Surface Micro Machined Accelerometers

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

366

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

This paper concentrates on polysilicon surface microstructures integrated with CMOS and Bipolar/MOS technologies. These technologies have been used for various accelerometer and resonant gyroscopes [5]-[7], as well as micro-resonator oscillators [8]. Other applications are being explored, such as micromechanical IF filters for signal processing and spectrum analysis [9], [lo]. The prospects for multisensing IC's using several sensing elements or possibly arrays are particularly interesting. Accelerometers serve as a vehicle in this paper to investigate the various aspects of surface micromachined sensors. These devices have evolved quite far, with commercial parts being available for some time now [ll], [12]. First generation I. INTRODUCTION devices achieve a noise floor around 10 m g / a over an VER the past decade, surface micromachining has be- input range of f 5 0 g or more and with shock survival in excess come established as a versatile solution for a wide of 2000 g. These specifications are compatible with automotive variety of sensing problems. Only a few additional processing applications such as airbag release. Second generation devices steps compatible with standard fabrication techniques and ma- achieve an order of magnitude better resolution and are well terials are required to cofabricate mechanical sensing elements suited for a very large range of needs, including active car and the associated electronic interface circuits on a single suspension, shock detection and monitoring, computer input die. 'Surface micromachined sensors are used, for example, devices, toys, and short term navigation. Substantial future in the automotive market as crash detectors and for dynamic improvements down to a noise level as low as 1 p g / a vehicle control. Applications as vibration and shock detectors can be expected based on analysis of the fundamental limits range from monitoring mechanical stress in airplane wings to of this technology. This level of performance is adequate for recording mechanical shock of fragile shipping goods. The all but the most demanding inertial navigation applications of technology is expected to have an even greater impact in acceleration sensors. We begin with a brief analysis of sensing elements for prospective applications such as head-mounted displays, where the small size and weight, combined with sophisticated on-chip linear acceleration and their implications for the measurement signal processing capability, are enabling features. The need in system performance. An overview of the several strategies this application for several different sensors, including linear for cofabrication of polysilicon microstructures and CMOS and angular accelerometers and gyroscopes, in a very small provides the context for a description of a modular approach volume, illustrates the advantages of monolithic fabrication of developed at Berkeley. The paper then discusses the electronic interface of the sensing element. Special attention is given to the sensing elements and associated electronics. Mechanical structures fabricated in surface micromachining factors limiting the achievable resolution of micromachined technologies consist of deposited thin films of polysilicon [ 13, sensors. Particularly, the small size and consequent low mass aluminum [2], silicon nitride 131, [4], and other materials. of the sensing element results in an elevated thermal noise Integration of surface microstructures with MOS electronics floor. It will be shown how this limitation can be overcome is relatively straightforward and economically attractive, with by vacuum packaging and embedding the sensing element in the microstructure typically occupying only a small fraction of an electronic force-feedback loop, resulting in a performance the die area. By bringing the sense element onto the integrated corresponding to a sensing element that is several orders of circuit, surface micromachining leverages the experience and magnitudes larger. The various aspects will be demonstrated sophisticated processes of IC manufacturing and brings about with examples of commercial accelerometer IC's [6], [12], all the customary advantages of IC solutions: batch fabrication, [13] and research prototypes [5], [7]. A detailed description high yield, small size, low power, and low cost. Elimination of an experimental device appears also in an earlier issue of of a separate sense element miniaturizes the package and also this journal [14]. results in improved reliability. 11. SENSWGELEMENTS

Abstract- Surface micromachining has enabled the cofabrication of thin-film micromechanical structuries and CMOS or BipolarMOS integrated circuits. Using linear, single-axis accelerometers as a motivating example, this paper discusses the fundamental mechanical as well as the electronic noise floors for representative capacitive position-sensing interface circuits. Operation in vacuum lowers the Brownian noise of a polysiticon For improved sensor perforaccelerometer to below 1 pg/&. mance, the position of the microstructure should be controlled using electrostatic force-feedback.Both analog and digital closedloop accelerometers are described and contrasted, with the latter using high-frequency voltage pulses to apply force quanta to the microstructure and achieve a very linear resplonse.

