MicromachinedHigh-Resolution Accelerometers survey of the literature suggests that the research in this area ... The emergence of silicon microfabrication ... high-resolution accelerometers

1Mechanical Engineering,Indian Institute of Science,Bangalore, India2Electrical CommunicationEngineering, IndianInstitute of Science,Bangalore, [email protected]

REVIEWS

MicromachinedHigh-ResolutionAccelerometers

Girish Krishnan1, Chaitanya U. Kshirsagar2, G. K. Ananthasuresh1 ANDNavakanta Bhat2

Abstract | In this paper, we review the high-resolution, micromachined accelerometers by

enunciating the development of their mechanical components, the electronic circuitry and the

microfabrication processes. A survey of the literature suggests that the research in this area is

mostly focused on improving microfabrication and electronic circuitry. The resolution of the

accelerometers is dependent on the sensitivity of the mechanical components as well as the

ability of the electronic circuitry in suppressing the noise. The noise comes from mechanical

components as well as electronic circuitry and plays a detrimental role in achieving high

resolution. This is reviewed in this paper after explaining the working principles of selected

high-resolution micromachined accelerometers. Sensing acceleration by measuring the change

in capacitance is widely used but it has many complications. These are discussed in detail by

presenting various techniques for sensing capacitance. The microfabrication processes that

dictate the design are reviewed by tracking the evolution of high-resolution accelerometers

found in the literature and comparing their sensitivities and resolutions. This paper gives an

account of the significant accomplishments, some challenges for the future, and a few novel

concepts in achieving high resolution of the order of a millionth of g, the acceleration due to

gravity.

1. IntroductionThe emergence of silicon microfabricationtechniques has enabled miniaturization and batchfabrication of a number of sensors. This has resultedin an increase in their market viability and reductionin costs in addition to enhanced performance inmost cases1. Accelerometers are some of the firstsensors, whose miniaturization and subsequentcost-reduction have paved the way for their use ina number of applications2,3. From being usedas sensors to deploy air bags in automobilesto inertial navigation and other consumerapplications, micromachined accelerometers ofvarying specifications are found in differentapplications today. They all differ, as required

by the specifications of their applications, in theirresolution, range, and bandwidth as well as theprinciple of transduction and microfabrication.

Typical ranges of acceleration for differentapplications are shown in Fig. 1 against the requiredbandwidth. High resolution may be needed inall of these applications depending on a specifictask at hand. For example, as high a resolution asmicro-g (i.e., one millionth of the acceleration dueto gravity, i.e., 10−6

× 9.81 m/s2) is needed notonly in space and inertial navigation applicationsbut also in vibration monitoring to detect humanfootsteps for surveillance applications and forprecise guidance in ballistics. Accelerometers oflow to medium resolution have been reviewed in

Journal of the Indian Institute of Science VOL 87:3 Jul–Sep 2007 journal.library.iisc.ernet.in 333

Resolution, range, andbandwidth: Resolution of asensor is the smallest signalthat it can detect in thepresence of noise. The rangerefers to the maximum valueof the signal that the sensorcan detect. And, thebandwidth is the range offrequency over which thesensor can perform.

Transduction: Conversion ofone form of energy toanother form is calledtransduction. This is the basicprinciple of sensors whereinthey convert a signal in oneenergy domain to, usually, anelectrical signal.

Proof-mass: The bulk portionof an accelerometer that actslike the primary mass sourceto experience the effect ofthe applied acceleration.

Compliant suspension:Restrained mass is the basicmode in which anaccelerometer works. Therestraint comes from acompliant (i.e., flexible)spring. The proof-mass issuspended by this spring.Without the compliantsuspension, the mass wouldsimply accelerate and movefreely instead of oscillating asit does when coupled withthe suspension.

Sensitivity: The outputvoltage of a sensor per unitsignal. The sensitivity of anaccelerometer is expressed asV/g. The larger the sensitivitythe better.

Offset: A non-ideality thatshifts a quantity from its(usually) zero value. An offsetvoltage occurs in a realoperational amplifier(OpAmp) because ofmismatch between thetransistors inside.

REVIEW Girish Krishnan et al.

Figure 1: Range vs. bandwidths required for various applications. (An enhanced version of the data givenin S. Reyntjens PhD thesis at Katholieke Universiteit Leuven, 2002.)

10 -3 10 -2 10 -1 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7

10-3

10-2

10-1

100

101

102

103

104

105

10-4

10-5

10-6

Micro-gravityand space

Machine vmonitoring

ibration

Acousticemission

Seismometry

Shock waves and ballistics

Airbag

Biomedical

Inertialnavigation

Bandwidth in Hz

Miniaturizationsuccessfullyachieved; well reviewed in literature.

Ran

ge in

g

Challenges are faced in miniaturizationand economicviability.

the literature4,5. Such accelerometers have beensuccessfully miniaturized for applications suchas air-bag sensors in automobiles, seismometry,vibration monitoring in machines, etc. On theother hand, miniaturization of high-resolutionaccelerometers has not been easy. There have beenrelatively few micro-g accelerometers reported inthe literature6–19. There are many trade-offs indesigning high-resolution accelerometers. Oncesuccessfully miniaturized and commercialized, theseaccelerometers might find widespread applicationin inertial navigation, in detecting micro-gravityin space, in achieving enhanced tilt-stabilization,and in consumer products such as camcorders,laptops, toys, etc. This paper gives an account of therelevant issues towards obtaining high resolution inaccelerometers and attempts to present the evolutionof microfarbication to the present state-of-the-artin fabricating them.

The basic working principle of mostmicromachined accelerometers is the same. Theyconsist of a proof-mass attached to a compliantsuspension. When an acceleration is applied,the proof-mass experiences an inertial force (D′

Alembert’s force), which is equal to the masstimes the value of the applied acceleration. Thisforce causes the proof-mass to move and therebyelastically deforming the suspension. The amountof deflection is decided by the stiffness of thesuspension. This deflection can be converted to aproportional change in the voltage by a number oftransduction principles such as piezo-resistive20,21,

capacitive22, piezo-electric23, electromagnetic24,tunneling15,16,25–29, optical11,12,30, resonant31,32,thermal33, and others. Each of these types of sensingtechniques needs electronic signal conditioning toconvert the signal to an amplified analog or digitaloutput. An overview of the different transductionprinciples, their applications, advantages, andshortcomings is given in Table 1. Many commerciallyavailable micromachined accelerometers usecapacitive or piezoresistive principle. A fewrepresentative ones are shown in Figs. 2 and 3in which the resolution is plotted against bandwidthand sensitivity, respectively. Note that the resolutionis indicated as g/

√Hz. This is because noise, which

is an important factor that determines resolution,varies with the frequency of the acceleration signalin this manner. This will be discussed later in thepaper. As can be seen in Fig. 3, high sensitivitydoes not automatically imply high resolution. Thehigh-resolution accelerometers seen in Figs. 2 and 3are currently very expensive.

High-resolution accelerometers need to behighly sensitive so that the signal produced dueto the slightest change in the acceleration canbe detected34. Sensitivity of the accelerometeris dependent on both the mechanical and theelectronic sensitivities. Increasing the mechanicalsensitivity by increasing the proof-mass size andreducing the suspension stiffness leads to lowdynamic performance (bandwidth) and workingrange. The sensitivity of the electronic circuitdepends upon the amplifier used and the techniques

334 Journal of the Indian Institute of Science VOL 87:3 Jul–Sep 2007 journal.library.iisc.ernet.in

Micromachined High-Resolution Accelerometers REVIEW

Figure 2: Resolution Vs. bandwidth of some commercial micromachinedaccelerometers. FSC = Freescale semiconductors, C = Colybris, CB =

Crossbow, ST = ST Microelectronics, and AD = Analog devices. The datashown is based on the specifications published by the respective companiesat the time of this publication.

105

104

103

102

101

100

101 102 103 104 10510-1

Noi

se f

loor

(ug

/sqr

t(H

Z))

Bandwidth (HZ)

FSCCCBSTAD

used to cancel the offsets and noise signals. For high-resolution accelerometers, usually capacitive sensingtechniques are used because of their advantages suchas high sensitivity, low temperature-dependence,low drift, and amenability for feedback35. The basicworking principle of the mechanical componentsand the principle of capacitance detection are

explained in detail in Section 2. That section alsogives a brief description of other principles suitablefor high-resolution sensing.

The noise aspects that limit the mechanical andelectronic resolution of the accelerometer13 aredetailed in Section 3. It is found that the electronicnoise is more dominant than the mechanical noisein the accelerometer system. Hence, it is sensible tomake the mechanical components more sensitive inaddition to efforts in improving the electronic noisefloor. Sensitive mechanical components, with theirhuge proof-mass and long, thin, and narrow (andhence flexible) suspensions require a large chip area.It is shown in Section 3 that a resolution of milli-gto several gs can be achieved by most fabricationprocesses within a chip area of 500 µm × 500 µm.Most capacitive micro-g resolution accelerometersoccupy an area of at least 1 mm × 1 mm.

To improve the dynamic behavior of high-resolution accelerometers, they are operated inthe force re-balance mode (i.e., in the closed loopmode), wherein the output voltage is applied as afeedback force to the accelerometer proof-massin a direction opposite to that of the appliedacceleration36. Some compensators or proportional-integral-derivative (PID) controllers are used inthe feedback path to make this system stable37.The closed-loop mode improves the bandwidth,linearity, and range of the accelerometer, however,the sensitivity and resolution decrease. These andvarious other implementation aspects of closed-loopoperation are detailed in Section 4.

Figure 3: Resolution Vs. sensitivity of some commercial micromachined accelerometers. FSC = Freescalesemiconductors, C = Colybris, CB = Crossbow, ST = ST Microelectronics, and AD = Analog devices. Thedata shown is based on the specifications published by the respective companies at the time of thispublication.

104

103

102

101

100

101 102 103 10410-1

Noi

se f

loor

(ug

/sqr

t(H

Z))

Sensitivity (mV/g)

FSCCCBSTAD



Figure 4: Principle of operation of an accelerometer.

x

a

ma

m

y

k

b

z = y – x

One of the major aspects of achieving highmechanical sensitivities is the ability of thefabrication process to form a bulky proof-mass to increase the force and sufficiently thinand narrow suspension beams to reduce thestiffness. Both of these help increase the measureddisplacement of the proof-mass. The mechanicalcomponents of the accelerometers are fabricatedby either surface or bulk micromachining. Surfacemicromachining38 gives very fine feature sizes, butthin proof-mass, whereas bulk micromachining22

gives huge wafer-thick proof-mass but also thicksuspensions. The evolution of the fabricationprocess towards a combination of both surface andbulk micromachining processes to get thick proof-mass and compliant suspensions14 is described inSection 5.

