modeling and simulation of the capacitive acceleration sensor

Figure 1.2: Piezoresistive acceleration sensor 7Figure 2.2 Frequency response and phase response with various damping 19 Figure 2.5 Variety o f capacitor structures used for position sen

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CHAPTER 2 ACCELEROMETER: FROM THEORY TO DESIGN

2.1 Operational Principles

2.1.1 Open-Loop Design2.1.2 Force-Balance Design2.1.3 Comparisons

3.2 Nodal Analysis Approach

3.3 Simulation Program based on SUGAR

3.4 Simulation Result

3.4.1 Single Capacitive Accelerometer3.4.2 Differential Capacitive Accelerometer with a Single Beam3.4.3 Differential Capacitive Accelerometer with Two Symmetric Beams3.4.4 Two Parallel Beams Accelerometer

3.4.5 Four Symmetric Beam Accelerometers3.5 Experimental Calibration Set-up and Experimental Results

3.6 Comparison o f the simulation and experimental results

6

7

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9141415

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ASIC Application-Specific Integrated Circuit

ITIMS International Training Institute for Materials Science

MOEMS M icroO ptoE lectroM echanical System s

ODEs Ordinary Differential Eqations

T able 1.1 A v ailab le M E M S sim u latio n tools, by level and view 13

T able 3.1 P hysical param eters o f the sim ple capacitive accelerom eter 47

T able 3.2 G eom etry p aram eters o f the single capacitive accelerom eter 48

T able 3.3 R elatio n b etw een b e a m ’s thickness and resonant frequency 50

T ab le 3.4 R elatio n b etw een b e a m ’s length and resonant frequency 51

T able 3.5 R elation b etw een b e a m ’s th ick n ess and resonant frequency 55

T ab le 3.6 R elation b etw een b e a m ’s th ick n ess and reso n an t frequency 58

Figure 1.2: Piezoresistive acceleration sensor 7

Figure 2.2 Frequency response and phase response with various damping 19

Figure 2.5 Variety o f capacitor structures used for position sensing 27

Figure 2.8 Transimpedance amplifier capture the capacitor current 29

Figure 2.10 M easurement the output voltage of a differential capacitor 31

Figure 3.2 A bent beam showing nodal forces, moments, and coordinates 39

Figure 3.6 Steady responses of the single capacitive accelerometers with different beam

Figure 3.7 Relation between beam’s thickness and resonant frequency 50

Figure 3.11 A differential capacitive accelerometer with two symmetric beams 54Figure 3.12 Relation between the voltage and proof mass’s displacement of the

Figure 3.14 Relation between voltage and acceleration of two parallel beams

Figure 3.16 Relation between voltage and acceleration of the four symmetric beams

Figure 3.18 Differential capacitive acceleration sensors with: double beams and (b)

Figure 3.19 Calibration set-up consisting of the rotating disk, the CVC, wireless

Figure 3.20 T he C VC circuit and interface of the calibration set-up 61

Figure 3.21 Relation between voltage and acceleration of the different sensors with the

Figure 3.22 R elation betw een the voltage and the acceleration: com parison

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 o f technology has only been studied several years ago Following this trend, the College of Technology o f VNUH started research on MEMS devices and their applications in 2003 This thesis is a continuation o f 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

m icrosignal processing units M icrocom ponents m ake the system faster, more reliable, cheaper, and capable o f incorporating m ore com plex functions.

In the beginning o f 1990s, M EM S appeared w ith the aid o f the developm ent

o f integrated circuit fabrication processes, in w hich sensors, actuators, and control functions are co-fabricated in silicon [1] Since then, rem arkable research progresses have been achieved in M EM S under the strong prom otions from both governm ent and industries In addition to the com m ercialization o f some less integrated M E M S devices, such as m icroaccelerom eters, inkjet printer head,

m icrom irrors for projection, etc., the concepts and feasibility o f m ore com plex

M E M S devices have been proposed and dem onstrated for the applications in such varied fields as m icrofluidics, aerospace, biom edical, chem ical analysis, wireless com m unications, data storage, display, optics, etc Som e branches o f M EM S appearing as m icrooptoelectrom echanical system s (M O E M S ), m icro total analysis system s, etc., have attracted a great research since their potential applications'

m icrom achining, and lithography, galvanoform ing, m oulding (L1GA) processes [ 2 1

T hree-dim ensional m icrofabrication processes incorporating m ore m aterials were presented for M EM S recently because o f specific application requirem ents (e.g biom edical devices) and higher output pow er m icroactuators

M icrom achining has becom e the fundam ental technology for the fabrication

o f M EM S devices and, in particular, m iniaturized sensors and actuators Silicon

m icrom achining is the m ost advanced o f the m icrom achining technologies, and it allow s for the fabrication o f M EM S that have dim ensions in the subm illim eter range It refers to fashioning m icroscopic m echanical parts out o f silicon substrate

or on a silicon substrate, m aking the structures three dim ensional and bringing new principles to the designers Em ploying m aterials such as crystalline silicon, polycrystalline silicon, silicon nitride, etc., a variety o f m echanical m icrostructures including beam s, diaphragm s, grooves, orifices, springs, gears, suspensions, and a great diversity o f other com plex m echanical structures have been conceived

