Integrated charge and position sensing for

Integrated charge and position sensing for feedback control of electrostatic MEMS Robert C. Andersona, Balasaheb Kawadeb, Kandiah Ragulanc, D. H. S. Maithripalab, Jordan M. Berg*b, Richard O. Galec, and W. P. Dayawansad a MDLO, Sandia National Laboratories, PO Box 5800-1084, Albuquerque, NM 87185 b Dept. of Mechanical Engineering, Texas Tech University, Lubbock, TX 79409-1021 c Dept. of Electrical Engineering, Texas Tech University, Lubbock, TX 79409-3102 d Dept. of Mathematics & Statistics, Texas Tech University, Lubbock, TX 79409-1042 ABSTRACT Closed-loop control of electrostatic MEMS requires sensing to provide a feedback signal. We present an integrated sensor for charge and position that negligibly affects the open-loop dynamics, does not increase the device footprint, and may be easily fabricated. Numerical finite-element simulation, incorporating a realistic electrostatic field model, and experimental results validate the functionality of the sensor. Simulations show how the sensor may be used in conjunction with nonlinear control to provide full gap operation and improved transient performance. Nonlinear control is often considered too complex for convenient implementation, however the controller presented may be implemented using on-chip, local, integrated circuit components. Keywords: electrostatic MEMS, nonlinear control, charge sensor, position sensor, integrated sensor

1. INTRODUCTION Electrostatic actuation of microelectromechanical systems (MEMS) exploits the attractive coulomb forces that develop between capacitively-coupled conductors differing in voltage. Electrostatically-actuated MEMS are popular because they are simple in structure, flexible in operation, and may be fabricated from standard, well-understood, materials1. Roughly speaking, electrostatic motion control is either gap-controlled—governed by varying the distance between two electrodes—or area-controlled—where it is the overlapping region between electrodes that changes. In the first, which is the focus of this paper, the electrostatic force is highly nonlinear, and open-loop control over a large operating range is correspondingly difficult. Furthermore, it has long been recognized2 that when controlled through the electrode voltage, the nonlinearity gives rise to a saddle-node bifurcation called “snap-through” or “pull-in” that results in severe operational limitations3–9. Analysis of pull-in instabilities have been extended to rigid electrodes of complex geometry by Nemirovsky and Bochobza-Degani5, and to flexible membranes by Pelesko and Triolo7. Further discussion of pull-in may be found in the literature5,7,8,9,10. One approach to the problem of snap-through is to design bi-stable digital devices that exploit the bifurcation. This is a successful strategy, demonstrated in numerous research prototypes and commercial devices11–15. However analog devices with continuously variable positioning enhance functionality in many applications, including optical switching15– 17 ; spatial light modulators for image projection11, data storage, and image recognition18; and reconfigurable diffraction gratings for biosensors19. Open-loop analog operation is possible if the device is restricted to a safe operating region20. However, the inter-electrode gap is then considerably larger than the range of motion, and much higher actuation potentials are required. Another approach is to use electrode charge instead of electrode voltage to adjust position. Several researchers have noted that the bifurcation associated with voltage control disappears when charge is considered as the control input instead of voltage. Furthermore, the high control voltages associated with capacitive stabilization are no longer required. Charge control has been most directly exploited by Nadal-Guardia, et al., who present open- and closed-loop switching circuits for injecting and maintaining the appropriate amount of charge on the device21. This scheme is globally stabilizing but, as discussed below, does not address transient behavior. A version of charge control, implemented by means of a switched capacitor circuit is used by Seeger and Boser to stabilize a double-sided electrostatic MEMS22, and an electrostatic MEMS with both rotational and translational degrees of freedom23. In that *[email protected]; phone: 806-742-3563; fax: 806-742-3540

