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In most MEMS devices, the moving micro-structures are surrounded by air which significantly affects their dynamics behaviors. The correct prediction of the squeeze-film air damping ratio is essential in MEMS (Micro-Electro-Mechanical System) devices design. In the paper, a static test is proposed to measure the squeeze-film air damping ratios of capacitive MEMS accelerometer under different pressures. The unsealed chip of capacitive accelerometer is placed in vacuum extraction equipment and an open loop circuit is developed to apply step signal in the test. By charging the pressure and measuring the overshoot
*M _{p}* and the settling time

*t*from the time response of the system, the damping ratio ξ under different pressures can be calculated. Finite element method (FEM) based on the modified Reynolds equation is utilized to simulate the transient response of the micro-structure. Good correlation between experiment and FEM analysis is obtained. The proposed static test in this paper provides a new method to more easily measure the dynamic performances of micro-structures under various pressures.

_{s}With the development of micro fabrication, Micro-Electro-Mechanical-System (MEMS) devices such as MEMS sensors, MEMS actuators and micro optical devices are widely used in aviation, aerospace, biomedicine and so on. The most notable characteristic of MEMS devices is that their structures are a few microns in size and separated by micron-sized gaps which are filled with fluid (mostly is air). As to micro-structure, the decrease in size leads to high surface-mass ratios and high surface forces that can mostly be neglected in macro-structures. Therefore, the dynamic behaviors of micro-parts in MEMS devices are tremen- dously affected by the fluid damping. The dominant damping in MEMS devices is the squeezed film air damping causing by air squeezed in and out of the gaps between the moving parts and the stationary parts. As the surface-mass ratios increase, viscous damping of the ambient fluid increases rapidly leading to significant energy dissipation. Namely, the performance or the quality factor of MEMS devices is largely determined by the squeezed film air damping. As a result, the design of the squeezed film damping becomes a crucial problem in MEMS.

Newell [

When a MEMS chip is packaged under a certain pressure, the dynamic response of it is generally aimed at defining its behavior in the working conditions and its performance in terms of the undamped natural frequency, quality factor (Q). The Q of a MEMS device is the most important properties which provide immediate quantification of energy dissipations and it is largely determined by the squeeze film damping. Hence, how to easily and accurately examine the squeeze film damping becomes essential.

In this paper, a static test based on an open loop circuit is presented to measure the squeeze film damping ratio of a capacitive MEMS accelerometer with a 2 μm air gap under different air pressures. FEM, used to solve the modified Reynolds equation, is applied to simulate the transient response. There is good agreement between experimental result and finite element analysis.

Based on a bulk micromachining technology, the capacitive MEMS accelerometer that consists of three silicon waters is illustrated in

where x is the displacement of the mass, M is the proof mass, C is the damping coefficient and K is the stiffness of spring.

By applying Laplace transform is Equation (1), the transfer function of the MEMS accelerometer is obtained

where

The motion of the mass-spring-damping system in MEMS accelerometer is equivalent to a second-order system. For a second-order underdamped system (

Then, the overshoot M_{p} which is defined as the maximum amount the system overshoots its final value divided by its final value can be calculated when the derivative is zero. From calculus we have this formula

From Equation (4), the damping ratio can be calculated by taking the logarithm on both sides

As the damping ratio increases, the overshoot does not exist. Therefore, the damping ratio can still be calculated from its settling time. For a second-order underdamping linear system, the envelope curve of its step response is

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The settling time (t_{s}) is that the system takes to settle down to steady-state and it is define as

・ When

・ When

Therefore, by measuring M_{p} and t_{s} of the step response of the mass-spring- damping system, the damping ratios caused by squeeze film air under different pressures are obtained. Furthermore, the quality factor of the MEMS accelerometer can be approximatively calculated by

When

where

The overdamped step response of second-order system is shown in

・ When

・ When

・ When

Let

The settling time can be measured from the step response of the accelerometer and the squeeze film air damping ratio is obtained using Equation (9).

In order to measure the step response of the MEMS accelerometer, an open loop circuit is designed shown in

When testing, the voltage signal from the signal generator is applied to the top electrode. The vacuum test bench is used to change the ambient pressure in the capacitive MEMS accelerometer. By pulling the proof mass to its equilibrium position and releasing under different air pressures, the step response of the MEMS accelerometer is tested. From the plots of the transient response of the system recorded by the oscilloscope, the overshoot M_{p} and the settling time t_{s} can be measured. And then, the squeeze film air damping ratio can be calculated. In this way, the electrostatic stiffness is also efficiently avoided. Most notably, due to the application of phase sensitive detection, the excitation signal frequency must be larger enough than the frequency of signal generator. In the test the excitation signal frequency is 100 kHz.

