modeling and simulation of the capacitive acceleration sensor

VIETNAM NATIONAL UNIVERSITY, HANOI
CO LLEG E O F TE C H NO LO G Y
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Tran Đ ức Tan
M O DEL ING AN D SIM U LA TIO N
OF THE C A PA C ITIVE A C C EL E R AT IO N SE N SO R
Field: Electronics and Telecommunication
Code: 2.07.00
MASTER THESIS
Scientific Advisor:
Prof. D r N g uy en Phu Thuy
Hanoi, 2004
ĐẠ I HỌ C Q U Ố C G IA H À N ỘI
TRƯ ỜNG ĐẠI HỌC CÔNG NGH Ệ
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Tran Đ ức lâ n
MÔ HÌNH HÓA VÀ MÔ PHỎNG
SENSOR GIA TỐC KIỂU TỤ
C hu y ên ngành: K ỹ thu ật vô tuyến điện tử và thô n g tin liên lạc
M ã số : 2.07.00
L U Ậ N V Ă N T H Ạ C SỸ
C án bộ hướng dẫn:
G S .TS K H N g uy ễn Phú Thuỳ
H à N ội, 2004

TABLE OF CONTENTS
ASSURAïvCE
ACKNOWLEDGMENTS
LI ST OF TABLES
LI ST OF FIGURES
FOREWORD
CHAPTER 1. INTRODUCTION
1.1 Overview of MEMS
1.2 Silicon Micro Accelerometers
1.2.1 Electromechanical Accelerometers
1.2.2 Piezoelectric Accelerometers
1.2.3 Piezoresistive Accelerometers
1.2.4 Electrostatic Accelerometers
1.2.5 Resonant Accelerometer
1.3 MEMS Modeling and Simulation
CHAPTER 2. ACCELEROMETER: FROM THEORY TO DESIGN
2.1 Operational Principles
2.1.1 Open-Loop Design
2.1.2 Force-Balance Design
2.1.3 Comparisons
2.2 Capacitive Accelerometer
2.2.1 Position Measurement with Capacitance
2.2.2 Noise Analysis
CHAFTER 3. MODELING AND SIMULATION OF THE ACCELEROMETER
3.1 Overview of SUGAR
3.2 Nodal Analysis Approach
3.3 Simulation Program based on SUGAR
3.4 Simulation Result
3.4.1 Single Capacitive Accelerometer
3.4.2 Differential Capacitive Accelerometer with a Single Beam

3.4.3 Differential Capacitive Accelerometer with Two Symmetric Beams
3.4.4 Two Parallel Beams Accelerometer
3.4.5 Four Symmetric Beam Accelerometers
3.5 Experimental Calibration Set-up and Experimental Results
3.6 Comparison of the simulation and experimental results
CHAPTER 4. CONCLUSIONS
4.1 Concluding Remarks
4.2 Future Work
4.3 List of Publications
REFERENCE
APPENDIX
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35

36
45
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52
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67
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74
Abbreviation
ASIC Application-Specific Integrated Circuit
CAD
Computer-Aided Design
CTS Clear To Send
Ctf
Capacitance to Mtage Converter
FEA
Finite Element Analysis
FHSS Freqency Hoping Spread Spectrum
ICs Integrated Circuits
ITIMS
International Training Institute for Materials Science
LIGA Lithography Galvanoforming Moulding Processes

