Circuit Theory Laboratory Report Series CT-39Equivalent Circuit Models for MicromechanicalInertial SensorsTimo Veijola

Thesis for the degree of Doctor of Technology to be presented with due permissionfor public examination and debate in Auditorium S4 at the Helsinki University ofTechnology (Espoo, Finland) on the 29th of January 1999, at 12 o'clock noon.

Helsinki University of TechnologyDepartment of Electrical and Communications EngineeringCircuit Theory Laboratory Espoo 1999

Distribution:Helsinki University of TechnologyCircuit Theory LaboratoryP.O. Box 3000FIN-02015 HUTTel. +358-9-451 2293Fax. +358-9-451 4818E-mail: timo@aplac.hut.�

ISBN 951{22{4403{9ISSN 1239-8233Picaset OyHelsinki 1999

T. Veijola,Equivalent Circuit Models for Micromechanical Inertial Sensors,Circuit Theory Laboratory Report Series, No. CT-39, Espoo 1999, 84 pp.,ISBN 951{22{4403{9,ISSN 1239-8233.AbstractNumerical simulation models for capacitive, micromechanical inertial sensorsare presented. These dynamic, nonlinear models include the contributionsof electrical, mechanical, and uidic energy domains by means of electricalequivalent circuits consisting of lumped, frequency-independent components.The mechanical resonance modes, gap capacitances, gas �lm damping forcesand the electrostatic actuation forces are accounted for in the model. Spe-cial attention is paid to modelling the rare�ed gas ow in narrow air-gaps,including the e�ect of gas-surface interaction. These gas-rarefaction e�ectsare included in the e�ective gas viscosity. Novel, compact design equationsare presented for the e�ective viscosity in narrow air-gaps, valid in viscous,transitional and molecular ow regions.Inertial sensor models for an accelerometer and an angular rate sensor havebeen built from electrical equivalent circuit blocks that consist of elementaryvoltage-controlled current and charge sources. The veri�cation show very goodagreement with the simulated and measured frequency responses over a widepressure range. The gas-surface interaction model has been tested by extract-ing a numerical value of the surface accommodation coe�cient. Compactequivalent circuit models allow both sensor and sensor system simulationswith any general purpose circuit simulation program. The models derivedhave been implemented and simulated with the circuit simulation programAPLAC.Keywords: Accelerometer model, Electromechanical analysis, Squeezed-�lm e�ect,E�ective viscosity, Accommodation coe�cient, Electrical equivalent circuit.

ContentsAcknowledgements 1List of publications 31 Introduction 52 Building blocks for micromechanical capacitive inertial sensors 72.1 Electrical equivalencies . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Mass-spring system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Air-gap capacitance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Electrostatic force in the air gap . . . . . . . . . . . . . . . . . . . . . 92.5 Gas �lm forces in the air gap . . . . . . . . . . . . . . . . . . . . . . 102.5.1 Modi�ed Reynolds equation . . . . . . . . . . . . . . . . . . . 112.5.2 Flow rate coe�cient and e�ective viscosity . . . . . . . . . . . 122.5.3 Gas-surface interaction . . . . . . . . . . . . . . . . . . . . . . 132.5.4 Linearized Reynolds equation . . . . . . . . . . . . . . . . . . 152.5.5 Electrical equivalent circuit implementation . . . . . . . . . . 162.5.6 Finite-di�erence equivalent circuit implementation . . . . . . . 182.5.7 Large-displacement model . . . . . . . . . . . . . . . . . . . . 193 Accelerometer model 213.1 Structure and operation of the device . . . . . . . . . . . . . . . . . . 213.2 Electrical equivalent circuit model . . . . . . . . . . . . . . . . . . . . 223.3 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.4 Model validity and the error sources . . . . . . . . . . . . . . . . . . . 233.4.1 Tilting motion of the mass . . . . . . . . . . . . . . . . . . . . 253.4.2 Systematic errors in the parameter extraction . . . . . . . . . 263.5 Extraction of the accommodation coe�cient . . . . . . . . . . . . . . 264 Angular rate sensor model 294.1 Structure and operation of the device . . . . . . . . . . . . . . . . . . 294.2 Electrical equivalent circuit model . . . . . . . . . . . . . . . . . . . . 304.3 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.4 Model validity and the error sources . . . . . . . . . . . . . . . . . . . 315 Summary of the publications 356 Conclusions 37References 43Errata 45

Paper reprints 47

AcknowledgementsThe work for this thesis has been carried out during 1994{1999 at the Circuit TheoryLaboratory of the Helsinki University of Technology.I wish to express my deep gratitude to Professor Martti Valtonen, the head ofthe Circuit Theory Laboratory, for providing me with the opportunity to do thiswork, and for his encouragement. His work with the circuit simulation programAPLAC has laid the foundation for the modelling and simulation work reported inthis thesis.I wish to express my gratitude to the sta� of the Circuit Theory Laboratory,especially to Jarmo Virtanen for helping me on various mathematical subjects dur-ing these years, and Ville Karanko for his help in the numerical solution of theBoltzmann equation presented in [P5]. Also, special thanks go to Sakari Aaltonenand Luis Costa for correcting the English language of all publications included thisthesis, including this survey.I am grateful to the industry partners and co-authors of the papers, HeikkiKuisma, Juha Lahdenper�a and Tapani Ryh�anen for setting challenging, practicalmodelling problems, and providing me with sensor speci�cations and measurementresults. The role of Tapani Ryh�anen has been of special importance; he saw thepossibilities o�ered by combining circuit simulation and micromechanical modellingand he also guided me into the world of micromechanics in 1994, when the modellingproject began.Professor Stephen Senturia from MIT contributed to the paper during the reviewprocess. I am grateful for the literature references he pointed out and for his excellentsuggestions on how to improve the quality of the thesis. The criticism on the thesisby Heikki Sepp�a from VTT Automation, also during the review process, forced meto analyze the error sources in the models. Thanks to him, a systematic error wasfound in the tilting motion model.This work has been �nanced by Helsinki University of Technology and VTIHamlin (formerly Vaisala Technologies).

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List of publicationsThis thesis is a compilation of the following papers published in international jour-nals and conference proceedings:[P1] T. Veijola, H. Kuisma, J. Lahdenper�a, and T. Ryh�anen, \Equivalent CircuitModel of the Squeezed Gas Film in a Silicon Accelerometer," Sensors andActuators A, vol. 48, pp. 239{248, 1995.[P2] T. Veijola and T. Ryh�anen, \Model of Capacitive Micromechanical Accelerom-eter Including E�ect of Squeezed Gas Film," in Proceedings of the 1995 IEEEInternational Symposium on Circuits and Systems, (Seattle), pp. 664{667, May1995.[P3] T. Veijola, T. Ryh�anen, H. Kuisma, and J. Lahdenper�a, \Circuit SimulationModel of Gas Damping in Microstructures with Nontrivial Geometry," in Pro-ceedings of Transducers'95 and Eurosensors IX, vol. 2, (Stockholm), pp. 36{39,June 1995.[P4] T. Veijola, H. Kuisma, J. Lahdenper�a, and T. Ryh�anen, \Simulation Modelfor Micromechanical Angular Rate Sensor," Sensors and Actuators A, vol. 60,pp. 113{121, 1997.[P5] T. Veijola, H. Kuisma, and J. Lahdenper�a, \The In uence of Gas-SurfaceInterface on Gas-Film Damping in a Silicon Accelerometer," Sensors and Ac-tuators A, vol. 66, pp. 83{92, 1998[P6] T. Veijola, H. Kuisma, and J. Lahdenper�a, \Dynamic Modelling and Simula-tion of Microelectromechanical Devices With a Circuit Simulation Program",Proceedings of the 1st International Conference on Modeling and Simulation ofMicrosystems, Semiconductors, Sensors and Actuators, (Santa Clara), pp 245{250, April 1998.All publications result from co-operation with VTI Hamlin. The contribution ofthe industry partners and co-authors Heikki Kuisma, Juha Lahdenper�a, and TapaniRyh�anen have been important in the micromechanical modelling project. The mea-surements presented in the publications were performed by them.All papers were primarily organized and written by the author. However, thecontribution of Tapani Ryh�anen was essential to the manuscript preparation of [P1].Heikki Kuisma included valuable background information and literature on the an-gular rate sensor, its operation principles, and the Coriolis force equations includedin [P4].3

