Assignment Material for Topic 6 - Engineering and information ...

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Sensors and Actuators A 3189 (2002) 1±6

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O®r Bochobza-Degani*, Eran Socher, Yael Nemirovsky

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Electrical Engineering Department, Kidron Microelectronics Research Center, TechnionÐIsrael Institute of Technology, Haifa 32000, Israel

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Received 11 June 2001; received in revised form 5 November 2001; accepted 7 November 2001

Abstract

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In this paper a quantitative model for the effect of residual charges, located in dielectric coating layers, upon the pull-in parameters of electrostatic actuators is presented. Applying voltages higher than the pull-in voltage across the electrodes of an electrostatic actuator results in the collapse of the movable electrode of the actuator on the ®xed one. In order to avoid short circuit, one or both of the electrodes can be coated by a dielectric isolating layer. Residual charges can accumulate in this dielectric coating and affect the behavior, and more speci®cally the pull-in parameters, of the electrostatic actuator. The model derived in this paper considers a general electrostatic actuator with a general charge distribution in the dielectric coating. The main interesting new results derived from the model are: (i) the pull-in displacement is unaffected by the residual charge and the travel range is only extended due to the series dielectric capacitor, (ii) the pull-in voltage is signi®cantly reduced due to the residual charge, independent of the residual charge polarity and distribution. # 2002 Published by Elsevier Science B.V. Keywords: Pull-in; Residual charge; Electrostatic; Actuator; Model

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On the effect of residual charges on the pull-in parameters of electrostatic actuators

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1. Introduction

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Electrostatic actuators exhibit an inherent instability, known as the pull-in phenomenon [1±5]. A typical electrostatic actuator is constructed from two conducting electrodes, where one is typically ®xed and the other is suspended via a mechanical spring. By applying a voltage difference between the electrodes, an electrostatic force is formed, which tends to reduce the gap between the electrodes. A stable equilibrium is established by the mechanical restoring force of the suspensions. By increasing the voltage difference, new stable equilibrium states are formed, which are characterized by a decreasing gap. At a certain voltage the actuator ceases to be stable and the gap between the electrodes rapidly decreases, until the two electrodes adhere. The voltage and deformation of the actuator at this state are referred to as the pull-in voltage and pull-in deformation, respectively, or shortly as the pull-in parameters of the actuator. In order to avoid short circuit, one of the electrodes (or both) is typically covered by a dielectric isolating layer. Residual charges, which can accumulate in this dielectric layer, affect the actuator performance and its pull-in parameters. The more critical issue is that these residual charges

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Corresponding author. Tel.: 972-4-829-2763; fax: 972-4-832-2185. E-mail address: [email protected] (O. Bochobza-Degani). 1 2

can vary in time. This effect was recently measured by Chan et al. [2]. Thus, for any device that uses pull-in for its operation, the effect of residual charges on these parameters is highly important. Recently, the authors have suggested a generalized model and governing equations for the pull-in parameters of electrostatic actuators [4]. Following the same methodology, the effect of residual charge on the pull-in parameters is now analyzed. The case of a parallel-plate actuator with a charge sheet located at the interface of the dielectric layer and air is ®rst considered. This case is used to illustrate the qualitative results from the model as well as to illustrate the effect quantitatively. The results are then extended to a general electrostatic actuator and an arbitrary distribution of charge in the dielectric.

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2. The parallel-plate actuator

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In the following section the case of a parallel-plate actuator, shown in Fig. 1, is ®rst considered. In order to analyze the effect of residual charges, the energy stored in the actuator is ®rst calculated. The bottom plate of the actuator is assumed to be ®xed and is covered by a dielectric layer. A sheet of charge Qres is assumed at the interface of the dielectric and air.

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0924-4247/02/$ ± see front matter # 2002 Published by Elsevier Science B.V. PII: S 0 9 2 4 - 4 2 4 7 ( 0 1 ) 0 0 8 7 0 - 6

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O. Bochobza-Degani et al. / Sensors and Actuators A 3189 (2002) 1±6

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the actuator are derived s a 8Ka3 x3max Q2res VPI zPI ; 3 27e0 A ed A=xd 2

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(6) 106

Two main conclusions can be derived from Eq. (6):

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where e0, ed and E0, Ed are the dielectric constant and electrical ®elds in the air and dielectric, respectively, x0 is the air gap, xd the dielectric thickness and A the area of the conducting plates. The resulting ®elds are then given by 1 xd Qres V E0 ; e0 x0 =e0 xd =ed ed A 1 x0 Qres V (2) Ed ed x0 =e0 xd =ed e0 A and the electrical energy stored in the actuator is UE 12 e0 Ax0 jE0 j2 12 ed Axd jEd j2

