Undergraduate Graduation Project Documentation - Faculty of Engineering, Ainshams University, Egypt.

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 by
Abdel 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. ABSTRACT
The report presents our surveys and work about MEMS Vibratory Gyroscope Devices and
Systems 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. The
discussion 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 physical
principle of operation, driving and sensing, mechanical and electrical design basics, and
fabrication 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, design
concept, and the equation of motion. This chapter will contain also a hand analysis for a TFG
design 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). The
features of the architecture, the vibrating modes of the structure and some of the vibration
induced errors in the vibrating ring gyroscope. At the end of the chapter, a famous design done
by Dr. Najafi and Dr. Ayazi (A HARPSS Polysilicon VRG) is redesigned using another model, finite
element analysis (FEA) verified the design.
Owner: A. M. El-Sayed.
Chapter 5 explores the interfacing system and its main loops and types, also this chapter
contain 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 capacitive
interface, the analog to digital converting, then the demodulation, the drive loop automatic
control 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 Contents
1. INTRODUCTION ................................................................................................................... 1
1.1. MEMS Technology............................................................................................................................... 1
1.1.1. MEMS Applications...................................................................................................................... 1
1.2. Gyroscopes.......................................................................................................................................... 1
2. BACKGROUND AND THEORY ............................................................................................ 3
2.1. Coriolis Force....................................................................................................................................... 3
2.2. Vibrating Two Modes Gyroscope ........................................................................................................ 3
2.2.1. Drive Mode Operation ................................................................................................................. 4
2.2.2. Sense Mode Operation ................................................................................................................ 4
2.3. Mechanical Design of MEMS Gyroscopes ............................................................................................ 5
2.3.1. Flexure Elements ......................................................................................................................... 5
2.3.2. Mode Coupling ............................................................................................................................ 9
2.4. Electrical Design of MEMS Gyroscopes .............................................................................................. 10
2.4.1. Capacitive Detection.................................................................................................................. 10
2.4.2. Electrostatic Actuation............................................................................................................... 12
2.5. 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 ............................................................................................................................... 16
3. TUNING FORK GYROSCOPE ............................................................................................ 21
3.1. Features of Tuning Fork Architecture ................................................................................................ 21
3.2. Design Concept ................................................................................................................................. 21
3.3. Equations of Motion.......................................................................................................................... 22
3.3.1. Drive-Mode Actuation ............................................................................................................... 24
3.3.2. Sense-Mode Detection: ............................................................................................................. 24
3.4. Mechanical Design ............................................................................................................................ 25
3.5. Finite Element Simulation ................................................................................................................. 25
3.6. Summary of Tuning Fork Gyroscope .................................................................................................. 26
3.6.1. Advantages of Tuning Fork Gyroscope ....................................................................................... 26
3.6.2. Disadvantage of Tuning Fork Gyroscope..................................................................................... 26
4. 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 .................................................................................................................. 29
4.2. 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............................................. 32
4.3. 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 ........................................................................................................................ 35
4.4. Evaluation of Vibrating Ring Gyroscope ............................................................................................ 36
4.4.1. Advantages of Ring Architecture ................................................................................................ 36
4.4.2. Disadvantages of Ring Architecture............................................................................................ 36
5. INTERFACE ELECTRONICS ............................................................................................. 37
5.1. System Components ......................................................................................................................... 37
5.1.1. Drive loop.................................................................................................................................. 37
5.1.2. Sense loop................................................................................................................................. 38
5.2. Overview of sense and drive electronics ........................................................................................... 38
6. BUILDING BLOCKS ........................................................................................................... 43
6.1. 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 ......................................................... 52
6.2. 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 ...................................................................................................... 71
6.3. Demodulation block .......................................................................................................................... 76
6.3.1. Coherent detection ................................................................................................................... 76
6.4. 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 ......................................................................................................................... 102
6.5. Automatic Gain Control Loop (AGC) ................................................................................................ 105
6.5.1. Design 1 .................................................................................................................................. 106
6.5.2. Design 2 .................................................................................................................................. 111
6.5.3. AGC Sum up ............................................................................................................................ 112
6.6. Design Flow will be used ................................................................................................................. 113
6.6.1. Analog ..................................................................................................................................... 113
6.6.2. Digital...................................................................................................................................... 114
7. CONCLUSION AND FUTURE WORK ............................................................................ 115
BIBLIOGRAPHY ....................................................................................................................... 116
APPENDICES ............................................................................................................................ 119
Appendix A – ANSYS Script to Calculate the Stiffness of a Fixed-Guided Straight Beam ............................... 119
Appendix B - ANSYS Script to Calculate the Stiffness of Fixed Guided Curved Beam .................................... 121
B.1 Horizontal and Vertical Stiffness ......................................................................................................... 121
B.2 45o Stiffness ....................................................................................................................................... 122
Appendix C - ANSYS Script for Modal Analysis of Tuning Fork Gyroscope .................................................... 123
Appendix D - Plotting Vibrations of the Ring Structure using SciLAB ............................................................ 125
Appendix E - ANSYS Script for Modal Analysis of Ring Gyroscope ................................................................ 126
Appendix F: Project Timeline ....................................................................................................................... 128
[V]

