  Two critical characteristics for any MEMS resonator are the quality factor
($Q$) and the temperature coefficient of frequency ($TCF$). The connection
between $Q$ and $TCF$ is demonstrated here with a phenomenological anharmonic
oscillator model. Specifically, it is shown that the same nonlinear terms
responsible for the $TCF$ set an upper limit on the resonator's $Q$. A concise
formula is found to estimate this loss and is shown to be closely related to
Woodruff's formula for Akhiezer damping. The use of this formula is illustrated
by extending the model to an important class of MEMS; piezoelectric resonators.
Finally, the model is applied to published data for an AlN-on-diamond
piezoelectric resonator. The focus here is on MEMS resonators, but the method
should apply broadly to any resonance with non-zero $TCF$.
