Operational characteristics of microfabricated electric motors

Sensors and Actuators A, 35 (1992) 3 3 -44

33

Operational characteristics of microfabricated electric motors* Lee S Tavrow**, Stephen F Bartt and

Jeffrey H Lang

Microsystems Technology Laboratories, Laboratory Jar Electromagnetic and Electronic Systems, Department and Computer Science, Massachusetts Institute of Technology, Cambridge, MA 02139 (USA)

of Electrical Engineering

(Received January 2, 1992, accepted March 31, 1992)

Abstract We report on the operational characteristics of LOCOS-based mtcrofabncated radial-gap electric motors through lifetime tests, transient measurements, modeling, and parameter extraction We have found that the reduction of static friction (suction) in the bearings by the incorporation of a silicon nitride film permits these micrometers to spin in normal air ambients Frictional drag from the bearing, which results from the electric-based side-pull of the rotor, is found to be the dominant rotor-retarding force and to lead to motor wearout after approximately 10 000 rotor revolutions Furthermore, the frictional coefficient of the nitride-on-polysilicon micrometer bearing is determined to be 0 36 + 0 04

Introduction Since the first mtcrofabncated electric motors (micromotors) were demonstrated by Fan, Tai, and Muller in 1988 [1], and shortly after by Mehregany et al [2] and Tavrow et al [3], great improvements in micromotor performance, lifetime, measurement ability, characterization, and understanding have occurred While early efforts centered on the rotor-stator gap of the micromotor in hopes of improving the drive torque [4], micromotor performance has proven to be largely determined by bearing design [5, 6] Static friction (stiction) and dynamic friction (which leads to wear) in the bearing can prevent motor operation, reduce motor speed, and curtail lifetime Unlike the scaling advantages in the rotor-stator gap of the mtcromotor, which lead to high energy densities from electric-field enhancement, the micromotor bearing is adversely affected by both scaling and fabrication ability Conventionalsized bearings are already constructed with tolerances approaching the limits of machining, employ

-Paper presented at the 6th international Conference on Solid . State Sensors and Actuators (Transducers '9l), San Francisco, CA, USA, June 24-28, 1991 **Present address Sun Microsystems Inc, 2550 Garcia Avenue, Mountain View, CA 94043, USA tpresent address Bose Corporation, The Mountain, Framingham, MA 01701, USA

0924-4247/92/$5 00

varied materials, ball-bearings, and lubricants, and encompass many designs In contrast, the micromotor bearings are quite coarse in relation to the rest of the motor, are based on a restricted set of materials, and are mechanically unoptimized Furthermore, geometric scaling to micron scales greatly increases the surface-area/volume ratio of these microstructures, so that inertial and gravitational forces are often second order to surface forces, such as frictional forces In this paper, we thus focus on the micromotor bearing design and characterization A typical mtcromotor in this study is shown in Fig 1 and has the dimensions given in Table 1 It has six stators and four rotor poles and is referred to as a 6 4 motor Three variations of this motor with bearing radii of 5 9, 7 3, and 8 8 gm as shown in Fig 2 were tested The various torques, forces, and dimensions are schematically illustrated in Fig 3

Mccromotor design and analysis Micromotor drive torque and electrically based rotor side-pull, including angular dependence, are modeled here using full three-dimensional (3-D) finite-difference method (FDM) simulations developed for this application [6] While 2-D simulations can provide adequate torque shapes, they

© 1992 -

Elsevier Sequoia All rights reserved

34

Fig I SEM of a typical 6 4 LOCOS micromotor which was used it operation, lifetime, and strobe tests Motor dimensions are listed in Table I The rotor-pole and hub shape were designed to minimize the underetch, and hence the release time, of the rotor

TABLE I SEM-measured dimensions of operational LOCOS motors Parameter

Description

Measured value (pm)

G T R, G., Rs Gy Tc

Gap spacing Rotor-stator thickness Rotor radius Shield-plate spacing Bearing radius Bearing clearance Bushing height