(3)

Substituting Eq. (2) into Eq. (3) and using x0 xmax x, where xmax is the rest gap of the actuator and x the displacement of the upper electrode, one can derive that Ae0 1 xd 1=a z 2 V2 Q UE 2axmax 1 z 2Aed 1 z res

(4)

where a 1 e0 xd =ed xmax and z x=axmax . The total energy stored in the actuator, UT is the sum of the electrical energy UE and the mechanical energy stored in the suspension, UM (i.e. UT UE UM ) [1,4]. Using the co-energy formulation, which is more adequate for voltagecontrol [1,4], the total co-energy, UT , of the actuator, de®ned as UT QV V UT , is given by UT

Ae0 V 2 2axmax 1 z

xd Q2res 1=a z 2Aed 1 z

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(1)

Q2res;cri

Ka2 x2max 2 z 2

8 a3 KAe0 xmax 27 a 12

(5)

where QV is the charge accumulated in the capacitors due to the applied voltage source, V, and the last term in Eq. (5) is the mechanical energy stored in the suspensions, UM, assuming linear suspensions. The pull-in point is a critically stable equilibrium point of the actuator, where the two conditions, @UT =@zjV 0 and @ 2 UT =@z2 jV 0, are simultaneously achieved [4]. Substituting Eq. (5) into both conditions the pull-in parameters of

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Thus, for this critical value of residual charge the actuator collapses even without applying any voltage across the electrodes. This critical value is the value of the pull-in applied charge if considering a charge-controlled actuation, as discussed in [4,8,9].

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Fig. 2 exhibits the forces acting on the movable electrode. The solid lines represent the combination of the mechanical force and electrostatic force applied due to the residual charge, for various residual charges, where b is the ratio between the applied residual charge and the critical residual charge from Eq. (7). The dashed lines are the electrostatic forces due to the applied voltage, for several pull-in voltages calculated from Eq. (6) for the same values of residual charge. The ®gure clearly exhibits that the pull-in displacement of the actuator remains the same for the various residual charges. The above discussion implies that this con®guration of the voltage-controlled actuator has the same pull-in displacement parameter as the charge-controlled con®guration of the same actuator, when applying the charge at the dielectric/air node. Another effect that can be observed from Fig. 2 is the reduction of the total travel range of the actuator. A preliminary displacement is observed even at zero applied voltages for non-zero residual charges. The force formed by the residual charge attracts the movable electrode towards the ®xed one and thus reduces the travel range. This preliminary displacement can be calculated by considering the balance between the electrostatic and mechanical forces at zero applied voltage, i.e. @UE =@zjV0 @UM =@z, which

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e d Ed

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xd Ed x0 E0 V;

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Applying a voltage, V, across the plates and using the Gauss and Kirchoff laws [6] the following relations are derived:

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Fig. 1. A schematic view of a cross-section of a parallel-plate actuator with a charged dielectric coating.

(i) The pull-in displacement is independent of the residual charge:When the dielectric layer vanishes, i.e. xd 0, a 1 and the well-known result of zPI 13 is reconstructed [1,4,5]. The dielectric layer adds a capacitor in series with the actuator. Using a constant capacitor in series with the actuator for extending the travel range of the actuator and pushing the pull-in state beyond the (13)xmax was first suggested by Seeger and Crary [5] and later experimentally shown by Chan and Dutton [7]. A more detailed discussion on this effect for a more general actuator can be found in [4]. (ii) The pull-in voltage is reduced due to the residual charge, independent of the charge polarity:The critical residual charge value, which reduces the pull-in voltage to zero is given by

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(8)

In order to further exhibit the effect of the residual charge on the pull-in voltage, Fig. 3 presents the pull-in voltage shift due to various typical residual charge density and typical dielectrics thickness. This shift can result in a relatively large shift of the pull-in voltage, few tenths of volts, as

3. The general actuator

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In what follows, the conclusions derived for the parallelplate actuator are now generalized for any electrostatic actuator. Fig. 4 exhibits a general electrostatic actuator.

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z2

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clearly observed. Moreover, as the dielectric thickness is increased this shift is even more profound.

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results in the following:

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Fig. 2. Forces acting on the movable electrode, where b is the ratio between the applied and critical charge and a 1 e0 xd =ed xmax .

Fig. 3. The pull-in voltage shift due to residual charges, where VPI(0) is the pull-in voltage for zero residual charge, Nres Qres =A=q and q is the electron charge.

