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366 IEEE JOURNAL OF SOLID-STATE CIRCUITS, VOL. 31, NO. 3, MARCH 1996

Surface Micromachined AccelerometersBernhard E. Boser and Roger T. Howe, Fellow, IEEE

Abstract— Surface micromachining has enabled the cofabri-

cation of thin-film micromechanical structures and CMOS orBipolar/MOS integrated circuits. Using linear, single-axis ac-celerometers as a motivating example, this paper discusses thefundamental mechanical as well as the electronic noise floorsfor representative capacitive position-sensing interface circuits.Operation in vacuum lowers the Brownian noise of a polysilicon

accelerometer to below 1 g =

p

Hz. For improved sensor perfor-mance, the position of the microstructure should be controlledusing electrostatic force-feedback. Both analog and digital closed-loop accelerometers are described and contrasted, with the latterusing high-frequency voltage pulses to apply force quanta to themicrostructure and achieve a very linear response.

I. INTRODUCTION

OVER the past decade, surface micromachining has be-come established as a versatile solution for a widevariety of sensing problems. Only a few additional processingsteps compatible with standard fabrication techniques and ma-terials are required to cofabricate mechanical sensing elementsand the associated electronic interface circuits on a singledie. Surface micromachined sensors are used, for example,in the automotive market as crash detectors and for dynamicvehicle control. Applications as vibration and shock detectorsrange from monitoring mechanical stress in airplane wings torecording mechanical shock of fragile shipping goods. Thetechnology is expected to have an even greater impact inprospective applications such as head-mounted displays, where

the small size and weight, combined with sophisticated on-chipsignal processing capability, are enabling features. The need inthis application for several different sensors, including linearand angular accelerometers and gyroscopes, in a very smallvolume, illustrates the advantages of monolithic fabrication of the sensing elements and associated electronics.

Mechanical structures fabricated in surface micromachiningtechnologies consist of deposited thin films of polysilicon [1],aluminum [2], silicon nitride [3], [4], and other materials.Integration of surface microstructures with MOS electronicsis relatively straightforward and economically attractive, withthe microstructure typically occupying only a small fraction of the die area. By bringing the sense element onto the integratedcircuit, surface micromachining leverages the experience andsophisticated processes of IC manufacturing and brings aboutall the customary advantages of IC solutions: batch fabrication,high yield, small size, low power, and low cost. Eliminationof a separate sense element miniaturizes the package and alsoresults in improved reliability.

Manuscript received October 25, 1995; revised December 19, 1995. Thiswork was supported by ARPA and the California PATH program.

The authors are with the Electrical Engineering and Computer ScienceDepartment, University of California Berkeley, Berkeley, CA 94720-1770USA.

Publisher Item Identifier S 0018-9200(96)02451-1.

This paper concentrates on polysilicon surface microstruc-

tures integrated with CMOS and Bipolar/MOS technologies.These technologies have been used for various accelerometerand resonant gyroscopes [5]–[7], as well as micro-resonatoroscillators [8]. Other applications are being explored, such asmicromechanical IF filters for signal processing and spectrumanalysis [9], [10]. The prospects for multisensing IC’s usingseveral sensing elements or possibly arrays are particularlyinteresting.

Accelerometers serve as a vehicle in this paper to investigatethe various aspects of surface micromachined sensors. Thesedevices have evolved quite far, with commercial parts beingavailable for some time now [11], [12]. First generationdevices achieve a noise floor around 10 m Hz over an

input range of 50 or more and with shock survival in excessof 2000 . These specifications are compatible with automotiveapplications such as airbag release. Second generation devicesachieve an order of magnitude better resolution and are wellsuited for a very large range of needs, including active carsuspension, shock detection and monitoring, computer inputdevices, toys, and short term navigation. Substantial futureimprovements down to a noise level as low as 1 Hzcan be expected based on analysis of the fundamental limitsof this technology. This level of performance is adequate forall but the most demanding inertial navigation applications of acceleration sensors.

