Surface Micro Machined Accelerometers

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

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

Abstract- Surface micromachining has enabled the cofabri-cation of thin-film micromechanical structuries and CMOS orBipolarMOS 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 polysiticonaccelerometer to below 1 pg/&. 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 resplonse.

I. INTRODUCTIONVE R 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, wherethe 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 ofthe sensing elements and associated electronics.Mechanical structures fabricated in surface micromachiningtechnologies consist of deposited thin films of polysilicon [13,aluminum [2], silicon nitride 131, [4], and other materials.Integration of surface microstructures with MOS electronicsis relatively straightforward and economically attractive, withthe microstructure typically occupying only a small fraction ofthe die area. By bringing the sense element onto the integratedcircuit, surface micromachining leverages the experience andsophisticated processes of IC manufacturing and brings about

all 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 Science'Department, University of California Berkeley, Berkeley, CA 94720-1770USA.Publisher Item Identifier S 0018-9200(96)02451-1.

0018-9200/96$05

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], [lo]. 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 [ll], [12]. First generationdevices achieve a noise floor around 10 m g / a ver aninput range of f 5 0 g or more and with shock survival in excessof 2000 g. 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 p g / acan 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 ofacceleration 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 ho w 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 ofmagnitudes 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 ofthis journal [14].

11. SENSWG LEMENTSFor 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

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BOSER AND H O W SURFACE M ICROMACHINEDACCELEROMETERS

displacementX A proof-mass

F=ma

XQ F=kx

reference frame(a )

current c-) orcevoltage C-C velocityflux - isplacement(b)Fig. 1. Concept of an accelerometer (a) and the equivalent electrical model(b).

a simple single-axis linear accelerometer is shown in Fig. l(a).The inertia of the proof mass restrains the motion of thiselement in the presence of an external force Fextcting 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 x as afunction of Fe,, is obtained from Newtons law

(1)d2x dxd t 2 d tm- + b- + kx = Fext ma.In this equation k and b are the spring constant and dampingcoefficient, respectively, and linear relations are assumed.Solving for x using the Laplace transform yields the second-order transfer function

with resonant frequency wT = and quality factorQ = w,m/b. Critical or under-damping (Q 5 0.5) of thesensor is assumed since this condition minimizes the thermalnoise of the sensor, as will be seen later.At frequencies well below resonance, the displacementx M alw:, which is proportional to the acceleration. Thisrelationship 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.

367

t

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 k and highmass m. The mass of typical surface micromachined sensorsis below one microgram. A one micron displacement due toa 1 g (lg = 9.8 m/s2) acceleration would require a springconstant k less than 10 mN/m. Material properties, geometricconstraints, and self resonance of the spring set a lower boundon 5 that is well above this value.A further complication is that the resonance frequenciesin different directions cannot be chosen independently. Forexample, the ratio of resonant frequencies about the y and zaxis for the cantilever beam shown in Fig. 2with thicknesst and width w is equal to the ratio t / w [15], [16]. Typicaldimensions for the suspension of a z axis accelerometer are athickness of 2 pm and a width of 5 pm, corresponding to aratio of resonant frequencies of only 2.5. Therefore, in practicalsituations, the suspension alone can provide only a portion ofa typical requirement for rejection of off-axis accelerations.Accelerometer suspensions based on folded trusses haveseveral 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 [161, [171 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 z axis. Typical polysilicon surface micromachinedaccelerometers have resonance frequencies in the kilohertzrange and a mass of between 0.1 and 1 pg.This low mass gives rise to another design challenge facedin high-sensitivity accelerometers. According to the laws ofthermodynamics, the thermal energy of a system in equilib-rium is k ~ T / 2or each energy storage mode, where k13 isBoltzmanns 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. l(a). In the electricalequivalent, currents represent forces, and voltage correspondsto velocity. The electrical component values are proportional



