496 IEEE SENSORS JOURNAL, VOL. 7, NO. 4, APRIL 2007

Characterization and Experimental Verification of theNonlinear Distortion in a Technique for Measuring

the Relative Velocity Between MicromachinedStructures in Normal Translational Motion

Robert Neal Dean, Jr., Member, IEEE, George T. Flowers, Roland Horvath, Nicole Sanders,A. Scottedward Hodel, Senior Member, IEEE, John Y. Hung, Senior Member, IEEE, and

Thaddeus Roppel, Member, IEEE

Abstract—Applications exist for microelectromechanical sys-tems (MEMS) devices where the measurement or estimation ofthe relative velocity, or at least the direction of instantaneousrelative velocity, between two microstructures in normal trans-lational motion is required. A technique for directly measuringthe relative velocity has not been available. This paper presents atechnique for directly measuring the relative velocity between twomicrostructures in normal translational motion. The techniqueconsists of measuring the current flowing into the capacitanceformed between the two microstructures when a constant voltageis applied across them. The technique and the resulting nonlineardistortion in the velocity measurement are characterized. A pro-totype relative velocity sensor is fabricated and evaluated to verifythe measurement technique.

Index Terms—Electrostatic actuators, microelectromechanicaldevices, microsensors, velocity measurement.

I. INTRODUCTION

NUMEROUS applications have been developed for parallelelectrode microstructures that experience relative transla-

tional motion normal to their opposing faces. Many of these ap-plications also require the measurement or estimation of relativevelocity or at least the instantaneous direction of relative velocitybetween the two electrodes. Examples include accelerometers[1]–[3], vibration control in a beam [4]–[6], enhancement of thestableoperatingrangeofparallelplateactuators[7], [8],advancedcontroller design for microelectromechanical systems (MEMS)

Manuscript received September 28, 2006; revised November 2, 2006; ac-cepted November 7, 2006. This work was supported in part by Morgan Re-search Corporation through the National Science Foundation under STTR GrantDMI-0339954. The associate editor coordinating the review of this manuscriptand approving it for publication was Dr. Babak.

R. N. Dean, Jr., A. Scottedward Hodel, J. Y. Hung, and T. Roppel are with theElectrical and Computer Engineering Department, Auburn University, Auburn,AL 36849 USA (e-mail: rdean@ eng.auburn.edu; scotte@eng.auburn.edu;hungjoh@eng.auburn.edu; roppeth@auburn.edu).

G. T. Flowers is with the Mechanical Engineering Department, Auburn Uni-versity, Auburn, AL 36849 USA (e-mail: gflowers@eng.auburn.edu).

R. Horvath is with Structural Integrity Associates, Inc., Centennial, CO80112-3868 USA (e-mail: RHorvath@structint.com).

N. Sanders is with Auburn University, Auburn, AL 36849 USA (e-mail: nl-sanders1018@yahoo.com).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JSEN.2007.891920

devices [9]–[12], and active vibration isolation control laws thatcould be applied to microsystems [13]. In all of these exampleswhere real systems were implemented, the relative velocity termwas estimated using either an observer or recorded displacementmeasurements.Relativedisplacementmeasurementbetweentwomicromachined structures in normal translational motional canreadily be accomplished using a number of methods, such as op-tical techniques [14], piezoresistive strain gauges [15], ac voltagedivision [16], diode-quad sensing [17], synchronous detection[18], and various switched capacitor techniques [19], [20]. How-ever, a suitable technique for directly measuring the relative ve-locity between two micromachined structures in normal trans-lational motional, outside of a laboratory setting, has not beenavailable [8].

Senturia [21] presented a technique for measuring relative ve-locity between two electrodes in relative motion by applyinga constant voltage between the two electrodes and then mea-suring the current flowing through the resulting capacitor, whichis modeled by

(1)

where is the displacement variable, is the capacitancebetween the two electrodes as a function of displacement, isthe constant voltage applied across the two electrodes, andis the current flowing through .