We begin with a brief analysis of sensing elements forlinear acceleration and their implications for the measurementsystem performance. An overview of the several strategiesfor cofabrication of polysilicon microstructures and CMOSprovides the context for a description of a modular approachdeveloped at Berkeley. The paper then discusses the electronicinterface of the sensing element. Special attention is given tofactors limiting the achievable resolution of micromachinedsensors. Particularly, the small size and consequent low massof the sensing element results in an elevated thermal noisefloor. It will be shown how this limitation can be overcomeby vacuum packaging and embedding the sensing element inan electronic force-feedback loop, resulting in a performancecorresponding to a sensing element that is several orders of magnitudes larger. The various aspects will be demonstratedwith examples of commercial accelerometer IC’s [6], [12],[13] and research prototypes [5], [7]. A detailed descriptionof an experimental device appears also in an earlier issue of this journal [14].

II. SENSING ELEMENTS

For sensing physical quantities such as acceleration, angularrate, or pressure, a mechanical sense element converts theunknown quantity into a displacement that is then detectedand converted to an electrical signal. A conceptual diagram of

0018–9200/96$05.00 © 1996 IEEE



BOSER AND HOWE: SURFACE MICROMACHINED ACCELEROMETERS 367

(a)

(b)

Fig. 1. Concept of an accelerometer (a) and the equivalent electrical model(b).

a simple single-axis linear accelerometer is shown in Fig. 1(a).The inertia of the proof mass restrains the motion of thiselement in the presence of an external force acting on areference frame to which the proof mass is attached by meansof a spring. The proof mass is further subject to damping fromthe surrounding gas ambient or from internal dissipation in thespring. The differential equation for the displacement as afunction of is obtained from Newton’s law

(1)

In this equation and are the spring constant and dampingcoefficient, respectively, and linear relations are assumed.Solving for using the Laplace transform yields the second-order transfer function

(2)

with resonant frequency and quality factor. Critical or under-damping of the

sensor is assumed since this condition minimizes the thermalnoise of the sensor, as will be seen later.

At frequencies well below resonance, the displacement, which is proportional to the acceleration. This

relationship implies a trade-off between sensitivity and band-width of the sensor: low resonant frequency results in largedisplacements and hence, good sensor resolution but restrictsthe bandwidth of the sensor. This trade-off can be eliminatedwith feedback, as will be discussed later.

Fig. 2. Cantilever beam with concentrated load.

In micromechanical systems, the choice of the resonantfrequency is also constrained by other considerations. Lowresonant frequency implies a low spring constant and highmass . The mass of typical surface micromachined sensorsis below one microgram. A one micron displacement due toa 1 m/s ) acceleration would require a springconstant less than 10 mN/m. Material properties, geometric

constraints, and self resonance of the spring set a lower boundon that is well above this value.A further complication is that the resonance frequencies

in different directions cannot be chosen independently. Forexample, the ratio of resonant frequencies about the andaxis for the cantilever beam shown in Fig. 2 with thickness

and width is equal to the ratio [15], [16]. Typicaldimensions for the suspension of a axis accelerometer are athickness of 2 m and a width of 5 m, corresponding to aratio of resonant frequencies of only 2.5. Therefore, in practicalsituations, the suspension alone can provide only a portion of a typical requirement for rejection of off-axis accelerations.

Accelerometer suspensions based on folded trusses have

several advantages, including independence of the spring con-stant on residual stress in the film and the capability of lowspring constants in a small area. Torsional resonant modes arepotentially near the fundamental mode of the sense element,making careful design essential. Analytical models of suchsuspensions [16], [17] have been developed that provideinsight. Fig. 3 shows the result of a finite element analysissimulation [18] of a proof mass suspended with four single-folded trusses that reveals a rotational resonance about eitherdiagonal at less than twice resonance for the desired motionalong the axis. Typical polysilicon surface micromachinedaccelerometers have resonance frequencies in the kilohertzrange and a mass of between 0.1 and 1 g.