1 65 2 55 50 1 35 59, 73, 8 8 0 35 09 Fig 2 SEMs of the bearings of the three micromotors tested, which have 5 9, 7 3, and 8 8 pin bearing radii

cannot provide accurate estimates of absolute torque magnitudes, as exemplified by Fig 4 in which 2-D and 3-D simulations are shown to differ by 25% due to the large rotor-stator gap axial fringing fields Even though FDMs can be formulated to account for the cylindrical shape of the rotary micromotor, a simpler Cartesian formulation is employed The errors resulting from neglecting the curvature of the rotor are minimal, as verified by special 2-D FEM simulations using the commercial ANSYS finite-element method (FEM) program [71, which are also included in Fig 4 The special-purpose 3-D FDM program has been developed for micromotor torque optimization by

computing torque curves for a variety of motor geometnes and drive conditions This program is also compared to similar 3-D ANSYS FEM simulations (Fig 4) The raggedness of the FEM torque curve in Fig 4 results from the barely adequate mesh density, which is limited by the ANSYS program (extended university version) Figure 5 displays the 3-D simulated effect of varying the gap spacing G on the micromotor drive torque T d The torque improvement from increasing the number of rotor stator pole pairs, which is desirable when gap widths decrease, is displayed elsewhere [6] A 1-D parallel-plate approximation predicts a torque that scales as 11G,



35

Fig 3 A 6 4 motor with accentuated rotor offset to illustrate the subsequent effects on drive and drag torque

I -o w--e--e

P-

-

-

I

Fig 5 The 3-D FDM-stimulated dove torque of a 24 16 motor plotted against the gap spacing (other motor dimensions are given in Table 1) The idealized torque relationship based on a parallel-plate approximation and an empirical best fit far 6 z 2 pin are also displayed for comparison

I

ARSYS2OFEr1CNNIAn AMSYSN7FEII,eiiMbl ANSY84S riiiiin FE n 30 FMt, iWflhi

"or POFlllon(4Fgivla) Fig 4 Simulated drive torque curves of a 12 8 motor with the rotor and stator level The zero-degree rotor position corresponds to rotorstator alignment The ANSYS 2-D FEM simulations, one to Cartestan and one in cylindrical coordinates, agree to within 1% but differ from the 3-D solutions by 25%

plotted for comparison in Fig 5 By curve fitting the simulated drive torque relationship to the gap spacing, we instead conclude that the torque scales as G - ' 9 for G x 2 pin as shown in Fig 5 The accelerated decay of the torque with increasing gap spacing results from a combination of the shield plate (the conductor underneath the rotor) [5, 6] and the axial fringing fields, which emphasizes the need for 3-D simulations From our experience, adequate rincromotor drive torque requires gap spacings below 2 pin

Drag torque from bearing friction

Electrical simulations can also be effective in describing certain field-associated friction mechanisms, such as the rotor side-pull in the variablecapacitance motor The electrically generated rotor side-pull results from the radial instability of the radial-gap rotor [3, 6] Electric-based drag torque is particularly troublesome for micromotor operation, since it scales as the square of the drive voltage, similar to the drive torque Hence, the electric-based factional torque must be minimized by design to create an operational nucromotor Rotor side-pull results from the inherent radial instability of the radial-gap rotor, which causes a mismatched radial attraction due to rotor runout on the bearing [8, 9], as illustrated in Fig 3 for a 6 4 motor Ideally, the rotor is centered, the radial forces from driven oppositely located laterally overlapping rotor-stator pairs cancel, and the smaller angular drive torque then dominates In practice, the non-zero bearing clearance will mismatch the various radial forces, with the result that the vector difference must be mechanically supported by the bearing We model the ensuing frictional drag torque T, as the product of this force times the bearing radius (i e , the lever arm at which it acts) and the frictional coefficient between the rotor material and bearing material Furthermore, the rotor partially rolls around the bearing, which is also modeled here [6]



36

0

4

of

os

Boring Clwnnr tom) Fig 6 The rotor angle at which the drive and frictional retarding torques balance as plotted against the bearing clearance for a frictional coefficient of 0 36, a bearing radius of 7µm, and different rotor-stator gaps (other motor dimensions are given in Table I)

