Acceleration sensitivity of crystal resonators affected by the mass and location of electrodes

An estimate of the second-order in-plane acceleration sensitivity of a Y-cut quartz thickness-shear resonator

We perform a theoretical analysis of the secondorder in-plane acceleration sensitivity of a Y-cut quartz thickness- shear mode resonator. The second-order nonlinear theory of elasticity for anisotropic crystals is used to determine the biasing fields in the resonator under in-plane acceleration. The acceleration-induced frequency shift is determined from a perturbation analysis based on the plate equations for small-amplitude vibrations superposed on a finite bias. We show that, whereas the first-order acceleration-induced frequency shift is zero for a structurally symmetric resonator under in-plane acceleration, the second-order frequency shift is nonzero and is quadratic in the acceleration. As the fourth-order nonlinear elastic constants of quartz have never been measured, we can only estimate the magnitude of the second-order frequency shift. For a particular case of interest, we find Δω/ω0 ~ 10(-18), 10(-16), and 10(-14) when the acceleration is 1, 10, and 100 g, respectively.

Effect of transverse force on the performance of quartz resonator force sensors

This paper discusses the effect of transverse force on the performance of quartz crystal resonator (QXR) force sensors that use symmetrical incomplete circular QXRs as sensing elements. Based on Lee's vibration theory and the transverse force-induced stress distribution, frequency changes are derived. This result and finite element method (FEM) are applied to investigate the performance of two metal-quartz combined sensors under the action of transverse force. Analytical and experimental results show that transverse force affects the sensitivity and linearity of QXR force sensors.

Theory and design of piezoelectric resonators immune to acceleration: present state of the art

A typical low noise oscillator uses a crystal resonator as the frequency-determining element. An understanding of the fundamental nature of acceleration sensitivity in crystal oscillators resides primarily in understanding the behavior of the crystal resonator. The driving factor behind the acceleration-induced frequency shift is shown to be deformation of the resonator. The deformation drives two effects: an essentially linear change in the frequency-determining dimensions of the resonator and an essentially nonlinear effect of changing the velocity of the propagating wave. In this paper, the fundamental nature of acceleration sensitivity is reviewed and clarified, and attendant design guidance is developed for piezoelectric resonators. The basic properties of acceleration sensitivity and general design guidance are developed through the simple examples of "bulk acoustic wave (BAW) in a box" and "surface transverse wave (STW) in a box." These examples serve to clarify a number of concepts, including the role of mode shape and the basic difference between the BAW and STW cases. The design equations clarify the functional dependencies of the acceleration sensitivities on the full range of crystal resonator design and fabrication parameters.

Designing for low acceleration sensitivity

Recent advances in acceleration sensitivity measurement and modeling are discussed, with an emphasis on what these advances indicate in terms of designing for low acceleration sensitivity. The design suggestions are separated into two parts, namely, those that use the crystal resonator as a mechanical vibrator and those that use the crystal oscillator as an electronic circuit. Such topics as symmetry considerations, metallization, mounting etc., in both bulk acoustic wave (BAW) and surface acoustic wave (SAW) devices are discussed for the former, while for the latter the equivalent circuit modeling of crystal resonators and other loop components is addressed.

An observation on quartz resonator aging

Aging and acceleration sensitivity of certain fabrication lots of quartz resonators show a trend that may arise from a common predominating mechanism, such as stresses arising from the mounting structure. The presence or absence of such a trend is discussed. As part of the work on bulk-acoustic-wave (BAW) resonators, a database of all known parameters of the resonators in question was assembled. Resonator processing and electrical parameters were examined for correlations with the acceleration sensitivity. A similar effort was being undertaken in regard to resonator aging, and possible correlations between the aging and acceleration sensitivities of the various resonators were examined. The results for two such fabrication lots are shown. The magnitude of the observed aging was typically in the 10(-10) per day range. The identity of the common contributor is unknown at the present time. It is likely that mounting related stresses are the common contributor.