This low mass gives rise to another design challenge facedin high-sensitivity accelerometers. According to the laws of thermodynamics, the thermal energy of a system in equilib-rium is for each energy storage mode, where isBoltzmann’s constant. The minuscule mass of the microma-chined device implies substantial agitation due to this thermalenergy, a process known at the molecular level as Brownianmotion [19]. The extent of this disturbance can be appreciatedreadily from the equivalent second-order system presented inFig. 1(b), which is described by the same differential (1) asthe mechanical system shown in Fig. 1(a). In the electricalequivalent, currents represent forces, and voltage correspondsto velocity. The electrical component values are proportional




(a)

(b)

Fig. 3. Finite-element analysis of the resonant frequencies of a proof mass suspended with four single-fold trusses. The plots illustrated an exaggerated viewof the motion corresponding to the desired first mode (a) at 7.5 kHz. The parasitic second and third modes occur at 12.6 kHz (b).

to the respective mechanical quantities. The current noisefrom the resistor corresponds to an

equivalent force noise source associated with the mechanicaldamping element. It is in parallel with the external input

, corresponding to the measured force in the mechanicaloriginal, and has a white power spectrum. The equivalentacceleration spectral density is obtained by back-substitutingmechanical for electrical quantities and is

(3)

For a typical micromachined sensor with ,kHz, and , the input referred noise density is

approximately 200 Hz at room temperature.For many commercial applications (e.g., in the automotive

sector), this noise level is acceptable, but inertial navigationand other precision applications such as tracking systems for

head-mounted displays demand better performance. Lowernoise can be achieved by either increasing the mass or in-creasing the quality factor by reducing the damping of thesensing element.

The first approach offers only limited improvement becauseof the relatively modest increase of the mass of the sensorobtained, for example, by substituting a material with higherdensity such as tungsten for polysilicon. The lateral dimensionsof the proof mass are restricted by economic constraintsthat call for minimizing silicon area as well as processingdifficulties. Polysilicon films typically have small gradientsin the residual stress that cause warpage of large-area plateswhich limits lateral dimensions to a few hundred microns intypical technologies.

By contrast, vacuum packaging of the sensor results in aseveral orders of magnitude decrease of the thermal noiseof the sensor and does not require any modification of thesensor fabrication process. Quality factors of 50 000 have




Fig. 5. Detail of a polysilicon micromachined structure [22].

integration are the key to maximizing the performance with

this technique. Researchers are also investigating alternativetransduction methods such as tunneling tips in surface mi-crostructures [25]–[27].

Fig. 6 schematically shows quarter-sections of two sensordesigns with sensitivity to accelerations perpendicular andparallel to the silicon surface. In the axis structure, the

capacitance between the proof-mass and the substrate servesas the mechanical to electrical interface [5]; the axis design

uses a comb-like structure [6]. The advantage of the first styleinterface is the usually larger capacitor area and value, but

the interface is asymmetric when a top electrode is missing.Over the finger structure, the parallel plate arrangement hasthe advantage of a much larger sense capacitance for a givenarea of up to 1 pF, compared to less than 200 fF for typical

axis sensing elements.For position measurement, the variation of this

capacitance due to displacement must be maximized. For the

axis structure, a parallel plate approximation is usuallyappropriate and for small displacements, the capacitor change

is approximately equal to . Maximizing the sensitivitytherefore calls for minimizing the capacitor gap, . In the

second structure, a substantial fraction of the total capacitanceis due to fringing fields and does not change substantiallyin the presence of small displacements [16], [28], resulting

in a somewhat lower sensitivity of this arrangement for agiven electrode spacing. Exact values can be obtained fromnumerical simulation [29].

The challenge in the design of a capacitive position mea-surement circuit consists in detecting extremely small capac-itance changes in the presence of much larger parasitics. Forexample, a sensing element with resonant frequency

kHz experiences a displacement of only 0.1 A in the presenceo f a 1 m constant acceleration signal. A parallel plate

(a)

(b)

Fig. 6. Accelerometers with z (a) and y axis (b) sensitivity.

capacitor with a 1 m gap and nominal value of 1 pF changesby a mere 10 aF due to this displacement, an amount that is100 000 times smaller than the capacitor itself. Flicker noise,offsets, and parasitic capacitances represent further difficulties.