The resolution of accelerometers is mainlylimited by the noise and the offsets in the electroniccircuitry. Capacitance detection circuitries differfrom each other in the manner in which they cancelthese offsets and reduce the noise levels. A numberof analog and digital techniques are reviewed inSection 6. Section 7 summarizes the main pointsof the paper and discusses further opportunitiesfor improving the resolution of micromachinedaccelerometers.

In the next section, we present the basic workingprinciples of mechanical and electronic componentsof micromachined accelerometers.

2. Working Principles of High-resolutionAccelerometers

2.1. Mechanical componentsFor acceleration to be sensed, most accelerometersconvert the effect of acceleration to displacement

and then transduce it to a voltage so that it can besignal-conditioned and worked upon further. Asnoted earlier, accelerometers have a proof-mass,which experiences an inertial force in the oppositedirection of the acceleration that is to be measured.The deflection of the proof-mass caused by thisforce is determined by the suspension’s stiffness andis converted to a voltage using a suitable principle oftransduction (see Table 1). For frequencies of theinput acceleration that are sufficiently smaller thanthe natural frequency of the device, the deflection islinear with the acceleration.

A lumped spring–mass–damper (k–m–b) modelschematically represents an accelerometer as shownin Fig. 4. The acceleration of the frame (a), which isto be sensed, manifests as the inertial force (ma) onthe mass. The equations below show the derivationof the sensitivity of the accelerometer using thefundamentals of mechanics and vibration theory39.

The movement of the frame is denoted by xand that of the mass by y. The net extension ofthe spring and the damper will thus be z = y − x.The equation of motion, by summing the forces—inertial, damping, and spring forces—to zero, isthen given by:

my +b(y − x)+ k(y −x) = 0 (1)

By substituting z = y − x and assuming thatframe’s excitation is harmonic, x = X sinωt withamplitude, X, and frequency, ω, Eq. (1) can bere-written as follows.

m(z + x )+bz + kz = 0

⇒ mz +bz + kz = −mx = mω2X sinωt (2)

By denoting the amplitude of z as Z , Eq. (2) canbe solved to obtain the following result.

Z

X=

mω2√(k−mω2)2 + (bω)2

(3)

Now, by defining terms of the form ω2n =

km and ξ=

bbc

where bc =√

4km, we recast Eq. (3) as

Z

X=

ω2

ω2n√

(1−ω2

ω2n)2 + (2ξ ω

ωn)2

(4)

When operated at frequencies much less thanthe resonance frequency, i.e., when ω ωn,the denominator approaches unity. We thenget Z = ω2X/ω2

n. But ω2X = a (equal to theacceleration) gives:

Z =a

ω2n

(5)



Table 1: Various types of accelerometers and their sensing techniques

Type of sensing, Range of sensing, and Description Applications Advantages Shortcomings

1) Piezo-resistive (0.001-50 g)20,21

Bending of suspension beams due to proof-mass displacementproduces strain in them. This is measured as a change inthe resistance of a piezoresistor placed at the support endof the beam. Change in resistance is measured by using aWheatstone bridge configuration.

Air bag deployment inautomobiles; High-gaccelerometers used forimpact testing etc.

a) Simplicity in structure andfabrication.b) Simple readout circuitrygenerating a low-impedancevoltage.

a) Large temperaturesensitivity.b) Small overallsensitivity of 1–2 mV/gand thus requires ahuge proof-mass.

2) Capacitive (2µg- several gs)22

Displacement of a mass due to applied acceleration leads to achange in capacitance between two plates, one fixed andthe other one attached to the moving mass. The capacitancechange can be due to a change in the overlapping area ordue to a change in the gap.

Air bag deployment inautomobiles, Inertialnavigation, micro gravitydetection, etc.

High sensitivity, good dcresponse, good noiseperformance, low drift, lowpower dissipation, lowtemperature sensitivity,amenability for feedback.

Parasitic capacitance,electromagneticinterference.

3) Tunneling (10 ng- 5 g)15,16,25–29

This accelerometer consists of a proof-mass with a sharp tipseparated from a bottom electrode. As the tip is broughtsufficiently close to its counter-electrode (within a few A)a tunneling current is established. This tunneling current isused as a measure for acceleration. This is operated in theclosed-loop mode.

Inertial navigation,micro-gravitymeasurement in space.

High sensitivity, excellentresolution, linearity can bemaintained by operating inthe closed-loop mode.

Difficult to fabricatesharp tips; nonlinearvariation of tunnelingcurrent with distance;high noise levels.

4) Piezoelectric (High g)23

In this type, the sensing element is a crystal, which has the property of creating a charge when subjected to mechanicalstrain. In the accelerometer, this crystal is bonded to a masssuch that when the accelerometer is subjected to the inertialforce, the mass strains the crystal that emits a signal. Thissignal is related to the imposed ‘g’ force.

Vibration sensors; activevibration control.

High bandwidth; simple inoperation and can beincorporated in anyapplication; amenable foractuation and thus vibrationcontrol.

Low resolution,low sensitivity,and high voltagerequirements.

5) Optical (Nano-g to high g)11,12,30

When light from a Light emitting Diode (LED) source isprojected onto a moving membrane, the reflected light fromit has a lower intensity. The intensity loss is a measure of theacceleration. Similarly, a change in wavelength, polarizationas well as diffraction could be used.

Inertial navigation,detecting small vibrationsin machines.

High resolution, low noise,can operate in places whereother principles fail (e.g.,when electromagnetic fieldsare present).

Large weight, complexfabrication process anddetection circuitry.

6) Resonant (5 – 5000 g)31,32

This type of acceleration measurement makes use of theshift in the natural frequency of the structure with appliedacceleration.

Vibration sensors inmachine tools.

High dynamic range,sensitivity, bandwidth,adaptable to digital circuits.

Leakage of signal,high noise levels.

7) Thermal (0.5 mg to high g)33

It consists of a substrate which is sealed with air or any othergas with a heater exactly in the middle. Two thermocouples arepresent at the ends of the substrate. When the acceleration isapplied, the hot air around the heater is pushed to one of theends by denser cold air thus giving a temperature difference inthe thermocouples which is proportional to the acceleration.

Inclination sensing (Dualaxis), automotive,electronic and gamingapplications.

Low cost, low noise, less drift,simple signal conditioningcircuitry.

Low resolution, batchfabrication can bedifficult, temperature-sensitivity, etc.

8) Electromagnetic (0-50 g)24

It consists of two coils, one on the proof-mass and the otheron the substrate. A square pulse is given to the primary and avoltage is induced in the secondary, which is proportional tothe distance between the proof-mass and the substrate

Air bag deployment. Good linearity, simple signalconditioning.

High power,low resolution,low sensitivity.

By observing the above formulae, it is evidentthat the natural frequency has to be low for highsensitivity. But this has the side-effect of reducingthe bandwidth, which is dependent on both thefundamental frequency and the damping factor.This is shown in Fig. 5 where the frequencyresponse is schematically illustrated. For under-or over-damped situations, the amplitude ratioremains constant for shorter range of frequencies

than the critically damped frequency. Thus, thegreater the sharpness at resonance the greaterthe available operating range for the frequencyof the accelerometer (see Fig. 5). The sharpnessat resonance is quantified by the quality factor,Q = 0.5/ξ. However, the under-damped conditionsresult in a ringing response, where the transients donot die down quickly. Thus, most accelerometersare operated at critical damping conditions36 so



Figure 5: The frequency response curve of an accelerometer18.

as to have a high bandwidth and good sensitivity.Closed-loop operation, which will be discussed indetail in Section 4, makes it possible to have under-damped or even vacuum-packaged accelerometers(i.e., extremely low damping) but the problemsassociated with transients need to be dealt withcarefully.

The displacement of the proof-mass istransduced into an electrical quantity (e.g.,capacitance in capacitive accelerometers). The sizeof the proof-mass and the stiffness and damping ofthe suspension decide the sensitivity, which is the

change in capacitance per unit acceleration. Theresolution depends on the noise, both mechanicaland electronic, in the system. Figure 6 shows thenoise floor against the bandwidth of differenttypes of accelerometers. Of the various sensingtechniques summarized in Table 1, only capacitive,tunneling, and interferometric methods give highsensitivity and fine resolution. The dashed orangeline is the best linear fit to the data. The solidmagenta line is based on a metric defined to bethe ratio of the noise floor and the bandwidthraised to 1.5 (i.e., anf /BW 3/2) that can be used to

Figure 6: A plot of the accelerometer noise with the sensor natural frequency. While the performance ofmost of these accelerometers, which were demonstrated in research labs, nicely align with the theoreticalrelationship (the solid line with a slope of 1.5), the commercial ones shown in Fig. 3 do not This impliesthat many factors such as packaging and environment significantly affect the performance. (Note: the twocommercial ones shown here (ADXL05, 105) do lie close to the line).

105

104

103

102

101

100

10-2 10-1 100 101 102

10-1

10-2

Acc

eler

omet

er N

oise

flo

or (

in u

g/rt

Hz)

Sensor Resonant frequency (in KHz)

Tunneling

Capacitive

Piezoresistive

Piezoelectric



Figure 7: Differential Capacitance arrangement (a) differential capacitance arrangement in for an out ofplane Z-axis accelerometer, (b) Differential capacitance arrangement for an in-plane accelerometer withpositive and negative static electrode along the same side of the proof-mass, (c) for the in-planeaccelerometer with positive and negative electrodes on either side of the proof-mass.

+

− d x−

d x+

(a)

Out-of-planearrangement

Moving combs

Staticmbsco

In-planearrangement

_

++

d x−

32

+

−

d x+d x−d x+

(b)1

3

(c)

3

21d x−

1

compare different accelerometers. The rationale forthis metric can be understood from Eq. 5, whichshows the amplitude of the proof-mass motion asthe ratio of the acceleration signal to the squareof the natural frequency. This means that, wecan obtain the amplitude of motion (and hencethe ensuing electrical signal) for the minimumresolvable acceleration, ar .

Zmin =ar

ω2n

'ar

BW 2'

anf√

BW

BW 2=

anf

BW 3/2(6)

This metric has a slope of 1.5 in the log-logplot shown in Fig. 6. Furthermore, the lower themetric the better the accelerometer. Next, we brieflydescribe the three transduction principles.

2.2. Capacitance detectionThe deflection of the proof-mass is converted to achange in capacitance. This is usually done witha differential capacitance arrangement as it giveshigh sensitivity and good linearity. Many otherarrangements are possible40. In the differentialcapacitance arrangement, the electrode attachedto the proof-mass moves between two other staticelectrodes of opposite polarity. Figure 7a shows az-axis accelerometer, where the proof-mass actsas the moving electrode with two static electrodesabove and below it.

In-plane accelerometers usually use comb-fingers (see Fig. 7b-c), which protrude from theproof-mass to measure capacitance-change. Twokinds of arrangements for these comb-drives areexplored here.

