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- 1. [I]
- 2. MEMS Vibratory Gyroscope Graduation Project Thesis (1) Faculty of Engineering - Ain Shams University In Partial Fulfillment of the Requirements for the Degree Bachelor of Science in Communication Systems Engineering Submitted byAbdel Rahman El-Naggar, Ahmed Magdy, Ahmed M. Hussien, Amr Atef, Haidy El-Medany, Islam Ayoub, Shady Ashraf Communication Systems Engineering Credit Hours Engineering Programs (CHEP) Fall 2012 Supervisors Dr. Mohamed El-Sheikh Dr. Maged Ghoneima ECE Department [I]
- 3. ABSTRACTThe report presents our surveys and work about MEMS Vibratory Gyroscope Devices andSystems as part of the graduation project, a brief outline for the thesis is shown below:Chapter 1 is an introduction to MEMS technology, MEMS applications and gyroscopes. Thediscussion is based on the early survey we did when we started the project.Chapter 2 describes the back ground and theory of Coriolis effect gyroscopes; the physicalprinciple of operation, driving and sensing, mechanical and electrical design basics, andfabrication technologies.Owner: A. M. El-Sayed, A. M. Hussein, A. A. Hussein.Chapter 3 discusses the Tuning Fork Gyroscope (TFG). The main features of TFG, designconcept, and the equation of motion. This chapter will contain also a hand analysis for a TFGdesign and a verification of this design with a finite element analysis (FEA) using ANSYS.Owner: A. A. Hussein.Chapter 4 discusses another architecture which is the vibrating ring gyroscope (VRG). Thefeatures of the architecture, the vibrating modes of the structure and some of the vibrationinduced errors in the vibrating ring gyroscope. At the end of the chapter, a famous design doneby Dr. Najafi and Dr. Ayazi (A HARPSS Polysilicon VRG) is redesigned using another model, finiteelement analysis (FEA) verified the design.Owner: A. M. El-Sayed.Chapter 5 explores the interfacing system and its main loops and types, also this chaptercontain an overview on previous work done regarding the interfacing system.Owner: A. Y. El-Naggar, S. A. El-Sayed.Chapter 6 in this chapter the system is divided into main blocks starting from the capacitiveinterface, the analog to digital converting, then the demodulation, the drive loop automaticcontrol and the clock generation of the system.Owner: A. Y. El-Naggar, A. M. Hussein , H. K. El-Medany, S. A. El-Sayed. I. A. Ayoub.Chapter 7 briefly summarizes the thesis and shows our conclusions from this semester’s work.Finally we explain the next semester plan (future work). [II]
- 4. Table of Contents1. INTRODUCTION ................................................................................................................... 11.1. MEMS Technology............................................................................................................................... 1 1.1.1. MEMS Applications...................................................................................................................... 11.2. Gyroscopes.......................................................................................................................................... 12. BACKGROUND AND THEORY ............................................................................................ 32.1. Coriolis Force....................................................................................................................................... 32.2. Vibrating Two Modes Gyroscope ........................................................................................................ 3 2.2.1. Drive Mode Operation ................................................................................................................. 4 2.2.2. Sense Mode Operation ................................................................................................................ 42.3. Mechanical Design of MEMS Gyroscopes ............................................................................................ 5 2.3.1. Flexure Elements ......................................................................................................................... 5 2.3.2. Mode Coupling ............................................................................................................................ 92.4. Electrical Design of MEMS Gyroscopes .............................................................................................. 10 2.4.1. Capacitive Detection.................................................................................................................. 10 2.4.2. Electrostatic Actuation............................................................................................................... 122.5. Fabrication Technologies................................................................................................................... 14 2.5.1. Microfabrication Techniques ..................................................................................................... 14 2.5.2. Bulk-Micromachining Processes ................................................................................................. 15 2.5.3. Surface-Micromachining Processes ............................................................................................ 16 2.5.4. SOI-MUMPs ............................................................................................................................... 163. TUNING FORK GYROSCOPE ............................................................................................ 213.1. Features of Tuning Fork Architecture ................................................................................................ 213.2. Design Concept ................................................................................................................................. 213.3. Equations of Motion.......................................................................................................................... 22 3.3.1. Drive-Mode Actuation ............................................................................................................... 24 3.3.2. Sense-Mode Detection: ............................................................................................................. 243.4. Mechanical Design ............................................................................................................................ 253.5. Finite Element Simulation ................................................................................................................. 253.6. Summary of Tuning Fork Gyroscope .................................................................................................. 26 3.6.1. Advantages of Tuning Fork Gyroscope ....................................................................................... 26 3.6.2. Disadvantage of Tuning Fork Gyroscope..................................................................................... 264. VIBRATING RING GYROSCOPE ...................................................................................... 27 [III]
- 5. 4.1. Vibration Modes of the Ring Structure .............................................................................................. 27 4.1.1. Normal Mode Model ................................................................................................................. 27 4.1.2. Mode Shapes............................................................................................................................. 27 4.1.3. Equations of Motion .................................................................................................................. 294.2. Vibration Induced Errors in Ring Gyroscopes..................................................................................... 31 4.2.1. Vibration Induced Errors due to Non Proportional Damping ....................................................... 31 4.2.2. Vibration Induced Errors due to Non Linearity of Sense Electrodes............................................. 324.3. A HARPSS Polysilicon Vibrating Ring Gyroscope ................................................................................ 32 4.3.1. Mechanical Design..................................................................................................................... 33 4.3.2. Finite Element Simulation .......................................................................................................... 34 4.3.3. Electrical Design ........................................................................................................................ 354.4. Evaluation of Vibrating Ring Gyroscope ............................................................................................ 36 4.4.1. Advantages of Ring Architecture ................................................................................................ 36 4.4.2. Disadvantages of Ring Architecture............................................................................................ 365. INTERFACE ELECTRONICS ............................................................................................. 375.1. System Components ......................................................................................................................... 37 5.1.1. Drive loop.................................................................................................................................. 37 5.1.2. Sense loop................................................................................................................................. 385.2. Overview of sense and drive electronics ........................................................................................... 