Jointly with the 17th European Frequency and Time Forum Proceedings of the 2003 IEEE International Frequency Control Symposium and PDA Exhibition 0-7803-7688-9/03/$17.00 © 2003 IEEE This work is supported by EU, contract n° G5RD-CT-2002-00709. TEMPERATURE-COMPENSATED CUTS FOR VIBRATING BEAM RESONATORS OF GALLIUM ORTHOPHOSPHATE GaPO 4 L. Delmas, F. Sthal, E. Bigler, B. Dulmet, R. Bourquin Laboratoire de Chronométrie, Electronique et Piézoélectricité Ecole Nationale Supérieure de Mécanique et des Microtechniques 26, Chemin de l'Epitaphe - 25030 BESANÇON CEDEX – France Abstract - A theoretical investigation of rectangular cross- section GaPO 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 4 . The goal is then to investigate the possibility of temperature- compensated cuts for all three kinds of vibrations in GaPO 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 4 for length extensional modes and flexural modes. For vibrating beams, some of temperature-compensated cuts of GaPO 4 exhibit inversion points at high temperatures. Keywords - GaPO 4 , vibrating beam, temperature-compensated cuts, extensional, flexural, and torsional modes. I. INTRODUCTION GaPO 4 is new quartz-homeotypic piezoelectric crystal. Its excellent thermal stability up to 970°C opens new fields of applications for high temperature sensors. Extensive studies on rotated Y-cut thickness shear resonators have been performed [1][2][3]. However, extensional and flexural vibrating beams of GaPO 4 used as resonators have never been reported yet. Vibrating modes could be used in GaPO 4 for physical sensors like vibrating beam accelerometers, pressure or temperature sensors. Modeling the frequency dependence vs temperature is a pre-requisite for any kind of physical sensors thus it is necessary to find, if they exist, temperature-compensated cuts. This paper reports on analytical models of vibrating beam resonators with a rectangular cross-section in extensional modes and flexural modes. Temperature effects are modeled by an approximate but very classical method of varying elastic constants, beam dimensions and crystal mass density versus temperature. It is shown that temperature- compensated cuts for vibrating beam resonators in extensional modes and flexural modes exist for GaPO 4 . Comparison with quartz shows some advantages of GaPO 4 over quartz and particularly the existence of high temperature inversion points. II. ANALYTICAL MODEL OF LENGTH EXTENSIONAL MODES A. Resonant frequencies [4][5] We determine resonant frequencies from the equation of motion (1) although the analytical model of length extensional is built without taking into account the piezoelectric effect as a first approach. Hypothesis used in this model are: length >> width and length >> thickness. 0 = y u t u c 2 2 2 2 2 ∂ ∂ - ∂ ∂ (1) with ρ 22 S 1 = 2 c The analytical formulae, giving the values of resonant frequencies (2, 3), are the same for all materials. For a fixed-free beam: c 4y 1 2n = f o n - (2) For a fixed-fixed or free-free beam: c 2y n = f o n (3) Different material properties as mass density, elastic constants and beam dimensions allow a simple comparison between GaPO 4 and quartz. In this study, we use only the rotated X-cut defined in Fig. 1. This simple rotation is sufficient to find temperature compensated cuts for GaPO 4 and quartz. The angle of rotation is called θ. 663