MEMS Microelectromechanical Systems
MOEMS M icroOptoElectroM echanical Systems
ODEs Ordinary Differential Eqations
PDM Pulse Density M odulation
P-P Peak to Peak
PWM Pulse Width M odulation
RTS Reqest To Send
List of Tables
Table 1.1. A vailable M EM S sim ulation tools, by level and view 13
Table 3.1 Physical param eters o f the simple capacitive accelerom eter 47
Table 3.2 Geometry param eters o f the single capacitive accelerometer 48
Table 3.3 Relation between beam ’s thickness and resonant frequency 50
Table 3.4 Relation between beam ’s length and resonant frequency 51
Table 3.5 Relation between beam ’s thickness and resonant frequency 55
Table 3.6 Relation between beam ’s thickness and resonant frequency 58
List of Figures
Figure 1.1 A compression type piezoelectric accelerometer arrangement 6
Figure 1.2: Piezoresistive acceleration sensor 7
Figure 1.3 Capacitive measurement of acceleration 8
Figure 1.4 Resonant accelerometer 9
Figure 1.5 Cantilever beam and beam - capacitor options 11
Figure 1.6 Nodal analysis and finite elements analysis 12
Figure 1.7 Ideal and actual cantilever beams (side view) 12
Figure 2.1 Open loop accelerometer 15
Figure 2.2 Frequency response and phase response with various damping 19
Figure 2.3 Transient responses with various damping 20
Figure 2.4 Force - balance accelerometer 21
Figure 2.5 Variety of capacitor structures used for position sensing 27
Figure 2.6 Variety of differential capacitor structures 27
Figure 2.7 Typical circuit use of a differential capacitor 28

Figure 2.8 Transimpedance amplifier capture the capacitor current 29
Figure 2.9 Feedback capacitor is added to circuit of Fig 2.8 30
Figure 2.10 Measurement the output voltage of a differential capacitor 31
Figure 3.1. A simple MEMS structure 37
Figure 3.2 A bent beam showing nodal forces, moments, and coordinates 39
Figure 3.3 the level-2 model of an electrostatic actuator 44
Figure 3.4 Flow chart of the simulation program 46
Figure 3.5 Single capacitive accelerometer 48
Figure 3.6 Steady responses of the single capacitive accelerometers with different beam
thicknesses 49
Figure 3.7 Relation between beam’s thickness and resonant frequency 50
Figure 3.8 Relation between beam’s width and resonant frequency 51
Figure 3.9 Differential capacitive accelerometer with a single beam 53
Figure 3.10 Relation between the voltage and the proof mass’s displacement of the
differential capacitive accelerometer with single beam 53
Figure 3.11 A differential capacitive accelerometer with two symmetric beams 54
Figure 3.12 Relation between the voltage and proof mass’s displacement of the
differential capacitive accelerometer with two symmetric beams 55
Figure 3.13 Two Parallel Beams Accelerometer 56
Figure 3.14 Relation between voltage and acceleration of two parallel beams
accelerometers 57
Figure 3.15 Four symmetric beam accelerometer 57
Figure 3.16 Relation between voltage and acceleration of the four symmetric beams
accelerometer with different beam’s thickness 58
Figure 3.17 The response frequency o f system under damping 59
Figure 3.18 Differential capacitive acceleration sensors with: double beams and (b)
four symmetrical beams 60
Figure 3.19 Calibration set-up consisting of the rotating disk, the CVC, wireless
circuits and the capacitive acceleration sensor under test 61
Figure 3.20 The CVC circuit and interface of the calibration set-up 61

Figure 3.21 Relation between voltage and acceleration of the different sensors with the
configurations noted in the figures 62
Figure 3.22 Relation between the voltage and the acceleration: comparison
between simulation and experimental results 65
Figure 4.1 A newly suggested structure with four symmetric beams 68
Figure 4.2 A comb structure proposed for future work 69
Foreword
MEMS technology has been developed since 1960 and MEMS products
have been commercialized and widely used around the world since 1980. In
Vietnam, however, this new field of technology has only been studied several
years ago. Following this trend, the College of Technology of VNUH started
research on MEMS devices and their applications in 2003. This thesis is a
continuation of this effort and is the first attempt to investigate and design
MEMS sensor by modeling and simulation.
My thesis includes four chapters. The first chapter introduces an overview of
MEMS and discusses some types of accelerometers. Capacitive accelerometer has
been chosen to be the object of my thesis because of its high sensitivity, good dc
response, noise performance, low drift, low temperature sensitivity, low-power
dissipation, and simple structure. Chapter 2 discusses operational principles of open-
loop and force-balance accelerometers. In addition, results of position measurement
and noise analysis of the capacitive accelerometer are given. Chapter 3 focuses on
modeling and simulation of different structures using SUGAR language in MATLAB
environment. In particular, the simulation results are compared to experimental
results. Finally, the conclusions of this research and proposal for future study are
presented in chapter 4 of this thesis.
Chapter I
CHAPTER 1
INTRODUCTION
1.1 Overview of MEMS
Microelectromechanical systems (MEMS) are collection of microsensors and

