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We present a novel way of using a double-sided comb-drive resonator, by disconnecting the rotor from any dc bias, and leaving it electrostatically floating. To demonstrate this, we first characterize the response of the resonator when the rotor is subjected to a dc bias, and demonstrate that as is widely known, increasing the dc bias on the rotor increases the resonator response to a small ac input signal. Then we demonstrate that even when the rotor is disconnected from any dc bias voltage, and it is left electrostatically floating, the resonator still responds to the input signal, and this response is considerably large. The explanation for this surprising phenomenon is that by disconnecting the rotor and keeping it electrostatically floating, we turn the common resonator into a parametric resonator. When the system is thus configured, small ac signals that drive the system are sufficient to produce a large amplitude of harmonic response, without requiring any dc bias. To validate the parametric response, we demonstrate that the system can be driven to resonance, by driving it at unit-fractions of the fundamental frequency. The relevance of this work is that it demonstrates that electrostatic resonators can be operated without requiring any dc bias. This may be beneficial for low-power sensors necessary for IoT technology. [2020-0066]

Vibrational wineglass modes in disk and rings that are made from (100) silicon are considered. It is shown that all odd-ordered wineglass modes are necessarily frequency matched, each with its orthogonal conjugate. In contrast, all even-ordered wineglass modes are likely not to be frequency matched with their orthogonal conjugate. Some specific cases are presented that show that many wineglass modes do not have a rotational periodic symmetry. This is relevant to disks and rings that are driven and sensed by surrounding gap-closing electrostatic actuators.

We present a novel piezoelectric bulk-unimorph beam actuator that can be directly driven in twisting. The bulk-unimorph beam is driven by interdigitated electrodes (IDEs), deposited over its top and bottom surfaces. These IDEs are oriented at 45° relative to the beam axis, and they are used for both poling and actuation. When subjected to a driving voltage, the IDEs induce both axial and shear stresses in the vicinity of the surface over which they are deposited. These stresses give rise to both internal bending and internal torsion moments. When one set of electrodes, either on the top or the bottom surface, is used to drive the lead zirconate titanate beam, a coupled bending/twisting response is induced. However, when both the top and bottom electrodes are used, we can achieve either a pure torsion mode or a pure bending mode, with no cross coupling, depending on the driving scheme. [2016-0313]

We present a thermoelastic micromotor for driving a rigid plate in an out-of-plane motion. The plate is suspended on spiral thermoelastic bimorph arms, made of a nitride layer coated by gold. Heating of the thermoelastic arms is achieved by driving electric current, either through polysilicon resistors that are embedded in the nitride layer, or through the gold layer itself. Heating generates isotropic internal bending moments in the bimorphs. We show that this bending produces only small deflections of the plate. To improve the performance of the actuator, we use selective stiffeners to convert bending into torsional deformation of the spiral arms. In one orientation of the stiffeners, this conversion may be used to achieve a seven-fold increase of the thermoelastic response of the actuator. In a different orientation of the stiffeners, a complete reversal of the enhanced thermoelastic response is achieved. [2016-0137]

We present a mechanism that converts in-plane to out-of-plane motions, which is fully compatible with mass-fabrication processes. By using selective stiffening, we induce coupling between in-plane and out-of-plane responses of the mechanism. The conversion ratio is constant (i.e., linear response), and it can be easily tailored by the number of stiffeners used in an otherwise unchanged design planform. The linearity of the motion conversion and the possibility to tailor it, are demonstrated experimentally using dedicated test devices. As a specific example of application, we use the motion conversion mechanism to achieve a parallel out-of-plane motion of a flat stage, which is driven by in-plane comb-drives. [2015-0154]

We present an electrostatic transducer in which voltage remains constant while charge is stored or extracted from the device. We achieve this by designing a nonlinear mechanical spring, which exactly counteracts the nonlinearity associated with electrostatic attraction forces. In essence, the new device responds like a rechargeable battery, only that here we store electric energy in elastic deformation, whereas in common batteries, electric energy is stored as chemical potential. The capacitance of the presented test devices is currently too small to be practical, and we are not, by any means, suggesting that this transducer can replace chemical rechargeable batteries. Nevertheless, some necessary steps toward constructing a more viable device with larger energy density are discussed. [2014-0093]

We present a piezoelectric twisting actuator for tilting micromirror applications. The actuator is made of two in­line bulk piezoelectric beams, which are directly driven in a pure twisting mode. This is in contrast to existing piezoelectric devices, in which bending must be converted to tilting motion. Proof-of-concept devices were fabricated from single layer of bulk PZT, which was machined using a cheap and simple sand-blasting process. Interdigitated electrodes, oriented at 45° to the beam axis, were deposited on the top and bottom surfaces of the beams, and they are used for both poling and driving. The quasi-static and resonance responses are experimentally measured.