In som e applications, stresses and strains to w hich the structure is subjected

to may pose a problem for conventional cabling In others, environm ental effects may affect system perform ance A dvances in ultra flat antenna technology coupled

w ith M EM S sensors and actuators seem to be an efficient solution The integration

o f m icrom achining and m icroelectronics on one chip results in so-called sm art sensors [3], In sm art sensors, sm all sensor signals are am plified, conditioned, and transform ed into a standard output form at They m ay include m icrocontroller, digital signal processor, application-specific integrated circuit (A SIC ), self-test, self-calibration, and bus interface circuits sim plifying their use and m aking them

m ore accurate and reliable

Silicon m icrom achining has been a key factor for the vast progress o f M EM S

in the last decade This refers to the fashioning o f m icroscopic m echanical parts out

o f silicon substrates and, m ore recently, other m aterials It is used to fabricate such features as clam ped beam s, m em branes, cantilevers, grooves, orifices, springs, gears, suspensions, etc These can be assem bled to create a variety o f sensors Bulk

m icrom achining is the com m only used m ethod, but it is being replaced by surface

m icrom achining that offers the attractive possibility o f integrating the m achined

device w ith m icroelectronics that can be patterned and assem bled on the same

w afer Thus po w er supply circuitry and signal processing using A SICs can be incorporated It is the efficiency o f creating several such com plete packages using existing technology that m akes this an attractive approach

M icrom achined inertial sensors, consisting o f acceleration and angular rate sensors are produced in large quantities m ainly for autom otive applications |4J, where they are used to activate safety system s, including air bags, and to im plem ent vehicle stability system s and electronic suspensions Besides these autom otive applications accelerom eters are used in m any other applications w here low cost and small size are im portant, e.g in biom edical applications for activity m onitoring and

in consum er applications such as the active stabilization o f cam corder pictures

M iniaturized acceleration sensors are also o f interest to the air and space industries and for m any o th er applications

Silicon acceleration sensors generally consist o f a p ro o f m ass which is suspended to a reference frame by a spring elem ent A ccelerations cause a displacem ent o f the p ro o f mass, w hich is proportional to the acceleration This displacem ent can be m easured in several w ays, e.g capacitively by m easuring a change in capacitance betw een the p ro o f m ass and an additional electrode or pie2oresistively by integrating strain gauges in the spring elem ent [3], To obtain large sensitivity and low noise a large p ro o f m ass is needed, w hich suggests the use

o f bulk m icrom achined techniques For less dem anding applications surface

m icrom achined devices seem to be m ore attractive because o f the easy integration with electronic circuits and the fact that bulk m icrom achining requires the use o f wafer bonding step [5] R ecently, som e designs have been presented w hich com bine bulk and surface m icrom achining to realize a large p ro o f m ass in a single w afer process

T he tech nology can be classified in a num ber o f w ays, such as m echanical or electrical, active or passive, deflection or null-balance accelerom eters, etc

Modeling and simulation o f the capacitive accelerometer

This thesis review ed follow ing type o f the accelerom eters:

> E lectrom echanical

> Piezoelectric

> P iezoresistive

> C apacitive and electrostatic force balance

> R esonant accelerom eter

D epending on the p rinciples o f operations, these accelerom eters have their own subclasses

1.2.1 E le c tro m e c h a n ica l A ccelero m eters

E lectrom echanical accelerom eters [6], essentially servo or null-balance types, rely on the p rin cip le o f feedback In these instrum ents, an acceleration- sensitive m ass is k ep t very close to a neutral position or zero displacem ent point by sensing the disp lacem en t and feeding back the effect o f this displacem ent A proportional m agnetic force is generated to oppose the m otion o f the m ass displaced from the neutral po sitio n , thus restoring this position ju st as a m echanical spring in a conventional accelero m eter w ould do The advantages o f this approach are better linearity and elim ination o f hysteresis effects, as com pared to the mechanical springs A lso, in som e cases, electrical dam ping can be provided, w hich is much less sensitive to tem p eratu re variations O ne very im portant feature o f electrom echanical accelero m eters is the capability o f testing the static and dynam ic perform ances o f the devices by introducing electrically excited test forces into the system This rem o te self-checking feature can be quite convenient in com plex and expensive tests w h ere accuracy is essential These instrum ents are also useful in acceleration control system s, since the reference value o f acceleration can be introduced by m ean s o f a proportional current from an external source They are used for g en eral-p u rp o se m otion m easurem ents and m onitoring low -frequency vibrations T here are a n u m b er o f different electrom echanical accelerom eters: coil- and-m agnetic types, in duction types, etc

1.2.2 P iezoelectric A ccelero m eters

Piezoelectric accelerom eters are w idely used for general-purposeacceleration, shock, and vibration m easurem ents They are basically m otion transducers w ith large output signals and com paratively sm all sizes and they are sell generators not requiring external pow er sources They are available w ith very high natural frequencies and are therefore suitable for high-frequency applications and shock m easurem ents These devices utilize a m ass in direct contact w ith the piezoelectric com ponent or crystal as show n in Fig 1.1 W hen a varying m otion is applied to the accelerom eter, the crystal experiences a varying force excitation (F

m a), causing a proportional electric charge q to be developed across it So,

W here q is the charge developed and dy is the piezoelectric coefficient o f the

m aterial

A s this equation show s, the output from the piezoelectric m aterial is dependent on its m echanical properties, djj Tw o com m only 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 tim es that o f quartz As a result PZT s are m uch m ore sensitive and sm aller in size than quartz counterparts These accelerom eters are useful for high-frequency applications These active devices

h av e no DC response Since piezoelectric accelerom eters have com paratively low

m echanical im pedances, their effect on the m otion o f m ost structures is negligible