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1 IntroductionA wide range of design tools are used in engineering generally. They range fromcalculations made by hand in the corner of a paper napkin and o�ce spreadsheetprograms to supercomputers. This is especially true in the �eld of microelectrome-chanics where physical phenomena in several engineering branches, viz., mechanical,thermal and electrical, are involved. One of the tools is a circuit simulation program,since microelectromechanical devices usually contain an electric interface connectedto the controlling and measuring electronics.Circuit simulators today o�er the possibility to simulate the behaviour of large,dynamic, nonlinear systems in the frequency and time domains. A wide rangeof simulation methods is available for the circuit designer including, e.g., noise,statistical and sensitivity analyses, all combined with optimization tools. Simulationmodels and their parameters for electrical components are available, but models forthe other energy domains are rare. When using circuit simulators, the natural wayto build models for other energy domains is the well-known method of electricalequivalencies and electrical equivalent circuits.This thesis presents electrical equivalent circuit models for micromachined ca-pacitive inertial sensors. Two component models are discussed in detail: a siliconcapacitive accelerometer and an angular rate sensor, gyroscope. Both sensors mod-elled are industrial products or prototypes and they are used in motion control invehicles and industry. The dynamic modelling of these components requires the con-sideration of electrical, mechanical and uidic energy domains and their interaction.The uidic domain, modelling the gas �lm damping, is treated extensively becauseof the lack of published models and because of the complex derivation of the model.Results from lubrication technology and the theory of rare�ed gas physics are ap-plied in the model. A novel, compact model for the gas �lm damping is derivedand implemented as an electrical equivalent circuit. Comparisons with frequency-domain measurements show that model accuracy is very good over a wide pressurerange in small-displacement conditions.The circuit simulation tool APLAC [1], under development in the Circuit Theorylaboratory, has good modelling facilities for nonlinear dynamic components. Bothsensor models were implemented in APLAC just as any other parameterized elec-trical component. The nonlinear modelling features and the optimization tools ofAPLAC were utilized extensively.This survey summarizes the work presented in publications [P1{P6]. The mi-cromechanical building blocks for inertial sensors are �rst presented in Section 2.Then, a model for an accelerometer is discussed in Section 3. Its error sources areconsidered, and due to a systematic error in the published results, the parameterextraction presented in [P5] is repeated. Section 4 shortly summarizes a model forthe angular rate sensor published in [P4]. A summary of the publications [P1]{[P6]is given in Section 5. Finally, the conclusions are presented in Section 6.5

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2 Building blocks for micromechanical capacitiveinertial sensorsIn the modelling of inertial sensors, four energy domains have to be considered:thermal, mechanical, uidic and electrical. Dynamic temperature changes can beomitted from the model when the use of a static temperature parameter is su�cient.The sensor models are built from separate blocks, each implementing the partialoperation in a single energy domain and interfacing with other energy domains.The building blocks discussed are the mechanical resonator and the phenomena inthe small air gap: the electrical attractive force, capacitance, and the gas �lm forces.2.1 Electrical equivalenciesThe use of electrical equivalencies is a widely applied approach in the �eld of, e.g.,mechanical engineering [2], thermal design [3] and acoustics [4]. Recently, several pa-pers in the �eld of micromechanics have also been published [5, 6, 7, 8, 9, 10, 11, 12].The equivalent circuits of sensors are usually built from linear resistors, capaci-tors, inductors and transformers, that is, models for real-world passive components.Here, on the contrary, the equivalent circuits are built from fundamental elementsof the circuit simulator used, namely, nonlinear or linear, current or charge sourcesthat have one or several controlling voltages. The circuits that result contain onlylumped, frequency-independent components and they can be simulated in both thefrequency and the time domain.Using the equivalencies, or inverse equivalencies, shown in Table 1, each physicalphenomenon in the sensor, instead of the whole sensor, is written separately as a cur-rent or charge source that depends on the controlling voltages. The complete sensormodel is then constructed by connecting these circuit blocks electrically. A gyratorconsisting of two controlled current sources, can be used to invert the equivalencies.This approach results in simple circuits that are e�cient to simulate.Mechanical Fluidic Electricalvelocity v pressure p voltage umass M volume V capacitance Celasticity k - inductance�1 L�1viscosity � viscosity �=A conductance Gforce F volume ow Q current idisplacement z - ux Table 1: Relations between mechanical, uidic and electrical quantities.In practice, scaling between the quantities is needed to keep the electrical equiv-alent voltages at approximately the same levels as the electrical voltages. This mayaid convergence in the iterations required in the nonlinear circuit simulation.7

2.2 Mass-spring systemThe operation of an inertial sensor is often based on a mass-spring system. Theoperation bandwidth depends on the fundamental resonance frequency determinedby the mass and the supporting spring. For proper operation, damping is usuallyrequired. In practice, this is accomplished with the internal friction of the gas owin the narrow air gap.If the mass is not exible it can be modelled with a lumped model such thatthe mass or the moment of inertia is concentrated in a single point, as an e�ectivemass. The harmonic motion of the mass-spring system can then be modelled witha second-order di�erential equationM @2z@t2 + @z@t + kz = Fext; (1)where t is time, z is the displacement of the e�ective mass M , is the dampingcoe�cient of the spring and k is the spring coe�cient. The driving force Fext is asum of forces created by the applied acceleration M � a, possible electrostatic forces,and the force due to the damping gas. After replacing the mechanical quantitieswith electrical equivalencies, Eq. (2) in [P1] models the behaviour of an electricalresonance circuit. Due to equivalencies shown in Table 1, the equivalent circuitrepresentation for the mass-spring system is a parallel resonator, shown in Fig. 1.Here the resonator inductance is replaced with a gyrator and a capacitor. Thegyrator coe�cients are such that the voltage uz across the capacitor Cz is equivalentto mass displacement z in �m (in [P1] symbol x is used instead of z).C G Cz

u z

iext

u v

Figure 1: Equivalent circuit for the mass-spring system. The spring is modelled witha gyrator and capacitance Cz, mass is equivalent to capacitance C and the losses inthe spring are modelled with conductance G.The method of implementing the resonator makes it easy to build the remainingequivalent-circuit blocks presented in this Section as voltage controlled sources, sinceall blocks are dependent on the displacement. As a matter of fact, this controllingvoltage is the heart of the modelling approach used. For example, a nonlinear force8

acting on the mass in the collision to the �xed electrode is very straightforward toimplement as a current source controlled by the displacement voltage.A similar modelling approach is applied in the case when another mass is at-tached with a spring to the �rst mass. Coupled di�erential equations modelling thissituation are given in [P4] in Eqs. (1{6). In this case the equivalent circuit consistsof two coupled LC resonators (Fig. 5 in [P4]).If oscillation modes other than the fundamental modes have to be considered inthe model, a separate resonator for each mode is required. Such a model is validonly for small amplitudes, that is, for cases where the deformation of the springsand the nonlinearity of the gas �lm are small enough to prevent coupling betweenthe modes.2.3 Air-gap capacitanceThe mass displacement in the sensor is measured by means of the change in thecapacitance between the moving mass and the �xed electrode. In symmetrical struc-tures, there are air gaps on both sides of the mass, and the capacitance di�erenceis measured instead. If motion normal to the surfaces is assumed and the fringing�elds are ignored, the capacitances in air gaps 1 and 2 areC1 = �Ad1 � z + C01; (2)C2 = �Ad2 + z + C02; (3)where d1 and d2 are the static air gap heights, z is the mass displacement, A is thesurface area and � is the permittivity of the gas. C01 and C02 are the constant straycapacitances. A more complicated expression for the capacitances results (Eqs. (9)and (10) in [P2]) if the small tilting angle due to the nonsymmetrical mass supportis accounted for in the model. An improved model for the tilting motion is discussedin Section 3.4.1, which suggests that the cantilever beam length b should be replacedwith a smaller, e�ective value.Since the capacitances C1 and C2 depend on the displacement z, the equivalent-circuit model consists of a nonlinear charge source, controlled by the capacitor volt-age and displacement voltage uz. The charges are q1 = C1u1 and q2 = C2u2, whereu1 and u2 are the voltages across air gaps 1 and 2, respectively.2.4 Electrostatic force in the air gapThe electric �eld across the air gap causes an electrical attractive force to act on thesurfaces. This force can be exploited in several ways depending on the operatingprinciple of the sensor. With a periodic electrical excitation, the mass can be set intoan oscillating motion, or the electrical attractive force can be used in a feedback loopto keep the mass centered. Symmetric biasing techniques for symmetrical structures9

can also be used to control the e�ective elasticity of the mass-spring system. If thegap height is small compared with the other dimensions, the fringing �elds can beneglected and simple equations for the force and torque acting on the mass can bederived. For a structure consisting of two air gaps, the force isFel = �A2 � u21(d1 � z)2 � �A2 � u22(d2 + z)2 : (4)In small-displacement conditions the force acting on the surface is the sum ofthe spring force �k ��z and the contribution of the electrical attractive force�F = @Fel@z ����z=0 ��z � k ��z = � k � 2�Au2d3 ! ��z; (5)where a symmetrical structure and symmetrical biasing conditions (d = d1 = d2 andu = u1 = u2) are assumed. According to Eq. (5), the spring constant is reduced toa smaller, e�ective value due to the DC bias voltage u.When the small tilting angle due to the nonsymmetrical support in the practicalstructure is considered, a more complicated expression for the force (or torque)results (Eqs. (11) and (12) in [P2]). See Section 3.4.1 for the e�ective cantileverbeam length.If the mass displacement is large and the gap is small, a pull-in phenomenon mayoccur due to the large electrical attractive force. Modelling of this phenomenonis di�cult due to its strong nonlinear characteristics and because of the multiplesolutions of the system state. Also, additional modes of movement will be generatedin the collision, making the model more complicated.2.5 Gas �lm forces in the air gapThe desired frequency or step response of the sensor can be achieved by controllingthe damping caused by the viscosity of the gas in the gap. Due to the compressibilityof the gas, spring forces also exist, leading to a complicated frequency response. The�rst gas �lm damping model for accelerometers was published in 1990 by Starr [13].He used the lubrication equation, Reynolds equation, and solved it with FEM-techniques using an analogy with the heat conduction equation. Andrews et al. [14,15, 16] used, in 1992{1995, the Reynolds equation in their accelerometer design.They utilized an analytic expression for damping and spring forces that can bederived for rectangular surface geometry [17].In capacitive sensors, the gap between the moving elements is very small. Forbulk micromachined devices, the gap is a few micrometers; for surface microma-chined devices, below one micrometer. In the viscous ow region, that is, at largegap heights, the velocity of the gas is zero at the surfaces and the ow pro�le inthe gap is parabolic. However, when the mean free path � of the gas moleculesis comparable with the gap dimensions, the ow velocity at the surface is greater10

