Effect of Environmental Conditions on Quality Factors of MEMS Cantilever Beam Resonator in Gas Rarefaction

Abstract

This paper discussed the effect of environmental conditions (moisture and temperature) on the quality factors (Q-factor) of micro-electro-mechanical systems (MEMS) cantilever beam resonators in wide range of gas rarefaction (pressure (p), and accommodation coefficients (ACs)), and flexural mode of resonator. The modified molecular gas lubrication (MMGL) equation is applied for modeling the dominant squeeze film damping (SFD) problem on the quality factor of MEMS cantilever beam resonators to discuss the effect of environmental conditions. The external SFD and the internal structure damping (thermoelastic damping) and support loss) are accurately taken into account. Effective viscosity, which is ratio of dynamic viscosity and Poiseuille flow rate of moist air, is utilized to modify the MMGL equation to consider the environmental effects of moisture and temperature in gas rarefaction. In low pressures, mean free path changes more significantly with relative humidity and temperature than that of dynamic viscosity of moisture in gas rarefaction. Thus, effect of environmental conditions such as moisture and temperature must be discussed to improve Q-factors of MEMS cantilever beam resonators in wide range of gas rarefaction (p and ACs) and flexural modes of resonator. The results showed that Q-factor of SFD decreases significantly as moisture and temperature increase at higher gas rarefaction (lower p, and ACs), while Q-factor of SFD decreases and then increases slightly as moisture and temperature increase at lower gas rarefaction (higher p, and ACs). The total Q-factor is highly sensitive to the relative humidity and temperature in higher gas rarefaction (lower p and ACs) and lower flexural modes of resonator.

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Acknowledgments

This research was supported by the Institute for Computational Science and Technology (ICST), Contract Number: 08/2019/HĐ-KHCNTT in October 24th, 2019 and series number: 082019-311.

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Appendices

Appendix A

In Fig. 10, the Q factor of TED (QTED) is calculated as function of temperature (T) for various flexural modes of cantilever beam resonator. The result showed that QTED decreases as T increases for different modes of resonator. Also, QTED decreases more significantly as flexural modes of resonator increases because the TED increases with T and becomes dominantly in higher flexural mode of resonator. The calculated results of QTED from the present LR model [11] (Eq. 15 in [35]) showed good agreement with those obtained results from the Zener models [19, 20] (Eq. 14 in [35]), and those obtained results with COMSOL Multiphysics 5.5 [55] (Sect. "Quality Factors of MEMS Cantilever Beam Resonators" in [35]) in wide range of temperatures and resonator modes. Thus, the obtained results of QTED from the LR model, which are used in the present analysis, can be applied to calculate the total Q factor (QT) of MEMS cantilever beam resonators in wide range of temperature and flexural mode of resonator.

Fig. 10
figure10

The Q-factor of TED (QTED) versus temperature (T) for different flexural mode of resonator

Appendix B

In Table 2, \(Q_{\sup }\) is calculated by the model of Hao et al. [23] (Eq. 18 in [35]) for various flexural mode of MEMS cantilever beam resonator. The validation of this model has been proved by the assumption that the width of cantilever beam (\(w_{p}\)) is much less than the transverse elastic wavelength (\(\lambda_{T}\)) (\(\lambda_{T} /w_{p} > > 1\)). The result showed that \(Q_{\sup }\) decreases significantly as the mode of resonator increases because the support loss becomes a dominant source of energy loss on MEMS resonators in higher flexural mode of resonator. Thus, the results of \(Q_{\sup }\) can be used to calculate the total Q-factor (\(Q_{T}\)) of MEMS cantilever beam resonator in wide range of flexural mode of resonator conditions.

Table 2 The quality factor of support loss (\(Q_{\sup }\)) for various flexural mode of MEMS cantilever beam resonator

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Phan, M.T., Trinh, X.T., Le, Q.C. et al. Effect of Environmental Conditions on Quality Factors of MEMS Cantilever Beam Resonator in Gas Rarefaction. Sens Imaging 22, 6 (2021). https://doi.org/10.1007/s11220-020-00329-9

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Keywords

  • Quality factor
  • MEMS cantilever beam resonators
  • Temperature
  • Relative humidity
  • Gas rarefaction
  • Flexural mode of resonator