of parameters based on the principle of strong thickness-shear mode and minimal presence of other modes to enhance energy trapping through maintaining the strong and pure thickness-shear vibrations insensitive to some complication factors such as thermal and initial stresses., Comment: Paper to be presented to the 2015 IEEE International Frequency Control Symposium and European Frequency and Time Forum, Denver, CO, USA. April 12-16, 2015

We present a new method in which a direct current (DC) bias field is used to control the resonant frequency of resonator subjected to body forces. The plate electrodes were used to create the DC bias field. Sensing electrodes were used to measure the body forces that cause the acceleration sensitivity. Finite element models were developed using the theory of small deformations superposed on finite initial deformations in Lagrangian formulation. The model results compared consistently well with the measured values for the force sensitivity coefficient K f of a circular plate subjected to a pair of diametrical forces; hence validating our model for acceleration sensitivity. For our 1 GHz AT-cut quartz plate resonator with the crystal digonal X-axis perpendicular to plate X-axis, the in-plane acceleration sensitivity is negligible, while the Y-axis acceleration sensitivity is maximum. A DC bias field with an appropriate DC bias voltage could yield a reduction in acceleration sensitivity from 10−10/g to 10−12/g in the Y-axis

Langasite and langatate materials are gaining in popularity for use in piezoelectric sensors because of their material properties. Both materials have similar crystal properties as the quartz crystal which is widely used in BAW applications such as resonators and a piezoelectric vibratory gyroscopes. Quartz, langasite, and langatate are trigonal class crystals belonging to the same point group 32. In this paper, we simulate and compare the gyro-characteristics of a piezoelectric doubled-ended tuning fork gyroscope using Comsol 4.2a software. Eigenfrequency analyses are performed to determine an optimal geometry of the gyroscope. Frequency response analyses are used to simulate the sensitivity of the gyroscope to angular velocity about the z-axis. The sensitivity of the quartz, langasite and langatate gyroscope are found to be 0.191 mV/(deg./s), 0.469 mV/(deg./s) and 0.784 mV/(deg./s), respectively. The results show that langasite and langatate crystal are more effective than quartz crystal for high precision piezoelectric gyroscopes.

High drive levels of the fundamental mode of a quartz shear-mode resonator can result not only in a frequency shift of the fundamental mode (amplitude-frequency effect) but also in nonlinear interactions with the third overtone mode. One such interaction reported here is a frequency shift of the third overtone mode. In addition, the third harmonic of the fundamental driving signal is generated which can appear close to, or directly at the third overtone resonance, depending on the relationship between the fundamental mode and third overtone mode frequencies. We examine these effects in 33-MHz fundamental mode AT-cut and SC-cut quartz resonators in which the fundamental mode and third overtone mode frequencies were closely matched by design.

For crystal resonators, it is always desirable to calculate the electric properties accurately for application purposes. Such calculations have been done with analytical solutions from approximate equations and simplified models with good results, but for better consideration of the actual resonators, finite element method has been used for the free vibration analysis with excellent results to aid the analysis and design. The finite element analysis based on the higher order Mindlin plate theory is particularly effective and easy to implement and expand. As an extension of the Mindlin plate theory based finite element analysis of crystal resonators, a new theory for the electroded plates is derived and the piezoelectrically forced vibrations are formulated and implemented in this paper in a manner similar to our previous work. The effect of the electrodes and the electric boundary conditions are taken into consideration through the modification of the higher order plate equations by changing the expansion function of the electric potential for this particular problem. Through the conventional discretization of the new plate theory, the linear equations for the piezoelectric plate under thickness excitation are constructed and solved with efficient numerical computation techniques such as the sparse matrix handling. The solutions of mechanical displacement and electric potential are then used for the computation of the capacitance ratio of the electroded plate with emphasis on its derivation with the two-dimensional plate theory. The applications of these results in crystal resonator modeling are discussed and demonstrated in detail. Numerical examples showing good predictions of the resonance frequency and capacitance ratio of electroded crystal plates of AT-cut quartz are presented with experimental data.

