Modelling and simulation of MEMS gyroscope

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 Acknowledgement There are so many people to thank for their help in my successful completion of this thesis I am immeasurably grateful for the help I have received from them I would like to thank my project supervisor Assoc Prof Liang Yung Chii He had introduced me into the interesting world of MEMS and provided me valuable technical guidance, support and encouragement He is always vibrant and full of fresh ideas and has helped to greatly improve my insight into MEMS microsensors and actuators 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 i 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: FEA models of balanced and unbalanced comb drive, and displacement-sensing structures (Section 3.1 and 3.2) Equivalent circuit models for the comb drive microgyroscope shown in Figs 4.2, 4.4 and 4.14 (Section 4.2 and 4.3) New actuation scheme for the push-pull comb drive microgyroscope (Section 5.1) Signature : ………………… ii Table of Contents Acknowledgement i Statement of Original Contribution .ii Summary v List of Tables vi List of Figures .vii List of Symbols xi Introduction 1.1 Conventional Gyroscopes Overview 1.2 MEMS Gyroscopes Overview 1.3 Research Objectives 1.4 Thesis Structure Operating Principles of Vibratory Microgyroscope 2.1 Coriolis Force 2.2 Operational Principles of Microgyroscope 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 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 iii 3.3 Summary 44 Modelling of Microgyroscope Using PSpice 45 4.1 Circuit Representation of Mechanical System 45 4.2 Simulation 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 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 Fabrication Process 82 6.1 MUMPS Surface Micromachining Fabrication 82 6.2 Summary 85 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 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 iv 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 v 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 vi List of Figures Figure 2.1 General motion of a mass in fixed frame OXYZ and moving frame oxyz 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, lf=20µm and lo=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, lo=50µm and h=4µm) 39 vii 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, lo=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 Qx 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 Qy 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 viii 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 Qy-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 ix [17] F.C.Moon, Applied Dynamics, With Application to Multibody and Mechatronic Systems, John Wiley & 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data fits a theoretical function The algorithm is: (1) Write an objective function which computes the "distance" between measured y(x) and that obtained by a parameter guess (2) Let Fminsearch() "search" for a set of parameters which "locally" minimize the error To implement the algorithm, first we write a function M-file that accepts curve parameters as inputs and outputs the fitting error The name of this file is "myfit_comb_ideal.m" % Matlab file with name ‘myfit_comb_ideal.m’ saved in the current directory function sse= myfit_comb_ideal(coeff, t, Actual_Output) C1 = coeff (1); C2 = coeff (2); C3 = coeff (3); C4 = coeff (4); C5 = coeff (5); Fitted_Curve=C1 + (C2+C3.*tanh(1.5+t)).*t + C4.*t.^2 + C5./(1-t); Error_Vector=Fitted_Curve - Actual_Output; % When curve fitting, a typical quantity to minimize is the sum of squares error sse=sum(Error_Vector.