Type 1: In the first arrangement, as shownin Fig. 7b, the two static electrodes of oppositepolarity are situated along the same side of theproof-mass41. In this arrangement, alternative staticcombs, which are of opposite polarity (i.e., 1 and2), need to be isolated from each other to preventshorting. Isolating these combs may not be possiblein most fabrication processes because a number ofconnections need to cross one another as shown inFig. 8a.

Type 2: In this type (see Fig. 7c), the stationarynegative combs are connected on the one side andthe positive combs on the other42. This means thattwo combs on either side make one differentialcapacitance arrangement. There will be someparasitic capacitance between the moving electrodeand the static electrodes marked 1 and 2 in Fig. 7c.There is a loss in sensitivity due to this type ofarrangement, which is explained later in this sub-section. The advantage here is that the positive andthe negative static electrodes are connected togetheras shown in Fig. 8b without having to interweavewith each other.

The change in capacitance between the movingcombs and the two static electrodes is measured asan voltage-change. The circuit diagram to measurethis voltage-change is shown in Fig. 9 40. The twovariable capacitors indicated in the figure are thecapacitances between the static and the movingcombs.

High-frequency pulses V of opposite phases areapplied to nodes 1 and 2 as shown in Fig. 9. This



means that in any cycle, if the voltage at node 1 is V ,the voltage of node 2 is −V . The output is taken atnode 3. This output voltage is given by

Vout = V −(C0 −1C2)2V

2C0 −1C2 +1C1(7)

which simplifies to

Vout =(1C1 +1C2)

2C0 −1C1 +1C2V ≈

1C1 +1C2

2C0V

=1C1 +1C2

CsV for 1C1 −1C2 2C0(8)

where Cs = 2C0 is the sense capacitance.Changes in capacitance 1C1 and 1C2 are

obtained by change in the distance between theelectrodes d. Referring to Fig. 7b, the output voltageis given by

Vout =

ε0Ap

d−x −ε0Ap

d+x2ε0Ap

d

V

=xd

d2 −x2V ≈

x

dV for x d (9)

where Ap is the area of overlap between theelectrodes and ε0 is the permittivity of air.

1C = 1C1 −1C2 =ε0Ap

d −x−

ε0Ap

d +x

=ε0Ap

d

(2xd

1−( x

d

)2

)= C0

( xd

)1−

( xd

)2 (10a)

For small displacements, i.e., when x d, weget

1C = C0

( xd

)1−

( xd

)2 ≈ C0x

d⇒

(1C

C0

)a=

x

d

(10b)

In the case of Fig. 7c, the change in capacitanceis given by

1C =ε0Ap

d −x−

ε0Ap

d1 −x−

ε0Ap

d +x+

ε0Ap

d1 +x

= C0

( xd

)1−

( xd

)2 −C1

(x

d1

)1−

(x

d1

)2 (11a)

For small displacements, i.e. x d, we get

C0

( xd

)1−

( xd

)2 −C1

(x

d1

)1−

(x

d1

)2 ≈ C0x

d−C1

x

d1

(11b)where C0 = 2ε0Ap/d is the base capacitancebetween a pair of combs. The expression C1 =

2ε0Ap/d1 is the capacitance between the stationarycomb of one pair and the moving comb of the otherpair, which can be termed as the cross-capacitance.The ratio of the change in capacitance of this typeto the base capacitance can then be given as

(1C

C0

)b

=C0

xd −C1

xd1

C1 +C2= x

(1

d1

)2−(

1d

)2

1d1

+1d

=x(d1 −d)

d1d(12)

So, by comparing the two types of sensing, the ratioof the fractional change in capacitance in type 2 tothat of type 1 is given by

ra/b =(1C/C0)b

(1C/C0)a=

d1 −d

d1

by taking

d1

d= α, ra/b = 1−

1

α(13)

Figure 8: Arrangement of the differential capacitance configurations (a) for the type 1 arrangement withpositive and negative static electrodes on the same side of the proof-mass (b) type 2 arrangement wherethe positive and the negative electrodes are on either side of the proof-mass.

++ –

–

(a) (b)


Noise floor: The noise, usuallypeak-to-peak, found in theoutput voltage in theabsence of a real signal. It isexpressed as g/

√Hz for an

accelerometer. Thus, thenoise-floor is inherentlydependent on the frequency.The lower the noise floor thebetter.


Figure 9: Circuit diagram to measure the differential capacitance change.

outV

V + V −

3

1 2

From Eq. 13, it can be seen that for a large valueof α, the loss in sensitivity for type 2 is minimized.In both cases, if the sensing distance d is reduced,the sensitivity of the accelerometer increases. Thisis evident from Eq. 10b. However, a small sensegap warrants a tight tolerance in the fabricationprocess, which is usually hard to achieve. This isfurther elaborated in Section 5.

2.3. Tunneling principleThe tunneling principle is one of the most sensitiveof all the available transduction principles forconverting the displacement to voltage. It consistsof two electrodes separated by a very small distanced. When a bias voltage Vb is given between theelectrode plates, there is a current flowing betweenthem, which is given by25

I ∝ Vbe(−αd√

φs) (14)

where α = 1.025(A)−1eV−1/2, φs is the workpotential between the two surfaces through whichtunneling takes place. The current increases by anorder of magnitude for a change correspondingto 1 A displacement25. Using this principle foraccelerometers, the moving electrode is attachedto a proof-mass, while the static electrode is keptbeneath it. For tunneling to be effective, the movingelectrode has to have a very sharp tunneling tipas shown in Fig. 10. A number of tunnelingaccelerometers are presented in the literature25–29.These accelerometers are sensitive and have a lownoise floor. But they are also extremely difficult tofabricate because they require a sharp conductingtip. Furthermore, there is a chance of the tunnelingtip to wear out if there is a constant or intermittentcontact. To overcome this, tunneling accelerometersare operated in the force re-balance mode. Thisis illustrated for capacitive accelerometers in latersections.

2.4. Interferometric principleThe optical interferometric principle gives adisplacement resolution that matches that ofthe tunneling method11,12. In this, a pair of

interdigitated comb fingers is attached to the fixedsubstrate and the moving proof-mass; one set to thesubstrate and the other to the proof-mass. Whenthe suspension-restrained proof-mass moves upand down, one set of fingers move along withit. The laser light that falls on the fingers thencreates a diffraction pattern, whose intensity ismeasured and calibrated for acceleration. This ispictorially shown in Fig. 11. Some are of the theMichelson type43 and the others Fabry–Perot cavitytype11,12. In one embodiment of the latter type,a laser diode and photo detector were integratedalong with the micromachined chip in an acrylicpackage of 8.6 cm3 volume. Because the intensity ofthe diffracted light varies linearly with the relativemotion between the stationary and moving fingers,this accelerometer can be operated in the open-loopmode. This simplifies the control and electroniccircuitry. However, the size is not in favor of thistype of accelerometer. But opportunities do exist forminiaturization12.

3. Noise sources in accelerometersNoise in accelerometers occurs due to themechanical components as well as the electroniccomponents. The white-noise floor is expressed interms of the spectral densities of the noise signal,which is constant for all frequencies40. The actualnoise which limits the resolution of the system thusdepends upon its spectral density and the range ofthe operating frequencies, i.e., the bandwidth ofthe system. The spectral density dsp of the noise-floor at the output of the accelerometer is expressedin V/

√Hz. If BW denotes the bandwidth of the

accelerometer in Hz, then the rms value of the noisefloor is given by44

Vrms = dsp ×√

BW × c (15)

where c is a number chosen based on experience∗.Therefore, for a large bandwidth, the minimumdetectable signal is large. This limits the resolutionfor a given noise-floor. It can be seen from Fig. 6

∗Analog Devices Inc. recommends 1.6 for c.


Fluctuation-dissipationtheorem: Any dissipativeprocess has fluctuations evenin equilibrium. For example,an electrical resistor has afluctuating voltage across itsterminals even when it doesnot carry any current.Similarly, a constrained massvibrates in the absence of aforce. Thefluctuation-dissipationtheorem states that theresponse of thermodynamicsystem in equilibrium for aperturbation is the same asthat due to a fluctuatingforce.

Nyquist relation: Nyquistrelation gives the value of thefluctuating force on adissipative mechanical systemin equilibrium in terms ofdamping and temperature. Itis the mechanical analogueof the Johnson’s noise foundin electrical resistors.


that the noise-floor is low for devices with lowbandwidth. This relationship is evident fromthe concentration of the previously publishedaccelerometers along the dashed orange line in thelog-log plot. If the bandwidth of the system is small,then the resolution of the device improves. Thismeans that the devices with very low bandwidthwould be able to resolve minute changes inacceleration better than those with large bandwidth.However, this is not consistently true with thecommercial accelerometers shown in Fig. 3. Possiblereasons for this might be that packaging andenvironmental factors are different and the reportedvalues of noise floor might not be applicable for thefull range of the bandwidth in all cases. There arealso other sources of noise as explained ahead.

3.1. Mechanical noiseTo probe the possible noise sources in a mechanicalsystem, we make use of the Fluctuation-Dissipationtheorem46, which states that if there is a mechanismfor dissipation in a system, then there will also bea component of fluctuation directly related to thedissipation. This is because any random motiongenerated within the system decays if there is anenergy dissipating mechanism. This might lead tothe temperature of the system becoming less thanthat of the surroundings. To account for this, thereis an associated fluctuating force. This force acts asnoise. In a spring–mass–damper model (see Fig. 12),the energy is dissipated through the damper andhence there should be a component of the force atthe input due to fluctuations.

Figure 10: Working principle of a tunneling accelerometer25.

Proof-massSuspension

Read-out circuitry

Tunneling tip

Top deflection electrode

V

Bottom deflectionelectrode

Figure 11: Schematic illustration of a quarter of the micromachined Fabry–Perot type interferometricaccelerometer12. The interdigitated fingers (shown in different colors) attached to the vertically movableproof-mass are exposed to laser light. In the presence of acceleration, when the movable fingers displaceto a different height, the diffracted light intensity changes linearly with acceleration.

Interdigitatedfingers Fixed to t

subsSuspensionbeams for vertical motion

trate

Laser light

he

Proof-mass


Signal-to-noise ratio: Theratio of the mean-squaresignal to mean-square noiseis called the signal-to-noiseratio. A large of value of thisratio is important for ahigh-resolutionaccelerometer.

Quality factor: Quality factorof a vibrating structure refersto the sharpness of itsresonance as seen in itsfrequency response. It isdirectly proportional to themass and natural frequencyand is inversely proportionalto damping. The larger thequality factor of a sensor thebetter.

Johnson’s noise: Johnson’snoise refers to the fluctuatingvoltage seen across theterminals of a resistorregardless of the appliedvoltage and current. It is aconsequence ofthermal-mechanicalvibrations of the chargecarriers in the lattice of thesolid resistor.

Flicker noise: A low-frequencynoise component found inelectronic elements such asdiodes and field effecttransistors. It is also called1/f noise due to the shape ofits spectral density function.It occurs because of captureand release of electrons inlocalized traps inside asemiconductor material.