The bearing force is proportional to the rotorstator overlap (with tails at the extrema [6]) and hence is angle dependent The square-law dependence of the radial force necessitates a very small ratio of bearing clearance to gap spacing which, in addition to a small rotor-stator gap, is a prerequisite of nucromotor operation Figure 6 displays the simulated [6] zero-net-torque angle (the angle at which Td -Tt = 0) of a 6 4 micromotor as a function of the bearing clearance for different rotor-stator gaps Large negative zero-net-torque angles will prevent complete rotor rotation, as observed in some motors [10] where the rotor inertia cannot sufficiently align the rotor to the next stator position Such a situation can be avoided by using Fig 6 as a design guide Net-torque maximization By simulating in three dimensions the effects of all micromotor geometry variations on drivetorque production, optimal motor geometries have been determined considering fabrication limits and electrically based frictional drag torque [6] These geometries are summarized in brief as follows First, the bearing clearance should be minimized and the rotor-stator gap should be minimized as long as the zero-net-torque angle remains small, as determined by Fig 6 Secondly, the ratio of stator poles to rotor poles should be 3 2, the ratio of the stator-pole width to stator-pole pitch should be 0 6, and the stator width should equal the rotor-

pole width as explained below Thirdly, the rotor radius should be limited by rotor warpage and the bearing radius should be reduced to fabrication limits Fourthly, the shield-plate distance and bushing height should be maximized Fifthly, the number of active rotor-stator gaps should be maximized until dominated by fringing fields, which occurs when the ratio of the stator width to the gap is approximately five [6] (not currently performed) Ordinarily, in variable-reluctance motors, the rotor-pole width is larger than the stator-pole width to prevent back pull on the rotor during phase commutation By widening the rotor pole, the rotor can coast by the stator pole during the finite time needed to shift the phase currents In the micromotor case, the rotor inertia can be virtually ignored and the switching time is often negligible compared to the mechanical time-constant of the system Consequently, rotor blades in the micromotor are drawn with the same width as stators The stator-pole width/pitch ratio (is (see Fig 3) also affects micromotor drive torque and electrically generated rotor side-pull Figure 7 displays the simulated drive torque as a function of S s For fl, > 0 6, the effect of $ on the motor drive torque is less than 3% For variable-reluctance motors, a small value of /h is required to leave space for the stator windings and consequently the value f, = 0 6 is often selected [ 11] A similar value of /3, for the micromotor was chosen based on this

22

2

is IMF

o

4

12

Fig 7 The 3-D FDM-simulated maximum drive torque of a 24 16 motor plotted as a function of (:, (other motor dimensions are given in Table 1)



37

experience with variable-reluctance motors The minimization of electrically generated rotor sidepull further suggests minimizing fi, in micromotors Note, however, the more severe effect of 0, on the starting dnve torque (i e , the torque on the rotor after phase commutation when exactly aligned to the previous rotor pole), and hence /3, should not be reduced below 0 6 Viscous drag For the micromotors tested, we can approximate the viscous drag B of the spinning rotor by considering a thin disk in a single-sided housing under laminar flow conditions [12, 13] B = afR4 - 2 x 10- ib N m s

(a)

(b)

(1)

2Tb

-3 where q is the viscosity of air (1 83 x 10 kg/m s) and the rotor is assumed to slide on its bushings (see Table 1) The viscous drag from the open micromotor top surface and from the rotor edge are negligible in comparison to the lower surface Furthermore, because of the rotor saliency (see Fig 1) eqn (1) is assumed to be an overestimate For a crude estimate of the lower bounds of the viscous drag to account for the rotor saliency, the viscous drag is scaled by the rotor pole width/pitch ratio fl, = 0 4 to give B ,& 08 x 10 -16 N m s

The LOCOS-based micromotor fabrication process Micromotors were fabricated using a six-mask LOCOS-based process described in detail previously [3, 6] A simplified micromotor process flow is illustrated in Fig 8 and final device dimensions are listed in Table 1 Briefly, regions of recessed sacrificial oxide layers are planarized to a 200 nm LPCVD nitride insulating layer using a modified LOCOS process, with the result shown in Fig 8(a) Bushing holes are time etched into the recessed oxide and a thick structural/conducting LPCVD polysilicon layer is deposited and POC13 doped to form the rotor, stator, and interconnect, and to fill the bushing `molds' Patterning and anisotropic etching of large-aspect-ratio rotor-stator gaps is required for sufficient torque production and is greatly simplified by the planar surface produced by the LOCOS process (see Fig 8(b)) A self-aligned bearing is created by depositing and patterning a sacrificial low-temperature un-

(c)