^2); % End of Matlab file ‘myfit_comb_ideal.m’ 107 Next, write a Matlab file to call the above function and obtain the best coefficients The name of the file is "Fit_comb_ideal.m" % Matlab file with name ‘Fit_comb_ideal.m’ % It is a Matlab script file to find the best fitting coefficients A1-A5 using Fminsearch(), and then plot the fitting error and derivative of the function It is believed that the data conform to a relation of the form C=C1 + (C2 + C3.*tanh(1.5+t)).*t + C4.*t.^2 + C5./(1-t) % Simulated data was obtained using IntelliSuite Comb drive n=10, g=w=2um, h=2um, initial overlap L0=10um, length of finger=20um, without ground plane % Input the simulation data L_2=[-10 -9 -7 -5 -4 -3 -2 -1 10 11 12 13 14 15 16 17 18 18.6 19 19.3 19.5 19.7 19.8]; % Overlap of the comb fingers in um C_2=[1.014 1.055 1.153 1.285 1.37 1.475 1.609 1.782 2.003 2.268 2.558 2.862 3.173 3.49 3.81 4.132 4.457 4.782 5.109 5.436 5.765 6.095 6.427 6.764 7.113 7.49 7.949 8.34 8.736 9.221 9.773 10.88 12.12]; % Simulated capacitance in fF % End of data input L0=10.0; % Initial overlap in um x=(L_2-L0)./L0; % Normalized x displacement t=x(:); % To make t a column vector Data=C_2(:); % To make t a column vector options=optimset('Display', 'iter', 'MaxIter', 1.0e4, 'TolX', 1.0e-6,'TolFun', 1.0e-6); Fs=[6 0]; % Starting fitting coefficients given randomly F=fminsearch('myfit_comb_ideal', Fs, options, t, Data) % F is the best coefficients vector ystart=Fs(1) + (Fs(2)+Fs(3).*tanh(1.5+t)).*t + Fs(4).*t.^2 + Fs(5)./(1-t); yfit=F(1) + (F(2)+F(3).*tanh(1.5+t)).*t + F(4).*t.^2 + F(5)./(1-t); 108 figure(1) % To output the curve-fitted results plot(t,Data,'bx', t, ystart,’k:‘, t,yfit,'r-'); hold on legend ('Original Data', 'Start Fitted curve', 'Final Fitted Curve', 2); title (sprintf ('Curve Fitting versus x displacement, h=2um’)); xlabel ('Normalized displacement x/L0') ylabel ('Capacitance C (fF)') figure(2) % To output the fitting percentage errors plot (t, 100.*(yfit-Data)./Data, 'rx:'); hold on legend ('Fitting Error', 2); title (sprintf ('Curve Fitting Error versus x displacement, h=2um’)); xlabel ('Normalized displacement x/L0') ylabel ('Percentage Fitting Error (%)') e_max=max(abs(100.*(yfit-Data)./Data)) figure(3) % Maximum fitting error % To output approximate derivative of C Step=0.0001; x2=-2:Step:0.88; C2= F(1) + (F(2)+F(3).*tanh(1.5+x2)).*x2 + F(4).*x2.^2 + F(5)./(1-x2); dC=1/(Step*L0).*diff(C2); % approximate derivative of capacitance C t2=-2+Step:Step:0.88; plot(t2, dC, 'r-');hold on legend(' dC/dx ',2); title(sprintf('Derivative of Capacitance versus x displacement, h=2um’)); xlabel('Normalized displacement x/L0') ylabel('Derivative of Capacitance dC/dx (fF/um)') % End of the Matlab file ‘Fit_comb_ideal.m’ 109 Appendix B New Driving Voltages for Push-Pull Comb Drive In Appendix B, the possible new forms of driving voltages for the push-pull electrostatic comb drive actuator are theoretically derived For the push-pull comb drive, assuming the driving signals have the following arbitrary format: V1 = Vd + Va1 sin(ωt + θ ) (B.1) V2 = Vd + Va sin(ωt ) (B.2) where Vd1 and Vd2 are the dc biases, Va1 and Va2 are the ac amplitudes, θ is the phase difference The net electrostatic force in the x direction will be Fx (t ) = = ∂C (V2 − V12 ) ∂x { ∂C [Vd + Va sin(ωt )]2 − [Vd1 + Va1 sin(ωt + θ )]2 ∂x } = f + f (ωt ) + f ( 2ωt ) (B.3) where f0, f1(ωt) and f2(2ωt) are the constant term, the first harmonic and the quadraticfrequency harmonic of the electrostatic force respectively if ∂C is considered constant ∂x The harmonics can be described in detail as follows: f0 = ∂C [(V d22 − V d21 ) + (V a22 − V a21 )] ∂x f (ω t ) = ∂C θ [ 4V d 2V a sin sin(ω t − ϕ ) + 2(Vd 2Va − Vd 1Va1 ) sin(ωt + θ )] ∂x (B.4) (B.5) 110 f (2ωt ) = − V − Va21 ∂C [V a sin θ sin( 2ωt + θ ) + a sin( 2ωt + 2θ )] ∂x where ϕ = tan −1 (B.6) sin θ Normally we should make f = so that the proof-mass