Modelling and simulation of MEMS gyroscope

Based on the models developed, the gyroscope and the quadrature error due to unbalanced comb drive are then simulated by the numerical tool of SPICE.. 22 Figure 3.4 Plot of simulated ca

MODELLING AND SIMULATION OF

MEMS GYROSCOPE

ZHAO TAO

(B.Eng., M.Eng., XJTU)

A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING DEPARTMENT OF ELECTRICAL AND COMPUTER

ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE

2003

My appreciation also goes to Assoc Prof Xu Yong Ping for guidance in circuit design and to Mr Logeeswaran V.J., Mr Tan Yee Yuan and Miss Chan Mei Lin from MEMS Lab for their help in the test and the SEM micrographs I wish to express my gratitude to the related staffs of Institute of Microelectronics (Singapore), TIMA of France for collaboration

In addition, my thanks go out to Mr Yeh Bao Yaw for the help of test of MUMPs comb drive microgyroscope, Mr Teo Thiam Teck and Ms Jassica Mah Wai Lim for the assistance provided in the laboratory I would like to extend my appreciation to Ms Ren Changhong, Ms Ong Peek Hong, Mr Lim Chow Yee, Mr Li Yue, Ms Yang Xin, and Miss Boh Siau Shuan from the Center for Power Electronics for their help and friendship

I must also acknowledge my family’s love, care and concern Most importantly, I would thank my wife Yajun for her genuine understanding and encouragement and for her help

in the tedious deduction of formulae

Statement of Original Contribution

The author would like to declare the following original contribution based on his research work in the National University of Singapore:

1 FEA models of balanced and unbalanced comb drive, and displacement-sensing structures (Section 3.1 and 3.2)

2 Equivalent circuit models for the comb drive microgyroscope shown in Figs 4.2, 4.4 and 4.14 (Section 4.2 and 4.3)

3 New actuation scheme for the push-pull comb drive microgyroscope (Section 5.1)

Signature : ………

Table of Contents

Acknowledgement i

Statement of Original Contribution ii

Summary v

List of Tables vi

List of Figures vii

List of Symbols .xi

1 Introduction 1

1.1 Conventional Gyroscopes Overview 1

1.2 MEMS Gyroscopes Overview 3

1.3 Research Objectives 4

1.4 Thesis Structure 5

2 Operating Principles of Vibratory Microgyroscope 7

2.1 Coriolis Force 7

2.2 Operational Principles of Microgyroscope 9

2.2.1 Lagrange's Equations of Motion 11

2.2.2 Motion Eqations of Microgyroscope (Exact Model) 12

2.2.3 Constant-, Low-Rate and Low-Rate-Change Simplification (Approximate Model) 14

2.3 Summary 17

3 Modelling and Simulation of Driving and Sensing Structures 18

3.1 Model of Electrostatic Comb Drive 18

3.1.1 Model of Balanced Comb Drive 20

3.1.1.1 Full-Range Model 24

3.1.1.2 Linear-Range Model 25

3.1.2 Model of Unbalanced Comb Drive 30

3.2 Model of Capacitive Displacement-Sensing Structure 37

3.3 Summary 44

4 Modelling of Microgyroscope Using PSpice 45

4.1Circuit Representation of Mechanical System 45

4.2Simulation of Microgyroscope with Balanced Comb Drive 47

4.2.1 Simulation Models of Microgyroscope 47

4.2.2 Simulation Results and Discussions 52

4.3 Simulation of Microgyroscope with Unbalanced Comb Drive 61

4.3.1 Modelling of Quadrature Error 62

4.3.2 Simulation Results 64

4.4 Summary 73

5 New Actuation Scheme for Momentum Enhancement 74

5.1 Derivation of Driving Voltages 74

5.2 Comparison of Actuation Schemes 76

5.3 Summary 81

6 Fabrication Process 82

6.1 MUMPS Surface Micromachining Fabrication 82

6.2 Summary 85

7 Experimental Results and Discussion 86

7.1 Experimental Setup 86

7.2 Laboratory Measurements 89

7.2.1 Test of Microgyroscope with Balanced Comb Drive 89

7.2.2 Test of Microgyroscope with Unbalanced Comb Drive 93

7.2.3 Comparison of Actuation Schemes 97

7.3 Summary 99

8 Conclusion 100

References 101

Appendix A Matlab Script File for Curve Fitting 107

Appendix B New Driving Voltages for Push-Pull Comb Drive 110

List of Publications 116

Summary

Micromachined gyroscopes have received much attention for their small dimensions, low cost, low power consumption and yet possible high sensitivity In this thesis, the modelling and simulation of the vibratory rate MEMS gyroscope and the critical phenomena on quadrature movement are presented At first, modelling of the driving and sensing structures of the gyroscope are proposed and verified by the commercial finite element software IntelliSuite Based on the models developed, the gyroscope and the quadrature error due to unbalanced comb drive are then simulated by the numerical tool of SPICE On the other hand, in order to improve the sensitivity, a new driving scheme is proposed to enhance the linear momentum of the gyroscope Experiments for the MUMPS gyroscope are done to verify the theoretical predictions, and good agreements between the theory and experimental results are obtained

