Resonant Frequency and Electrical Characteristics for Flexural Mode Quartz Crystal Resonators of Tuning Fork-Type with Stepped Vibration Bars

Abstract

This paper describes the flexural mode quartz crystal resonators of tuning fork-type with stepped vibration bars.The object of this paper is to clarify resonant frequency and electrical characteristics for the resonators. A frequency equation and a transcendental equation are derived from the boundary condition of "clamped-free" and the continuation condition, respectively. From the equations, theoretical relationships of frequency constant and resonant frequency to a width ratio W_21 and a length ratio L_21 are shown. The results are then compared with the measured data. In addition, series resistance R_l and a quality factor versus resonant frequency are examined experimentally, as a result, the resonators of lower frequency sufficiently satisfactory as a resonator are found to be obtained without enlarging the resonators' size.