Effects of stress on the temperature coefficient of frequency in double clamped resonators

This paper presents a theoretical framework for evaluating the temperature coefficient of frequency (TCf) of double clamped resonators due to stresses induced by the die and through die packaging. It is desirable to have a zero TCf such that the resonator frequency is stable over a broad temperature range. The TCf depends on how the resonator's material properties, dimensions, and stresses change with temperature. A passive method of using thin film induced stresses in an encapsulated resonator to compensate for material softening is explored. By using a combination of finite element and analytical models it is possible to predict the TCf and improve thermal frequency stability of micromachined resonators.