(d) Fig 8 Simplified cross sections of the LOCOS-based micromotor fabrication process

doped LPCVD oxide layer whose sidewall thickness determines the rotor bearing clearance A 130 rim LPCVD nitride layer is then deposited and patterned to undercoat the bearing, which permits the motors to spin in normal air ambients as described below Finally, a 1 pin LPCVD structural polysilicon layer is deposited, doped, and patterned to complete the bearing A cross-sectional SEM of the bearing (released) in Fig 9 reveals the nitride film and 0 35 µm bearing clearance Micromotor wafers are then diced, cleaned, and individually released (i e , the sacrificial oxide layers are removed) by a 6 mm immersion in 49 wt HF followed by a short 3-5 min DI rinse and nitrogen blow dry Released rotors rest on their bushings as shown in Fig 8(d) and can be electncally operated by applying phased voltages to their stators Note that because of the non-zero etch rate of nitride in HF, the bearing clearance is slightly enlarged beyond the oxide sidewall thickness mentioned above

35

Fig 10 Optical micrograph of an operating 6 4 LOCOS micromotor The camera shutter was open during the step transient Because the rotor transient is short (100 list compared to the frame rate (17 ms), the rotor motion blur cannot be seen but only the starting and stopping rotor positions

Fig 9 Cross-sectional SEM of the pin-bearing of a LOCOS micromotor, which clearly shows the nitride solid-lubricant on the bearing inner wall The self-aligned hearing clearance is 0 35 pin

perform the dynamic stroboscopic micromotor measurements, the stroboscope flashes were synchronized to a single drive-circuit phase by means of a selectable delay Spin-down (undriven) rotor transients were also captured, as illustrated in Fig 11(b), by shortening one drive phase to 85 is

Mccromotor testing Operation and lifetime

A typical LOCOS micromotor is shown in Fig 1 and its step transient is shown in Fig 10 Because the rotor transients are too fast for conventional video cameras (see Fig 10), we have used stroboscopic dynamometry [14, 9] to measure the rotor dynamics, from which we can quantify the motor performance The shape of the motor in Fig I (i e , the rotor-stator gap, rotor-pole width, stator-pole width, stator-pole width/pitch ratio, slot depth, and number of rotor poles per phase) was selected for maximum net drive torque based on magnetic motor experience, verified and extended where appropriate through 3-D simulations [6] In contrast, the purpose of the rotor-pole cutouts is primarily to reduce rotor release times Data were collected from 6 4 nucromotors with bearing radii of 5 9, 7 3, and 8 8 pin (see Fig 2) For motors with dimensions given in Table 1, the zero-net-torque angle is approximately -2 degrees as determined from Fig 6, which is consistent with complete motor-stepping behavior, as was venfied An off-chip, high-voltage, three-phase bipolar, square-wave, non-resonant drive circuit was developed to drive the micromotors by generating the waveforms displayed in Fig 11 In order to

LOCOS micromotors have spun for up to 30 min in room air at typical speeds of between 200 and 2000 rpm, corresponding to approximately 10 000 revolutions Furthermore, released motors stored in room air ambients were operated

B

c (b) Fig 11 The drivecircuit wavefonns including the stroboscopic tngger for (a) normal motor operation and (b) the spin-down test

39

over two weeks after release without any apparent ill effect Typical operational LOCOS motors often spun a single turn at voltages in the range 30-40 V before stopping Drive voltages above 50 V were then required to reinstate operation Fine-particle generation in the micrometer beanng may account for this as yet unexplained effect Above 50 V, micromotors would stop at approximately 5 min intervals and required a short puff of air to unstick them and get them to resume operation After spinning for 10-20 min, the motors made more frequent stops and by 25-30 min, the rotors only turned a few times between stops By raising the drive voltage, the length of time that the rotor operated between stops increased, however, the overall motor lifetime could not be extended Strobe tests were thus often performed at voltages well above 50 V to insure that the motors did not stop during the approximately 10 min required for the capture of one averaged transient Unoperated and electrically operated rotors were examined for wear and wear particles Figure 12 displays the bearing and rotor of a motor operated until failure The wear particles on the bearing were found primarily in regions aligned to