We present a bulk-unimorph PZT actuator that can achieve large piston motions, and small angle adjustments. The actuator is constructed from four spiral arms supporting a central stage, and each arm can be actuated separately. The spiral arms are driven by both internal bending moments and torsion moments. We experimentally show that the torsional deformation in the arms dominates the response, and that the internal bending moments have a negligible effect. The presented device achieves a piston motion of up to 75μm, and small tilting angles around two perpendicular in-plane axes.

We present a novel motion conversion mechanism which is fully compatible with standard mass-fabrication micromachining. The new mechanism is designed to linearly convert in-plane motion induced by electrostatic comb-drive actuators, to out-of- plane motion. The conversion mechanism is constructed from pairs of beams with two different heights. Motion conversion is achieved by fastening these beams together using a small number of rigid connectors. The rigid connectors stiffen the structure in-plane response, but induce an out-of-plane motion. We fabricated new test devices and demonstrated parallel out-of-plane motion of a flat stage.

We present a gap-closing electrostatic actuator with a linear extended range. We achieve this by designing a nonlinear spring that exactly counteracts the nonlinear effects of electrostatic attraction forces in gap-closing actuators. We demonstrate this on parallel-plates actuators with a nominal gap of g=21 μm. The initial response up to a deflection of g/4 is stable and is inevitably nonlinear, but beyond this point we demonstrate good linearity up to a displacement of 85% of the nominal gap. By introducing the nonlinear spring, we also avoid the pull-in instability, which would have occurred at a voltage of 31 V. Instead, the system is stable and the driving voltage may be increased up to 130 V where the maximal displacement is achieved.

We present for the first time ever, a piezoelectric beam actuator that can be directly driven in pure torsion. The beam actuator is designed with interdigitated electrodes (IDEs), on both its top and bottom surfaces. The IDEs are used for both poling and driving. When the IDEs are driven by a voltage with the same polarity as used for poling, they induce a combination of shear and expansion strains, close to the top and bottom surfaces. The expansion strains at the top and bottom surfaces are similar, and hence they produce no net bending. However, the shear strains at the top and bottom surfaces are in opposite directions, and they produce a pure twist. If the IDEs on the top and bottom surfaces are driven by the same voltage but in opposite polarity, then a pure bending response is produced. We experimentally demonstrate that the actuator can be driven in either pure torsion or in pure bending, with negligible cross-coupling.

The electromechanical response of a symmetric electret parallel-plates actuator is analyzed. The actuator is constructed from a dielectric plate that is suspended by a linear spring between two electrodes of a planar capacitor. The dielectric plate is loaded with fixed charge, and is displaced due to the interaction between this charge and the electrostatic field. The electrostatic field that displaces the charged dielectric has three components: a field induced by the voltage difference across the capacitor plates, a field induced by the anti-symmetric part of charge, and a field induced by the combined effect of the symmetric part of charge and the displacement of the plate itself. The analysis shows that the displacement of the electret actuator is a linear function of the driving voltage, and that a full range of stable motion can be achieved. Furthermore, it is shown that all important characteristics of the fixed charge can be deduced from simple measurements.

The undamped dynamic response of electrostatic MEMS switches that are driven by a step-function voltage is investigated. A systematic analysis using energy methods is presented. An analytic expression for switching time is derived and this expression may be used as a design rule for electrostatic switches. The analytic predictions are validated experimentally using test structures with parallel-plates actuators. It is shown that the analysis is also applicable for switches with more general geometry.

We present a two axis actuator which can be controlled both in-plane, and out-of-plane. Previously we demonstrated a conversion mechanism in which in-plane motion was translated to out-of-plane motion. That mechanism harnesses comb-drive actuation and converts the in-plane motion of the comb-drive shuttle to a parallel out-of-plane motion of a rigid stage. In the present work we show a two axis control of the actuator, one axis inplane and one out-of-plane. We also show a new test device that converts in-plane motion to a tilting motion, in which both axes may be separately controlled. Presented experimental data is in good agreement with model predictions. Specifically, we demonstrate a linear relation between in-plane electrostatic force, and the tilt angle.