The circuit shown in Fig. 7 uses chopper stabilizationand bootstrapping to minimize the noise and maximize thesensitivity. The voltage at the midpoint of the dividerformed by the two sense capacitors and excited with the acsignal is proportional to the capacitor mismatch and, hence,




Fig. 7. Sensor interface using a unity-gain buffer.

Fig. 8. Sensor interface based on correlated double sampling.

position of the proof mass. The measurement is performedat a sufficiently high frequency to suppress offset and flickernoise and the result demodulated. The parasitic capacitancein node must be minimized to avoid signal attenuation.Many designs reduce the attenuation by shielding interconnectcapacitances and tying the shield to the output of a unity gain

buffer as shown. Shields are also needed to avoid electricalfields between the sensor and surrounding conductors sincethe resulting electrostatic forces are indistinguishable from anexternal force and, hence, corrupt the acceleration measure-ment. Special provisions are needed to control the dc potentialat node . The solution shown here relies on a resistorthat typically must be in the megohm range to minimize itsnoise contribution.

An alternative solution presented in Fig. 8 eliminates boththe demodulator and the need for a resistor or similar elementto set the dc potential [7], [30]. This circuit is based onan amplifier with an auxiliary input with reduced gainand operates in two phases. First, the sense capacitors areprecharged to a constant voltage. At the same time, the offsetand flicker-noise of the amplifier are stored on the holdingcapacitor . During the second phase, the voltage acrossthe sense capacitors is changed, causing a charge that isproportional to the mismatch between the two sense capacitorsto flow into node . Since the amplifier input is now avirtual ground, this charge flows onto the integrating capacitor

unattenuated by the parasitic , producing an output ,which is proportional to the position of the proof-mass.

Like Brownian motion of the sensing element, the noisefloor of the position measurement interface sets a limit onthe achievable sensitivity of the sensor. To assess the ultimateperformance achievable with surface micromachined sensors,

Fig. 9. Electronic noise as a function of transistor size.

it is important to derive relationships relating this noise tobasic physics and technology parameters.

At least in principle, all noise sources except the input

referred thermal noise from the amplifier can be madeinsignificantly small. For either the circuit shown in Figs. 7 or8, this noise appears across all capacitors connected to node

and produces an equivalent current noise that is added tothe signal current . In an appropriately designedamplifier, the input transistor is the dominant noise contributor,and hence

(4)

assuming a differential input. Apparently, the noise can bereduced by increasing the transconductance . Increasing

, however, requires increasing either the saturation voltage

of the input devices, which in practice is limited by thesupply voltage, or enlarging the gate capacitance . This,in turn, increases the capacitance at node and hence, thecurrent noise, thus calling for a trade-off between and

. For a given saturation voltage, these two quantities arerelated as by the cutoff frequency of thetechnology. Fig. 9 shows the normalized current noise as afunction of the ratio of to the sum of all other capacitances

connected to the amplifier input. Clearly, thenoise is minimized for , with a small penalty forsomewhat larger values of as might be required to meetother circuit requirements, such as bandwidth.

Based on these considerations and assuming that the ampli-

fier input capacitance is chosen optimally, the mean squarederror of the capacitance change divided by the sense capaci-tance is

(5)

For a parallel plate sense capacitor with gap , the corre-sponding input referred acceleration noise floor is

(6)




This expression demonstrates how the sensitivity of an ac-celerometer that is limited by electronic noise in the positionmeasurement circuit can be maximized. Upon first inspection,the biggest improvement appears to result from minimizing theresonant frequency. Accelerometers described in the literaturefeature resonant frequencies in the range of 5 kHz [14]to over 20 kHz [11]. Unfortunately, substantially loweringthese frequencies compromises the reliability of the device,

in particular its robustness to mechanical shock, and increasesthe probability of the proof mass sticking to the substrate.Better suppression of undesired resonances is needed as wellbefore the resonant frequency can be reduced substantially.Lowering the capacitor gap also decreases the noise floorof the sensor but, of course, depends on technology. Inaxis structures, the spacing is dictated by the thickness of the sacrificial oxide and must be chosen sufficiently large toprevent the sensing element from touching the substrate dueto warpage and mechanical excitation. For current polysiliconsurface micromachining technologies, minimum vertical orlateral gaps are around 1 m.

