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Procedía Engineering 64 (2013) 292 - 301

International Conference On DESIGN AND MANUFACTURING, IConDM 2013

A PSpice Model for the study of Thermal effects in Capacitive

MEMS Accelerometers

C. Kavithaa* and M. Ganesh Madhana

aDepartment of Electronics Engineering, Anna University, M.I.T Campus, Chennai, 600 044, India

Abstract

An electrical equivalent circuit model is developed to study the thermal effects in capacitive MEMS accelerometer. The mechanical system of the MEMS is implemented as an analogous electrical system and analyzed under different temperature conditions in the range of 100K to 400K. The variation in elongation, spring constant, damping coefficient of the MEMS cantilever are incorporated in the model. The entire analysis is carried at a constant pressure of 30Pa. The transient and frequency response is determined by simulating the equivalent circuit using PSpice® circuit simulator.

© 2013TheAuthors.Published by ElsevierLtd.

Selection and peer-review under responsibility of the organizing and review committee of IConDM 2013

Keywords:Thermal effect; capacitive accelerometer; MEMS; PSpice; circuit model.

1. Introduction

MEMS (Micro Electro Mechanical System) comprises of micro sensors, micro actuators, microelectronics and microstructures. MEMS accelerometer is one of the most popular MEMS devices for acceleration sensing. This signal to be detected may be static or dynamic due to gravity or motion respectively. These are used to detect seismic activity, angles of inclination, dynamic distance as well as speed, and rate of vibration. Accelerometer applications include medical, navigation, transportation, consumer electronics, and structural integrity. The capacitive type is preferred among piezoelectric, piezoresistive, hot air bubbles, and light based devices, due to high sensitivity, low noise and power saving features. Other characteristics include high resolution, accuracy and reliability. Accelerometers refer specifically to a mass-displacer [1] that can translate external forces such as

* Corresponding author. Tel.: 9786205893; fax: +91-44-22232403. E-mail address:kaviphd2011@yahoo.co.m

1877-7058 © 2013 The Authors. Published by Elsevier Ltd.

Selection and peer-review under responsibility of the organizing and review committee of IConDM 2013 doi:10.1016/j.proeng.2013.09.101

gravity into kinetic motion. The sensing part of the accelerometer usually consists of some type of spring force in order to balance the external pressure and displace its mass, thus leading to the motion. Since the structure consist of a proof mass, as well as a cantilever beam, residual gas is sealed inside the device, which can cause damping. Timo Veijola et. al. [2-4] has developed a circuit model that involves damping and spring forces created by squeezed film in a MEMS accelerometer. They have determined the response of the device using linear and nonlinear frequency dependent components, under different pressures. They have used a self developed circuit simulation program APLAC [5] for their analysis. We report an equivalent circuit model based on Timo Veijola's approach, to study the effect of temperature [6-9] on capacitive MEMS accelerometer. As equivalent circuits are developed for MEMS structure, a single domain approach to study the mechanical and electrical effects is possible. Incorporating thermal effects and implementing in electrical equivalent circuit model will help to provide an integrated approach to evaluate the MEMS performance under electrical, mechanical and thermal domains. In our approach, the parallel resonator circuit model is replaced by variable inductor and resistor implemented as controlled current sources [10]. This scheme also includes pressure dependent squeezed film and thermo elastic damping models. Based on this model, the transient and frequency response analysis are carried out at different temperatures, using a commercial circuit simulator PSpice.

2. MEMS capacitive accelerometer

2.1. MEMS structure

Fig. 1. Structure of Micromechanical Accelerometer [1].

A micromechanical accelerometer consists of mass suspended with two cantilever beams [1, 3]. The structure is implemented by three wafers in which top and bottom wafers are thick and developed as fixed electrodes and the middle one is thin and movable. The structure is shown in Fig. 1. The electrode is deposited by metal film which is formed on top surface of insulating glass layer. The sealed cavity is filled with the gas for damping of the system. At one side the contact pads are deposited in chip.

