Pull-in Conditions Research Articles

A scaling law approach to rate fabrication tolerances of double-sided electrostatic actuators

Abstract Symmetric double-sided electrostatic actuators in push-pull configuration are particularly suitable for linear actuation with low harmonic distortion. However, their motion still is largely determined by pull-in instabilities that are sensitive to geometry variations. A considerable simulation effort is therefore required when assessing manufacturing tolerances during the design process or determining the optimal operating point. Recently, an accurate method was demonstrated, allowing for the numerically inexpensive and experimentally non-destructive extraction of the full quasi-static performance of a clamped-free beam-like electrostatic micro-mechanical actuator with complex 3D design. The key step was to determine the voltage scaling related to the pull-in voltage based on data collected far away from pull-in conditions. This relates a dimensionless ansatz to the physical input voltages as well as the output like e.g. the actuator’s tip deflection. For the chosen approach, however, the relationship between the model and the geometry parameters is unknown. In this paper we propose a method to enable quantifying the impact of geometry parameter variations. In particular, we adapt the model equation for the case of symmetry-breaking tolerances on the basis of few FEM-simulations. The quasi-static pull-in instability, as well as the nonlinear deflection, are consistently reproduced over the full range of relevant combinations of signal and bias voltages. Our analysis was developed in the context of a specific electro-acoustic transducer. However, we find indications that the underlying method is in fact applicable to a much broader range of micro-mechanical actuators.

Analysis of Electrostatic MEMS Using Energy-Charge Landscape

A common way to analyze electrostatic microelectromechanical systems (MEMS) actuators is to use their energy-displacement landscape. Here, we describe an alternative approach to analyze electrostatic MEMS actuators using their energy-charge landscape. This technique involves coordinate transformation from displacement to charge, thereby formulating the Hamiltonian of electrostatic MEMS actuators in terms of charge. We investigate the use of the energy-charge landscape to analyze static pull-in, dynamic pull-in, and pull-out phenomena. The voltage expressions derived using this method are identical with those derived using the conventional energy-displacement landscape. In addition, we also obtain the expressions for charge under static and dynamic pull-in conditions. This work can aid in the design and analysis of electrostatic MEMS devices. As a case study, the analysis of a feedback capacitor-MEMS actuator system is presented to illustrate the application of the energy-charge landscape.

A Numerical Improvement in Analyzing the Dynamic Characteristics of an Electrostatically Actuated Micro-beam in Fluid Loading with Free Boundary Approach

Electrostatically actuated microbeams have been studied by many researchers in the last few years. The aim of this study is to present an improved numerical analysis of the dynamic instability of a cantilever microbeam immersed in an incompressible viscous fluid. The finite element method is used for solving the vibrational equation of the microbeam and the potential functions of the fluids in the coupled system. The proposed model is more stable for higher fluid densities and changes the area of the pull-in voltage. Finally, it has been shown that involving the free boundary condition affects the dynamic responses and pull-in conditions of the structure in comparison with pertinent literature.

Resonant Pull-in of High-Q MEMS Oscillators with Arbitrary Closed-loop Phase Shift

In this article, we develop a simple closed-form expression of the resonant pull-in instability of high-Q MEMS resonators actuated with arbitrary periodic waveforms and arbitrary closed-loop phase shifts. The instability of MEMS oscillators operated under a low bias voltage is shown to be independent of the dimensions and physical properties of the resonator and fully-determined by the quadrature component of the actuation. The results are illustrated on parallel-plate capacitive MEMS actuated with sine-, square- and pulsed-wave voltages. Input-design methods optimizing the pull-in conditions of sensors and switches can be built based on our conclusions.

Dynamic Response of an Electrostatically Actuated Micro-Beam in an Incompressible Viscous Fluid Cavity

This paper studies the dynamic instability of cantilever micro-beam submerged in an incompressible viscous fluid cavity and actuated by electrostatic force. Equivalent squeeze film damping is incorporated in the vibrational equation of the micro-beam to obtain the natural frequencies of the coupled system. Then, imposing various step voltages, dynamic responses, and pull-in conditions of the micro-beam are studied. A parametric study is conducted to evaluate the effect of fluidic confinement on the instability voltage. Dielectric constant of the fluid proves dominantly influential compared with viscosity and density. In addition, values of pull-in voltage are seen to be highly dependent on the vertical position as well as the length of the micro-beam.

