Mihir S. Patel

Estimation of quartz resonator Q and other figures of merit by an energy sink method

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 substrate 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. A scaled boundary finite element method is employed to model a semi-infinite base. The frequency spectra of the quartz resonator with and without the base are presented. In addition to the loss of energy via the base, there are other factors which affect the resonator Q, such as, for example, material dissipation, and damping at the interfaces of quartz and electrodes. The energy dissipation due to material damping increases with the resonant frequency and the reduction of resonator size; hence material damping becomes important in the current and future miniaturized resonators operating at very high frequencies. An energy sink model along with material dissipation would provide realistic Q, motional capacitance, motional resistance, and other figures of merit useful for designing resonators. The model could be used for evaluating resonator and mountings designs of microelectromechanical systems and miniaturized devices. The effect of the mountings, and plate and electrode geometries on the resonator Q and other electrical parameters are presented for AT-cut quartz resonators. Model results from the energy sink method were compared with experimental results and were found to be good.

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.

Application of a DC bias to reduce acceleration sensitivity in quartz resonators

When a quartz resonator is subjected to acceleration, the inertial effects cause stresses and strains in the plate which in turn causes the resonant frequency to shift. A method is proposed to actively minimize the inertial stresses and strains by piezoelectric stiffening using a DC electric field bias across the electrodes. Finite element models are created to across calculate the effects of acceleration on resonant frequency and to compare the results with experimental data. The model results are shown to compare well with the experimental results. The models are then used to calculate the effects of the DC electric field bias and to demonstrate the reduction of acceleration sensitivity.

Effects of electromagnetic radiation on the Q of quartz resonators

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 Qx provided by the formula Qx = 16 times 106/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.

Temperature compensation of Longitudinal Leaky SAW with silicon dioxide overlay

Longitudinal leaky surface acoustic wave (LLSAW) is attracting considerable attention for its high velocity and reasonable coupling coefficient. The intrinsic temperature coefficient of frequency (TCF) for these waves for different longitudinal cuts with metal gratings is in the range of -110 to -90 ppm/degC. However, for certain filter applications the TCF of the LLSAW waves must be quite low in the range of ~ 20 to -20 ppm/degC. The introduction of a positive TCF overlay material (generally SiO2) alters the LLSAW wave characteristics, and also changes the coupling coefficient. The modified LLSAW mode retains the same high velocity characteristics with an estimated improved TCF. Thus, a systematic study has been attempted by us to evaluate the effect of SiO2 on some interesting longitudinal cuts, such as, YZ and 128deg lithium niobate. In this paper, we show the possibility to realize a good TCF value for LLSAW mode using a three dimensional (3-D) periodic finite element (FE) model. The predicted results show an improvement in the TCF value to -20 ppm/degC for the modified LLSAW mode which is in excellent agreement with the measurement results.

Effects of non-homogeneous thermal stresses on the frequency-temperature behavior of AT-cut quartz resonators

The frequency-temperature (f-T) behavior of a quartz resonator can be predicted quite accurately if the resonator crystal is under an ideal stress-free condition. Under such a condition, the steady state temperature change then induces a homogeneous field of thermal strains in the crystal. The ideal thermal stress-free condition is however seldom achieved practically. In practical devices, thermal stresses are often present with temperature changes. We study the effects of non-homogeneous thermal stresses on the f-T behavior of AT-cut quartz resonators by employing a novel method of superposing the results from three existing methods for calculating (1) thermal stresses, (2) acceleration effects, and (3) f-T curves under a homogeneous thermal strain condition. We assume that for a steady state temperature change, the crystal resonator undergoes not only a homogeneous thermal strain field but also a nonhomogeneous thermal stress field. We present numerical results compared with experimental results for an AT-cut quartz resonator mounted on glass. The difference between the thermal expansion coefficients of glass and quartz give rise to the thermal stresses.

Force-frequency effect on the Q-factor of thickness- shear mode quartz and langasite resonator at high temperatures

A theoretical study was conducted for a 20 MHz circular resonator to investigate the force-frequency effect on the Q-factor value for the slow (C-) and fast (B-) thickness-shear mode using the 3-D Lagrangian finite-element model (FEM), and the Sinha-Tiersten perturbation method. The calculations were carried out for doubly rotated quartz cuts and single rotated langasite (LGS) cut. Based on the frequency spectrum analysis, the best aspect ratio for each crystal cut angle was investigated for the force-frequency effect at 25 and 300°C. The Q-factor calculations are based on the intrinsic viscosity of crystalline quartz and LGS. This paper provides the foremost computational result for the force-frequency constants and the Q-factor at 300°C for doubly rotated quartz cuts, and single rotated LGS cut.

A new angular velocity sensor using the temperature stable AT-cut quartz

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.