Manuscript received October 25, 1995; revised December 19, 1995. This work 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 94720-1770 USA. Publisher Item Identifier S 0018-9200(96)02451-1.

For sensing physical quantities such as acceleration, angular rate, or pressure, a mechanical sense element converts the unknown quantity into a displacement that is then detected and converted to an electrical signal. A conceptual diagram of

0018-9200/96$05 .00 0 1996 IEEE

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BOSER AND H O W SURFACE MICROMACHINEDACCELEROMETERS

displacement X A

367

proof-mass

F=ma

t F=kx

XQ

Fig. 2. Cantilever beam with concentrated load.

reference frame (a)

current voltage flux

c )

C-C

force velocity displacement

(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 this element in the presence of an external force Fext acting on a reference frame to which the proof mass is attached by means of a spring. The proof mass is further subject to damping from the surrounding gas ambient or from internal dissipation in the spring. The differential equation for the displacement x as a function of Fe,, is obtained from Newton’s law d2x

mdt2

+ b-dx + kx = Fext= ma. dt

(1)

In this equation k and b are the spring constant and damping coefficient, respectively, and linear relations are assumed. Solving for x using the Laplace transform yields the secondorder transfer function

with resonant frequency wT = and quality factor Q = w,m/b. Critical or under-damping (Q 5 0.5) of the sensor is assumed since this condition minimizes the thermal noise of the sensor, as will be seen later. At frequencies well below resonance, the displacement x M alw:, which is proportional to the acceleration. This relationship implies a trade-off between sensitivity and bandwidth of the sensor: low resonant frequency results in large displacements and hence, good sensor resolution but restricts the bandwidth of the sensor. This trade-off can be eliminated with feedback, as will be discussed later.

In micromechanical systems, the choice of the resonant frequency is also constrained by other considerations. Low resonant frequency implies a low spring constant k and high mass m. The mass of typical surface micromachined sensors is below one microgram. A one micron displacement due to a 1 g ( l g = 9.8 m/s2) acceleration would require a spring constant k less than 10 mN/m. Material properties, geometric constraints, and self resonance of the spring set a lower bound on 5 that is well above this value. A further complication is that the resonance frequencies in different directions cannot be chosen independently. For example, the ratio of resonant frequencies about the y and z axis for the cantilever beam shown in Fig. 2 with thickness t and width w is equal to the ratio t / w [15], [16]. Typical dimensions for the suspension of a z axis accelerometer are a thickness of 2 pm and a width of 5 pm, corresponding to a ratio of resonant frequencies of only 2.5. Therefore, in practical situations, 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 constant on residual stress in the film and the capability of low spring constants in a small area. Torsional resonant modes are potentially near the fundamental mode of the sense element, making careful design essential. Analytical models of such suspensions [ 161, [ 171 have been developed that provide insight. Fig. 3 shows the result of a finite element analysis simulation [18] of a proof mass suspended with four singlefolded trusses that reveals a rotational resonance about either diagonal at less than twice resonance for the desired motion along the z axis. Typical polysilicon surface micromachined accelerometers have resonance frequencies in the kilohertz range and a mass of between 0.1 and 1 pg. This low mass gives rise to another design challenge faced in high-sensitivity accelerometers. According to the laws of thermodynamics, the thermal energy of a system in equilibrium is k ~ T / 2for each energy storage mode, where k13 is Boltzmann’s constant. The minuscule mass of the micromachined device implies substantial agitation due to this thermal energy, a process known at the molecular level as Brownian motion [19]. The extent of this disturbance can be appreciated readily from the equivalent second-order system presented in Fig. 1(b), which is described by the same differential (1) as the mechanical system shown in Fig. l(a). In the electrical equivalent, currents represent forces, and voltage corresponds to velocity. The electrical component values are proportional