A 1 GHz AT-cut quartz thickness shear mode resonator is modeled for the first time with thermally induced bonding stresses and their effect on the device frequency-temperature (f-T) characteristic. Without the details of the bonding configuration, modeling indicates the f-T characteristic slightly rotates as a function of the change in stiffness of a simplified absorbing mount. However, if details of the bonding configuration are included, our modeling predicts the potential for a significant distortion in the f-T curve. High or varying stress over temperature in the device active region is found to lead to an undesirable increase in the f-T slope. The origin of the active region stress can be varied, but in practice it frequently originates from a temperature dependent bonding stress, or from fabrication steps such as metal depositions. In this paper we highlight the magnitude of the thermal stress effect on the f-T curve, and offer design methods that mitigate the thermally induced bonding stress by de-coupling the active resonator area from high stress regions of the quartz device.

Finite element matrix equations are derived from a two-dimensional, piezoelectric high frequency laminated plate theory. Two-layer ZnO/Si and three-layer ZnO/SiO/sub 2//Si thin film resonators vibrating at the fundamental thickness shear mode are investigated using the finite element models. Straight crested waves propagating in a resonator are studied. Resonant frequency, mode shapes and electromechanical coupling coefficient of the fundamental thickness shear mode are calculated and presented with respect to the C-axis orientation of the ZnO layer, and thickness ratios of the layers in a resonator. The results from the laminated plate model are checked against the three dimensional finite element solution and found to be accurate for the fundamental thickness shear and its anharmonic overtones. The plate model does not yield as accurate a frequency spectrum for flexure, face shear and extensional modes. >

Ultra high frequency resonators have higher noise levels due to their greater miniaturization and higher power density. This paper investigates a new method using nonlinear acoustic coupling for improving the resonator Q and hence reducing noise levels in third overtone thickness shear resonators. For trapped energy resonators such as the AT-cut quartz resonators the fundamental and third overtone thickness shear modes are well behaved. At high drive levels, the fundamental mode may be described by a Duffing equation that has a nonlinear cubic term in displacement. This cubic term in displacement has a third overtone thickness shear frequency component that could be used to improve the Q of third overtone thickness shear resonator. The coupling of the Duffing equation for the fundamental thickness shear mode to the third overtone thickness shear mode was solved using a MATLAB Simulink model. At higher drive levels, the mechanical nonlinearities of the fundamental mode will drive the third overtone thickness shear mode if its resonant frequency is sufficiently close to three times the fundamental thickness shear frequency. This nonlinear coupling will improve the Q of the third overtone thickness shear mode by as much as 15-fold. The Q increase is dependent on (1) frequency matching of the third overtone mode to three times the fundamental mode, (2) the drive level of the fundamental mode, and (3) the relative phase of the fundamental drive to the third overtone drive.

Characterization of a quartz resonator operating at 553 MHz in a closed loop oscillating system has been performed under various drive levels to characterize its nonlinear behavior. The voltage and current were measured across the resonator using differential active probes and used to calculate the magnitude of the admittance under drive. Our results show that nonlinear resonator operation widens the bandwidth of low phase noise operation. At the low frequency end of the operating range, where overall phase noise is degraded, phase noise improvement is ∼ 20dB. However, the overall phase noise levels do not improve significantly over the linear case.

A high frequency, piezoelectric, laminated plate theory is developed and presented for the purpose of modeling and analyzing piezoelectric thin‐film resonators and filters. The laminated plate equations are extensions of anisotropic composite plate theories to include piezoelectric effects and capabilities for modeling harmonic overtones of thickness‐shear vibrations. Two‐dimensional equations of motion for piezoelectric laminates were deduced from the three‐dimensional equations of linear piezoelectricity by expanding the mechanical displacements and electric potential in a series of trigonometric function, and obtaining stress resultants by integrating through the plate thickness. Relations for handling the mechanical and electrical effects of platings on the top and bottom surfaces of the laminate are derived. A new matrix method of correcting the cutoff frequencies is presented. This matrix method could also be used to efficiently correct the cutoff frequencies of any nth order plate laminate theories...