actuators that have the ability to sense its environment and react to changes in that
environment with the use of a microcircuit control. They also include the
conventional microelectronics packaging, integrating antenna structures for
command signals into microelectromechanical structures for desired sensing and
actuating functions. The system may also need micropower supply, microrelay, and
microsignal processing units. Microcomponents make the system faster, more
reliable, cheaper, and capable of incorporating more complex functions.
In the beginning o f 1990s, MEMS appeared with the aid o f the development
o f integrated circuit fabrication processes, in which sensors, actuators, and control
functions are co-fabricated in silicon [1]. Since then, remarkable research
progresses have been achieved in MEM S under the strong promotions from both
governm ent and industries. In addition to the commercialization of some less
integrated M EM S devices, such as microaccelerometers, inkjet printer head,
micromirrors for projection, etc., the concepts and feasibility of more complex
M EM S devices have been proposed and demonstrated for the applications in such
varied fields as microfluidics, aerospace, biomedical, chemical analysis, wireless
comm unications, data storage, display, optics, etc. Some branches of MEMS.
appearing as microoptoelectrom echanical systems (MOEM S), micro total analysis
systems, etc., have attracted a great research since their potential applications'
market.
At the end of 1990s, most of MEMS devices with various sensing or
actuating mechanisms were fabricated using silicon bulk micromachining, surface
Modeling and simulation o f the capacitive accelerometer
1
Chapter I
micromachining, and lithography, galvanoforming, moulding (L1GA) processes [21.
Three-dim ensional microfabrication processes incorporating more materials were
presented for MEMS recently because o f specific application requirements (e.g
biomedical devices) and higher output power microactuators.
M icromachining has become the fundamental technology for the fabrication

o f MEMS devices and, in particular, miniaturized sensors and actuators. Silicon
micromachining is the most advanced of the micromachining technologies, and it
allows for the fabrication o f MEMS that have dimensions in the submillimeter
range. It refers to fashioning microscopic mechanical parts out o f silicon substrate
or on a silicon substrate, making the structures three dimensional and bringing new
principles to the designers. Employing materials such as crystalline silicon,
polycrystalline silicon, silicon nitride, etc., a variety o f mechanical m icrostructures
including beams, diaphragms, grooves, orifices, springs, gears, suspensions, and a
great diversity o f other complex mechanical structures have been conceived.
In some applications, stresses and strains to which the structure is subjected
to may pose a problem for conventional cabling. In others, environmental effects
may affect system performance. Advances in ultra flat antenna technology coupled
with MEMS sensors and actuators seem to be an efficient solution. The integration
o f microm achining and microelectronics on one chip results in so-called smart
sensors [3], In smart sensors, small sensor signals are amplified, conditioned, and
transform ed into a standard output format. They may include microcontroller,
digital signal processor, application-specific integrated circuit (ASIC), self-test,
self-calibration, and bus interface circuits simplifying their use and making them
more accurate and reliable.
Silicon m icromachining has been a key factor for the vast progress of MEMS
in the last decade. This refers to the fashioning of microscopic mechanical parts out
o f silicon substrates and, more recently, other materials. It is used to fabricate such
features as clam ped beams, membranes, cantilevers, grooves, orifices, springs,
gears, suspensions, etc. These can be assembled to create a variety o f sensors. Bulk
m icrom achining is the commonly used method, but it is being replaced by surface
m icrom achining that offers the attractive possibility o f integrating the machined
Modeling and simulation o f the capacitive accelerometer
2
Chapter I
device with m icroelectronics that can be patterned and assembled on the same

