Simple design rules for optimal design of dielectric temperature-compensated sapphire resonators

It has been shown that the use of two dielectric crystals with opposite temperature coefficient of permittivity allows the realization of a resonator with a zero temperature coefficient of frequency. By using sapphire and rutile materials, which have low-loss tangents, some compensated resonators with very high Q-factors have been realized. In this work we develop rules that greatly simplify the design of a dielectric-compensated resonator. We show that the optimum design for compensation at a specific temperature may be determined by simply selecting the aspect ratio of the sapphire resonator.