Geometry and charge carrier induced stability in Casimir actuated nanodevices

In this work we demonstrate, that in Casimir actuated nanodevices, geometry and charge carriers concentration change the stability and the pull-in conditions that cause stiction. The stability is analyzed by calculating the bifurcation diagram of the capacitive switch as a function of plate thickness for Au and Si showing that previous calculations based on Lifshitz formula for half-spaces underestimated the stability conditions. Taking into account the size effect, we recalculate the bifurcation diagram for different metals and for Si with different carrier concentrations showing the change in the stability conditions.

STATIC AND DYNAMIC RESPONSE OF CARBON NANOTUBE-BASED NANO-TWEEZERS

In this paper static and dynamic response of nanotweezers composed of two carbon nanotube (CNT) arms are investigated. Taking into account a continuum model and considering the electrostatic actuation as well as the presence of the van der Waals forces, the static nonlinear equations are solved by a step by step linearization and Galerkin projection method. Simulating the closing dynamics of a specific nanotweezer, the effective diameters of its nanotube cantilevers are specified and compared with existing experimental data. Then by imposing a step DC voltage and taking into account the inertia effects, dynamic responses and pull-in conditions of nanotweezers are studied. In the analyses, the effects of various parameters such as initial gap, nanotube length and diameter on the pull-in conditions are investigated. Effects of damping and asymmetric stiffness's of the arms on the pull-in voltages of a nanotweezer are also reported. Comparing the results with the published experimental data shows that the use of continuum model and employing the Galerkin based step by step linearization method (SSLM) could effectively simulate the response of nanotweezers.

Resonant pull-in of a double-sided driven nanotube-based electromechanical resonator

We theoretically investigate the electromechanical dynamics of a double-sided driven cantilevered nanotube-based electromechanical resonator. Closed-form analytical solutions capable of predicting the steady-state resonant oscillation of the device and its resonant pull-in conditions are derived using an energy-based method and are verified through a comparison with numerical simulations. Our closed-form formulas clearly reveal the complex relationship among the device geometry, driving voltages, and the device’s electromechanical dynamics. Our results show that the stable steady-state spanning range of the resonating cantilever can reach up to 90% of the gap between the actuation electrodes, which substantially exceeds the previously reported quasistatic pull-in limit for cantilevered nanotube-based nanoelectromechanical systems and the resonant pull-in limit for double-sided driven microelectromechanical gyroscopes. Our results also reveal that the processes of tuning the resonant frequency of the resonator and controlling its stable steady-state oscillation amplitude can be decoupled and controlled separately by controlling the dc and ac components in the driving signal. The unique behavior of the large stable steady-state resonant oscillation range, which is independent of the electrostatic-force-induced resonant frequency tuning, makes this double-sided driven resonator attractive for many applications, such as tunable sensors for detecting ultratiny mass and force and tunable electronics. The results reported in this paper are useful to the optimal design of novel nanotube- or nanowire-based double-sided driven electromechanical resonators.

Resonant Pull-In Condition in Parallel-Plate Electrostatic Actuators

Electrostatic parallel-plate actuators are a common way of actuating microelectromechanical systems, both statically and dynamically. In the static case, the stable actuation voltages are limited by the static pull-in condition, which indicates that the travel range is approximately limited to 1/3 of the initial actuation gap. Under dynamic actuation conditions, however, the stable voltages are reduced, whereas the travel range can be much extended. This is the case with the dynamic pull-in and the resonant pull-in conditions (RPCs). Using energy analysis, this paper extends the study of pull-in instability to the resonant case and derives the analytical RPC. This condition predicts snapping or pull-in of the structure for a given domain of dc and ac actuation voltages versus quality factor, taking into account the nonlinearities due to large amplitudes of oscillation. Experimental results are presented to validate the analytically derived RPC.