List of Tables

Table 3.1 Summary of driving voltage and electrostatic force for comb drive 20

Table 3.2 Curve-fitted results for model Eq.(3.4) with different thickness 24

Table 3.3 Curve-fitted results for model Eq (3.10) with different thickness 30

Table 3.4 Curve-fitted results for model Eq (3.14) with different thickness 34

Table 3.5 Curve-fitted results for model Eq (3.13) with different thickness 35

Table 3.6 Comparison of electrostatic force models for comb drive 37

Table 3.7 Curve-fitted results for model Eq (3.17) with different thickness 40

Table 4.1 Analogies of mechanical variables and circuit variables 46

Table 7.1 Key parameters of the microgyroscope 95

Table B.1 Summary of push-pull driving voltages and electrostatic force 114

List of Figures

Figure 2.1 General motion of a mass in fixed frame OXYZ and moving frame oxyz 7

Figure 2.2 The schematic structure of vibratory microgyroscope 10

Figure 2.3 Forced-spring- mass-damper model of microgyroscope 12

Figure 3.1 Schematic of a balanced comb drive cell 18

Figure 3.2 The model of a balanced comb drive generated by IntelliSuite(n=10, g=w=2µm, h=4µm, l f=20µm and l o=10µm) 22

Figure 3.3 Close-up of the comb drive model with refined surface mesh 22

Figure 3.4 Plot of simulated capacitance of the balanced comb drive versus the normalized x displacement 23

Figure 3.5 Plot of the capacitance derivative of balanced comb drive versus the normalized x displacement 25

Figure 3.6 Electric field distributions in a comb drive cell with (a) smaller and (b) larger overlaps 26

Figure 3.7 A sketch plotting the approximate flux pattern for a parallel-plate capacitor (the gap between the plates is exaggerated for clarity) 27

Figure 3.8 Plot of simulated capacitance of unbalanced comb drive versus the normalized x displacement at different y offsets (h=4µm with ground plane) 31

Figure 3.9 Derivative of the capacitance of unbalanced comb drive versus the normalized y offset 33

Figure 3.10 Plot of simulated capacitance of the comb drive versus the normalized y displacement (x=0) 35

Figure 3.11 Plot of capacitance derivative of unbalanced comb drive versus the normalized y displacement (with ground plane and x=0) 36

Figure 3.12 Schematic of a sensing structure cell 38

Figure 3.13 The model of a sensing structure generated by IntelliSuite (n=2, g1=2µm, g2=4µm, w=2µm, l o=50µm and h=4µm) 39

Figure 3.14 Plot of simulated capacitance of the sensing structure versus the y

displacement (the magnification of the circled section is illustrated in Fig 3.15) 40

Figure 3.15 Plot of simulated capacitance of the sensing structure versus small y

displacement (magnified view of the circled section of Fig 3.14, h=4µm with ground plane) 42

Figure 3.16 The model of a symmetric sensing structure generated by IntelliSuite

(g1=2µm, g2=4µm, l o=50µm and h=4µm) 43

Figure 3.17 Plot of simulated capacitance of the symmetric sensing structure versus the

x displacement 43

Figure 4.1 Analogy between a (a) spring-mass-damper mechanical system and a (b)

linear RLC electric circuit 45

Figure 4.2 An equivalent circuit for comb drive microgyroscope (approximate model)48 Figure 4.3 Analog Behavioural Modelling (ABM) parts of PSpice used in this thesis 49 Figure 4.4 An equivalent circuit for comb drive microgyroscope (exact model) 51 Figure 4.5 Simulated step response of the microgyroscope (exact model) 54

Figure 4.6 Spectrums of the y displacements of the gyroscope (exact model) 54

Figure 4.7 Lissajous figure of x and y displacements of the microgyroscope with step

input (exact model) 55

Figure 4.8 Comparison of the exact and approximate models of the microgyroscope by

plotting the simulated x and y displacement amplitudes versus rotation rate 55

Figure 4.9 Simulated frequency response for x displacement of the gyroscope with

different Q x 57

Figure 4.10 Frequency response diagrams for the y displacement of the gyroscope with

different natural frequency mismatches 58

Figure 4.11 Normalized output amplitude versus driving frequency for different Q y 60

Figure 4.12 Normalized y displacement amplitude versus excitation frequency for

different driving voltages 60

Figure 4.13 Comparison of (a) balanced comb drive cell and (b) unbalanced comb drive

cell 62

Figure 4.14 An equivalent circuit for the microgyroscope with quadrature error 63

Figure 4.15 x and y displacements of the microgyroscope with balanced comb drive (Y0/g=0) 65