Fig 12 SEM of the underside of a LOCOS mu romotor operated until wear failure The wear particles on the beanng are seen primarily in the regions aligned to the stator poles (not shown) where the radial force is maximum No bushing wear or particle generation is visible

the stator poles where the radial force is maximum Furthermore, no wear particles were ever observed on the bushing surfaces, which corroborates the negligible bushing faction value extracted from the transient data below By operating the motors at both high and low speeds, we have concluded that the motor lifetime is dependent on the number of rotor revolutions and not on the length of excitation SEM analysis of motors operated until failure (see Fig 12) further suggests that conventional wear is responsible for LOCOS micromotor failure in air, as opposed to field-assisted hydration, contamination, or native-oxide growth, which depend on motor excitation Other techniques that might lengthen, reinstate, or improve motor performance, such as operation in dry nitrogen [15], sophisticated release processes [15], or dehydration bakes [5], had no measurable effect on LOCOS motor operation Very short DI rinses during the release ( < 2 mm), however, were often found to lead to lower rotorstator gap breakdown strengths Suction The addition of the nitride layer in the bearing (see Fig 8(c) and Fig 9) appears to be critical to micromotor operation without special ambients While a polysilicon-on-nitride contacting surface has less dynamic friction [ 16] and probably less wear than a polysilicon-on-polysilicon contact, the stiction between the two sets of surfaces appears to be dramatically different, which permits polysilicon-on-nitride motors to spin while others do not (unless aided by air levitation) [2, 3] Motors without nitride films reported by Mehregany et al [ 15] operated only in dry nitrogen or immediately after release, and thus they show an ambient/time dependence not observed in polysilicon-on-nitride motors Hydrofluoric-acid (HF) etched silicon surfaces become hydrophilic in a period of minutes due to hydration and hydrocarbon contamination [17] Attraction of such surfaces may cause micromotors with a polysilicon-to-polysilicon rotor-bearing contact to stick Hydration/contamination of nitride-coated polysilicon differs, which might explain why ntnde-on-polysilicon micromotors spin even after a two-week exposure to normal air ambients Bushing friction was found to be a minor component of friction and wear in the LOCOS



40

mtcromotors and hence independent of materials in contact Electric levitation forces inherent to the planar radial-gap motor could account for the lack of bashing wear/friction [6], however, this levitation has not been observed visually Electricbased vertical pull-down rotor forces from rotor charging do not appear to hamper micromotor performance

1E4

1E7

Electric breakdown

Breakdown in the LOCOS motors often occurred at drive voltages at or above 90 V and originated in the rotor-stator gap For a grounded rotor offset by the bearing clearance, the minimum effective gap is 1 3 µm Assuming a maximum field-enhancement factor of 1 6 as calculated by 3-D simulations [6], maximum field strengths of 1 2 x 10 8 V/m are regularly attainable in operating micromotors in accordance with Paschen's curve [ 18], which is well above the breakdown strength of air, 3 x 10 6 V/m, in conventional-sized gaps In contrast, breakdown test structures with fixed gaps withstood much higher fields Applied voltages as large as 340 V across 1 65 lint gaps were applied before breakdown, which correspond to breakdown fields in excess of 3 x 10' V/m When breakdown did occur (usually above 300 V or ± 150 V on a single gap), the cause could not be differentiated between gap arcing and nitride insulation breakdown from the pad to the substrate (the final nitride thickness is approximately 150 nm and is thus supporting its breakdown strength of l0' V/m at 150V) Even considering rotor runout, which is not present in the fixed test structures, the discrepancy between the breakdown voltages of the fixed-electrode test structures and the movable rotor-stator gap is presently inexplicable, but may be due to particle generation from bearing wear (sec Fig 12), hence practical issues may limit the field strengths in an operating micromotor Figure 13 displays the measured gap currents in air in two fixed-electrode test structures The different orientations of the electrodes in the two test structures caused different currents (see Fig 13) as would be expected, because the fieldenhancement factor of the misaligned electrodes is larger than the field-enhancement factor of the aligned electrodes While these gaps do conduct a small amount of current in air (in the range nA to µA for potentials over 200 V), they do not break

ME 9

IE'0 140

160

1N

]aa

220 240 V~(V)

290

79a

IN

Fig 13 Measured gap currents in air to two fixed-electrode test structures, corresponding to full rotor- stator alignment and complete rotor-stator misalignment (the leading edge of the rotor is aligned to the trailing edge of the stator) Both structures have 1 65 pm gaps and other relevant dimensions identical to 6 4 micromotors as given in Table I Best-fit lines are also included, ignoring data below the resolution of the measurement apparatus (nA range)

down because the gap potential is below Paschen's minimum in air, which is 340 V [18] By extrapolating the currents in Fig 13 to 50-100V, the standard potentials applied to the gaps in the micromotors studied here, we conclude that pA currents flow through active rotor-stator gaps during micromotor operation These currents are probably comparable to leakage currents across hydrated surfaces and consequently should not affect present micromotor operation