7. 1. Introduction
1.1.MEMS Technology
Micro-electromechanical systems (MEMS) technology is a process technology used to create
tiny integrated devices or systems that combine mechanical and electrical components. They
are fabricated using integrated circuit (IC) batch processing techniques and can range in size
from 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 device
electronics are fabricated using ‘computer chip’ IC technology, the micromechanical components
are fabricated by sophisticated manipulations of silicon and other substrates using
micromachining processes. Processes such as bulk and surface micromachining, selectively
remove parts of the silicon or add additional structural layers to form the mechanical and
electromechanical components.
MEMS technology has several distinct advantages as a manufacturing technology. First, the
interdisciplinary nature of MEMS technology and its micromachining techniques, as well as its
diversity of applications has resulted in an unprecedented range of devices and synergies across
previously unrelated fields (for example biology and microelectronics). Second, MEMS with its
batch fabrication techniques enables components and devices to be manufactured with
increased performance and reliability, combined with the obvious advantages of reduced
physical size, volume, weight and cost. These factors make MEMS potentially a far more
pervasive technology than integrated circuit microchips.
1.1.1. MEMS Applications
MEMS applications are diverse; the oldest application is pressure sensors [1]. The other major
sensor market is the inertial sensors. This market has been dominated by the automotive
industry, but recently the reduction in price has enabled adoption of MEMS inertial sensors
(accelerometers and gyroscopes) in consumer devices like digital cameras, mobile phones, and
Laptops.
For average consumers, the inkjet print heads may be the most familiar micro-device. Each
replacement inkjet cartridge has a micromachined inkjet nozzle head. The inkjet print heads are
frequently regarded as the largest MEMS market in terms of revenue. Texas Instruments holds
the key patents of the field of digital micro-displays (DMD). In projection displays, the high
contrast ratio of mechanically actuated mirrors enables the micro-mirrors to compete against
the common LCD technology.
Silicon microphones are the latest entry to the mass market. The growth is driven by cell phone
industry that is increasing rapidly. The microphones are an encouraging example of a MEMS
product that only a few years ago was deemed too expensive, but now gaining a market share
rapidly. [1]
1.2.Gyroscopes
The word gyroscope was coined by the French scientist Leon Foucault and is derived from the
Greek words “Gyros” meaning rotation, and “Skopien” meaning to view. Simply, gyroscope is the
sensor that measures the rate of rotation of an object. It can be used for example for inertial
navigation, image stabilization, and automotive chassis control and rollover detection.
[1]

8. Historically, the angular rate has been measured with rotating
wheel gyroscope. The spinning wheel conserves the angular
momentum resisting the change in the rotation axis orientation.
The angular velocity can now be sensed by measuring the force
on the spinning wheel due to rotation [2].
Mechanical gyroscopes are comprised of a spinning wheel
mounted on two gimbals which allow rotation along all three
axes. Due to conservation of angular momentum, the spinning
wheel will resist change in orientation. Hence when a
mechanical gyroscope is subjected to a rotation, the wheel will
remain at a constant global orientation and the angles between
the adjacent gimbals will change. To measure the orientation of
the device, the angles between the adjacent gimbals is read Figure ‎ -1 One of the first examples of
1
the gyrocompass
using angle pick-offs. It must be noted that a mechanical
gyroscope measures orientation directly. The disadvantage of mechanical gyroscopes is that
they comprise
of 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 which
states that a certain rate of rotation induces a small difference between the time it takes light to
traverse the ring in the two directions. These gyroscopes are not subject to a mechanical wear
and are the most precise ones [1]. Consequently, they are the most expensive gyroscopes, and
are used in aircraft navigation systems and missile guidance.
MEMS gyroscopes, fabricated using silicon micromachining technology, have low part counts
and are relatively cheap to manufacture in commercial quantities. They enable new applications
that are not possible with the classic optical or mechanical gyroscopes. Nearly all MEMS
gyroscopes are based on two orthogonal vibration modes. The drive-mode is orthogonal to the
sense-mode meaning that the two modes do not normally interact and the drive-mode
movement does not result in movement in sense-mode direction. The resonator is excited to
vibrate 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 is
proportional to the angular rotation rate. [2]
Figure ‎ -2 The Operation Principle of Vibrating Two Mode Gyroscope
1
[2]

9. 2. Background and Theory
The underlying physical principle of vibratory gyroscopes is that a vibrating object tends to
continue vibrating in the same plane as its support rotates. This device is also known as a
Coriolis vibratory gyroscope because it is based on the principle of “Coriolis Effect”.
2.1.Coriolis Force
The Coriolis force is the perpendicular deflection of a moving element that arises in connection
with 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 acted
on the particle, it would have reached the point ‘B’. However, the wheel has rotated, so the
particle will not reach the point ‘B’, but a point ‘A’. [2]
Figure ‎ -1 Illusration of Coriolis Effect
2
The vector formula for the magnitude and direction of the Coriolis acceleration is [1]:
⃗⃗⃗⃗ ⃗ 2-1
where ⃗⃗⃗⃗ is the acceleration of the particle in the rotating system (coriolis acceleration), is the
velocity of the particle in the rotating system, and ⃗ is the angular velocity vector of the wheel
which has magnitude equal to the rotation rate ω and is directed along the axis of rotation of the
rotating reference frame, Thus, the coriolis force ( ⃗⃗⃗ ) acting on a particle of mass is:
⃗⃗⃗ ⃗ 2-2
2.2.Vibrating Two Modes Gyroscope
The basic architecture of a vibratory gyroscope is comprised of a drive-mode oscillator that
generates and maintains a constant linear or angular momentum, coupled to a sense-mode
Coriolis accelerometer that measures the sinusoidal Coriolis force induced due to the
combination of the drive vibration and an angular rate input. The vast majority of reported
micromachined rate gyroscopes utilize a vibratory proof mass suspended by flexible beams
above a substrate. The primary objective of the dynamical system is to form a vibratory drive
oscillator, coupled to an orthogonal sense accelerometer by the Coriolis force. The drive mode is
orthogonal to the sense mode means that the two modes don’t normally interact [2].
[3]