386. BUILDING BLOCKS ........................................................................................................... 436.1. Capacitive interface........................................................................................................................... 43 6.1.1. Theory of Operation .................................................................................................................. 43 6.1.2. Capacitive sensing VS Resistive sensing ...................................................................................... 43 6.1.3. Main ruling Specs ...................................................................................................................... 45 6.1.4. Electronic Noise Sources ............................................................................................................ 45 6.1.5. Capacitive Sensing configurations .............................................................................................. 46 6.1.6. Comparison of Different Capacitive Sensing Architectures ......................................................... 526.2. Analog / Digital converter ................................................................................................................. 53 6.2.1. Introduction .............................................................................................................................. 53 6.2.2. General Specifications ............................................................................................................... 53 6.2.3. Analog to Digital converters....................................................................................................... 58 6.2.4. Decimation Filters...................................................................................................................... 64 6.2.5. Digital to Analog Converters ...................................................................................................... 716.3. Demodulation block .......................................................................................................................... 76 6.3.1. Coherent detection ................................................................................................................... 766.4. Frequency synthesizer (PLL) .............................................................................................................. 85 6.4.1. Theory of PLL ............................................................................................................................. 85 6.4.2. Terminology of PLL .................................................................................................................... 88 6.4.3. Types of PLL............................................................................................................................... 88 [IV]
- 6. 6.4.4. Non Ideal Effects in PLL .............................................................................................................. 89 6.4.5. Applications of PLL..................................................................................................................... 90 6.4.6. Limitations of Simple PLL architecture........................................................................................ 91 6.4.7. Phase frequency detector .......................................................................................................... 92 6.4.8. The Charge Pump ...................................................................................................................... 93 6.4.9. Voltage Controlled Oscillator ................................................................................................... 100 6.4.10. Frequency divider .................................................................................................................... 101 6.4.11. Mixed PLL/DLL ......................................................................................................................... 1026.5. Automatic Gain Control Loop (AGC) ................................................................................................ 105 6.5.1. Design 1 .................................................................................................................................. 106 6.5.2. Design 2 .................................................................................................................................. 111 6.5.3. AGC Sum up ............................................................................................................................ 1126.6. Design Flow will be used ................................................................................................................. 113 6.6.1. Analog ..................................................................................................................................... 113 6.6.2. Digital...................................................................................................................................... 1147. CONCLUSION AND FUTURE WORK ............................................................................ 115BIBLIOGRAPHY ....................................................................................................................... 116APPENDICES ............................................................................................................................ 119Appendix A – ANSYS Script to Calculate the Stiffness of a Fixed-Guided Straight Beam ............................... 119Appendix B - ANSYS Script to Calculate the Stiffness of Fixed Guided Curved Beam .................................... 121 B.1 Horizontal and Vertical Stiffness ......................................................................................................... 121 B.2 45o Stiffness ....................................................................................................................................... 122Appendix C - ANSYS Script for Modal Analysis of Tuning Fork Gyroscope .................................................... 123Appendix D - Plotting Vibrations of the Ring Structure using SciLAB ............................................................ 125Appendix E - ANSYS Script for Modal Analysis of Ring Gyroscope ................................................................ 126Appendix F: Project Timeline ....................................................................................................................... 128 [V]
- 7. 1. Introduction 1.1.MEMS Technology Micro-electromechanical systems (MEMS) technology is a process technology used to createtiny integrated devices or systems that combine mechanical and electrical components. Theyare fabricated using integrated circuit (IC) batch processing techniques and can range in sizefrom a few micrometers to millimeters. These devices (or systems) have the ability to sense,control and actuate on the micro scale, and generate effects on the macro scale. While the deviceelectronics are fabricated using ‘computer chip’ IC technology, the micromechanical componentsare fabricated by sophisticated manipulations of silicon and other substrates usingmicromachining processes. Processes such as bulk and surface micromachining, selectivelyremove parts of the silicon or add additional structural layers to form the mechanical andelectromechanical components.MEMS technology has several distinct advantages as a manufacturing technology. First, theinterdisciplinary nature of MEMS technology and its micromachining techniques, as well as itsdiversity of applications has resulted in an unprecedented range of devices and synergies acrosspreviously unrelated fields (for example biology and microelectronics). Second, MEMS with itsbatch fabrication techniques enables components and devices to be manufactured withincreased performance and reliability, combined with the obvious advantages of reducedphysical size, volume, weight and cost. These factors make MEMS potentially a far morepervasive technology than integrated circuit microchips. 1.1.1. MEMS ApplicationsMEMS applications are diverse; the oldest application is pressure sensors [1]. The other majorsensor market is the inertial sensors. This market has been dominated by the automotiveindustry, but recently the reduction in price has enabled adoption of MEMS inertial sensors(accelerometers and gyroscopes) in consumer devices like digital cameras, mobile phones, andLaptops.For average consumers, the inkjet print heads may be the most familiar micro-device. Eachreplacement inkjet cartridge has a micromachined inkjet nozzle head. The inkjet print heads arefrequently regarded as the largest MEMS market in terms of revenue. Texas Instruments holdsthe key patents of the field of digital micro-displays (DMD). In projection displays, the highcontrast ratio of mechanically actuated mirrors enables the micro-mirrors to compete againstthe common LCD technology.Silicon microphones are the latest entry to the mass market. The growth is driven by cell phoneindustry that is increasing rapidly. The microphones are an encouraging example of a MEMSproduct that only a few years ago was deemed too expensive, but now gaining a market sharerapidly. [1] 1.2.GyroscopesThe word gyroscope was coined by the French scientist Leon Foucault and is derived from theGreek words “Gyros” meaning rotation, and “Skopien” meaning to view. Simply, gyroscope is thesensor that measures the rate of rotation of an object. It can be used for example for inertialnavigation, image stabilization, and automotive chassis control and rollover detection. [1]