Observe that when the partial derivative of capacitance withrespect to electrode separation distance is constant,the current flowing through the capacitor is linearly proportionalto the relative velocity between the two electrodes. For tangen-tial electrostatic actuators, such as comb drive actuators,has the form

(2)

where is the displacement variable, is the fixed amount ofelectrode overlap, is the fixed separation distance betweenthe two electrodes, and is the permittivity of the dielectricmaterial between the electrodes. Therefore

(3)

which is a constant.

1530-437X/$25.00 © 2007 IEEE

DEAN et al.: CHARACTERIZATION AND EXPERIMENTAL VERIFICATION OF THE NONLINEAR DISTORTION 497

However, for parallel plate actuators and other microstruc-tures experiencing normal translational motion, has theform

(4)

where is the overlapping electrode area and is the displace-ment variable. Therefore

(5)

which is not a constant, resulting in nonlinear distortion of therelative velocity measurement. Therefore, this technique has notbeen utilized to measure relative velocity in these applications.Since another technique for measuring relative velocity betweenmicrostructures that can be directly integrated into the devicehas not been available, the relative velocity term is estimated [4],[22]. However, this paper demonstrates that this technique canbe applied to some applications involving two microstructuresexperiencing normal translational motion described by (5).

II. RELATIVE VELOCITY SENSOR THEORY

Consider a microstructure where two electrodes are in relativesinusoidal motion, as illustrated in Fig. 1, described by

(6)

and

(7)

Although the electrodes may experience nonsinusoidal motion,exploration of the effects of sinusoidal motion is warranted sincesome MEMS structures are designed to be highly underdamped[23] and have a tendency to ring at their natural frequency whenexcited. Therefore, the first-order model for the capacitance be-tween the two electrodes is

(8)

where is the rest distance between the two electrodes, andis the amplitude of the sinusoidal relative motion between

the electrodes with frequency . If a small voltage , whichis less than the pull-in voltage, is applied across the electrodes,then charge accumulates in the time-varying capacitor ac-cording to

(9)

The current flowing into can be obtained by taking thetime derivative of , resulting in

(10)

Let be a small, fixed dc voltage . Then, the current in(10) becomes

(11)

Fig. 1. Illustration of a microstructure with two electrodes in relative sinusoidalmotion.

The negative sign in the numerator indicates that current flowsout of the capacitor when the distance between the two elec-trodes is increasing, which is correct since, under this condition,the capacitance is decreasing. Likewise, when the distance be-tween the electrodes is decreasing, capacitance is increasing, re-sulting in current flow into the capacitor and a positive value for

. For small amplitudes of relative sinusoidal motion suchthat , (11) can be approximated by

(12)

where

(13)

The current flowing out of in (12) is , which isproportional to the relative velocity between the two electrodesin (7) for . Therefore, this technique for measuringrelative velocity can be applied to microstructures experiencingnormal translational motion between two electrodes if the rela-tive motion is small compared to the rest gap distance.

For many applications, a low impedance current to the voltageconversion amplifier can be utilized as the sensor interface cir-cuit [24]. The gain of the amplifier is , where is the am-plifier feedback resistor. The amplifier gain can be incorporatedinto (11) to obtain an equation for the sensor interface circuitoutput voltage , where

(14)

III. RELATIVE VELOCITY SENSOR CHARACTERIZATION

Consider a normalized version of the equation for the sensorinterface circuit output voltage such that

(15)

where (14) was modified such that

(16)

(17)

and

(18)

498 IEEE SENSORS JOURNAL, VOL. 7, NO. 4, APRIL 2007

Fig. 2. Graph of the performance of the normalized relative velocity sensor forvarious ratios of x =x .

Fig. 3. Plot of the THD as a function of x =x for the normalized sensor.