than zero and the ow pro�le will not be parabolic anymore. This happens at lowpressures or in extremely narrow channels at atmospheric pressures, as well.For an accurate damping model, it is important to take into account the gas rar-efaction e�ect, even if the gap height is 30 times larger than the mean free path. Forexample, at atmospheric pressures in 2�m air gap (�air = 64 nm), a typical clearancein micromechanical inertial sensors, the gas rarefaction e�ectively decreases the vis-cosity about 20%. At lower pressures, the gas rarefaction e�ect even dominates thedamping in the gap and it must be considered in sensor design and in the simulationmodel.Burgdorfer's classical slip- ow approximation [18] for the gas rarefaction e�ect insmall clearances, published in 1959, is generally in use. Unfortunately, this approx-imation is accurate only if the gap height is much larger than the mean free path.Andrews et al. [14, 15, 16] studied the gas rarefaction e�ect in accelerometers, andused the results of Knudsen's classical ow measurements in capillary tubes [19],published in 1909. However, as shown in the model comparison in [P1] (Table 2and Fig. 9), a superior model for the rare�ed gas ow is the one derived from thelinearized Boltzmann equation. That model is used here since it is valid over a widepressure range and includes the contribution of the surface conditions.2.5.1 Modi�ed Reynolds equationA nonlinear partial di�erential equation, the Reynolds equation [20, 21], is used inlubrication theory to determine the behaviour of a thin uid �lm between movingsurfaces. For very narrow gaps, the modi�ed Reynolds equation must be applieddue to the gas rarefaction, as shown by Fukui and Kaneko [22, 23]. It is applicablein modelling the gas ow between moving micromechanical structures too. Whenthe surfaces do not move in the x- or y-directions, the modi�ed Reynolds equationis @@x "�h3Qpr(h; p)@p@x# + @@y "�h3Qpr(h; p)@p@y# = 12�@(�h)@t ; (6)where �p�n is a constant, and gas density �, pressure p, and the gap separation h arefunctions of time t and place (x, y). � is the viscosity of the gas and Qpr(h; p) is therelative ow rate coe�cient. The contribution of the rare�ed gas ow is includedin Qpr(h; p) which is a function of h and p (Qpr = 1 for viscous ow). When anisothermal process is assumed (n = 1), density � can be replaced with pressure p.In the Reynolds equation, the gas inertia has been neglected. The equation isaccurate only if the contribution of the gas inertia is small. The condition for thisis that �!h2=� � 1 [20]. This condition is given only for the continuum ow region(Qpr = 1).11

2.5.2 Flow rate coe�cient and e�ective viscosityThe measure of the gas rarefaction e�ect in the air gap is the Knudsen number Kn(or the inverse Knudsen number D), the ratio between the mean free path � andthe gap height h,Kn = �h; D = p�2Kn : (7)Because the mean free path � is inversely proportional to the pressure (� =�0P0=p, �0 is the mean free path at pressure P0), small Knudsen numbers (Kn <0:01) model viscous, continuos ow. In practical inertial sensor structures, the gas ow is in the transitional (0:01 � Kn � 3) or in the molecular (Kn > 3) region.For an accurate damping analysis, it is essential to have a model that is valid forarbitrary Kn.The modi�ed Reynolds equation (6) includes the gas rarefaction e�ect in the rel-ative ow rate coe�cient Qpr(h; p). It is the ratio between the ow rate coe�cientQp(D) and the continuum ow rate D=6 (Qpr(h; p) = 6Qp(D)=D) [22]. It dependson the gap height, pressure and also on the surface and gas properties. The owrate coe�cient has been derived from the linearized Boltzmann equation by Cercig-nani et al. in the 1960s [24, 25]. It was solved using both variational analysis andnumerical methods, assuming di�use molecular re ections from the surfaces. Theresult of the variational analysis is a non-trivial expression, consisting of Abramowitzintegrals [25] (Eq. (6) in [P5]).In small-displacement conditions, the ow rate coe�cient depends on static dis-placement d and static pressure Pa only. In this case, the ow rate coe�cient can beincluded in the viscosity coe�cient. The resulting coe�cient is an e�ective viscosity�e�. It is related to the ow rate coe�cient as follows:�e� = �Qpr(d; Pa) = D�6Qp(D) : (8)[P1] presents a simple experimental approximation for this e�ective viscosity, theresult of �tting the respective ow rate coe�cient to the values tabulated by Fukuiand Kaneko in [22]:�e� = �1 + 9:638K1:159n : (9)Its relative accuracy is �5% and it is valid for 0 � Kn � 880 (0:0018 � D � 1).Based on the e�ective viscosity extracted from the accelerometer measurements atvarious pressures, this approximation was compared in [P1] with other publishedmodels including the gas rarefaction e�ect. The approximation Eq. (9) agrees mostclosely with the extracted values.12

2.5.3 Gas-surface interactionUnfortunately, the assumption of di�use molecular re ections seems not to hold formicromachined surfaces, as shown by measurements by Arkilic [26, 27]. The contri-bution of specular re ections increases the ow rate coe�cient. This is one of thereasons why there is a systematic deviation between the model and the measure-ment results in [P1]. By means of the surface accommodation coe�cient �, specularmolecular re ections are accounted for (� = 1 for di�use re ections, and � = 0 forspecular re ections). Its value cannot be exactly calculated as it depends on variousproperties of the surface and the gas. The coe�cient value must thus be extractedexperimentally.When � < 1 is considered, and an identical gas-surface interaction on both sidesof the air gap is assumed, a new model, presented by Fukui and Kaneko [23, 28], isapplicable. They solved and tabulated ow rate coe�cient values as a function ofthe inverse Knudsen number D and the accommodation coe�cient � (� = 0:7, 0.8,0.9, 1.0). The expression given by Fukui and Kaneko is too complicated to be usedin a practical model. The following approximate equation has been �tted [P5] tothe tabulated ow rate coe�cient values:~Qp(D;�; �) = D6 + 1�1:34p� ln� 1D + 4:1�+ �6:4 + 1:3 � (1� �)1 + 0:08 �D1:83 + 0:64 � �D0:171 + 1:12 �D0:72 : (10)It is valid for D � 0:01 and 0:7 � � � 1, and its relative error is less than � 1%.The tabulated values of the ow rate coe�cient together with Eq. (10) are depictedin Fig. 2. For comparison, Burgdorfer's �rst-order slip- ow approximation [18]~Qp,slip(D;�; �) = D6 "1 + 6� 2� � 1� p�2D # (11)is drawn in the same �gure.However, the air gap surfaces are not necessary identical in practice. For exam-ple, one surface in the accelerometer modelled in Section 3 is made of silicon andthe other is metallized. Another solution for the ow rate coe�cient results if it isassumed that the metallized surface has � = 1. Following the guidelines by Fukuiand Kaneko [28], the linearized Boltzmann equation has been derived in [P5] fornonsymmetrical boundary conditions. To verify the result, both variational analy-sis and numerical methods were applied in the solution. The ow rate coe�cientresulted from the variational analysis is given by the rather long equation (11) in[P5].Again due to the complicated expression, an approximate equation, �tted tothe calculated ow rate coe�cient, has been given in [P5]. It approaches the exactcontinuum ow value for large D, whereas the logarithmic term ensures correct13

0.01 0.1 1 10 100

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APLAC 7.10 User: HUT Circuit Theory Lab. Fri Sep 26 1997