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368

(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 view 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).

head-mounted displays demand better performance. Lower noise can be achieved by either increasing the mass or increasing the quality factor Q by reducing the damping of the sensing element. The first approach offers only limited improvement because of the relatively modest increase of the mass of the sensor obtained, for example, by substituting a material with higher density such as tungsten for polysilicon. The lateral dimensions of the proof mass are restricted by economic constraints that call for minimizing silicon area as well as processing V Q G m= (3) difficulties. Polysilicon films typically have small gradients in the residual stress that cause warpage of large-area plates For a typical micromachined sensor with m = 0.5 pg, w, = which limits lateral dimensions to a few hundred microns in 271.10 kHz, and Q = 0.5, the input referred noise density is typical technologies. By contrast, vacuum packaging of the sensor results in a at room temperature. approximately 200 pg/& For many commercial applications (e.g., in the automotive several orders of magnitude decrease of the thermal noise sector), this noise level is acceptable, but inertial navigation of the sensor and does not require any modification of the and other precision applications such as tracking systems for sensor fabrication process. Quality factors of 50 000 have

to the respective mechanical quantities. The current noise zz/Af = 4 k B T / R from the resistor corresponds to an equivalent force noise source associated with the mechanical damping element. It is in parallel with the external input Iekxt, corresponding to the measured force in the mechanical original, and has a white power spectrum. The equivalent acceleration spectral density is obtained by back-substituting mechanical for electrical quantities and is

/&y- ,,;,,-.

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369

BOSER AND HOWE SURFACE MICROMACHINED ACCELEROMETERS

U P wall

N substrate

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

been reported in the literature for in situ phosphorus-doped polysilicon [SI, [lo], corresponding to a reduction of the noise floor in above example from 200 pg/& to below 1 pg/&. By comparison, the ratio of the density of tungsten to that of polysilicon is less than an order of magnitude. Of course, the elimination of nearly all mechanical damping requires that the position of the suspended mass be controlled electrically. 111. FABRICATION TECHNOLOGY Monolithic integration of the proof mass and suspension for an accelerometer or other physical sensor with its associated electronics requires the merging of surface micromachining with an IC fabrication process. Polysilicon and aluminum are the most attractive candidates for the structural elements of micromechanical sensors. Both are already used in standard IC technology and can be deposited and patterned to very accurate dimensions. The negligible fatigue [20] and lack of memory of polysilicon make it the material of choice for the fabrication of high performance micromechanical sensors. The advantages of aluminum are its low processing temperatures, which makes integration with electronic processing much more straightforward. In order to integrate polysilicon microstructures with CMOS processing, the 600 "C deposition and 950 OC rapid-thermal annealing temperature needed for the former must be considered carefully. One approach is to replace the conventional aluminum metallization with tungsten, together with a titanium silicide diffusion barrier in the contacts, to enable the CMOS to withstand these high post-processing temperatures. Low pressure chemical vapor deposition (LPCVD) silicon nitride is used to isolate the CMOS from the micromachining process steps, such as the final etching of the sacrificial layer in hydrofluoric acid. The modular integration of CMOS and microstructures (MICS) process uses this integration strategy and is based on a conventional 3 pm CMOS technology [21]. Fig. 4 shows a cross-sectional view of an MICS wafer. The sensing structures can be seen on the right and consist of three layers of 0.5 pm, 1.5 pm, and 1 pm thick phosphorus-doped polysilicon. The second and third layer are deposited on top of 1 and 2 pm thick sacrificial phosphosilicate glass layers that are later removed. Fig. 5 shows an scanning electron microscopy (SEM) picture of a section of a multimode test structure for digital