work a nonlinear feedback of electrode position and velocity is effectively added to the linear charge feedback term. This nonlinear term arises as a by-product of the amplifier design in the switched capacitor circuit that generates the charge feedback and hence does not improve the transient behavior. This observation is supported by the simulations presented in those papers22,23. Further, guaranteed stability properties are only local. Finally, one may consider voltage control using feedback. Chu and Pister proposed closed-loop voltage control based on the electrode gap10. They show by analysis, and verify through simulation, that linear position feedback may be used to locally stabilize any point in the gap. Their implementation relies on local set-point control, and moves the electrode through large translations using a series of small step changes. They do not address the nature of the position measurement. A well-known result due to Seeger and Crary is that every point in the gap may be stabilized by an appropriately sized capacitor placed in series with the electrostatic MEMS8. The analysis was later extended to include discussion of parasitics and the rotational tip-in instability for rigid electrodes3,5, and for membrane electrodes7. From a control perspective, capacitive stabilization implements static charge feedback, and can be shown to semi-globally stabilize every point of the gap24–26. Again, static charge feedback cannot significantly alter transient behavior. Lu and Fedder use capacitance measurements to obtain displacement, and apply a classical linear, time-invariant controller design to approximately double the operational range of a parallel plate capacitor27. Transients are addressed indirectly, through an input-shaping pre-filter. This method extends the operational range of the device, but guaranteed stability properties are only local. Horenstein, Perreault, and Bifano present a integrated capacitive position sensor28. This sensor is similar in some ways to the one presented here, but it does not measure charge, and is therefore not suitable for use in the passivity-based feedback controller, and it requires a secondary AC signal to be superimposed on the electrode drive signal. Other control schemes presented in the literature include an inductor placed in series with the electrostatic MEMS29. This approach applies a sinusoidally-varying open-loop control voltage, and requires large time-scale separation between the mechanical and electrical subsystems. Its main advantage is a low control voltage requirement. The above-cited papers are concerned primarily with stabilization. Transients are fixed by the physical damping of the mechanical subsystem, which for MEMS can vary widely. This can be seen, for example, by comparing the highly damped TI DMD mirrors12 to the lightly damped structures reported by other researchers14,30. Digital device lifetimes are often limited by the incremental surface damage done at each contact14, and therefore analog devices promise improved reliability and longer lifetimes, and reduced packaging costs due to the elimination of anti-stiction treatments. However fully realizing this benefit depends upon the ability to modify device transient behavior, as mechanically under-damped structures will either suffer the undesirable effects of impact, or require remedial measures, such as controlling the cavity environment to produce the desired damping (which increases packaging costs), or pulse shaping to avoid overshoot (which increases switching times and hurts performance). If feedback control is used to augment stability, overshoot can be minimized without reducing performance, at only a small cost in increased compensator complexity. It has been shown that it is not possible to significantly affect mechanical transients using static charge feedback25,26. However, it has been shown that dynamic output feedback may be used to stabilize any point of the gap with good transient performance, as characterized by fast settling times and low overshoot. These passivity-based control laws are surprisingly simple, however they must contain a term incorporating the velocity of the movable electrode25,26. While this can be measured in the laboratory31,32, it is extremely difficult to sense directly during normal operation of the device. Improved transient performance of a MEMS device has been addressed by Wang for a PDE model of a cantilever beam with double-sided actuation33. Wang also applies an energy-based control method, and finds that velocity feedback is needed. That work applies a robust variable-structure approach to avoid a velocity measurement. Sane also applies variable-structure control to an electrostatically-actuated torsional mirror33. This work sensed mirror angle through large moment arm beam deflections, an approach impractical outside of the laboratory. The present analysis uses a dynamic velocity estimator, following earlier results by the authors24–26. It is the purpose of this paper to show that this control law can be practically realized using an integrated sensor, straightforward electrical measurements, and a control law implemented in integrated circuitry. Section 2 explains the operating principles of the sensor, and presents two models that can be used for extracting position and charge measurements. One is based on a simple circuit model, the other uses an electrostatic ANSYS analysis. Section 3 presents experimental results using an electrostatic test structure incorporating the sensor. Section 4 describes how the sensor may be used in conjunction with passivity-based control. Section 5 investigates an integrated circuit realization of the controller, and briefly discusses implementation. Section 6 concludes the paper with a brief discussion of the results.