wafer. Thus power supply circuitry and signal processing using ASICs can be
incorporated. It is the efficiency of creating several such complete packages using
existing technology that makes this an attractive approach.
1.2 Silicon Micro Accelerometers
M icromachined inertial sensors, consisting o f acceleration and angular rate
sensors are produced in large quantities mainly for automotive applications |4J,
where they are used to activate safety systems, including air bags, and to implement
vehicle stability systems and electronic suspensions. Besides these automotive
applications accelerom eters are used in many other applications where low cost and
small size are important, e.g. in biomedical applications for activity monitoring and
in consumer applications such as the active stabilization o f camcorder pictures.
Miniaturized acceleration sensors are also o f interest to the air and space industries
and for many other applications.
Silicon acceleration sensors generally consist o f a proof mass which is
suspended to a reference frame by a spring element. Accelerations cause a
displacement o f the pro o f mass, which is proportional to the acceleration. This
displacement can be measured in several ways, e.g. capacitively by measuring a
change in capacitance between the proof mass and an additional electrode or
pie
2
oresistively by integrating strain gauges in the spring element [3], To obtain
large sensitivity and low noise a large proof mass is needed, which suggests the use
o f bulk m icrom achined techniques. For less demanding applications surface
micromachined devices seem to be more attractive because o f the easy integration
with electronic circuits and the fact that bulk micromachining requires the use of
wafer bonding step [5]. Recently, some designs have been presented which combine
bulk and surface m icromachining to realize a large proof mass in a single wafer
process.
The technology can be classified in a number o f ways, such as mechanical or
electrical, active or passive, deflection or null-balance accelerometers, etc.

Modeling and simulation o f the capacitive accelerometer
Chapter 1
This thesis reviewed following type o f the accelerometers:
> Electromechanical
> Piezoelectric
> Piezoresistive
> Capacitive and electrostatic force balance
> Resonant accelerometer
Depending on the principles o f operations, these accelerometers have their own
subclasses.
1.2.1 Electrom echanical Accelerom eters
Electromechanical accelerometers [6], essentially servo or null-balance
types, rely on the principle o f feedback. In these instruments, an acceleration-
sensitive mass is kept very close to a neutral position or zero displacement point by
sensing the displacem ent and feeding back the effect o f this displacement. A
proportional m agnetic force is generated to oppose the motion o f the mass displaced
from the neutral position, thus restoring this position just as a mechanical spring in a
conventional accelerom eter would do. The advantages of this approach are better
linearity and elim ination o f hysteresis effects, as compared to the mechanical
springs. Also, in some cases, electrical damping can be provided, which is much
less sensitive to tem perature variations. One very important feature of
electromechanical accelerometers is the capability o f testing the static and dynamic
performances of the devices by introducing electrically excited test forces into the
system. This rem ote self-checking feature can be quite convenient in complex and
expensive tests where accuracy is essential. These instruments are also useful in
acceleration control systems, since the reference value o f acceleration can be
introduced by means o f a proportional current from an external source. They are
used for general-purpose motion measurements and monitoring low-frequency
vibrations. There are a num ber of different electromechanical accelerometers: coil-
and-magnetic types, induction types, etc.

Modeling and simulation o f the capacitive accelerometer
4
Chapter 1
1.2.2 Piezoelectric Accelerom eters
Piezoelectric accelerometers are widely used for general-purpose
acceleration, shock, and vibration measurements. They are basically motion
transducers with large output signals and comparatively small sizes and they are sell
generators not requiring external power sources. They are available with very high
natural frequencies and are therefore suitable for high-frequency applications and
shock measurements. These devices utilize a mass in direct contact with the
piezoelectric component or crystal as shown in Fig. 1.1. When a varying motion is
applied to the accelerometer, the crystal experiences a varying force excitation (F
ma), causing a proportional electric charge q to be developed across it. So,
Where q is the charge developed and dy is the piezoelectric coefficient o f the
material.
As this equation shows, the output from the piezoelectric material is dependent on
its mechanical properties, djj. Two com monly used piezoelectric crystals are lead-
zirconate titanate ceram ic (PZT) and crystalline quartz. They are both self-
generating m aterials and produce a large electric charge for their size. The
piezoelectric strain constant o f PZT is about 150 times that o f quartz. As a result.
PZTs are m uch more sensitive and smaller in size than quartz counterparts. These
accelerometers are useful for high-frequency applications. These active devices
have no DC response. Since piezoelectric accelerometers have comparatively low
mechanical impedances, their effect on the motion o f most structures is negligible.
q = d,jF = djina
( 1.1)
a
PZT crystal
Output
MASS

