R. Bourquin

Temperature-compensated cuts for length-extensional and flexural vibrating modes in GaPO/sub 4/ beam resonators

Flexural modes are the basic vibrating mode of tuning forks used in quartz wrist watches; they also can be used as the basis for sensors. Very little work, if any, has been done for vibrating beam resonators in GaPO/sub 4/. In this paper, the possibility of temperature-compensated cuts in GaPO/sub 4/ is investigated for length-extensional and flexural vibrating modes. A theoretical investigation of rectangular cross-section GaPO/sub 4/ vibrating beam resonators is accomplished by analytical methods. Modeling temperature effect is achieved by the approximate but classical method in which the effective elastic constants, beam dimensions, and crystal mass density are varied as a function of temperature. Temperature-compensated cuts are given in GaPO/sub 4/ for length-extensional and flexural modes. Some temperature-compensated cuts of GaPO/sub 4/ exhibit inversion points at high temperatures.

Theoretical and experimental studies of the force-frequency effect in BAW LGS and LGT resonators

It is well known that, for quartz crystal, mechanical and/or thermal stress sensitivities prevent the achievement of ultimate performances of bulk acoustic wave piezoelectric resonators. It is important to check these effects in newly developed Langasite (LGS) and Langatate (LGT) resonators. A theoretical and experimental study of the force-frequency effect (denoted by the K/sub f/ coefficient) in Y-cut LGS thickness-shear resonators, working at various overtone modes, is presented. Theoretical predictions are based on Tiersten theory of wave propagation in a pre-stressed medium. For LGS crystals, 3/sup rd/ order nonlinear elastic constants have been published and used here for modeling the behavior of thickness-shear resonators submitted to diametrical compression. A comparison between theoretical predictions and experimental measurements is presented. In the case of LGT crystal, numerical data of the third order elastic coefficients are not available at this time. Only experimental measurements of Kf coefficient are presented in order to compare the behaviors of LGS and LGT Y-cut resonators.

Temperature compensated cuts in LGT crystal microresonators using length extensional mode

In this letter, experimental investigation of frequency-temperature effects in langatate rectangular cross-section beams are presented. It is shown that a first-order temperature compensated cut exists for the first vibrating mode of length extension.

Frequency-temperature behavior of langasite rectangular beam resonators vibrating in length extension

This paper shows that first order temperature compensated cut exists in langasite rectangular cross-section beam vibrating in length extensional mode. Theoretical and experimental investigations of frequency-temperature effects are given.

Thermal sensitivity of elastic coefficients of langasite and langatate

Thermal coefficients of elastic constants of langasite and langatate crystals have been determined from frequency-temperature curves of contoured resonators operating in thickness modes. The effect of the trapping of the vibration has been taken into account to improve the accuracy. In a first step, the thermal sensitivities of stiffness coefficients in Lagrangian description are obtained. Thermal sensitivities of the usual elastic constants are further deduced. Predictions of thermally compensated cuts are given.

Study of LGS crystal micro-resonators using flexure mode: Temperature-compensated cuts

In this paper, new experiments have been carried out on LGS crystal to highlight the existence of temperature compensated cuts for flexure vibration. The micro-resonators are square cross section tuning forks whose arms are vibrating in flexure-mode with clamped-free boundary conditions. Frequency versus temperature behaviors have been measured for several cut angles. The results of our experiments show that there is a first order temperature-compensated cut for Langasite crystal resonators vibrating in flexion at room temperature.

Temperature-compensated cuts for vibrating beam resonators of gallium orthophosphate GaPO/sub 4/

A theoretical investigation of rectangular cross-section GaPO/sub 4/ vibrating beam resonators is proposed. Flexural modes are the basic vibrating mode of tuning forks used in quartz wrist watches, and can also be used as sensors. Very little work, if any, has been done for vibrating beam resonators in GaPO/sub 4/. The goal is then to investigate the possibility of temperature-compensated cuts for all three kinds of vibrations in GaPO/sub 4/: extensional, flexural, and torsional modes by analytical methods. Modeling temperature effects is achieved by the approximate but classical method of varying effective elastic constants, beam dimensions and crystal mass density versus temperature. Temperature-compensated cuts are found in GaPO/sub 4/ for length extensional modes and flexural modes. For vibrating beams, some of temperature-compensated cuts of GaPO/sub 4/ exhibit inversion points at high temperatures.

SC-cut resonator with reduction of B-mode electrical response

Excitation, by the use of two pairs of cross-connected half electrodes, of SC-cut resonator operating on first anharmonic modes is considered. In order to obtain resonators having the best electrical response of the C-mode, it is shown that two cases can be encountered, depending on the shapes and orientation of modes, i.e. on overtone number. The computation of electrical parameters is based on Lewis formula and use of analytical expressions for the vibration and the electric field.

Theoretical and experimental investigations of 1/f noise in quartz crystal resonators

Fluctuation dissipation theorem allows to recover a 1/f noise spectral density at low frequency. The level of this 1/f noise is governed by a single fitting parameter that can be connected to the onset frequency of this 1/f regime. It constitutes the fundamental intrinsic limit of quartz crystal. Some preliminary considerations on the physical origin of this parameter in terms of microscopic processes in the crystal are given. Experimentally, quartz crystal resonators have been cut from a quartz crystal block supplied specifically for this study on 1/f noise. The short-term stabilities of several resonators have been measured to be lower than 8-10-1. A comparison of these resonators is given and the results are discussed according to the position of the resonators inside the crystal block.

Frequency-temperature behavior of flexural quartz resonators by means of Timoshenko's model

The frequency of a flexural resonator and its frequency-temperature behavior usually are computed by Bernoullis classical approximation. This approach is valid for beams with a large length-over-thickness-ratio. For shorter beams, the effects of shear stress and rotary inertia may play a significant role for temperature-compensated resonators. These effects have been taken into account for isotropic beams. The aim of this paper is to discuss the extension of the shear coefficient in the case of an anisotropic material and to compute the frequency-temperature characteristic of an (XYt)thetas cut resonator when the shear stress and the rotary inertia have been taken into account. Comparisons between the classical approximation arid this treatment are given for quartz. Furthermore, the numerical predictions obtained by means of different sets of data available for thermal sensitivities of elastic coefficients are compared