TT-cut torsional quartz crystal resonator

A novel cut torsional quartz crystal resonator with an absolute value of the second-order temperature coefficient smaller than that of the tuning fork-type flexural quartz crystal resonator, which is designated the TT-cut, is proposed. Its frequency characteristics, frequency temperature behavior, and electrical characteristics are discussed. First in an analysis procedure, an equation of motion is derived from an energy method. By solving the equation of motion under the boundary conditions of free-free bar or clamped-free bar, the frequency equations are derived. From the equation, a relationship between thickness-to-width ratio and cut angles where the first-order temperature coefficient reaches zero is shown. A tuning-fork-type torsional quartz crystal resonator with excellent frequency temperature behavior is theoretically obtained. The analysis results are then compared with the measured values. Both results are found to agree comparatively well