PZT crystal
o

Figure 1.1 A compression type piezoelectric accelerometer arrangement.
Modeling and simulation o f the capacitive accelerometer
5
Chapter!
The low-frequency response is limited by the piezoelectric characteristic,
while the high frequency response is related to mechanical response. The damping
factor is very small and it is usually less than 0.01 or near zero. Accurate low-
frequency response requires large damping factor, which is usually achieved by use
o f high-impedance voltage amplifiers. At very low frequencies thermal effects can
have severe influences on the operation characteristics. Piezoelectric accelerometers
are available in a wide range of specifications and are offered by a large number o f
manufacturers.
1.2.3 Piezoresistive Accelerom eters
Piezoresistive accelerometers (see Fig. 1.2) are essentially semiconductor
strain gauges with large gauge factors. High gauge factors are obtained since the
material resistivity is dependent primarily on the stress, not only on the dimensions.
The sensitivity of a piezoresistive sensor comes from the elastic response of its
structure and resistivity o f the material. Wire and thick or thin film resistors have
low gauge factors, that is, the resistance change due to strain is small. Piezoresistive
accelerometers are useful for acquiring vibration information at low frequencies, for
example, below 1 Hz. In fact, they are inherently true non-vibrational acceleration
sensors. They generally have wider bandwidth, smaller nonlinearities and zero
shifting, and better hysteresis characteristics compared to piezoelectric counterparts.
They are suitable to measure shocks well above 100,000g. Typical characteristics o f
piezoresistive accelerometers may be listed: 100 mV/g as the sensitivity, 0-750 Hz
as the frequency range, 2500 Hz in resonance frequency, 25g as the amplitude
range, 2000g as the shock rating, and 0-95°C as the temperature range. The total

mass is about 25 g. Most contemporary piezoresistive sensors are manufactured
from a single piece of silicon. This gives better stability and less thermal mismatch
between parts. In a typical monolithic sensing element a 1-mm silicon chip
incorporates the spring, mass and four-arm bridge assembly. The elements are
formed by a pattern of dopant in the originally flat silicon. Subsequent etching of
channels frees the gauges and simultaneously defines the masses as regions of
silicon o f original thickness.
Modeling and simulation of the capacitive accelerometer
6
Chapter 1
Figure 1.2: Piezoresistive acceleration sensor.
1.2.4 Electrostatic Accelerom eters
Electrostatic accelerometers are based on Coulom b’s law between two
charged electrodes; therefore, they are capacitive types. Depending on the operation
principles and external circuits they can be broadly classified as (a) electrostatic-
force-feedback accelerometers, and (b) differential-capacitance accelerometers.
1.2.4.1 Electrostatic-Force-Feedback Accelerometers
An electrostatic-force-feedback accelerometer consists o f an electrode, with
mass m and area S, mounted on a light pivoted arm that moves relative to some
fixed electrodes. The nominal gap h between the pivoted and fixed electrodes is
maintained by means o f a force-balancing servo system, which is capable o f varying
the electrode potential in response to signals from a pickoff mechanism that senses
relative changes in the gap.
Hence, if the bias potential is held constant and the gain of the control loop is
high so that variations in the gap are negligible, the acceleration becomes a linear
function o f the controller output voltage. The principal difficulty in mechanizing the
electrostatic force accelerometer is the relatively high electric field intensity
required to obtain an adequate force. Damping can be provided electrically or by
viscosity o f the gaseous atmosphere in the inter-electrode space if the gap h is
sufficiently small. The scheme works best in micromachined instruments.