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

to he respective mechanical quantities. The current noisezz/Af = 4 k B T / R from the resistor corresponds to anequivalent force noise source associated with the mechanicaldamping element. It is in parallel with the external inputIekxt,orresponding 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/ & y -QGm = , ( 3 )For a typical micromachined sensor with m = 0.5 pg , w, =271.10 kHz, and Q = 0.5, the input referred noise density isapproximately 200 pg/& at room temperature.For many commercial applications (e.g., in the automotivesector), 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 Q 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



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BOSER AND H O W E SURFACE MICROMACHINED ACCELEROMETERS 369

UwallN substrate

Fig. 4. Cross-section of a CMOS wafer with added micromechanical structures [16].

been reported in the literature for in situ phosphorus-dopedpolysilicon [SI, [lo], corresponding to a reduction of thenoise floor in above example from 200 pg/& to below 1pg/&. By comparison, the ratio of the density of tungstento that of polysilicon is less than an order of magnitude.Of course, the elimination of nearly all mechanical dampingrequires that the position of the suspended mass be controlledelectrically.

111. FABRICATIONECHNOLOGYMonolithic integration of the proof mass and suspension foran accelerometer or other physical sensor with its associatedelectronics requires the merging of surface micromachiningwith an IC fabrication process. Polysilicon and aluminum arethe most attractive candidates for the structural elements ofmicromechanical sensors. Both are already used in standardIC technology and can be deposited and patterned to veryaccurate dimensions. The negligible fatigue [20] and lack of

memory of polysilicon make it the material of choice for thefabrication of high performance micromechanical sensors. Theadvantages of aluminum are its low processing temperatures,which makes integration with electronic processing much morestraightforward.In order to integrate polysilicon microstructures with CMOSprocessing, the 600 "C deposition and 950 OC rapid-thermalannealing temperature needed for the former must be con-sidered carefully. One approach is to replace the conventionalaluminum metallization with tungsten, together with a titaniumsilicide diffusion barrier in the contacts, to enable the CMOSto withstand these high post-processing temperatures. Lowpressure chemical vapor deposition (LPCVD) silicon nitrideis used to isolate the CMOS from the micromachining processsteps, such as the final etching of the sacrificial layer inhydrofluoric acid. The modular integration of CMOS andmicrostructures (MICS) process uses this integration strategyand is based on a conventional 3 pm CMOS technology [21].Fig. 4 shows a cross-sectional view of an MICS wafer. Thesensing structures can be seen on the right and consist of threelayers of 0.5 pm, 1.5 pm, and 1 pm thick phosphorus-dopedpolysilicon. The second and third layer are deposited on top of1 and 2 pm thick sacrificial phosphosilicate glass layers thatare later removed.Fig. 5 shows an scanning electron microscopy (SEM) pic-ture of a section of a multimode test structure for digital

control that was fabricated in the MICS technology [ 2 2 ] . n thisdevice, the proof mass and spring suspension are fabricatedin poly level two. The third layer serves as a limit stopand electrostatic actuator. The etch holes in the proof massand top actuator are required for complete removal of thesacrificial oxides and to further reduce squeeze film damping,thus improving the Q of the structure at atmospheric pressure.The technology also permits the fabrication of polysiliconfuses that hold the structure in place during the release process.They are cut electrically with a current pulse.Low strain gradient in the polysilicon films is importantto prevent overhanging structures from warping after release.Polysilicon cantilever beams fabricated in the MICS technol-ogy typically warp less than 0.1 pm over 400 pm in length.Stiction of the sense element or its suspension to the limit stopsor to the substrate due to overrange forces poses an additionalchallenge for the fabrication technology.The first commercial integrated surface machining mi-crosensor technology, the BiMEMS process of AnalogDevices, Inc. [23] uses a different integration strategy fromthe MICS technology. The sacrificial oxide and structuralpolysilicon film are deposited and annealed prior to a standardaluminum metallization. In BiMEMS, a diffusion is used toconnect the polysilicon microstructure to the circuit, ratherthan the poly level one and gate polysilicon levels used inMICS. Finally, BiMEMS is a linear bipolar/MOS technologythat includes thin film resistors, making it very well suited toimplementing analog closed-loop control of the sense element.