10. 2.2.1. Drive Mode Operation
The Coriolis Effect is based on conservation of momentum; every gyroscopic system requires a
mechanical subsystem that generates momentum. In vibratory gyroscopes, the drive-mode
oscillator, which is comprised of a proof-mass driven into a harmonic oscillation, is the source of
momentum. 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 drive
damping consisting of viscous and thermoelastic damping. With a sinusoidal drive-mode
excitation force, the drive equation of motion along the x-axis becomes:
̈ ̇ 2-3
The scale factor of the gyroscope is directly proportional to the drive-mode oscillation
amplitude. Therefore, the drive mode is usually excited at resonance to obtain maximum
displacement with small driving force (lower actuation voltage).
It is extremely critical to maintain a drive-mode oscillation with stable amplitude, phase and
frequency. Self-resonance by the use of amplitude regulated positive feedback loop (Figure ‎ -2)
2
is a common and convenient method to achieve a stable drive-mode amplitude and phase. The
positive feedback loop destabilizes the resonator, and locks the operational frequency to the
drive-mode resonant frequency. An Automatic Gain Control (AGC) loop detects the oscillation
amplitude, compares it with a reference amplitude signal, and adjusts the gain of the positive
feedback to match the reference amplitude. Operating at resonance in the drive mode also
allows 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 Operation
The Coriolis response in the sense direction is best understood starting with the assumption that
the 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 that
excites the sense-mode oscillator is:
2-4
where is the portion of the driven proof mass that contributes to the Coriolis force. Similar to
the drive-mode oscillator, the sense-mode oscillator is also often a 1-DOF resonator, the sense
mode equation of motion is:
̈ ̇ 2-5
Thus, 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 [ ] are
equal to zero, this means that no mode coupling happens except by the influence of the Coriolis
effect.
2.3.Mechanical Design of MEMS Gyroscopes
2.3.1. Flexure Elements
In linear micromachined gyroscopes, the suspension systems are usually designed to be
compliant along the desired motion direction, and stiff in other directions. Most suspension
systems utilize narrow beams as the primary flexure elements, aligning the narrow dimension of
the beam normal to the motion axis.
2.3.1.1. Fixed Guided Linear Beam
In purely translational modes, the boundary conditions of the beams that connect the
components of the gyroscopes are most commonly the fixed-guided end configuration (Figure
2
‎ -3), in which the moving end of the beam remains parallel to the fixed end. Many complete
gyroscope 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 Condition
If 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 along
the three principle axes become (assuming a linear case) [2]:
[5]

12. 2-7
2-8
2-9
For 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 from
equation 2.7 to be 1.92N/m. However this stiffness changes with the amount of deflection
practically 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 is
F/x approximately equals 1.92N/m. Performing nonlinear analysis, the stiffness reached
4.86N/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 5.00
Displacement (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
2
An ANSYS script for plotting the load-deflection graph is shown in appendix A.
2.3.1.2. Curved Beam
Curved (or semicircular) beams are widely used in vibrating ring gyroscopes (to be explained in
chapter 4. The stiffness of a curved beam is highly dependent on the direction of the applied
force. 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 is
calculated from equation 2.10 to be 16.573N/m, in vertical direction from equation 2.11 is
3.139N/m, and that in the 45o direction is 9.856N/m. However this stiffness also changes with
the amount of deflection practically, as in the case of straight fixed-guided beams, 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 FEA (Figure ‎ -6), using a
2
linear 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 applied
force value was 452.59μm leading to KVA = 2.21N/m, and in 45o direction the deflection was
106.445 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
2
ANSYS scripts to generate the previous plots are found in appendix B.
2.3.2. Mode Coupling
There are two types of mode coupling in MEMS gyroscopes; the first is the desired one which
arises from Coriolis force, and the designer aims to magnify it, the second is an undesired one
which arises from non-idealities. In reality, fabrication imperfections result in non-ideal
geometries in the gyroscope structure, which in turn causes the drive oscillation to partially
couple into the sense-mode. Considering the relative magnitudes of the drive and sense
oscillations, even extremely small undesired coupling from the drive motion to the sense-mode
could completely mask the Coriolis response.
Equation 2.6 describes an ideal gyroscope, where the mode coupling happens only due to
Coriolis force, the practical equation of motion of a vibratory two modes gyroscope can be
written as:
̈ ̇
[ ][ ] * +[ ] [ ]* + [ ] 2-13
̈ ̇
Where and in the damping matrix represents the coefficients of the anisodamping forces
in 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 elements
in real implementations of vibratory gyroscopes have elastic cross-coupling between their
principal axes of elasticity). [2]
[9]

