The dynamic performance of a micro-resonator depends on its energy loss mechanism which is quantified by Q-factor (Quality factor). This paper presents numerical and analytical modelling techniques to understand the air damping on energy loss and Q-factor in enclosed micro-resonators. A complete finite element based numerical model is presented that can recreate a vacuum packaged MEMS resonators operating condition and capture its Q-factor for various design and pressure conditions. The finite element model was then compared with analytical models available in the literature. In this work, various vacuum regions relevant to encapsulated MEMS resonators are investigated using analytical and finite element approaches for pressure ranges of < 1 Pa, 1–10 Pa, and 10–100 Pa. When the pressure decreases, the Q factor from finite element analysis and the analytical model exponentially increases until it levels off in intrinsic damping. The modelling techniques described in this paper are compared with previously reported experimental work showing good qualitative agreement of the change in Q-factor with pressure. Air damping is divided into squeeze film damping and slide film damping to further explore damping effects and squeeze film damping is found to be the dominant energy loss mechanism in the studied device. In a gap-closing structure, the air gaps between moving structure and fixed fingers create squeeze film damping and cause energy loss while the smaller air gaps between them generate large forces, increasing the damping. The modelling techniques presented in this paper can be applied generically to MEMS resonators to mitigate air damping losses.

微谐振器的动态性能取决于其能量损失机制，该机制由Q因子（品质因子）量化。本文介绍了数值和分析建模技术，以了解空气阻尼对封闭式微谐振器中能量损失和Q因子的影响。提出了一个完整的基于有限元的数值模型，可以重建真空封装的 MEMS 谐振器的工作条件并捕获其Q- 各种设计和压力条件的系数。然后将有限元模型与文献中可用的分析模型进行比较。在这项工作中，使用解析和有限元方法研究了与封装 MEMS 谐振器相关的各种真空区域，压力范围为 < 1 Pa、1-10 Pa 和 10-100 Pa。当压力降低时，来自有限元的Q因子分析和分析模型呈指数增长，直到其固有阻尼趋于平稳。本文描述的建模技术与先前报道的实验工作进行了比较，显示出Q变化的良好定性一致性- 压力因素。空气阻尼分为挤压膜阻尼和滑膜阻尼，以进一步探索阻尼效应，发现挤压膜阻尼是研究装置中的主要能量损失机制。在间隙闭合结构中，移动结构和固定指之间的气隙会产生挤压膜阻尼并导致能量损失，而它们之间的较小气隙会产生较大的力，从而增加阻尼。本文介绍的建模技术可以普遍应用于 MEMS 谐振器，以减轻空气阻尼损失。