The low -frequency response is lim ited by the piezoelectric characteristic, while the high frequency response is related to m echanical response The dam ping factor is very small and it is usually less than 0.01 or near zero A ccurate low- frequency response requires large dam ping factor, w hich is usually achieved by use

o f high-im pedance voltage am plifiers A t very low frequencies therm al effects can have severe influences on the operation characteristics P iezoelectric accelerom eters are available in a wide range o f specifications and are offered by a large num ber o f

m anufacturers

1.2.3 P iezo resistiv e A ccelerom eters

Piezoresistive accelerom eters (see Fig 1.2) are essentially sem iconductor strain gauges with large gauge factors H igh gauge factors are obtained since the

m aterial resistivity is dependent prim arily on the stress, not only on the dim ensions The sensitivity o f a piezoresistive sensor comes from the elastic response o f its structure and resistivity o f the m aterial W ire and thick or thin film resistors have low gauge factors, that is, the resistance change due to strain is small Piezoresistive accelerom eters are useful for acquiring vibration inform ation at low frequencies, for exam ple, below 1 Hz In fact, they are inherently true non-vibrational acceleration sensors They generally have w ider bandw idth, sm aller nonlinearities and zero shifting, and better hysteresis characteristics com pared to piezoelectric counterparts They are suitable to m easure shocks well above 100,000g Typical characteristics o f piezoresistive accelerom eters m ay be listed: 100 m V /g as the sensitivity, 0 -7 5 0 Hz

as the frequency range, 2500 Hz in resonance frequency, 25g as the am plitude range, 2000g as the shock rating, and 0 -9 5 °C as the tem perature range The total

m ass is about 25 g M ost contem porary piezoresistive sensors are m anufactured from a single piece o f silicon This gives better stability and less therm al m ism atch betw een parts In a typical m onolithic sensing elem ent a 1-mm silicon chip incorporates the spring, m ass and four-arm bridge assem bly The elem ents are form ed by a pattern o f dopant in the originally flat silicon Subsequent etching o f channels frees the gauges and sim ultaneously defines the m asses as regions o f silicon o f original thickness

Figure 1.2: Piezoresistive acceleration sensor.

1.2.4 E lectro sta tic A ccelero m eters

E lectrostatic accelerom eters are based on C o u lo m b ’s law betw een tw o charged electrodes; therefore, they are capacitive types D epending on the operation principles and external circuits they can be broadly classified as (a) electrostatic- force-feedback accelerom eters, and (b) differential-capacitance accelerom eters

1.2.4.1 Electrostatic-Force-Feedback Accelerometers

An electrostatic-force-feedback accelerom eter consists o f an electrode, w ith

m ass m and area S, m ounted on a light pivoted arm that m oves relative to som e fixed electrodes T he nom inal gap h betw een the pivoted and fixed electrodes is

m aintained by m eans o f a force-balancing servo system , w hich is capable o f varying the electrode potential in response to signals from a p ic k o ff m echanism that senses relative changes in the gap

H ence, i f the bias potential is held constant and the gain o f the control loop is high so that variations in the gap are negligible, the acceleration becom es a linear function o f the controller output voltage The principal difficulty in m echanizing the electrostatic force accelerom eter is the relatively high electric field intensity required to obtain an adequate force D am ping can be provided electrically or by viscosity o f th e gaseous atm osphere in the inter-electrode space if the gap h is sufficiently sm all The schem e w orks best in m icrom achined instrum ents

M onlinearity in the voltage break dow n phenom enon perm its larger gradients in

very small gaps The m ain advantages o f electrostatic accelerom eters are their extrem e m echanical sim plicity, low pow er requirem ents, absence o f inherent sources o f hysteresis errors, zero tem perature coefficients, and ease o f shielding from stray fields.

1.2.4.2 Differential -Capacitance Accelerometers

D ifferential-capacitance accelerom eters are based on the principle o f the change o f capacitance in proportion to applied acceleration In one type, the seism ic

m ass o f the accelerom eter is m ade as the m ovable elem ent o f an electrical oscillator The seism ic m ass is supported by a resilient parallel-m otion beam arrangem ent from the base The system is set to have a certain defined nominal frequency w hen undisturbed I f the instrum ent is accelerated, the frequency varies above and below the nom inal value depending on the direction o f acceleration I'he seism ic m ass carries an electrode located in opposition to a num ber o f base-fixed electrodes that define variable capacitors The base-fixed electrodes are resistances coupled in the feedback path o f a w ideband, phase-inverting am plifier

1.2.5 Resonant Accelerometers

R esonant accelerom eters are attractive for their high sensitivity and frequency output M ost o f the conventional, high precision accelerom eters are o f this type T he structure o f resonant accelerom eters is quite different from other sensors, as show n in Fig 1.4 The p ro o f m ass is suspended by relatively stiff suspension to prevent large displacem ent due to acceleration U nlike other types o f accelerom eters, resonators are attached to the p ro o f m ass U pon acceleration, the

p ro o f m ass changes the strain in the attached resonators, w hich causes a shift in those resonant frequencies The frequency shift is then detected by the electronics and the output can be m easured easily by digital counters R esonant accelerom eters are still in the early stages o f research and developm ent N evertheless, the use o f resonant strain gauges is a com petitive approach for high precision sensing and can

be developed into a key technology for inertial grade accelerom eters

Figure 1.4 Resonant accelerometer

1.3 M E M S M o d e lin g and Sim u lation

A ccurate m odeling and efficient sim ulation, in support o f greatly reduced developm ent cycle tim e and cost, are w ell established techniques in the

m iniaturized w orld o f integrated circuits (ICs) [7-9] Sim ulation accuracies o f 5% or less for param eters o f interest are achieved fairly regularly, although even much less accurate sim ulations (2 5 -3 0 % , e.g.) can still be used to obtain valuable inform ation