Transconductance: It is acontraction of “transferconductance” that refers tothe ratio of the outputcurrent to the input voltageacross the terminals of anelectronic element. It isexpressed in Siemens (1ampere per volt).

Quantization noise: A noisemodel to account for theinevitable error that occurswhen a continuous analogsignal is converted into adiscrete digital signal. Itdepends on the bit resolution,sampling interval, the loadresistance and the analogvoltage.


By using the Nyquist relation, we can determinethe spectral densities of the fluctuating force asshown below.46

Ff =

√4KBbT

where

Ff = Fluctuating force in N/√

Hz

KB = Boltzmann’s constant in N–m/K (16)

b = Damping coefficient in N–s/m

T = absolute temperature in K

Thus, any complex mechanical system can beanalyzed for thermo-mechanical noise by addinga force-generator along with a damper. If thefrequency components in the noise signal arewithin the operating range, then we can express thedisplacement of the mass as follows.

Xn =

√4KBbT

kin m /

√Hz (17)

The detected displacement of the proof-mass due toa signal expressed in the spectral density form, canbe given by

Xs =As

ω20

in m /√

Hz (18)

Where As is the amplitude of the spectral densityof the acceleration noise-floor. Now, the signal-to-noise ratio (SN R) is as shown below.

SN R =

(Xs

Xn

)2

=k2A2

s

4KBbTω40

=M2A2

s

4KBbT

=MA2

s

4KBTω0

(Mω0

b

)=

MA2s Q

4KBTω0(19)

From the above equations, it can be inferredthat increasing the mass and the quality factoralong with decreasing the natural frequency can

reduce the noise considerably. But this limits thebandwidth of the system. In general we can concludethat increasing the sensitivity as well as decreasingthe noise in an accelerometer involves decreasingthe natural frequency, thus limiting the range ofoperating frequencies.

3.2. Electronic noise sourcesThe main source of electronic noise is due to theamplifier used to amplify the signal obtained fromthe differential capacitance setup. An amplifierhas a number of stages of amplification. Thus,the noise in the first stage is amplified in thesuccessive stages of the op-amp40. This noise ismainly limited by the Johnson’s noise due toresistors in the circuitry. Another equally importantnoise source is the 1/f or the flicker noise. Thisis predominant in low frequency signals andoccurs due to trapping and subsequent release ofelectrons along the semiconductor40–oxide interfacein circuits implemented using CMOS technology.The front-end amplifier thermal noise spectraldensity is given by

Sn−amp = 4kBT

(2

3gm

)(1+Fn) (20)

where, gm is the transistor transconductance at theoperating point of the op-amp, and Fn the noisefactor. The 1/f noise is given by

Sn−1/f =Kf

W LCox f(21)

where, Kf is a scale factor, W LCox is the gate-to-channel capacitance. These terms are well definedin a book by Senturia40. In digital circuits, there isan additional noise. This is called the quantizationnoise6. It is given by

nrms = ermsπ2

√5M2.5

(22)

Figure 12: Lumped characterization of the mechanical noise.

b

k

M


Force re-balance mode:When the proof-mass isbrought back to itsundeformed state byapplying a feedback force,the accelerometer is said tobe operating in the forcere-balance mode orclose-loop mode. Thefeedback force is then usedto estimate the acceleration.The force re-balance methodhelps improve the bandwidthof an accelerometer andkeeps the mechanism sensorelement (especially thecompliant suspension) staywithin the linear (i.e., smalldeformation) regime.


Figure 13: Block diagram of the capacitive detection circuit including mechanical and electronic noisesources.

2 /g Hzµ

0.5 /g HzµMechanical noise

Acceleration'a'

K-

MassM

+

+

–

–

K-

1/mass

1/s 1/s

Integrator Integrator 1

-K

-K

Damping 'D'

Spring Const 'K'

Electronic noise

displacement tocapacitance

Op-ampAmplification

K- ++ K-

Output voltage

Mechanical system Capacitance detection circuit

where erms is the unshaped value of the quantizationnoise for a 6 − 1 converter and M is theoversampling ratio13. These are elaborated inSection 6.

Kulah et al.13 showed that the electronic noise(≈ 2µg/

√Hz) dominates when compared to the

mechanical noise (≈ 0.5 µg/√

Hz). This meansthat the mechanical signal should be more sensitivethan the electronic noise floor for small changesin acceleration to be detected. Kulah et al.13 alsonote two other sources of noise. The first is theSource Charging Reference Voltage (SCRV) noise.This occurs due to the noise in the reference voltagethat is used to charge the sense capacitor in eachcycle. This becomes dominant at high samplingfrequencies. The second is the residual motion ofthe proof-mass in the closed-loop operation. Here,the proof-mass will not be completely stationaryin the changing acceleration environment. Thiscontributes to additional noise in the measurementcircuitry. This noise is inversely proportional to thesquare of the frequency. Hence, it is dominant atthe low frequencies. This is attributed to decreasedresolution in the closed-loop mode as comparedwith the open-loop resolution13.

3.3. Achieving µ-g resolution: an exampleIt is now clear that the resolution of theaccelerometer is limited by the mechanical andelectronic noise sources. This means that thesmallest signal that can be detected would have to begreater than the noise floor in the circuit. These twonoise sources can be included in the circuit modelof the accelerometer system as shown in Fig. 13. For

most capacitive detection circuits, a resolution of 1part per million can be safely assumed. This meansthat if the base capacitance is C0, a capacitancechange 10−6 can be detected. However, there aresome capacitance detection circuits, which candetect a minute capacitance change as low as 10 aF.Furthermore, the sense-capacitance C0 should bemore than the parasitic capacitance in the circuit,which is usually around 0.5 pF. For detecting 1 µgacceleration, these constraints can be represented asfollows.

1C

C0≈

x

d> 1e−6 (23)

For a sense-gap of around 1.5 µm 27, the deflectionrequired for the proof-mass of the accelerometeris around 1.5 × 10−6 µm. A proof-mass of size2.4 mm × 1 mm, with a suspension stiffness of17 N/m 27 is able to give a displacement of1.2×10−6 µm, which promises µg resolution. Theoverall chip-area of such an accelerometer is around3 mm×2 mm. The proof-mass thickness is 460 µm.If the area is reduced to 1 mm×1mm, the resolution,with the same electronic circuitry falls to 16 µg.With a 500 µm area, the minimum accelerationthat can be detected is around 400µg. Thus, wecan see that a large chip area is needed to make themechanical components sufficiently sensitive.

It is further seen from the above analysis that themaximum detectable acceleration is less than 1 gbecause, at this acceleration, the moving electrodecomes into contact with one of the static electrodes.Furthermore, a large proof-mass and a flexiblesuspension reduce the natural frequency and thus


State-space transfer function:The function relating theLaplace transforms of twoquantities (usually output andinput) in the state-spacerepresentation.


the bandwidth of the accelerometer. To circumventthese problems, these accelerometers are operatedin force re-balance (closed-loop) mode. This isdescribed in detail in the next section.

4. Closed-loop OperationHigh-resolution accelerometers are usually operatedin the closed-loop mode to increase their bandwidthand the range of operation. In this mode, the outputvoltage of the capacitance detection circuit is fedback to the proof-mass and an electrostatic forcein a direction opposite to that of the accelerationis applied14. In other sensing modes, a suitableactuation is necessary for providing the feedback.The feedback ensures that the deflection of theproof-mass is very small and thus it increasesthe range of operation. Additionally, the minuteproof-mass movement improves the linearity ofthe system. Feedback improves the bandwidth by afactor equal to loop gain45. Furthermore, feedbackimproves the dynamic range, drift, and temperaturesensitivity. Sensitivity of an open-loop accelerometeris inversely proportional to the spring constant ofits suspension which in turn is proportional tothe cube of the length as well as the thickness ofsuspension beams. So, a small change in processingcauses a large change in the sensitivity. By operatingsensing element in electronic force feedback loop,strong dependence of accelerometer on processingparameters can be reduced substantially47. However,operating in the closed-loop mode may make thesystem unstable for large feedback gains. Theremight also be a decrease in the sensitivity of thesystem. In this section, we study the effects ofsome controllers used for feedback, their effecton the bandwidth, sensitivity, and the stability ofthe system.

Figure 14 shows a block diagram of the ofa closed-loop accelerometer, where each blockdepicts a state-space transfer function of the variouscomponents of the accelerometer. The mechanicalcomponent is represented by a second order transferfunction, while the amplifier and the modulationare represented by simple gains in the circuit (Ka

and Kν respectively). The feedback is represented bya gain Kf b.

If the feedback was just a gain factor Kf b =K , theclosed-loop transfer function of the accelerometerbecomes

TFk =V (s)

A(s)=

mKv KaK

ms2 +ds+ k+Kv KaK(24)

By substituting s = 0 in Eq. (24), it can beseen that a simple proportional feedback has asteady-state error associated with it. With a largefeedback gain, the system response tends to oscillate

as it moves farther away from the real axis, thusdecreasing the system’s damping.

By using a controller with an extra pole alongthe feedback path, i.e., by having Kf b = K/(1+ τs),we get an additional degree of freedom in designingthe circuit. However, it can be seen from the transferfunction below that the steady state error is still notovercome. The farther the electronic pole from theorigin (−1/τ) the better the bandwidth and thedynamic behavior of the system40.

TFk =V (s)

A(s)=

mKv KaK

(ms2 +ds+ k)(1+ τs)+Kv KaK(25)

To overcome the problem of the steady-state error, aPID controller of the form Kf b = K (1+β/s+γ s)37

is used. The closed-loop transfer function of thesystem is then given as follows.

TFk =V (s)

A(s)

=mKv KaK (s+β+γ s2)

(ms2 +ds+ k)s+Kv KaK (s+β+γ s2)

(26)

The integral term β helps reduce the steady stateerror. But increase in its value leads to oscillations.The differential term γ helps reduce the ringingresponse of the system. A value of β ≈ 100γ ischosen for good system response. The PID controlleris widely used in most closed-loop accelerometercircuits. In the above analysis, it is assumed thatthe amplifier is ideal and there is no noise in thesystem. However, in reality, the amplifier will havesome bandwidth of its own and the noise in thecircuit does affect the closed-loop response of thesystem40. It is further seen that the closed-loopnoise floor is more than the open-loop noise floor13.In practice, it is seen that the closed-loop noisefloor is sometimes ten times more than that of theopen-loop noise floor. The authors attribute thisto the environmental factors13. This issue needsfurther investigation.

4.1. Feedback to the proof-massThe proof-mass is subjected to a feedback forceby applying the output voltage of the electroniccircuit to a set of feedback combs. However, it is tobe noted that the forces produced by the voltageapplied between two electrodes is always attractivein nature. Assuming an area of overlap of A and asense gap of d, the electrostatic force between twoelectrodes is given by

Fel =εoAV 2

d2(27)


Pull-in voltage: The voltage atwhich two conductors withan electrostatic forcebetween them (usually twoplates in parallel) experiencea catastrophic instability andpull themselves to each other.