16. Since the oscillation amplitudes in the sense-mode are orders of magnitude smaller than the
drive-mode, the coupling due to and in the drive dynamics is negligible. The impact of
anisodamping 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 Coriolis
response ( ̇ ) and the mechanical quadrature ( ), therefore, the quadrature signal
can 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 same
frequency and phase). However, the ansiodamping component is in phase with the Coriolis
response, therefore it can’t be removed during demodulation, and it should be minimized in the
design of the gyroscope itself or by vacuum packaging of the device.
2.4. Electrical Design of MEMS Gyroscopes
Micromachined gyroscopes are active devices, which require both actuation and detection
mechanisms. Various vibratory MEMS gyroscopes have been reported in the literature
employing a wide range of actuation and detection methods. For exciting the gyroscope drive
mode oscillator, the most common actuation methods are electrostatic, piezoelectric, magnetic
and 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 offer
several benefits compared to other sensing and actuation means, especially due their ease of
implementation. Capacitive methods do not require integration of a special material, which
makes them compatible with almost any fabrication process. They also provide good DC
response and noise performance, high sensitivity, low drift, and low temperature sensitivity
2.4.1. Capacitive Detection
Parallel-plate capacitors can be mechanized in several ways to detect deflection. For a generic
parallel-plate electrode plate with a gap d and overlap area Aoverlap , the capacitance is
2-14
Figure ‎ -7: Variable Gap Capacitor
2
where is the dielectric constant of the material between the plates. Each parameter in this
expression 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 with
deflection. In variable area capacitors, the motion is parallel to the plane, which results in a
change 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 are
variable gap and variable area capacitors, which are summarized below.
2.4.1.1. Variable Gap Detector
Variable-gap capacitors are the most widely used electrode type
for detection of small displacements. When the parallel plates are
oriented normal to the motion direction, deflections cause a
change in the gap .
It should be noticed that capacitance is a nonlinear function of
displacement in variable-gap capacitors. However, for very small
deflections relative to the initial gap, the capacitance change is
linearized. Denoting the displacement in the motion direction as
and assuming << , the capacitance change in a variable-gap Figure ‎ -8 Variable Gap Detector
2
capacitor with an overlap area becomes:
2-15
Thus, small gap changes could result in high capacitance changes, providing very large
sensitivity.
2.4.1.2. Variable Area Detector
Variable area capacitors are ideal when the detected motion magnitudes are larger, especially
either when variable gap capacitors become significantly nonlinear, or deflections are larger
than a minimum gap. Since the overlap area is proportional to both dimensions in the plate
plane, capacitance change is purely linear with respect to motion parallel to the plates. Denoting
x as the displacement in the motion direction parallel to the plates, the capacitance change
becomes
2-16
Figure ‎ -9: Variable Area Detector
2
Table 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 capacitor
The change in capacitance is a result of change The change in capacitance is a result of change
in 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 a
large displacement and get a high sensitivity. However, we can use variable area capacitor for a
larger travelling distance in the expense of sensitivity.
2.4.2. Electrostatic Actuation
Electrostatic or capacitive actuation is based on the attraction of electric charges [2]. As the
device size is reduced to the micro-scale, this force become significant. The capacitive actuators
are easily fabricated and consume no DC power. There are two main types of capacitive
actuators; the closing gap actuator, and the variable area actuator. The variable area actuators
are much common in gyroscopes, because the electrostatic force varies linearly with the moved
distance. However, in some cases (like the vibrating ring gyroscope), the closing gap actuator is
used.
2.4.2.1. Closing Gap Actuator
Consider the closing gap actuator in Figure ‎ -10 Closing Gap Actuator, the electrostatic force of a
2
parallel plate capacitor (which is a good model for many MEMS actuators) is derived from the
energy stored, and is given by:
2-17
Where is the polarization voltage applied on the actuator, is the overlapping area between
the two electrodes, is the initial gap between the electrodes. The restoring force generated in
the spring of stiffness due to displacement is given by:
2-18
At equilibrium, neglecting the weight of the electrodes, the electrostatic force is
equal to the spring force which makes the voltage required to move a distance
equals:
√ 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 the
movable electrode accelerates till it sticks with the fixed one, and the structure fails. This
condition is called: the pull-in condition, and the pull-in distance is considered the maximum
distance that the actuator can move, it’s proved in [1] that , which means that the
closing gap actuator can only move one third of the gap between the electrodes. It
should be considered by the designers that the drive mode amplitude shouldn’t exceed
the pull-in distance. Figure ‎ -11 shows the electrostatic and spring forces of an actuator of
2
area = 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
2
2.4.2.2. Variable Area Actuators
Variable-area actuators aim to linearize the capacitance change versus displacement, in order to
achieve constant electrostatic force with respect to displacement. The inter digitated comb-drive
structure is based on generating the actuation force through a series of parallel plates sliding
parallel to each other, without changing the gap between the plates. The electrostatic force
generated in the x-direction for two parallel plates as in Figure ‎ -12 is
2
2-20
It should be noticed that this force is independent of displacement in the x-direction and the
overlap length of the capacitor plates, x0.
[13]

20. Figure ‎ -12: Variable-area electrostatic actuator model.
2
Inter digitated comb-drives based on variable-area actuation are one of the most common
actuation structures used in MEMS devices. The primary advantages of comb-drives are long-
stroke actuation capability and the ability to apply displacement-independent forces, which
provide highly stable actuation.
In a comb-drive structure made of N fingers, each finger forms two parallel-plate pairs, and the
total electrostatic force generated in the x-direction becomes
2-21
Where z0 is the structure thickness and y0 is the distance between the fingers.
2.4.2.3. Balanced Actuation
In MEMS gyroscope, we always want to induce harmonic motion. Therefore, when a sinusoidal
net actuation force is desired, the drive force can be linearized with respect to the actuation
voltages by appropriate selection of voltages applied to the opposing electrode sets [2]. The net
electrostatic force generated by two opposing capacitors C1 and C2 is:
2-22
A balanced actuation scheme is a common method to linearize the force with respect to a
constant bias voltage and a time-varying voltage . The method is based on applying
to one actuator, and to the opposing actuator. Assuming two
electrodes are identical, and the DC voltage is much greater than the time varying component,
the net electrostatic force reduces to:
[ ] 2-23
2.5.Fabrication Technologies
2.5.1. Microfabrication Techniques
Microfabrication describes processes of fabrication of miniature structures, of micrometer sizes
and smaller. Integrated Circuit (IC) fabrication is the earliest microfabrication processes used.
Inertial sensors require moving parts to detect inertial phenomena. Micromachining
technologies have revolutionized inertial sensing by allowing fabricating moving mechanical
systems 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 of
thin structural layers on a substrate [2].
2.5.1.1. Deposition
The process flow of micromachining fabrication starts with a
blank wafer. The intention is to pattern a multiple structural
layer for moving structures, interconnect, electrode areas, or
dielectric layers for electrical isolation using successive
deposition and patterning of these layers.
Depending on the material, layer thickness, or conformal
coverage requirements, different deposition techniques may be
used such as Chemical Vapor Deposition (CVD), Physical Vapor
Deposition (PVD), or electroplating.
2.5.1.2. Photolithography
Photolithography, also known as lithography, is the process of
patterning parts of a thin film or the bulk of a substrate. Prior to Figure 2-14: Microfabriaction
processing, a photolithography mask that carries the wafer-level Process
layout of a layer is generated. Then the image on the mask is
projected onto a photosensitive material deposited on the wafer, commonly known as
photoresist.
2.5.1.3. Etching
Etching transfers the pattern formed by photolithography into the actual structural materials
and 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 vapor
phase etchant or reactive ions. In MEMS, to determine the required etching method many factors
are involved such as the desired sidewall and bottom surface profiles, isotropy, or stiction
issues. [2]
2.5.2. Bulk-Micromachining Processes
Micromachining processes are usually divided, depending on the structural layer forming
technique, into two main categories: Surface micromachining and Bulk micromachining.
Traditionally, bulk micromachining implies the use of subtractive processes to pattern thick
structural layers. In most bulk micromachining process, two or more wafers are bonded, and the
moving structures are made out of the whole thickness of a silicon wafer.
Bulk micromachining offers many advantages for inertial micromachined devices, since it
provides thick structural layers. Larger device thickness increases the mass and overlap area of
capacitive electrodes, directly improving gyroscope performance. Thicker suspension beams
provide higher out-of-plane stiffness, which reduces shock and vibration susceptibility, and
minimizes the risk of stiction to the substrate. It also allows the use of single crystal silicon as the
device material, which provides excellent mechanical stability. [2]
The implementing of bulk micromachining can be done by many different fabrication
technologies:
[15]