In the IC w orld, sim ulation can be used to predict the perform ance o f a design, to analyze an already existing com ponent, or to support autom ated synthesis o f a design E ventually, M EM S sim ulation environm ents should also be capable o f these three m odes o f operation The M EM S developer is, o f course, m ost interested in quick access to particular techniques and tools to support the system currently under developm ent In the long run, how ever, consistently achieving acceptably accuratc

M EM S sim ulations w ill depend both on the ability o f the CA D (com puter-aided design) com m unity to develop robust, efficient, user-friendly tools w hich will be

w idely available both to cutting-edge researchers and to production engineers and

on the existence o f readily accessible standardized processes

W e need to look specifically at the tools and techniques the M EM S designer has available for the m odeling and sim ulation tasks because all m odels are not created equal T he developer must be very clear about w hat param eters are o f greatest interest and then m ust choose the m odels and sim ulation techniques (including im plem entation in a tool or tools) that are m ost likely to give the most accurate values for those param eters in the least am ount o f sim ulation time

Let us look at a sim ple exam ple that com bines electrical and m echanical parts The cantilever beam in Fig 1.5(a), fabricated in m etal, polysilicon, or a com bination, m ay be com bined w ith an electrically isolated plate to form a parallel plate capacitor I f a m echanical force or a varying voltage is applied to the beam (Fig 1,5 ( b l)), an accelerom eter or a sw itch can be obtained

Figure 1.5 Cantilever beam and beam - capacitor options (a) cantilever dimension

(b) Basic - capacitor designs

To obtain an accurate m odel o f the beam w e can use the m ethod o f nodal analysis that treats the beam as a graph consisting o f a set o f edges or “devices”

Nodes

1/

linked together at "nodes” [10] N odal analysis assum es that at equilibrium the sum

o f all values around each closed loop (the “ across” quantities) will be zero, as will the sum o f all values entering or leaving a given node (the “through” quantities) Thus, for exam ple, the sum o f all forces and m om ents on each node must be zero, as must th e sum o f all currents flow ing into or out o f a given node This type o f

m odeling is som etim es referred to as “ lum ped param eter,” since quantities such as resistance and capacitance, w hich are in fact distributed along a graph edge, are

m odeled as discrete com ponents In the electrical dom ain K irc h h o ffs laws are exam ples o f these rules

Since nodal analysis is based on linear elem ents represented as the edges in the underlying graph, it cannot be used to model m any com plex structures and phenom ena such as fluid flow or piezoelectricity Even for the cantilever beam , if the beam is com posed o f layers o f tw o different m aterials (e.g., polysilicon and

m etal), it cannot be adequately m odeled using nodal analysis The technique o f finite elem ent analysis (F E A ) m ust be used instead [11-12] Finite elem ent analysis for the beam begins w ith the identification o f sub elem ents, as in Fig 1.5(a), but each elem ent is treated as a true three-dim ensional object Elem ents need not all have the sam e shape, for exam ple, tetrahedral and cubic “brick” elem ents could be

m ixed together, as appropriate In FEA , one cubic elem ent now has eight nodes, rather than tw o (Fig 1.6), so com putational com plexity is increased Thus, developing efficient com puter softw are to carry out FEA for a given structure can

be a difficult task in itself B ut this general m ethod 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)

M odeling and simulation o f the capacitive accelerometer

(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

In the past fifteen years, m uch progress has been m ade in providing M EM S designers w ith sim ulators and other tools w hich will give them the ability to make

M EM S as useful and ubiquitous W hile there is still m uch to be done, the future is bright for this flexible and pow erful technology Table 1 listed several sim ulation tools and th eir supported levels:

Table 1.1 Available MEMS simulation tools, by level and view

M athem atics, M atlab All

In this thesis I used SU G A R tool w hich applies m odified nodal method to

im plem ent sim ulation program s M ore details o f this tool w ill be discussed in chapter 3

w here k is the spring co n stan t o f the suspension T he d isp lacem en t can be detected and converted into an electrical signal by several sensing techniques This simple principle underlies the o p eration o f all accelerom eters.

From a system p o in t o f view , there are tw o m ajo r classes o f silicon m icro­accelerom eters; open-loop and force-balanced accelero m eters [13-14] In open-loop accelerom eter design, the suspended p ro o f m ass displaces from its neutral position and the displacem ent is m easured either p iezoresistively or capacitively In force- balance accelerom eter design, a feedback force, typically an electrostatic force, is applied onto the p ro o f m ass to co u n teract the disp lacem en t caused by the inertial force H ence, the p ro o f m ass is virtually stationary relative to the fram e The output signal is proportional to the feedback signal In th is section, the first order behavior

o f open-loop accelerom eters w ill be described Steady state, frequency, and transition response w ill be studied analytically T he p erfo rm an ce o f force balance

accelerom eters will be then considered Finally, the operational characteristics o f the tw o types o f accelerom eters w ill be com pared.