A theoretical study has been carried out for a 1.0–3.0 GHz rectangular monolithic BAW filter configuration for different Y-cut angles of LN and LT using 3-D finite element (FEM) models. The frequency-temperature characteristics for cut angles varying from 0 deg to 180 deg for Y-cut LN and Y-cut LT is evaluated using 3-D FE analysis. The performance of the monolithic filters as a function of resonator geometry, and acoustic gap is studied. FEM results for different Y-cut angles of LT and LN are obtained with respect to the optimal resonator geometry. FEM results show that Y-cut LN has a higher band-width as compared to that of the Y-cut LT, however LT shows a better TCF and a lower loss as compared to that of LN. Based on the 3D-FEM analysis cut angle in the range of 170 to 178 deg for Y-cut LN and 158 to 165 deg for Y-cut LT are identified as the promising cut angles for possible UHF applications.

A novel analytical/numerical method for calculating the resonator Q, and its equivalent electrical parameters due to viscoelastic, conductivity and mounting supports losses was presented. The method presented will be quite useful for designing new resonators, and reducing their time and costs of prototyping. There was also a necessity for better and more realistic modeling of the resonators due to miniaturizations, and the rapid advances in the frequency ranges of telecommunication. We present new three-dimensional finite elements models of quartz resonators with viscoelasticity, conductivity, and mounting support losses. For quartz the materials losses attributed to electrical conductivity and acoustic viscosity were obtained from Lee, Liu and Ballato, and Lamb and Richter, respectively. The losses at the mounting supports were modeled by perfectly matched layers (PML's). The theory for dissipative anisotropic piezoelectric solids given by Lee, Liu and Ballato was formulated in a weak form for finite element applications. PML's were placed at the base of the mounting supports to simulate the energy losses to a semi-infinite base substrate. FE simulations were carried out for free vibrations and forced vibrations of quartz tuning fork and AT-cut resonators. Results for quartz tuning fork and thickness shear AT-cut resonators were presented and compared with experimental data. Results for the resonator Q and the equivalent electrical parameters were compared with their measured values. Good comparisons were found. Results for both low and high Q AT-cut quartz resonators compared well with their experimental values. A method for estimating the Q directly from the frequency spectrum obtained for free vibrations was also presented. An important determinant of the quality factor Q of a quartz resonator is the loss of energy from the electrode area to the base via the mountings. The acoustical characteristics of the plate resonator are changed when the plate is mounted onto a base substrate. The base affects the frequency spectra of the plate resonator. A resonator with a high Q may not have a similarly high Q when mounted on a base. Hence, the base is an energy sink and the Q will be affected by the shape and size of this base. A lower bound Q will be obtained if the base is a semi-infinite base since it will absorb all acoustical energies radiated from the 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. >

Equations for the calculation of the equivalent circuit parameters of the piezoelectric resonator are presented. The results from both eigen-frequency and frequency response analyses could be used for calculating the electrical parameters of the Butterworth Van Dyke model of the piezoelectric resonator. The values of the electrical parameters C 1 , and L 1 obtained from both types of analyses were found to be similar, and were found to compare reasonably well with the measured values of two 12 MHz AT-cut quartz resonators. The values of R 1 , and Q were less well predicted when the resonator has a lower Q. The static capacitance C 0 could be calculated accurately by its frequency response at 1 Hz. The eigen-frequency analysis is better suited than the frequency response analysis for the study of electrode and plate geometry designs, hence the eigen-frequency equations for calculating the equivalent circuit parameters are more useful for parametric designs.

The finite element analysis of quartz crystal resonators has been gradually adopted in the design and product improvement for the advantage in predicting the vibration frequency, energy trapping, and calculation of device properties. The analysis can be done with 3D theory of elasticity or piezoelectricity or the Mindlin plate theory by general purpose software or custom development, which has the advantage of reducing the size of analysis significantly for typical thickness-shear vibrations of quartz crystal plates. While linear finite element analysis are adequate for the vibration frequency and frequency-temperature relations, further analysis on important phenomena and electrical parameters require the consideration of nonlinear material properties of quartz crystal. The finite element analysis of quartz crystal resonators has been making great contribution to the design and improvement facing the fast shrinkage of resonator size and elevated precision requirements. The full advantage of the finite element analysis can be taken if electrical parameters and performance behavior can be predicted with the improved analytical model and consideration of nonlinear material properties. The current approach based on the nonlinear theory will meet these objectives since the advantage of the finite element analysis on parallel platforms have been well understood and widely implemented. Our nonlinear analysis will utilize the nonlinear Mindlin plate theory and provide important and essential details on the nonlinear behavior of quartz crystal resonators.