Micromotor dynamics We have used a low-cost stroboscope to visualize the micromotor rotor dynamics [ 14, 9] Motor parameters were then extracted from rotor dynamics in conjunction with a suitable dynamic model By synchronizing the strobe to a single drive phase, the rotor image was effectively stopped The addition of a selectable delay between the drive phase and the stroboscope allowed us to sweep out an entire transient (see Fig 11) This transient, however, is not a single motor transient, as would be recorded with a high-speed camera, but a statistically averaged transient Consequently, any variations in rotor transients, due to rotor sticking, for example, were manifested as noise in the measurement

41

Measurement procedure and apparatus In order to measure micromotor dynamics, the normal microscope light source on a standard microprobe/microscope station was replaced by a General Radio model 1531-AB Strobotac The microscope objectives focus and direct the fight once the strobscope has been accurately aligned to the microscope At the brightest setting, the Strobotac flash has a rated persistence of 3 µs, which was verified with a photodiode and which is short compared to the 100-150 µs transients observed The rotor position was recorded by a standard CCD camera attached to the microscope, which was connected to a video tape recorder and monitor for later frame-by-frame analysis and data extraction The Strobotac was triggered by the micromotor drive circuit as illustrated in Fig 11 To prevent two stroboscope flashes from overlapping on the same 16 ms video frame, motors were operated at 20 ms per phase (250 rpm) To extract each averaged rotor transient, 10 rotor-angle readings for each strobe trigger delay were measured from individual frames The accuracy of these measurements was approximately ± I degree The rotor motion was clearly stopped on each frame and no motion blurring was apparent Typical step and spin-down rotor transients for LOCOS motors are shown in Figs 14-16 (The error bars represent plus and minus one standard deviation )

Fig 15 The measured step transient of a 6 4 LOCOS motor with a 5 9 pm radius bearing at 75 V (other motor dimensions are given m Table 1) The solid line corresponds to the numerically determined best-fit curve

Fig 16 The measured spin-down transient of a 6 4 LOCOS motor with a 7 3 pin radius bearing at 60 V (other motor dimensions are given in Table 1) The drive phase was terminated after 85 ps The solid Ime corresponds to the numerically determined best-fit curve

Fig 14 The measured step transient of a 6 4 LOCOS motor with an 8 8 pin radius bearing at 75 V (other motor dimensions are given in Table 1) The solid line corresponds to the numerically determined best-fit curve

The dynamic model Unlike the micromotor transients measured by Bart et al [ 14, 9], which had large overshoots and multiple rings, the LOCOS motor transients overshoot little and do not ring Furthermore, the starting and ending angles, which should differ by 30 degrees, are offset appreciably from the full alignment position, also unlike the motors measured by Bart Clearly, the relative frictional component is larger in the LOCOS motors and thus must be more accurately modeled



42

Based on the work of Tai et al [8] and Bart et al [9, 13], we propose the following dynamic model of micromotor behavior, which contains only an angle-dependent factional-retarding force that results from the rotor side pull,

J,B = - BA- r r (O) V 2 sgn(B) +

rd (0) V-

(2)

where 0 is the rotation angle of the rotor and 0 = 0 implies the rotor is aligned to the driven stator, Jr is the rotor moment of inertia, which can be calculated from the rotor geometry to be 1 8 x 10 - '-" kg m 2 for the motor in Fig 1 with the dimensions given in Table 1, B is the coefficient of viscous drag, Y,(0) is the angle-dependent friction per voltage squared resulting from the electrically based rotor side pull, and T d (0) is the drive torque per voltage squared Additional friction terms, as for example the constant friction term due to gravity, were also considered but were found to be negligible [9, 61 Note that the model in eqn (2) is further simplified when considering undrivcn rotor dynamics, as for example for spin-down transients, where the viscous drag alone balances the rotor inertia The frictional drag torque Y r (0) was further decomposed into Yr(0)

