Vibration analysis of the new shape extensional mode quartz crystal resonator with barbell-type vibration bar

A novel shape length extensional mode quartz crystal resonator with a barbell-type vibration bar, which makes it possible to provide lower frequency, is described. Frequency characteristics, frequency temperature behavior, and electrical characteristics of the resonator are theoretically and experimental clarified. A frequency equation is derived by applying a variational principle whereby a vibration analysis for the resonator is performed, taking into account lateral motions and including a vibrational portion and two supporting portions. Frequency characteristics, frequency-temperature behavior, and electrical characteristics are analyzed. The results are found to agree well with the measured values. The resonator is also miniaturized, is shock-resistant, and has a small series resistance in the frequency range of 200 kHz to 500 kHz, because it is incorporated with the vibrational portion and the two supporting portions by a chemical etching process