Modeling Of Microelectromechanical Systems Research Articles

Mathematical modeling of an electrostatic MEMS with tilted elastomeric micro-pillars

The aim of this study is to develop a comprehensive mathematical model for an electrostatic MEMS (Micro-Electro-Mechanical Systems) with gap-filling tilted micro-pillars. Elastomeric pillars are used due to their high dielectric constant, resulting in a higher equivalent permittivity of the gap medium. This leads to increased sensitivity and decreased required voltage. Although this system has gained popularity in recent years, a thorough understanding and accurate predictions of its behavior require a detailed mathematical model. Regrettably, the existing literature does not offer adequate information. The present model incorporates three coupled set of nonlinear differential equations that govern the longitudinal and transverse vibrations of the PDMS (Polydimethylsiloxane) pillars, as well as the transverse vibrations of the moving electrode. It demonstrates how the device's behavior is affected by varying tilt angles and numbers of micro-pillars. The viscoelastic properties of the PDMS pillars are also taken into account, following the Kelvin-Voigt model with non-linear strains. Numerical methods are used to solve the governing equations, enabling investigation of static deformation and vibrational responses under different DC (Direct Current) and DC+AC (Alternating Current) actuations. The nonlinear equations proposed in this study provide a valuable tool for accurately predicting the behavior of capacitive MEMS devices that use microstructured gaps with tilted micro-pillar arrays.

Numerical Simulation of Highly Sensitive Ga2O3 Pressure Sensor

This article presents beta‐gallium oxide (β‐Ga2O3) micro electro mechanical systems (MEMS) strain/pressure sensors as a way to enhance sensitivity. The model consists of four piezoresistive strain gauges connected in a Wheatstone bridge configuration. The MEMS model is simulated from 0 Pa to 50 kPa, resulting in an output signal range of −3–16 mV and a responsivity of 0.38 mV kPa−1. The simulation also shows that as temperature increases, the resistance of the piezoresistive material in the MEMS decreases, leading to changes in the output signals. The reliable device effectively utilizes the full Wheatstone bridge configuration to compensate for temperature‐related influences. These early results suggest that Ga2O3‐based MEMS devices have great potential for use in high‐temperature pressure sensor applications in the future.

A stochastic parabolic model of MEMS driven by fractional Brownian motion.

In this paper, we study a stochastic parabolic problem that emerges in the modeling and control of an electrically actuated MEMS (micro-electro-mechanical system) device. The dynamics under consideration are driven by an one dimensional fractional Brownian motion with Hurst index [Formula: see text]. We derive conditions under which the resulting SPDE has a global in time solution, and we provide analytic estimates for certain statistics of interest, such as quenching times and the corresponding quenching probabilities. Our results demonstrate the non-trivial impact of the fractional noise on the dynamics of the system. Given the significance of MEMS devices in biomedical applications, such as drug delivery and diagnostics, our results provide valuable insights into the reliability of these devices in the presence of positively correlated noise.

Extended and Generic Higher-Order Elements for MEMS Modeling

State-dependent resistors, capacitors, and inductors are a common part of many smart engineering solutions, e.g., in MEMS (Micro-Electro-Mechanical Systems) sensors and actuators, Micro/NanoMachines, or biomimetic systems. These memory elements are today modeled as generic and extended memristors (MR), memcapacitors (MC), and meminductors (ML), which are more general versions of classical MR, MC, and ML from the infinite set of the fundamental elements of electrical engineering, known as Higher-Order Elements (HOEs). It turns out that models of many complex phenomena in MEMS cannot be constructed only from classical and state-dependent elements such as R, L, and C, but that other HOEs with generalized behavior should also be used. Thus, in this paper, generic and extended versions of HOEs are introduced, overcoming the existing limitation to MR, MC, and ML elements. The relevant circuit theorems are formulated, which generalize the well-known theorems of classical memory elements, and their application to model complex processes of various physical natures in MEMS is shown.

