New cuts for torsional mode quartz crystal resonators

Abstract

This paper describes new cuts for torsional mode quartz crystal resonators, which are called lTT(Y)-cutr. 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 Ct. Namely, the problem of a vibration for torsional mode is substantially equivalent to that of torsional rigidity Ct, and it is derived from stress function P 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 a reaches zero are found to exist between thickness-to-width ratio Rzy and cut angle (p,t), especially, the second order temperature coefficient b has a small value of -1.25t10-8/dC2 whose absolute value is approximately one third of the well-known flexural mode quartz crystal resonator. The value of b is then compared with the measured data -1.00t10-8/dC2, 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 o and a large Q value of 241,000 to 272,000 in a frequency range of about 300 to 600 kHz