Virtual modes in AT-cut resonators

Although studied for more than 40 years, drive-level dependent modes apparently remain an unresolved problem in the design and manufacture of high-performance AT-cut quartz crystal resonators. Our intention is to give a clear presentation of this phenomenon and what is known about these modes. In particular, we cover the following topics: (1) the problems they cause in the crystal's operation over temperature; (2) their characterization using an impedance analyzer; (3) a model for their impedance as a function of drive level (4) their identification as being virtual images of real modes at twice the frequency caused by the inherent non-linearity of quartz; and (5) a discussion of those factors that lead to the existence of these modes. The hope is to renew an interest in their study with the aim of developing design rules that minimize the strength or entirely eliminate these modes. There is a good chance that this can be done as past experience has shown that problem designs can be modified to eliminate (or at least greatly reduce) the existence of these modes. However, we still have no clear guidelines for carrying this out in general and the problem appears even more difficult for higher frequency crystals, which are expected to play an important role in future communications systems.