Figure 4.16 Electrostatic forces of the unbalanced comb drive (offset Y0/g=0.2) 66

Figure 4.17 x and y displacements of the asymmetric comb drive 67

Figure 4.18 Spectrums of x and y displacements of the microgyroscope 67

Figure 4.19 Lissajous figure of x and y displacements of the microgyroscope 68

Figure 4.20 Spectrum of y displacement with different y offset 69

Figure 4.21 Spectrum of y displacement with different resonant frequency mismatches (Y0/g=0.2) 70

Figure 4.22 Spectrum of y displacement with different Q y-factors 71

Figure 4.23 Step response of the microgyroscope with quadrature error 72

Figure 4.24 Spectrum of y displacement with quadrature error 72

Figure 4.25 Lissajous figure of x and y displacements of the gyroscope with quadrature error for step input rotation rate 73

Figure 5.1 Schematic of circuit for conventional driving signals 77

Figure 5.2 Schematic of circuit for proposed driving scheme 77

Figure 5.3 Theoretical curves and PSpice simulated results on displacement ratio at different voltage gains 79

Figure 5.4 Theoretical curves and PSpice simulated results on percentage system voltage reduction for the same resonant amplitude at different voltage gains 81

Figure 6.1 Main processing steps of MUMPs 84

Figure 6.2 Photo of a die compared with a 10-cent coin 84

Figure 6.3 Close-up of the springs 84

Figure 6.4 Micrograph of a comb drive 85

Figure 6.5 Close-up of the comb drive finger 85

Figure 7.1 Experimental setup to test the MUMPS comb drive gyroscope 87 Figure 7.2 Photo of the actuation circuit 87 Figure 7.3 Waveforms of small input and output of the absolute value circuit 88 Figure 7.4 (a) Waveforms of the new driving voltages and (b) spectrum of one of the

new driving voltages 88

Figure 7.5 Waveforms of conventional driving voltages 89

Figure 7.6 Micrographs of the MUMPS comb drive driven by different voltage

schemes with the same system voltage V s = V s' =12.0V 90

Figure 7.7 Frequency response of the push-pull comb drive driven by both voltage

schemes For the conventional scheme, V d = 6.0V, V a = 5.0V; for the proposed scheme,

V q = 11.0V 91

Figure 7.8 Plot of the normalized displacement amplitude versus the applied variable

voltage (V d or V a ) for the conventional driving scheme V1,2 = V d ± V asin(2πf a t) with one

voltage component fixed while the other varies, driving frequency is fixed at

f a=10.02kHz 92

Figure 7.9 Plot of the normalized displacement amplitude versus the applied voltage

squared for the proposed driving scheme V1 =Vqsin(2πf a t) , V2 =Vqcos(2πf a t)  with

f a=5.05kHz 93

Figure 7.10 Comparison of microgyroscope with balanced and unbalanced comb drive95

Figure 7.11 Measured and PSpice simulated results of quadrature error of the

microgyroscope 97

Figure 7.12 Measurement results and theoretical curve on displacement ratio between

the conventional and the proposed driving schemes at system voltage of 12.0V For the

conventional scheme, f a =10.02kHz; for the proposed scheme f a=5.05kHz 98

Figure 7.13 Measurement results and theoretical curve on percentage system voltage

reduction between the conventional and the proposed driving schemes for the same

resonant amplitude For the conventional scheme, f a=10.02kHz; for the proposed

scheme f a=5.05kHz 99

List of Symbols

A, A1, A2, A3 Constants in equivalent circuit models for the microgyroscope

av Acceleration vector

b x, y Damping factors in the x and y direction, respectively

C Capacitance of the MEMS comb structures

f x, f y Resonant frequencies in x and y directions, respectively

G Voltage gain of the non-inverting amplifier of new driving circuit

g1, g2 Small and larger capacitor gap of sensing comb

h Thickness of the MEMS structures

I x Linear momentum of the proof mass in the x direction

k x, k y Spring constants in the x and y directions, respectively

L Lagrangian of a dynamic system

L, C, R Inductance, capacitance and resistance

L x, C x, C y, R x, R y Equivalent circuit parameters for the mechanical system

l f, l o Length and overlap of the comb fingers, respectively

m Effective mass of the proof mass

n Number of unit cells per half of the comb drive or sensing structure

Q x, Q y Quality factors in the x and y direction, respectively

Rv

,Rvo The displacement vectors of particle m and the origin of a moving

reference frame oxyz observed from the fixed reference frame OXYZ

rv Relative position vector of mass in the moving reference frame oxyz

T, V The total kinetic and potential energy of a system, respectively

V1, V 2 Driving voltages applied on each side of the push-pull actuator

respectively

V d, V a dc bias and ac voltage amplitude of the traditional driving voltages

V q Amplitude of the new driving voltages

V s Minimum system voltage of the driving circuits

X, Y, Z The inertial fixed axis

X0, Y0 Excitation and sense amplitude of the proof mass, respectively

x, y, z Deflections in Cartesian coordinate system (non-inertial)