= µdFF(0)Rb

(3)

where µ d is the dynamic coefficient of friction between the rotor and the bearing, F is the magnitude of the rotor radial force, t(O) is the normalized angle dependence of the radial force, and Rb is the radius of the bearing that provides the lever arm at which this force acts The triangularshaped F(0) and the normalized torque shape (required to extract the maximum drive torque) were simulated [6] as described above Note that small variations in these shapes produce correspondingly small errors in extracted parameter values The angular dependence of the electrical-based frictional term differs from the model proposed by Bart et al [13, 9] The more dominant friction in the LOCOS motors, however, warrants the added complexity of the model, which in Bart et al 's case [ 13, 9] made little difference In addition, the rotor does not come to rest in perfect alignment with the stator (see Figs 14 and 15), unlike in [9], as would be expected with less friction This effect, which formed the basis of the model of Tai et al [8], is not an artifact of measurement error, and is in fact predicted by eqn (2) Note that the requirement

that the rotor starting and stopping positions differ by 30 degrees, a periodic boundary condition in a rotary 6 4 motor, is not explicitly enforced by eqn (2) but is nonetheless satisfied in all simulations [6] (see Figs 14 and 15), which further validates the model Parameter extraction

By performing parameter extraction on the transient and spin-down data from the strobe measurements of the LOCOS motors with the model proposed above, the drive and drag torque magnitudes were determined We have used the Marquardt gradient-expansion non-linear search routine [ 19] as implemented by Bart et at [ 13, 9], but containing the updated dynamic model of eqn (2) This algorithm minimizes the energy function x2

f

X = n Z 62[t]

[e[t] -O([tl, P)] 2}

( 4)

as a function of the parameters P = {T d , B, lt d } for the n data points, thereby determining the closest fit of these parameters to the model In the residual [0[t] - O([t], P)], 0[t] corresponds to the mean of the angular position of the rotor as measured at discrete intervals, the function ' is the numerically integrated model evaluated at those intervals, and the residual is weighted by the inverse of the standard deviation v squared at each point, which emphasizes more statistically valid data Note that the function sgn(6) in eqn (2) is non-analytic and thus degrades convergence of the gradient search algorithm This function was replaced by the analytic inverse-tangent function with an appropriately sharp transition around zero [13] A total of 10 data sets, comprising 8 step and 2 spin-down transients and consisting of a total of 1790 measured data points (i e , n = 179, since we have averaged 10 frames per rotor position), were used together and in groups to estimate the parameters and to determine the validity of the model Typical transients are given in Figs 14-16, which include the numerically determined best-fit curves based on all of the data sets The extracted parameters are given in Table 2, in addition to comparable values in the literature or computed herein Note that the radial-force value used to compute µ d was corrected by the variation in the computed versus extracted drive torque magnitude

43

TABLE 2 Extracted micromotor parameters and comparable values found in the literature or computed herein Parameter

Yd (N m7 2 ) R(Nms) 4d

Estimate

43x10 -16 12xto 1 A 036

Standard deviation

Comparable value

15x10' 12x10 -^ 004

47x10 -1 °16] 08-2x10 -in 03±009181, 015±01 p6]

achievable Reduction in the radial-force-based friction is also required to decrease micromotor wear and improve the lifetime In contrast, viscous drag consumes less than 15 11/a of the total drive torque at the maximum rotor velocity during a step transient ( ;z: 5000 rad/s) If limited by viscous forces alone, micromotors could operate at speeds exceeding 100 000 rpm (they have been operated above 10 000 rpm for short duration) At such high speeds especially, without

and by the partial rotor rolling behavior [6] The accuracy of ua is limited by the precision of the bearing clearance measurement, not through

improved bearing designs, materials, and/or lubricants, short micrometer lifetimes are unavoidable

parameter extraction, which returns a standard deviation for Pd which is 1/14 of its value in the full parameter estimation The mean-square error X'=0002, which indicates a good fit, as is also verified by studying Figs 14-16 Finally, as seen

Acknowledgements This work was supported in part by the US

in Table 2, the extracted parameters agree well

National Science Foundation under grant ECS8614328 and the Charles Stark Draper Labora-

with computed and otherwise established values

tory, Inc under grant DL-H-315269 The authors wish to thank Martin F Schlecht, Martin A Schmidt, Stephen D

Conclusions This paper has studied the electric simulation,

Senturia, and Mehran

Mehregany for their contributions to this research, in particular, Mehregany for the use of his air levitation apparatus

optimized design, and characterization of LOCOS micromotors 3-D electric-field simulation is an essential component of this work, which, in concert with stroboscopic-based motor transient measurements, has not only been verified but has also allowed us to derive key micrometer characteristics such as bearing friction and windage Finally, through these simulations a set of geometric relationships was derived to aid future micrometer design efforts Without considering windage, only 60% of the micromotor torque is available to perform useful work, 1-((Tf>l
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