2.1.1 O p en -L oop D esign

An open-loop accelerom eter can be m odeled as a p ro o f m ass suspended elastically on a fram e, as show n in Fig 2.1 The fram e is attached to the object whose acceleration is to be m easured The p ro o f m ass m oves from its neutral position relative to the fram e w hen the fram e starts to accelerate For a given acceleration, the p ro o f m ass displacem ent is determ ined by the m echanical suspension and the dam ping C apacitive sensing is used here

:Vo=Tv x(t) — O Vo

-777

Figure2.1 Open loop accelerometer

As show n in Fig 2.1, y and z are the absolute displacem ent (displacem ent relative to the earth) for the fram e and the p ro o f m ass, respectively The acceleration y is o f the interest o f m easurem ent Let x be the relative displacem ent

o f the p ro o f m ass w ith respect to the frame The relative displacem ent is the difference betw een the absolute displacem ent o f the fram e and the p ro o f mass, or

x = z - y

In the follow ing analysis, the displacem ent refers to the relative displacem ent

o f the p ro o f m ass to the fram e (x), unless otherw ise specified The low er case x y, and z denote the d isp lacem en t in the tim e dom ain, and the upper case X Y and Z are their Laplace transform s in the s-dom ain, respectively W hen the inertial force displaces the p ro o f m ass, it also experiences the restoring force from the m echanical spring and the dam ping force from the viscous dam ping The equation o f m otion o f the p ro o f m ass can be w ritten as:

The negative sign indicates that the displacem ent o f the p ro o f m ass is always

in the opposite direction o f the acceleration E quation (2.3) can also be re-writtcn as:

This is the g overning equation for an open loop accelerom eter relating the

p ro o f m ass displacem ent and the input acceleration The perform ance o f an open- loop accelerom eter can be characterized by the natural resonant frequency con and

the dam ping factor C, T he dam ping is determ ined by the viscous liquid or the

cham ber pressure F o r silicon m icro accelerom eters, gas dam ping is most com m only used and the dam ping factor is controlled by the cham ber pressure and the gas properties C ritical dam ping is desired in m ost designs in order to achieve

m axim um ban d w id th and m inim um overshoot and ringing

The natural reso n an t frequency is another im portant param eter in open loop accelero m eter design [15] It is designed to satisfy the requirem ents on the sensitivity and the bandw idth The natural resonant frequency can be m easured

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:

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:

of the structure can be increased by increasing the spring constant and decreasing the proof mass, while the quality factor o f 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

Modeling and simulation o f the capacitive accelerometer 17

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ỈRUNG TÂM THÕNG TIN THƯ VIỆN

Ế á o j_ 4 í_

Note that magnitude o f accelerator is-Y co 2 The motion governing equation,

eq 2.4 becomes:

The frequency response can be obtained by solving this equation either in the time domain or in the s-domain using Laplace transforms To solve it in the time domain, assuming that the initial velocity and displacement are both zero, we can transform Eq 2.10 into s domain and obtain:

(2.13)

The sensitivity o f an accelerometer can be defined as S(ja>) = ■

a{j(o)

Substituting jco for s in Eq 2.11, the amplitude response can be plotted with various

damping factors and is presented in Fig 2.2 (a) It shows, that there are big overshoot and ringing for under-damped accelerometers, and the cut-off frequency for over-damped accelerometers is lower than for critically damped accelerometers The phase lag ^ can also be plotted for various damping factors, as shown in the Fig 2.2 (b)

At low frequency (co « con), we can obtain S 0 =-co~2 = - — from Eq 2.12,

k

which agrees w ith the steady state response state (Eq 2.7) At high

Modeling and simulation o f the capacitive accelerometer 18

frequency (co » con), the m echanical spring cannot respond to the high frequency

vibration and relax its elastic energy Therefore, for a given acceleration, the proof mass displacem ent decrease as the frequency increases From Eq 2.12, we can

obtain S(jco) = so the slope o f the asym ptote is - ^ - a t high frequency

Figure 2.2 Frequency response and phase response with various damping

The accelerom eter can also be used to m easure velocity and displacem ent in addition to acceleration, although accelerom eters used to m easure the velocity have very lim ited applications The displacem ent is proportional to the acceleration when the frequency is below natural resonant frequency, as show n in Fig 2.2 The accelerom eter can be used as a vibro-m eter (or displacem ent m eter) for frequencies

w ell above the resonant frequency From Eq 2.12, w e find that

In other words, the response o f the vibrometer is the ratio o f the Vib ration

amplitude o f the proof m ass and the amplitude o f applied Vibration.

Sim ilar to the analysis for the frequency response, w e can obtain the transient response in the tim e dom ain:

where (/> = tan 1 ^ is the phase lag

The transient responses in the tim e dom ain w ith various dam ping are shown

signal is then used to generate a feedback force onto the p ro o f m ass to counteract the displacem ent due to the acceleration, so the p ro o f m ass is virtually stationary and its displacem ent is negligible The output is now proportional to the feedback signal, rather than to the p ro o f m ass displacem ent, as in the open loop design.

In m any m acro-force-balance accelerom eters, m agnetic feedback is used The feedback force is g enerated by passing the output cu rren t through a restoring coil w ith the m agnetic force linearly proportional to the feedback current

C apacitive sensing and m agnetic feedback are often used to g eth er in order to avoid any effect at the sensing node from the feedback signal, know n as signal feed through D ue to the incom patibility o f m agnetic m aterials and coils w ith IC processes, m agnetic feedback is rarely used in silicon accelerom eters Electrostatic feedback has been the m ost po p u lar technique in recent designs H ow ever, unlike

m agnetic feedback w h ere the force is linearly p roportional to the feedback current, the electrostatic force is given by:

F = - ^ J L = - — V 2

w here d is the gap betw een the tw o electrodes, C is the feedback capacitance, and V

is the applied feedback voltage

The feedback force is linearly prop o rtio n al to the square o f the feedback voltage, rather than to the feedback voltage directly T herefore, this nonlinearity

m ust be corrected in order to have a linear feedback system A furtherm ore difficulty is that electrostatic forces are alw ays contractive - regardless o f the polarity o f the applied voltage.