We present for the first time ever, a method for reversing the response of thermoelastic bimorph actuators, without changing the order of their layers. Our device is constructed from a flat stage that is suspended on three thermoelastic bimorph spiral arms. We demonstrate that the thermoelastic response of the spiral bimorphs can be affected by using directional stiffeners. We show that with no stiffeners, the thermoelastic response naturally displaces the stage downwards by 7µm. Stiffeners that are angled at 45° relative to the spiral tangent, enhance this response by a factor of 3, and increase the downward displacement to 23µm. Surprisingly, stiffeners which are angled at −45°, completely reverse the direction of the response, and result in an enhanced upward displacement of 23µm.

We show that nonlinear elastic springs can be used to counteract the nonlinearity of electrostatic forces in gap-closing electrostatic actuators. We demonstrate this in two types of devices. In the first, we use a nonlinear spring to extend the stable range of the parallel-plates actuator, and to ensure that the response in this extended range is linear by design. In the second device, we use a nonlinear spring to ensure that beyond what would have been the pull-in point, voltage remains constant and independent of charge. In effect, this second device is a rechargeable mechanical battery.

This study considers the dynamic response of electrostatic actuators with multiple degrees of freedom that are driven by multiple voltage sources. The critical values of the applied voltages beyond which the dynamic response becomes unstable are investigated. A methodology for extracting a lower bound for this dynamic pull-in voltage is proposed. This lower bound is based on the stable and unstable static response of the system, and can be rapidly extracted because it does not require time integration of momentum equations. As example problems, the dynamic pull-in of two prevalent electrostatic actuators is analyzed.

This work considers microresonators that are driven by a temperature-gradient actuator. The driving forces in the temperature-gradient actuation scheme are internal mechanical moments induced by temperature gradients. In this work, a model for analyzing the dynamic thermomechanical response of temperature-gradient actuators is presented. In this model, the two-dimensional nature of the temperature field is fully considered. Several test structures were fabricated and their dynamic thermomechanical response was found to be in agreement with the model predictions. Some examples of parametric analysis are presented that show how the two-dimensional model can be used to optimize the system parameters under specific constraints of the operation conditions.

This work presents a systematic analysis of electrostatic actuators driven by multiple uncoupled voltage sources. The use of multiple uncoupled voltage sources has the potential of enriching the electromechanical response of electrostatically actuated deformable elements. This in turn may enable novel MEMS devices with improved and even new capabilities. It is therefore important to develop methods for analyzing this class of actuators. Pull-in is an inherent instability phenomenon that emanates from the nonlinear nature of the electromechanical coupling in electrostatic actuators. The character of pull-in in actuators with multiple uncoupled voltage sources is studied, and new insights regarding pull-in are presented. An analytical method for extracting the pull-in hyper-surface by directly solving the voltage-free K-N pull-in equations derived here, is proposed. Solving simple but interesting example problems illustrate these new insights. In addition, a novel /spl alpha/-lines numerical method for extracting the pull-in hyper-surface of general electrostatic actuators is presented and illustrated. This /spl alpha/-lines method is motivated by new features of pull-in, that are exhibited only in electrostatic actuators with multiple uncoupled voltage sources. This numerical method permits the analysis of electrostatic actuators that could not have been analyzed by using current methods.

We present a mechanism that converts in-plane to out-of-plane motion, which is fully compatible with standard mass-fabrication methods. The mechanism harnesses the well-established in-plane actuation achieved by comb-drives, and converts it to out-of-plane motion. The motion conversion ratio is constant (i.e. linear conversion), and it can be easily tuned by adding or subtracting modular elements, in an otherwise unchanged design planform. We experimentally demonstrate the linearity of the mechanism, and use dedicated test devices to show the tunability of the conversion ratio. With a different test device, we demonstrate parallel out-of-plane motion of a flat stage. The measurements of this device show good agreement with model predictions.

Pull-in parameters are important properties of electrostatic actuators. Efficient and accurate analysis tools that can capture these parameters for different design geometries, are therefore essential. Current simulation tools approach the pull-in state by iteratively adjusting the voltage applied across the actuator electrodes. The convergence rate of this scheme gradually deteriorates as the pull-in state is approached. Moreover, the convergence is inconsistent and requires many mesh and accuracy refinements to assure reliable predictions. As a result, the design procedure of electrostatically actuated MEMS devices can be time-consuming. In this paper a novel Displacement Iteration Pull-In Extraction (DIPIE) scheme is presented. The DIPIE scheme is shown to converge consistently and far more rapidly than the Voltage Iterations (VI) scheme (>100 times faster!). The DIPIE scheme requires separate mechanical and electrostatic field solvers. Therefore, it can be easily implemented in existing MOEMS CAD packages. Moreover, using the DIPIE scheme, the pull-in parameters extraction can be performed in a fully automated mode, and no user input for search bounds is required.

