INTRODUCTIONSqueeze-Film DampingBasic TheorySecondary ConsiderationsContinuum LimitsEdge-effectsCompressibility and squeeze numberSubstrate Proximity EffectsSimulation resultsConclusionsAcknowledgmentsReferences1 Abstract— Squeeze film damping is a major source of noise in MEMS structures. Since damping limits the sensing accuracy of a given MEMS structure, a model relating design parameters to the damping coefficient is critical so that the system may be optimized. Several successful squeeze-film damping models exist [1-3]. While certain models are able to accommodate complicated edge effects [2] and perforations [3], to the best of our knowledge, none have addressed substrate proximity effects. This paper introduces a simplified squeeze-film macro-model for a lateral parallel-plate sensing structure (Fig. 1) that takes into account substrate proximity effects. Second order considerations such as edge effects, compressibility effects, Couette flow damping (between the substrate and the laterally moving mass), and non-zero slip conditions are not included here as they have been successfully addressed elsewhere [1-4]. Substrate proximity effects are approximated by modeling the gap separating the mass and substrate as a channel. Despite its many simplifications, our model delivers sufficient accuracy for hand-analysis as needed during initial design steps. Our results were validated against simulations obtained via the CoventorWare MemDamping module. Index Terms—Damping model, Squeeze film, Viscous Damping. I. INTRODUCTION amping is a critical consideration in the design of many planar MEMS devices such as accelerometers and gyroscopes. While Couette-type damping models typically take into account the interaction between a substrate layer and a moving mass [4], present squeeze-film damping models do not. The main motivation of this research stemmed from the fact that squeeze-film damping is affected by the substrate in many common situations. As illustrated in Fig. 1, the substrate tends to constrict the air being squeezed out of the gap. If the gap in question is located sufficiently close to the substrate, the pressure gradient between the two plates is directly affected. In modern MEMS processes, the distance separating a metal/poly layer and the substrate is typically between 1 and 10µm. Hence such effects can easily contribute non-negligible damping components. Section II of this paper begins with a treatment of some of the established theory describing squeeze-film damping. This is then used as the foundation for our modeling of substrate proximity effects. Several secondary modeling considerations (treated elsewhere) are discussed to provide the reader with a practical sense of what must be assessed in order to achieve accurate predictions of squeeze-film damping effects. In section III, simulation results are presented and compared to hand-analysis models derived in the previous section. II. SQUEEZE-FILM DAMPING A. Basic Theory In general, Navier-Stokes equations describe viscous, pressure, and inertial mechanisms in fluids. Under certain flow conditions, these much-complicated Navier-stokes equations can be simplified into the Reynolds equations. Its underlying assumptions are as follows: 1) the film is isothermal, 2) inertial effects are negligible, 3) amplitude motion and pressure changes are small, 4) fluid velocity normal to the surface is negligible, and 5) Substrate Effects in Squeeze Film Damping of Lateral Parallel-Plate Sensing MEMS StructuresAxel Berny, Student Member, IEEE DsubgfixedLmasshWFig. 1. Typical lateral parallel-plate sensing structure.2the gap is small compared to lateral dimensions (ho << L, W). The isothermal Reynolds equation is: tPHhPPHh ∂∂=∂∂⋅∂∂ )(3σ, (1) with P=∆p/Po and σ =12µL2/(Poho2), and where P is the normalized pressure, ∆p is the small variation in pressure, Po is the ambient pressure, H is the normalized gap thickness h/ho, µ is the fluid viscosity, and L is the length of the moving mass. B. Secondary Considerations 1) Continuum Limits Both Navier-Stokes and Reynolds are derived under the assumption that the fluid medium is continuous, implying that energy transfer is only achieved through molecular interaction within the fluid. This condition is only satisfied when the ratio of the fluid particle mean free path to the characteristic dimension of the system is less than one tenth or so. Violation of this condition leads to non-zero slip-conditions (momentum is transferred as fluid molecules collide with the oscillating plates), which in turn results in a higher fluid flow rate and a reduced damping coefficient. This effect can be modeled by an effective viscosity given in [8]: 159.196381⋅+=LPPoeffλµµ. (2) 2) Edge-effects Finite element simulations have been used to take effects due to finite size, edges, and perforations [2, 3]. As pointed out in [2], the use of trivial boundary conditions where the gauge pressure is set to zero at edges leads to large errors, especially as plate dimensions decrease (see Fig. 5 in [2]). This deficiency steams from the fact that the gauge pressure become zero only at a certain distance away from any edge. The authors in [2] demonstrated that applying trivial boundary conditions to a control volume that extends beyond the plate edges yields accurate (simulation) results. 3) Compressibility and squeeze number In squeeze-film analysis, it is common to introduce a non-dimensional squeeze number defined [1,3,5] as: ooPhL2212µωσ= , (3) where ω is the oscillation frequency. For squeeze numbers less than 0.2 [1], the gas behaves as if it were incompressible. Otherwise, the film stiffness increases as the squeeze number increases and the damping coefficient falls approximately as 1/σ 0.4 [5]. C. Substrate Proximity Effects For the sake of clarity/simplicity, let us consider the case where Reynolds equations apply. Furthermore, let us limit our analysis to geometries for which a 1-dimensional flow dominates (L>>W). This is only a modest restriction since this condition is satisfied for many structures such as ADI’s ADXL50 [6]. In addition, low excitation frequencies are assumed so that the term ∂P/∂t in (1) can be set to zero. Hence equation (1) is linearized as follows: ()thhxxPo∂∂⋅=∂∂32212µ, (4) where P(x) is the pressure departure from the nominal ambient pressure Po, as a function of x. The above
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