22. 2.5.2.1. SOI-Based Bulk Micromachining
Silicon-on-Insulator (SOI) wafers are excellent starting materials for bulk micromachining. The
silicon device layer comes bonded on an insulator layer. Electrically isolated and mechanically
anchored free-standing structures can be formed simply by patterning the device layer and the
oxide layer underneath.
Figure ‎ -13: An SOI-based bulk-micromachined gyroscope, diced and released.
2
2.5.3. Surface-Micromachining Processes
While bulk micromachining uses subtractive processes to pattern thick structural layers, surface
micromachining is in essence an additive technique. It relies on successive deposition and
patterning of thin structural layers on the surface of a substrate, rather than etching thick bulk
layers.
In surface micromachining, complex three-dimensional devices are built by depositing multiple
stacks of alternating structural layers and sacrificial layers. Each sacrificial layer supports the
structural layer above it during fabrication, and separates it from the other layers below. At the
end of the process, the sacrificial layers are selectively etched away, releasing the structural
layers. [2]
2.5.4. SOI-MUMPs
The following is a general process description and user guide for Silicon-On-Insulator Multi-User
MEMS Processes (SOIMUMPs). It is a simple 4-mask level SOI patterning and etching process
derived from work performed at MEMSCAP.
The process begins with 150mm n-type double-side polished Silicon on Insulator wafers. A
phosphosilicate glass layer (PSG) is deposited, and the wafers are annealed at 1050°C for 1 hour
in Argon to drive the Phosphorous dopant into the top surface of the Silicon layer. The PSG layer
is subsequently removed using wet chemical etching [3].
[16]

23. Figure ‎ -14
2
The wafers are coated with negative photoresist and lithographically patterned by exposing the
photoresist with light through the first level mask (PADMETAL), and then developing it.
Figure ‎ -15
2
The wafers are coated with UV-sensitive photoresist and lithographically patterned by exposing
the photoresist to UV light through the second level mask (SOI), and then developing it. The
photoresist in exposed areas is removed, leaving behind a patterned photoresist mask for
etching.
[17]

24. Figure ‎ -16
2
A front side protection material is applied to the top surface of the patterned Silicon layer. The
bottom side of the wafers are coated with photoresist and the third level (TRENCH) is
lithographically patterned.
Figure ‎ -17
2
The 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
2
A separate silicon wafer is used to fabricate a shadow mask for the Metal pattern. The shadow
mask wafers are coated with photoresist and the fourth level (BLANKETMETAL) is
lithographically patterned.
Figure ‎ -19
2
The shadow mask is aligned and temporarily bonded to the SOI wafer. The Blanket Metal layer is
deposited through the shadow mask.
[19]

26. Figure ‎ -20
2
The shadow mask is removed, leaving a patterned Metal layer on the SOI wafer.
Figure ‎ -21
2
[20]

27. 3. Tuning Fork Gyroscope
3.1.Features of Tuning Fork Architecture
For many applications, gyroscopes are subject to a wide variety of changing environmental
conditions such as temperature, pressure, and ambient vibrations. The robustness of the sensor
to these external influences during operation is critical for adequate performance. A level of
robustness is commonly achieved through electronic control systems, such as a temperature
compensation circuit which post-processes the output of the mechanical sensor depending upon
temperature or an automatic mode matching controller. Robustness to ambient vibrations,
however, is generally addressed by the mechanical design through the use of tuning fork driving
architectures. 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 Concept
The mechanical architecture of the tuning fork gyroscope, Figure ‎ -1, comprises of two proof-
3
masses, supported by a network of flexural springs and anchored at a central post.
Figure ‎ -1: Schematic diagram of the TFG
3
The 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 is
formed by the two linearly coupled tines moving in anti-phase Figure ‎ -2 [5]. The gyroscope is
3
electro-statically driven into anti-phase motion using driving voltages imposed across the
differential 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-mode
vibrations which are capacitively detected using differential parallel plate electrodes on the
sense-mode shuttles [6].
[21]

28. Figure ‎ -2: In-plane operating flexural modes. (Left) Drive resonant mode along the x-axis.
3
The anchor design of the TFG satisfies two critical properties: mechanical coupling and resonant
mode 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.
3
Due 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
3
two other in-plan modes.
Figure ‎ -3: In-plane pseudo-operating flexural modes. (Left) Pseudo-drive resonant mode
3
The flexural spring must be designed to ensure large mobility along both axes. To this effect, a
fish-hook architecture was adopted which ensures that the mode shapes have two-directional
flexibility.
3.3.Equations of Motion
The TFG can be conceptualized as a coupled resonator system, with the rotation induced Coriolis
force being the coupling agent between the two resonant operating modes. The dynamics of the
device 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, which
can be modeled as a mass-spring-damper system consisting of the drive proof-mass m, the
drive-mode suspension system providing the drive stiffness k, and the drive damping c
consisting 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
‎ -1
With the definition of the drive-mode resonant frequency wd and the drive-mode Quality factor
Qd the amplitude and phase of the drive-mode steady-state response
becomes:
3
‎ -2
√ ( )
at w = wd the amplitude becomes
3
‎ -3
where
√ 3
‎ -4
and
3
‎ -5
A rotation signal along the normal axis (z-axis) of the results in a Coriolis induced acceleration
on the individual proof-masses along the sensitive axis (y-axis). The magnitude of the Coriolis
acceleration is given by the vector cross product of the input rotation rate vector and the
velocity of the proof mass (2Ω x V).
Considering that the proof-masses are oscillating in a sinusoidal fashion at the drive-mode
resonance, the expression for the Coriolis acceleration along the sense-axis is given by:
3
‎ -6
Where Ωz is the input rotation rate, ‘VDrive-x’ is the velocity of the drive resonant mode, ‘XDrive’ is
the amplitude of drive-mode oscillation and ‘ωDrive’ is the drive-mode resonant frequency. And
from Newton’s second law we know that so we can describe the equation of motion of
the 1-DOF sense mode oscillator by the following:
̈ ̇ 3
‎ -7
The 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
‎ -9
and
[23]