Positivity-preserving and unconditionally energy stable numerical schemes for MEMS model

In this paper, we propose and analyze the positivity-preserving, unconditionally energy stable, and linear second order fully discrete schemes for Micro-Electromechanical system (MEMS). More precisely, we use the first order backward difference formulation (BDF1) and second order Crank-Nicolson (CN) formulation for the temporal discretization, and the central finite difference method for spatial discretization. A variant of the exponential scalar auxiliary variable (ESAV) approach is involved in our numerical schemes to deal with the singular nonlinear term. The unconditional energy stability of the numerical schemes is rigorously proved, without any restriction for the time step sizes. Furthermore, we derive that the numerical solutions always preserve the positivity property of the MEMS model, that is the distance variable is always between 0 and the steady solution, at a point-wise level. A series of numerical simulations are presented to demonstrate the positivity and energy stability of our numerical schemes.

Radial single point rupture solutions for a general MEMS model

We study the initial value problem $$\begin{aligned} {\left\{ \begin{array}{ll} r^{-(\gamma -1)}\left( r^{\alpha }|u'|^{\beta -1}u'\right) '=\frac{1}{f(u)} &{} \text {for}\ 0<r<r_0,\\ u(r)>0 &{} \text {for}\ 0<r<r_0,\\ u(0)=0, \end{array}\right. } \end{aligned}$$for \(\gamma>\alpha >\beta \ge 1\) and \(f\in C[0,{{\bar{u}}})\cap C^2(0,{{\bar{u}}})\), \(f(0)=0\), \(f(u)>0\) on \((0, {{\bar{u}}})\) and f satisfies certain assumptions which include the standard case of pure power nonlinearities encountered in the study of Micro-Electromechanical Systems (MEMS). We obtain the existence and uniqueness of a solution \(u^*\) to the above problem, the rate at which it approaches the value zero at the origin and the intersection number of points with the corresponding regular solutions \(u(\,\cdot \,,a)\) (with \(u(0,a)=a\)) as \(a\rightarrow 0\). In particular, these results yield the uniqueness of a radial single point rupture solution and other qualitative properties for MEMS models. The bifurcation diagram is also investigated.

Design of a fuzzy PID controller for a MEMS tunable capacitor for noise reduction in a voltage reference source

This study presents a conventional Ziegler-Nichols (ZN) Proportional Integral Derivative (PID) controller, having reviewed the mathematical modeling of the Micro Electro Mechanical Systems (MEMS) Tunable Capacitors (TCs), and also proposes a fuzzy PID controller which demonstrates a better tracking performance in the presence of measurement noise, in comparison with conventional ZN-based PID controllers. Referring to importance and impact of this research, the proposed controller takes advantage of fuzzy control properties such as robustness against noise. TCs are responsible for regulating the reference voltage when integrated into Alternating Current (AC) Voltage Reference Sources (VRS). Capacitance regulation for tunable capacitors in VRS is carried out by modulating the distance of a movable plate. A successful modulation depends on maintaining the stability around the pull-in point. This distance regulation can be achieved by the proposed controller which guarantees the tracking performance of the movable plate in moving towards the pull-in point, and remaining in this critical position. The simulation results of the tracking performance and capacitance tuning are very promising, subjected to measurement noise.Article HighlightsThis article deals with MEMS tunable capacitor dynamics and modeling, considering measurement noise.It designs and applies fuzzy PID control system for regulating MEMS voltage reference output.This paper contributes to robustness increase in pull-in performance of the tunable capacitor.