2. AUXILIARY ELECTRODE SENSOR As shown by the authors24–26, the series capacitor stabilization scheme3,8 implements charge feedback control to extend the operating region of the movable parallel plate capacitor. In that scheme the series capacitor serves as a combined charge sensor and voltage actuator. By decoupling the sensing and actuation functions, controllers with significantly improved performance may be obtained. Furthermore, augmenting the basic sensor with a capacitance model Figure 1: Device schematic with auxiliary sense allows both charge and position signals to be obtained. In earlier work electrode. the size of the capacitor required for control made fabrication difficult3. When only sensing is required, it is possible to integrate the device directly into the drive electrode fabrication, in the form of an auxiliary electrode separated from the drive electrode by an insulating layer. Figure 1 illustrates this configuration. 2.1 Sensor equations The charge on the drive electrode, Qd, and the charge on the auxiliary electrode, Qaux, are related to the electrode voltages by the following equations:

Qd = C dd (x)Vd + C da (Vd " Va ) Qa = C aa (x)Va + C da (Va " Vd ) = 0

(1) (2)

where Vd is the drive electrode voltage, Va is the auxiliary electrode voltage, Cdd(x)is the self-capacitance of the drive ! electrode, Caa(x) is the self-capacitance of the auxiliary electrode, and Cda is the mutual capacitance between the drive ! and auxiliary electrodes. Note all capacitances are given as lumped values for equivalent circuit analysis [CapacRef]. Because the auxiliary electrode is assumed to be “floating,” that is, isolated from any currents, it can attain no net charge. We further assume that since it is fixed with respect to the drive electrode, Cda is independent of x. Denoting the voltage drop from the drive electrode to the auxiliary electrode by Vm = Vd – Va, (2) may be re-written as

C aa (x) =

Vm C da Vd " Vm

(3)

Assuming that the self-capacitance of the auxiliary electrode is an invertible function of x,

!

# Vm & "1 x = C aa %C da ( Vd " Vm ' $

(4)

Then the charge on the drive electrode (which is equal and opposite to the charge on the movable electrode) is obtained by substituting (4) into (1) to yield !

) # V &, "1 m Qd = C dd *C aa C da (-Vd + C da Vm % $ Vd " Vm '. +

(5)

Together (4) and (5) give a transformation from variables Qd and x to variables Vm and Vd. This transformation allows us to re-write control laws formulated using charge and position in terms of the measured electrode voltages instead. This ! should be distinguished from sensors in which an AC signal at frequency much higher than the natural frequency of the movable plate is used to estimate Cdd.(x). In the present case no additional signal is needed; the sensor function arises from the normal operation of the device.

2.2 Parallel plate capacitor model Capacitance models of the drive and sense electrodes are needed to use (4) and (5). Estimates may be derived from parallel plate approximations of the geometry shown in Fig. 1. We assume the areas of the drive electrode and movable electrode immediately below and above the auxiliary electrode, respectively, form the components of two parallel plate capacitors in series. We further assume that this series capacitance is itself in parallel with the parallel plate capacitor formed by the remaining area of the device. Figure 2 is a circuit diagram illustrating this approximation.

Figure 2: Parallel plate capacitor model for analysis.

Each of the three lumped capacitances may be approximated as follows:

! !

" A C˜ da = aux aux # " gap Aaux C˜ aa (x) = x #$ # gap (A $ Aaux ) C˜ "dd (x) = x C˜ C˜ (x) C˜ dd (x) " C˜ #dd (x) + ˜ da aa . C da + C˜ aa (x)

(6) (7) (8) (9)

! As seen from Fig. 2 these correspond to the capacitances appearing in (1) and (2). Inverting (7) we obtain,

!