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(9)

where V12, V23 are the voltage across the air gap and voltage across the dielectric coating, respectively. The total electrical energy is given by the sum of the energies stored at each of the capacitors 1 1 1 Cd Cw 2 2 2 V Cd V23 UE CwV12 2 2 2 Cd Cw 1 1 Q2 2 Cd Cw res

where UM(w) is the energy stored in the suspensions. Combining the pull-in conditions, introduced in the preceding section, and eliminating V, the following algebraic equation is derived: 2 2 @ UM @C @UM @ 2 C @C @UM C Cd 0 2 2 2 @w @w @w @w @w @w (12)

The solution of Eq. (12) yields the pull-in displacement of the actuator, wPI. It is clearly observed from Eq. (12) that the residual charge has no in¯uence on the pull-in displacement. On the other hand, the series capacitor Cd does in¯uence the pull-in displacement and can be shown to increase the travel range of the actuator [4,5]. The pull-in voltage can also be derived and is given by " # @UM =@w C 2 Q2res 2 1 (13) VPI 2 @C=@w Cd Cd2 wwPI

Again the pull-in voltage is reduced due to the residual charge, and both of the conclusions derived for the parallelplate actuator are shown to hold also for the general actuator.

(10)

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In order to further generalize this result to a charge sheet with an arbitrary location in the dielectric, only a slight modi®cation should be considered in the above derivation. Fig. 6 exhibits the modi®ed equivalent circuit of the actuator with the charge sheet at an arbitrary location in the dielectrics, where Cd1 is the capacitance between the charge sheet and the air interface and Cd2 the capacitance between the charge sheet and the ®xed electrode.

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It should be noted that, as for the parallel-plate actuator, no mixed Qres and V term appears in the total electrical energy. The total electrical energy is the sum of the contributions of

Fig. 5. Equivalent electrical circuit of the general actuator, where C(w) is the air gap capacitance and Cd the dielectric layer capacitance.

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4. Generally located charge sheet

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Cd 1 V Qres ; Cw Cd Cw Cd Cw 1 V Qres Cw Cd Cw Cd

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(11)

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UM w

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The movable electrode is free to move along a general trajectory, which is denoted by the degree of freedom w. A dielectric isolating layer is assumed to coat the ®xed electrode, having a residual charge at its interface with the air. An applied voltage, V, is assumed between the electrodes. The equivalent electrical circuit of the actuator with the voltage and charge sources is shown in Fig. 5, where C(w) denotes the capacitance function of the air gap and Cd the capacitance of the dielectric layer with respect to the ®xed electrode. Using the superposition theorem the voltage difference across each of the capacitors is calculated

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1 1 Q2 2 Cd Cw res

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Fig. 4. A general electrostatic actuator with charged isolating dielectric layer, where UM is the mechanical energy stored in the suspension, V the applied voltage between the electrodes, Qres the residual charge at the dielectric/air interface and w the generalized degree of freedom of the actuator.

1 Cd Cw 2 V 2 Cd Cw

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each energy source with its equivalent capacitor. This is not generally the case if multiple sources are considered (one may consider the case of an electrostatic actuator with two voltage sources connected in series to observe this fact). The total co-energy of the actuator is then given by

Fig. 6. Electrical equivalent circuit of the general actuator with arbitrary located charge sheet, where C(w) is the air gap capacitance, Cd1 the capacitance of the dielectric layer between the charge sheet and air and Cd2 the capacitance of the dielectric layer between the charge sheet and grounded fixed electrode.

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wwPI

From Eq. (14), it is observed that the residual charge in¯uence is reduced as the capacitance between the charge sheet and the ®xed electrode is increased, i.e. the distance of the charge sheet from the ®xed electrode is reduced. 5. General distribution of charge

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Following the results derived in Sections 3 and 4, the case of a general distribution of charge is now considered. First consider the same general electrostatic actuator with two charge sheets, one located at dielectric/air interface and the second at an arbitrary layer in the dielectric. The equivalent electrical circuit of this con®guration is shown in Fig. 7. Following the arguments in Section 3, the voltage across each capacitor is given by

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CV Q1 Q2 Cd1 V ; Cw CQ1 CQ2 Cw Cd1 CV Q1 Cd2 Q2 Cw V ; Cd1 CQ1 Cd1 Cd2 CQ2 Cw Cd1 CV Q1 Cd1 Q2 V Cd2 CQ1 Cd1 Cd2 CQ2

