Estimating Temperature-Dependent Glass Substrate CTE via Frequency Measurements in a MEMS Gyroscope

We present a novel physics-based optimal parameter estimation framework to extract the coefficient of thermal expansion (CTE) of the glass substrate from temperaturefrequency measurements in a MEMS gyroscope. We set forth a parametrization of the gyroscope's stiffness that distinguishes stress-induced components from other terms. This formulation enables an inverse problem approach to identify the substrate CTE through stiffness perturbation caused by the anchors' thermal expansion. As the average estimate from two temperature tests, we obtain the glass's CTE as 3.0510 + 0.0018 (T-20) - 1.4393×10<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−5</sup> (T-20)<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> (ppm/°C) for T E [20, 100]°C. Compared to silicon, glass has a larger CTE with significantly less temperature variation (1% versus Si's 20% over 20-100°C).