Equation (15) can be used to investigate the nonlinear effects inthe sensor performance due to the term in the de-nominator. Fig. 2 presents a graph of the normalized sensor re-sponse from (15) for , compared to the ideal re-sponse of the sensor for various ratios of . Observe thatthe sensor response quickly deviates from the desired responseas the ratio falls below 100. However, notice that thepoints where the velocity changes direction do not deviate fromthe desired sensor response as the ratio varies. This indi-cates that the velocity sensor serves well as a velocity directionsensor, where the polarity of the sensor interface circuit outputvoltage indicates the direction of relative velocity between thetwo electrodes. One application for a velocity direction sensoris active vibration isolation where a variable structure controllaw, such as semiactive damping ON–OFF skyhook control [13],[25], is implemented.

Another measure of the performance of the relative velocitysensor is the total harmonic distortion (THD) of the normalizedsensor response as the ratio of varies. A plot of the THDas a function of for the normalized sensor is presentedin Fig. 3. Observe that the THD can be approximated byfor ratios greater than 1.5. Therefore, the THD estimatecan be used to determine the minimum ratio that canbe tolerated for a particular application to maintain a minimumlevel of quality in the relative velocity measurement.

IV. RELATIVE VELOCITY SENSOR PROTOTYPING

A prototype relative velocity sensor has been designed, fab-ricated and tested. The sensor consisted of a micromachined Siplanar spring-mass-damper platform that was suspended over asecond planar electrode. A photograph of the micromachined

Fig. 4. Photograph of the micromachined Si spring-mass-damper platform.

Si spring-mass-damper platform is presented in Fig. 4. It con-sisted of a square proof mass attached to a surrounding frame byeight springs. The bottom side of the proof mass plate formedthe upper electrode. A nominally 375- m-thick (100) n-type Sidouble-side polished wafer with a resistivity of 1–10 cm wasutilized to fabricate the spring-mass-damper platform. A front-side dry etch was performed in a Surface Technology Systems’Advance Silicon Etcher (STS ASE) plasma-etching system todefine the spring thickness. Then, a backside dry etch was per-formed to release the proof mass from the surrounding frame.The top surface of the micromachined device was uniformlycoated with 1000A Ti, followed by 5000A Cu, followed by1000A Au through electron beam evaporation. This allowed theelectrical connection to the proof mass plate electrode throughthe metallization on top of the surrounding frame structure. Themass of the proof mass and the spring dimensions were utilizedto set the natural frequency of the passive spring-mass-damperplatform. Utilizing a proof mass plate of 1 cm 1 cm with anattached Si die (5 mm 5 mm by 560 m thick) and springsof 1.3 mm long 225 m wide 40 m thick resulted in a de-signed natural frequency of 1.39 KHz. After micromachiningwas completed, the 5 mm 5 mm Si proof mass die was at-tached on top of the proof mass plate using cyanoacrylate glue.

The second electrode consisted of a round, 3.4-mm-diametermetallized landing on the backside of the printed circuit board(PCB) containing the sensor interface circuitry. An approxi-mately 100- m-thick nonconductive polyester spacer was usedto mount the micromachined Si spring-mass-damper platformover the PCB with the second electrode. This architecture hasbeen utilized in micromachined vibration isolation platformsfor advanced packaging of MEMS angular rate sensors [25].After assembly, the 3.4-mm-diameter bottom electrode restedapproximately under the center of the micromachined top elec-trode, with a rest gap of nearly 100 m. A capacitance of 1.2pF was measured between the two electrodes while using airas the dielectric material. Using the equation for capacitanceof the sense capacitor (8) with a fixed rest gap, the calculatedrest gap distance was 66.99 m. The difference between thecalculated rest gap distance and the nominal thickness of thepolyester spacer can be attributed to the measurement of straycapacitance in the capacitance measurement setup, compression

DEAN et al.: CHARACTERIZATION AND EXPERIMENTAL VERIFICATION OF THE NONLINEAR DISTORTION 499

of the spacer during assembly, and fringing effects between thetwo electrodes that are not accounted for in (8).