IV. POSITION ENSINGAcceleration sensors translate the external signal in a cor-responding displacement that can be measured by severalmeans. Piezo-resistive strain gauges are used widely in sen-sors because of the simple interfacing to off-chip electroniccircuits. Ion-implanted piezo-resistors have been used in un-doped polysilicon resonant microbeams [24] and polysiliconpiezo-resistors have also been embedded in silicon nitridemembranes for pressure sensing [3]. Position sensors thatmeasure the capacitance between a conducting polysiliconproof mass and a fixed electrode, on the other hand, requireno additional processing and can be extremely sensitive, aswill be shown later. The negligible temperature coefficientis another important advantage of a capacitive sensor read-out. The low parasitics that are characteristic of monolithic



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ac

mt

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

integration are the key to maximizing the performance withthis 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 z axis structure, thecapacitance between the proof-mass and the substrate servesas the mechanical to electrical interface [ 5 ] ; he y axis designuses a comb-like structure [ 6 ] .The advantage of the first styleinterface is the usually larger capacitor area and value, butthe 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 typicaly axis sensing elements.For position measurement, the variation dC/dx of thiscapacitance due to displacement must be maximized. For thez axis structure, a parallel plate approximation is usuallyappropriate and for small displacements, the capacitor changeis approximately equal to C / X O . aximizing the sensitivitytherefore calls for minimizing the capacitor gap, XO. n thesecond structure, a substantial fraction of the total capacitanceis due to fringing fields and does not change substantiallyin the presence of small displacements [l6], [as], resultingin 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 fr = 5kH z experiences a displacement of only 0.1 A in the presenceof a 1 mg constant acceleration signal. A parallel plate

anchor suspension,

anchor tether proofmass

substrate(b)

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

capacitor with a 1 pm gap and nominal value of 1 pF changesby a mere 10 aF due to this displacement, an amount that is100000 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 the Isensitivity. The voltage V, at the midpoint of the dividerformed by the two sense capacitors and excited with the acsignal V, is proportional to the capacitor mismatch and, hence,



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BOSER AND H O W E SURFACE MICROMACHINED ACCELEROMETERS 37 1+vsysl27t

-vs

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

-vs

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 V, 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 gainbuffer 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 V,. The solution shown here relies on a resistor R d cthat typically must be in the megohm range to minimize itsnoise contribution.An alternative solution presented in Fig. 8eliminates 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 A2 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 Ch. 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 V,. Since the amplifier input is now avirtual ground, this charge flows onto the integrating capacitorC, unattenuated by the parasitic C,, producing an output V,,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,

C&TElectronic 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 inputreferred thermal noise 3 rom the amplifier can be madeinsignificantly small. For either the circuit shown in Figs. 7or

8, this noise appears across all capacitors connected to nodeV, and produces an equivalent current noise that is added tothe signal current i, = V,AC. In an appropriately designedamplifier, the input transistor is the dominant noise contributor,and hence -3

assuming a differential input. Apparently, the noise can bereduced by increasing the transconductance gm . Increasinggm , however, requires increasing either the saturation voltageof the input devices, which in practice is limited by thesupply voltage, or enlarging the gate capacitance C,,. This,in turn, increases the capacitance at node V, and hence, thecurrent noise, thus calling for a trade-off between gm andC,,. For a given saturation voltage, these two quantities arerelated as gm M W T C ~ ,y the cutoff frequency WT of thetechnology. Fig. 9 shows the normalized current noise as afunction of the ratio of C,, to the sum of all other capacitancesCT = C, + C, connected to the amplifier input. Clearly, thenoise is minimized for C,, = CT, with a small penalty forsomewhat larger values of C,, 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 C, is

For a parallel plate sense capacitor with gap 20, he corre-sponding input referred acceleration noise floor is

(6)3wTcs



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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 [113. 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 xo also decreases the noise floorof the sensor but, of course, depends on technology. In zaxis structures, the spacing is dictated by the thickness ofthe 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 pm.

