Equivalent circuit model for the damping of micro-oscillators from molecular to viscous flow regime

Abstract

This paper presents an equivalent circuit (EQC) model for describing the damping mechanism of micro-oscillators from the molecular flow regime up to the viscous flow regime. The EQC model consists of two parallel lines, representing the molecular and the viscous damping mechanism, respectively. The molecular line consists of an R–L circuit, which describes the intrinsic losses and the transfer of kinetic energy from the micro-oscillator to the surrounding gas molecules. The viscous line is formed by a resistor and a newly introduced inductive constant phase element. These elements represent the thermoviscous losses and the collision processes in the viscous flow regime. This EQC model was applied to the first fundamental bending mode of an oscillator vibrating in seven different types of gases (He, Ne, Ar, N2, CO2, N2O, SF6) as well as to the first five bending modes for another type of oscillator vibrating in nitrogen gas, only. In all cases, variations of an adjustable gap width to a fixed plate in the range of 150–3500 µm were also taken into account. The results from the presented EQC model are found to be in excellent agreement with the experimental data. Additionally, the electronic components of the EQC can be correlated to real physical properties of the setup, i.e. to the viscosity and density of the gas, as well as to the adjusted gap width and to the resonance frequency of the micro-oscillator. This allows the prediction of the quality factor curve of an oscillator over a large pressure range in a simple way for various configurations, concerning gas type, gap width, and bending mode/resonance frequency. Based on these results, an optimization model and design rules for enhancing the quality factor were derived.