We present for the first time ever, a perfect electrostatic anti-spring. This electrostatic actuator is a symmetric double-sided comb-drive with uniform-width fingers and a linear tapering of the lengths of the rotor fingers. When both stators are grounded and a constant dc bias is applied to the rotor, this actuator induces a diverging electrostatic force which is a perfectly linear function of displacement. This force is exactly the opposite of the restoring force in a linear spring, and hence we refer to it as a linear anti-spring. The new design is superior in terms of design robustness and sensitivity to fabrication imperfections, relative to comb-drives with shaped fingers which have been used for similar purposes. We demonstrate the performance of the new anti-spring, by using it to continuously and reversibly down-tune the natural frequency of a linear mechanical system. Experimental data exhibit excellent fit between predictions and measurements.

In this letter, a novel adjustable micromirror device using multiple electrodes is presented. The novel design allows post fabrication continuous and reversible adjustment of the electromechanical response and pull-in instability. This property is demonstrated on a fabricated device and is used for analog adjustment of the stable angular travel range of the micromirror. In addition, the inherent properties of the device enable simple duty cycle modulation of the dynamic response that can be used for gray-scale control. These properties are in contrast to prevalent torsion actuators in which the angular travel range and pull-in instability are set and cannot be changed once the device is fabricated.

Pull-in instability is a crucial effect in electrostatically-actuated microelectromechanical devices. This phenomenon was recently shown to occur under both voltage and charge control. So far, for all known cases, it has been shown that, under charge excitation, electrostatic actuators have a larger range of stability than under voltage excitation. No counter-examples are known, and whether this is a general rule for electrostatic actuators is an open question. In this work, a general proof is presented that shows this is indeed a fundamental rule for general electrostatic actuators.

This paper considers the electromechanical response of electrostatic actuators that are driven by both voltage and charge. The model system is an electrostatic actuator in which the suspended electrode is subjected to a driving voltage and the fixed electrode, which is electrostatically floating, is loaded by charge. The response of the system is analyzed using energy methods, and it is shown that the system has two distinct pull-in voltages. It is also shown that the amplitude of charge on the floating electrode is proportional to the average of these two pull-in voltages. Test-actuators were designed, fabricated, and characterized, and their measured response validates the theoretical predictions. A nondisruptive measurement of charge is proposed and demonstrated which enables to monitor charge decay over time.

Introduction Electrostatic actuation is seemingly the most prevalent method of applying forces to deformable elements in microsystems. Two factors make electrostatic actuation appealing in systems with micron-scale dimensions. First, it is associated with the physics of the system, when practical voltages (e.g. –100 V) are applied across micron-scale gaps; the resulting electrostatic fields are sufficiently high to deform elements which, in turn, are sufficiently flexible due to their micron-scale thickness. Second, electrostatic actuation is perfectly compatible with microfabrication technology used for fabricating ICs. The same fabrication processes may be used to produce isolated conducting elements that are supported by flexible suspensions. Electrostatic actuation was introduced over four decades ago [1, 2], and has found many applications in actuation and sensing [3, 4]. Electrostatic forces are inversely proportional to the square of the distance between the electrodes and are, therefore, inherently non-linear [5, 6]. Because of this non-linearity, the electromechanical response of electrostatic actuators may become unstable. This instability, known as the pull-in phenomenon, is an unwarranted effect in applications where a large controllable dynamic range is wanted. In such cases, special designs (e.g. comb-drives [7]) or special operation modes (e.g. charge actuation [8]) may alleviate the difficulty. In other applications, the pull-in instability may be utilised to achieve a fast transition between two states of an electromechanical switch [9]. Electromechanical switches have many uses in optical MEMS [10, 11], RF MEMS [12, 13], nanoelectronics [1, 14] and nanologic [15, 16].

This paper presents a new efficient technique for simulating the dynamic response of electrostatic micro switches. It is demonstrated that the dynamic response of a switch can be well predicted based on only a few static states of the system. Using energy methods the deformation of the system is approximated by a single adaptive mode, such that time integration is performed on a scalar equation of motion. Due to the high computational efficiency of the proposed method, it can be used to motivate the functional form of damping forces and extract damping parameters, by fitting experimentally measured dynamic responses.

In this work, new analytic expressions for the frequency of large vibrations and for the switching time, of electrostatic actuators, are presented. These expressions predict that the frequency and switching time are linearly related to specific measures of the applied voltage, on a logarithmic scale. Test devices were designed, fabricated, and characterized, and measurements are shown to be in very good agreement with predictions. These linear relations are important and relevant as design rules for development of fast and accurate switches.