V23 V34

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where CV CwCd1 Cd2 =CwCd1 CwCd2 Cd1 Cd2 is the equivalent capacitance observed by the voltage source V, CQ1 Cw Cd1 Cd2 =Cd1 Cd2 is the equivalent capacitance observed by the charge Q1 at the dielectric/air interface, CQ2 Cd2 Cd1 Cw=Cd1 Cw is the equivalent capacitance observed by the charge Q2 in the

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Fig. 7. Electrical equivalent circuit of the general actuator with two charge sheets, Q1 and Q2, required for the analysis of any charge distribution.

wwPI

and the pull-in displacement still satis®es Eq. (12), thus it is independent of the residual charges. This result can be easily extended to any charge distribution, observing that each charge sheet is weighted by the capacitance between the charge sheet and the ®xed electrode. Assuming a charge distribution in the dielectric layer of r(y), where y is the location coordinate of the charge sheet, the pull-in voltage is given by " 2 # Z ymax @UM =@w C 2 ry 2 1 dy VPI 2 @C=@w Cd Cd y 0

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wwPI

(18)

where ymax is the dielectric thickness and Cd(y) the capacitance between a charge sheet located at y and the ®xed electrode. Even in this general case, it is clearly observed that the pull-in voltage is reduced by the residual charges in the dielectrics, regardless of their distribution.

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6. Discussion

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The signi®cant effect of residual charges on actuation characteristic was recently measured by Chan et al. [2]. Chan et al. measured the charge build up in an isolating silicon nitride layer due to electrostatically actuated ®xed± ®xed beams, which came into contact with the nitride. Furthermore, they have measured the magnitudes of the VPI's in quick succession, i.e. less than 1 min between measurements, showing progressively lowered VPI's. This indicates that the charge, which accumulates in the nitride with each measurement, reduces the pull-in voltage, as predicted by the above model. Wu et al. [3] have recently designed and characterized a torsion actuator with SiO2 dielectric coated silicon electrodes. They have measured an increase in the travel range and pull-in, which was larger than expected considering only the series dielectric capacitance. They have suggested that the reason for the further extended travel range was the residual charge in the dielectrics. This is in complete contradiction with the theory presented above. Seeger and Crary [5] showed that the depletion capacitance in the silicon has

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Combining again the pull-in conditions, one can derive the pull-in voltage " # @UM =@w C 2 Q1 Q2 2 2 1 VPI 2 (17) @C=@w Cd Cd1 Cd2

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dielectric, and V12, V23, V34, are the voltages across C(w), Cd1 and Cd2, respectively. The total co-energy of the actuator can be then derived 1 1 1 1 2 CV UT CV V Q1 Q2 2 Q2 2 2 CwCd2 CwCd1 1 1 2 Q (16) UM w Cd1 Cd2 2

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This problem is easily solved when considering the transformation CwCd1 =Cw Cd1 ! Cw, i.e. the actuator capacitance is rede®ned as the series capacitance of the dielectric above the charge sheet and the capacitance of the actuator. Since in the above derivation no assumption was made for the actuator capacitance function, one can easily derive the same conclusions again using this simple transformation. However, in order to evaluate the pull-in parameters some modi®cation should be made in Eqs. (12) and (13). Eq. (12) can be used as is, considering that Cd Cd1 Cd2 =Cd1 Cd2 , i.e. the series capacitance of Cd1 and Cd2. The pull-in voltage is also modi®ed and is given by " # 2 2 @U =@w C Q M 2 res 1 2 (14) VPI 2 @C=@w Cd Cd2

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7. Conclusions

Haifa, Israel. He is currently working toward the PhD degree in electrical engineering at the Technion. He investigates motion sensing and actuation mechanisms in microopto-electro-mechanical systems (MOEMS). His research focuses on the coupled energy domain modeling as well as noise modeling of MOEMS including: electrostatic actuation, magnetostatic actuation and optical motion sensing. He is involved in the development of micromachined inertial (acceleration and rate) sensors employing integrated optical sensing. Other fields of interest are: analog readout and control interfaces, silicon optical benches and thermal sensors.

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Residual charges can have large effect on the behavior of electrostatic actuators. A model characterizing the effect of residual charge on the pull-in parameters of electrostatic actuators is presented. The model is derived in the case of a general actuator with a general charge distribution in the dielectric layer. The model exhibits that the pull-in voltage is always reduced due to the existence of residual charges in the dielectric layer, independent of the residual charge polarity. However, the pull-in displacement is unaffected by the residual charge.