α, β, θ Phase shift

∆y Small deflection in the y direction

ε0 Permittivity of free space

η, ζ Excitation amplitude ratio and the percentage system voltage

reduction, respectively

λ Voltage ratio and λ=V a/V d

Ω, Ωz Rotation rate

ωa Excitation circular frequency and ωa=2πf a

ωx, ωy Resonant circular frequencies in x and y direction, respectively

Chapter 1 Introduction

The purpose of this thesis is to model the vibratory rate MEMS (Microelectromechanical Systems) gyroscope and the critical phenomena on quadrature movement Based on the models developed, the gyroscope and the quadrature error due to unbalanced comb drive are then simulated by the numerical tool of SPICE (Simulation Program with Integrated Circuit Emphasis) On the other hand, in order to improve the sensitivity, a new driving scheme is also proposed to enhance the linear momentum of the gyroscope Finally, experimental results for the MUMPS (Multi-User MEMS Process Service) gyroscope are presented to verify the theoretical predictions made before

To start at the first chapter, an overview of the micromachined gyroscopes together with

a brief description of conventional gyroscopes is presented The quadrature error of the microgyroscope is also highlighted

1.1 Conventional Gyroscopes Overview

A gyroscope is an inertial sensor that measure angular rotation with respect to inertial space about its input axis(es) [1] and gyroscopes can be used in any application that requires the measurement of a rotation or rotation rate, such as navigation, guidance and control Three types of conventional gyroscopes are briefly described as follows [2] [3]

A large class of conventional gyroscopes are mechanical gyroscopes that take advantage of the properties of angular momentum to detect rotation and the core of them

is a spinning rotor By the principle of conservation of momentum, the angular momentum vector of the rotating rotor remains fixed with respect to an inertial frame unless acted upon by a torque For a gimballed gyroscope, the rotation angle of the inertial frame relative to the rotating frame is simply the orientation of the angular momentum relative to the rotating frame For a strap-down gyroscope, the rotation rate is related to the torque applied to force the orientation of the rotor to change with the rotating reference frame However, these mechanical gyroscopes have a common shortcoming of high cost, large in dimension and a short working lifetime

Another group of gyroscopes are optical gyroscopes such as ring-laser gyroscopes (RLGs) and fibre-optic gyroscopes (FOGs) Optical gyroscopes have different operating characteristics from the conventional mechanical gyroscopes and they base on the inertial property of light (Sagnac Effect) While have good long-term stability and extremely sensitivity, the optical gyroscopes have a relatively high threshold input rotation rate When the input rotation rate is below the threshold, the optical gyroscope will produce no output

The third kind of gyroscopes are vibratory rate gyroscopes including Foucault’s pendulum, turning-fork gyroscopes, vibrating shell gyroscopes, etc These gyroscopes use vibrating rather than rotating bodies to provide gyroscopic torques from the Coriolis acceleration and thus are more reliable The vibratory bodies are driven into oscillation in one mode, in the presence of a rotation, the direction of the oscillation shifts so that some oscillation is detected in the sense mode, from which the rotation rate can be calculated The micromachined gyroscopes to be discussed in the next section have the similar operational principles

1.2 MEMS Gyroscopes Overview

Conventional gyroscopes mentioned before are all too expensive and too large for use

in most emerging application [4] Recent advances in micromachining technology have made the design and fabrication of MEMS gyroscopes possible The MEMS components and systems may be fabricated by techniques used for integrated circuit manufacturing, which could reduce costs It is expected that these techniques will allow miniaturization and mass production of MEMS components in a fashion similar to what has been done with integrated circuits Micromachining not only drastically reduce size, price and electric power consumption, it also produces structures that can withstand rough environment for long period of time, implying a long lifetime MEMS has been identified

as a key enabling technology and already has numerous successful commercial products including accelerometers, pressure sensors, ink-jet pens, and digital micromirror device For these products, MEMS technology was used to enable functionality not otherwise possible In general, MEMS technology offers the possibility of low cost, high performance micro-machines to be used in a wide range of domestic and industrial applications Today, MEMS technology is driving innovation in the telecommunication, biomedical, automotive, and aerospace industries

The limitation in present micromachining technology limits the types of gyroscope that can be built [4] [5] High quality rotational bearings have not been demonstrated and it is difficult to build angular momentum based gyroscopes using the available micro-machining technologies Optical gyroscopes need a cavity with perimeter of ten centimetres or more, or fibre coil in the order of one kilometre [2], so optical gyroscopes are not readily implemented in MEMS devices which have typical dimensions less than

one centimetre Vibratory gyroscopes are readily implemented in MEMS Thus all the MEMS gyroscopes are vibratory gyroscopes and operate by using the conservation of linear momentum of a resonant structure with the orthogonal Coriolis force to detect the angular rotation rate of a moving system