Qp

D

α=0.7

α=1.0

Figure 2: Flow rate coe�cient Qp(D;�; �) values from Ref. [23] (2) for symmetricalsurfaces, and the approximate function Eq. (10) (||) for � = �0 = �1 = 0.7, 0.8,0.9, 1. The �rst-order slip- ow approximation in Eq. (11), ({ { {) is drawn forcomparison.asymptotic behaviour when D approaches zero. The approximate equation for the ow rate coe�cient is~Qp(D;�0; 1) = D6 + 2� �0p� ln� 1D + 2:18�+ �00:642+ (1� �0)(D + 2:395)2 + 1:12 � �0D � 1:26 + 10�0D1 + 10:98 �D + e�D=58:77 : (12)The relative error of this approximation is less than �1% for all values of D and0 � �0 � 1. Figure 3 shows the ow rate coe�cient calculated from the approximatefunction Eq. (12) and from the variational analysis solution. For comparison, the ow rate coe�cient for the �rst-order slip- ow approximation, derived in [P5], isalso drawn in Fig. 3.~Qp,slip(D;�0; �1) = D6 a0a1 + 4a0 + 4a1 + 12a0a1 + a0 + a1 ; (13)where a0 = �02� �0 2Dp� 14

0.01 0.1 1.0 10.0 100.0

1

3

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APLAC 7.10 User: HUT Circuit Theory Lab. Mon Sep 29 1997

Qp

D

α0 = 1

α0 = 0

Figure 3: Flow rate coe�cient values calculated with the variational analysis (2)and the approximate function (||) for �0 = 0, 0.25, 0.5, 0.75, 1 (�1 = 1). The�rst-order slip- ow approximation ({ { {), derived for the nonsymmetrical case inEq. (13), is drawn for comparison.a1 = �12� �1 2Dp� :The general solution, valid for arbitrary D, �0 and �1, can be estimated witha linear combination of the ow rate coe�cients previously presented in Eqs. (10)and (12).~Qp(D;�0; �1) � �1 � �01� �0 ~Qp(D;�0; 1) + 1� �11� �0 ~Qp(D;�0; �0); (14)when �0 < �1.2.5.4 Linearized Reynolds equationAssuming a small pressure change �p compared with the ambient pressure Pa anda small displacement z compared with the static height d of the gap (h = d + z),a linearized form of the modi�ed Reynolds equation (6) results. The linearizedequation for isothermal conditions is:Pad212�e�r2 ��pPa �� @@t ��pPa � = @@t �zd� : (15)The gas rarefaction e�ect is included in the e�ective viscosity �e�.15

The form of the solution of the linearized Reynolds equation depends on theboundary conditions, that is, the surface and gap geometry. Expressions for thepressure distribution and the resulting force acting on the surfaces are discussedin the literature [13, 17, 20, 21, 29, 30, 31]. For rectangular, rigid plates movingin normal to their surfaces, the pressure distribution for steady-state sinusoidalexcitation can be found analytically. The resulting force acting on the surfaces canbe expressed with an in�nite series [17] that converges rapidly,Frz = 64�PaA�6d Xm;n=odd �=�2 + j(m2 + c2n2)(mn)2 [(m2 + c2n2)2 + �2=�4] ; (16)where A = wl is the surface area, c = w=l, m and n are odd integers and � is thesqueeze number� = 12�e�w2!Pad2 ; (17)where ! is the angular frequency. The damping and spring forces acting on thesurfaces areFr0z = 64�PaA�6d Xm;n=odd m2 + c2n2(mn)2 [(m2 + c2n2)2 + �2=�4] ; (18)Fr1z = 64�2PaA�8d Xm;n=odd 1(mn)2 [(m2 + c2n2)2 + �2=�4] : (19)Analytic solutions for the linearized Reynold equation have been presented byDarling et al. [30] for other simple boundary conditions. For example, the solutionfor tilting surfaces, discussed also by Pan et al. [31], is very similar to that of Eq. (16),with the exception that one of the indices m or n is even, not odd.2.5.5 Electrical equivalent circuit implementationWhen, now, electrical equivalencies for the force and velocity in Table 1 are applied(in the periodic steady-state z = j!v), Eq. (16) is equivalent toir = Xm;n=odd im;n = uv Xm;n=odd 1Zm;n : (20)After a short manipulation, each impedance Zm;n can be written asZm;n = �6d!(mn)264�PaA �(m2 + c2n2) + j ��2 � : (21)It is easy to see that the impedance in Eq. (21) can be implemented with an RLcircuit section, see Fig. 4a. Fig. 4b shows an alternate equivalent circuit implemen-tation. 16

Cm,n

u v

Gm,nRm,n

L m,n

im,n

im,n

u v

a) b)Figure 4: a) A single RL-section in the squeezed-�lm equivalent circuit model. Volt-age uv is equivalent to the velocity of the mass and the current im;n is equivalent toforce. b) An alternate realization using a gyrator and a GC-section.The component values areRm;n = Gm;n = (mn)2 m2w2 + n2l2 ! �6d3768A�e� ; (22)Lm;n = Cm;n = (mn)2 �4d64APa : (23)The current in Eq. (20) is approximated with several parallel connected sections,each representing a single term in the series. In practice, only a few of these sectionsare required for su�cient accuracy. From Eq. (22), it can be seen that for squaresurfaces, the ratio R3;1=R1;1 = 45. At low frequencies, where the damping part ofthe impedance dominates, the accuracy of a single section (w = l) is about 5%; forthree sections the accuracy is about 0.6%.In [P1], the equivalent circuit implementation was derived by comparing thespring and damping forces of the analytic solution with the real and imaginary partsof the current in the equivalent circuit. The electrical equivalent circuit realizationcan be also easily derived from the solution by Darling et al., Eq. (21) in [30],published recently.The electrical equivalent circuit model was derived from the frequency response,but it is valid in the time domain, too, since the components in Eqs. (22) and (23)are frequency-independent. The model is accurate only if the requirements for lin-earization are met. The large-displacement model is discussed in Section 2.5.7.Due to the nonsymmetrical mass support the surface may also contain a smalltilting component, as is the case in the modelled accelerometer in Section 3. Due tothe linearity assumptions, the total gas �lm force can be calculated as a superpositionof forces by both components. Since the expression for the tilting motion is verysimilar to the expression for the perpendicular motion, the same equivalent circuit17

topology is applicable.If the supporting bar length is b and the mass length is l, it is easy to show thatthe force of the tilting component, Eq. (16), is reduced by a factor of (1 + 2b=l)2.The model for damping due to the tilting motion is implemented in the form ofadditional parallel RL- or GC-sections, where the index m is odd and n is even andthe component values in Eqs. (22){(23) are multiplied by (1 + 2b=l)2.The damping caused by the tilting motion is much smaller than that caused bythe perpendicular motion. This can be seen by comparing the �rst-order dampingapproximations of both modes. The ratio is R2;1=R1;1 = 40 for square surfaces andb = l=2, predicting a contribution of about 2.5% in the damping force.2.5.6 Finite-di�erence equivalent circuit implementationThe solution for rectangular surfaces is not always su�cient. Rectangular holes inthe mass can be used, in practice, for damping control. Moreover, the corners ofthe surface are not precisely rectangular, and due to nonsymmetrical mass support,the gap displacement depends on the surface coordinates. To model these cases,a �nite-di�erence model was generated in [P3] that satis�es Eq. (15) and the newboundary conditions. The model was again implemented with an electrical equiva-lent circuit [P3]. Each element in the two dimensional grid is implemented with aconductance that is controlled by adjacent elements, and with a controlled capaci-tance. The elements are basically identical to the GC-section shown in Fig. 4b. Thevoltage-controlled current source models the gradient term in Eq. (15). The secondterm is equivalent to a constant capacitor. The source term in Eq. (15) is equivalentto an external displacement voltage uz controlling the charge in the capacitance.Alternatively, if velocity excitation is used, the conductance is controlled by thevelocity voltage uv instead.Here, again, the approach of using voltage-controlled blocks leads to a verystaightforward implementation of the pressure gradient. The boundary conditionof zero pressure di�erence at the surface borders is satis�ed by simply setting therespective controlling voltages to zero. Thus no special treatment for the borders orcorners is required.Accurate simulation using the model presented requires a large amount of nodesin the equivalent circuit. However, for a �xed surface topology, a simple laddercircuit, the same as used in the rectangular solution in [P1], can be used instead.The values of the ladder components can be found by means of curve �tting andparameter extraction. The resulting simple model is valid for any displacement d,pressure Pa and e�ective viscosity �e� and its solution is found much faster than thesolution of the large mesh circuit.With this �nite-di�erence approach, a model for true large-signal conditions,where the pressure caused by the signals is comparable to or greater than the ambientpressure, can be constructed, in which case each element in the mesh should obeythe nonlinear Reynolds equation (6). 18

