An Analysis of Torsional Mode Quartz Crystal Resonators with Galerkin's Method and Its Application to Research of New Cuts

Abstract

This paper describes new cuts for torsional mode quartz crystal resonators, which are called "TT(Y)-cut". The object of this paper is to propose new cuts for torsional quartz crystal resonators with a zero temperature coefficient and to clarify their frequency constants, frequency temperature behavior and electrical equivalent circuit parameters. First, torsional rigidity Ct is derived from stress function ψ obtained by solving approximately a partial differential equation by Galerkin's method. Next, from the frequency equation numerous relationships where the first order temperature coefficient a reaches zero exist between thickness-to-width ratio Rzy and cut angle (ψ, θ). Especially, the second order temperature coefficient β has a small value of -1.25×10-8/°C2, whose absolute value is approximately one third of the flexural mode quartz crystal resonators. Finally, it is shown that torsional quartz crystal resonators of tuning fork-type are successfully obtained with a small R1 of 3.5 to 4.6 kΩ and a large Q value of 241000 to 272000 in a frequency range of about 300 to 600 kHz.