- 8. Historically, the angular rate has been measured with rotatingwheel gyroscope. The spinning wheel conserves the angularmomentum resisting the change in the rotation axis orientation.The angular velocity can now be sensed by measuring the forceon the spinning wheel due to rotation [2].Mechanical gyroscopes are comprised of a spinning wheelmounted on two gimbals which allow rotation along all threeaxes. Due to conservation of angular momentum, the spinningwheel will resist change in orientation. Hence when amechanical gyroscope is subjected to a rotation, the wheel willremain at a constant global orientation and the angles betweenthe adjacent gimbals will change. To measure the orientation ofthe device, the angles between the adjacent gimbals is read Figure -1 One of the first examples of 1 the gyrocompassusing angle pick-offs. It must be noted that a mechanicalgyroscope measures orientation directly. The disadvantage of mechanical gyroscopes is thatthey compriseof moving/spinning parts, which lead to friction. This eventually causes drift over time.Optical gyroscopes encompass more recent technology. They are based on Sagnac effect whichstates that a certain rate of rotation induces a small difference between the time it takes light totraverse the ring in the two directions. These gyroscopes are not subject to a mechanical wearand are the most precise ones [1]. Consequently, they are the most expensive gyroscopes, andare used in aircraft navigation systems and missile guidance.MEMS gyroscopes, fabricated using silicon micromachining technology, have low part countsand are relatively cheap to manufacture in commercial quantities. They enable new applicationsthat are not possible with the classic optical or mechanical gyroscopes. Nearly all MEMSgyroscopes are based on two orthogonal vibration modes. The drive-mode is orthogonal to thesense-mode meaning that the two modes do not normally interact and the drive-modemovement does not result in movement in sense-mode direction. The resonator is excited tovibrate in the drive-mode in the x-direction. The Corilois force due to a rotation around z-axis,excites the resonator sense-mode in y-direction. Thus, the sense-mode vibration amplitude isproportional to the angular rotation rate. [2] Figure -2 The Operation Principle of Vibrating Two Mode Gyroscope 1 [2]
- 9. 2. Background and TheoryThe underlying physical principle of vibratory gyroscopes is that a vibrating object tends tocontinue vibrating in the same plane as its support rotates. This device is also known as aCoriolis vibratory gyroscope because it is based on the principle of “Coriolis Effect”.2.1.Coriolis ForceThe Coriolis force is the perpendicular deflection of a moving element that arises in connectionwith rotation. Figure 2-1 illustrates how rotation affects the travel path of a freely moving object:A particle is thrown from the center of a rotating wheel in the radial direction. If no forces actedon the particle, it would have reached the point ‘B’. However, the wheel has rotated, so theparticle will not reach the point ‘B’, but a point ‘A’. [2] Figure -1 Illusration of Coriolis Effect 2The vector formula for the magnitude and direction of the Coriolis acceleration is [1]: ⃗⃗⃗⃗ ⃗ 2-1where ⃗⃗⃗⃗ is the acceleration of the particle in the rotating system (coriolis acceleration), is thevelocity of the particle in the rotating system, and ⃗ is the angular velocity vector of the wheelwhich has magnitude equal to the rotation rate ω and is directed along the axis of rotation of therotating reference frame, Thus, the coriolis force ( ⃗⃗⃗ ) acting on a particle of mass is: ⃗⃗⃗ ⃗ 2-22.2.Vibrating Two Modes GyroscopeThe basic architecture of a vibratory gyroscope is comprised of a drive-mode oscillator thatgenerates and maintains a constant linear or angular momentum, coupled to a sense-modeCoriolis accelerometer that measures the sinusoidal Coriolis force induced due to thecombination of the drive vibration and an angular rate input. The vast majority of reportedmicromachined rate gyroscopes utilize a vibratory proof mass suspended by flexible beamsabove a substrate. The primary objective of the dynamical system is to form a vibratory driveoscillator, coupled to an orthogonal sense accelerometer by the Coriolis force. The drive mode isorthogonal to the sense mode means that the two modes don’t normally interact [2]. [3]
- 10. 2.2.1. Drive Mode OperationThe Coriolis Effect is based on conservation of momentum; every gyroscopic system requires amechanical subsystem that generates momentum. In vibratory gyroscopes, the drive-modeoscillator, which is comprised of a proof-mass driven into a harmonic oscillation, is the source ofmomentum. The drive-mode oscillator is most commonly a 1 degree-of-freedom (1-DOF)resonator, which can be modeled as a mass-spring-damper system consisting of the drive proof-mass , the drive mode suspension system providing the drive stiffness , and the drivedamping consisting of viscous and thermoelastic damping. With a sinusoidal drive-modeexcitation force, the drive equation of motion along the x-axis becomes: ̈ ̇ 2-3The scale factor of the gyroscope is directly proportional to the drive-mode oscillationamplitude. Therefore, the drive mode is usually excited at resonance to obtain maximumdisplacement with small driving force (lower actuation voltage).It is extremely critical to maintain a drive-mode oscillation with stable amplitude, phase andfrequency. Self-resonance by the use of amplitude regulated positive feedback loop (Figure -2) 2is a common and convenient method to achieve a stable drive-mode amplitude and phase. Thepositive feedback loop destabilizes the resonator, and locks the operational frequency to thedrive-mode resonant frequency. An Automatic Gain Control (AGC) loop detects the oscillationamplitude, compares it with a reference amplitude signal, and adjusts the gain of the positivefeedback to match the reference amplitude. Operating at resonance in the drive mode alsoallows minimizing the excitation voltages during steady-state operation [2]. Figure -2 A typical implementation of an Automatic Gain Control (AGC) loop, which drives the drive-mode 2 oscillator into self-resonance and regulates the oscillation amplitude. 2.2.2. Sense Mode OperationThe Coriolis response in the sense direction is best understood starting with the assumption thatthe drive-mode is operated at drive resonant frequency , and the drive motion is amplitude [4]
- 11. regulated to be of the form with a constant amplitude . The Coriolis force thatexcites the sense-mode oscillator is: 2-4where is the portion of the driven proof mass that contributes to the Coriolis force. Similar tothe drive-mode oscillator, the sense-mode oscillator is also often a 1-DOF resonator, the sensemode equation of motion is: ̈ ̇ 2-5Thus, the system’s equations of motion can be written in matrix form: ̈ ̇ [ ][ ] [ ][ ] [ ]* + [ ] 2-6 ̈ ̇We notice that the off diagonal elements on the matrices of damping [ ] and stiffness [ ] areequal to zero, this means that no mode coupling happens except by the influence of the Corioliseffect.2.3.Mechanical Design of MEMS Gyroscopes 2.3.1. Flexure ElementsIn linear micromachined gyroscopes, the suspension systems are usually designed to becompliant along the desired motion direction, and stiff in other directions. Most suspensionsystems utilize narrow beams as the primary flexure elements, aligning the narrow dimension ofthe beam normal to the motion axis.2.3.1.1. Fixed Guided Linear BeamIn purely translational modes, the boundary conditions of the beams that connect thecomponents of the gyroscopes are most commonly the fixed-guided end configuration (Figure2 -3), in which the moving end of the beam remains parallel to the fixed end. Many completegyroscope suspension systems can be modeled as a combination of fixed-guided end beams. Figure -3 The fixed-guided end beam under translational deflection – (a) Beam Dimensions (b) Guided 2 Boundary ConditionIf we define the length of a beam (L) as the x-axis dimension, width (w) as the y-axis dimension,and the thickness (t) as the z-axis dimension, the stiffness values of the fixed-guided beam alongthe three principle axes become (assuming a linear case) [2]: [5]
- 12. 2-7 2-8 2-9For example, for a fixed-guided beam with the dimensions L = 500μm, w = 4μm, and t = 25μm.Assuming an elastic modulus of E = 150 GPa, the stiffness in the y direction is calculated fromequation 2.7 to be 1.92N/m. However this stiffness changes with the amount of deflectionpractically due to the increase in reaction forces causing a nonlinear behavior by the beam.The stiffness of the beam in the previous example was verified by Finite Element Analysis (FEA),using a linear solution, and the deflection for a force of 10μN was about 5.208μm, therefore isF/x approximately equals 1.92N/m. Performing nonlinear analysis, the stiffness reached4.86N/m at a load of 10μN and deflection of 3.428μm as shown in Figure -4. 2 Load-Deflection Plot K - displacement Plot 6.00 6.00 5.00 5.00Displacement (μm) 4.00 4.00 k (Nm) 3.00 3.00 2.00 2.00 1.00 1.00 0.00 0.00 0.00 5.00 10.00 0.00 2.00 4.00 6.00 Load (μN) Y-Discplacement (μm) Figure -4 FEA Results - (a) Load - Deflection Plot (b) K - Displacement Plot 2An ANSYS script for plotting the load-deflection graph is shown in appendix A.2.3.1.2. Curved BeamCurved (or semicircular) beams are widely used in vibrating ring gyroscopes (to be explained inchapter 4. The stiffness of a curved beam is highly dependent on the direction of the appliedforce. We will consider the 3 main stiffness in the horizontal, vertical, and 45o directions (KHA,KVA, and K45) as shown in Figure -5. 2 [6]