2.5.7 Large-displacement modelIn large-displacement conditions, an analytic solution for the modi�ed Reynoldsequation (6) cannot be generally formulated. FEM or �nite-di�erence methods arein this case needed. A �nite-di�erence equivalent circuit implementing Eq. (6) can beformed [32] using the same techniques as used in the linear circuit in [P3]. However,when the gap height is uniform, a simple, approximate large-displacement modelcan be derived [33]. For a uniform gap height and linearized pressure (p = Pa+�p,�p� Pa), Eq. (6) reduces toPah2Qpr(h; Pa)12� r2 ��pPa �� @@t ��pPa � = 1h @h@t : (24)Linearization with respect to the gap height was not required here and thus Eq. (24)models accurately the gas ow at any large-displacement condition, assuming neg-ligible pressure variations compared with the ambient pressure. This equation isvery similar to the linearized one in Eq. (15); only two di�erences can be seen: inEq. (24) h and �=Qpr(h; Pa) replace d and �e� in Eq. (15), respectively.The nonlinear solution is now approximated by replacing d and �e� in the com-ponent values solved from the linearized equation. The modi�ed component valuesare Rm;n = Gm;n = (mn)2 m2w2 + n2l2 ! �6h3Qpr(h; Pa)768A� ; (25)Lm;n = Cm;n = (mn)2 �4h64APa : (26)In contrast to linear components, they are implemented with nonlinear current andcharge sources controlled by the displacement voltage uz. For the best accuracy, itis essential that the relative ow rate coe�cient depends on the large-displacementgap height h. Since the components depend on the dynamic displacement, a correctDC solution always results, also when the mass is initially displaced (h 6= d), causedby, e.g., an electrostatic force. The resulting approximation is surprisingly good inlarge-displacement transient analysis, too. The nonlinear model consisting of threesections has been veri�ed by comparing it [33] with the more accurate nonlinear�nite-di�erence model [32]. Simulations show that the maximum displacement errorin the step response of an accelerometer with at frequency response, uniform gapheight and square surfaces is only 1% and 2.7% for displacements of 0:6 d and 0:75 d,respectively.In [P1], the large displacement model was derived from the linearized equa-tion (15). A similar derivation is not generally justi�ed, but in this case, the resultis quite good, as explained above. However, it was not emphasized that the e�ectiveviscosity in Eqs. (20){(23) of [P1] should depend on the dynamic displacement d�z,not on d only. 19

20

3 Accelerometer modelAccelerometers are used in industrial and automotive applications where detectionof motion is required. Small-sized and high-performance acceleration sensors canbe fabricated by micromachining [34, 35]. A model for the capacitive accelerometerproduced by VTI Hamlin is discussed here.Besides the accelerometer models published by Starr [13] and Andrews et al. [14,15, 16], discussed in the beginning of Section 2.5, several other accelerometer sim-ulation models have been published [12, 36, 37, 38, 39, 40]. These models use verysimple models for the gas �lm damping or ignore it. In the accelerometer model [9],published recently, the damping circuit presented in [P1] is applied.3.1 Structure and operation of the deviceAn accelerometer measures acceleration by means of the change in capacitance ratioof the air gaps on both sides of a moving mass. The mass and the elasticity of thecantilever beams form a mechanical resonator and the gas �lm damping producesthe desired frequency response. Figure 5 shows the structure of the modelled device.

Figure 5: The structure of the micromechanical accelerometer produced by VTIHamlin. A thin silicon wafer containing the mechanical structure is anodicallybonded between two glass-covered thick silicon wafers.21

3.2 Electrical equivalent circuit modelThe equivalent circuit is easily constructed by connecting the various force generat-ing blocks, presented in Section 2, in parallel. The simulation model contains a sin-gle resonance circuit at the fundamental resonance frequency, two gap capacitances,electrostatic force source and several gas �lm sections. The complete equivalentcircuit model is shown in Fig. 6. Capacitances C1 and C2, the electrical attractiveforce iel and the gas �lm components are controlled by the displacement (voltageuz) of the mass center position.

u z

u 1

u 2

iext

u v

Resonator Gap capacitancesElectrostatic force

iel

Gas film (section 1)

Gas film (section 2)

C1

C2

Figure 6: Equivalent circuit for the accelerometer. The number of gas �lm sectionsdepends on the desired accuracy.The accelerometer model was implemented in APLAC in 1994 [41] with twoimplementation levels: level 1 for linear motion and level 2 for tilting motion. Itwas built in a way similar to other implemented parameterized models for electri-cal components. In addition to the capacitance nodes, external nodes are providedfor connecting an external force (current iext) source, and for monitoring the massdisplacement (uz). The model has about 30 parameters including, e.g., sensor di-mensions and gas constants.An improved implementation was released in 1998 [42] introducing new model22

levels 3 and 4 including, e.g., equations for the e�ective viscosity, Eq. (14), tem-perature-dependent gas parameters, and bias potential. Level 4 includes also themoment of inertia of the mass and the contribution of the small tilting motion tothe damping model.The �nite-di�erence gas �lm model, discussed in Section 2.5.6, is not imple-mented in the accelerometer model. It is a separate component block that can beconnected in accelerometer's velocity nodes. However, this requires velocity controlinstead of the displacement control used in [P3]. At the present, since the circuitis linear, the displacement is not needed at all in the model. An elecrical equiv-alent circuit mesh of any size is generated in a straightforward manner using theprogramming capabilities of APLAC.Three applications for the equivalent circuit model for the accelerometer can bedistinguished:� Design of an electrical sensor system containing one or several accelerometers.� Design of the accelerometer itself.� Platform for research work on the various phenomena in the accelerometerand in microelectromechanical devices generally. New results achieved areimplemented in the model.3.3 MeasurementsThe displacement of the mass is measured, in practice, by measuring the capacitancedi�erence of the gap capacitances using a carrier frequency (1MHz). However, whenthe accelerometer model is simulated, the capacitance measurement circuit can beignored, since the model gives directly the displacement as a node voltage uz.In all comparisons with measurement results, curve �tting techniques are usedhere because not all model parameters, e.g., the gas pressure, are known. Thismethod easily leads to an excellent �t between the model and the measurements.Unfortunately, erroneous extracted parameters may result, since all errors in themodel will accumulate in these extracted parameters. This is the case especially ifthere are several unknown parameters and little measured information. This is thereason why measurements at several bias conditions [P1] or at several pressures [P5]were �tted simultaneously to the model.Figure 7 demonstrates the �t between the accelerometer model and measuredfrequency responses at four pressures. The simulated capacitance-voltage charac-teristics are shown in Fig. 8.3.4 Model validity and the error sourcesIn the devices modelled, the gap height is a few micrometers and the surface di-mensions are a few millimeters. Due to large surface dimensions compared with the23

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Figure 7: Measured frequency responses (2) of Sensor #3 and �tted model responses(||) at four pressures.

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Figure 8: The simulated (||) and measured (2) accelerometer capacitance-voltagecharacteristics, and the simulated mass displacement ({ { {). The minimum of thecurve is not at zero volts but at about 0.5V.24

gap height, and due to the very small tilting angle, the error of the capacitance andelectrostatic force can be considered negligible. The springs are also in their linearoperating region. The second torsional mode appears at a frequency 10 times higherthan the fundamental mode. Its in uence on the measured frequency responses canbe considered negligible. The basic requirement for very narrow gas �lm is also ful-�lled. The condition for the negligible gas inertia seems also to hold, since !�d2=�is about 0.02{0.07, depending on the sensor, at frequency 10 kHz at atmosphericpressures.The measured capacitance-voltage characteristics of an accelerometer, Fig. 8,show an antisymmetric response with respect to the DC bias voltage. This is proba-bly due to the work function di�erence between the silicon and Ti/Al metallization.This phenomenon is included in the model as a constant bias voltage, substractedfrom the capacitor voltage in the electrical attractive force equation (4). This modelis not good and it can be used only when measuring sensor capacitances with acarrier frequency.3.4.1 Tilting motion of the massThe in uence of the tilting motion due to the nonsymmetric mass support has beenassumed negligible and ignored in [P1], and a very simple model for the tiltingmotion has been used in [P2], [P3], [P5] and [P6]. In this simple model it wasassumed that the spring de ection is linear, suggesting a gap displacement pro�leof z = 2x+ 2bl + 2b zc; (27)where zc is the displacement of the reference position of the center of the mass andb is the length of the spring (the surface coordinates are 0 < x < l). However, amore realistic static displacement pro�le, calculated from the beam de ection theory,suggests that the tilting of the mass is larger than assumed in the simple model.Fortunately, a better model results by simply replacing the physical cantilever beamlength, a parameter of the model, with an e�ective length be�. It can easily bederived from the formulas for the spring de ection pro�le given in [9]:be� = 4b1 + 3l + 6b26(b1 + l + 2b2)b1 + b2; (28)where b1 is the actual length of the exible spring and b2 is the non- exible partof the beam. In the case of the accelerometers modelled, the ratio between thee�ective beam length and the length of the beam in the simple model, b = b1 + b2,varies approximately between 0.6 and 0.7. This will sligtly change the capacitance,electrostatic force, and the damping since they depend all on be�.Because of the tilting motion of the mass, the e�ective mass should be calculatedfrom the moment of inertia of the moving structure. However, the real mass was25

used instead in all simulations and curve �ttings presented in [P1] and [P5]. Assum-ing rectangular mass surfaces and a height smaller than the other dimensions, thee�ective mass reduced to the center of the surface isMe� = "1 + l23(2be� + l)2 #M: (29)The reduced mass will be about 15% higher than the real mass. This correctionincreases the extracted spring constant and e�ective viscosity approximately by15%.3.4.2 Systematic errors in the parameter extractionIn two of the papers, [P1] and [P5], e�ective viscosity and ow rate coe�cient valueshave been extracted from measurements. In [P1], the in uence of the tilting massmotion is ignored in the model. Since the model is �tted to the measurement results,the additional error caused by the tilting motion is included in the extracted e�ectiveviscosity. The extracted e�ective viscosity was 25% smaller than predicted by themodel. Using the error estimations above, the contribution of various error sourcescan now be estimated: tilting motion ignored in the gas �lm model (�5%), error inthe e�ective mass (17%), and the contribution of the accommodation coe�cient byabout 0.9 (13%).In the measurements presented in [P5], the contribution of the tilting motion isincluded in the model. However, due to the simpli�ed tilting model the extracted ow rate coe�cient will be about 13% smaller, increasing the extracted accommo-dation coe�cient of about 0.1.3.5 Extraction of the accommodation coe�cientThe extraction of the accommodation coe�cient originally presented in [P5] is re-peated here assuming 13% smaller ow rate coe�cients than in [P5]. Four di�erentaccelerometers were measured at 23 controlled pressures altogether. The ow ratecoe�cient was extracted separately at each pressure by �tting the model response tothe measured frequency response. Figures 9 and 10 depict the new approximatelycorrected ow rate coe�cients. The results of �tting accommodation coe�cients� = 0:912 and �0 = 0:828 are also shown in the �gures. The corrected result iscredible, since the �t is now even better than that reported in [P5].26