2.2. MEMS Accelerometer Modeling

The mechanical model developed as mass spring dashpot is shown in Fig. 2(a). According to Hooke's Law, extension is directly proportional to load. However, this proportionality holds up to a certain limit called the elastic limit. If the mass displaced by a distance of x from its rest position, spring causes the restoring force as Fr = —kx . Where k is the spring constant. Assuming that damping is purely viscous, and then the mass moves with the velocity v = dx/dt, the force created by the damper is FD = —Dv . Where D is the damping coefficient. As an external acceleration is applied, proof mass tends to move in opposite direction of the acceleration, due to Newton's law of motion. The force by the acceleration is F = Ma. Where M is the mass, a is the acceleration. The dynamics of a simple accelerometer is characterized by the following equation

Fig. 2. (a) schematic diagram of single axis accelerometer [2]; (b) electrical equivalent circuit of simple accelerometer [3].

dx dx M—^ + D — + kx = FEXT dt2 dt EXT

The force F is the sum of an external mechanical force FEXT and an internal electrical attractive force Fel.

electrical attractive force on the mass is caused by the potential difference across the capacitor plates. Assuming the motion is perpendicular to the plate surfaces, the electrostatic force is given by

eAU_ 2d2

Where £ is the dielectric constant of the gas, and d is the gap width. The sensing element typically consists of seismic mass which can move freely between two fixed electrodes, each forming a capacitor with the seismic mass used as the common centre electrode. The differential change in capacitance between the electrodes is proportional to the deflection of the seismic mass from the centre position. The mechanical accelerometer model is realized by the equivalent electrical model of a parallel resonator. The mass is considered as capacitance, cantilever or spring and dashpot are equated to inductance and resistance [3-5] respectively.

The system is given by the differential equation as

d < 1 dç 1 _ I

~dt2 + ~R~dt+~L<P~ EXT

Where ç is the flux in inductance is L , C is the capacitance, R is the resistance and IEXT is the external current. The displacement x equals LIL , where IL is the current through the inductor. The electrical equivalent circuit of mass spring damper system with displacement is shown in Fig. 2(b). The accelerometer physical parameters used in this work are listed in Table 1.

Table 1. Specifications of the MEMS accelerometer

Parameters

Values

Mass M

Width of the moving mass w Length of the moving l Length of the cantilever beam Gap widths dA and dB Mean free path A at 1 atm Viscosity coefficient n Temperature T

4.9 ßg 2.96 mm 1.78 mm 520 ßm

3.95 ± 0.05 ßm 70.0 ± 0.7 nm 22.6 ± 0.2 ßN.s.m-2 100-400 K

2.3. Thermal modeling

It is well known that as temperature increases, the thermal expansion coefficient also increases [6-9] and Young's modulus decreases, as shown in Fig. 3. This affects the length and spring constant of the cantilever. The change in cantilever length is given by the following relation

L = L0 (1 + (T2- T))

Where L is the final length of cantilever beam Lo is the initial length of the cantilever, aL is the thermal expansion coefficient of silicon material, T and T2 are the initial and final temperatures respectively.

100 ISO 200 250 300 3»

Temperature ( К )

100 150

Temperature i к )

Fig. 3. Thermal effect on accelerometer parameters (a) length, thermal expansion coefficient; (b) young's modulus, spring constant.

The length of cantilever increases due to thermal expansion. The original length of cantilever beam is obtained as 519.6^m atO°C . The total expansion of cantilever beam is calculated by 5 = LulAT . Thermal strain and stress can be estimated as £T - a/L and aT —EeT . Where ДT is the change in temperature, £T is the Strain, aT is the stress, E is the Young's Modulus of silicon material. Moment of inertia of a beam is given by I = bh3/12 .Where b is the width of the cantilever beam and h is the thickness of the cantilever beam. As temperature increases mass of cantilever beam also increases by bhlp [12, 13]. Where p is the density of silicon for the beam.