does − cos θ not have a net displacement in the x direction Moreover, in order to make the steady-state motion of the proof-mass be a simple harmonic, the electrostatic force Fx(t) should have only one simple harmonic: the first harmonic or the quadratic-frequency harmonic This will be discussed in the following two subsections B.1 Only the First Harmonic If the electrostatic force has only the first harmonic, the following three equations should be satisfied at the same time f0 = (B.7) f (ωt ) ≠ (B.8) f (2ωt ) = (B.9) Solving the equations, we find that the corresponding driving voltages should have the following characteristics: V d = ±V d , V a1 = mV a and θ = ± nπ ( n =0, 2, 4, 6…) (B.10) or V d = ±V d , V a1 = ±V a and θ = ± nπ ( n =1, 3, 5…) (B.11) It is found that the net electrostatic forces in the x direction excited by these voltages are all the same In fact, all these voltages can be summed up as the conventional format of 111 driving voltages represented by the voltages shown in Eqs (B.12) and (B.13) and they are used by most of the push-pull comb-drive actuator users V1= Vd -Vasin(ωt) (B.12) V2= Vd +Vasin(ωt) (B.13) B.2 Only the Quadratic-frequency Harmonic If the electrostatic force has only the quadratic-frequency harmonic, the following three equations should be satisfied at the same time f0 = (B.14) f (ωt ) = (B.15) f ( 2ωt ) ≠ (B.16) We can find two new forms of driving voltages by solving the equations The first new form of driving voltages is characterized by V d = V d = ,V a1 = ±V a and θ ≠ ± nπ ( n =0, 1, 2, 3…) (B.17) The second new form meets the conditions of Vd = , Va = , V d = ± V a1 (B.18) or V d = , V a1 = , V d = ± Va (B.19) B.2.1 The First New Form: For the first new format of driving voltages, assuming V1 = V sin(ωt ) (B.20) 112 V2 = V sin(ωt + θ ) (B.21) The excited net electrostatic force will be F x (t ) = ∂C V sin θ sin( 2ω t + θ ) ∂x (B.22) To maximize the amplitude of the electrostatic force, we should let θ = ± nπ (n=1, 3, 5…) If we select θ = π , the driving voltages are actually the first new format of driving voltages we want to obtain: V1′ = Vq sin( ωq V2′ = Vq cos( t) ωq t) (B.23) (B.24) where Vq>0 The voltages are quadrature signals For the convenience of discussion, the frequency of the driving voltages is ωq/2 instead of ωq It should be noted that the absolute value of Eqs (B.23) and (B.24) are also the correct solution and they are just the new driving voltages discussed in Chapter Similarly, the electrostatic force excited by this new form of driving voltages can be expressed as Fx1 (t ) = ∂C V q cos(ω q t ) ∂x (B.25) B.2.2 The Second New form According to Eqs (B.18) and (B.19), the second new form of driving voltages can be represented by V1′′ = V r sin( ωr t) (B.26) 113 V 2′′ = Vr (B.27) where Vr>0 Because the driving voltages are a dc voltage and an ac voltage, they can be obtained easily by using any handy oscillator and a dc voltage reference The electrostatic force excited by this new form of driving voltages can be expressed as Fx1 (t ) = ∂C V r cos(ω r t ) ∂x (B.28) As a summary, Table B.1 lists the push-pull driving voltages and the corresponding electrostatic forces Table B.1 Summary of push-pull driving voltages and electrostatic force Push-pull driving voltages Conventional V1= Vd -Vasin(ωt) V2= Vd +Vasin(ωt) V1= Vq sin(ωt ) Fx = ∂C Vd Va sin(ωt ) ∂x Fx = ∂C Vq cos(2ωt ) ∂x V1= Vqsin(ωt) V2= Vqcos(ωt) Fx = ∂C Vq cos(2ωt ) ∂x V1= Vrsin(ωt) V2 = Vr Fx = ∂C Vr cos(2ωt ) ∂x V2= Vq cos(ωt ) New Electrostatic Force B.3 Other Driving Voltages It should be noted that the forms of driving voltages that make the net electrostatic force Fx(t) have only one simple harmonic are not restricted to the above-mentioned four forms For example, if the driving voltages