New cuts for torsional mode quartz crystal resonators

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 characteristics, frequency temperature behavior and electrical equivalent circuit parameters. First, one shows a frequency equation that is given as a function of torsional rigidity C_t. Namely, the problem of a vibration for torsional mode is substantially equivalent to that of torsional rigidity C_t, and it is derived from stress function Ψ obtained solving a partial differential equation with respect to y and z, including elastic compliance constant s₅₆. Next, from the frequency equation numerous relationships where α reaches zero are found to exist between thickness-to-width ratio Rzy and cut angle (φ,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 well-known flexural mode quartz crystal resonator. The value of β is then compared with the measured data -1.00×10^-8/°C², so that both results are found to agree sufficiently well. Finally, it is shown that torsional quartz crystal resonators to tuning fork-type are successfully obtained with a small R₁ of 3.5 to 4.6 k ω and a large Q value of 241,000 to 272,000 in a frequency range of about 300 to 600 kHz