Electrothermal frequency tuning of folded and coupled vibrating micromechanical resonators

The use of constrained thermal expansion to tune the resonant frequency of vibrating micromechanical resonators is explored. A simple model is developed to predict the power sensitivity obtained with folded and unfolded geometries, including the effects of electrothermal heating, and conduction and convection cooling. It is shown that the sensitivity of folded structures can change sign as the ambient gas pressure is lowered in contrast to the behavior of unfolded structures. Tuning is then by tensile axial stress rather than compressive stress. Using folded laterally resonant bulk-micromachined comb-drive electrostatic actuators, tuning ranges of -25% and +50% are obtained (at atmospheric pressure and at 10 mTorr, respectively, with powers of 10 and 1.5 mW, respectively). A nested coupled resonator is then presented in which the frequency of one resonator may be tuned without effecting that of the other, thus allowing frequency matching to be obtained.