- 13. Figure -5 Stiffnesses of a semicircular spring in three directions 2 (a) Horizontal stiffness (KHA), (b) vertical stiffness (KVA), (c) stiffness along 45◦ direction (K45).As proved in [3] the stiffness for a beam of radius , and moment of area ⁄ , in the three directions are given by: 2-10 ( ) 2-11 ( ) 2-12 ( )( )Where w is the width of the beam, t is the thickness of the structure.For example, for semicircular beam with the radius r = 235μm, w = 4μm, and t = 80μm as in [4]Assuming an elasticity modulus of E = 150 GPa, the stiffness in the horizontal direction iscalculated from equation 2.10 to be 16.573N/m, in vertical direction from equation 2.11 is3.139N/m, and that in the 45o direction is 9.856N/m. However this stiffness also changes withthe amount of deflection practically, as in the case of straight fixed-guided beams, due to theincrease in reaction forces causing a nonlinear behavior by the beam.The stiffness of the beam in the previous example was verified by FEA (Figure -6), using a 2linear solution, the deflection for a force of 1mN was about 61.91μm in the horizontal direction,thus the value of KHA was 16.15N/m. The deflection in the vertical direction at the same appliedforce value was 452.59μm leading to KVA = 2.21N/m, and in 45o direction the deflection was106.445 and the value of K45 was 9.40N/m. [7]
- 14. (a)(b)[8]
- 15. (c) Figure -6 FEA of curved beam - (a) Horizontal Deflection (b) Vertical Deflection (c) 45 o Deflection 2ANSYS scripts to generate the previous plots are found in appendix B.2.3.2. Mode CouplingThere are two types of mode coupling in MEMS gyroscopes; the first is the desired one whicharises from Coriolis force, and the designer aims to magnify it, the second is an undesired onewhich arises from non-idealities. In reality, fabrication imperfections result in non-idealgeometries in the gyroscope structure, which in turn causes the drive oscillation to partiallycouple into the sense-mode. Considering the relative magnitudes of the drive and senseoscillations, even extremely small undesired coupling from the drive motion to the sense-modecould completely mask the Coriolis response.Equation 2.6 describes an ideal gyroscope, where the mode coupling happens only due toCoriolis force, the practical equation of motion of a vibratory two modes gyroscope can bewritten as: ̈ ̇ [ ][ ] * +[ ] [ ]* + [ ] 2-13 ̈ ̇Where and in the damping matrix represents the coefficients of the anisodamping forcesin y and x directions as a result of motion in y and x directions respectively. The terms and in the stiffness matrix represents the anisoelasticity forces coefficients (suspension elementsin real implementations of vibratory gyroscopes have elastic cross-coupling between theirprincipal axes of elasticity). [2] [9]
- 16. Since the oscillation amplitudes in the sense-mode are orders of magnitude smaller than thedrive-mode, the coupling due to and in the drive dynamics is negligible. The impact ofanisodamping and anisoelasticity is primarily on the sense-mode dynamics due to and ,which couples the drive-mode displacement into the sense-mode accelerometer.In Equation 2.13, we notice that there is always a 90o phase difference between the Coriolisresponse ( ̇ ) and the mechanical quadrature ( ), therefore, the quadrature signalcan be separated from the Coriolis signal during amplitude demodulation at the drive frequency(using coherent detection In which we multiply the sense signal by a carrier with samefrequency and phase). However, the ansiodamping component is in phase with the Coriolisresponse, therefore it can’t be removed during demodulation, and it should be minimized in thedesign of the gyroscope itself or by vacuum packaging of the device.2.4. Electrical Design of MEMS GyroscopesMicromachined gyroscopes are active devices, which require both actuation and detectionmechanisms. Various vibratory MEMS gyroscopes have been reported in the literatureemploying a wide range of actuation and detection methods. For exciting the gyroscope drivemode oscillator, the most common actuation methods are electrostatic, piezoelectric, magneticand thermal actuation. Most common Coriolis response detection techniques include capacitive,piezoelectric, piezoresistive, optical, and magnetic detection.In many MEMS applications, capacitive detection and electrostatic actuation are known to offerseveral benefits compared to other sensing and actuation means, especially due their ease ofimplementation. Capacitive methods do not require integration of a special material, whichmakes them compatible with almost any fabrication process. They also provide good DCresponse and noise performance, high sensitivity, low drift, and low temperature sensitivity2.4.1. Capacitive DetectionParallel-plate capacitors can be mechanized in several ways to detect deflection. For a genericparallel-plate electrode plate with a gap d and overlap area Aoverlap , the capacitance is 2-14 Figure -7: Variable Gap Capacitor 2where is the dielectric constant of the material between the plates. Each parameter in thisexpression can be modulated by a deflection to result in a capacitance change. In variable gap [10]
- 17. capacitors, the motion is normal to the plane of parallel plates, and the gap d changes withdeflection. In variable area capacitors, the motion is parallel to the plane, which results in achange in Aoverlap. By placing a moving media between the parallel plates, the dielectric constant can be modulated by deflection. The most common electrode types in inertial sensors arevariable gap and variable area capacitors, which are summarized below.2.4.1.1. Variable Gap DetectorVariable-gap capacitors are the most widely used electrode typefor detection of small displacements. When the parallel plates areoriented normal to the motion direction, deflections cause achange in the gap .It should be noticed that capacitance is a nonlinear function ofdisplacement in variable-gap capacitors. However, for very smalldeflections relative to the initial gap, the capacitance change islinearized. Denoting the displacement in the motion direction asand assuming << , the capacitance change in a variable-gap Figure -8 Variable Gap Detector 2capacitor with an overlap area becomes: 2-15Thus, small gap changes could result in high capacitance changes, providing very largesensitivity.2.4.1.2. Variable Area DetectorVariable area capacitors are ideal when the detected motion magnitudes are larger, especiallyeither when variable gap capacitors become significantly nonlinear, or deflections are largerthan a minimum gap. Since the overlap area is proportional to both dimensions in the plateplane, capacitance change is purely linear with respect to motion parallel to the plates. Denotingx as the displacement in the motion direction parallel to the plates, the capacitance changebecomes 2-16 Figure -9: Variable Area Detector 2Table 2-1 shows a brief comparison between variable area and variable gap detectors [11]
- 18. Table -1: Comparison between Detectors 2 Variable gap Capacitor Variable area capacitorThe change in capacitance is a result of change The change in capacitance is a result of changein the gap d between the two plates. in the overlap area between the two plates.Higher sensitivity. Lower sensitivity.Non-linear for large displacement. The change in capacitance is linear.From the above table, we can conclude that we can use variable gap capacitor if we don’t need alarge displacement and get a high sensitivity. However, we can use variable area capacitor for alarger travelling distance in the expense of sensitivity.2.4.2. Electrostatic ActuationElectrostatic or capacitive actuation is based on the attraction of electric charges [2]. As thedevice size is reduced to the micro-scale, this force become significant. The capacitive actuatorsare easily fabricated and consume no DC power. There are two main types of capacitiveactuators; the closing gap actuator, and the variable area actuator. The variable area actuatorsare much common in gyroscopes, because the electrostatic force varies linearly with the moveddistance. However, in some cases (like the vibrating ring gyroscope), the closing gap actuator isused.2.4.2.1. Closing Gap ActuatorConsider the closing gap actuator in Figure -10 Closing Gap Actuator, the electrostatic force of a 2parallel plate capacitor (which is a good model for many MEMS actuators) is derived from theenergy stored, and is given by: 2-17Where is the polarization voltage applied on the actuator, is the overlapping area betweenthe two electrodes, is the initial gap between the electrodes. The restoring force generated inthe spring of stiffness due to displacement is given by: 2-18At equilibrium, neglecting the weight of the electrodes, the electrostatic force isequal to the spring force which makes the voltage required to move a distanceequals: √ 2-19 Figure -10 Closing Gap Actuator 2 [12]
