Electrical Equivalent Circuit and a Theoretical Analysis on Dependence of Load Capacitance for the Coupled Flexural-torsional Resonator

Abstract

This paper describes a theoretical analysis on the electrical equivalent circuit and the dependence of load capacitance for the coupled flexural-torsional quartz crystal resonator. It is well known that the flexural-torsional quartz crystal resonator has a cubic curve in frequency temperature characteristics. First of all, it is conceivable that a flexural-torsional quartz crystal resonator is a resonator where electrical coupling and elastic internal coupling coexists, and the coupling of both vibrations is examined as capacitance coupling. Next, the load capacitance C_L is connected to the obtained electrical equivalent circuit in series, and furthermore the electrical equivalent constants are calculated as a function of C_L by performing circuit transformation. Particularly, the examination of C_L dependence for coupling capacitance is made, and also the detailed examination of the influence in frequency temperature characteristics is made. As a result, as C_L increases, the first order temperature coefficient α increases. In particular, as C_L becomes small, change rate of α against C_L becomes large. From this, it is shown that α can be made equal to almost zero by the choice of C_L.