Micromachined gyroscopes can be used either as a low-cost miniature companion with micromachined accelerometer to provide heading information for inertial navigation purpose or in other areas, including automotive application for ride stabilization and roll-over detection; some consumer electronic applications, such as video-camera stabilization, virtual reality, inertial mouse for computer; robotics application; and a wide range of military applications [4] Existing commercial gyroscopes are expensive and have yet to fulfil the high performance, low-cost, reliable and mass production vision The increasing need for small and cheap new MEMS gyroscopes has initiated research and development projects at many places A lot of efforts are currently underway for large volume production of micromachined gyroscope and many companies are still largely in the R&D stages

1.3 Research Objectives

Micromachined gyroscopes are probably the most challenging type of transducers ever attempted to be designed in micro-world There are many different approaches to the microgyroscopes and a number of micromachined vibratory gyroscopes have been demonstrated [5]-[16] But even today, MEMS gyroscopes have not yet seriously broached the market [3] Multi-degrees of freedom dynamics, quadrature error due to fabrication imperfections, higher performance, and low power consumption remain

fundamentally challenging issues in the design, analysis, and control of micromachined gyroscopes

In this thesis, modelling and simulation of the gyroscope is presented to gain in-depth understanding of the electromechanical dynamics and the quadrature error due to the unbalanced comb drive is discussed In addition, an actuation scheme to enhance the linear momentum of the comb drive vibratory microgyroscope, or reduce the system voltage is presented The goals of this research are to:

1) Develop models of the vibratory microgyroscope with comb drive electrostatic actuation

2) Demonstrate the validity of the models through simulations

3) Model the quadrature error due to the unbalanced comb drive and confirm the theory predictions experimentally

4) Improve the performance of the microgyroscope by enhancing the linear momentum

Chapter 3 models the driving and the capacitive sensing structures A commercial finite element software IntelliSuite is used to benchmark the models and equations

Based on the models developed in Chapter 2 and Chapter3, Chapter 4 details the analysis and simulation of microgyroscope using PSpice electronic simulator Different errors associated with the structure, especially quadratue error, are analyzed and simulated

Chapter 5 presents a new actuation scheme to enhance the linear momentum of the push-pull comb drive microgyroscope

Chapter 6 describes the fabrication of the MEMS microgyroscope devices used in this thesis The main process steps are explained and fabrication results are shown

Chapter 7 presents a collection of the test results for the MUMPS microgyroscope with the push-pull comb drive It is found that the observed quadrature errors are close to the theoretical predictions, and the proposed new actuation scheme is more efficient than the conventional scheme as theory predicted

Chapter 8 briefly summarises the conclusions of this thesis

Appendix A gives an example of the Matlab script file to use the Fminsearch() function

to find the fitting coefficients of a theoretical function for a particular set of data

Appendix B shows the derivation of the new driving voltages for the push-pull comb drive to obtain a single harmonic motion

Chapter 2 Operating Principles of Vibratory Microgyroscope

Forming a mathematical model that represents the characteristics of the microgyroscope

is crucially important for the further analysis of the system In this chapter, the motion equations of the microgyroscope are derived and the operational principles of the vibratory microgyroscope are briefly discussed

Figure 2.1 General motion of a mass in fixed frame OXYZ and moving frame oxyz

2.1 Coriolis Force

Unlike the conventional mechanical gyroscope that functions on the conservation of angular momentum of a spinning rotor, microgyroscope operates by using the conservation of linear momentum of a resonant structure with the orthogonal Coriolis

force to detect the rotation rate of a moving system Coriolis force, named after the French engineer and mathematician G.G.de Coriolis (1772-1843), is an apparent force that arises

in a rotation reference frame The effect of Coriolis acceleration is best explained by examining the expression for acceleration in terms of rates observed in a moving

(translating and rotating) reference frame [17] In Fig 2.1, OXYZ is a global inertial fixed frame of reference and oxyz is a moving reference frame, which has a position vector Rvo

to its origin The Coriolis force can be determined when a mass, m, is observed from the

non-inertial system The absolute position vector of the mass m is

r

R

Rv = vo +v (2.1) where rv is the relative position vector of m in the moving reference coordinate system

The velocity of the mass observed in the fixed frame of reference can be derived as

r dt

d R dt

d R

dt

d

vvOXYZ = v = vo + v (2.2) Further, note that

r v

r

dt

d r dt

d v

a R

dt

d

r r

o

dt

d r a

represents the acceleration of the origin of oxyz relative to OXYZ, av ris the

acceleration of m as observed in the moving reference oxyz and

oxyz r

measured in the fixed system Ωv ×vv r

2 is the Coriolis acceleration of particular interest

We can find that the Coriolis acceleration is given by the vector cross product of the external rotation rate vector and the velocity of the mass as measured in the moving reference frame oxyz So the Coriolis acceleration is perpendicular to the angular rotation

and the velocity The Coriolis force can be written as:

r

Fv = 2 Ωv ×v (2.9)