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Sensor #1

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Sensor #3

Sensor #4

Figure 9: Extracted and approximately corrected ow rates at 23 gas pressures. Theresult of curve �tting when identical surfaces are assumed (||) giving � = �0 =�1 = 0:915 is also shown.

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Sensor #1

Sensor #2

Sensor #3

Sensor #4

Figure 10: Extracted and approximately corrected ow rates at 23 gas pressures.Di�use re ections from the metal surfaces are assumed (�1 = 1). The result of curve�tting (||) giving �0 = 0:828 is also shown.27

28

4 Angular rate sensor modelA micromechanical angular rate sensor has applications in industrial and automo-tive motion sensing. It is used, for example, to measure the yaw rate of a vehicleor angular velocity about the vehicle's vertical axis. Angular rate sensors, or gyro-scopes, can be divided into two categories based on the operation principle: rotatingand vibrating gyroscopes. A review article [43] describes the operating principles ofvarious possible designs.Here, a vibrating gyroscope based on a dual pendulum system is discussed. Therehave been no other published models describing the operation of a complete angularrate sensor.4.1 Structure and operation of the deviceThe angular rate sensor is assembled similarly to the accelerometer with three siliconwafers bonded together. The mechanical system in this case is more complicatedconsisting of two vibrating beams, as shown in Fig. 11. Its structure and operationare discussed in mode detail in [P4]. When operating, the outer mass is electricallyDual pendulum

Frame actuation

Beam sensing

contacts

contacts

system

Figure 11: Structure of angular rate sensor prototype being developed by VTI Ham-lin. The center silicon wafer contains the mechanical system.actuated into a vibrating motion. When the system is rotated, the vibrating motion29

is coupled to the inner mass due to the Coriolis force. The vibrating motion ofthe inner mass is sensed capacitively, in a manner similar to the accelerometer.Symmetrical, adjustable bias voltages are utilized to match the resonance frequenciesof the vibrating masses through the e�ective spring elasticity in accordance withEq. (5). The operation of the sensor system is quite complex. Several phase-lockedloops are usually used to control the operation of the device. Therefore, a simulationmodel is useful in the system design.4.2 Electrical equivalent circuit modelIn the model implemented, three di�erent modes of coupled motion of both mov-ing elements are modelled, two of them are vibrational and one is linear. Eachmode is implemented with a coupled resonance circuit, as shown in Fig. 5 in [P4].Torsional equivalencies are applied in vibrational resonators: voltage and currentare equivalent to angular velocity and twisting moment, respectively. Instead ofa controlling voltage relative to the displacement, a voltage relative to the tiltingangle is available in these resonators. There are, altogether, eight air-gaps withcapacitive acting/sensing electrodes. The Coriolis force is modelled with a singlevoltage-controlled current source. As the device operates in near-vacuum condi-tions, the damping is small and the gas compressibility is negligible. In this casethe gas �lm model is reduced to a single damping conductance. The block diagramof the model is shown in Fig. 12. These blocks are discussed in detail in [P4].The electrical equivalent circuit model presented here was implemented as amacro component using APLAC's modelling language. The use of the model isidentical to the simulator's build-in models. There are 28 model parameters and 23interfacing nodes in the model.4.3 MeasurementsThe frequency response was measured using electrostatic excitation on one top-bottom electrode pair and capacitance half bridge detection on the second similaropposite electrode pair, see Fig. 11 in [P4]. A 1MHz test signal (VS) was used onthe bridge and the ampli�er output was tuned by an inductor. Linear voltage-force(and torque) conversion is obtained with a DC bias voltage (Vb) and opposite phaseAC drive.A simulated frequency response using the extracted parameters is comparedwith measurement results of the beam mass displacement. A nonsymmetrical elec-trostatic force signal was used to excite the mass in both linear and torsional modes.The beam mass was actuated with an AC voltage from 1kHz to 20 kHz and a sym-metrical 10V DC bias voltage Vb was applied to air gaps 1 and 2. The measuredand simulated frequency responses are compared in Fig. 13. The extracted torsionaland linear viscosities were used to estimate the remaining gas pressure inside thesealed structure. 30

Resonator

Resonator

Resonator

ωF,x

ωB,x

ωF,y

ωB,y

vF

vB

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cor

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τB,y,el

B,elF

Beam air gaps

Frame air gapsθF,x

z F

θB,y

z B

τ

z B

AB

C

B

A

Figure 12: Block diagram of the angular rate sensor model.The measured and simulated frequency responses in Fig. 14 show how the in-creasing symmetrical DC bias e�ectively changes the spring constants through thenon-linear force function and shifts the beam resonance to a lower frequency. Thismeasurement is similar to the previous one with the exception that the bias voltageis varied (1, 3, 10, 20 and 30V).Capacitance-voltage characteristics of the simulated model are also in good agree-ment with measured characteristics, as can be seen from Figs. (9) and (10) in [P4].4.4 Model validity and the error sourcesThe modes of motion of the moving parts are, in principle, easier to model thanthe nonsymmetrically supported mass in the accelerometer. However, the mass ofthe accerelometer is rigid, but the whole gyroscope structure is more or less exible.Since a simple model is desired, the structures must be assumed to be non- exibleand the vibrating modes are de�ned by the torsional and exural springs only. It isdi�cult to estimate the amount of error due to this simpli�cation.Since the surface dimensions are not especially large compared with the gapheight, the fringing �elds should be considered in an accurate model. In practice,31

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Figure 13: Relative amplitude (|{) and phase ({ { {) responses of simulated andmeasured (2) frequency responses of beam mass.

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Figure 14: Simulated (|) and measured (2) frequency response of the beam masswith di�erent bias voltages. 32

both beams rock about two axes. However, in the capacitance and force equationsonly the vibration about the main axis is modelled.The incorrect signs in the Coriolis force equation (25) in [P4] are not due to aprinting error. The prototype of the device shown in Fig. 11 was designed accordingto this erroneous equation. The corrected signs have been taken into account in thenext design step of the gyroscope.

33

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5 Summary of the publicationsThe Circuit Theory Laboratory report CT-18 \Accelerometer Model in APLAC" [41]was the �rst paper documenting the micromechanical modelling performed by theauthor. It is not included in this thesis for several reasons: it is rather a referencemanual than a scienti�c publication, it is too extensive (62 pages), and the mostimportant aspects are summarized in papers [P1] and [P2]. Moreover, its contentsare partially out of date due to the development of the model over the years.Paper [P1] concentrates on the modelling of the forces caused by the squeezedgas �lm in the silicon capacitive accelerometer. The model is based on the analyticsolution of the linearized Reynolds equation for a rectangular geometry [17] andthe simulation model is implemented with a simple electrical equivalent circuit.The results from the gas-lubrication area, published by Fukui and Kaneko [22], areapplied in modelling the pressure dependence of the gas �lm damping. This gas�lm model is compared with other models and with accelerometer measurements intransitional and molecular damping regions. It is con�rmed to be superior, in spiteof the systematic error between the measurement results and the model that wecould not explain in the paper at the time of the publication. A simple approximateequation for the pressure dependence of the damping in the form of e�ective viscositywas given. The expression for the e�ective viscosity is useful in all calculations ofgas �lm damping in any structures built with narrow air-gaps. Transient responsesof an accelerometer are demonstrated by simulations. The systematic error in theextracted e�ective viscosity has been explained in Section 3.4.2. This paper hasbeen cited by other authors [9, 29, 30, 31, 44, 45, 46, 47, 48, 49] in internationaljournals and conferences.Paper [P2] discusses the complete accelerometer simulation model including non-linear e�ects in the electrical attractive force and capacitances in the accelerometer'sair gaps, in addition to the damping model presented in [P1]. The paper empha-sizes the electrical equivalent circuit approach, its use in simulating sensor systems,and the practical model implementation in APLAC. The capacitance-voltage char-acteristics and the frequency responses with various symmetrical bias voltages of anaccelerometer are simulated and compared with the measured responses. Also, atransient simulation example of a simple sensor system is given.In [P3] the gas �lm damping model is extended to more complicated sensorsurface geometries. A new approach is required because the gas- ow model presentedin [P1] is valid only for rectangular, solid surfaces. The paper presents a �nite-di�erence equivalent circuit model for the gas ow in the accelerometer's air gap.Again, measurements are used to verify the model. The paper has been cited byother authors [36].Paper [P4] presents a complete dynamic model for a micromechanical angularrate sensor, a gyroscope. The previously used modelling technique is applied, butthe sensor structure is more complicated than the accelerometer structure. Thevibrational modes of the system are modelled with coupled resonance circuits. The35

