Yook-Kong Yong

A new theory for electroded piezoelectric plates and its finite element application for the forced vibrations of quartz crystal resonators

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.

Resonator surface contamination-a cause of frequency fluctuations?

The mass loading effects of adsorbing and desorbing contaminant molecules on the magnitude and characteristics of frequency fluctuations in a thickness-shear resonator are studied. The study is motivated by the observation that the frequency of a thickness-shear resonator is determined predominantly by such mechanical parameters as the thickness of the resonator, elastic stiffnesses, mass loading of the electrodes, and energy trapping. An equation was derived relating the spectral density of frequency fluctuations to: (1) rates of adsorption and desorption of one species of contaminant molecules; (2) mass per unit area of a monolayer of molecules: (3) frequency constant; (4) thickness of resonator; and (5) number of molecular sites on one resonator surface. The induced phase noises were found to be significant in very-high-frequency resonators and are not simple functions of the percentage of area contaminated. The spectral density of frequency fluctuations was inversely proportional to the fourth power of the thickness if other parameters were held constant. >

A MEMS-based quartz resonator technology for GHz applications

We report on the development of a new MEMS quartz resonator technology that allows for the processing and integration of VHF to UHF high-Q oscillators and filters with high-speed silicon or III-V electronics. The paper describes the successful demonstration of new wafer bonding and dry plasma etching processes that make quartz-MEMS technology possible. We present impedance, Q, and temperature sensitivity data along with comparison to 3D harmonic and thermal analysis of VHF-UHF resonators. We also show Coventor simulation data of our first two- and three-pole monolithic crystal filter designs as well as a filter array layout which facilitates integration with front-end RF electronics and switches. Finally, we demonstrate a mechanical tuning technique for our resonators utilizing focused-ion-beam (FIB) technology.

Analysis of periodic structures for BAW and SAW resonators

A bulk acoustic wave (BAW) or a surface acoustic wave (SAW) resonator is an acoustic device with the frequency characteristic of standing acoustic waves in the substrate and electrodes. Sometimes, the standing waves are almost spatially periodic depending on the geometry of the electrodes and the substrate. This is especially true in SAW resonators with long arrays of inter-digital transducers (IDTs) on the surface of the substrate, and may also be true for BAW resonators with similar arrays of IDTs. In these situations where the standing waves are essentially periodic, substantial analytical and design information on the frequency characteristics of the acoustic resonator itself can be obtain from the analysis of one period of the standing waves. The advantage of analyzing just one period of the standing wave is the significant reduction in the size of the numerical model. Sometimes, it is not possible to perform an finite element (fem) analysis of the entire acoustic resonator. This is especially true for SAW resonators, and we have to resort to a periodic analysis. This paper discusses the finite element periodic analysis of BAW and SAW resonators. The boundary conditions and finite element meshing of a periodic structure are presented. Examples are provided which shows the results of the periodic analysis compared with the experimental results of the actual resonators or the results of the full analysis which include the frequency-temperature behavior, mode shapes and frequency spectra. Examples are drawn from BAW resonators and SAW resonators including leaky SAW resonators.

Love wave propagation in piezoelectric layered structure with dissipation.

We investigate analytically the effect of the viscous dissipation of piezoelectric material on the dispersive and attenuated characteristics of Love wave propagation in a layered structure, which involves a thin piezoelectric layer bonded perfectly to an unbounded elastic substrate. The effects of the viscous coefficient on the phase velocity of Love waves and attenuation are presented and discussed in detail. The analytical method and the results can be useful for the design of the resonators and sensors.

Drive level dependency in quartz resonators

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

Effects of Thermal Stresses on the Frequency-Temperature Behavior of Piezoelectric Resonators

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 temperature derivatives of the material constants. Numerical results were compared with the experimental results for a 50 MHz AT-cut quartz resonator mounted on a glass package. Good comparisons between the experimental results and numerical results from our new model were found. The differences between the thermal expansion coefficients of glass and quartz gave rise to the thermal stresses that had adverse effects on the frequency stability of resonators. Different optimal crystal cut angles of quartz, and resonator geometry were found to achieve stable frequency-temperature behavior of the resonator in a glass package. The dynamic frequency-temperature model was used in the theoretical analyses and designs of high Q, 3.3 GHz, quartz thin film resonators.

Theory and experimental verifications of the resonator Q and equivalent electrical parameters due to viscoelastic, conductivity and mounting supports losses

A novel analytical/numerical method for calculating the resonator Q and its equivalent electrical parameters due to viscoelastic, conductivity, and mounting supports losses is presented. The method presented will be quite useful for designing new resonators and reducing the time and costs of prototyping. There was also a necessity for better and more realistic modeling of the resonators because of miniaturization and the rapid advances in the frequency ranges of telecommunication. We present new 3-D finite elements models of quartz resonators with viscoelasticity, conductivity, and mounting support losses. The losses at the mounting supports were modeled by perfectly matched layers (PMLs). A previously published theory for dissipative anisotropic piezoelectric solids was formulated in a weak form for finite element (FE) applications. PMLs 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 equivalences 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 because it will absorb all acoustical energies radiated from the resonator.

Lagrangian temperature coefficients of the piezoelectric stress constants and dielectric permittivity of quartz

Piezoelectric, Lagrangian equations for the frequency-temperature behavior of quartz are presented. From the solutions of the third order temperature perturbations of these Lagrangian equations for the thickness resonances of infinite quartz plates with air-gap electrodes, regression equations for determining the temperature derivatives of elastic constants, piezoelectric constants and dielectric permittivities am developed. By using these regression equations, and the measured data on temperature coefficients of frequency by Bechmann, Ballato and Lukaszek [1962] for doubly rotated cuts, the first, second, and third temperature derivatives of elastic constants, piezoelectric constants and dielectric permittivities are obtained. The second and third temperature derivatives of the piezoelectric constants and dielectric permittivities were not available in the literature and are published here for the first time. These temperature derivatives will provide a more accurate map of the temperature stable cuts for bulk wave and surface acoustic wave quartz resonators.

Third-order Mindlin plate theory predictions for the frequency-temperature behavior of straight crested wave modes in AT- and SC-cut quartz plates

The frequency-temperature behavior of straight crested wave modes in AT- and SC-cut quartz plates are studied using Mindlin first- and third-order plate equations. The first order Mindlin plate theory with shear correction factors was previously found to yield an inaccurate frequency spectra of the modes in the vicinity of the fundamental thickness shear frequency. The third order Mindlin plate equations without correction factors, on the other hand, predict well the frequency spectrum in the same vicinity. In general, the frequency-temperature curves of the fundamental thickness-shear obtained from the first-order Mindlin plate theory are sufficiently different from those of the third-order Mindlin plate theory that they raise concerns. The least accurately predicted mode of vibration is the flexure mode which results in discrepancies in its frequency-temperature behavior. The accuracy of other modes of vibrations depends on the degree of couplings with the flexure mode. Mindlin first-order plate theory with only the shear correction factors is not sufficiently accurate for high frequency crystal vibrations at the fundamental thickness-shear frequency. Correction factors for the flexure and face-shear branches may be needed. Hence, a total of five correction factors may be needed for the first-order plate theory to yield accurate frequency spectra and frequency-temperature curves.