The remaining parameters in (6) characterize the electronic

interface of the sensor. Accordingly, the sense capacitanceshould be maximized, a condition that favors axis overaxis designs. The parasitic capacitance should not exceedthe value of for not significantly degrading the sensorperformance. High cutoff frequency and, hence, short channellength also improve sensor sensitivity. Increasing the sensevoltage results in an increased signal current and, hence,better signal-to-noise ratio. In practice, is often limitednot only by the supply voltage, but also by the maximumacceptable electrostatic force exerted on the sensor. Thisforce can degrade the linearity of the sensor (see Section V-B).

Typical numbers for surface micromachined sensors,

pF, , MHz, V,kHz, and m give a noise floor of 1 Hz at roomtemperature. This figure is of the same order as the Browniannoise of a vacuum-packaged structure. While this performancelevel has not yet been achieved with micromachined sensors,these numbers demonstrate the potential of this technology forvery sensitive inertial measurement applications.

The 1 Hz noise floor corresponds position measure-ment noise of only 10 A Hz. This means that displace-ments as small as the classical diameter of an electron can bedetected in a 10 Hz bandwidth. In an actual circuit, the noisewould be somewhat larger due to additional noise sourcesneglected in this analysis.

V. FORCE-FEEDBACK

Fig. 10 shows an acceleration sensor embedded in a feed-back loop. A compensator and force-feedback transducer areadded to the open loop sensor consisting of the proof mass andposition measurement circuitry. The feedback force opposesdisplacements of the proof mass from its nominal position.Compensation is required for stability and will be discussedlater.

Feedback improves many important characteristics of a sen-sor, including bandwidth, dynamic range, and, in certain cases,linearity and drift. Increasing the bandwidth is particularly

Fig. 10. Accelerometer with analog force-feedback servo loop.

important in accelerometers because of the trade-off betweensensitivity and bandwidth imposed by the sensing element.Feedback increases the useful bandwidth by a factor equalto the loop gain, which, thanks to the electronic circuitry,can be made large. Consequently, the resonant frequency of the sensing element can be optimized for sensitivity alone,regardless of the desired sensor bandwidth.

Controlling the displacement of the proof mass is equallyimportant, particularly for vacuum-packaged high- deviceswhich can exhibit motion at the resonant frequency that

exceeds the small spacings between the electrodes of the sensecapacitor. Finally, imperfections, for example, due to nonlinearor temperature sensitive springs, are attenuated provided thatthe force-transducer does not introduce similar errors.

Electrostatic actuation is the simplest means for generatingthe feedback force in a micromechanical sensor. The electrodesof a capacitors with a constant voltage across are attractedwith a force

(7)

in the case of a parallel plate capacitor with value and gap. The same capacitor can be used for force-feedback and

sense [11], or separate electrodes can be added [5].Because of the quadratic dependence of the force on voltage,

electrostatic actuators cannot be used directly, but must becombined with some means of linearization. For symmetricsensors [Fig. 6(b)], a simple solution is to apply a voltage

across and across . Since thetwo resulting forces are in opposite directions, the quadraticterms cancel and the net difference isproportional to the controlling voltage . In practice, thelinearity of this technique is limited by the matching accuracyof and . More importantly, this approach cannot beused at all with asymmetric sensors such as the one depictedin Fig. 6(b).

A more general and potentially more accurate solutionconsists in pulse modulating the feedback signal. Fig. 11shows a system that employs a clocked comparator to quantizethe feedback force to only two levels [5], [7], [14]. Thissystem is equivalent to a sigma-delta modulator as used inA/D conversion [31], except that the noise shaping filterhas been replaced by the mechanical sensor. If all feedback pulses are kept equal in length, imbalance merely results inan offset and/or gain error, but does not cause distortion.The pulse-density of the one-bit output stream tracks theinput acceleration, which is obtained by low-pass filtering anddecimating the pulse-density code. Because of the inherent




Fig. 11. Accelerometer with digital force-feedback loop.