- 19. From Equation 2.14, we notice that the force is a nonlinear function with the displacement x,which means that the electrostatic force increases rapidly as the two electrodes gets closer.When the electrode moves a distance , the electrostatic force grows fast and themovable electrode accelerates till it sticks with the fixed one, and the structure fails. Thiscondition is called: the pull-in condition, and the pull-in distance is considered the maximumdistance that the actuator can move, it’s proved in [1] that , which means that theclosing gap actuator can only move one third of the gap between the electrodes. Itshould be considered by the designers that the drive mode amplitude shouldn’t exceedthe pull-in distance. Figure -11 shows the electrostatic and spring forces of an actuator of 2area = 150μm x 60μm, initial gap of 1.4μm at different polarization voltages. Electrosatic and Spring forces Vs displacement 200 180 160 140 Force (μNewton) 120 Fspring 100 Fe(Vlow) 80 Fe(Vhigh) 60 Fe(Vpull-in) 40 pull- 20 in 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 displacement x (μm) Figure -11 Electrostatic Force and spring force of a closed gap actuator 22.4.2.2. Variable Area ActuatorsVariable-area actuators aim to linearize the capacitance change versus displacement, in order toachieve constant electrostatic force with respect to displacement. The inter digitated comb-drivestructure is based on generating the actuation force through a series of parallel plates slidingparallel to each other, without changing the gap between the plates. The electrostatic forcegenerated in the x-direction for two parallel plates as in Figure -12 is 2 2-20It should be noticed that this force is independent of displacement in the x-direction and theoverlap length of the capacitor plates, x0. [13]
- 20. Figure -12: Variable-area electrostatic actuator model. 2Inter digitated comb-drives based on variable-area actuation are one of the most commonactuation structures used in MEMS devices. The primary advantages of comb-drives are long-stroke actuation capability and the ability to apply displacement-independent forces, whichprovide highly stable actuation. In a comb-drive structure made of N fingers, each finger forms two parallel-plate pairs, and thetotal electrostatic force generated in the x-direction becomes 2-21Where z0 is the structure thickness and y0 is the distance between the fingers.2.4.2.3. Balanced ActuationIn MEMS gyroscope, we always want to induce harmonic motion. Therefore, when a sinusoidalnet actuation force is desired, the drive force can be linearized with respect to the actuationvoltages by appropriate selection of voltages applied to the opposing electrode sets [2]. The netelectrostatic force generated by two opposing capacitors C1 and C2 is: 2-22A balanced actuation scheme is a common method to linearize the force with respect to aconstant bias voltage and a time-varying voltage . The method is based on applying to one actuator, and to the opposing actuator. Assuming twoelectrodes are identical, and the DC voltage is much greater than the time varying component,the net electrostatic force reduces to: [ ] 2-232.5.Fabrication Technologies 2.5.1. Microfabrication TechniquesMicrofabrication describes processes of fabrication of miniature structures, of micrometer sizesand smaller. Integrated Circuit (IC) fabrication is the earliest microfabrication processes used.Inertial sensors require moving parts to detect inertial phenomena. Micromachiningtechnologies have revolutionized inertial sensing by allowing fabricating moving mechanicalsystems at the micro scale. Originated from semiconductor fabrication techniques,micromachining technologies have made it possible to merge micro-scale mechanical and [14]
- 21. electrical components. The essence of all micromachining techniques is successive patterning ofthin structural layers on a substrate [2].2.5.1.1. DepositionThe process flow of micromachining fabrication starts with ablank wafer. The intention is to pattern a multiple structurallayer for moving structures, interconnect, electrode areas, ordielectric layers for electrical isolation using successivedeposition and patterning of these layers.Depending on the material, layer thickness, or conformalcoverage requirements, different deposition techniques may beused such as Chemical Vapor Deposition (CVD), Physical VaporDeposition (PVD), or electroplating.2.5.1.2. PhotolithographyPhotolithography, also known as lithography, is the process ofpatterning parts of a thin film or the bulk of a substrate. Prior to Figure 2-14: Microfabriactionprocessing, a photolithography mask that carries the wafer-level Processlayout of a layer is generated. Then the image on the mask isprojected onto a photosensitive material deposited on the wafer, commonly known asphotoresist.2.5.1.3. EtchingEtching transfers the pattern formed by photolithography into the actual structural materialsand defines the geometry of the device by selective material removal.There are two primary categories of etching: wet etching and dry etching. As the name implies,wet etching uses a liquid chemical solution. On the other hand, dry etching uses either a vaporphase etchant or reactive ions. In MEMS, to determine the required etching method many factorsare involved such as the desired sidewall and bottom surface profiles, isotropy, or stictionissues. [2] 2.5.2. Bulk-Micromachining ProcessesMicromachining processes are usually divided, depending on the structural layer formingtechnique, into two main categories: Surface micromachining and Bulk micromachining.Traditionally, bulk micromachining implies the use of subtractive processes to pattern thickstructural layers. In most bulk micromachining process, two or more wafers are bonded, and themoving structures are made out of the whole thickness of a silicon wafer.Bulk micromachining offers many advantages for inertial micromachined devices, since itprovides thick structural layers. Larger device thickness increases the mass and overlap area ofcapacitive electrodes, directly improving gyroscope performance. Thicker suspension beamsprovide higher out-of-plane stiffness, which reduces shock and vibration susceptibility, andminimizes the risk of stiction to the substrate. It also allows the use of single crystal silicon as thedevice material, which provides excellent mechanical stability. [2]The implementing of bulk micromachining can be done by many different fabricationtechnologies: [15]
- 22. 2.5.2.1. SOI-Based Bulk MicromachiningSilicon-on-Insulator (SOI) wafers are excellent starting materials for bulk micromachining. Thesilicon device layer comes bonded on an insulator layer. Electrically isolated and mechanicallyanchored free-standing structures can be formed simply by patterning the device layer and theoxide layer underneath. Figure -13: An SOI-based bulk-micromachined gyroscope, diced and released. 2 2.5.3. Surface-Micromachining ProcessesWhile bulk micromachining uses subtractive processes to pattern thick structural layers, surfacemicromachining is in essence an additive technique. It relies on successive deposition andpatterning of thin structural layers on the surface of a substrate, rather than etching thick bulklayers.In surface micromachining, complex three-dimensional devices are built by depositing multiplestacks of alternating structural layers and sacrificial layers. Each sacrificial layer supports thestructural layer above it during fabrication, and separates it from the other layers below. At theend of the process, the sacrificial layers are selectively etched away, releasing the structurallayers. [2]2.5.4. SOI-MUMPsThe following is a general process description and user guide for Silicon-On-Insulator Multi-UserMEMS Processes (SOIMUMPs). It is a simple 4-mask level SOI patterning and etching processderived from work performed at MEMSCAP.The process begins with 150mm n-type double-side polished Silicon on Insulator wafers. Aphosphosilicate glass layer (PSG) is deposited, and the wafers are annealed at 1050°C for 1 hourin Argon to drive the Phosphorous dopant into the top surface of the Silicon layer. The PSG layeris subsequently removed using wet chemical etching [3]. [16]
- 23. Figure -14 2The wafers are coated with negative photoresist and lithographically patterned by exposing thephotoresist with light through the first level mask (PADMETAL), and then developing it. Figure -15 2The wafers are coated with UV-sensitive photoresist and lithographically patterned by exposingthe photoresist to UV light through the second level mask (SOI), and then developing it. Thephotoresist in exposed areas is removed, leaving behind a patterned photoresist mask foretching. [17]
- 24. Figure -16 2A front side protection material is applied to the top surface of the patterned Silicon layer. Thebottom side of the wafers are coated with photoresist and the third level (TRENCH) islithographically patterned. Figure -17 2The front side protection material is then stripped using a dry etching process. The remaining“exposed” oxide layer is removed from the top surface using a vapor HF process. [18]
- 25. Figure -18 2A separate silicon wafer is used to fabricate a shadow mask for the Metal pattern. The shadowmask wafers are coated with photoresist and the fourth level (BLANKETMETAL) islithographically patterned. Figure -19 2The shadow mask is aligned and temporarily bonded to the SOI wafer. The Blanket Metal layer isdeposited through the shadow mask. [19]
- 26. Figure -20 2The shadow mask is removed, leaving a patterned Metal layer on the SOI wafer. Figure -21 2 [20]