control that was fabricated in the MICS technology [22].In this device, the proof mass and spring suspension are fabricated in poly level two. The third layer serves as a limit stop and electrostatic actuator. The etch holes in the proof mass and top actuator are required for complete removal of the sacrificial 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 polysilicon fuses 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 important to prevent overhanging structures from warping after release. Polysilicon cantilever beams fabricated in the MICS technology typically warp less than 0.1 pm over 400 pm in length. Stiction of the sense element or its suspension to the limit stops or to the substrate due to overrange forces poses an additional challenge for the fabrication technology. The first commercial integrated surface machining microsensor technology, the BiMEMS process of Analog Devices, Inc. [23] uses a different integration strategy from the MICS technology. The sacrificial oxide and structural polysilicon film are deposited and annealed prior to a standard aluminum metallization. In BiMEMS, a diffusion is used to connect the polysilicon microstructure to the circuit, rather than the poly level one and gate polysilicon levels used in MICS. Finally, BiMEMS is a linear bipolar/MOS technology that includes thin film resistors, making it very well suited to implementing analog closed-loop control of the sense element. IV. POSITIONSENSING Acceleration sensors translate the external signal in a corresponding displacement that can be measured by several means. Piezo-resistive strain gauges are used widely in sensors because of the simple interfacing to off-chip electronic circuits. Ion-implanted piezo-resistors have been used in undoped polysilicon resonant microbeams [24] and polysilicon piezo-resistors have also been embedded in silicon nitride membranes for pressure sensing [3]. Position sensors that measure the capacitance between a conducting polysilicon proof mass and a fixed electrode, on the other hand, require no additional processing and can be extremely sensitive, as will be shown later. The negligible temperature coefficient is another important advantage of a capacitive sensor readout. The low parasitics that are characteristic of monolithic

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370

ac

mt

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

integration are the key to maximizing the performance with this technique. Researchers are also investigating alternative transduction methods such as tunneling tips in surface microstructures [25]-[27]. Fig. 6 schematically shows quarter-sections of two sensor designs with sensitivity to accelerations perpendicular and parallel to the silicon surface. In the z axis structure, the capacitance between the proof-mass and the substrate serves as the mechanical to electrical interface [5];the y axis design uses a comb-like structure [6].The advantage of the first style interface 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 has the advantage of a much larger sense capacitance for a given area of up to 1 pF, compared to less than 200 fF for typical y axis sensing elements. For position measurement, the variation dC/dx of this capacitance due to displacement must be maximized. For the z axis structure, a parallel plate approximation is usually appropriate and for small displacements, the capacitor change is approximately equal to C / X OMaximizing . the sensitivity therefore calls for minimizing the capacitor gap, XO. In the second structure, a substantial fraction of the total capacitance is due to fringing fields and does not change substantially in the presence of small displacements [l6], [as], resulting in a somewhat lower sensitivity of this arrangement for a given electrode spacing. Exact values can be obtained from numerical simulation [29]. The challenge in the design of a capacitive position measurement circuit consists in detecting extremely small capacitance changes in the presence of much larger parasitics. For example, a sensing element with resonant frequency fr = 5 kHz experiences a displacement of only 0.1 A in the presence of a 1 mg constant acceleration signal. A parallel plate

anchor

suspension,

anchor

tether

proof mass

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 changes by a mere 10 aF due to this displacement, an amount that is 100000 times smaller than the capacitor itself. Flicker noise, offsets, and parasitic capacitances represent further difficulties. The circuit shown in Fig. 7 uses chopper stabilization and bootstrapping to minimize the noise and maximize the I sensitivity. The voltage V, at the midpoint of the divider formed by the two sense capacitors and excited with the ac signal V, is proportional to the capacitor mismatch and, hence,

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BOSER AND HOWE SURFACE MICROMACHINED ACCELEROMETERS

371

+vsy 7 Csl2

t

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

C&T Electronic noise as a function of transistor size.