# gap Aaux "1 C˜ aa (y) = +$ . y

(10)

Now using (10) in (4), then substituting (9), (10), and (6) into (5) and simplifying gives the following desired explicit expression for x and Qd, respectively: !

$V ' & d # 1) + * % Vm ( " (A # Aa )C da Vm Vd Q˜ d = C da Vm + 0 d , " 0 Aa (Vd # Vm ) + C da Vm$ x˜ =

" 0 Aa C da

(11) (12)

! where Cda is assumed known, εr = εgap/εaux, and ε0 is the permittivity of free space.

! 2.3 ANSYS model Finite-element simulation using a more realistic electrostatic field model should give more accurate formulas for Cdd(x) and Caa–1(x) than (9), and (10). An ANSYS simulation was developed for a parallel plate capacitor of dimensions appropriate to the test structure discussed below in Section 3. The dimensions are given in Table 1. Figure 3 shows the outlines of the components. The ANSYS CMATRIX macro was used to compute the capacitances at 22 values of x, ranging from 3 µ m to 0.35 µm, corresponding to the full gap and 12% of full gap. Simulations were not performed below 12% of full gap, due to the difficulty of forming a good mesh. We denote the ANSYS computed capacitances as CAdd, CAaa, and CAda. As expected, CAda was essentially constant at all values of x. Functions approximating Cdd(x) and Caa–1(x) are obtained by linear interpolation on the CAdd-x and x-CAaa curves, respectively. We hereafter refer to

Figure 3. ANSYS device model.

Table 1. ANSYS simulation geometry parameters. Length of movable electrode (µm) Width of movable electrode (µm) Thickness of movable electrode (µm) Length of auxiliary electrode (µm) Width of auxiliary electrode (µm) Thickness of auxiliary electrode (µm) Auxiliary electrode dielectric gap (µm) Dielectric constant of auxiliary electrode gap

these functions as CAdd(x) and CAaa–1(x). These functions may be used in expressions (4) and (5) to yield estimates of position and charge, which we denote xA and QdA, respectively.

200 100 0.3 20 100 0.1 0.11 3.8

The snap-through voltage of the ANSYS model was 23V. This is very close to the design value of the test device described in the next Section, but as discussed below, far from the experimentally-observed behavior, due to the much higher stiffness of the fabricated structure. The electrode stiffness affects only the dynamic behavior. The function of the auxiliary electrode for position and velocity sensing depends only on the electrostatics, and should be modeled reasonably accurately by ANSYS.

Figure 4. Simulated sensor performance: x.

Figure 5. Simulated sensor performance: Qd.

The ANSYS model described above was used to test the use of the auxiliary electrode for position and charge estimation. A fixed Vd of 10V was applied to the drive electrode, and the displacement x(t) of the movable electrode was constrained to follow a sinusoidal path from full gap to one-third of full gap. At each step Vm and Qd were computed by ANSYS. The values of Vd and Vm were used in (11) and (12) to compute x˜ and Q˜ d , and as described above to compute xA and QdA. These are plotted in Figs. 4 and 5 along with the “truth” values x and Qd. From the results it is clear that either capacitance model gives good results, but that the more accurate ANSYS model notably improves performance. !

!

3. EXPERIMENTAL VALIDATION An electrostatically-actuated test structure was designed to investigate the functionality of the auxiliary electrode for charge and displacement sensing. A series of voltages was applied to the drive electrode. At each step the voltage drop between the drive and auxiliary electrode was measured, as was the capacitance of the drive electrode. Charge and position of the movable electrode can be estimated from the capacitance and voltage. Due to instrument limitations, these measurements were made separately and represent equilibrium, rather than transient, values. The displacement of the movable electrode was also measured at two drive voltages using an optical interferometer.

Figure 6. Major steps in test structure fabrication process flow.