The temperature dependent parameters of linear type accelerometer in the mechanical domain and its equivalent in electrical domain are given in Table 2.

Table 2. Temperature dependent parameters

Mechanical Element Electrical Element Temperature

Damping Coefficient Conductance Dependent

Spring constant Inductance Dependent

Proof Mass Capacitance Independent

The damping coefficient is dependent on the dynamic viscosity and length, which depends on temperature and thermal expansion coefficient. It is incorporated as an inverse of resistance. The spring constant that depends on young's modulus, length and inertia is implemented as an inverse of inductance. This implementation is shown in Fig. 4. The effect of temperature on mechanical and electrical parameters of an MEMS accelerometer is shown in Fig. 5.

Fig.4. Electrical implementation of thermal effects (a) resistance; (b) inductance.

100 125 150 175 200 225 250 275 300 325 350 375 400

Temperature (K}

100 125 150 175 200 225 250 T

Temperature (K)

Fig. 5. Thermal effect on accelerometer (a) mechanical parameters; (b) electrical parameters. The increase in cantilever length affects the spring constant as per the following equation

and in turn affects the resonant frequency of the MEMS accelerometer as per the equation (6)

f= - K

Jr 2nV M

Viscosity of the gas is affected by temperature as follows

/■ N / \3/2

0.555T + C

Where n is dynamic viscosity at input temperature T , no is reference viscosity at reference temperature To, C is the Sutherland constant for argon gas (133). The mean free path equation for the gas medium is given by

y/2nda 2 LP

Where R is gas law constant 8.314510 JK_1 mole"1 , L is the Avogadro's number 6.0221367*10+23mole'1, T is the temperature in Kelvin, da is the collisional cross section3.57 *10"10m, P is the Pressure. As temperature increases the viscosity increases as per the Sutherlands formula. The effective viscosity depends on the absolute viscosity. The damping coefficient is determined by

D = (9)

2.4. Electrical Equivalent Circuit Model

In the proposed model, inputs are temperature, force and the bias voltage applied to the parallel plates. The resultant displacement is the output obtained from the system. The general block diagram is shown in Fig. 6(a).

Fig. 6. (a) Proposed model; (b) equivalent electrical circuit for capacitive accelerometer with thermal effects.

The electrical equivalent circuit incorporating thermal effects is normally developed from the parallel RLC circuit with two air gap sections implemented in the form of parallel LR sections (squeezed film damping effects) [2-4]. The completed equivalent circuit is shown in Fig. 6(b).

The temperature is modeled as a voltage source VTE in PSpice which has shown in Fig. 7(a). The pressure of the argon gas, considered as the damping medium, is also fixed as 30 Pa. The pressure parameter affects the Knudsen number, dynamic viscosity and effective viscosity.

Fig. 7. (a) Temperature as independent voltage source; (b) VCVS for thermal expansion coefficient and young's modulus.

Temperature is varied in the range of 100K to 400K for the device and the performance is studied. The thermal expansion coefficient ETEC and young's modulus Em values are evaluated by a second order polynomial, using curve fitting technique by the constant a and b . They are implemented as a Voltage controlled voltage source (VCVS) and is shown in Fig. 7(b). RTE , RTEC , Rm represent the resistance used for enabling the calculation.

Resistance and inductance in the analogous parallel resonator are implemented as a nonlinear voltage controlled current sources (VCCS) and are shown in Fig. 8. By considering Table 1 specifications, the electrical parameters are calculated under different temperatures. The inductance and resistance values are found to vary in the range of 3.43mH to 3.49mH and 23.06MQ to 5.423MQ respectively, for temperature range of 100K to 400K. The equation is formed by second order polynomial curve fitting method for the values of damping coefficient and spring

constant. These equations are implemented as a voltage controlled voltage source (VCVS) in the simulator to get as a voltage in a particular node. To implement temperature controlled resistance model, the above said node voltage of damping coefficient EG is considered with one independent voltage source VINR for developing Voltage controlled current source GINR, which replaces the resistance element in a parallel resonator model reported in the literature [1].