it is important to derive relationships relating this noise to basic physics and technology parameters. At least in principle, all noise sources except the input from the amplifier can be made referred thermal noise insignificantly small. For either the circuit shown in Figs. 7 or 8, this noise appears across all capacitors connected to node V, and produces an equivalent current noise that is added to the signal current i, = V,AC. In an appropriately designed amplifier, the input transistor is the dominant noise contributor, and hence

3

-vs

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

position of the proof mass. The measurement is performed at a sufficiently high frequency to suppress offset and flicker noise and the result demodulated. The parasitic capacitance in node V, must be minimized to avoid signal attenuation. Many designs reduce the attenuation by shielding interconnect capacitances and tying the shield to the output of a unity gain buffer as shown. Shields are also needed to avoid electrical fields between the sensor and surrounding conductors since the resulting electrostatic forces are indistinguishable from an external force and, hence, corrupt the acceleration measurement. Special provisions are needed to control the dc potential at node V,. The solution shown here relies on a resistor R d c that typically must be in the megohm range to minimize its noise contribution. An alternative solution presented in Fig. 8 eliminates both the demodulator and the need for a resistor or similar element to set the dc potential [7], [30]. This circuit is based on an amplifier with an auxiliary input A2 with reduced gain and operates in two phases. First, the sense capacitors are precharged to a constant voltage. At the same time, the offset and flicker-noise of the amplifier are stored on the holding capacitor Ch. During the second phase, the voltage across the sense capacitors is changed, causing a charge that is proportional to the mismatch between the two sense capacitors to flow into node V,. Since the amplifier input is now a virtual ground, this charge flows onto the integrating capacitor C, 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 noise floor of the position measurement interface sets a limit on the achievable sensitivity of the sensor. To assess the ultimate performance achievable with surface micromachined sensors,

-

3

assuming a differential input. Apparently, the noise can be reduced by increasing the transconductance gm. Increasing gm, however, requires increasing either the saturation voltage of the input devices, which in practice is limited by the supply voltage, or enlarging the gate capacitance C,,. This, in turn, increases the capacitance at node V, and hence, the current noise, thus calling for a trade-off between gm and C,,. For a given saturation voltage, these two quantities are related as gm M W T C ~by, the cutoff frequency WT of the technology. Fig. 9 shows the normalized current noise as a function of the ratio of C,, to the sum of all other capacitances CT = C, + C, connected to the amplifier input. Clearly, the noise is minimized for C,, = CT, with a small penalty for somewhat larger values of C,, as might be required to meet other circuit requirements, such as bandwidth. Based on these considerations and assuming that the amplifier input capacitance is chosen optimally, the mean squared error of the capacitance change divided by the sense capacitance C, is

For a parallel plate sense capacitor with gap 20,the corresponding input referred acceleration noise floor is

3wT cs

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(6)