3.1 Test structure fabrication Figure 6 shows the major steps in the process flow for surface micromachining of the test structure. In step (a) 650 nm of oxide is grown in a wet thermal process on a 50 mm diameter Si wafer. In step (b) a custom-built, multi-crucible e-beam chamber is used to deposit a 15 nm Ti adhesion layer, followed by 100 nm of Al, 15 nm of Ti for adhesion, and 100 nm of insulating oxide, all without

breaking vacuum. The drive electrode and its contact pad is patterned in this layer using S1813 and a wet Al/Ti etch of DIW:H2O2:HF (20:1:1). In step (c) a layer of Shipley S1813 photoresist is patterned, with open areas defining the auxiliary electrode and contact pad. 15 nm of Ti and 100 nm of oxide are e-beam deposited, a) b) followed by the same Ti/Al/Ti/SiO2 sequence as in step (b). The preliminary extra Ti/ SiO2 layer is needed for two reasons; first because the previous etch may strip insulating oxide from the wafer surface, and second to prevent shorting where the auxiliary electrode structure overlaps the edge of the drive electrode. In step (d) a 3 µm thick sacrificial layer of c) d) Figure 8. Sensor test structure at increasing magnification. Shipley S1827 is spun onto the wafer, and patterned to define the bridge structure. In step (e) the wafer is baked at 120º C for 45 minutes. The resist reflows, and exhibits the sloping profile shown in Fig. 7. In step (f) a 15 nm Ti adhesion layer is deposited, followed by 300 nm of Al to form the main bridge structure. Another 15 nm of Ti is deposited on top of the Al layer to reduce deformation caused by asymmetric stress. All layers in this step are deposited using e-beam evaporation, though sputter would be preferred, to minimize stress. The layer is patterned using S1813 and a DIW:H2O2:HF (20:1:1) etch. Finally, step (g) is a 45-minute O2/CHF3 isotropic plasma etch to remove the sacrificial layer. Further details of the processing are given by Anderson34. Figure 8a–d shows the test structure at progressively increasing magnification, starting with an overview of the entire structure, including the 1 mm square bond pads, and ending with details of the auxiliary electrode. The roughness of the pattern at high magnification is due to the low-resolution photomasks, which were created on a 3600 dpi film printer. The Ti/Al/Ti bridge structure is suspended over the 200 x 100 µm drive electrode and the 20 x 100 µm auxiliary electrode. The design snap-down point of the device was 20V, but, as discussed below, at a drive voltage of 20V the actual device deflected only a small fraction of the total gap. The difference is explained by the wrinkles visible in Fig. 8c and d, which act to stiffen the structure. 3.2 Auxiliary electrode voltage versus drive voltage The experiment shown schematically in Fig. 9 was performed to determine the voltage drop across the auxiliary electrode voltage, Vm as a function of the drive electrode voltage Vd. A voltage from 4 to 28 volts was applied across the driving electrodes using a Keithley 2400 SourceMeter. An Agilent 34401A digital multimeter measured the voltage drop between the drive and auxiliary electrodes. Figure 10 shows the resulting voltage-voltage curve. Since the time-scale of the measurement was long compared to the natural frequency of the device, the values are assumed to represent the equilibrium state. Only measurements in the stable voltage range were made. 3.3 Capacitance versus drive voltage By measuring the steady-state self-capacitance of the drive electrode, Cdd, versus drive voltage, Vd, we can estimate position and charge at equilibrium, as a function of drive voltage. This can then be compared to the position and charge predicted by the Vm and Vd data gathered as described in Section 3.4. In this Section we describe the measurement. Comparison to analytical predictions and other data are given in Section 3.5.

Figure 9 Measurement of auxiliary versus drive voltage.

Figure 10. Measured voltage drop Vm versus drive voltage Vd.