Monlinearity in the voltage break down phenom enon permits larger gradients in
Modeling and simulation of the capacitive accelerometer
7
Chapter 1
very small gaps. The main advantages of electrostatic accelerometers are their
extreme m echanical simplicity, low power requirements, absence o f inherent
sources of hysteresis errors, zero temperature coefficients, and ease of shielding
from stray fields.
1.2.4.2 Differential -Capacitance Accelerometers
Differential-capacitance accelerometers are based on the principle of the
change of capacitance in proportion to applied acceleration. In one type, the seismic
mass of the accelerom eter is made as the movable element of an electrical
oscillator. The seismic mass is supported by a resilient parallel-motion beam
arrangement from the base. The system is set to have a certain defined nominal
frequency when undisturbed. If the instrument is accelerated, the frequency varies
above and below the nom inal value depending on the direction of acceleration. I'he
seismic mass carries an electrode located in opposition to a number o f base-fixed
electrodes that define variable capacitors. The base-fixed electrodes are resistances
coupled in the feedback path o f a wideband, phase-inverting amplifier.
1.2.5 Resonant Accelerometers
Resonant accelerometers are attractive for their high sensitivity and
frequency output. M ost o f the conventional, high precision accelerometers are of
this type. The structure o f resonant accelerometers is quite different from other
sensors, as shown in Fig. 1.4. The proof mass is suspended by relatively stiff
suspension to prevent large displacement due to acceleration. Unlike other types o f
accelerometers, resonators are attached to the proof mass. Upon acceleration, the
m ass b e a m
g o ia
Figure 1.3 Capacitive measurement of acceleration.
' Silicon

Modeling and simulation o f the capacitive accelerometer
8
Chapter I
proof mass changes the strain in the attached resonators, which causes a shift in
those resonant frequencies. The frequency shift is then detected by the electronics
and the output can be measured easily by digital counters. Resonant accelerometers
are still in the early stages of research and development. Nevertheless, the use of
resonant strain gauges is a competitive approach for high precision sensing and can
be developed into a key technology for inertial grade accelerometers.
Figure 1.4 Resonant accelerometer
1.3 M EM S M odeling and Simulation
Accurate modeling and efficient simulation, in support o f greatly reduced
developm ent cycle time and cost, are well established techniques in the
miniaturized world of integrated circuits (ICs) [7-9]. Simulation accuracies of 5% or
less for parameters o f interest are achieved fairly regularly, although even much less
accurate simulations (25-30% , e.g.) can still be used to obtain valuable information.
In the IC world, simulation can be used to predict the performance of a design, to
analyze an already existing component, or to support automated synthesis o f a
design. Eventually, M EM S simulation environments should also be capable of these
three modes o f operation. The MEMS developer is, of course, most interested in
quick access to particular techniques and tools to support the system currently under
development. In the long run, however, consistently achieving acceptably accuratc
MEMS sim ulations will depend both on the ability of the CAD (computer-aided
design) comm unity to develop robust, efficient, user-friendly tools which will be
Modeling and simulation of the capacitive accelerometer
9
Chapter 1
widely available both to cutting-edge researchers and to production engineers and
on the existence of readily accessible standardized processes.
We need to look specifically at the tools and techniques the MEMS designer

has available for the modeling and simulation tasks because all models are not
created equal. The developer must be very clear about what parameters are of
greatest interest and then must choose the models and simulation techniques
(including implementation in a tool or tools) that are most likely to give the most
accurate values for those parameters in the least amount o f simulation time.
Let us look at a sim ple example that combines electrical and mechanical
parts. The cantilever beam in Fig. 1.5(a), fabricated in metal, polysilicon, or a
combination, may be combined with an electrically isolated plate to form a parallel
plate capacitor. If a mechanical force or a varying voltage is applied to the beam
(Fig. 1,5 (b l)), an accelerom eter or a switch can be obtained.
Figure. 1.5 Cantilever beam and beam - capacitor options (a) cantilever dimension
(b) Basic - capacitor designs
To obtain an accurate model o f the beam we can use the method of nodal
analysis that treats the beam as a graph consisting o f a set o f edges or “devices”.
Nodes
1/
Modeling and simulation of the capacitive accelerometer
10
Chapter 1
linked together at "nodes” [10]. Nodal analysis assumes that at equilibrium the sum
o f all values around each closed loop (the “across” quantities) will be zero, as will
the sum of all values entering or leaving a given node (the “through” quantities).
Thus, for example, the sum o f all forces and moments on each node must be zero, as
must the sum o f all currents flowing into or out o f a given node. This type of
modeling is sometimes referred to as “lumped parameter,” since quantities such as
resistance and capacitance, which are in fact distributed along a graph edge, are
modeled as discrete components. In the electrical domain K irchhoffs laws are
examples o f these rules.
Since nodal analysis is based on linear elements represented as the edges in
the underlying graph, it cannot be used to model many complex structures and