2.2 Operational Principles of Microgyroscope

All vibratory gyroscopes are based on the transfer of energy between two vibration modes of a structure caused by Coriolis force [4] [5] A schematic of vibratory fishhook microgyroscope is shown in Fig.2.2 This fishhook microgyroscope is implemented using

a micromachined mechanical proof mass, which can be deemed as a rigid body and is

suspended by arched fishhook spring Although the structure has many modes of

vibration, we consider a model with only two degrees of freedom denoted by x(t) and y(t)

that describe the lateral and vertical motion of the proof mass The fishhook spring can provide an almost equal compliance in both lateral and vertical directions (or modes) [13] The proof mass is driven into lateral oscillations with electrostatic force generated by the comb drives and this is called drive mode From the discussion about Coriolis force before, we can find that a rotation rate perpendicular to the plane of the substrate will result in Coriolis force, which will make the proof mass vibrate in the vertical direction (sense mode) By measuring the displacements in the sense mode, the external rotation velocity is determined The differential capacitive sensing structures in Fig 2.2 are used to detect the Coriolis-induced deflection in the vertical direction

Figure 2.2 The schematic structure of vibratory microgyroscope

The operation principle discussed above is only a first-glance theory To analyze the dynamics of the microgyroscope accurately, a mathematical model of the microgyroscope should be derived Two methods available for deriving the motion equations: application

of Newton’s laws to free-body diagrams and energy method [18] The application of

Newton’s laws to free-body diagram is straightforward but tedious and the resulting equations may have to be manipulated to be put into a usable form Energy method is

based on the use of Lagrange’s equations, which will be discussed in the next subsection

2.2.1 Lagrange’s Equations of Motion

Lagrange’s equations are general equations derived from energy methods that are used

to formulate differential equations for possible nonlinear systems The Lagrangian of a dynamic system is defined as the difference between its kinetic and potential energy at an arbitrary instant

V

T

L= − (2.10)

where T denotes the total kinetic energy of the system, V denotes the potential energy of

the system arising from conservative forces The Lagrangian is a function of the generalized coordinates and their time derivatives

),,,,

,

(x1 x2 x n x1 x2 x n

L

L= L & & L& (2.11)

It should be remarked that the Lagrangian is a function of 2n independent variables; the

time derivatives of the generalized coordinates are viewed as being independent of the generalized coordinates The energy equation can be manipulated to yield Lagrange’s equations for a nonconservative system

i i i

Q x

where δx i is the virtual displacement For a nonconservative system, Q i ≠0 (i=1,2,…, n)

and for a conservative system, Q i =0 (i=1,2,…, n)

Figure 2.3 Forced-spring- mass-damper model of microgyroscope

2.2.2 Motion Equations of Microgyroscope (Exact Model)

Although the design of a vibratory gyroscope can take many different physical forms, they all can be modelled by a forced-spring-mass-damper system as illustrated in Fig 2.3

In the model, a particle mass m represents the proof mass and the springs and dampers

represent the elasticity and viscous effects of the supporting structures and surrounding

air The total kinetic energy T of the system is

2 2

2

2

1)(

2

1)(

2

1

z z

x

m

T = &−Ω + &+Ω + Ω (2.14)

where J is the moment of inertia The MEMS devices have a size scale on the order of one

micron (10-6m) and the gravitational force can be neglected Assuming the springs are

linear (spring deformation is proportional to force), the potential energy V of the system

arising from conservative forces is

2 2

2

12

1

y k x

k

V = x + y (2.15)

where k x and k y are the spring constants in the x and y direction, respectively By

reference to Eq (2.10), the Lagrangian is

2 2

2 2

2

2

12

12

1)(

2

1)(

2

1

y k x k J

x y m y

x

m

L= &−Ωz + &+Ωz + Ωz − x − y (2.16) Viscous damping is assumed in the system and the damping force produced when a rigid body is in contact with a viscous liquid is proportional to the velocity of the body So the virtual work δW is

y y b y F x x b

where F x , F y , b x and b y are the forces and damping factors in the x and y directions,

respectively From Eqs (2.13) and (2.17), the generalized forces are determined as

& (2.20)

we obtain the equations of motion for the microgyroscope in terms of the x and y

displacements (exact model):

)(2

find that in the absence of a rotation rate, the modes of vibration are ideally uncoupled However, when the sensing element is rotated at a rate Ωz, the modes become coupled to each other and the displacements of the proof mass are not a linear function of Ωz One can also find that the excitation mode amplitude is affected by Ωz for open actuation loop (This is to be verified in Chapter 4 by PSpice simulation) So it is difficult to obtain a solution of Eqs (2.21) and (2.22) without recourse to numerical integration Fortunately, it

is found that for the motions in a certain range, the motion equations can be linearized This is further discussed in the next subsection