The experiment is shown schematically in Fig. 11. The test setup consisted of a HP 4275A Multi-Frequency LCR Meter that was controlled by a separate computer running LabView. The LabView program allows the operator to vary bias voltage and test frequency and to plot the data real time on screen. The test frequency was fixed at 1 Mhz. The time-scale of the measurement was long compared to the natural frequency of the device, and so the values are assumed to represent the equilibrium state. Only measurements in the stable voltage range were made. Figure 12 shows the results. As expected, the capacitance of the test structures increases with drive voltage. Several tests were run in succession to test the repeatability of the measurement. The first three tests on this device show nearly identical results. The fourth test appears to have been biased upwards, possibly by a change in the test probe configuration, though the capacitance behavior over the voltage range is similar to the other results. The expected capacitance of the test device can be estimated using a parallel plate approximation. For an area of 100 x 200 µm, and an air gap of 3 µm, this gives about 0.06 pF. As seen from Fig. 12, the measured values were on the order of 28 pF. We hypothesize a parasitic capacitance between the 1 mm square drive voltage bond pad and the silicon substrate beneath the insulating oxide layer. A Grove-Deal calculation of the oxide thickness based on wet oxidation for 1 hour at temperature and pressure of 1100 °C and 1 atmosphere gives an oxide thickness of 650 nm. By a parallel plate approximation, the effect of this capacitance would be a constant bias of about 50 pF. For the positive biases applied, accounting for the depth of the depletion layer in the p-doped silicon following reduces this to about 35 pF. Given the relatively good agreement between this estimate and the observed bias, we proceed on the assumption that this is the source of the bias, and in Section 3.5 shift the measured C-V data by an appropriate constant. Possible sources of the 7 pF discrepancy include errors in our estimate of the oxide thickness. Future devices will use smaller bond pads to reduce the effect.

Figure 11. Measurement of capacitance of drive electrode versus drive voltage.

Figure 12. Test structure Cdd-Vd curves

3.4 Optical displacement measurement Displacement of the movable electrode was measured with a Veeco NT1100 Optical Profileometer. Two measurements were taken, one with no voltage applied to the drive electrode, the other at drive voltage of 20V. The bridge structure is not uniform, so the average displacement was computed over the 100 x 100 µm central area of the bridge structure. Figure 13 shows the measurements. The results were 2.41 µm at 0V and 1.92 µm at 20V. These results are discussed further in Section 3.5.

The same device as in the previous Sections was used for the optical displacement measurements, but was damaged after the 20V test. Therefore no further testing was possible with this particular device, and due to differences in behavior between structures, testing was suspended at this point.

a)

b)

Figure 13. NT1100 optical profileometer measurements of electrode displacement.

3.5 Discussion of experimental results The performance of the integrated charge and position sensor is evaluated in two ways. First, the Vm-Vd data described in Section 3.2 are processed using (4) to yield x˜ -Vd and xA-Vd curves, where

# Vm (Vd ) & "1 ˜ x˜ (Vd ) = C˜ aa %C da ( Vd " Vm (Vd ) ' $ !

(13)

# A Vm (Vd ) & A "1 x A (Vd ) = C aa %C da (. Vd " Vm (Vd ) ' $

(14)

and

!

Then C˜ dd -Vd and CAdd-Vd curves are computed from C˜ dd (Vd ) = C˜ dd ( x˜ (Vd )) and CAdd(Vd) = CAdd(xA(Vd)).

!

!

!

Figure 14. Predicted and measured displacement vs Vd. Solid line is the ANSYS-based calculation, dashed line is analytical calculation, triangles are NT100 measurements. Lighter lines without symbols show uncertainty ranges.

Figure 15. Predicted and measured Cdd vs Vd. Solid line is measured data, dashed line is ANSYS calculation, dash-dot is theoretical calculation. Lighter lines without symbols show uncertainty ranges.