A/D conversion, this approach is referred to as “digital”feedback, while the technique illustrated in Fig. 10 is called“analog” feedback in this paper.

The feedback loops will now be analyzed with respect tostability, residual motion of the proof mass, and, for the caseof digital feedback, quantization noise.

A. Stability

The force-feedback accelerometers shown in Figs. 10 and11 are unstable without compensation because of the 180degree phase delay from the sensing element for frequencies

above the resonance. Several solutions are used in accelerom-eter designs and will be described below: over-damped proof mass [32], limiting the loop bandwidth electronically [11],[33], and compensation with a lead filter [5], [7], [34].

The first solution is simple but because of the high Browniannoise floor of an over-damped low-mass sensing element,practical only for low performance applications or when muchlarger and thus heavier sensing elements than those built withsurface micromachining technology are used.

The second approach uses an electronic low-pass filter toreduce the loop bandwidth to frequencies well below theresonance of the proof-mass. Obviously, this technique isapplicable only in situations where either a low measurement

bandwidth or low sensitivity can be tolerated, since in this casethe sensor bandwidth is actually reduced to a value less thanthe resonant frequency of the sensing element. Stabilitydemands that the loop gain is lower than the ratio of tothe sensor bandwidth. A commercial surface micromachinedaccelerometer implements this technique [11]. The sensor hasa resonant frequency of 24 kHz and 1 kHz signal bandwidthand a low-frequency loop gain of about 10, thus diminishingsomewhat the benefits of feedback.

A compensation filter requires additional circuitry butavoids the aforementioned problems. The basic strategy isto add a left half-plane zero to the loop transfer function inorder to decrease the phase delay at the unity-gain frequency.The analysis is straightforward for analog feedback, butcomplicated by the nonlinearity in the digital case.

The conventional definitions of stability involving theboundedness of states and absence of limit cycles are notuseful for sigma-delta modulators which use oscillations asa means for A/D conversion. A more appropriate criterionfollows from analyzing the spectrum of the pulse-densityoutput. For illustration purposes, consider the digital feedback loop in Fig. 11 with zero input. Then the comparator outputand feedback signal switch rapidly between positive andnegative values. To minimize the residual motion of the proof mass, it is important to maximize the rate of this signal:motion is reduced by a factor four for every doubling of the

(a)

(b)

Fig. 12. Stability of accelerometer with digital feedback loop for two dif-

ferent actuation delays t

d .

frequency of the feedback signal. The maximum possible rateis , constrained by the clocking and second-order natureof the feedback loop, and is achieved when the total phasedelay is less than 180 at this frequency. This condition ismet when the phase lead from the compensator is larger thanapproximately . Here, is the delay from sensingthe position of the proof mass to applying the feedback signal.

Simulation results for two loops operating under this con-dition are shown in Fig. 12. In both cases, a compensator

is used to add a zero at andapproximately 27 phase lead at . In the first case,short feedback pulses are issued almost immediately afterthe position of the proof mass has been measured. Thecompensator output leads despite a small lag of ,suggesting that a smaller amount of lead (e.g., ) wouldsuffice in this case. Simulation confirms that this is indeed thecase. In the second case, the feedback pulses are elongatedand delayed by causing to arrive just in timeto ensure a correct decision by the comparator. The longerfeedback pulses round the edges of the position signal, but forstability, only the delay from position measurement to themidpoint of the feedback pulse is relevant.

B. Residual Motion of the Proof Mass

The sense and feedback capacitors exert a position depen-dent force on the proof-mass. Residual motion consequentlyresults in an error force that cannot be distinguished fromthe accelerometer input. Assuming a zero external input andproper compensation of the feedback loop, the feedback signalof the accelerometer is pulse-train at one quarter the samplingfrequency and amplitude equal to the input range of the device. This feedback signal causes the proof mass to moveup and down at the frequency . Owing to the second-ordernature of the proof mass, the amplitude of the fundamental




of the motion is . Assuming a full-scale input range and MHz,

A. Noise or an external acceleration signal randomizethe spectrum of the proof mass displacement and give rise tolow-frequency components falling into the signal-band of thesensor.