There are various techniques to overcom e disadvantages (nonlinearity and one-directional force) o f electrostatic feedback The m ost com m on approach is to build tw o sym m etrical feedback electrodes on both side o f the p ro o f m ass so that the push-pull force can be generated by changing the m agnitudes o f the voltage applied to the tw o electrodes By applying a differential v oltage to the electrodes, the capacitive feedback nonlinearity can be corrected (to first order) and a linear system can be realized For exam ple, let V be the DC bias v oltage and v be the differential feedback voltage applied to the tw o feedback electrodes, w e can have the feedback force:

F = i c , r f l( r + v)2 - I c 2d 2( ^ - v ) i = 2C0d 0Vv (2.17)

W hen C] = C 2 = C 0 and di = d2 = d0 T herefore, w ith this technique, the feedback force is linearly proportional to the DC bias v oltage and the differential feedback voltage The push-pull feedback force can be generated by changing the polarity o f the feedback voltage O ther approaches, such as pulse w idth m odulation (P W M ) and pulse density m odulation (PD M ), can also be em ployed w here the

m agnitude o f the feedback voltage is a constant and the feedback force is proportional to the pulse w idth and pulse density o f the feedback voltage

Steady State Response: the output o f the accelerom eter is proportional to the input acceleration:

w h ere T f is the gain in the feedback path, as show n in Fig 2.4

The electrical feedback force is typically m uch larger than the m echanical restoring force; thus, m echanical spring can be n eglected in the analysis

R earranging the equation, w e obtain:

In force-balance accelerom eter design, the disp lacem en t o f the p ro o f m ass is now the error signal o f the feedback system and is determ in ed by the electrical control loop For a given acceleration, the d isplacem ent decreases as the feedback loop gain increases w hile the steady state output rem ains the sam e As show n in Eq 2.19, the output voltage is dependent o f the am p lifier gain in the forw ard path and is inversely proportional to the gain in the feedback path T herefore, the steady-state response is independent o f the displacem ent o f the p ro o f m ass

Dynamic Response: F o rce-b alan ce accelerom eters can be analyzed sim ilarly to the open loop accelerom eter W e can w rite the eq u atio n o f m otion as:

w here a = T J f is the electrical feedback force p er u n it displacem ent (N/m),

P = Tvy is the electrical dam ping force p er unit speed (N /(m /s))

The first tw o term s on the right hand side o f the eq u atio n are the restoring forces from m echanical suspension and dam ping T he last tw o term s are the electric feedback force and are often term ed as the “ electrical sp rin g ” and the “ electrical dum per” R earranging the equation, w e can obtain:

Again, the mechanical spring and damping are usually dominated by their electric counterparts and can be neglected in the analysis Eq 2.21 can be rewritten as:

x + 2^'a>0x + a>lx = - ÿ ( 2 2 2 )

where co0 = ^{k + a) / m - yfa Im is the resonant frequency for the feedback system,

= —— — = ——— is the electrical damping factor

2mcu0 2 mct)0

Comparing Eq 2.3 and 2.22, it can be seen that open loop and force-balance accelerometers have very similar dynamic response when the extra poles and zeros introduced by the electronic feedback are at much higher frequencies than the frequencies o f interests In this case, the analysis for the open loop accelerometers, such as frequency and transient response, are all applicable to the force-balance accelerometers, except that the resonant frequency and damping are controlled by the electronic feedback control loop However, there are some subtle differences between the two classes o f accelerometers and it will be discussed in the next section

2.1.3 Com parisons

Open-loop and force-balance accelerometers have similar responses to the input acceleration It can be easily seen from Eq 2.3 and 2.22 that both o f them are second-order systems Thus the analysis for the open loop accelerometers can all be applicable for the force-balance accelerometers The difference is that the spring constant and damping are determined by the mechanical design in open loop accelerometers, and the “electrical spring” and “electrical dumper” are determined

by the electrical feedback system in force- balance accelerometers Open loop accelerometers tend to be low cost and robust for its simplicity In force-balance accelerometers, the proof mass displacement signal is sensed and amplified The output signal is then used to generate a feedback force onto the proof mass to counter-act the displacement due to the acceleration The proof mass is “virtually

stationary” and its displacem ent is negligible Thus the m echanical suspension can

be m ade m ore com pliant so that high sensitivity can be achieved The dynam ic range is usually determ ined by the linearity o f the electronic feedback control [17-

18].