Figure 14 shows x˜ and xA plotted against Vd, along with the displacements measured using the Veeco NT1100 as described in Section 3.4. Uncertainties of ±10% in the Vm and Vd measurement are assumed. Because the baseline displacement of the NT1100 measurement may be shifted with respect to the top of the drive electrode, x˜ and xA are biased to agree with the measured 0V point. With the bias applied, the NT1100 measurement of relative displacement should be!accurate to within 0.1 µm. As seen from the Figure, the agreement at the 20V measurement point is good.

!

Figure 15 shows C˜ dd and CAdd plotted versus Vd, along with the Cdd-Vd data obtained as described in Section 3.3. As discussed in that Section, a parasitic capacitance due to the bond pads is assumed to bias the measured results, and the measured data was therefore!shifted to coincide with CAdd at 0V. The agreement is imperfect, but promising. While further improvement is certainly necessary—by reducing parasitic capacitance and improving the capacitance model, for example—the experiments described in this Section are considered Figure 16. 1D electrostatic MEMS model. to provide preliminary validation of the functionality of the integrated charge and position sensor. Subsequent Sections show how the sensor may be used as part of a closed-loop control system, to stabilize the movable electrode and to improve transient performance.

4. CLOSED-LOOP CONTROL SIMULATIONS The sensor presented in this paper is intended to be part of a closed-loop control scheme. The previous Sections presented the principles of operation of the sensor, and validated its use for position and charge sensing. This Section presents two passivity-based, nonlinear output-feedback control designs. The first uses static feedback of Qd, is purely stabilizing, and is equivalent to charge control schemes presented by other researchers3,8,24,25. The second uses dynamic feedback of Qd, and x, incorporates a velocity estimator, and may be used to improve transient performance25,26. These control designs have been discussed elsewhere in considerable detail24–26. Here we are chiefly concerned with how the controllers behave when Vm and Vd are substituted for Qd, and x. 4.1 Nonlinear passivity-based control Figure 16 shows a 1D model of an electrostatically-actuated MEMS. The corresponding dynamic equations are given in (15)–(18), where ω2 = K/M, and 2ζω = C/M.

1 Q˙ d = " (Vd " u(t)) r x˙ = v

(15) (16)

Qd2

v˙ = "2#$v " $ 2 ( x " x 0 ) "

(17) 2% 0 AM ! ! Previously we have shown that by first input-output linearizing the system and then applying output feedback globally asymptotically stabilize any point of the gap24–26. The output of the system is device charge. The input-output linearization can ! be achieved using only the measurement and feedback of the voltage drop Vd across the electrodes. The resulting control law is given by (19)

u = Vd " k(Qd (Vd ,Vm ) " Q d )

It can be shown that this scheme inherently suffers from poor transient performance, especially if the mechanical system damping is very low24–26. This shortcoming may be avoided by employing a passivity-based control design. Nominally, such a controller!requires the measurement of the velocity of the movable electrode24–26. Since the measurement of the velocity is not typically feasible, a velocity observer may be used based on the measurement of the gap and the voltage drop across the device24–26. The composite dynamic feedback controller is

u = Vd " r

(Q

d

(Vd ,Vm ) + Q

) vˆ " k Q (

2# 0 Aa z˙ = Tz + VLyˆ + V"( yˆ ) + VBu

where,

! !

d

(Vd ,Vm ) " Q d

)

(20) (21)

" % $ ' "1% 0 " xˆ % "C % " yˆ % "Qd (Vd ,Vm )% "1 0 0% $ ' $ ' 0 $ ˆ ' = $ ' $ ' ; yˆ = $ '; C = $ ' ; B = $ 0' ; )( yˆ ) = $ ' 2 # v & #V & # z & # x(Vd ,Vm ) & # 0 1 0& $( Qd (Vd ,Vm ) + + 2 x ' $# 0'& 0 $# '& 2* 0 A (1

More detail on the derivation and significance of these terms may be found in previous work26. The performance of the static output feedback controller (19) and the dynamic output feedback controller (20)–(21) using Vd and Vm as !measurements in place of x and Q are shown in Fig. 17, which plots gap versus time for a setpoint change from full gap d to 1 µm gap.