In this study, we analyze the thickness-shear vibration of a simple resonator model by means of complex material constants with consideration of the electrode inertia. The effects of viscosity on frequency behavior of piezoelectric plate are presented and discussed. The results can be useful for design of the relevant resonators.

The nonlinear effects of material constants and initial stresses and strains in quartz crystal resonators is well known f on the frequency-temperature curves, drive-level dependency, acceleration sensitivity, and stress compensation. Consequently, accurate predictions on resonator behavior and their electrical circuit parameters require the use of nonlinear vibration equations and their solutions. The effectiveness of nonlinear analyses has been shown by a few researchers with the finite element and perturbation methods. The Mindlin plate theory, which has been used extensively for understanding plate modes and their coupling effects in plate vibrations analysis, is not enough in the study of the nonlinear behavior of quartz resonators. We have followed the Mindlin plate theory to derive the nonlinear equations with the inclusion of large displacements and higher order elastic constants. The coupling of vibration modes due to nonlinearity is clearly observed and it is quite different from linear cases that we are familiar with. We start from the equations of vibration for the thickness-shear mode to validate the solution techniques, which could be the perturbation method and the latest Homotopy Analytical Method (HAM). Then the methods are applied to the coupled equations of thickness-shear and flexural vibrations which are the two dominant modes of quartz crystal resonators of the thickness-shear type. These solutions, in the absence of the strong electrical field, can be used to study the frequency, deformation, and mode conversion in nonlinear vibrations. We hope the frequency spectra and spatial variations of the thickness-shear and flexural displacements from the accurate solutions of nonlinear equations will provide insights on the changes in each mode when compared with their linear vibrations. The further extension of nonlinear plate equations with the inclusion of piezoelectric effects will also provide useful examination of nonlinear behavior of quartz crystal resonators.

The accuracy of the finite element analysis for thickness shear quartz resonators is a function of the mesh resolution; the finer the mesh resolution, the more accurate the finite element solution. A certain minimum number of elements are required in each direction for the solution to converge. This places a high demand on memory for computation, and often the available memory is insufficient. Typically the thickness of the electrode films is very small compared with the thickness of the resonator itself; as a result, electrode elements have very poor aspect ratios, and this is detrimental to the accuracy of the result. In this paper, we propose special methods to model the electrodes at the crystal interface of an AT cut crystal. This reduces the overall problem size and eliminates electrode elements having poor aspect ratios. First, experimental data are presented to demonstrate the effects of electrode film boundary conditions on the frequency-temperature curves of an AT cut plate. Finite element analysis is performed on a mesh representing the resonator, and the results are compared for testing the accuracy of the analysis itself and thus validating the results of analysis. Approximations such as lumping and Guyan reduction are then used to model the electrode thin films at the electrode interface and their results are studied. In addition, a new approximation called merging is proposed to model electrodes at the electrode interface.

The feasibility of a new angular velocity sensor using the temperature stable AT-cut quartz was presented. The sensor consisted of one pair of electrodes for driving the fundamental thickness shear mode, and another pair of electrodes for sensing the angular velocity. Two tines extended outward from resonator, and in the plane of the plate. The tines were designed to be sensitive to angular velocity of the resonator. The Coriolis body force caused by the cross product of the angular velocity with the linear momentum of the vibrating tines changes the their mode shapes that in turn perturbed the thickness shear mode, and changed the voltage at the sensing electrodes. A vibratory gyroscope with a trapped energy thickness mode as the main driving mode offers good improvements in terms of frequency stability and less dependence on the mounting supports and lead installation of the sensing element. Since the AT-cut resonators were known to have good f-T curves and long term aging, such a gyroscope would have the advantages of a stable quartz AT-cut resonator. Results for two angular velocity sensors were presented: (1) a 5 MHz sensor with a sensitivity of 5.8 mV/deg./s angular velocity about the X-axis, and (2) a 37 MHz sensor with a sensitivity of 0.38 mV/deg./s angular velocity about the Z-axis.