In the open loop design, the p ro o f m ass deflects from its neutral position upon acceleration and the acceleration is sensed indirectly by the m easurem ent of the displacem ent The accelerom eter p erfo rm an ce is then determ ined by the

m echanical design o f the suspension and the dam ping m ethod The dam ping is controlled by the air pressure in the cham ber and critical dam ping is usually desired for m axim um bandw idth T o achieve high sensitivity, a com pliant suspension is desired for large displacem ent The dynam ic range is lim ited by the linearity o f the

m echanical spring and a stiffer suspension w ill im prove the linearity by lim iting the displacem ent at the cost o f low sensitivity T herefore, tradeoffs should be made

am ong sensitivity, dynam ic range, and bandw idth for open loop accelerom eters

The detection electronics is usually not the m ajo r lim iting factor for the sensitivity in the open loop design T he accuracy is m ostly lim ited by the

im perfection in the m echanical suspension, such as hysteretic and the tem perature sensitivity in the spring constant Thus, the m echanical suspension is the key to open-loop accelerom eter designs T hese problem s lead to the force-balance accelerom eters w here the m echanical suspension is dom inated by the better controlled electrical feedback The p erform ance o f force-balance accelerom eters is

m ostly determ ined by the design o f the interface electronics and the servo loop

A nd the electronic characteristics can be m uch b etter controlled than those in the

m echanical system The feedback design and system p artitio n in g are the key in force-balance accelerom eter design D am ping fo r silicon accelerom eters is often achieved by squeeze-film dam pers, w hich depend on the gas pressure in the package It is difficult to achieve a linear dam ping co efficien t due to the com pressibility o f the gas T he dam ping coefficient can also be sensitive to the tem perature variations In force-balance designs, the dam p in g factor can be determ ined by the electrical system F o rce-balance accelerom eters provide high sensitivity and large dynam ic range w hich can be o ptim ized by electronic design Since the p ro o f m ass displacem ent is negligible, the linearity o f the accelerom eter is

determ ined by the electronic feedback loop It is reported that 120dB dynam ic range can be achieved The m inim um detectable signal is lim ited by the noise level at the input, such as the electronic therm al noise, 1/f noise, B row nian noise, etc and the dynam ic range is determ ined by the interface circuitry and the feedback loop

M ultiple w orking ranges can also be achieved by using variable gains in the feedback path

The bandw idth o f an open loop accelerom eter is set by the ratio o f the spring constant and the p ro o f m ass, w hich has to com prom ise w ith the sensitivity The desired dynam ic response o f a force-balanced accelerom eter can be achieved by tailoring the electronic design, w ithout m odifying the m echanical designs For exam ple, the bandw idth can be doubled by increasing the loop gain by a factor o f four To overw helm the m echanical properties, it is desirable to have high gain in the feedback loop, w hich also increases the bandw idth o f the accelerom eter For som e applications, accelerom eters are operated in a noisy vibration environm ent and the signal o f interest is in the low frequency spectrum The force-balance accelerom eters m ay not be w ell-suited for these applications because the unw anted high frequency vibration m ay introduce errors and distortions in low frequencies

M ounting o f the accelerom eter is also a design consideration The low cu t-o ff frequency feature o f the open loop accelerom eter is now becom e very am ative for these applications The high frequency vibration noise is conveniently restricted and only the low frequency signal is m easured Stability is very critical in the force- balance design, w hereas it is not an issue for the open-loop design Incorporation o f active netw ork in the feedback loop often introduces additional poles and zeros

w hich cause phase distortion and rim e lags in the loop The accelerom eter is then no longer a second-order system Therefore, the feedback m ust be carefully designed to ensure the stability o f the loop

2.2 Capacitive Accelerometer

In chapter 1, w e review ed several types o f accelerom eter A m ong them, the capacitive accelerom eter is m ost w idely used in m icroaccelerom eter [19-20] My thesis focuses on this type o f accelerom eters because o f its advantages as high

sensitivity, good dc response and noise perform ance, low drift, low tem perature sensitivity, low -pow er dissipation, and a sim ple structure.

2.2.1 P osition M ea su re m en t w ith C ap acitan ce

C apacitance m easurem ent is one o f the m ost flexible m ethods o f position

m easurem ent Figure 2.5 illustrates several types o f capacitors that are in com m on use

Figure 2.5 Variety o f capacitor structures that can be used for position sensing

■ The parallel plate capacitor can vary either w ith vertical m otion o f a m ovable plate, m odifying the gap, or by transverse m otion o f one plate relative to another, m odifying the effective area o f the capacitor

■ Interdigital capacitor varies w ith the degree o f engagem ent o f the lingers Also, displacem ent o f one o f the electrodes out o f the plane o f the figure

w ould m odify the capacitance, but this is not a configuration in the com m on use

■ The fringing capacitance deploys an interdigital capacitance as the electrodes are brought into proxim ity w ith a third electrode I f this electrode is grounded, proxim ity reduces the interdigital capacitance I f this electrode is floating, proxim ity increases the interdigital capacitance

Figure 2.6 illustrate differential capacitors that can be used for position sensing There are three electrodes used for the m easurem ent w ith tw o capacitors that are nom inally o f equal size w hen the m oveable com ponent is center M otion o f the m ovable com ponent in the indicated direction increases one capacitance and decreases the other

Parallel plate Interdigitated Fringing

Figure 2.6 Variety of differential capacitor structures

D ifferential capacitors (see Fig 2.7) are useful for canceling m any effects to the first order, providing a signal that is zero at the balance point and carries a sign that indicates the direction o f m otion From a system point o f view , a differential capacitor accom plishes linearization about the balance point L e t's consider the parallel plate exam ple, w ith the gap o f the upper capacitor G | and that o f the lower capacitor G2 We assum e equal area o f both capacitors A voltage +V is applied to the upper plate Sim ultaneously, a voltage - V s is applied to the low er plate This voltage that appears at the output is:

Figure 2.7 Typical circuit use o f a differential capacitor

B ecause w e assum e the areas are equal, this can be rearranged as:

0 g, + g2

I f tw o gaps are equal, the output voltage is zero H ow ever, if the m iddle plate

m oves so that one gap is larger than the other, the output voltage is a linear function

o f this change

Circuits for capacitance measurements: First o f all, we find the charge voltage relationship for the capacitor We can assum e this relations is linear (there is no nonlinear dielectric m edium involved), but the capacitance w ill, in general, depend

on displacem ent [21] T hus w e can write:

where Q is capacitor charge, V is the voltage across the capacitor, and C(x) is the capacitance that depends on one or m ore displacem ent coordinates The current in the capacitor is the tim e - derivative o f the charge:

r f

Figure 2.8 Transimpedance amplifier capture the capacitor current

T he sim plest circuit for m easuring capacitance is show n in Fig 2.8 Both the sensor capacitance C (x) and the parasitic capacitance to ground C P m ust be included because the interconnection betw een the sensor and the am plifier alw ays adds som e

am ount o f parasitic capacitance H ere, a transim pedance am plifier is used to capture the current though the capacitor C(x) T he advantage o f this configuration is that, because o f the virtual ground at the op-am p input, there is negligible charge on the

parasitic capacitance and it does not affect the m easurem ent The output is

V o= -Rf

ic-I f V s is a DC source, V 0 is proportional to the velocity dx/dt W hile an output proportional to the velocity is very useful w hen co n stru ctin g oscillators or force- feedback system s, a m easu rem en t o f velocity is not eq u iv alen t to a m easurem ent o f position To obtain position, one m ust either use a properly initialized integrator to reconstruct the po sitio n from the velocity output, or use a tim e-v ary in g source

w aveform , as discussed below

I f we use a sinusoid as the source V s in F ig 2.8 w e can d eten n in e capacitance directly For exam ple, i f the position, hence the capacitance, is constant and

Vs = Vs0 coscot then the output o f the am plifier is -a )V S0C (x) cos cot T he value o fC(x) can be determ ined from the am plitude o f the output sine w ave H ow ever, if x

is tim e varying, there is a second term in the ou tp u t pro p o rtio n al to dx/dt I f these two term s w ere o f com parable size, the ou tp u t w o u ld be a m ix o f direct position inform ation v ia the C (x) term and velocity inform ation, v ia dx/dt term Therefore, this approach is m o st useful w hen the dx/dt term can be p resu m ed to be negligible,

a situation that can usually be achieved by m aking the frequency o f the V s sufficiently large

W hen using a high frequency A C source, so that the velo city - dependent com ponent o f the cu rren t can be ignored, the circu it o f Fig 2.9 is used We assum e that the value o f R F is chosen such th at at the m easu rem en t frequency, gjRfCf is large com pared to unity T he output is then:

F

Vs

Cp

-o

Figure 2.9 Feedback capacitor is added to circuit o f Fig 2.8

The function o f the resistor is to provide DC feedback to the op-am p input so that the DC value at the inverting input is clam ped at zero A lternatively, this resistor could be co n n ected betw een the inverting input and ground W ithout the resistor, the potential at the input node could drift aw ay from zero, and the am plifier output could saturate

W hen a differential capacitor is used, the voltage on the shared term inal is

m easured Figure 2.10 illustrates the m ost direct approach using a unity - gain buffer to sense the ou tp u t voltage, labeled here as V x H ow ever, now the parasitic capacitance contributes to the output A ssum ing sym m etric positive and negative sinusoidal or pulse signal ± V S applied to the outer term inal, V (x) is given by:

The parasitic capacitance reduces the signal, and also affects the calibration

o f the m easurem ent O ne w ay to m itigate this problem in fully - integrated designs

C - C

Figure 2.10 Measurement the output voltage of a differential capacitor

is to fabricate a guard electrode beneath the interconnection that is driven by the output V 0 Since V 0 is alm ost exactly equal to V x, the net voltage across the parasitic interconnect capacitance is very sm all, ju st like in the virtual-ground exam ples H ow ever, this does add fabrication com plexity, and it is difficult to cancel all o f the parasitic capacitance this w ay An alternate m ethod is to use oppositely phased sinusoidal sources for +V S and -V s , and to replace the C(x) connection o f Fig 2.8 w ith a connection to the shared term inal o f the differential capacitor.

2.2.2 N oise A n a ly sis

This section describes the effect o f noise to signal - to - noise ratio or achievable accelerom eter sensitivity The com putation o f noise is very necessary for design and fabrication [22], In practice, the sensitivity is dom inated by these effects: the intrinsic noise due to dam ping, the noise contributed by the position m easuring circuit, the need to build additional stiffness into structure to prevent either sticking

o f parts during fabrication or excess fragility, residual calibration errors and drift

p ro b lem s

The noise voltage for the detection circuit consists o f therm al (Johnson) noise and 1/f noise The 1/f noise is a low frequency noise and is rem oved by chopper stabilization

I f the voltage used in any capacitance is larger m easurem ent, the signal is larger, too H ow ever, along w ith this voltage goes an actuating force that can disturb the position o f the m oveable elem ent o f the sensor This force is proportional to the square o f the applied voltage (or the tim e average o f the square if

the vo ltag e is high frequency AC w ave form ) and also to the gradient term dC / dx

This gradient term determ ines the sensitivity o f the capacitive sensor The more sensitive device, the low er the voltage that can be used w ithout w hat may prove to

be an unacceptable applied force

T he offset o f the sensor has an interesting origin First o f all, the device m ust live in 1 g w orld Therefore, the orientation o f the device relative to the earth 's gravitational field affects its apparent offset Further, i f the gaps betw een fixed and