electrostatic and Coriolis forces, as well as variable capacitances in the small air-gapsare modelled with nonlinear controlled current sources. Again, the model is veri�edagainst measured frequency responses and capacitance-voltage characteristics.In [P5] the gas-surface interaction model presented in [P1] is improved by includ-ing the in uence of the surface accommodation coe�cient in the damping model.Two di�erent interaction models are discussed: symmetrical surfaces and non-symmetrical surfaces in the air gap. The solution for the symmetrical case canbe found in the literature [23]. In the non-symmetrical case it is assumed that themetallized surface is rougher than the silicon surface and a new solution is presentedin [P5] for a problem of nonsymmetrical air-gap surface properties. To verify themodel, two di�erent solution methods are applied to solve the linearized Boltzmannequation. Both methods are di�cult to evaluate computationally. The resultingexpression is too complicated to be used in practical design, and thus a simple ap-proximation including the in uence of the accommodation coe�cient is given. Itreplaces the approximation for the e�ective viscosity given in [P1].New measurements with four di�erent accelerometers are used to con�rm thevalidity of the model and to extract the numerical value of the accommodationcoe�cient. The ow rate coe�cient extracted from the measurement results clearlyreproduces the shape given by the mathematical model. The extracted value of theaccommodation coe�cient is compared with published values found in the literature,showing a good agreement. The systematic error in the extracted accommodationcoe�cient has been explained in Section 3.4.2Paper [P6] presents elementary equivalent circuit blocks of inertial microme-chanical sensors. The building blocks are basically the same as discussed in papers[P1{P5]. Here, the blocks are arranged as a sample library consisting of param-eterized component blocks. Several features of a circuit simulation program arediscussed and their use in the simulation of micromechanical sensors is emphasized.As an example, an accelerometer and an angular rate sensor model are constructedwith these blocks.Papers [P4] and [P5] were presented at international conferences [50] [51] by theauthor.

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6 ConclusionsModels for micromechanical sensors have been presented, implemented as electricalequivalent component models, and veri�ed with measurements. The circuit modelspresented can be, in principle, implemented in any circuit simulation tool withnonlinear dynamic modelling capabilities. In addition to APLAC, there are severalcommercial tools, e.g., Saber and the numerous variations of Spice, where the modelspresented could be implemented. However, APLAC has proved to be an excellentenvironment where to develop, implement and test these models mainly due itsversatile programmable netlist and optimization methods.Gas �lm forces acting on the surfaces in the air gap are modelled and im-plemented with electrical equivalencies. Results from lubrication and rare�ed-gasphysics are utilized in this model. A new mathematical model for the e�ective vis-cosity in narrow channels for nonsymmetrical surfaces is presented and a relativelygood approximate equation for practical design is given. This model can be gen-erally used in calculating the damping of a gas �lm between parallel surfaces inviscous, in transitional and molecular damping regions.Two industrially manufactured sensors have been modelled with electrical-equiv-alent circuits. Their simulated frequency responses are in good agreement withmeasured responses at small-displacement conditions. The model predicts correctlyalso several special features of these devices: the capacitance-voltage characteristics,and the change of the e�ective spring constant vs. capacitor bias voltages. Theapproximate large-displacement damping model for parallel surfaces presented herehas not yet been veri�ed with measurements; this will be done in the near future [33].The accelerometer model is under development and new features are included in itwhen needed. A proper model for the bias potential, that can be detected in thecapacitance-voltage characteristics, is one of the features that shoud be included inthe model.The error caused by several simpli�cations in the accelerometer model are dis-cussed and veri�ed with measurements. In the accelerometer model development,most of the e�ort has been put into the development of the gas �lm model. Un-fortunately, the simple model for the tilting motion, adopted in the beginning ofthe modelling work, was not veri�ed in papers [P2], [P3], [P5] and [P6]. A moreaccurate model, shortly discussed in Section 3.4.1, suggests that the canlilever beamlength and the mass should be replaced with their e�ective values in the model.The in uence of the corrections is estimated, predicting a damping 15% smaller.This systematic error does not invalidate the conclusions presented in [P5], since thecorrected value, estimated to be 0.88{0.92 for symmetrical surfaces and 0.8{0.84 fornonsymmetrical surfaces, is still in agreement with the published values. The newextracted values do not yet reveal whether the symmertrical surface model is betterthan the nonsymmetrical one. However, since the new corrected values are larger,a better agreement with the published values is achieved using the nonsymmetricalmodel. The in uence of the the corrected tilting model is only approximated here37

and thus the parameter extraction should be repeated using e�ective values for thecantilever beam length and mass.Since these models have been implemented using electrical equivalencies, theycan be simulated together with electronic control circuits in all analysis modes, bothin the frequency and time domains. The inertial sensor models are in use in thedesign of the interfacing electronics of the sensor system. The accelerometer modelis in use in the design of new accelerometers and the interfacing control circuits.The building blocks presented in this thesis are useful in simulating the operationof similar sensors and actuators [52].

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References[1] M. Valtonen et al., APLAC { An Object{Oriented Analog Circuit Simulator andDesign Tool. Helsinki University of Technology, Circuit Theory Laboratory andNokia Research Center, Hardware Design Technology, 7.0 Reference Manualand 7.0 User's Manual, Otaniemi, Jan. 1996.[2] J. Scholliers and T. Yli-Pietil�a, \A SPICE-based library for mechatronic sys-tems," in Proceedings of the 1995 IEEE International Symposium on Circuitsand Systems, (Seattle), pp. 668{671, 1995.[3] Bell Telephone Laboratories, Physical Design of Electronic Systems, vol. I.Prentice-Hall, 1971.[4] S. N. Rschevkin, The Theory of Sound. Pergamon Press, 1963.[5] S. Senturia, \CAD for microelectromechanical systems," in Proceedings ofTransducers'95 and Eurosensors IX, vol. 2, (Stockholm), pp. 5{8, June 1995.[6] S. D. Senturia, \CAD challenges for microsensors, microactuators, and mi-crosystems," Proceedings of the IEEE, vol. 86, pp. 1611{1626, 1998.[7] H. A. C. Tilmans, \Equivalent circuit representation of electromechanical trans-ducers: I. Lumped-parameter systems," Journal of Micromechanics and Micro-engineering, vol. 6, pp. 157{176, Sep 1996.[8] H. A. C. Tilmans, \Equivalent circuit representation of electromechanical trans-ducers: II. Distributed-parameter systems," Journal of Micromechanics andMicroengineering, vol. 7, pp. 285{308, Dec 1997.[9] R. P. van Kampen and R. F. Wol�enbuttel, \Modeling the mechanical behav-ior of bulk-micromachined silicon accelerometers," Sensors and Actuators A,vol. 64, pp. 137{150, 1998.[10] L. Lin, R. T. Howe, and A. P. Pisano, \Microelectromechanical �lters for signalprocessing," Journal of Microelectromechanical Systems, vol. 7, no. 3, pp. 286{294, 1998.[11] T. T.-C. Nguyen, L. P. B. Katehi, and G. M. Rebeiz, \Micromachined devicesfor wireless communications," Proceeding of the IEEE, vol. 86, pp. 1756{1768,Aug. 1998.[12] M. Kraft, C. P. Lewis, and T. G. Hesketh, \Closed-loop silicon accelerometer,"IEE Proc. Circuits Devices Syst., vol. 145, no. 5, pp. 325{331, 1998.[13] J. B. Starr, \Squeeze-�lm damping in solid state accelerometers," in Solid-StateSensor and Actuator Workshop, IEEE, (Hilton Head Island), pp. 44{47, June1990. 39