A novel AT-cut quartz vibratory gyroscope with a sensitivity to angular velocity of 1.3 mV/deg/s was presented. The gyroscope employed the thickness shear mode as the driving mode and the flexure mode of tines as the sensing mode. The tines are placed normally on the AT-cut quartz plate surface. These tines were tuned to resonate with the harmonic Coriolis body force caused by the angular rotation of the plate about its plate thickness axis. The flexure mode of the tines changed with the angular velocity about the thickness axis of the plate. The change in the flexure mode of the tines could be detected by the sensing electrode separated from the driving electrode by an acoustic gap. Since the AT-cut resonators are known to have good f-T curves and long term aging, the new AT-cut gyroscope would have the advantages of a highly stable quartz AT-cut resonator.

Two finite element models of a resonator were employed to investigated electromagnetic radiation on the resonator Q. The two models employed the governing equations for viscopiezoelectricity and viscopiezoelectromagnetism, respectively. The models showed decreasing Q with frequency increase. The Q of the viscopiezoelectric model compared well with the empirical formula for the Q of quartz AT-cut resonators. At frequencies in excess of 2 GHz, the Q of the viscopiezoelectromagnetic model showed increasing difference from the Q of the viscopiezoelectric model. The electromagnetic radiation loss took up an increasing percentage in the loss of Q with frequency increase in excess of 2 GHz. At 3.3 GHz, it took up about 14% of the loss in Q.

Theoretical analyses and designs of high Q, thin film solidly mounted resonators are presented. The resonators analyzed in this paper operate at an ultra-high frequency (UHF) of 1.9 GHz for application to high frequency filter devices. A new accurate novel approach using three dimension periodic finite element analyses of solidly mounted resonators (SMR) is presented. The frequency spectra for the 1.9 GHz SMR which serve as a design aid in selecting the resonator dimensions/configurations for two different Bragg configurations are presented. Good agreement of the new approach with 2-D FE and 3-D FE analysis is demonstrated The frequency response analysis is carried out in order to calculate the Q's value and k2 and is compared with the measured data. Further, the static frequency-temperature (f-T) behavior for the SMR's is calculated using the Lagrangian method.

A new thickness shear mode vibratory gyroscope which employed the thickness shear mode at both the driving and sensing electrodes was presented. The thickness shear mode shape changed slightly with the Coriolis force, and this change could be detected by the sensing electrode separated from the driving electrode by an acoustic gap. Vibrating tines normal to the plate surface were placed along the acoustic gap. These tines were tuned to resonate with the harmonic Coriolis force caused by the angular rotation of the plate about its plate thickness axis. The change in vibration of the tines with the Coriolis force caused a change in charge or voltage at the sensing electrodes. Since the AT-cut resonators are known to have good f-T curves and long term aging, the new AT-cut gyroscope would have the advantages of a highly stable quartz AT-cut resonator.

The quartz resonator is the most common piezoelectric resonating structure designed to provide a stable frequency source for electronic devices and applications. An important figure of merit for the quartz resonator structure is the quality factor Q which measures the “sharpness” of its frequency response curve and its frequency stability. This quality factor Q is inversely proportional to the energy loss per cycle of oscillation. The Q is hence also inversely proportional to the resonator resistance or impedance. Currently there are no analytical tools for estimating the Q without an apriori assumption of the resonator damping or impedance. In order to get good numerical agreement with the measured data, all the current finite element software requires an assumption of either the resonator resistance or resonator Q value. We propose and present a new analytical tool for estimating the quartz resonator Q and other figures of merit by an energy sink method. Results for the thickness shear mode, AT-cut quartz resonators are presented. Experimentally measured material constants of mechanical dissipation and conductivity of quartz was included in our models. Our energy sink method is more realistic than assuming the crystal impedance or Q. One of the most important factors affecting the Q in the design of a new quartz resonator structure of a given frequency is the loss of energy from the electrode area to the base via the mounting supports. The acoustical characteristics of the plate resonator are changed when the plate is mounted onto a base. This is analogous to a soil-structure interaction problem with a semi-infinite boundary. The substrate base affects the frequency spectra of the plate resonator. A resonator with a high Q may not have a similarly high Q when mounted on a substrate base; hence the base is an energy sink. We present the frequency spectra of the quartz resonator with and without the substrate base. A scaled boundary finite element method is employed to model a semi-infinite base. Since a semi-infinite base will absorb all acoustical energies radiated from the resonator, a forced vibration analysis of such a model will provide the lower bound Q values. The model with a semi infinite base could be used for evaluating resonator and mountings designs. The effect of the mountings, plate and electrode geometries on the resonator Q and other figures of merit are presented. Comparisons of the experimental measurements of the admittance and Q with the simulated results using the semi-infinite energy sink method were performed. The measured admittance compared very well with the simulated results. The measured Q showed the same trend as the simulated Q.