are 114 V1 = V [sin(ωt ) − cos(2ωt ) − 1] (B.29) V2 = V [sin(ωt ) + cos( 2ωt ) + 1] (B.30) then the net driving force has only one harmonic, i.e Fx (t ) = ∂C V sin(3ωt ) ∂x (B.31) For another example, if the driving voltages are V1 = V [cos(ωt ) − cos(2ωt ) + 1] (B.32) V2 = V [cos(ωt ) + cos(2ωt ) − 1] (B.33) then the net electrostatic force will have only one harmonic, too, i.e F x (t ) = ∂C V cos( 3ω t ) ∂x (B.34) It is not easy to obtain the driving voltages given by Eqs (B.29), (B.30), (B.32) and (B.33) In addition, similar analyses as those discussed in Chapter show that these two forms of driving voltages cannot improve the drive efficiency of the push-pull comb drive satisfactorily 115 List of Publications [1] Yung C.Liang, Tao Zhao, Yong P.Xu and S.S.Boh, A CMOS Fully-integrated Lowvoltage Vibratory Microgyroscope, Proceedings of IEEE Region 10 International Conference on Electrical and Electronic Technology, 2001 TENCON, Singapore, 19-22 Aug 2001, pp 825-828 [2] Y.C.Liang, T.Zhao, Y.P.Xu and S.S.Boh, A Low-voltage Vibratory Micro-gyroscope with ASIC Control, Proceedings of Design, Test, Integration and Packaging of MEMS/MOEMS, Cannes, France, 25-27 Apr., 2001, pp.40-49 [3] Tao Zhao and Yung C Liang, New Actuation Method for Push-Pull Electrostatic MEMS Comb Drive, accepted by IEEE Transactions on Industrial Electronics 116 End of the Thesis 117 [...]... 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... equations governing the motions the gyroscope are derived and linearized, making the further analysis of the system possible The simplified solution of the Coriolisinduced y displacement is also given, which clarify the operational principles of the gyroscope 17 Chapter 3 Modelling and Simulation of Driving and Sensing Structures In a vibratory rate microgyroscope, the proof mass must be driven into oscillation... 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... introduction of the gyroscopes and work that has been done Chapter 2 derives the motion equations of the microgyroscope and briefly describes the operational principles of the vibratory microgyroscope Chapter 3 models the driving and the capacitive sensing structures A commercial finite element software IntelliSuite is used to benchmark the models and equations 5 Based on the models developed in Chapter 2 and. .. 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... al [19] [20], is one of the most important microactuators in microelectromechanical systems Fig 3.1 18 shows the schematic of a comb drive cell with a thickness of h Each finger has a length of lf and a width of w The overlap of the movable and stationary fingers is lo and the air gap is g The electrostatic forces of the comb drive can be obtained from conservation of energy [21] and are respectively... frequencies in x and y directions, respectively G Voltage gain of the non-inverting amplifier of new driving circuit g Gap of the comb g1, g2 Small and larger capacitor gap of sensing comb h Thickness of the MEMS structures Ix Linear momentum of the proof mass in the x direction i Current K Constant K1, K2 Scale factors Kd Capacitive distortion factor xi kx, ky Spring constants in the x and y directions,... The micromachined gyroscopes to be discussed in the next section have the similar operational principles 2 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... 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... 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 ... 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... Representation of Mechanical System 45 4.2 Simulation of Microgyroscope with Balanced Comb Drive 47 4.2.1 Simulation Models of Microgyroscope 47 4.2.2 Simulation Results and Discussions... 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