2.2.3 Constant-, Low-Rate and Low-Rate-Change Simplification (Approximate Model)

In practice, the angular velocity needed to be measured lies between the order of 0.1deg/h and up to 1000deg/s (or 17.5rad/s) [11], while the natural frequencies of the microgyroscope are of the order of a few kilohertz, normally be kept above environment noise (>2kHz) [4] Thus, the following condition can be assumed:

ω are the lateral and vertical natural frequencies of the

gyroscope Also, the rotation rate Ωz usually change very slowly relative to the natural frequency of the structure, the angular velocity can be regarded as a constant during a

particular time interval In addition, a basic assumption is made that the y displacement is much smaller compared to the x displacement So, we can neglect the angular and

centrifugal acceleration Here it is also assumed that the comb drive is balanced and the

electrostatic force in the y direction is zero (The unbalanced comb drive that has y

direction force will be discussed in Chapter 3) The resulted simplified motion equations (approximate model) are then

)

(t

F x k

approximate model agrees with the exact model well in most of the practical conditions, which will be proved in Chapter 4 Noticing that the linear momentum of the proof mass

in the x direction is I x(t)=m x&, Eq.(2.25) can be rewritten as

)(

2 I t y

)sin(

)sin(

)

(

2 2

2 2

ωωω

−

Q m

F t

x

a x a

x

x

=X0sin(ωa t−α) (2.28)

where

x

x x

tan

a x

a x x

Q ω ω

ωω

phase shift,

0 0

−

=

x

a x a

x

x

Q m

F X

ωωω

ω

is the actuation amplitude in the x direction

With an excitation displacement of the form of Eq.(2.28), the corresponding displacement

in the y direction of Eq (2.26) can be expressed as

)cos(

2)

(

2 2

2 2

ωωω

I t

y

a y a

y

x z

=−Y0cos(ωa t−α −β) (2.29)

where

y

y y

tan

a y

a y y

ωωβ

are sensing quality factor and phase

shift, respectively,

2 2

2 2

0 0

y

x z

Q m

I

ωωω

ω

in the y direction I x0 is the amplitude of the linear momentum given by I x0 =mX0ωa It is

clear from Eq.(2.29) that displacement in the y direction is proportional to the angular rate

Ωz, thus the external rotation rate can be easily determined by measuring the

Coriolis-induced y displacement The sensitivity of the gyroscope is derived as

2 2

2 2

0 0

)()(

2

y

a y a

y

x z

Q m

I Y

ωωω

=

Ω (2.30)

From Eq (2.30) one can find that the sensitivity is proportional to the linear momentum of

the comb actuator in the x direction, and one of the methods to improve the performance

of the gyroscope is to increase the linear momentum, that is, to increase the excitation amplitude for the same excitation frequency, which is to be discussed in Chapter 5

2.3 Summary

In this chapter, the operating principles of the vibratory microgyroscope are discussed The differential equations governing the motions the gyroscope are derived and linearized, making the further analysis of the system possible The simplified solution of the Coriolis-

induced y displacement is also given, which clarify the operational principles of the

gyroscope

Figure 3.1 Schematic of a balanced comb drive cell

3.1 Model of Electrostatic Comb Drive

The electrostatic comb drive actuator, firstly demonstrated by Tang et al [19] [20], is

one of the most important microactuators in microelectromechanical systems Fig 3.1

shows the schematic of a comb drive cell with a thickness of h Each finger has a length of

l f and a width of w The overlap of the movable and stationary fingers is l o and the air gap

is g The electrostatic forces of the comb drive can be obtained from conservation of

energy [21] and are respectively given by

the MEMS devices is neglected

It is clear from Eqs (2.1) to (2.3) that electrostatic forces can be calculated easily once the capacitance as a function of position is given So in the following discussions of this chapter, the capacitances of the MEMS structures are in particular interest A detailed analysis of the field lines using field theory can yield the actual capacitance of the comb drive [22] [23] Due to the computational burden of calculating this for the complex structures, approximations were proposed and finite element analysis (FEA) was used to check the models The capacitance solver used in this thesis is IntelliSuite (version 4.0) (http://www.intellisense.com) IntelliSuite can extract capacitance using Electrostatic

Analysis or Electromechanical Analysis modules Electromechanical Analysis allows the users to apply Elec_mesh function to refine the critical electrostatic surface mesh and remove the unimportant surfaces from electrostatic analysis, thus increase the accuracy of results and reduce computational time and memory expense [24] So all the simulated 3-dimensional capacitances in this chapter were extracted by the Electromechanical Analysis module, and refined surface mesh was used unless otherwise noted