The remaining parameters in (6) characterize the electronicinterface of the sensor. Accordingly, the sense capacitance C,should be maximized, a condition that favors z axis over yaxis designs. The parasitic capacitance C, should not exceedthe value of C, for not significantly degrading the sensorperformance. High cutoff frequency and, hence, short channellength also improve sensor sensitivity. Increasing the sensevoltage V, results in an increased signal current and, hence,better signal-to-noise ratio. In practice, V 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, C, =0.5 pF, C, = C,, WT = 2.ir x 500 MHz, V,= 0.5 V, f r = 5kHz, and xo = 1 pm give a noise floor of 1 p g / a t 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 p g / z / H z noise floor corresponds position measure-ment noise of only A/z/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-FEEDBACKFig, 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

Iproof mass position sen se compensatorFfb

force transducerFig. 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 ofthe 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-Q deviceswhich can exhibit motion at the resonant frequency thatexceeds 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 V, across are attractedwith a force

d C(x(j)V,2- v:c__, = -dx o 2 2x0in the case of a parallel plate capacitor with value C and gap2 0 . The same capacitor can be used for force-feedback andsense 1111, 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 voltageVi + AV across C+ and V - AV across C-. Since thetwo resulting forces are in opposite directions, the quadraticterms cancel and the net difference AF = ~ V O A V C / ~ Osproportional to the controlling voltage AV. In practice, thelinearity of this technique is limited by the matching accuracyof C+ and C-. 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 feedbackpulses 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



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BOSER AND HOWE: SURFACE MICROMACHINED ACCELEROMETERS 373

voI -c(z)I ProofmaSS pasitionsense compensator comparator I

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

A D conversion, this approach is referred to as digitalfeedback, while the technique illustrated in Fig. 10 is calledanalog 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 frequenciesabove the resonance. Several solutions are used in accelerom-eter designs and will be described below: over-damped proofmass [32], limiting the loop bandwidth electronically [111,[331, and compensation with a lead filter 151, [71, 1341.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 measurementbandwidth or low sensitivity can be tolerated, since in this casethe sensor bandwidth is actually reduced to a value less thanthe resonant frequency f T of the sensing element. Stabilitydemands that the loop gain is lower than the ratio of f r tothe sensor bandwidth. A commercial surface micromachinedaccelerometer implements this technique [111. 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 feedbackloop in Fig. 11with zero input. Then the comparator outputand feedback signal switch rapidly between positive andnegative values. To minimize the residual motion of the proofmass, it is important to maximize the rate of this signal:motion is reduced by a factor four for every doubling of the

I I---I i I i

X~

(b)Fig. 12.ferent actuation delays t d .Stability of accelerometer with digital feedback loop for two dif-

frequency of the feedback signal. The maximum possible rateis f s 14, onstrained 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 t d f s x 4 5 O . Here, t d is the delay from sensingthe position of the proof mass to applying the feedbacksignal.Simulation results for two loops operating under this con-dition are shown in Fig. 12. In both cases, a compensatorH,(x)= 2 - z - l is ased to add a zero at fs/9 andapproximately 27 phase lead at fs/4. In the first case,short feedback pulses are issued almost immediately afterthe position of the proof mass has been measured. Thecompensator output V, leads V, despite a small lag of V,,suggesting that a smaller amount of lead (e.g., 5 - - l ) wouldsuffice in this case. Simulation confirms that this is indeed thecase. In the second case, the feedback pulses are elongatedand delayed by t d = T / 2 causing V, 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 t d from position measurement to themidpoint of the feedback pulse is relevant.B. Residual Motion of the Proof MassThe 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 f s and amplitude amaxequal to the input range ofthe device. This feedback signal causes the proof mass to moveup and down at the frequency f s / 4 .Owing to the second-ordernature of the proof mass, the amplitude of the fundamental