Electrostatic actuation of micro-devices was introduced four decades ago in the pioneering work of Nathanson et al. [1], and many applications have been found since [2, 3]. Electrostatic actuation is prevalent in MEMS due to its compatibility with microfabrication technology, and its low power consumption. Many studies of electrostatic actuators have been presented in the literature, and many different methods for modeling their static and dynamic response have been proposed (e.g. [4]). A review of the different modeling methods is not within the scope of this chapter. The purpose of this chapter is to present a coherent and systematic approach for modeling the static electromechanical response of electrostatic actuators, and to present some design rules. The examples discussed in this chapter are models of actual systems. These models have simple geometry, linear elastic suspensions, and they do not consider fringing field capacitance. Nevertheless, the analysis of the electromechanical response of these model problems provides useful tools for preliminary design.

In this work systematic approaches for extracting parameters of the static and dynamic response of electrostatic switches are presented. A novel strategy enables accurate and efficient extraction of the static pull-in state of voltage and charge driven actuators. Another strategy enables to extract parameters of the dynamic pull-in while only considering static states of the system.

In prevalent torsion actuators the angular travel range and pull-in instability are set and cannot be changed once the device is fabricated. The current work presents a novel adjustable micromirror device. By using multiple electrodes, the novel design allows continuous and reversible adjustment of the electromechanical response and pull-in instability of a fabricated device. This response adjustment is demonstrated on a torsion micromirror device and it is used for analog adjustment of the stable angular travel range of micromirror. In addition, the inherent properties of the device enable simple duty cycle modulation of the dynamic response that can be used for gray-scale control.

The pull-in state of electrostatic actuators that are driven by a single varying voltage is the equilibrium state for which the applied voltage is maximal. In contrast, electrostatic actuators driven by multiple voltage sources have an infinite number of critical states that constitute the pull-in hyper-surface. This work reports on a novel methodology for experimental characterization of the electromechanical response, and specifically of the pull-in hyper-surface, of electrostatic actuators driven by multiple voltages. This methodology is demonstrated on several fabricated devices.

Gap-closing electrostatic actuators are inherently nonlinear and their dynamic range is often limited by the pull-in instability. To overcome this, we propose a nonlinear spring that counteracts the nonlinear effects of electrostatic attraction. The nonlinear spring is designed to extend the stable range of the actuator and to enforce a linear electromechanical response. We present a method for designing elastic springs with monotonically increasing stiffness. The mechanism we propose is effective shortening of a straight clamed-guided beam flexure, by wrapping it over a cam. We consider two specific cases. The first case assumes the wrapped section of the beam flexure fully conforms to the cam shape. The second case assumes that there is a single contact point between the beam flexure and the cam. To validate the concept we have designed and measured the response of a nonlinear spring with a prescribed force–displacement law. Experimental measurements of a macro-scale spring are in good agreement with the model predictions.

In this work we demonstrate that thermoelastic actuation can be used to drive tilting micromirrors to large angles at high frequencies. We present a Single Crystalline Silicon (SCS) micromirror with 1 mm diameter that achieves a scanning deflection angle of ± 8.5 ° at 9.5 kHz, which is driven by a 1.5 Volt source. The key feature that enables the high frequency of the thermoelastic actuator is that it uses temperature gradient as the driving force rather than using temperature itself as in prevalent thermoelastic actuators.

Pull-In parameters are important properties of electrostatic actuators. Efficient and accurate analysis tool that can capture these parameters for different design geometries, are therefore essential. Current simulation tools approach the Pull-In state by iteratively adjusting the voltage applied across the actuator electrodes. The convergence rate of this scheme gradually deteriorates as the Pull-In state is approached. Moreover, the convergence is inconsistent and requires many mesh and accuracy refinements to assure reliable predictions. As a result, the design procedure of electrostatically actuated MEMS devices can be time-consuming. In this paper a novel Displacement Iteration Pull-In Extraction (DIPIE) scheme is presented. The DIPIE scheme is shown to converge consistently and far more rapidly than the voltage iterations (VI) scheme (>100 times faster!). A relaxation based DIPIE scheme that requires separate mechanical and electrostatic field solvers is suggested. Therefore, it can be easily implemented into existing MOEMS CAD packages. Moreover, using the DIPIE scheme, the Pull-In parameters extraction can be performed in a fully automated mode, and no user input for search bounds is required.