312

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

This expression demonstrates how the sensitivity of an accelerometer that is limited by electronic noise in the position measurement circuit can be maximized. Upon first inspection, proof mass position sense compensator the biggest improvement appears to result from minimizing the resonant frequency. Accelerometers described in the literature Ffb feature resonant frequencies in the range of 5 kHz [14] to over 20 kHz [ 113. Unfortunately, substantially lowering force transducer these frequencies compromises the reliability of the device, in particular its robustness to mechanical shock, and increases Fig. 10. Accelerometer with analog force-feedback servo loop. the probability of the proof mass sticking to the substrate. Better suppression of undesired resonances is needed as well before the resonant frequency can be reduced substantially. important in accelerometers because of the trade-off between Lowering the capacitor gap xo also decreases the noise floor sensitivity and bandwidth imposed by the sensing element. of the sensor but, of course, depends on technology. In z Feedback increases the useful bandwidth by a factor equal axis structures, the spacing is dictated by the thickness of to the loop gain, which, thanks to the electronic circuitry, the sacrificial oxide and must be chosen sufficiently large to can be made large. Consequently, the resonant frequency of prevent the sensing element from touching the substrate due the sensing element can be optimized for sensitivity alone, to warpage and mechanical excitation. For current polysilicon regardless of the desired sensor bandwidth. Controlling the displacement of the proof mass is equally surface micromachining technologies, minimum vertical or important, particularly for vacuum-packaged high-Q devices lateral gaps are around 1 pm. which can exhibit motion at the resonant frequency that The remaining parameters in (6) characterize the electronic interface of the sensor. Accordingly, the sense capacitance C, exceeds the small spacings between the electrodes of the sense should be maximized, a condition that favors z axis over y capacitor. Finally, imperfections, for example, due to nonlinear axis designs. The parasitic capacitance C, should not exceed or temperature sensitive springs, are attenuated provided that the value of C, for not significantly degrading the sensor the force-transducer does not introduce similar errors. Electrostatic actuation is the simplest means for generating performance. High cutoff frequency and, hence, short channel the feedback force in a micromechanical sensor. The electrodes length also improve sensor sensitivity. Increasing the sense of a capacitors with a constant voltage V, across are attracted voltage V, results in an increased signal current and, hence, better signal-to-noise ratio. In practice, V, is often limited with a force not only by the supply voltage, but also by the maximum v:c F, = d C(x(j)V,2- __ acceptable electrostatic force exerted on the sensor. This dxo 2 2x0 force can degrade the linearity of the sensor (see Section VB). in the case of a parallel plate capacitor with value C and gap Typical numbers for surface micromachined sensors, C, = 20. The same capacitor can be used for force-feedback and 0.5 pF, C, = C,, WT = 2.ir x 500 MHz, V, = 0.5 V, fr = 5 sense 1111, or separate electrodes can be added [5]. kHz, and xo = 1 pm give a noise floor of 1 p g / a at room Because of the quadratic dependence of the force on voltage, temperature. This figure is of the same order as the Brownian electrostatic actuators cannot be used directly, but must be noise of a vacuum-packaged structure. While this performance combined with some means of linearization. For symmetric level has not yet been achieved with micromachined sensors, sensors [Fig. 6(b)], a simple solution is to apply a voltage these numbers demonstrate the potential of this technology for Vi + AV across C+ and V, - AV across C-. Since the very sensitive inertial measurement applications. two resulting forces are in opposite directions, the quadratic The 1 p g / z / H z noise floor corresponds position measureterms cancel and the net difference A F = ~ V O A V C /is~ O ment noise of only A/z/Hz. This means that displace- proportional to the controlling voltage AV. In practice, the ments as small as the classical diameter of an electron can be linearity of this technique is limited by the matching accuracy detected in a 10 Hz bandwidth. In an actual circuit, the noise of C+ and C-. More importantly, this approach cannot be would be somewhat larger due to additional noise sources used at all with asymmetric sensors such as the one depicted neglected in this analysis. in Fig. 6(b). A more general and potentially more accurate solution V. FORCE-FEEDBACK consists in pulse modulating the feedback signal. Fig. 11 Fig, 10 shows an acceleration sensor embedded in a feed- shows a system that employs a clocked comparator to quantize back loop. A compensator and force-feedback transducer are the feedback force to only two levels [5], [7], [14]. This added to the open loop sensor consisting of the proof mass and system is equivalent to a sigma-delta modulator as used in position measurement circuitry. The feedback force opposes A D conversion [31], except that the noise shaping filter displacements of the proof mass from its nominal position. has been replaced by the mechanical sensor. If all feedback Compensation is required for stability and will be discussed pulses are kept equal in length, imbalance merely results in later. an offset and/or gain error, but does not cause distortion. Feedback improves many important characteristics of a sen- The pulse-density of the one-bit output stream tracks the sor, including bandwidth, dynamic range, and, in certain cases, input acceleration, which is obtained by low-pass filtering and linearity and drift. Increasing the bandwidth is particularly decimating the pulse-density code. Because of the inherent

1

I

~

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BOSER AND HOWE: SURFACE MICROMACHINED ACCELEROMETERS

vo

Hc(z)

I

ProofmaSS

pasitionsense

compensator

373

comparator

II

-

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

AD 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 to stability, residual motion of the proof mass, and, for the case of digital feedback, quantization noise.