Figure 17. Gap versus time for static and dynamic output feedback controllers. Solid line is with velocity estimation, dashed line is without.

Figure 18. Computed control voltage for static and dynamic output feedback controllers using P-SPICE circuit simulation.

5. ANALOG CONTROLLER IMPLEMENTATION Off-chip computation of the feedback control voltage can cause implementation problems. Bond pads and wiring can give rise to parasitic effects, as previously observed. These can be greatly reduced by implementing the controller using integrated circuitry local to the electrostatic device. Other deterministic error sources that may be helped by local CMOS implementation are reduced amplifier leakage currents and offset voltages, better transistor matching in the multiplier and divider core, and reduced thermal effects35. 5.1 P-SPICE Simulation results The static and dynamic output feedback controllers, (19) and (20)–(21) respectively, can be implemented using analog electronics consists of amplifiers and a multiplier/divider circuit. Fig. 18 shows the control output from one such circuit, compared to Matlab output. It can be seen from the figure that the circuit closely follows the Matlab control, and hence that the circuit will effectively realize the nonlinear control law. Q_bar

Q_bar

+

U1

R15

k_1Q_bar

OUT

R1

-

R16

R2

R17

R19

0

Amplifier

R18

R4

V_m

V_m

-

R3

Q1

U9

OUT

-u

-

U2

-k_2V_m

OUT

+

0

0

+

+

R11

Inverting Amplifier

0

U5

0

OUT

Adder

R14

Q2 + U3 -

OUT

k_4 V_m

R7

+

R8

U4

R9 R6

OUT

V_d-k_4V_m

R12

0

U6

OUT

Q3

+ U8

0

+

R5

OUT

-

U10

OUT

u

-

Inverter

Q4

+ R10

0 V_d

V_d

0 +

0 Subtractor

R13

U7

0 -

OUT Multiplier and Divider

Figure 19. Circuit implementation of the transformation from Vd and Vm to Qd and x.

A portion of the circuit itself is shown in Fig. 19. This is the component that converts the actual measurements Vd and Vm to the controller inputs Qd and x. Q is the set point and will be given as a voltage input. Non-inverting amplifier U1 provides the multiplying constant k1 = (R1 + R2) / R2. Inverting amplifier U2 provides −k2Vm and !

k2 = R2 / R1. U3 provides k4Vm and k4 = (R5 + R6) / R6. U4 is configured with R7 = R9 and R8 = R10 as a subtractor to provide Vd − k4Vm. Multiplier and divider block produces VmVd / (Vd − k4Vm). Finally, U9 provides the addition of all the quantities with a gain of one (R15 = R16 = R18 = R19) except for the multiplier divider output with a gain of k3 = R17 / R19. Then output of U9 is inverted by U10 to give the required quantity u. The area required for this circuit will be about 500 x 500 µm23. This is about twice the footprint of the 100 x 200 mm device, but still reasonable.

6. CONCLUSION We have presented an integrated sensor for charge and position sensing in electrostatically-actuated MEMS. The sensor is suitable for feedback control of position. Experiments have shown reasonable agreement between the sensor output and independent measurements involving position and charge, although a significant amount of work remains to be done. Matlab and ANSYS simulations predict that use of the sensor for feedback control will result in extension of the operating range through the snap-through bifurcation, and in improvement of transient performance. P-SPICE simulations also show that the controller may be implemented using integrated circuitry in an area no more than twoand-a-half times the device footprint.

ACKNOWLEDGEMENTS This work is partially supported by the National Science Foundation under grant ECS-0218245. The authors would like to thank Mr. Peter van Kessel of Texas Instruments DLP Division for helpful discussions. The authors would also like to recognize the efforts of Mr. Michael Xiong on developing fabrication processes.

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