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374 IEEE JOURNAL OF SOLID-STATE CIRCUITS, VO L 31, NO 3, MARCH 1996

of the motion is Ax = ~ ~ ~ / ( 2 x f ~ / 4 ) ~ .ssuming a full-scale input range amax = 2 g and f s = 10 MHz, Ax Ml0W3A. 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 isproportional to the position of the proof mass. Residual motionmodulates these forces, resulting in a spurious accelerationof

1 -T = 4/fsFig. 13.digital feedback.Block diagram illustrating the origin of a dead-zone in sensors with

(8)AX2 0Aa M (asense + amax)-- VI. CONCLUSIONUsually, the acceleration due to the sense capacitor dominatesbecause of the need to maximize the sense voltage V, inorder to minimize the noise floor of the position measurementcircuit. A typical asense= 50 g results in an error A u M8 pg. 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 fs.A sensor with analog feedback and identical loop bandwidthwill 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 [ 3 5 ] .Because of the typically high ratio of thesampling frequency f s to the signal bandwidth of interest,this error is usually insignificant. However, the finite resonantfrequency of the proof mass causes a dead-zone for inputsless than

(9)

For am= = 2 g and f s / f r = 1000, ad M 16 pg. Like errorsdue to residual motion of the proof mass, the dead-zone widthcan be decreased by raising the sampling rate, f s . Smallerdead-zones exist also for other inputs that are rational fractionsThe cause of the dead-zone can be appreciated from theblock diagram in Fig. 13,where the proof mass is modeled asa linear filter H ( s ) and the order of the linear filtering actionand feedback summing node have been reversed. Assumingfor a moment that the input of the accelerometer labeled aminis zero, the feedback signal is a square wave with amplitude

amax nd frequency fs/4. To be detected, an input signal mustdisturb this idle pattern of the modulator. This occurs only fordisplacements U that are at least equal in amplitude to the idlechannel residual motion w.In oversampled AID 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.

of amax 1311.

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 p g / G 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.

REFERENCES[l ] R. T. Howe, Surface micromachining for microsensors and microactu-ators, J. Vacuum Sci. Technol. B, vol. B6, pp. 1809-1813, 1988.[2 ] L. Hombeck, Current status of the digital micromirror device (DMD)for projection television applications, in IEDM Tech. Dig., Dec. 1993,[3] S . Sugiyama et al., Microma chined sensors using polysilicon sacrificiallayer etching technology, in IEDMTech. Dig., Dec. 1994, pp. 127-130.[4] B. E. Cole et al., 512 x 512 infrared scene projector array for low-