3.1.1 Model of Balanced Comb Drive

The comb drive used in our gyroscope is mainly for actuation in the x direction and it is

expected to be balanced, i.e.: the comb gaps are equal on both sides of the finger and there

is no electrostatic force in the y direction There are different driving configurations for the

comb drive: drive at one side or drive differentially (push-pull) Usually the push-pull methods using two combs are used in microgyroscopes Table 3.1 summarizes the

configurations and force functions in the x direction

Table 3.1 Summary of driving voltage and electrostatic force for comb drive

t V

V t V

V x

The capacitances of the balanced comb drive at different x displacements were

simulated with IntelliSuite Fig.3.2 shows an IntelliSuite-generated polysilicon comb drive model with maximum mesh size of 10µm and Fig 3.3 shows the close-up of the model with refined mesh The comb drive model has 10 fingers and each finger is 20µm long,

2µm wide The gaps between the fingers are 2µm and the non-deflect overlap of the fingers is 10µm The thickness of the model shown in Fig.3.2 is 4µm Usually the silicon substrate (which is normally grounded) under the comb drive cannot be neglected and we can model the substrate as a ground plane Also, some comb drives have silicon ground planes [19] [20] So the same comb drive as shown in Fig 3.2 with an underlying ground plane (ground plane not shown in the figure for clarity) are also considered and simulated The ground plane cannot be very large due to the memory limit of the workstation that running the IntelliSuite In this thesis, the rectangle ground planes of the simulated MEMS devices (including comb drives and capacitive displacement-sensing structures) have an extension of 50µm to the simulated structures on four sides in the directions parallel to the wafer surface, and the ground planes have a thickness of 1µm and lie 2µm beneath the simulated structures unless otherwise noted During simulation, the thickness of the comb drive structure was selected to be 2µm, 4µm and 6µm respectively and the capacitances of the comb drive at different displacement were obtained

Figure 3.2 The model of a balanced comb drive generated by IntelliSuite (n=10, g=w=2µm,

h=4µm, l f=20µm and l o=10µm)

Figure 3.3 Close-up of the comb drive model with refined surface mesh

Fig 3.4 plots the simulated capacitance of the balanced comb drive versus the

normalized x displacement For clarity, for those comb drive without ground plane, only

the results of the structure with a thickness of 4µm are shown The plot shows the

non-linearity near full insertion (x/l o =1.0) and disengagement (x/l o<–1.0) When the comb is almost entirely inserted, the effects of the end of the comb begin to become important and

capacitance increases fast with x/l o When comb is disengaged, the capacitance is almost

the fringing capacitance and it changes slowly with changing x/l o For those structures with ground planes, some of the electric flux lines will end at the ground plane and thus their capacitances are smaller than those of their counterparts without ground plane It is also true for the comb drive structures to be discussed in the following sections

Figure 3.4 Plot of simulated capacitance of the balanced comb drive versus the normalized x

displacement

3.1.1.1 Full-Range Model

In this thesis, an approximate full-range capacitance model proposed for the balanced comb drive is

1 5

2 4 3

+

=

o o

o

x C

l

x C l

x l

x K C

C

C

x

where C1, C2, C3, C4 and C5 are fitting coefficients, which have the same unit as C(x) K is

a constant and K=1.6 for the comb drive with ground plane and K=1.5 for that without

ground plane This nonlinear fit was calculated with Matlab (version 6.1) in the least squares fitting sense using the function Fminsearch() (A detailed example is given in Appendix A.) The curve-fitted results are listed in Table 3.2, and Fig 3.4 also shows the curve-fitted model superimposed on top of the actual data The relative fitting errors of Eq.(3.4) against the FEA results computed by IntelliSuite are at most 4.7% It should be noted that this analysis and model are for a specific comb drive To determine the capacitance for a comb drive with different size, this analysis must be performed again

Table 3.2 Curve-fitted results for model Eq.(3.4) with different thickness

Coefficients of the model [fF]

Thickness

h [µm] C

Maximum fitting error

From Eq (3.4) the derivative of the capacitance with respect to the x displacement can

be obtained and results are shown in Fig 3.5 It is found that

x

C

∂

∂ can only be considered

constant for displacement in the region

To obtain a significant region of

constant capacitance derivative

the movable and stationary fingers The approximate capacitance of parallel-plate

capacitors is derived in simple electrostatics for the case in which the electric charge

density on the plates is uniform and the fringing fields at the edge are negligible The

capacitance of the parallel-plate capacitors of the comb drive is

)(

2 0

x l g

h n

hl n

Eqs (3.5) and (3.6) only hold when the gap g is far smaller than other dimensions (h

and l o ) of the plates As g becomes large compared to the smallest dimension (h here) of

the plates, the equations do not provide accurate results To accurately model the

capacitance of the comb drive, Eq (3.5) needs corrections

(a) (b)

Figure 3.6 Electric field distributions in a comb drive cell with (a) smaller and (b) larger overlaps

Fig 3.6 shows the 2-dimensional electric filed distribution of the comb drive with two

different finger overlaps One can find that the fringing capacitance, which results from