Viscothermal acoustic waves in micro scale resonators

Accurate analytical air damping models for the transducers operating in the radio frequency range or for relatively large scaled devices like microphones are not available until now. Therefore the present paper aims at providing a full model wherein the compressibility, viscosity and heat conductivity of the air are considered. The solution of the model can be interpreted by “three wave theory” with coupled viscous, entropic and acoustic waves. Furthermore a self-developed FEM solver has been used to solve and simulate the full model equations. An aluminum nitride based piezoelectric resonator was fabricated and tested under controlled pressure from ultra high vacuum to normal atmosphere, using a custom-built vacuum chamber. The comparison of the air damping coefficient from the measurement and simulation confirms the accuracy of the full model prediction.

INTRODUCTION

Estimating the air damping characteristic of micro devices is one of the most important steps in the design process, because it determines the dynamic performance of the system. When oscillators are operated in a fluid environment, the Quality (Q) factor is typically dominated by the dissipation mechanisms associated with the surrounding fluid medium: squeeze film and viscous losses. The squeeze film damping has been traditionally modeled with the Reynolds equation [1], [2], by assuming a small air gap compared with the length or width of the oscillating structure surface. The damping coefficient due to squeeze film is (squeeze model):

$$c_{sque.}={64\sigma Pl_{x}l_{y}\over\pi^{6}\omega h}\sum\limits_{{m,n}\atop{odd}}{m^{2}+(n/\beta)^{2}\over(mn)^{2}\left\{\left[m^{2}+(n/\beta)^{2}\right]^{2}+\sigma^{2}/\pi^{4}\right\}},\eqno{\hbox{(1)}}$$ where and are the width and length of the beam, the aspect ratio of the beam, the air gap height, the air pressure and the resonance angular frequency, is the squeeze number, defined as , herein is the dynamic viscosity of the fluid. Viscous dissipation in resonators has been studied for many years, including the work of Blom et al. [3] and Hosaka et al. [4]. In the first approach the resonating beam is modelled as a sphere, however, the method for obtaining the sphere diameter is not clear. The second approach improved the method by visualizing the beam as a string of coherently spheres, whose diameter is the width of the beam. Based on the damping of each sphere, the total air damping is calculated as (viscous model): $$c_{visc}=3\pi\mu_{a}l_{y}\left(1+{l_{x}\over 2} \sqrt{{\rho \omega\over 2\mu_{a}}}\right), \eqno{\hbox{(2)}}$$ with the density of the fluid.