30. 3
‎ -10
The rotation-induced proof-mass displacement along the y-axis causes the gap between the
parallel plate sense electrode and the proof-mass to change. This change in capacitive gap is
proportional to the input rotation rate, and is detected by means of transimpedance front-end
electronics.
3.3.1. Drive-Mode Actuation
A key parameter that determines both the resolution and the sensitivity of a micromachined
vibratory gyroscope is the drive amplitude. For this reason, comb-drive electrodes were chosen
ahead 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 displacement
before pull-in.
The overall capacitance of the comb-drive electrode is expressed as:
( ) 3
‎ -11
where ‘N’ indicates the number of combs, ‘h’ refers to the
comb-thickness, ‘wo’ is the initial overlap, ‘g’ is the adjacent
gap size, ‘x’ is ‘y’ represents the transversal displacement
along the sense axis (y-axis) which may be caused either by
Coriolis or quadrature errors [7].
We also know that
3
‎ -12
From equation 3-11 and equation 3-12 we can get the Figure ‎ -4: comb-drive electrode
3
following:
3
‎ -13
We can neglect y with respect to g as y << g, equation 4-13 becomes:
3
‎ -14
3.3.2. Sense-Mode Detection:
Based on the comparison in Table ‎ -1 and as we are not in need for a large displacement,
2
variable-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
‎ -15
where ’ls’ is the overlap length between the two electrodes, ‘t’ is the thickness, ‘g s’ is the initial
gap between the electrodes, and ‘y’ is the lateral displacement amplitude along the sense axis (y-
axis).
[24]

31. 3.4.Mechanical Design
Consider the design shown in Figure ‎ -1; we need to determine all the dimensions of the proof-
3
masses, actuators, and detectors. After that, we can calculate the change in the actuation and
detection capacitances. The intention of this design is to get a change in the drive mode
capacitance differentially and change in the sense mode capacitance
differentially 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 specifications
and design rules. Then determine the drive and sense frequencies at which the device will
operate ( = 7.5 KHz and = 7.7 KHz).
We can now calculate the stiffness of the flexure beams along the drive axis (x-axis) using
equation 3-4 we get .
Assuming the gap between the fingers in the comb-drive actuators to be g = 3 um and the
thickness of one finger b = 3 um taking into consideration that they must be greater than the
minimum feature length given in the design rules.
From the previous assumed dimensions of the comb-actuator we find that the maximum
allowable number of fingers is N = 38 finger. We use Na = 27 finger for detection and Nd = 11 for
actuation.
Now by applying a volt V = 9.5 V we get the drive force using equation 3-13
and 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 the
following equation:
3
‎ -16
From 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 causes
the gap between the parallel plate sense electrode and the proof-mass to change. This change in
the gap causes a change in the sense capacitance. To calculate this change we need first to
calculate the displacement in y-direction. After that, we found the change in capacitance by using
the following equation:
3
‎ -17
where and .
After some calculation using equation 3-8 with we get y = 0.0037 um. As a result we
get differentially.
3.5.Finite Element Simulation
After the previous first order analysis, a 2-D Finite Element Analysis (FEA) was carried out using
the values calculated in the previous section, the FEA revealed 4 in-plane modes. The first mode
was pseudo drive mode at about 7.48 KHz; the second mode was the anti-phase drive mode at
about 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
3
between the model and FEA might have happened due to the approximations done when
calculating the masses and stiffness. An ANSYS script that animates the mode shapes of the
tuning 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
3
3.6.Summary of Tuning Fork Gyroscope
The 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 GYROSCOPE
The 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 shell
gyroscope can be explained as the ring vibrates in an elliptical (flexural) manner that have two
nodal diameters. When the structure is rotated, the node lines lag behind the rotation (Figure
4
‎ -1) [8]. Therefore, the principle of operation of a vibrating ring gyroscope will be: exciting the
ring to vibrate elliptically (Drive mode), then monitoring the lag of the nodes capacitively.
Figure ‎ -1 Vibrating Ring and lagging nodes
4
4.1.Vibration Modes of the Ring Structure
The 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 shaped
vibration modes. Because of their mode shapes, the locations of maximum motion or antinodes
for 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 Model
Any general vibration-induced displacement of an elastic body ( ⃗ can be expressed by the
linear combination of its normal vibration modes :
⃗ ∑ 4-1
where p is the independent position coordinate which can be expressed by Cartesian
coordinates (x and y) or by cylindrical coordinates (radial and tangential coordinates). The
equation includes generalized (modal) coordinates (mode amplitudes) (i.e. ) and mode
shape functions (i.e. ). [2]
4.1.2. Mode Shapes
There are several modes for the ring structure (out of plane, torsional, translational, and flexure
modes). The most important mode shapes, as mentioned by [9], are four. The first two modes
are translation modes in the x and y directions ( ), and their radial/tangential
components are:
X-axis translation mode:
[27]