- 27. 3. Tuning Fork Gyroscope3.1.Features of Tuning Fork ArchitectureFor many applications, gyroscopes are subject to a wide variety of changing environmentalconditions such as temperature, pressure, and ambient vibrations. The robustness of the sensorto these external influences during operation is critical for adequate performance. A level ofrobustness is commonly achieved through electronic control systems, such as a temperaturecompensation circuit which post-processes the output of the mechanical sensor depending upontemperature or an automatic mode matching controller. Robustness to ambient vibrations,however, is generally addressed by the mechanical design through the use of tuning fork drivingarchitectures. Tuning fork designs have the ability to reject common mode inputs due to anti-phase forcing which results in anti-phase Coriolis responses [4].3.2.Design ConceptThe mechanical architecture of the tuning fork gyroscope, Figure -1, comprises of two proof- 3masses, supported by a network of flexural springs and anchored at a central post. Figure -1: Schematic diagram of the TFG 3The drive-mode of the gyroscope is formed by the two masses forced into anti-parallel, anti-phase motion synchronized by the integrated mechanical lever system. The sense-mode isformed by the two linearly coupled tines moving in anti-phase Figure -2 [5]. The gyroscope is 3electro-statically driven into anti-phase motion using driving voltages imposed across thedifferential lateral comb electrodes on the drive-mode shuttles. During rotation around the z-axis, the Coriolis acceleration of the proof masses induces linear anti-phase sense-modevibrations which are capacitively detected using differential parallel plate electrodes on thesense-mode shuttles [6]. [21]
- 28. Figure -2: In-plane operating flexural modes. (Left) Drive resonant mode along the x-axis. 3The anchor design of the TFG satisfies two critical properties: mechanical coupling and resonantmode isolation. The mechanical coupling allows synchronization of the phases of the proof-masses. Hence, the central beam is designed as ladder-shape structure as shown in Figure -2. 3Due to the non idealities other modes are excited such as pseudo drive and pseudo sense modes,Figure -3, so the anchor should also be able to isolate the in-plane operating modes from the 3two other in-plan modes. Figure -3: In-plane pseudo-operating flexural modes. (Left) Pseudo-drive resonant mode 3The flexural spring must be designed to ensure large mobility along both axes. To this effect, afish-hook architecture was adopted which ensures that the mode shapes have two-directionalflexibility.3.3.Equations of MotionThe TFG can be conceptualized as a coupled resonator system, with the rotation induced Coriolisforce being the coupling agent between the two resonant operating modes. The dynamics of thedevice are governed by Newton’s second law of motion [7].The drive-mode oscillator is most commonly a 1 degree-of-freedom (1-DOF) resonator, whichcan be modeled as a mass-spring-damper system consisting of the drive proof-mass m, thedrive-mode suspension system providing the drive stiffness k, and the drive damping cconsisting of viscous and thermoelastic damping. With a sinusoidal drive-mode excitation force,the drive equation of motion along the x-axis becomes [22]
- 29. ̈ ̇ 3 -1With the definition of the drive-mode resonant frequency wd and the drive-mode Quality factorQd the amplitude and phase of the drive-mode steady-state responsebecomes: 3 -2 √ ( )at w = wd the amplitude becomes 3 -3where √ 3 -4and 3 -5A rotation signal along the normal axis (z-axis) of the results in a Coriolis induced accelerationon the individual proof-masses along the sensitive axis (y-axis). The magnitude of the Coriolisacceleration is given by the vector cross product of the input rotation rate vector and thevelocity of the proof mass (2Ω x V).Considering that the proof-masses are oscillating in a sinusoidal fashion at the drive-moderesonance, the expression for the Coriolis acceleration along the sense-axis is given by: 3 -6Where Ωz is the input rotation rate, ‘VDrive-x’ is the velocity of the drive resonant mode, ‘XDrive’ isthe amplitude of drive-mode oscillation and ‘ωDrive’ is the drive-mode resonant frequency. Andfrom Newton’s second law we know that so we can describe the equation of motion ofthe 1-DOF sense mode oscillator by the following: ̈ ̇ 3 -7The amplitude and phase of the steady-state sense-mode Coriolis response in a linear system,defining the sense-mode resonant frequency ws and the sense-mode Quality factor Qs, become[2]: 3 -8 √ ( )where √ 3 -9and [23]
- 30. 3 -10The rotation-induced proof-mass displacement along the y-axis causes the gap between theparallel plate sense electrode and the proof-mass to change. This change in capacitive gap isproportional to the input rotation rate, and is detected by means of transimpedance front-endelectronics.3.3.1. Drive-Mode ActuationA key parameter that determines both the resolution and the sensitivity of a micromachinedvibratory gyroscope is the drive amplitude. For this reason, comb-drive electrodes were chosenahead of parallel-plate electrodes as the choice of actuation for the drive resonant mode. Comb-drive actuation offers greater linear range of operation as well as larger drive displacementbefore pull-in.The overall capacitance of the comb-drive electrode is expressed as: ( ) 3 -11where ‘N’ indicates the number of combs, ‘h’ refers to thecomb-thickness, ‘wo’ is the initial overlap, ‘g’ is the adjacentgap size, ‘x’ is ‘y’ represents the transversal displacementalong the sense axis (y-axis) which may be caused either byCoriolis or quadrature errors [7].We also know that 3 -12From equation 3-11 and equation 3-12 we can get the Figure -4: comb-drive electrode 3following: 3 -13We can neglect y with respect to g as y << g, equation 4-13 becomes: 3 -143.3.2. Sense-Mode Detection:Based on the comparison in Table -1 and as we are not in need for a large displacement, 2variable-gap detection will be used as the choice of detection for the sense mode detection.The capacitance of the sense electrodes is expressed as: 3 -15where ’ls’ is the overlap length between the two electrodes, ‘t’ is the thickness, ‘g s’ is the initialgap between the electrodes, and ‘y’ is the lateral displacement amplitude along the sense axis (y-axis). [24]
- 31. 3.4.Mechanical DesignConsider the design shown in Figure -1; we need to determine all the dimensions of the proof- 3masses, actuators, and detectors. After that, we can calculate the change in the actuation anddetection capacitances. The intention of this design is to get a change in the drive modecapacitance differentially and change in the sense mode capacitancedifferentially with a biasing voltage V < 10V. We first select the width ‘W’, length ‘L’, thickness ‘t’of the proof-masses (W = 0.4 mm, L = 0.27 mm, and t = 25 um) sticking to the size specificationsand design rules. Then determine the drive and sense frequencies at which the device willoperate ( = 7.5 KHz and = 7.7 KHz).We can now calculate the stiffness of the flexure beams along the drive axis (x-axis) usingequation 3-4 we get .Assuming the gap between the fingers in the comb-drive actuators to be g = 3 um and thethickness of one finger b = 3 um taking into consideration that they must be greater than theminimum feature length given in the design rules.From the previous assumed dimensions of the comb-actuator we find that the maximumallowable number of fingers is N = 38 finger. We use Na = 27 finger for detection and Nd = 11 foractuation.Now by applying a volt V = 9.5 V we get the drive force using equation 3-13and from equation 3-3 we calculate Xo = 4.83 um at QD = 500.As mentioned above we need a change in the drive mode capacitance > 10 fF, from equation 3-11 neglecting ‘y’ with respects to ‘g’ we can deduce the change in capacitance as described in thefollowing equation: 3 -16From equation 3-16 and all the previous calculations we compute differentially.As mentioned before that the rotation-induced proof-mass displacement along the y-axis causesthe gap between the parallel plate sense electrode and the proof-mass to change. This change inthe gap causes a change in the sense capacitance. To calculate this change we need first tocalculate the displacement in y-direction. After that, we found the change in capacitance by usingthe following equation: 3 -17where and .After some calculation using equation 3-8 with we get y = 0.0037 um. As a result weget differentially.3.5.Finite Element SimulationAfter the previous first order analysis, a 2-D Finite Element Analysis (FEA) was carried out usingthe values calculated in the previous section, the FEA revealed 4 in-plane modes. The first modewas pseudo drive mode at about 7.48 KHz; the second mode was the anti-phase drive mode atabout 7.52 KHz the fourth and fifth was sense mode and pseudo sense mode at about 7.7 KHz [25]
- 32. and 8.8 KHz, respectively as shown in Figure -5. The difference in the resonant frequencies 3between the model and FEA might have happened due to the approximations done whencalculating the masses and stiffness. An ANSYS script that animates the mode shapes of thetuning fork structure can be found in Appendix C. (a) (b) (c) (d) Figure -5: (a) Drive-mode; (b) Sense-mode; (c) Pseudo drive-mode; (d) Pseudo sense-mode 33.6.Summary of Tuning Fork GyroscopeThe tuning fork gyroscope has important features compared to other vibratory gyroscopes.3.6.1. Advantages of Tuning Fork Gyroscope Tuning Fork Gyroscope (TFG) is designed with a symmetrical structure. It employs two masses that vibrate out of phase. This differential operation cancels common-mode errors. It also doubles the amplitude of the output signal. High sense capacitance.3.6.2. Disadvantage of Tuning Fork Gyroscope Small displacement in the sense mode. Large zero bias errors caused by the slight misalignment of the mass centers of the individual tines. If the electrostatic drives for the individual tines are not preciously matched, an out of plane vibration response is introduced. [26]