Capacitive position sensing and feedback both exert anelectrostatic force on the proof mass. To first order, this force is

proportional to the position of the proof mass. Residual motionmodulates these forces, resulting in a spurious accelerationof

(8)

Usually, the acceleration due to the sense capacitor dominatesbecause of the need to maximize the sense voltage inorder to minimize the noise floor of the position measurementcircuit. A typical results in an error

. Simulations and measurements show that this erroris sufficiently random to raise the noise floor of the sensor.

It is most easily suppressed by choosing a sufficiently highsampling rate .A sensor with analog feedback and identical loop bandwidth

will exhibit the same error, however, because of the absenceof limit cycles, it will not result in an increased noise floor butinstead in a slightly increased nonlinearity.

C. Quantization Noise

Digital feedback adds quantization noise as an additionalerror source [35]. Because of the typically high ratio of thesampling frequency to the signal bandwidth of interest,this error is usually insignificant. However, the finite resonant

frequency of the proof mass causes a dead-zone for inputsless than

(9)

For and . Like errorsdue to residual motion of the proof mass, the dead-zone widthcan be decreased by raising the sampling rate, . Smallerdead-zones exist also for other inputs that are rational fractionsof [31].

The cause of the dead-zone can be appreciated from theblock diagram in Fig. 13, where the proof mass is modeled asa linear filter and the order of the linear filtering actionand feedback summing node have been reversed. Assumingfor a moment that the input of the accelerometer labeledis zero, the feedback signal is a square wave with amplitude

and frequency . To be detected, an input signal mustdisturb this idle pattern of the modulator. This occurs only fordisplacements that are at least equal in amplitude to the idlechannel residual motion .

In oversampled A/D converters with electronic noise shap-ing filters, the resonant frequency of the mechanical noiseshaper corresponds to the filter pole frequencies. These areusually at much lower frequencies; consequently, dead-zonesare not observed.

Fig. 13. Block diagram illustrating the origin of a dead-zone in sensors withdigital feedback.

VI. CONCLUSION

The advantages of monolithic fabrication of micromachinedsensors and associated electronics have been discussed. It hasbeen demonstrated that the limits of the technology are beyonda 1 Hz noise level, which is compatible with a largevariety of demanding applications.

ACKNOWLEDGMENT

The authors are very indebted to their students who didmost of the work reported here.

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Bernhard E. Boser received the Diploma in elec-trical engineering in 1984 from the Swiss FederalInstitute of Technology (ETH) in Zurich, Switzer-land, and the M.S. and Ph.D. degrees from StanfordUniversity, Stanford, CA, in 1985 and 1988, respec-tively.

From 1988 to 1991 he was a Member of TechnicalStaff at AT&T Bell Laboratories, working on VLSIimplementations of artificial neural networks andalgorithms for automatic learning. Since 1992 hehas been an Assistant Professor in the Department

of Electrical Engineering and Computer Sciences, University of California,Berkeley. His current research interests are in the areas of integrated circuitsfor data conversion and communication, and on the design and fabrication of micromechanical systems.

Roger T. Howe (S’79–M’84–SM’94–F’96) wasborn in Sacramento, CA on April 2, 1957. Hereceived the B.S. degree in physics from HarveyMudd College, Claremont, CA in 1979, and theM.S. and Ph.D. degrees in electrical engineeringfrom the University of California at Berkeley, in1981 and 1984, respectively.

He was on the faculty of Carnegie-Mellon Uni-versity during the 1984–85 academic year and wasan Assistant Professor at the Massachusetts Institute

of Technology from 1985 to 1987. In 1987, he joined the Department of Electrical Engineering and Computer Sciences atthe University of California at Berkeley, where he is now a Professor, as wellas a Director of the Berkeley Sensor & Actuator Center. His research interestsinclude silicon microsensors and microactuators, micromachining processes,and integrated-circuit design.

Dr. Howe served as Co-General Chairman of the 1990 IEEE Micro ElectroMechanical Systems Workshop and is General Chairman of the 1996 Solid-State Sensor and Actuator Workshop at Hilton Head, SC.

surface micromachined accelerometers

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