34. 4-2
Y-axis translation mode:
4-3
where θ is an independent spatial coordinate (angle) describing position around the ring. The
second two modes are elliptical-shaped flexural modes ( ), and their radial/tangential
components are:
Drive axis flexure mode:
4-4
Sense axis flexure mode:
4-5
Where determines the angle between the principle mode axis and the horizontal axis [8], to
simplify the math, we take , the radial components of the mode shapes are plottet in
Figure ‎ -2. It’s clear that the each of the two flexure modes has its nodes on the antinodes of the
4
other. Figure 4-3 shows the vibrations of the ring. A Scilab code for these plots can be found in
Appendix 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
4
4.1.3. Equations of Motion
We Consider the ring structure in Figure ‎ -4 Conceptual view of a MEMS ring gyroscope, the ring
4
structure is attached to an anchor by 8 symmetric curved beams, the ting is driven into flexure
mode horizontally by two electrodes at 0o and 180o, and the two electrodes at 45o and 225o are
used for sensing (Open Loop Operation).
Unlike the non-degenerate gyroscopes, like tuning fork gyros, ring gyroscopes cannot be
analyzed using simple lumped models because the mass and the stiffness of the ring gyro are
distributed along the ring. The equations of motion of the ring gyroscope structure can obtained
by deriving the kinetic energy, potential energy, and dissipated energy by viscous damping for
each mode, and substituting in Lagrange’s equation.
[29]

36. Figure ‎ -4 Conceptual view of a MEMS ring gyroscope
4
Considering the flexure and translational modes only, the equations of motion can be expressed
as shown below, a detailed derivation of these equations is done in [9], and the final results
were:
̈ ̇
̈ ̇
[ ][ ] [ ][ ] [ ][ ]
̈ ̇
̈ ̇
̇
̇ ̇ [
[ ][ ] ][ ]
̇
̇
[ ][ ] [ ][ ]
̇
[ ][ ̇ ] [ ][ ]
[ ]
4-6
where line (1) represents Mass, Damping, and Stiffness; M 1, M2, M3, M4 are the modal masses of
the 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 the
flexure modes. The terms in line (2) (γT and γF) represents the modal coupling terms induced by
the Coriolis forces and by angular acceleration, the angular acceleration is negligible because the
ratio between angular acceleration to Coriolis response is inversely proportional to the flexural
resonant frequency , Line (3) contains additional stiffness terms that arise from centripetal
acceleration and electrostatic effects (α, β, χ), line (4) contains the terms representing the
environmental excitation ( and vo), and it’s clear that the ambient vibrations affect only the
translational modes, line (5) contains a term from the electrostatic actuation. Details of
Calculation 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 of
equations in the absence of angular rotation; which independently govern the translation and
flexural modes. Therefore, the flexural modes, which are excited by the operation of the ring
gyroscope, are not influenced by the translation modes which are excited by the environmental
vibration. Thus, the flexural modes are not influenced by environmental vibrations (Ideally)
[9,10].
4.2.Vibration Induced Errors in Ring Gyroscopes
When looking at the previous equations (4.6), it seems that the output of a ring gyroscope is
insensitive to vibration due to the decoupled dynamics governing ring translation versus ring
flexure; however, this decoupling is violated in the presence of non-proportional damping and
capacitive nonlinearity at the sense electrodes [9,11,10].
4.2.1. Vibration Induced Errors due to Non Proportional Damping
Proportional Damping is the type of damping in which the modal damping matrix [C] is in the
form of:
[ ] [ ] [ ] 4-7
where [M], [K] are the modal masses and stiffness respectively, α, β are constants, usually
empirical. This type of damping is known as PROPORTIONAL, i.e proportional to either the mass
M 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, proportional
damping is not a rule but rather the exception. [9]
Considering Proportional Damping, the damping matrix is diagonal, since the matrices of mass
and 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-proportional
damping, 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 of
undesired mode coupling.
[31]