Analytical modeling to estimate the sensitivity of MEMS technology-based piezoresistive pressure sensor

Design and modeling of microelectromechanical system (MEMS)-based piezoresistive pressure sensor are main requirements to fabricate application-oriented pressure sensor devices for the industry, i.e., nuclear power plants, aerospace and avionics, oil and gas, Internet of Things, wearable electronics and consumer electronics. In this research work, analytical modeling is presented to estimate the overall sensitivity of the MEMS technology-based piezoresistive pressure sensor. The sensitivity of a piezoresistive pressure sensor is estimated using the thin plate theory and the theory of piezoresistivity in silicon. The mechanical responses of a thin plate in terms of deflection and induced stresses are presented and discussed. The effects of geometrical parameters on deflection and induced stresses are analyzed using a ratio of half-edge length with the thickness ( $$a/h$$ ) and a ratio of diaphragm edges ( $$a/b$$ ). These ratio parameters are responsible for the sensitivity of the piezoresistive pressure sensor. Moreover, a comparative assessment is presented for the current model with a model available in the literature. Further, calculations of average stresses are carried out for the piezoresistor geometry of a small rectangular area. Thereafter, a quantitative variation in the calculated sensitivity is presented based on calculation with maximum stress and average stress. The calculated difference in overall sensitivity is found to be 3%. However, a significant reduction in average stresses as compared to maximum induced stresses is obtained as 28% and 36% change for stress X and stress Y, respectively.

Singular Perturbation Analysis of a Regularized MEMS Model

Micro-Electro Mechanical Systems (MEMS) are defined as very small structures that combine electrical and mechanical components on a common substrate. Here, the electrostatic-elastic case is considered, where an elastic membrane is allowed to deflect above a ground plate under the action of an electric potential, whose strength is proportional to a parameter $\lambda$. Such devices are commonly described by a parabolic partial differential equation that contains a singular nonlinear source term. The singularity in that term corresponds to the so-called touchdown phenomenon, where the membrane establishes contact with the ground plate. “Touchdown" is known to imply the non-existence of steady-state solutions and blow-up of solutions in finite time. We study a recently proposed extension of that canonical model, where such singularities are avoided due to the introduction of a regularizing term involving a small “regularization" parameter $\varepsilon$. Methods from dynamical systems and geometric singular perturbation theory, in particular the desingularization technique known as “blow-up", allow for a precise description of steady-state solutions of the regularized model, as well as for a detailed resolution of the resulting bifurcation diagram. The interplay between the two main model parameters $\varepsilon$ and $\lambda$ is emphasized; in particular, the focus is on the singular limit as both parameters tend to zero.

Heterogeneous Dielectric Properties in Models for Microelectromechanical Systems

An idealized electrostatically actuated microelectromechanical system involving an elastic plate with a heterogeneous dielectric material is considered. Starting from the electrostatic and mechanical energies, the governing evolution equations for the electrostatic potential and the plate deflection are derived from the corresponding energy balance. This leads to a free boundary transmission problem due to a jump of the dielectric permittivity across the interface separating the elastic plate and free space. Reduced models retaining the influence of the heterogeneity of the elastic plate under suitable assumptions are obtained when either the elastic's plate thickness or the aspect ratio of the device vanishes.

A constrained model for MEMS with varying dielectric properties

A semilinear parabolic equation with constraint modeling the dynamics of a microelectromechanical system (MEMS) is studied. In contrast to the commonly used MEMS model, the well-known pull-in phenomenon occurring above a critical potential threshold is not accompanied by a breakdown of the model, but is recovered by the saturation of the constraint for pulled-in states. It is shown that a maximal stationary solution exists and that saturation only occurs for large potential values. In addition, the existence, uniqueness, and large time behavior of solutions to the evolution equation are studied.

Erratum to: Transient behavior and dynamic pull-in instability of electrostatically-actuated fluid-conveying microbeams

Fluid-conveying microelectromechanical systems (MEMS) are indispensable components of many sensors/actuators used in many systems ranging from resonators to atomic force microscopy. Due to their low cost, minimal power consumption, and uncomplicated architecture, electrostatically-actuated fluid-conveying microbeams have become a popular choice of MEMS in many industries. In this work, the dynamic behavior and pull-in instability of fluid-conveying micro-bridges and micro-cantilevers are modeled and validated. The microbeams are actuated through application of abrupt electrostatic forces. The equation of motion is derived with respect to a non-uniform profile for the fluid flow and nonlinear electrostatic actuation. The developed model is then implemented into a nonlinear finite-element model. Equation of motion is solved by employing Newmark’s time discretization method. The static pull-in behavior is investigated to validate the model. The static pull-in results are then compared and validated with different results in literature. The dynamic pull-in behavior of the aforementioned systems is examined through a parameter study on the applied voltage, fluid flow velocity, midplane stretching, and axial force. The results of this study are applicable in design and modeling of MEMS in sensing and resonators applications.