Eran Socher was born in Tel Aviv, Israel, on 12 December 1975. He received his BSc degree in electrical engineering, a BA degree in physics (both Summa Cum-Laude, 1996) and an MSc degree in electrical engineering in 1999, all from the TechnionÐIsrael Institute of Technology, Haifa, Israel. He is now pursuing his PhD degree in electrical engineering at the Technion. His research focuses on analysis, optimal design, fabrication and characterization of micromachined integrated electro-thermal devices, especially for uncooled IR sensing applications. Other fields of interest are micromachined inertial sensors, analog interface circuits for microsystems, integrated electrostatic and magnetic actuation for microsystems and fundamental noise phenomena in microsystems.

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Yael Nemirovsky received her BSc in 1966, DSc in 1971 (Technion), Senior Member (1984), IEEE Fellow (1999), IEE Fellow (1999). She joined the Department of Electrical Engineering in 1980. Prior to that she was a research scientist specializing in microelectronics in Rafael, a national R&D organization. She graduated from Technion in chemistry and her DSc thesis was in electrochemistry. For over 20 years she has been active in electro-optical devices in II±VI compound semiconductors and additional advanced semiconductor materials as well as infrared focal plane arrays. She has been involved in growth, processing, device design and modeling of detectors as well as VLSI circuits. She has a well-equipped MOCVD laboratory for growth of heterostructures, extensive facilities for device and interfaces processing and characterization. She has been a principal investigator in large funded research programs that ended in prototype infrared detectors and systems that were transferred to industry. Twice she was the head of the microelectronics research center of the Department of EE at Technion. Currently, her research focuses on micro-opto-electro-mechanical systems (MOEMS), CMOS compatible micromachining and microsystems implemented in CMOS technology and integrated with silicon devices. She has published over 130 papers in the open literature, has filed several patents and a large number of classified reports. She has collaborated with the microelectronics industry as a consultant in sensors and VLSI technology and has been quite active in national and international conferences. She has supervised over 40 graduate students for MSc and DSc. She is an IEEE Fellow, an IEE Fellow and has been the Chairperson of the Israeli Association for Crystal Growth. Currently, she is the Chairperson of the microelectronics and photonics section of URSI. In the past she received awards as a ``Best Teacher'' at Technion, a national award of high esteemÐ``The Award for the Security of Israel'' and a Technion award for ``Novel Applied Research''. She has received The Kidron Foundation award for ``Innovative Applied Research'' (a US$ 100,000 grant for research program). She is a distinguished Lecturer of the electron device society of IEEE.

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References

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[1] S.D. Senturia, Microsystems Design, Kluwer Academic Publishers, Boston, 2000. [2] E.K. Chan, K. Garikipati, R.W. Dutton, Characterization of contact electromechanics through capacitance±voltage measurements and simulations, JMEMS 7 (4) (1999) 208±217. [3] X.T. Wu, R.A. Brown, S. Mathews, K.R. Farmer, Extending the travel range of electrostatic micro-mirrors using insulator coated electrodes, in: Proceedings of the Opt. MEMS'2000, Kauai, Hawaii, August 21±24, 2000, pp. 151±152. [4] Y. Nemirovsky, O. Degani, A methodology and model for the pull-in parameters of electrostatic actuators, JMEMS 10 (4), in press. [5] J.I. Seeger, S.B. Crary, Stabilization of electrostatically actuated mechanical devices, in: Proceedings of the Transducers'97, Chicago, USA, June 16±19, 1997, pp. 1133±1136. [6] R.S. Elliot, Electromagnetics, IEEE Press, New York, 1993. [7] E.K. Chan, R.W. Dutton, Electrostatic micromechanical actuator with extended range of travel, JMEMS 9 (3) (2000) 321±328. [8] L.M. Castaner, S.D. Senturia, Speed-energy optimization of electrostatic actuators based on pull-in, JMEMS 8 (3) (1999) 290±298. [9] L. Castaner, J. Pons, R. Nadal-Guardia, A. Rodriguez, Analysis of the extended operation of electrostatic actuators by current-pulse drive, Sensors and Actuators A 90 (3) (2001) 181±190.

Ofir Bochobza-Degani was born in Ashkelon, Israel, on 17 April 1974. He received his BSc degree in electrical engineering and the BA degree in physics (both Summa Cum-Laude) in 1996 and MSc degree in electrical engineering in 1999, all from the TechnionÐIsrael Institute of Technology,

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even more signi®cance effect on the travel range of the actuator than the series dielectric capacitor. Since Wu et al. actuator consisted of silicon electrodes, this effect should also be considered in the analysis of this actuator when considering the extended travel range.

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