The fabricated prototype interface circuit consisted of threestages and used the Texas Instruments OPA445 FET-inputop amp. This amplifier was selected due to its high inputimpedance, wide power supply operating range ( 10 V to

45 V), high slew rate (15 V s), surface mount footprint,and commercial availability. The interface circuit was poweredwith 25 V, with 25 V applied across the velocity sensingcapacitor . The first stage of the sensor interface circuitconsisted of a low impedance current-to-voltage conversionamplifier with set at 100 k . Two additional gain stageswere utilized so that the total gain of the sensor interface circuitwas . Since ac coupling was utilized between thesecond and third amplifier stages to prevent the amplified offsetvoltage from the first stage from saturating the amplifier in thethird stage, the sensor interface circuit had a bandpass responsewith a center frequency of 5.6 KHz and a 3-dB bandwidth of11.2 KHz. A dc voltage less than 25 V could have been appliedacross the two electrodes while a higher gain was utilized in thesensor interface circuit to obtain the same sensor response.

V. RELATIVE VELOCITY SENSOR TESTING

The micromachined Si spring-mass-damper platform fromFig. 4 was integrated with the 100- m-thick polyester spaceronto the backside of the PCB. The resulting compilation wasintegrated into a machined plastic fixture for dynamic evalu-ation on an electromechanical shaker, as shown in Fig. 5. Aphotograph of the dynamic evaluation setup is presented in Fig,6, showing the assembled prototype sensor mounted onto theelectromechanical shaker, and two laser interferometric mea-surement systems. First, the transmissibility (i.e., the magnitudeof the measured mechanical frequency response of the proofmass divided by the measured mechanical frequency responseof the frame) of the spring-mass-damper platform was mea-sured, as shown in Fig. 7, in order to determine the mechanicalresonant frequency. The micromachined spring-mass-damperstructure had a second-order underdamped low-pass responsewith a resonant frequency at 1.34 KHz and a mechanicalfactor of approximately 20. Next, the shaker was programmedto excite the test article at the resonant frequency, 1.34 KHz,while the time responses of the relative displacement betweenthe two electrodes and the sensor interface circuit output voltagewere recorded, as presented in Fig. 8. Evaluating the prototypeat its resonant frequency takes advantage of the amplitude gainafforded by the highly underdamped response in order to max-imize the relative electrode motion, since the electromechan-ical shaker can only produce a few micrometers of motion whendriven at frequencies greater than about 1 KHz. The time traceof the relative electrode displacement was obtained using twoPolytec OFV 353 laser interferometric displacement measure-ment systems, where one measured the motion of the frame,and the other measured the motion of the proof mass. Then, thedifference was taken to determine the relative displacement be-tween the two electrodes as a function of time.

The relative electrode displacement, presented in Fig. 8, hasthe strongest component at 1.34 KHz with an amplitude of ap-proximately 3.5 m. Analysis of the test data reveals that the

Fig. 5. Photograph of the assembled prototype sensor mounted for dynamicevaluation.

Fig. 6. Photograph of the dynamic evaluation set up.

Fig. 7. Measured mechanical frequency response of the Si spring-mass-damperstructure.

fundamental frequency component of the sensor output signalleads the relative electrode displacement signal by 90 , whichis characteristic of the relative velocity between the two elec-trodes. This can be observed by numerically calculating thetime derivative of the measured relative displacement data, aspresented in Fig. 9, and comparing it with the theoretical andcalibrated output signal from the relative velocity sensor. Thetheoretical output of the sensor, calculated from (14), closely

500 IEEE SENSORS JOURNAL, VOL. 7, NO. 4, APRIL 2007

Fig. 8. Plot of the recorded relative electrode displacement and the sensor interface circuit output.

Fig. 9. Comparison of the theoretical and calibrated relative velocity sensor output and the calculated time derivative of the measured relative displacement forthe 1.34-KHz test.

matched the estimated velocity waveform. A sensor calibrationfactor of 7.33 was applied to the actual sensor output signal toobtain the relative velocity output. The differences between thetheoretical and actual sensor output signals were due to differentvalues for the sensor equation constants than modeled in (14)and fringing effects, which were not modeled in (14). Utilizingthe THD model from Fig. 3, the measured sensor output signaldata had a THD of approximately 0.035. Observe that althoughthe measured sensor output signal was somewhat distorted, itwas relatively clean at the crossover points where the relativevelocity changed directions.