X ~

A. Stability

The force-feedback accelerometers shown in Figs. 10 and 11 are unstable without compensation because of the 180 degree phase delay from the sensing element for frequencies above the resonance. Several solutions are used in accelerometer designs and will be described below: over-damped proof mass [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 Brownian noise floor of an over-damped low-mass sensing element, practical only for low performance applications or when much larger and thus heavier sensing elements than those built with surface micromachining technology are used. The second approach uses an electronic low-pass filter to reduce the loop bandwidth to frequencies well below the resonance of the proof-mass. Obviously, this technique is applicable only in situations where either a low measurement bandwidth or low sensitivity can be tolerated, since in this case the sensor bandwidth is actually reduced to a value less than the resonant frequency f T of the sensing element. Stability demands that the loop gain is lower than the ratio of f r to the sensor bandwidth. A commercial surface micromachined accelerometer implements this technique [ 111. The sensor has a resonant frequency of 24 kHz and 1 kHz signal bandwidth and a low-frequency loop gain of about 10, thus diminishing somewhat the benefits of feedback. A compensation filter requires additional circuitry but avoids the aforementioned problems. The basic strategy is to add a left half-plane zero to the loop transfer function in order to decrease the phase delay at the unity-gain frequency. The analysis is straightforward for analog feedback, but complicated by the nonlinearity in the digital case. The conventional definitions of stability involving the boundedness of states and absence of limit cycles are not useful for sigma-delta modulators which use oscillations as a means for A/D conversion. A more appropriate criterion follows from analyzing the spectrum of the pulse-density output. For illustration purposes, consider the digital feedback loop in Fig. 11 with zero input. Then the comparator output and feedback signal switch rapidly between positive and negative 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

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

frequency of the feedback signal. The maximum possible rate is f s 14,constrained by the clocking and second-order nature of the feedback loop, and is achieved when the total phase delay is less than 180’ at this frequency. This condition is met when the phase lead from the compensator is larger than approximately t d f s x 4 5 O . Here, t d is the delay from sensing the position of the proof mass to applying the feedback signal. Simulation results for two loops operating under this condition are shown in Fig. 12. In both cases, a compensator H,(x)= 2 - z-l is ased to add a zero at fs/9 and approximately 27’ phase lead at fs/4. In the first case, short feedback pulses are issued almost immediately after the position of the proof mass has been measured. The compensator output V, leads V, despite a small lag of V,, suggesting that a smaller amount of lead (e.g., 5 - z - l ) would suffice in this case. Simulation confirms that this is indeed the case. In the second case, the feedback pulses are elongated and delayed by t d = T / 2 causing V, to arrive just in time to ensure a correct decision by the comparator. The longer feedback pulses round the edges of the position signal, but for stability, only the delay t d from position measurement to the midpoint of the feedback pulse is relevant. B. Residual Motion of the Proof Mass

The sense and feedback capacitors exert a position dependent force on the proof-mass. Residual motion consequently results in an error force that cannot be distinguished from the accelerometer input. Assuming a zero external input and proper compensation of the feedback loop, the feedback signal of the accelerometer is pulse-train at one quarter the sampling frequency f s and amplitude amaxequal to the input range of the device. This feedback signal causes the proof mass to move up and down at the frequency f s / 4 . Owing to the second-order nature of the proof mass, the amplitude of the fundamental