Figure 14: Mathematical model of the analog closed-loop accelerometer.

A(s) Tm(s) =

V(s) Kfb

Kv Kam +–

1ms2 + ds + k

It can be seen from the above equation that theelectrostatic force varies nonlinearly with the appliedvoltage. To linearize the force vs. voltage relationship,the arrangement shown in Fig. 15 is used34,36. Here,a dc bias is applied to two static combs so that thenet force on the moving comb is zero. The feedbackvoltage is then added to the dc bias on one comband subtracted from it in the other comb.

The feedback force is given by

Ff b =ε0Anf b(Vb +Vf b)

2

2(d −x)2−

ε0Anf b(Vb −Vf b)2

2(d +x)2.

(28)By assuming d x, Eq. 28 can be simplified asfollows.

Ff b =2ε0nf bAVbVf b

d2(29)

where nf b is the number of feedback combs. Itcan be seen from Eq. 29 that the feedback force isproportional to the voltage Vf b. It is further seenfrom Eq. 28 that the maximum value of the feedbackvoltage Vf b is the bias voltage Vb. This is because thesecond term of the right-hand side of the equationbecomes zero at this voltage. This determines theoperating range of the accelerometer. Care shouldbe taken to see that the dc bias Vb is far from thepull-in voltage of the structure. Since, the maximumvoltage applied on combs (as seen from the left setof feedback combs in Fig. 15, when Vf b = Vb) is

2Vb, the bias voltage Vb should be less than half thepull-in voltage.

In the first four sections, we have outlinedthe basic components of the high-resolutionaccelerometers. In the next section, we describethe evolutionary development of the accelerometerstowards achieving inertial grade resolution byconsidering representative examples from theliterature.

5. Evolution of high-resolution,high-sensitivity accelerometers

It was mentioned in the previous section that high-resolution accelerometers require a large proof massand flexible suspensions, which in turn dependon the effectiveness of the fabrication processes.We chronologically illustrate the developments inthe design and fabrication innovations that leadto the high-sensitivity accelerometers and make acomparison of these in terms of their sensitivitiesand resolutions.

5.1. Bulk micromachined piezo-resistiveaccelerometer (Roylance and Angell, 1979)21

It appears that the first bulk-micromachinedaccelerometer was made by Roylance and Angell(1979). It was a z-axis accelerometer consisting ofa cantilever support holding a huge proof-mass.Piezo-resistors were embedded at the support endof the cantilever beam where the maximum strain is

Figure 15: The feedback combs with dc bias.


Torsional suspension: Themechanical restraint whichallows torsional (i.e., twisting)motion of the suspendingbeams and thus enabling therotation of the proof-mass.

Squeezed-film damping: Thedamping (and spring force)provided by a thin film offluid trapped in the gapsbetween two surfaces. It iswidely analyzed in lubricationtheory. It becomes importantin micromachined structureswhich have narrow gaps.


Figure 16: A bulk micromachined piezo-resistive accelerometer by Roylance and Angell21 with schematicviews from the side and the top.

Piezo resistor

Suspension Proof-masslm

lb

experienced. Figure 16 shows a schematic of thisaccelerometer.

By using the linear beam theory for thesuspension, the stiffness of this accelerometer canbe obtained as

Kst =12EIb

lb(4l2b +6lm lb +3l2

m)(30)

where Ib is the area moment of inertia of the beam,E the elastic modulus, and lm and lb are as shownin Fig. 16.

Further improvement on the structure couldhave two or more beams on either side of themass to reduce the cross-axis sensitivity. A similarstructure could be used for capacitive sensing41.Two electrodes were patterned on glass slabs andwafer-bonded on the top and the bottom of the mass.Since small gaps are required for large sensitivities,the distance between the electrode and the massshould be small. Through slits on the wafer reducedamping that arises due to small gaps below theproof-mass. Wet etching in silicon (110) wafershelps obtain straight vertical side-walls on the wafer.A schematic is shown in Fig. 17.

Fabrication:In Roylance an Angell’ accelerometer, silicon wafersof (110) orientation were taken and wet-etched todetermine the suspensions as well as the slits. Theentire substrate thickness was used for the mass toincrease its value. Finally, electrodes were depositedon the glass plates and they were wafer-bondedto the wet-etched silicon wafer so that differentialcapacitance could be sensed. The electrodes wereused for both sensing as well as actuation in aclosed-loop mode. The major processes requiredfor this fabrication were wet etching and waferbonding. Since these are simple processes, thesetypes of accelerometers were popular and werebatch-fabricated.

Advantages:(a) Simple design and fabrication process.(b) High base-capacitance of close to 20 pF, which

helps eliminate the parasitic capacitance to agreat extent.

(c) Low cross-axis sensitivity.(d) Ability to incorporate slits that reduce

damping.

Disadvantages:(a) Low stiffness for inertial sensing is tough to

obtain.(b) Small gaps below the proof-mass are also not

possible for inertial-grade sensitivity.(c) Temperature-sensitivity is high if there is a

mismatch in the coefficient of expansion of theglass plate and the silicon, which are bondedtogether.

5.2. A capacitive accelerometer using pressuresensor fabrication technology - Rudolf et al.(1983)22

This too is a z-axis accelerometer but uses a torsionalsuspension as shown in Fig. 18. As in the previouscase, the electrodes were deposited on the glassplates and the glass plates were wafer bonded to thesilicon chip. If d, h and L are the width, thicknessand the length of the torsion bars, respectively, thetorque required to twist the bar by unit radian isgiven by the formula:

T

α=

2Gdh3

3L(31)

The torque is related to the acceleration by theformula T = a

∫r dm where r is the distance of the

differential mass dm from the rotational axis. Themoving plate rotates about the torsional beams andthus creates a capacitance-change between the upperand lower electrodes.


Fringing fields: The crowdingof the electric (or other) fieldlines at the corners andedges.


Figure 17: Capacitive Z-axis accelerometer with quad-beam suspension.

Electrodes

Beams

SliSlits

Figure 18: Capacitive torsional silicon accelerometer22.

Electrodes

Mass

Suspension

Fabrication:This particular structure used the pressure

sensor technology. A slightly p-doped silicon (100)wafer was back-etched up to a certain depth andthen electrochemical etch-stop was used to thinthe membrane further down. The etch-stop wasobtained by doping (n-type) of the wafer before theprocess, which defines the suspension. Then, theelectrodes were deposited on the glass plates andwere wafer-bonded onto silicon and then diced. Theentire device was vacuum-packaged.

Advantages:(a) A very thin suspension is obtained which can

reduce the stiffness of the device.(b) Fabrication is easy and its repeatability is good

due to the use of the electrochemical etch stop.(c) Vacuum packaging reduces Brownian noise

and thus increases the resolution of the device.

Disadvantages:(a) The entire device is of uniform thickness, thus

the proof-mass is reduced.(b) Vacuum packaging reduces the damping factor

and thus even small vibrations do not die outquickly enough.

5.3. Surface Micromachined Accelerometers.In the mid-1990’s when the field of micro-electromechanical systems (MEMS) was attractingresearchers’ interest, there was a constanteffort to integrate electronics as well as theinertial sensing components in the samechip. Thus, the complementary metal oxidesemiconductor (CMOS) processes to fabricatesurface micromachined accelerometers started togain predominance because they were amenablefor monolithic integration of electronics on thesame chip. The monolithic integration had theadvantage of good capacitance resolution and lowparasitics. Furthermore, the power consumed bythe entire device is low. Surface micromachiningon the other hand, could give smaller gaps, andthus larger sensitivity, than the bulk micromachinedaccelerometers. However, these accelerometers couldnot be made to be of inertial-grade quality becauseof their poor sensitivity due to small size of theproof-mass. Furthermore, low stiffness and smallgaps decrease the pull-in voltage, which reduce thesense voltage that can be applied for sensing. Theseaccelerometers are best suited for batch-fabricationand for applications such as air-bag deployment inautomobiles.

Surface-micromachined accelerometers canbe for z-axis as well as x − y axis accelerationsignals. The z-axis accelerometers have highsense-capacitances, but suffer from squeeze-filmdamping48. The x − y accelerometers suffer fromlow damping but require a number of comb drivesto increase the base-capacitance. Fringing fieldscontribute to the majority of the capacitance in thex − y axis accelerometers. Some of the popularx − y as well as z-axis surface-micromachinedaccelerometers are discussed below.

5.3.1. A Z-axis surface-micromachined accelerometerwith integrated CMOS circuitry - Lu et al.(1995)49

The structure of this accelerometer consistsof a rectangular mass with L-shaped beams in apin-wheel arrangement as shown in Fig. 19. TheL-shaped beams give very low stiffness in the z-direction and low cross-axis sensitivity. If lb is thelength of the beam and lb its area moment of inertia,the stiffness of the entire structure with four beamsis given by:

Kst r =24EIb

l3b

(32)


Stiction: A contraction of“sticking friction”encountered in releasedmicromachined structuresthat tend to stick to thesubstrate underneath.

Surface and bulkmicromachining: Surface andbulk micromachining ofsilicon refer to twofundamentally differenttechniques in realizing shapesat the micron scale. In surfacemicromachining, thin filmsare deposited and etched ontop of a silicon wafer orsome other substrate tocreate layered microstructure.In contrast, in bulkmicromachining, the siliconwafer (or another substrate)is carved (by chemical etchingor by other means) to createdesired geometrical features.The thickness of themicromachined structure isan order of magnitude largerin bulk micromachining thanthat in surfacemicromachining.


Figure 19: Surface Micromachined z-axis accelerometer by Lu et al.(1995)49.

Suspension

Proof-mass

Fabrication:This structure was fabricated by surfacemicromachining. First, an oxide layer of thicknessequal to the desired gap between the electrodes wasdeposited. It is then patterned to define the anchorsof the suspension. Then a polysilicon layer of thedesired thickness was deposited and patterned toshape the mass and the suspension. Etch-holes,which help decrease damping, were then made. Thesacrificial oxide is then etched away to release theproof-mass and the suspension.

Advantages:(a) High sense-capacitance.(b) Low stiffness leading to high sensitivity.

Disadvantages:(a) Fabrication process needs to take care of the

stiction problem.

(b) Low stiffness leads to a low pull-in voltage.(c) Low z-axis stiffness may also lead to sagging of

the mass due to self-weight.(d) Small gaps lead to large squeeze-film damping.

5.3.2. Surface micromachined lateral (x − y)

commercial accelerometer (ADXL-50) -(Kuehnel and Sherman, 1994)41

This accelerometer, shown in Fig. 20, is one ofthe earliest of the accelerometers manufactured byAnalog Devices Inc. It uses a surface micromachinedproof-mass, suspension, and comb-drives for lateralsensing. The accelerometer was incorporated witha closed-loop circuit. The suspension used in thisconsisted of a pair of simple guided cantilevers oneither side. If lb is the length of each beam, and Ib

is the area moment of inertia of the cross-sectionof the beam then the stiffness of the suspension isgiven by

Kst r =48EIb

l3b

(33)

The fabrication process of this accelerometer issimilar to any typical surface micromachiningprocess. Advantages of these types of accelerometersare that low stiffness is achievable without theproblems associated with sagging due to self-weight.