phenomena such as fluid flow or piezoelectricity. Even for the cantilever beam, if
the beam is composed of layers o f two different materials (e.g., polysilicon and
metal), it cannot be adequately m odeled using nodal analysis. The technique of
finite element analysis (FEA) must be used instead [11-12]. Finite element analysis
for the beam begins with the identification o f sub elements, as in Fig. 1.5(a), but
each element is treated as a true three-dim ensional object. Elements need not all
have the same shape, for example, tetrahedral and cubic “brick” elements could be
mixed together, as appropriate. In FEA, one cubic element now has eight nodes,
rather than tw o (Fig. 1.6), so computational complexity is increased. Thus,
developing efficient computer software to carry out FEA for a given structure can
be a difficult task in itself. But this general method can take into account many
features that cannot be adequately addressed using nodal analysis, including, for
exam ple, unaligned beam sections, and surface texture (Fig. 1.7).
Modeling and simulation o f the capacitive accelerometer
Chapter 1
(a) Nodal analysis / Modified nodal analysis
("Linear" elements)
nodes
(b ) Finite element analysis
(Three - dimensional elements)
Figure 1.6 Nodal analysis and finite elements analysis
a) ideal beam
unaligned sections
^ \
rough surface
b) actual beam
Figure. 1.7 Ideal and actual cantilever beams (side view).
In the past fifteen years, much progress has been made in providing MEMS
designers with simulators and other tools which will give them the ability to make
MEMS as useful and ubiquitous. While there is still much to be done, the future is

bright for this flexible and powerful technology. Table 1 listed several simulation
tools and their supported levels:
Modeling and simulation of the capacitive accelerometer
12
Chapter 1
Table 1.1. Available MEMS simulation tools, by level and view
Simulation tool Levels supported
Mathematics, Matlab All
MEM CAD
Low
SPICE
Low to medium
APLAC
Low to medium
ANSYS, CFD Low to medium
SUGAR, NODAS
Low to medium
Memspro Low to medium
V H D L - AMS M edium to high
In this thesis I used SUGAR tool which applies modified nodal method to
implement simulation programs. More details o f this tool will be discussed in
chapter 3.
Modeling and simulation o f the capacitive accelerometer
13
Chapter 2
CHAPTER 2
ACCELEROMETER: FROM THEORY TO DESIGN
2.1 Operational Principles
The operational principle o f an accelerom eter is based on the N ewton's
second law. Upon acceleration, the p roof mass (seismic m ass) that is anchored on

the frame by mechanical suspensions experiences an inertial force F (= -ma)
causing a deflection o f the proof mass, where a is the frame acceleration. Under
certain conditions, the displacement is proportional to the input acceleration:
where k is the spring constant o f the suspension. The displacem ent can be detected
and converted into an electrical signal by several sensing techniques. This simple
principle underlies the operation o f all accelerom eters.
From a system point o f view, there are two m ajor classes o f silicon micro
accelerometers; open-loop and force-balanced accelerometers [13-14]. In open-loop
accelerom eter design, the suspended pro of mass displaces from its neutral position
and the displacement is m easured either piezoresistively or capacitively. In force-
balance accelerometer design, a feedback force, typically an electrostatic force, is
applied onto the proof mass to counteract the displacem ent caused by the inertial
force. Hence, the pro of mass is virtually stationary relative to the frame. The output
signal is proportional to the feedback signal. In this section, the first order behavior
o f open-loop accelerometers will be described. Steady state, frequency, and
transition response will be studied analytically. The performance o f force balance
Modeling and simulation of the capacitive accelerometer
14
Chapter 2
accelerometers will be then considered. Finally, the operational characteristics of
the two types o f accelerom eters will be compared.
2.1.1 Open-Loop Design
An open-loop accelerom eter can be modeled as a proof mass suspended
elastically on a frame, as shown in Fig. 2.1. The frame is attached to the object
whose acceleration is to be measured. The proof mass moves from its neutral
position relative to the frame when the frame starts to accelerate. For a given
acceleration, the proof mass displacement is determined by the mechanical
suspension and the damping. Capacitive sensing is used here.
:Vo=Tv x(t) — O Vo
-777