A review of the finite element method for the analysis and design of quartz resonators is presented. Finite element analysis of the factors which impact on the performance of high quality quartz resonators are discussed. Novel techniques for the visualization of frequency spectra, calculation of resonator Q, and modeling of the effects of thermal stresses on the f-T behavior are presented. Good comparisons of simulated finite element results with their respective experimental results were found.

Theoretical analyses and designs of high Q, quartz thin film resonators are presented. The resonators operate at an ultra-high frequency (UHF) of 3.4 GHz for application to high frequency timing devices such as cesium chip-scale atomic clocks. The frequency spectra for the 3.4 GHz thin film quartz resonators which serve as a design aid in selecting the resonator dimensions/configurations for the simple electrode, and ring electrode mesa designs are presented here for the first time. The thin film aluminum electrodes are found to play a major role in the resonators because the electrodes are as much as one third the thickness and mass of the active areas of the plate resonator. Hence, in addition to the material properties of quartz, the elastic, viscoelastic and thermal properties of the electrodes are included in the models. The frequency-temperature behavior is obtained for the best resonator designs. In order to improve the frequency-temperature behavior of the resonators, new quartz cuts are proposed to compensate for the thermal stresses caused by the aluminum electrodes and the mounting supports. Frequency response analyses are performed to determine the Q, motional resistance, capacitance ratio, and other figures of merit. The resonators have Q's of about 3800, resistance of about 1300 to 1400 ohms, and capacitance ratios of 1100 to 2800.

A method is presented in selecting the resonator configuration based on the results obtained from three- dimensional FEM analysis of monolithic filters. The primary concern in the design of monolithic filters for high frequency is the complexity involved in tuning control circuits in order to achieve small fractional bandwidths. The bandwidth of the resonating system depends on parameters such as the electrode thickness and external termination resistance. The performance of the monolithic filters as a function of the resonator geometry, acoustic gap and electrode thickness is studied. In addition to the electrical parameters for filters, vibration energies within the electrodes and the acoustic gap are used as relative measures for the filter performance. In particular, a 1 GHz AT-cut quartz, two-port monolithic filter was designed. Both free and forced vibration analyses were employed to select the physical parameters of the filter, such as the filter dimensions, electrode dimensions and acoustic gap. The frequency response from a forced vibration analysis was used to find the equivalent circuit parameters and S- parameters of the filter.

Common piezoelectric resonators such as the quartz resonators have a very high Q and ultra stable resonant frequency. However, due to small material nonlinearities in the quartz crystal, the resonator is drive level dependent, that is, the resonator level of activity and its frequency are dependent on the driving, or excitation, voltage. The size of these resonators will be reduced to one fourth of their current sizes in the next few years, but the electrical power which is applied will not be reduced as much. Hence, the applied power to resonator size ratio will be larger, and the drive level dependency may play a role in the resonator designs. We study this phenomenon using the Lagrangian nonlinear stress equations of motion and Piola-Kirchhoff stress tensor of the second kind. Solutions are obtained using FEMLAB for the AT-cut, BT-cut, SC-cut and other doubly rotated cut quartz resonators and the results compared well with experimental data. The phenomenon of the drive level dependence is discussed in terms of the voltage drive, electric field, power density and current density. It is found that the drive level dependency is best described in terms of the power density. Experimental results for the AT-, BT- and SC-cut resonators in comparison with our model results are presented. Results for new doubly rotated cuts are also presented