5.4. Micro-g resolution accelerometers fabricatedby a combination of surface and bulk micromachining- Chea et al. (2004-5)17,18

It is evident from the previous examples thatsurface micromachining can provide low gaps andsufficiently flexible suspensions. However, low massdecreases the overall sensitivity and increases noise.Bulk-micromachining, on the other hand, providesa huge mass, a large capacitance area, but has the

Figure 20: ADXL 50 surface micromachined lateral accelerometer by Analog Devices, Inc.

Static combs

Movingcombs

Suspension

Proof-mass



Figure 21: A high-sensitivity z-axis accelerometer using combined bulk and surface micromachining(re-drawn after reference [7]).

Top electrode

Proof-massSuspensionBottom electrode

Figure 22: A high-sensitivity In-plane accelerometer using combined bulk and surface micromachining byChae et al. (2004)42 (a) A perspective view showing the proof-mass, suspension beams, comb-fingers thatform the capacitors, and bond pads on the frame. (b) a cross-section view of the proof-mass, suspensionbeams, and the comb-fingers.

(a)

(b)

drawback of stiff suspensions and large minimumgaps. Then, the gradual evolution for accelerometerstowards high resolution and sensitivity led to thecombination of the advantages of surface and bulkmicro machining processes where huge proof-masscould be obtained by bulk-micromachining whilesmall gaps and suspension stiffness could be realizedby surface-micromachining.

5.4.1. A z-axis accelerometer using bulk and surfacemicromachining (Chae et al., 2004)7

The proof-mass of this accelerometer, which isshown in Fig. 21, has an area of 2000×2000µm2.The thickness of the proof mass is 450 µm having amass of 2.07 mg. There are eight suspension springs

each 700 µm long, 3 µm thick, and 40 µm wideproviding a stiffness of 14 N/m. The electrodes areat the top and bottom of the proof mass maintaininga gap of 1.5–2 µm. This gap was obtained bysurface micromachining, i.e., by depositing poly-silicon on a sacrificial oxide layer. The electrode hasconsiderable number of damping holes to reducethe mechanical noise. The electrode has verticalstiffeners to decrease its deflection under the internalforces.

5.4.2. An In-plane accelerometer using bulk andsurface micromachining (Chae et al., 2000)42

The fabrication procedure for this accelerometer,which is shown in Fig. 22, is the same as that of



Table 2: A comparison of various capacitive accelerometers discussed in Section 5

Accelerometer structure(with reference)

A torsionalcapacitiveaccelerometer(Section 5.2)

Bulk micromachinedcapacitiveaccelerometer(Section 5.1)

Surfacemicromachinedaccelerometer(Section 5.3)

Accelerometer withcombination of bulk andsurface micromachining(Section 5.4)

Accelerometer with highaspect ratio etching usingDRIE(Section 5.5)

Mass in kg 1e-9 4e-6 0.5e-9 2.76e-6 1.8216e-6

Stiffness N/m 0.158 572 0.94073 14 90.4

Natural frequency in Hz 2000 1835 7500 250 2140

Base Capacitance in pF 1 12.3 0.88 8 32.1

Sensitivity in (1C/C)/g 0.04 0.0148 0.0052 0.25 0.0249

Resolution (assuming a circuitwith 10 ppm resolution)

250µg 650µg 1.9mg 1.5µg 400µg

the above z-axis accelerometer shown in Fig. 21.Deep trenches were made on the proof mass andthe electrodes were deposited in these trenches. Thegap, obtained by sacrificial etching of the oxide, isaround 2 µm. The stiffness of the suspension isgiven by

Kst if f = 96EIb/l3b (34)

where lb is the length of the suspension length andIb is the area moment of inertia of the suspensionsprings. The capacitance obtained in the lateralcase is slightly less than that of the out-of-planecase. The sensitivities (i.e., the relative change incapacitance per applied g acceleration) of theseaccelerometers, expressed as the fractional change ofthe base capacitance, is reported to be 0.25, althoughcalculations indicate that it can be as high as five.

5.5. A bulk-micromachined accelerometer usingdeep reactive etching technique (DRIE) Chaeet al. (2005)17

The design of this accelerometer is very simple; itconsists of a mass with comb drives on both sidesof the mass as shown in Fig. 23. A gap of 2 µmass obtained within the comb drives, the aspectratio of 1:60 permits the thickness of the mass andthe comb drives to be 120 µm. To overcome thedisadvantage of large gaps obtained by traditionalwet etching and wafer bonding techniques, a highaspect ratio etching technique called the DRIE(Deep- Reactive Ion Etching) was used by Chaeet al. (2004)7. This process is a combination ofalternate etching and deposition. The etching createscavities into the wafer and deposition protects thesidewalls from being etched away. This processproduces deep trenches with aspect ration of around1:60. Thus, small but deep gaps between the fixedand the movable plates were realized in capacitiveaccelerometers.

Table 2 summarizes the physical andperformance parameters of the high-resolutionaccelerometers described above. The physicaldimensions determine the mass, stiffness and thebase-capacitance. The natural frequency, sensitivityand the resolution are decided by the overallperformance of the accelerometer. It can be noticedin Table 2 that less than 1 µg resolution is possiblebut its bandwidth is decreased to 250 Hz.

It was noted in Section 3 that the ability of theelectronics circuitry to cancel amplifier offsets andnoise determines the resolution of the electroniccircuit. In the next section, we review the differenttypes of capacitance measurement methods.

6. Capacitance Measurement TechniquesThe displacement of the proof-mass due to anapplied acceleration might be as small as a fewAngstroms. The electronics circuitry has to besensitive enough to detect such a small displacement.So, the capacitance-sensing circuit should be able tosense the capacitance-change in the range of femtoto atto Farad. Various techniques are investigatedfor measuring such a minute change in capacitance.In this section, various capacitive sensing techniquesare described first. This is followed by descriptionsof both digital and analog closed-loop systems usedin accelerometers.

6.1. Analog circuitsIn micromachined accelerometers, a sensing circuitis designed to obtain low noise levels and highsensitivity. Operational Amplifiers are used incircuits to obtain high gain and thereby increase thesensitivity. But an op-amp itself contains manynon-idealities such as input offset, 1/f noise,thermal noise, etc. Hence, challenge in designingsensing circuit for an accelerometer is in increasingthe sensitivity while keeping the noise level low.



Figure 23: A DRIE in-plane accelerometer.

-

Suspension

Proof-mass

Staticelectrode

+

There are several capacitive sensing techniques thatare used to detect changes in capacitance in anaccelerometer. Some of them are:

1. Switched capacitor19,47,49,50–56

2. Correlated double sampling45,52,57–59

3. Chopper stabilization methods50,60

Switched capacitance is widely used to reducethe offset of the op-amp. In this technique the circuitoffset is measured at a fixed time and is stored in acapacitor which can be later brought in series withthe input in order to cancel the offset. Chargingand discharging of the capacitor is controlled bythe switches. This technique is also known as autozeroing (AZ). This is a simple method to eliminatethe offset.

At higher operating frequencies, the value of1/f noise keeps on decreasing and it reduces belowthe thermal noise mark at some frequency. Thisfrequency is known as the corner frequency. Toremove all the noise, the auto zeroing frequencyshould be higher than the noise corner frequency.The residual noise is almost white. However, itis not equal to the thermal noise floor, as it isincreased by the ratio of the unity-gain bandwidthof the amplifier and the auto zeroing frequency.The reason for this is that due to the samplingaction, high-frequency components are folded backto the baseband. The higher the bandwidth ofthe amplifier, the more the sampled noise on thecapacitor. The advantages and disadvantages of theswitched capacitor technique are as follows:

Advantages:(a) Relatively easy to realize.(b) Good for discrete, sampling based system.

Disadvantages:(a) It suffers from noise folding.(b) Overall noise level is more than the thermal

noise floor.(c) High kT/C noise at low capacitance.(d) Suffers from switching noise.

Correlated double sampling (CDS) technique isan extension of the switched-capacitor technique.The CDS technique, which was originally introducedto reduce the noise produced in charged-coupleddevices (CCD’s)58,59, can be described as an AZoperation followed by a S/H (sample and hold).It also reduces the 1/f noise along with the offset.The offset of an Op Amp is independent of theinput signal, but 1/f noise depends on the inputfrequency. In this method, sampling is done twice ineach cycle. First, the 1/f noise and the offset of theOp-Amp are stored in a capacitor. Next, the storednoise and offset are subtracted from a sample signalwhich is corrupted with 1/f noise and offset. Onesuch circuit is shown below in Fig. 24. Here, thesense-capacitors (Cs1and Cs2) correspond to thedifferential capacitors of the accelerometer. Duringthe first phase – the offset store phase – φ1 is closedand φ2 is open resulting in the storage of 1/f noiseand offset onto the holding capacitor (Chold). Inthe second phase – the signal amplification phase –the signal proportional to the difference betweenthe two sense capacitors gets amplified and appearsat Vo. Overall, CDS is superior in terms of noisereduction compared to simple switched capacitorcircuit as it not only reduces the offset but alsohelps reduce the 1/f noise. But as it still relies onswitched capacitors, it suffers from the switchingnoise. The advantages and disadvantages of thecorrelated double sampling technique are as follows.



Figure 24: Correlated double sampling technique.

Ci Φ1

Φ2

Φ1

Φ1

Cp

Vx

Cs1 /2

Cs2 /2

Chold

Vo

Advantages:(a) kT/C noise is reduced significantly.(b) It can also be used to enhance the effective gain

of the op-amp.

Disadvantages:(a) It uses two phase switches which have to be

controlled and timed precisely.(b) It suffers from noise folding.(c) Suffers from switching noise.

Chopper stabilization is based on modulationand demodulation principle. Initially, the signalmodulates a high-frequency carrier (fc), also calledchopping, signal. The effect of 1/f noise and offsetis eliminated by up-converting the signal to a highfrequency. The sensed signal is then amplified at thishigh frequency. Finally, the signal is demodulatedto the baseband signal. The chopping frequencyis chosen such that Brownian noise instead of 1/fnoise will decide the noise floor. This frequency isgenerally in the range of 1–2 MHz60. Consequently,the baseband noise of the chopper amplifiers isalmost equal to Brownian white noise.