background simulations, in Tech. Dig. Solid-State Sensor and Actuato rWorkshop, June 1994, pp. 7-12.151 W. Yun, R. T. Howe, and P. R. Gray, Surface micromachined,digitally force-balanced accelerometer with integrated CMOS detectioncircuitry, in Proc. IEEE Solid-state Sensor and Actuator Workshop,June 1992, pp. 126-131.[6] S. J. Sherman et al., A low cost monolithic accelerometer: Prod-uct/technology update, in IEDM Tech. Dig., Dec. 1992, pp. 501-504.[7] C. Lu, M. Lemkin, and B. E. Boser, A monolithic surface microma-chined accelerometer with digital output, in ISSCC D ig. Tech. Papers.IEEE Int. Solid-state Circuits Con$, 1995, pp. 160-161.[8] C. T. Nguyen and R. T. Howe, CMOS micromechanical resonatoroscillator, in IEDM Tech. Dig., Dec. 1993, pp. 199-202.[9] L. Lin, C. T.-C. Nguyen, R. T. Howe, and A. P. Pisano, Microelectromechanical filters, in Proc. Fifth IEEE Micro ,ElectromechanicalSystems Workshop, Feb. 1992, pp. 226-231.[101 C. T.-C. Nguyen, Polysilicon microresonators for frequency referencesand signal processing, Ph.D. dissertation, University of California,Berkeley, Dec. 1994.[1 11 Analog Devices, ADXLSO-M onolithic accelerometer with signal con-ditioning, Datasheet, 1993, One Technology Way, Norwood, MA02062.[12] L. Ristic, R. Gutteridge, J. Kung, D. Koury, B . Dunn, and H. Zunino, Acapacitive type accelerometer with self-test feature based on a double-pinned polysilicon structure, in Dig. Transducers 93,June 1993, pp.8 10-8 13 .[13] R. S . Payne, S. Sherman, S. Lewis, and R. T. Howe, Surface microma-chining: From vision to reality to vision, in ISSCC Dig. Tech. Papers.IEEE Int. Solid-State C ircuits Con$, 1995, pp. 164-165.[14] C. Lu, M. Lemkin, and B. Boser, A monolithic surface micromachinedaccelerometer with digital output, IEEE J. Solid-state Circuits, vol. 30 ,

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pp. 903-909, May 1993.[20] D. W. Burns et al., A digital pressure sensor based on resonantmicrobeams, in Tech. Dig. Solid-State Sensor and Actuator Workshop,June 1994, pp. 221-228.1211 J. M. Bustillo, G . K. Fedder, C. T.-C. Nguyen, and R. T. Howe. Processtechnology for the modular integr atkn of C MOS and polysiliconmicrostructures, Microsystem Technol., vol. 1, pp. 3041, 1994.[22] G. K. Fedder and R. T. Howe, Integrated tested for multi-mode digitalcontrol of suspended m icrostructures, in Proc. IEEE Solid-State Sensorand Actuator Workshop, June 1994, pp. 145-150.[23] T. Core, W. Tsang, and S. Sherman, Fabrication technology for anintegrated surface-micromachined sensor. Solid State Technol.. DD._ _39-47, Oct. 1993.1241 H. Guckel et al., Polvsilicon resonant microbeam technology for high_.performance sensor applications, in Tech. Dig. IEE E Solid-State Sensorand Actuator Workshop, June 1992, pp. 153-160.P. M. Zavracky, F. Hartley, N. Sherman, T. Hansen, and K. Warner,A new force balanced accelerometer using tunneling tip positionsensing, in Dig. Int. Con$ Solid-state Sensors and Actuators, 1993,[26] H. K. Rockstad, T. W. Kenny, J. K. Reynolds, W. J. Kaiser, and T.B. Gabrielson, A miniature high-sensitivity broad-band accelerometerbased on electron tunneling transducers, in Dig. Int. Con$ Solid-stateSensors and Actuators, 1993, pp. 836-839.[27] N. C. MacDonald, Nanomechanisms and tips for microinstruments,in Tech. Dia. Seventh Int. C o d on Solid-state Sensors and Actuators.

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[34] W. Henrion, L. DiSanza, M. Ip, S. Terry, and H. Jerman, Wide dynam icrange direct digital accelerometer, presented at Proc. IEEE Solid-stateSensor and Actuator Workshop, June 1990.[35] B. E. Boser and B. A. Wooley, The design of sigma-delta modulationanalog-to-digital converters, IEEE J . Solid-state Circuits, vol. 23, no.6, pp. 1298-1308, Dec. 1988.

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 1 991 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 Departmentof 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 ofmicromechanical systems.

Roger T. Howe (S79-M84-SM94-F96) washorn 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 Califomia 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 M assachusetts Instituteof Technolow from 1985 to 1987. In 1987. he2joined the Department of Electrical Engineering and Computer Sciences atthe University of Califo mia at Berke ley, 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. How e 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.

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