38. 4.2.2. Vibration Induced Errors due to Non Linearity of Sense Electrodes
The parallel-plate sensing mechanism contributes a nonlinear behavior between sense
capacitance and the sense-axis displacement. This nonlinearity is negligible in normal operation
because the displacement produced by the Coriolis force is small. However, larger displacements
can be readily generated by vibration, and these displacements are subject to capacitive
nonlinearity. [11]
Vibration-induced errors are explained in [9] by subtracting the capacitive change by only
Coriolis force and no external vibration, from the capacitive change by both Coriolis force and
external vibration, and removing the signals produced having frequencies far from the resonant
frequency of the gyroscope (∼20 30 kHz), because they will be filtered out by the interface
circuit demodulation system. The resulting change in capacitance due to vibrations is given by:
[ ]
4-8
where / and / are the initial capacitance and the initial gap of the sense
electrode at 45◦/225◦. In an ideally fabricated symmetric ring structure, = and =
and becomes:
4-8
Therefore, another source of vibration-induced errors in ring gyroscopes arises from the high
order (cubic) terms in the capacitive nonlinearity at the sense electrodes.
There are other vibration induced error sources like those resulted from high frequency external
vibration or from imperfections that couple ring translation and flexure. High frequency
vibration (with spectral content frequency containing the flexural-mode resonant frequencies)
may directly excite the flexural modes leading to undesired responses that cannot be
distinguished from the desired responses excited by ring gyro operation. This error mechanism
obviously exists even for ideally fabricated ring gyroscopes.
On the other hand, vibration-induced errors by fabrication imperfection may occur when the
flexural modes are excited by translation modes. The decoupling of flexural and translation
modes can arise from the assumed perfect symmetry of the ring gyro. The symmetry may be
destroyed by a non-uniform or asymmetric distribution of ring mass and/or stiffness (inertial
and/or compliance coupling) as previously noted in analyses of degenerate gyroscopes.
4.3. A HARPSS Polysilicon Vibrating Ring Gyroscope
A famous design example about ring gyroscopes was the one made in [12]. The paper presents a
80-μm-thick, 1.1 mm in diameter high aspect-ratio (20:1) polysilicon ring gyroscope (PRG). A
detailed analysis has been performed to determine the overall sensitivity of the vibrating ring
gyroscope and identify its scaling limits. An open-loop sensitivity of 200 μV/deg/s in a dynamic
range of ±250 deg/s was measured under low vacuum level for a prototype device. The
resolution for a PRG with a quality factor (Q) of 1200, drive amplitude ( ) of 0.15 μm was
measured 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
4
support 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 flexural
modes that have equal natural frequencies and are 45o apart from each other. The ring is
electrostatically vibrated into the primary flexural mode with fixed amplitude.
Figure ‎ -5 The HARPSS Vibrating Ring Gyroscope – (a) SEM Image, (b) Electrode Voltages
4
In this section, we will apply the model suggested by the authors of [4] to redesign the HARPSS
Gyroscope.
4.3.1. Mechanical Design
General gyro specifications often include gyro size, environmental conditions (or applications),
or sensitivity. We first select the ring structure radius ( = 550 μm) from the size
specification and flexural and translation resonant frequencies ( = 29 KHz and =20 KHz)
from the environment conditions (or applications) or g-sensitivity (sensitivity to linear
acceleration). The g-sensitivity (in deg/s/(m/s2)2 is given by [9]:
4-10
Where 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 the
frequency spectrum of the environmental vibration. The support beam radius ( = 235
μm) is successively set to be from a half to a quarter of the as observed in Figure ‎ -4.
4
Next, we adjust the effective mass [ ] and stiffness [ ] matrices in equation 4.2 to match the
decided flexural and translation resonant frequencies. The flexure mode is concerned with only
4 springs, thus the flexure mode effective mass in the mass matrix is calculated as shown below:
4-11
Where and are the effective mass of the ring frame and the support
springs that is stretched horizontally for the flexure mode and can be considered as one third of
the 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 = 2328
Kg/m3. For the translational modes, the effective mass is approximately the sum of two
horizontal, two vertical and four 45o effective spring masses. In addition to the actual ring mass:
4-14
Where and , can be considered as half of the spring’s mass (since
they are very compliant).
The stiffness matrix is calculated from the curved beam stiffness equations 2.10, 2.11, and 2.12):
4-15
4-16
Where = 150 GPa, for , , and , = = 235 μm, and = at r = ,
because the ring frame is considered as 2 parallel curved beams [12]. By dividing by , and
equating the resulting expression to the square of the resonance angular flexure frequency, we
can obtain the width of the ring ( μm), substituting in mass and stiffness matrices to
get and . The height of the ring is still not calculated because
the resonant frequencies of the ring don’t depend on it [12]. is better to be set to a large
value to reject out of plane modes, but this will require higher driving voltage to maintain the
same drive amplitude as shown in the Electric design part in 4.3.3, we will set the height to 80
μm as the paper.
4.3.2. Finite Element Simulation
After the previous first order analysis, a 2-D Finite Element Analysis (FEA) was carried out using
the values calculated for ring and spring dimensions (Figure ‎ -6), by tuning the obtained values
4
for 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 its
center in the middle of the inner circular post), the second two modes were translational ones at
about 20 KHz the fourth and fifth were flexure at approximately 28 KHz. The difference in the
resonant frequencies between the model and FEA might have happened due to the
approximations done when calculating the effective masses and stiffness. The mode shapes
weren’t very accurate due to asymmetries in the mesh which caused rotation of the principle
mode axis. However, this won’t affect the resonant frequencies too much, and can be managed
by balancing electrodes. 2-D FEA didn’t show out of plane modes, however we shouldn’t worry
about them since they are minimized due to the high aspect ratio of the device. An ANSYS script
that 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 Mode
4.3.3. Electrical Design
The 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]:
4-17
where 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 for
sensing 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 assume
values for drive mode amplitude: , therefore the needed electrode capacitance
. The electrode capacitance is given by:
4-18
Therefore, 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. 4-19
Therefore, the AC drive signal is equal to 15.5 mV. The value is too far from what was
mentioned in the paper (5-8mV) because the equations of motion were derived based on the
usage of 2 driving electrodes at 0o and 180o as illustrated in Figure ‎ -4, while the operation
4
mode of the gyroscope in [12] was different (Force to rebalance mode, see Figure ‎ -5 The
4
HARPSS Vibrating Ring Gyroscope – (a) SEM Image, (b) Electrode Voltages - b). Table ‎ -1 4
Summary 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 KHz
Translational 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 mV
The results in Table ‎ -1 shows that the approximations done to calculate the effective masses of
4
the flexure and translational modes were acceptable for a first order hand analysis.
4.4.Evaluation of Vibrating Ring Gyroscope
4.4.1. Advantages of Ring Architecture
The 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 Electronics
5.1.System Components
Simple generalized model of a gyroscope with the electronic interface necessary to produce the
final output. An oscillator establishes the drive oscillation at the drive resonance frequency, and
the Coriolis readout interface detects and amplifies the Coriolis acceleration. A demodulator
demodulates the angular rate signal from the Coriolis acceleration, and a low-pass filter removes
other unwanted signals out-side the desired frequency band, from the final output. [14]
Figure ‎ -1 generalized model of a gyroscope with the electronic interface
5
5.1.1. Drive loop
The drive loop electronics are responsible for starting and sustaining oscillations along the
reference axis at constant amplitude. It is essential that a constant drive amplitude be
maintained, as any variation in the drive amplitude manifests itself as a change in velocity of the
mechanical structure (along the driven axis). Velocity fluctuations modulate the sensor output
and can result in false or inaccurate rate output.
There are two approaches to implement the drive loop, both of which have been implemented in
this work:
• An electromechanical oscillator: Here the drive mode oscillations are started and sustained
by using a positive feed-back loop that satisfies the Barkhausen’s criteria (Loop gain = 1, Loop
Phase 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-up
using a phase locked loop (PLL). The PLL center frequency and capture range are set close to the
drive resonant frequency of the gyroscope. On power up, the PLL locks on to the output of the
front-end. The PLL output is amplified or attenuated to achieve the desired voltage amplitude
and used to drive the microgyroscope. [15]
[37]