As the signal is modulated to a high frequencybefore passing trough Op Amp, it does not getaffected by offset of the Op Amp because the offsetis dc. When the signal is de-modulated, this dcoffset gets modulated to a high frequency whichcan be filtered out. This way, the signal is alwayskept away from the offset to reduce the effect ofthe offset on the signal. The principle of chopperstabilization is depicted pictorially in Fig. 25. Here uis input signal, n the noise, p(t) is the modulatingsignal and p(t − t0) is the de-modulating signalwhich is delayed version of p(t), to account forthe signal propagation delay through the Op-Amp.For ease of implementation the chopping signal issquare wave rather than sine wave. This results in

up-conversion of signal and subsequently the noiseat odd harmonics of the chopping signal. Howeverall the higher harmonics are filtered out using alow-pass filter60.

As shown in Fig. 25, modulating signal pcan take values between +1 or −1, If n is noisesignal at the low frequency, then p2

= 1 and theoutput y = (up + n)Ap = uA + A(np). Thus, theamplified signal is in the base-band but the amplifiednoise is up-converted to high frequency throughmultiplication by p. If p has period Tp with 50%duty cycle, its Fourier coefficient denoted by Pk isgiven by60.

Pk =1

Tp

∫ Tp

0P(t)e−jk(2π/Tp)t dt

=2

jkπif k = ±1,±3,±5, . . .

= 0 if k = 0,±2,±4, . . . (35)

If Sn(ω) denotes the power spectral density (PSD)of n, then PSD of pn becomes

Spn(ω)=4

π2

∞∑k=−∞

1

(2k+1)2Sn

(ω− (2K +1)

2π

Tp

)(36)

Thus, the effect of Sn is strongly reduced inbaseband.

The circuit realization of chopper stabilizationcan be done in various ways. One of them is byarranging switches at the input of amplifier toalternately route the signal to + and − inputsof the Op-Amp . Thus the input is alternatelymultiplied by +1 and −1. The output of this circuitis demodulated. Demodulation is done either byusing the same kind of switching arrangement orby using a multiplier circuit. Finally, the outputis filtered to remove the offset component of the


Limit cycle: A closedtrajectory in the state-spacerepresentation of a dynamicalsystem into which at leastone other trajectory spiralsinto it. It refers to a sustainedoscillation of a nonlineardynamic system.

Single-bit coding: The processof converting a continuousamplitude analog signal intobinary-valued digital signal.

ADC/DAC: Analog to DigitalConverter (ADC) transformsan analog signal into a digitalword. The higher the numberof bits in the digital word,the higher the efficiency oftransformation. Digital toAnalog Converter (DAC)performs the inversetransformation of ADC.

Delta modulator: The Deltamodulator uses themodulation scheme wherethe output of the modulatoris responsive to the differencein input signal samples attwo time instances, ratherthan the absolute magnitudeof the input signal.


Figure 25: Chopper stabilization principle.

p p(t- to)

Input Output

uy

SignalNoise

Noise

f f f0 0 0

Modulation and amplification Demodulation

n

fc fc

+

signal. High order low-pass filtering such as 3rd or5th order Butterworth or any other active filter isgenerally used. The cut off frequency of the filtershould be more than highest frequency expectedat the input but less than chopping frequency.The advantages and disadvantages of the chopperstabilization technique are as follows.

Advantages:(a) Noise floor is decided by white noise instead of

1/f noise.(b) Good choice for continuous time system

like MEMS based accelerometer where lowbaseband noise is an important requirement.

(c) Unlike the AZ technique, the choppermodulation does not introduce aliasing ofthe broadband noise.

Disadvantages:(a) Effective gain of the op-amp is usually slightly

reduced.(b) Suffers from switching noise.

6.2. Analog feedbackAlthough analog feedback has been discussed earlieras an example of a closed-loop system, in this sub-section the electronics aspect of the analog feedbackis discussed. In analog feedback, the feedback isproportional to the sensor output. The constant ofproportionality is such that it counters the forceapplied and tries to bring the proof-mass to nominalposition. This force acts as an equivalent electricalstiffness. All the forces—viz., damping, stiffness andelectrical feedback—co-exist and act simultaneouslyon the proof mass. But in order to get negligibledisplacement even when constant force is applied,the feedback force has to be integrated before beingapplied to the proof-mass. This can be done by

introducing proportional-integral (PI) controller inthe feedback. The block diagram of such an analogforce feedback system is as shown in Fig. 26 37.

Finding exact values of Kp and Ki is an iterativemethod. For tuning PID many methods are availableto start with better approximated values. One ofthem is Ziegler–Nichols Step–Response Method.The step-response of the system can be predictedusing a software tool such as Simulink61. Here,we get approximate values of Kp and Ki based onopen-loop step-response of the system. Based onthis approximate value, the PID gains can be fine-tuned to meet specific parameters such as the finaldisplacement of the proof-mass or the responsetime.

The main advantage of using analog feedbackloop is that it does not have the problem oflimit cycles that we see in digital feedback loops.Furthermore, unlike digital loop, analog closed-loop accelerometers do not require additional signalprocessing after the final sense-signal is obtained.Additionally, the proof-mass is not subjected to highfrequency oscillations as there are no limit cycles.The only disadvantage of analog closed-loop is thatthe final output signal is analog. Digital signal isalways a preferred form of final output as digitalsignal is easy to transmit and store. Furthermore, ananalog signal is more vulnerable to noise.

6.3. Digital feedback (Use of Sigma-deltamodulation)

In 1962–63, Inose and Yasuda published twoimportant papers62,63 in which they introducedthe concept of the single-bit-coding method. Thebasic idea was to have an integrator followed by adelta modulator with negative feedback. That wasthe first basic first-order sigma delta modulator.Though it was introduced as a coding method, it


One-bit quantizer: Analog toDigital Converter withone-bit resolution.

Over-sampling: For an ADC,the analog signal has to besampled at a minimumfrequency (Nyquist rate)which is twice the bandwidthof the signal. The process oftaking the samples at a ratefaster than the Nyquist rate iscalled Over-sampling.

Noise-shaping: Noise shapingis a technique to reduce thequantization noise in ADCs.The quantization error is putin a feedback loop tominimize the in-band noise.

Decimation filter: The SigmaDelta modulator produces abit stream at a very highfrequency. The purpose ofthe Decimation filter is toaverage the bit stream andreduce its rate to the requiredvalue.

Tyspkin or Hammel’s method:This is a technique to analyzethe Limit cycle instability in aclosed loop feedback system.


Figure 26: Block diagram of analog force feedback accelerometers (based on a reference37).

1/m

Integrator IntegratorSensing Circuit(Gain, ( )/ /C C gΔ )

PID

(Kp, Ki, Kd)

Damping

Force adder

b

k

Mechanical System

m Σ

ElectrostaticForce

Kf

Springforce

soon became popular as ADC/DAC at places wherelow or medium frequency and high resolution wererequired. Sigma-Delta ADC/DAC is an extension ofthe concept with the addition of a low pass filterafter the modulator, which may be analog or digitaldepending on whether required output is digital oranalog. As the VLSI technology progressed, the needfor integrating more blocks on chip arose whichmade Sigma-Delta method popular. As its buildingblocks are simple, the Sigma-Delta modulationmethod became the first choice for designers inmany low-frequency applications. With the adventof micromachined devices, the method found itsapplication in accelerometers. The Sigma-Deltafeedback was first applied to micromechanicalelement in a closed-loop accelerometer by Henrionin 199064. The Sigma-Delta modulation systemwas selected because of its wide dynamic rangepossibilities. Use of a Digital Sigma-Delta modulatorresulted in an all-digital closed-loop force-balancesensors.

The name Sigma-Delta modulator comes fromputting the integrator (sigma) in front of the deltamodulator65. As expected, there is an integratorahead of a comparator, which acts as normaldelta modulation, and there is a feedback throughDAC. The final signal is passed through a filter fordecimation. The filter can be either digital or analogdepending on the type of the signal we need at theoutput.

Figure 27 shows a simple block diagram of afirst order Sigma-Delta Analog-to-Digital Converter(ADC). The input signal X comes into the modulatorvia a summing junction. It then passes through theintegrator, which feeds a comparator that acts as aone-bit quantizer. The comparator output is fed

back to the input-summing junction via a one-bitdigital-to-analog converter (DAC). The signal alsopasses through the digital filter and emerges at theoutput of the converter. The feedback loop forcesthe average of the signal W to be equal to the inputsignal X (see Fig. 24). The in-band noise power andthe signal-to-noise ratio is given by29

σ2ey = σ2

eπ2

3

(2fB

fs

)3

(37)

And SNR in dB is given as follows.

SN R = 10log(σ2x)−10log(σ2

e )

− 10log

(π2

3

)+30log

(fs

2fB

)(dB)(38)

Assuming an over-sampling ratio, fs2fB

= 2r , Eq. (37)

becomes

SN R = 10log(σ2x)−10log(σ2

e )

− 10log

(π2

3

)+9.03r(dB) (39)

It can be seen from Eq. (39) that for every incrementin r, SNR improves by 9 dB, equivalent to aresolution increment by 1.5 bits. Similarly it can beshown that if second order sigma-delta modulationis used, SNR increases by 15 DB, equivalentto resolution increment by 2.5 bits. This is theadvantage of using sigma delta over all other digitalmodulation techniques.

The Sigma-Delta feedback was first appliedto Micromechanical element in a closed loop



Figure 27: First order sigma delta ADC.

Integrator Comparator

1 Bit DAC

+

-X

W

Modulator

YDigitalFilter

Σ

accelerometer by Henrion64. The basic blockdiagram an accelerometer that uses a Sigma-Deltamodulator can be represented as shown in Fig. 2837.

The Sigma-Delta modulation involves twoimportant techniques namely over-sampling andnoise-shaping. It is known that for a faithfulreproduction of any signal, it needs to be sampledat a frequency higher than the Nyquist rate, whichis twice the bandwidth of signal. The basic ideaof over-sampling is to trade-off resolution in timefor resolution in voltage. This means that whenthe signal is sampled at a rate much higher thanthe Nyquist rate, the samples are averaged to givebetter resolution at the output. The output of theSigma-Delta modulator, therefore, has to be down-sampled to the base-band frequency. That is donethrough a special type of filter called the decimationfilter. Decimation filter can be on-chip or it can beoff-chip using a signal processing algorithm. Various

approaches of decimation filter design are availablein the literature66,67.

The main advantage of the Sigma-Delta feedbackis that it gives digital output. In certain cases,if limit cycles are controlled properly, it alsogives better resolution than analog closed-loopaccelerometer. But as mentioned earlier, it requiresextra-decimation filter and it suffers from limitcycles which is a kind of instability. There area few ways to reduce the limit cycles to haveimproved minimum detectable displacement due tothe applied acceleration. In the next sub-section wewill see how to reduce the limit cycles.

6.3.1. Limit CyclesThe presence of nonlinear elements in the

feedback loop can produce stable boundedoscillation called a limit cycle, Here, the nonlinearityof one-bit quantiser gives rise to a limit cycle. Forcefeedback is unstable without compensation becauseof the 180 phase delay from sensing element forfrequencies above the resonant frequencies. Thereare three approaches to solve this issue, namely,electronically limiting the loop-bandwidth; over-damping the proof-mass69; and compensating witha lead filter49,64.