By changing the ambient pressure in capacitive MEMS accelerometer using the vacuum chamber, the squeeze film air damping ratio at 100 Pa, 300 Pa, 1600 Pa, 3000 Pa and room pressure. The plots of transient response of the capacitive MEMS accelerometer are shown is _{p}, settling time t_{s} and the squeeze film are damping ratio ξ calculated from the measured overshoot are listed in

The two basic damping mechanisms in inertial MEMS are slide and squeeze damping. Slide damping happens when a suspended plate is moving parallel to the substrate and it is not considered in this paper. Squeeze damping occurs when a parallel plate is moving vertical to the substrate. The motion increases the pressure and squeezes the gas out of the gap or decreases the pressure and draws the gas into the gap. The situation of capacitive MEMS accelerometer considered in this paper is demonstrated in

For macro-structures, the viscous and inertial effect both affects the behavior of squeeze film. However, for micro-structure in MEMS devices, inertial effect

Pressure | 100 | 300 | 1000 | 1600 | 3000 | 1atm |
---|---|---|---|---|---|---|

ξ | 0.172 | 0.189 | 0.344 | 0.457 | 1.003 | 16.038 |

can be negligible. Therefore, the squeeze film air damping problems encountered in the capacitive MEMS accelerometer is govern by the Reynolds equation which has the following formula [

Due to the small dimensions in MEMS devices, the temperature variation is negligible. With the isothermal condition, air density

where pressure P consists of two components:_{a} is the ambient pressure and P is the deviatory pressure caused by the squeeze air film effect; h is the height of the air gap and μ is the dynamic viscosity.

When a plate is moving perpendicular to the substrate, the air gap height h and dynamic viscosity μ are not functions of position. Equation (11) is simplified as

When the ambient pressure P_{a} is small, the rarefaction effects need to be taken into account. A few approaches are reported to model the rarefaction effects for squeezed lm damping problems under a wide range of ambient pressures [_{eff}. The commonly used effective viscosity is given by Fukui [

where K_{n} is the Knudsen number. In this paper, the formula of effective viscosity is defined as Equation (14):

where

there, _{0}; h_{0} is the gap height. For very narrow gap and low pressure, the Knudsen number increases and the fluid flow transits from continuum flow to rarefied flow.

In this paper, the transient response of the capacitive MEMS accelerometer was computed by using COMSOL Multiphysics v5.1 software.

For non-slip conditions at the proof mass and the bases, the modified Reynolds equation takes the form for a transient model:

where_{A} is the ambient pressure, p_{f} is the pressure developed as a result of the flow. h is the gap height. In the up gap,_{a}, h and μ need to be defined.

The finite element model of the spring-mass system is shown in

・ The air gap height: h = 2 μm

・ Length of spring: L_{s} = 1385 μm

・ Width of spring: W_{s} = 1452 μm

・ Thickness of spring: h_{s} = 38 μm

・ Length of proof mass: L_{m} = 1997 μm

・ Width of proof mass: W_{m} = 1680 μm

・ Thickness of proof mass: h_{m} = 380 μm

Fixed constraint was applied on one side of the spring and the step load was applied on the proof mass. The thin film flow was defined on the up and down surface of the proof mass.

The pressure distribution on the proof mass surfaces obtained from COMSOL Multiphysics is shown in

Pressure | 100 | 300 | 1000 | 1600 | 3000 | 1atm |
---|---|---|---|---|---|---|

ξ_{t} | 0.172 | 0.189 | 0.344 | 0.457 | 1.003 | 16.038 |

ξ_{c} | 0.181 | 0.205 | 0.351 | 0.478 | 1.058 | 16.791 |

standard earth gravity with different pressures. As the pressure increases, the oscillation times that the response requires to settle down to steady-state value decrease. When the pressure increases more than 3000 Pa, the response curve does not oscillate any more. The curves show that the higher pressure leads to a larger damping ratio.

The damping ratios under different pressures obtained from test and computation are listed in

The squeeze film air damping ratios in MEMS capacitive accelerometer is tested by using a special designed circuit and a vacuum chamber under different pressures in this paper. An assessment of dynamic response of the chip of a MEMS capacitive accelerometer is easily obtained by using this model.

The dynamic response of the chip of a MEMS capacitive accelerometer is analyzed by using finite element method. In the FEM model, the air viscosity is modified and numerical results are in good agreement with the test results within an acceptable error.

Mo, Y.M., Du, L.M., Qu, B.B., Peng, B. and Yang, J. (2017) Damping Ratio Analysis of a Silicon Capacitive Micromechanical Accelerometer. Wireless Sensor Network, 9, 178-188. https://doi.org/10.4236/wsn.2017.95010