- 33. 4. VIBRATING RING GYROSCOPEThe vibrating ring gyroscope is based on the transfer of energy between two identical modes,thus we can expect high sensitivity. The rotation sensing principles of the vibrating shellgyroscope can be explained as the ring vibrates in an elliptical (flexural) manner that have twonodal diameters. When the structure is rotated, the node lines lag behind the rotation (Figure4 -1) [8]. Therefore, the principle of operation of a vibrating ring gyroscope will be: exciting thering to vibrate elliptically (Drive mode), then monitoring the lag of the nodes capacitively. Figure -1 Vibrating Ring and lagging nodes 44.1.Vibration Modes of the Ring StructureThe vibration of the ring structure can be explained by the normal mode model. In this model,the elliptic vibration of the ring is considered to be a superposition of two identically shapedvibration modes. Because of their mode shapes, the locations of maximum motion or antinodesfor the two vibration modes are 45 o apart rather than 90o as in the tuning fork gyroscope.Coriolis effect causes energy transfer between the two modes. [8] 4.1.1. Normal Mode ModelAny general vibration-induced displacement of an elastic body ( ⃗ can be expressed by thelinear combination of its normal vibration modes : ⃗ ∑ 4-1where p is the independent position coordinate which can be expressed by Cartesiancoordinates (x and y) or by cylindrical coordinates (radial and tangential coordinates). Theequation includes generalized (modal) coordinates (mode amplitudes) (i.e. ) and modeshape functions (i.e. ). [2]4.1.2. Mode ShapesThere are several modes for the ring structure (out of plane, torsional, translational, and flexuremodes). The most important mode shapes, as mentioned by [9], are four. The first two modesare translation modes in the x and y directions ( ), and their radial/tangentialcomponents are:X-axis translation mode: [27]
- 34. 4-2Y-axis translation mode: 4-3where θ is an independent spatial coordinate (angle) describing position around the ring. Thesecond two modes are elliptical-shaped flexural modes ( ), and their radial/tangentialcomponents are:Drive axis flexure mode: 4-4Sense axis flexure mode: 4-5Where determines the angle between the principle mode axis and the horizontal axis [8], tosimplify the math, we take , the radial components of the mode shapes are plottet inFigure -2. It’s clear that the each of the two flexure modes has its nodes on the antinodes of the 4other. Figure 4-3 shows the vibrations of the ring. A Scilab code for these plots can be found inAppendix D. (a) (b) Figure -2 Mode amplitude plots of the ring vibrations 4(a) Horizontal and Vertical Translational Modes, (b) Primary and Secondary Flexural Modes [28]
- 35. (a) (b) (c) (d) Figure -3 Evolution of Translational (a, b) and Elliptic (c, d) vibrations of the ring structure 44.1.3. Equations of MotionWe Consider the ring structure in Figure -4 Conceptual view of a MEMS ring gyroscope, the ring 4structure is attached to an anchor by 8 symmetric curved beams, the ting is driven into flexuremode horizontally by two electrodes at 0o and 180o, and the two electrodes at 45o and 225o areused for sensing (Open Loop Operation).Unlike the non-degenerate gyroscopes, like tuning fork gyros, ring gyroscopes cannot beanalyzed using simple lumped models because the mass and the stiffness of the ring gyro aredistributed along the ring. The equations of motion of the ring gyroscope structure can obtainedby deriving the kinetic energy, potential energy, and dissipated energy by viscous damping foreach mode, and substituting in Lagrange’s equation. [29]
- 36. Figure -4 Conceptual view of a MEMS ring gyroscope 4Considering the flexure and translational modes only, the equations of motion can be expressedas shown below, a detailed derivation of these equations is done in [9], and the final resultswere: ̈ ̇ ̈ ̇[ ][ ] [ ][ ] [ ][ ] ̈ ̇ ̈ ̇ ̇ ̇ ̇ [ [ ][ ] ][ ] ̇ ̇ [ ][ ] [ ][ ] ̇ [ ][ ̇ ] [ ][ ] [ ] 4-6where line (1) represents Mass, Damping, and Stiffness; M 1, M2, M3, M4 are the modal masses ofthe modes 1, 2, 3, 4 respectively, C1, C2, C3, C4 are the viscous damping coefficients for each mode,K1, K2, K3, K4 are the stiffness seen by each mode due to support springs, for the translational [30]
- 37. modes, and due to a combination of the support springs and the ring structure stiffness for theflexure modes. The terms in line (2) (γT and γF) represents the modal coupling terms induced bythe Coriolis forces and by angular acceleration, the angular acceleration is negligible because theratio between angular acceleration to Coriolis response is inversely proportional to the flexuralresonant frequency , Line (3) contains additional stiffness terms that arise from centripetalacceleration and electrostatic effects (α, β, χ), line (4) contains the terms representing theenvironmental excitation ( and vo), and it’s clear that the ambient vibrations affect only thetranslational modes, line (5) contains a term from the electrostatic actuation. Details ofCalculation of each term in the previous matrices are explained in [8], [9]..From the equations of motion, it’s clear that the four modes form two decoupled sets ofequations in the absence of angular rotation; which independently govern the translation andflexural modes. Therefore, the flexural modes, which are excited by the operation of the ringgyroscope, are not influenced by the translation modes which are excited by the environmentalvibration. Thus, the flexural modes are not influenced by environmental vibrations (Ideally)[9,10].4.2.Vibration Induced Errors in Ring GyroscopesWhen looking at the previous equations (4.6), it seems that the output of a ring gyroscope isinsensitive to vibration due to the decoupled dynamics governing ring translation versus ringflexure; however, this decoupling is violated in the presence of non-proportional damping andcapacitive nonlinearity at the sense electrodes [9,11,10]. 4.2.1. Vibration Induced Errors due to Non Proportional DampingProportional Damping is the type of damping in which the modal damping matrix [C] is in theform of: [ ] [ ] [ ] 4-7where [M], [K] are the modal masses and stiffness respectively, α, β are constants, usuallyempirical. This type of damping is known as PROPORTIONAL, i.e proportional to either the massM of the system, or the stiffness K of the system, or both. Proportional damping is rather unique,since only one or two parameters, α, β, appear to fully describe the complexity of damping,irrespective of the system number of DOFs, n. This is clearly not realistic. Hence, proportionaldamping is not a rule but rather the exception. [9]Considering Proportional Damping, the damping matrix is diagonal, since the matrices of massand stiffness are supposed to be diagonal, which results in decoupled modes. However, Non-Proportional damping has been observed in MEMS gyroscopes. In case of non-proportionaldamping, the damping matrix contains non-zero off-diagonal elements as follows: [ ]where N is the number of modes of the structure, and this is considered as one of the causes ofundesired mode coupling. [31]
- 38. 4.2.2. Vibration Induced Errors due to Non Linearity of Sense ElectrodesThe parallel-plate sensing mechanism contributes a nonlinear behavior between sensecapacitance and the sense-axis displacement. This nonlinearity is negligible in normal operationbecause the displacement produced by the Coriolis force is small. However, larger displacementscan be readily generated by vibration, and these displacements are subject to capacitivenonlinearity. [11]Vibration-induced errors are explained in [9] by subtracting the capacitive change by onlyCoriolis force and no external vibration, from the capacitive change by both Coriolis force andexternal vibration, and removing the signals produced having frequencies far from the resonantfrequency of the gyroscope (∼20 30 kHz), because they will be filtered out by the interfacecircuit demodulation system. The resulting change in capacitance due to vibrations is given by: [ ] 4-8where / and / are the initial capacitance and the initial gap of the senseelectrode at 45◦/225◦. In an ideally fabricated symmetric ring structure, = and = and becomes: 4-8Therefore, another source of vibration-induced errors in ring gyroscopes arises from the highorder (cubic) terms in the capacitive nonlinearity at the sense electrodes.There are other vibration induced error sources like those resulted from high frequency externalvibration or from imperfections that couple ring translation and flexure. High frequencyvibration (with spectral content frequency containing the flexural-mode resonant frequencies)may directly excite the flexural modes leading to undesired responses that cannot bedistinguished from the desired responses excited by ring gyro operation. This error mechanismobviously exists even for ideally fabricated ring gyroscopes.On the other hand, vibration-induced errors by fabrication imperfection may occur when theflexural modes are excited by translation modes. The decoupling of flexural and translationmodes can arise from the assumed perfect symmetry of the ring gyro. The symmetry may bedestroyed by a non-uniform or asymmetric distribution of ring mass and/or stiffness (inertialand/or compliance coupling) as previously noted in analyses of degenerate gyroscopes.4.3. A HARPSS Polysilicon Vibrating Ring GyroscopeA famous design example about ring gyroscopes was the one made in [12]. The paper presents a80-μm-thick, 1.1 mm in diameter high aspect-ratio (20:1) polysilicon ring gyroscope (PRG). Adetailed analysis has been performed to determine the overall sensitivity of the vibrating ringgyroscope and identify its scaling limits. An open-loop sensitivity of 200 μV/deg/s in a dynamicrange of ±250 deg/s was measured under low vacuum level for a prototype device. Theresolution for a PRG with a quality factor (Q) of 1200, drive amplitude ( ) of 0.15 μm wasmeasured to be less than 1 deg/s in 1 Hz bandwidth, limited by the noise from the circuitry.The vibrating ring gyroscope, shown in Figure -5 [12], consists of a ring, eight semicircular 4support springs, and drive, sense and control electrodes. Symmetry considerations require at [32]