The first approach uses electronic low-pass filterto reduce the loop-bandwidth to a frequency wellbelow the resonance frequency of the proof-mass.But this technique is applicable only in situationswhere low sensitivity and limited bandwidth aretolerable. Thus, it reduces some benefits of feedback.The second solution is simple but, because ofhigh Brownian noise floor of an over-damped lowmass-sensing element, it is practical only for low-performance applications. The third approach of

Figure 28: Block diagram of force feedback digital accelerometers.

1/m Kpo

Integrator Integrator Comparator

Damping

Springforce

Force adder

b

k

Mechanical System

m

ElectrostaticForce

Kf DAC

DecimationFilter

Σ



Figure 29: Limit Cycle waveforms of linearised feedback force with compensation (solid lines) and verticaldisplacement (Dashed line)70.

Time0 T/2 T

+ F0

– F0

– Fv

+ Fv

+ z

Td

Ts

Displacement

– z

Vfeedback

using a compensation filter requires extra circuitrybut there are no other trade-offs involved in it. Thebasic strategy is to add a left-half plane zero to theloop transfer function in order to decrease the phasedelay at unity gain frequency. A limit cycle waveformwith lead compensation is shown in Fig. 29.

Kraft37 has predicted some tendencies in thecomparison of limit cycles, which imply that,for frequencies much smaller than that of thelimit cycle, flat frequency response and lineartransfer characteristic are expected; and higher thesampling frequency higher the limit cycle frequencyhence the higher the bandwidth of the transducer.Furthermore, for higher limit cycle modes a lowercut-off frequency is expected, and finally, an increasein the feedback voltage results in an increase in thebandwidth.

A limit cycle can be found by mathematicalanalysis by deriving the transfer function of thefeedback loop39. There is another digital signalprocessing approach for finding limit cycles used byColinet et al.68 where they used two-step methods.The first step is the determination of system’sresponse for a candidate limit cycle, which isdone by Tsypkin or Hamel’s Method. Softwareimplementations of these methods were donethrough FFT algorithms. In the second stage, theconsistency of sign-change at comparator inputwith those of candidate limit cycle is verified.

6.3.2. Recent DevelopmentsThe CMOS technology scaling in the last few

years has continuously improved the speed ofthe transistor and the circuits built thereof. The65 nm CMOS technology has enabled the teraHz transistors. Since the Sigma-Delta techniqueexploits the resolution in time to provide a very highresolution in voltage, the performance of Sigma-Delta converters has steadily improved over the last

several years with very high frequency and highresolution converters implemented in a variety ofapplications71,72. Recently a higher order Sigma-Delta micro-g accelerometer has been realized witha capacitive resolution of 2 aF and with a dynamicrange of 95 dB at 20 Hz73. It appears that the digitalfeedback with Sigma-Delta modulator will becomea preferred technique to implement the closed loopaccelerometers, with further improvements in theCMOS technology in future.

7. Summary, Recent Developments, andFuture Trends

In this paper, an attempt has been made to enunciatevarious aspects of micromachined accelerometersthat help in achieving a very fine resolution. Someof the salient points noted regarding achieving highresolution are summarized below.

(a) The resolution of the accelerometer is mainlylimited by the electronic and the mechanicalnoise floors.

(b) These noise floors have a strong dependenceon the frequency of operation (bandwidth) ofthe system.

(c) Electronic noise dominates the mechanicalnoise in most analog and digital accelerometers,and thus requiring a high mechanicalsensitivity to overcome the electronic offsetsand noise.

(d) A high mechanical sensitivity requires bulkyproof-mass and flexible suspensions, thusreducing the bandwidth of the system.

(e) Microfabrication processes which make useof a combination of surface and bulkmicromachining processes help in achieving ahigh mechanical sensitivity.

(f) Improved bandwidth, linearity, and range canbe obtained by operating the accelerometerin a force re-balance mode. Feedback can beeither analog or digital.


Dithering: Dithering is aprocess of adding a smallamount of random noise,with amplitude that is half ofthe least significant bit, tominimize the effect ofquantization noise. However,the white noise floorincreases slightly, which maybe tolerable in a variety ofapplications.


Figure 30: An Accelerometer with a displacement-amplifying compliantmechanism (DaCM)

External suspension

Sense-combs

Displacement-amplifying compliantmechanism (DaCM)

Proof-mass

Suspension

(g) Electronic noise is overcome by a numberof techniques such as chopper stabilization,co-related double sampling, etc. Capacitancechange as low as 2 aF can be detected.Electronic noise can be reduced to 100 nV.

Overall, miniaturization of micro-g resolutionaccelerometers is possible within the capabilitiesof the microfabrication technology. Better noisecancellation techniques and methods to increase themechanical sensitivity are worth investigating toachieve micro-g and sub micro-g resolutions over awide bandwidth.

Since an accelerometer involves many specializedfields such as structural mechanics, structuraloptimization, microfabrication, capacitive sensing,signal processing, and feedback control systems,new developments in any of them will contributeto improvement in its performance. Mechanicalsensitivity can be enhanced with a large proof-massand a flexible suspension. Recent developmentshave shown that a combination of surface andbulk micromachining processes are able to givethe required proof-mass thickness and suspensionstiffness, thus achieving micro-g resolution within achip area of 2 mm × 1 mm. Though the sensitivityof such structures is high, their natural frequency islow, thus decreasing their bandwidth. To overcomethis problem, a displacement-amplifying compliantmechanism (DaCM) can be used to amplify thedisplacement of the proof-mass to increase itssensitivity74,75. The displacement is sensed fromthe output portion of the DaCM, where the sense-combs are placed. The inertia at the output of the

DaCM, which displaces by a large amount, is verysmall compared to the proof-mass. Thus, its naturalfrequency is larger than that of the conventionalaccelerometers with the same sensitivity. However,the flexibility of the output of the DaCM leads tolarge cross-axis sensitivity, which can be limitedby adding a suspension at this location. Figure 30shows such an accelerometer with a DaCM. When,using such a device in the force re-balance mode,feedback has to be applied to the proof-mass andthe sense-combs at the output of the DaCM75.

On the electronics front, there have beenattempts to reduce offset up to 100 nV using nestedchopper stabilization76. Such a development interms of offset and noise reduction can be appliedto the accelerometer. Similarly, there have beenattempts to increase sensitivity by introducingdithering to increase resolution77. Recently, therehas been work on using floating gate transistorsin sensing circuits, with a claim to improveperformance in terms of improving SNR andreducing power consumption78,79. There has alsobeen a development of simple one-mask processusing thin-film SOI technology80. Thus, despiteseveral decades of research in this area, it is stilla dynamic field to work on and to apply variousnew ideas. Further research and increased use ofthe high-resolution accelerometers might help inbringing their cost down.

Received 5 August 2007; revised 11 September 2007.

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Girish Krishnan completed his Bachelors inMechanical Engineering from R. V. College ofEngineering, Bangalore in the year 2004 andmasters (MSc) in Mechanical Engineeringfrom Indian Institute of Science (IISc),Bangalore in 2006. He is currently pursuinghis PhD in the Mechanical engineeringdepartment, University of Michigan, Ann

Arbor. His areas of interests are Microsystems design, Compliantmechanisms and Topology optimization. He is a student memberof the ASME (American Society of Mechanical Engineers).

Chaitanya Kshirsagar is BE in electronicsand tele-communication from GovernmentEngineering College, Aurangabad. He hasworked as research scholar in microelectronicslab, electrical communication engineeringdepartment, Indian Institute of Science.His research interest includes capacitivesensing for MEMS devices, VLSI circuits and

nanoscale devices. He is currently pursuing the Ph.D. degree in


www.mathworks.com

http://www.elecdesign.com/Articles/Index.cfm?AD=1&ArticleID=14105

http://www.elecdesign.com/Articles/Index.cfm?AD=1&ArticleID=14105


electrical and computer engineering at the University of California,Santa Barbara.

G. K. Ananthasuresh obtained his B.Tech.degree in 1989 from IIT-Madras, M.Sfrom University of Toledo, OH, USA, andPh.D. in 1994 from the University ofMichigan, Ann Arbor, MI, USA. He wasa post-doctoral research associate in theMicrosystems Technology Laboratories at M.I. T., Cambridge, MA, USA, before he joined

the University of Pennsylvania’s Mechanical Engineering andApplied Mechanics department in 1996 as an Assistant Professorand then promoted with tenure to Associate Professor in 2002.He joined the Indian Institute of Science (IISc) in 2004. Hisresearch interests include compliant mechanisms, microsystemsmulti-disciplinary design optimization, topology optimization,microfabrication, mechanism design and kinematics, proteinmodeling and design, micromanipulation, and bio-design. He hasmore than 125 publications out of which 46 are journal papers. Heis also the author of six book-chapters and an edited book entitled“Optimal Synthesis Methods for MEMS”. He has three best paperawards at international conferences and three patents. He is therecipient of the Swarnajayanthi Fellowship from the Departmentof Science and Technology, India; National Science Foundations’Early Career (CAREER) award by the National Science Foundation,USA; and the Society of Automotive Engineers’ Ralph TeetorDistinguished Educator award. He served/serves on the editorialboards of four journals and is the editor of the Institute of SmartStructures and Systems (ISSS) newsletter, Sukshma.

Navakanta Bhat received his B.E. inElectronics and Communication fromUniversity of Mysore in 1989, M.Tech. inMicroelectronics from I.I.T. Bombay in1992 and Ph.D. in Electrical Engineeringfrom Stanford University, Stanford, CAin 1996. Then he worked at Motorola’sNetworking and Computing Systems Group

in Austin, TX until 1999. At Motorola he worked on logictechnology development and he was responsible for developinghigh performance transistor design and dual gate oxide technology.He joined the Indian Institute of Science, Bangalore in 1999where he is currently an Associate Professor in the ElectricalCommunication Engineering department. His current research isfocused Nano-CMOS technology and Integrated CMOS-MEMSsensors. The work spans the domains of process technology, devicedesign, circuit design and modeling. He has several researchpublications in international journals and conferences and 3 USpatents to his credit. He has received the Young Engineer Award(2003) from the Indian National Academy of Engineering. He isalso the recipient of the Swarnajayanti fellowship (2005) from theDepartment of Science and Technolgy, Govt. of India and Prof.Satish Dhavan award (2005) from the Govt. of Karnataka. He wasthe founding chair of the IEEE Electron Devices and Solid-StateCircuits society, Bangalore chapter which was recognized asthe Outstanding Chapter of the Year by the IEEE SSC society(2003) and IEEE EDS society (2005). He has been on the programcommittees of several international conferences. He was thetechnical program chair for the international conference on VLSIdesign and Embedded systems (2007). He is a DistinguishedLecturer of the IEEE Electron Devices Society.


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