[14] M. K. Andrews, G. C. Turner, P. D. Harris, and I. M. Harris, \A resonantpressure sensor based on a squeezed �lm of gas," Sensors and Actuators A,vol. 36, pp. 219{226, 1993.[15] M. Andrews, I. Harris, and G. Turner, \A comparison of squeeze-�lm theorywith measurements on a microstructure," Sensors and Actuators A, vol. 36,pp. 79{87, 1993.[16] M. Andrews and P. D. Harris, \Damping and gas viscosity measurements usinga microstructure," Sensors and Actuators A, vol. 49, pp. 103{108, 1995.[17] J. J. Blech, \On isothermal squeeze �lms," Journal of Lubrication Technology,vol. 105, pp. 615{620, Oct. 1983.[18] A. Burgdorfer, \The in uence of the molecular mean free path on the perfor-mance of hydrodynamic gas lubricated bearings," Journal of Basic Engineering,Trans. ASME, vol. 81, pp. 94{99, Mar. 1959.[19] S. Dushman, Scienti�c Foundations of Vacuum Technique. Wiley, New York,1949.[20] W. E. Langlois, \Isothermal squeeze �lm," Quarterly of Appl. Math., vol. 20,pp. 131{150, 1962.[21] W. A. Gross, Gas Film Lubrication. Wiley, New York, 1962.[22] S. Fukui and R. Kaneko, \Analysis of ultra-thin gas �lm lubrication basedon linearized Boltzmann equation: First report | derivation of a generalizedlubrication equation including thermal creep ow," Journal of Tribology, Trans.ASME, vol. 110, pp. 253{262, Apr. 1988.[23] S. Fukui and R. Kaneko, \A database for interpolation of Poiseuille ow ratesfor high Knudsen number lubrication problems," Journal of Tribology, Trans.ASME, vol. 112, pp. 78{83, Jan. 1990.[24] C. Cercignani and A. Daneri, \Flow of a rare�ed gas between two parallelplates," Journal of Applied Physics, vol. 43, pp. 3509{3513, Dec. 1963.[25] C. Cercignani and C. D. Pagani, \Variational approach to boundary-value prob-lems in kinetic theory," The Physics of Fluids, vol. 9, pp. 1167{1173, June 1966.[26] E. B. Arkilic, Measurement of the mass ow and tangential momentum ac-commodation coe�cient in silicon micromachined channels. PhD thesis, Mas-sachusetts Institute of Technology, Cambridge, Jan. 1997.[27] E. B. Arkilic, M. A. Schmidt, and K. S. Breuer, \Gaseous slip ow in longmicrochannels," Journal of Microelectromechanical Systems, pp. 167{178, June1997. 40

[28] S. Fukui and R. Kaneko, \Analysis of ultra-thin gas �lm lubrication basedon the linearized Boltzmann equation," JSME International Journal, vol. 30,pp. 1660{1666, 1987.[29] R. B. Darling, C. Hivick, and J. Xu, \Compact analytical models for squeezed�lm damping with arbitrary venting conditions," in Proceedings of Transduc-ers'97, (Chicago), pp. 1113{1116, June 1997.[30] R. B. Darling, \Compact analytical modeling for squeeze �lm damping witharbitrary venting conditions using a Green's function approach," Sensors andActuators A, vol. 70, pp. 32{41, 1998.[31] F. Pan, J. Kubby, E. Peeters, A. T. Tran, and S. Mukherjee, \Squeeze �lmdamping e�ect on the dynamic response of a MEMS torsion mirror," in Journalof Micromechanics and Microengineering, vol. 8, pp. 200{208, 1998.[32] T. Veijola, \Finite-di�erence large-displacement gas-�lm model," in Proceedingsof Transducers'99, (Sendai), June 1999. Manuscript in review.[33] T. Veijola, H. Kuisma, and J. Lahdenper�a, \Compact large-displacement modelfor capacitive accelerometers," in Proceedings of the 2nd International Confer-ence on Modeling and Simulation of Microsystems, Semiconductors, Sensorsand Actuators, (Puerto Rico), Apr. 1999. Manuscript in review.[34] H. H. Bau, N. F. deRooij, and B. Kloeck, Mechanical Sensors. VCH, Weinham,New York, 1994.[35] J. Lahdenper�a and U. Merihein�a, \A low cost high performance capacitiveaccelerometer," in Proceedings of Sensor '91, vol. 3, (N�urnberg), p. 235, 1991.[36] C. Bourgeois, F. Porret, and A. Hoogerwerf, \Analytical modeling of squeeze-�lm damping in accelerometers," in Proceedings of Transducers'97, (Chicago),pp. 1117{1120, June 1997.[37] H. Ahmad, A. J. Al-Khalili, L. Landsberger, and M. Kahrizi, \A 2D microma-chined accelerometer," in Proceedings of the Third International Conference onElectronics, Circuits, and Systems ICECS'96, (Rodos), pp. 908{911, Oct. 1996.[38] C. P. Lewis, M. Kraft, and T. G. Hesketh, \Mathematical model for a micro-machined accelerometer," Trans. Inst. MC, vol. 18, no. 2, pp. 92{98, 1996.[39] J. M. Gomez-Cama, O. Ruiz, S. Marco, J. M. Lopez-Villegas, and J. Samitier,\Simulation of a torsional capacitive accelerometer and interface electronicsusing an analog hardware description language," in MICROSIM II Simulationand Design of Microsystems and Microstructures, (Lausanne), pp. 189{198,Sept. 1997. 41

[40] M. Kraft and C. P. Lewis, \System level simulation of a digital accelerometer,"in Proceedings of the 1st International Conference on Modeling and Simula-tion of Microsystems, Semiconductors, Sensors and Actuators, (Santa Clara),pp. 267{272, Apr. 1998.[41] T. Veijola, \Accelerometer model in APLAC," Tech. Rep. CT-18, Helsinki Uni-versity of Technology, Circuit Theory Laboratory, Feb. 1994.[42] T. Veijola, \Accelerometer level 3 and 4 models in APLAC," Tech. Rep. CT-36,Helsinki University of Technology, Circuit Theory Laboratory, 1998.[43] J. S�oderkvist, \Micromachined gyroscopes," Sensors and Actuators A, vol. 43,pp. 65{71, 1994.[44] T. Corman, P. Enoksson, and G. Stemme, \Low pressure encapsulated res-onant structures excited electrostatically," in Proceedings of Transducers'97,(Chicago), pp. 101{104, June 1997.[45] T. Corman, P. Enoksson, and G. Stemme, \Gas damping of electrostaticallyexcited resonators," Sensors and Actuators A, vol. 61, pp. 249{255, 1997.[46] M. Fischer, M. Giousouf, J. Schaepperle, M. Weinmann, W. von M�unch, andF. Assmus, \Electrostatically de ectable polysilicon micromirrors { dynamicbehaviour and comparison with the results from FEM modelling with ANSYS,"in Proceedings of Eurosensors XI, vol. 1, (Warsaw), pp. 43{46, Sept. 1997.[47] J. Mehner, S. Kurth, D. Billep, C. Kaufmann, K. Kehr, and W. D�otzel, \Sim-ulation of gas damping in microstructures with nontrivial geometries," in Pro-ceedings of MEMS'98, (Heidelberg), pp. 172{177, Jan. 1998.[48] Y. J. Yang, M. A. Gretillat, and S. D. Senturia, \E�ect of air damping onthe dynamics of nonuniform deformations of microstructures," in Proceedingsof Transducers'97, (Chicago), pp. 1093{1096, June 1997.[49] Y. J. Yang and S. D. Senturia, \Numerical simulation of compressible squeezed-�lm damping," in Solid-State Sensor and Actuator Workshop, IEEE, (HiltonHead Island), pp. 76{79, June 1996.[50] T. Veijola, H. Kuisma, J. Lahdenper�a, and T. Ryh�anen, \Simulation modelfor micromechanical angular rate sensor," in Proceedings of Eurosensors X,(Leuven), pp. 1365{1368, Sept. 1996.[51] T. Veijola, H. Kuisma, and J. Lahdenper�a, \Model for gas �lm damping in asilicon accelerometer," in Proceedings of Transducers'97, (Chicago), pp. 1097{1100, June 1997. 42

[52] T. Veijola, T. Corman, P. Enoksson, and P. Stemme, \Dynamic simulationmodel for a vibrating uid density sensor," Sensors and Actuators A, 1998.Accepted to be published.

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ErrataPaper [P1]: Page 239, abstractThe model is realized with frequency-dependent resistors and inductors.should readThe model is realized with frequency-independent resistors and inductors.Paper [P3]: Page 37, Eq. (7) (the error has been corrected in the reprinted version)im;n = Pad312�e� um�1;n � 2um;n + 2um+1;n(�x)2+ um;n�1 � 2um;n + 2um;n+1(�y)2 !should readim;n = Pad312�e� um�1;n � 2um;n + um+1;n(�x)2+ um;n�1 � 2um;n + um;n+1(�y)2 ! :Paper [P4]: Page 117, Eq. (25)�B,y,cor = (�IB,x + IB,y + IB,z)!B,x!z,extshould read �B,y,cor = (IB,x + IB,y � IB,z)!B,x!z,extPaper [P5]: Page 92, references (the error has been corrected in the reprinted version)J. S. Starrshould readJ. B. Starr

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Paper reprintsThe papers [P2], [P3] and [P6] are not copies of the conference proceedings. Toachieve better printing quality, they have been reprinted from the original manuscripts.Also, the errors listed in Section Errata have been corrected in these papers.The articles [P1], [P4] and [P5], published in Sensors and Actuators A, arereprinted here with permission from Elsevier Science.

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