- 39. least eight springs to result in a balanced device with two identical elliptically-shaped flexuralmodes that have equal natural frequencies and are 45o apart from each other. The ring iselectrostatically vibrated into the primary flexural mode with fixed amplitude. Figure -5 The HARPSS Vibrating Ring Gyroscope – (a) SEM Image, (b) Electrode Voltages 4In this section, we will apply the model suggested by the authors of [4] to redesign the HARPSSGyroscope. 4.3.1. Mechanical DesignGeneral gyro specifications often include gyro size, environmental conditions (or applications),or sensitivity. We first select the ring structure radius ( = 550 μm) from the sizespecification and flexural and translation resonant frequencies ( = 29 KHz and =20 KHz)from the environment conditions (or applications) or g-sensitivity (sensitivity to linearacceleration). The g-sensitivity (in deg/s/(m/s2)2 is given by [9]: 4-10Where is the angular rate, is the gap between the capacitor’s electrode and the ring.The flexural resonant frequency (used for drive and sense modes) should lie well above thefrequency spectrum of the environmental vibration. The support beam radius ( = 235μm) is successively set to be from a half to a quarter of the as observed in Figure -4. 4Next, we adjust the effective mass [ ] and stiffness [ ] matrices in equation 4.2 to match thedecided flexural and translation resonant frequencies. The flexure mode is concerned with only4 springs, thus the flexure mode effective mass in the mass matrix is calculated as shown below: 4-11Where and are the effective mass of the ring frame and the supportsprings that is stretched horizontally for the flexure mode and can be considered as one third ofthe actual mass if the spring is stiff, and half of the mass if the spring is compliant [13]: 4-12 4-13 [33]
- 40. Where , are the width and the thickness of the structure, is the density = 2328Kg/m3. For the translational modes, the effective mass is approximately the sum of twohorizontal, two vertical and four 45o effective spring masses. In addition to the actual ring mass: 4-14Where and , can be considered as half of the spring’s mass (sincethey are very compliant).The stiffness matrix is calculated from the curved beam stiffness equations 2.10, 2.11, and 2.12): 4-15 4-16Where = 150 GPa, for , , and , = = 235 μm, and = at r = ,because the ring frame is considered as 2 parallel curved beams [12]. By dividing by , andequating the resulting expression to the square of the resonance angular flexure frequency, wecan obtain the width of the ring ( μm), substituting in mass and stiffness matrices toget and . The height of the ring is still not calculated becausethe resonant frequencies of the ring don’t depend on it [12]. is better to be set to a largevalue to reject out of plane modes, but this will require higher driving voltage to maintain thesame drive amplitude as shown in the Electric design part in 4.3.3, we will set the height to 80μm as the paper. 4.3.2. Finite Element SimulationAfter the previous first order analysis, a 2-D Finite Element Analysis (FEA) was carried out usingthe values calculated for ring and spring dimensions (Figure -6), by tuning the obtained values 4for the ring dimensions above, the FEA revealed 5 in-plane modes at μm and =235 μm. The first mode was torsional at about 10KHz (the outer ring is rotating about with itscenter in the middle of the inner circular post), the second two modes were translational ones atabout 20 KHz the fourth and fifth were flexure at approximately 28 KHz. The difference in theresonant frequencies between the model and FEA might have happened due to theapproximations done when calculating the effective masses and stiffness. The mode shapesweren’t very accurate due to asymmetries in the mesh which caused rotation of the principlemode axis. However, this won’t affect the resonant frequencies too much, and can be managedby balancing electrodes. 2-D FEA didn’t show out of plane modes, however we shouldn’t worryabout them since they are minimized due to the high aspect ratio of the device. An ANSYS scriptthat animates the mode shapes of the ring structure can be found in Appendix E. [34]
- 41. (a) (b) (c) (d)Figure -6 Finite Element Analysis of the HARPSS Ring Gyroscope – (a) X-Axis Translational Mode, (b) Y--Axis 4 Translational Mode, (c) Primary Flexural Mode, (d) Secondary Flexural Mode4.3.3. Electrical DesignThe driving specification of the electric design is the sensitivity (or may be the resolution)requirement. It is calculated from the capacitive change per angular rate which is given by [11]: 4-17where is the number of used sense electrodes, is the rest capacitance of the electrode, is the angular gain 0.37, is the quality factor, is the drive mode amplitude.Given that the required sensitivity is 0.12 fF/(deg/s), the number of electrodes to be used forsensing or driving is 2 (for each, see Figure -4), quality factor of 1200, electrode gap 4 (from equation 4.10), Polarization voltage of the ring , we can assumevalues for drive mode amplitude: , therefore the needed electrode capacitance . The electrode capacitance is given by: 4-18Therefore, we can let the height of the electrode , to have the angle . From the drive mode amplitude, the damping coefficient ( ) can be found: [35]
- 42. 4-19Therefore, the AC drive signal is equal to 15.5 mV. The value is too far from what wasmentioned in the paper (5-8mV) because the equations of motion were derived based on theusage of 2 driving electrodes at 0o and 180o as illustrated in Figure -4, while the operation 4mode of the gyroscope in [12] was different (Force to rebalance mode, see Figure -5 The 4HARPSS Vibrating Ring Gyroscope – (a) SEM Image, (b) Electrode Voltages - b). Table -1 4Summary of Design Parameters estimated by Model, FEA, and Achieved in the paper. Table -1 Summary of Design Parameters estimated by Model, FEA, and Achieved 4 Design Parameter Model FEA Achieved Flexure Mode Effective Mass 2.05x10-9 Kg 2.04x10-9 Kg Translational Mode Effective Mass 4.54x10-9 Kg Flexure Mode Stiffness 65.38 N/m 63.46 N/m Translational Mode Stiffness 74.84 N/m Flexure mode resonant frequency 28.38 KHz 28.08-28.17 KHz 28.3 KHzTranslational mode resonant frequency 20.44 KHz 19.26-19.36 KHz Ring and Spring Width 3.9 μm 4.0 μm 4 μm Electrode Gap Spacing 1.4 μm 1.4 μm Electrode Height 60.0 μm 60.0 μm Length of Electrode 148 μm 150.0 μm AC Signal Amplitude 15.5 mV 5-8 mVThe results in Table -1 shows that the approximations done to calculate the effective masses of 4the flexure and translational modes were acceptable for a first order hand analysis.4.4.Evaluation of Vibrating Ring Gyroscope4.4.1. Advantages of Ring ArchitectureThe vibrating ring structure has important features compared to other architectures. In a brief: It has a balanced symmetrical structure that is less sensitive to environmental vibrations. Since two identical flexural modes of the structure are used to sense rotation, the sensitivity of the sensor is amplified by the quality factor of the structure (Eq. 4.17). The vibrating ring is less temperature sensitive since the two flexural vibration modes are affected equally by temperature [8]. Any frequency mismatch between the drive and sense resonance modes that occurs during fabrication process (due to mass or stiffness asymmetries) can be electronically compensated by use of the tuning electrodes that are located around the structure [8]. More resistive to ambient vibrations [9,10].4.4.2. Disadvantages of Ring Architecture Lower pick off capacitance compared to tuning fork gyroscopes [8]. Requires a high aspect ratio fabrication process; thin tall structure is needed to obtain reasonable actuation voltages and low resonant frequencies [8]. [36]
- 43. 5. Interface Electronics5.1.System ComponentsSimple generalized model of a gyroscope with the electronic interface necessary to produce thefinal output. An oscillator establishes the drive oscillation at the drive resonance frequency, andthe Coriolis readout interface detects and amplifies the Coriolis acceleration. A demodulatordemodulates the angular rate signal from the Coriolis acceleration, and a low-pass filter removesother unwanted signals out-side the desired frequency band, from the final output. [14] Figure -1 generalized model of a gyroscope with the electronic interface 55.1.1. Drive loopThe drive loop electronics are responsible for starting and sustaining oscillations along thereference axis at constant amplitude. It is essential that a constant drive amplitude bemaintained, as any variation in the drive amplitude manifests itself as a change in velocity of themechanical structure (along the driven axis). Velocity fluctuations modulate the sensor outputand can result in false or inaccurate rate output.There are two approaches to implement the drive loop, both of which have been implemented inthis work:• An electromechanical oscillator: Here the drive mode oscillations are started and sustainedby using a positive feed-back loop that satisfies the Barkhausen’s criteria (Loop gain = 1, LoopPhase shift = 0o) based on the natural frequency of the mechanical gyrpscope.• A Phase-Locked Loop(PLL) based approach: Here the reference drive vibrations are set-upusing a phase locked loop (PLL). The PLL center frequency and capture range are set close to thedrive resonant frequency of the gyroscope. On power up, the PLL locks on to the output of thefront-end. The PLL output is amplified or attenuated to achieve the desired voltage amplitudeand used to drive the microgyroscope. [15] [37]

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