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374

of the motion is Ax = ~ ~ ~ / ( 2 x f ~Assuming / 4 ) ~ . a fullscale input range amax = 2 g and f s = 10 MHz, Ax M l0W3A. Noise or an external acceleration signal randomize the spectrum of the proof mass displacement and give rise to low-frequency components falling into the signal-band of the sensor. Capacitive position sensing and feedback both exert an electrostatic force on the proof mass. To first order, this force is proportional to the position of the proof mass. Residual motion modulates these forces, resulting in a spurious acceleration of

AX

Aa M (asense + amax)--

1

-

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

(8)

20

VI. CONCLUSION

Usually, the acceleration due to the sense capacitor dominates because of the need to maximize the sense voltage V, in order to minimize the noise floor of the position measurement circuit. A typical asense= 50 g results in an error Au M 8 pg. Simulations and measurements show that this error is sufficiently random to raise the noise floor of the sensor. It is most easily suppressed by choosing a sufficiently high sampling rate fs. A sensor with analog feedback and identical loop bandwidth will exhibit the same error, however, because of the absence of limit cycles, it will not result in an increased noise floor but instead in a slightly increased nonlinearity.

The advantages of monolithic fabrication of micromachined sensors and associated electronics have been discussed. It has been demonstrated that the limits of the technology are beyond a 1 p g / G noise level, which is compatible with a large variety of demanding applications. ACKNOWLEDGMENT

The authors are very indebted to their students who did most of the work reported here. REFERENCES

C. Quantization Noise Digital feedback adds quantization noise as an additional error source [35]. Because of the typically high ratio of the sampling frequency f s 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 inputs less than (9) For am= = 2 g and f s / f r = 1000, ad

M 16 pg. Like errors due to residual motion of the proof mass, the dead-zone width can be decreased by raising the sampling rate, f s . Smaller dead-zones exist also for other inputs that are rational fractions of amax 1311. The cause of the dead-zone can be appreciated from the block diagram in Fig. 13, where the proof mass is modeled as a linear filter H ( s ) and the order of the linear filtering action and feedback summing node have been reversed. Assuming for a moment that the input of the accelerometer labeled amin is zero, the feedback signal is a square wave with amplitude amaxand frequency fs/4. To be detected, an input signal must disturb this idle pattern of the modulator. This occurs only for displacements U that are at least equal in amplitude to the idle channel residual motion w. In oversampled AID converters with electronic noise shaping filters, the resonant frequency of the mechanical noise shaper corresponds to the filter pole frequencies. These are usually at much lower frequencies; consequently, dead-zones are not observed.

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Bernhard E. Boser received the Diploma in electrical engineering in 1984 from the Swiss Federal Institute of Technology (ETH) in Zurich, Switzerland, and the M.S. and Ph.D. degrees from Stanford University, Stanford, CA, in 1985 and 1988, respectively. From 1988 to 1991 he was a Member of Technical Staff at AT&T Bell Laboratories, working on VLSI implementations of artificial neural networks and algorithms for automatic learning. Since 1992 he has 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 circuits for data conversion and communication, and on the design and fabrication of micromechanical systems.

Roger T. Howe (S’79-M’84-SM’94-F’96) was horn in Sacramento, CA on April 2, 1957. He received the B.S. degree in physics from Harvey Mudd College, Claremont, CA in 1979, and the M.S. and Ph.D. degrees in electrical engineering from the University of Califomia at Berkeley, in 1981 and 1984, respectively. He was on the faculty of Carnegie-Mellon University during the 1984-85 academic year and was an Assistant Professor at the Massachusetts Institute of Technolow from 1985 to 1987. In 1987. he joined the Department of Electrical Engineering and Computer Sciences at the University of Califomia at Berkeley, where he is now a Professor, as well as a Director of the Berkeley Sensor & Actuator Center. His research interests include silicon microsensors and microactuators, micromachining processes, and integrated-circuit design. Dr. Howe served as CO-General Chairman of the 1990 IEEE Micro Electro Mechanical Systems Workshop and is General Chairman of the 1996 SolidState Sensor and Actuator Workshop at Hilton Head, SC. “2

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