An input voltage ( EL ) is applied with an inductance Ll , for determining the value of current in inductance. These are connected in series with small value of resistance RLL which is used to avoid the convergence problem. This current is taken out using current controlled current source [11] in the other node. The previously evaluated spring constant value as a voltage and CCCS node voltage FL are used for further calculations. They are considered as inputs for the second order polynomial element of Voltage controlled current source G

1NL ' R1NR '

Rn , Rm

Rl , Rinl are the resistance required to fix the sources.

Fig. 8. Voltage controlled (a) resistance model;(b) inductance model.

The voltage controlled current source model of resistance and inductance is used to replace the simple resistance and inductance elements of the earlier model. The complete equivalent circuit is shown in Fig. 9. The developed model can be used to determine the system performance at any temperature in the range of 100 - 400K.

Fig. 9. Temperature controlled equivalent circuit.

3. Simulation results

3.1. Transient Analysis

A step input current equivalent to an acceleration of 0.5g is applied to the movable mass and the plate moves from the centre position. The gap increases in one side and decreases in the other. It reflects in the capacitance change in both air gaps. Capacitance variations due to two different temperatures 100K and 400K at 30Pa pressure

are shown in Fig. 10. At lower pressures, the displacement exhibits large oscillations. These results are in accordance with the results of Timo Veijola [1] and thus validate our model.

Time (m s)

Time (m s)

Fig.10. Transient response on capacitance at the temperature of (a) 100K; (b) 400K.

The air gap (dA) capacitance voltage in one side is reduced and the other side is increased. Further, the settling time of both capacitances is reduced as the temperature increases. This phenomenon is shown in Fig. 11.

Temperature (K}

Temperature (K)

Fig.11. Temperature variation on capacitance (a) voltage;(b) settling time.

The transient response of displacement is shown in Fig. 12. As the temperature increases, the corresponding displacement and settling time decreases, which is evident from the simulation carried out at 100K and 400K.

Fig.12. Transient response on displacement at the temperature of (a) 100K; (b) 400K.

The displacement and settling time found to be 120nm, 30ms and 114nm, 17ms respectively, at 30Pa. The peak value and settling time variation with temperature is shown in Fig. 13.

Fig.13. Temperature variation on peak value and settling time of displacement.

3.2. Frequency Analysis

An external ±3g force is applied to the mass of accelerometer and the displacement at each frequency is evaluated.

Fig.14. Frequency response on displacement at the temperature of (a) 100K and (b) 400K.

Fig. 15. Temperature variation on normalized displacement.

The analysis is repeated for different temperatures at constant pressure of 30Pa. The response is plotted in Fig. 14. From the response, the normalized displacement value of 17.821 dB , 12.975 dB at the resonance frequency of 1.2589 KHz, for the temperature of 100K and 400K are obtained. It is observed that, as the temperature increases, the normalized displacement decreases as shown in Fig. 15. However, below the resonant frequency, the impact of temperature is not significant.

4. Conclusion

A nonlinear electrical equivalent circuit model is developed for the analysis of thermal effects on the damping coefficient and spring constant of MEMS accelerometer. The analysis is carried out in the range of 100K to 400K, at a constant pressure of 30Pa. It is found that, as temperature increases the normalized peak displacement, at resonance decreases. In the cases of transient analysis, the settling time for displacement and capacitance are found to reduce with increase in temperature. The model is compatible with any electronic circuit implemented in PSpice simulator.

Acknowledgements

The authors gratefully acknowledge Anna University, Chennai for providing financial support to carry out this research work under Anna Centenary Research Fellowship (ACRF) scheme. One of the authors, C. Kavitha is thankful to Anna University, Chennai for the award of Anna Centenary Research Fellowship.

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