Figure2.1 Open loop accelerometer
As shown in Fig. 2.1, y and z are the absolute displacement (displacement
relative to the earth) for the frame and the proof mass, respectively. The
acceleration y is o f the interest o f measurement. Let x be the relative displacement
of the proof mass with respect to the frame. The relative displacement is the
difference between the absolute displacement o f the frame and the proof mass, or
x = z - y .
In the following analysis, the displacement refers to the relative displacement
of the proof mass to the frame (x), unless otherwise specified. The lower case x. y,
and z denote the displacement in the time domain, and the upper case X. Y. and Z
are their Laplace transforms in the s-domain, respectively. When the inertial force
displaces the proof mass, it also experiences the restoring force from the mechanical
spring and the damping force from the viscous damping. The equation of motion of
the proof mass can be w ritten as:
Modeling and simulation of the capacitive accelerometer
15
Chapter 2
m — —= - k x - b — (2.2)
t,2 ». v
d 2z , . dx
— — = -k x - b —
d t2 dt
where k is the spring constant of the suspension and b is the damping coefficient of
the air and any structural damping.
Since the proof mass is usually sealed in the frame, the damping force is
proportional to the velocity relative to the frame, rather than to the absolute
velocity. Using x=y-z the following equation o f motion can be obtained:
d x b dx k d 'y ,
— r- +

+ —x =
f = -a lt) (2.3)
dt m dt m dt
The negative sign indicates that the displacement o f the proof mass is always
in the opposite direction o f the acceleration. Equation (2.3) can also be re-writtcn
as:
d 2x dx 2
— — + 2âû)n — + co x —
d t2 ^ " dt "
u
where con = /— is natural resonant frequency, £ =
V m
(2.4)
dt2
—— is damping factor.
2 mo
This is the governing equation for an open loop accelerometer relating the
proof mass displacement and the input acceleration. The performance o f an open-
loop accelerom eter can be characterized by the natural resonant frequency con and
the damping factor C,. The dam ping is determined by the viscous liquid or the
chamber pressure. For silicon micro accelerometers, gas damping is most
comm only used and the dam ping factor is controlled by the chamber pressure and
the gas properties. Critical dam ping is desired in most designs in order to achieve
maxim um bandwidth and m inimum overshoot and ringing.
The natural resonant frequency is another important parameter in open loop
accelerom eter design [15]. It is designed to satisfy the requirements on the
sensitivity and the bandwidth. The natural resonant frequency can be measured
Modeling and simulation of the capacitive accelerometer
16
Chapter 2

either dynamically by resonating the accelerometer or statically by measuring the
displacement for a given acceleration. From its definition, the natural resonant
frequency can be re-written as:
= ^ (2-5)
V/H V x
where a is the acceleration and x is the displacement. Therefore, the natural resonant
frequency can be determined conveniently by measuring the displacement due to
the gravitational field.
Steady-State Response: For a constant acceleration, the proof mass is stationaly
relative to the frame so that equation (2.4) becomes:
o)2nx = - ^ - = -a (2.6)
or
m
X = - — a
k
(2.7)
The static sensitivity of the accelerometer is shown to be:
X _ m _ 1
a k col
(2.8)
Therefore, the proof mass displacement is linearly proportional to the input
acceleration in steady state. The sensitivity is determined by the ratio m/k or the
inverse of the square of natural resonant frequency. Hence, the resonance frequency
of the structure can be increased by increasing the spring constant and decreasing
the proof mass, while the quality factor of the device can be increased by reducing
damping and by increasing proof mass and spring constant. Last, the static response
of the device can be improved by reducing its resonant frequency.
Frequency Response: Frequency response is the acceleration response to a
sinusoidal excitation. Let the frame be in harmonic motion
a(t) = y(t) = -Y(o2 sin cot (2 9)

Modeling and simulation o f the capacitive accelerometer
17
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