VI. CONCLUSION

Many micromachined devices utilize parallel plate electrodesthat experience normal translational motion. Relative displace-ment can be measured by a number of commonly used tech-niques. Obtaining a relative velocity measurement or the direc-tion of relative velocity between the two electrodes would beuseful in many applications. When the ratio of the rest gap dis-tance to the amplitude of relative motion is large, a closeapproximation to the relative velocity can be obtained by mea-suring the current flowing into the capacitance between the twoparallel electrodes when a dc voltage is applied across them.For ratios greater than 1.5, the THD of the sensor outputsignal can be estimated as the ratio . The polarity of thesensor output signal yields the direction of relative velocity, withno decrease in precision as the ratio is varied.

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Robert Neal Dean, Jr. (M’90) received the B.E.E.,M.S., and Ph.D. degrees in electrical engineeringfrom Auburn University, Auburn, AL, in 1988, 1991,and 2006, respectively.

He is currently an Assistant Research Professorin the Department of Electrical and Computer En-gineering, Auburn University. His research interestsinclude microelectromechanical systems (MEMS)devices, fabrication, packaging and testing, andmicrosystems interface electronics.

George T. Flowers received the Ph.D. degree inmechanical engineering from the Georgia Instituteof Technology, Atlanta, in 1988.

He is currently a Professor in the Departmentof Mechanical Engineering, Auburn University,Auburn, AL. His research interests include dynamicmodeling and vibration analysis/testing and controlfor mechanical and electromechanical systems.

Roland Horvath received the Ph.D. degree inmechanical engineering from Auburn University,Auburn, AL, in 2006.

He is currently a Senior Engineer at StructuralIntegrity Associates, Inc., Centennial, CO. Hisresearch interests include vibration analysis androtor-dynamics, including analytical, computational,and experimental methods.

Nicole Sanders received the B.E.E. degree fromAuburn University, Auburn, AL, in 2002.

She currently works for Auburn University, per-forming microfabrication and device packaging. Herresearch interests include the fabrication and pack-aging of MEMS devices.

A. Scottedward Hodel (SM’03) received thePh.D. degree from the University of Illinois atUrbana-Champaign in 1989.

Currently, he is an Associate Professor in theDepartment of Electrical and Computer Engineering,Auburn University, Auburn, AL. His research inter-ests include control systems, computer-aided controlsystem design, numerical linear algebra, and signalprocessing.

John Y. Hung (S’79–M’80–SM’93) received theB.S. degree from the University of Tennessee,Knoxville, in 1979, the M.S.E. degree fromPrinceton University, Princeton, NJ, in 1981, andthe Ph.D. degree from the University of Illinois,Urbana-Champaign, in 1989, all in electrical engi-neering.

He is a Professor in the Department of Electricaland Computer Engineering, Auburn University,Auburn, AL.

Prof. Hung served as an Associate Editor of theIEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY in 1997 and 1998,and the IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS from 1996 to 2005.He is the Technical Program Co-Chair for the 2007 IEEE International Sympo-sium on Industrial Electronics (ISIE-2007), Vigo, Spain, and will be the GeneralChair for the 2008 Annual Conference of the IEEE Industrial Electronics So-ciety (IECON-2008), Orlando, FL. He also serves as the Treasurer of the IEEEIndustrial Electronics Society. He has received several awards for his teachingand research, including a Best Paper Award from the IEEE TRANSACTIONS ON

INDUSTRIAL ELECTRONICS.

Thaddeus Roppel (M’86) received the Ph.D. degreein electrical engineering from Michigan State Univer-sity, East Lansing, in 1986.

He is an Associate Professor in the Departmentof Electrical and Computer Engineering, AuburnUniversity, Auburn, AL. He is also the Director ofthe Sensor Fusion Laboratory at Auburn University.His research interests include collaborative robotics,gesture recognition, video-based motion analysis,and wireless communications.