Electrical Control of Effective Mass, Damping, and Stiffness of MEMS Devices

We propose the use of electrostatic force feedback to control the effective mass, damping, or stiffness of micro electro mechanical system (MEMS). Our analysis suggests that if feedback force is proportional to sensed displacement, velocity, or acceleration of a MEMS proof mass, then feedback can be used to increase or decrease the apparent stiffness, damping, or mass of the system. Such feedback might be used to compensate the process variations, packaging stress, thermal drift, and damping. Prior efforts by others include position or velocity-based feedback for modifying frequency, bandwidth, quality factor, or sensitivity of resonators. However, we present a comprehensive means of controlling the response of MEMS. We develop an analytical steady-state MEMS model that includes feedback forces and circuit delay, and we develop a stability model. Our analytical models are verified using numerical simulations that include circuit delay, electrical feedback delay, and noise. Our results support the realization toward performance-on-demand MEMS (PODMEMS).

Self-similar solutions for a quenching problem with spatially dependent nonlinearity

This paper is devoted to the study of the self-similar solutions for a semilinear parabolic equation with spatially dependent nonlinearity which arises in the model of micro-electro mechanical system. We first provide a result on the non-existence of slow orbit for a certain range of parameters. Next, we prove the existence of backward solutions with the desired polynomial growth condition at infinity to the associated equation by a fixed point argument. Then we give a detailed analysis of the behavior of global solutions at the origin. Finally, as an application of the above results, we prove a uniqueness theorem.

3-D Printing as an Effective Educational Tool for MEMS Design and Fabrication

This paper presents a series of course modules developed as a high-impact and cost-effective learning tool for modeling and simulating the microfabrication process and design of microelectromechanical systems (MEMS) devices using three-dimensional (3-D) printing. Microfabrication technology is an established fabrication technique for small and high-precision MEMS devices; these processes typically take place in a cleanroom with the use of expensive high-vacuum equipment. These course modules were developed to provide engineering educators a more affordable and effective method for teaching MEMS modeling in settings without a cleanroom, as is the case in many undergraduate institutions. Feedback from student evaluations as well as course grades all support the efficacy of these course modules. In these hands-on modules, by designing and building the MEMS prototypes, the students learn by experiencing the process of building a MEMS device from the specifications given. The results are also compared to similar assessments made in the course before the introduction of these course modules to verify success. The detailed description of the modules, the evaluation methodologies adopted, and reflections on the implementation are discussed.

A fourth-order model for MEMS with clamped boundary conditions

The dynamical and stationary behaviors of a fourth-order equation in the unit ball with clamped boundary conditions and a singular reaction term are investigated. The equation arises in the modeling of microelectromechanical systems (MEMS) and includes a positive voltage parameter $\lambda$. It is shown that there is a threshold value $\lambda_*>0$ of the voltage parameter such that no radially symmetric stationary solution exists for $\lambda>\lambda_*$, while at least two such solutions exist for $\lambda\in (0,\lambda_*)$. Local and global well-posedness results are obtained for the corresponding hyperbolic and parabolic evolution problems as well as the occurrence of finite time singularities when $\lambda>\lambda_*$.

A free boundary problem modeling electrostatic MEMS: I. Linear bending effects

The dynamical and stationary behaviors of a fourth-order evolution equation with clamped boundary conditions and a singular nonlocal reaction term, which is coupled to an elliptic free boundary problem in a non-smooth domain, are investigated. The equation arises in the modeling of microelectromechanical systems and includes two positive parameters $$\lambda $$ and $$\varepsilon $$ related to the applied voltage and the aspect ratio of the device, respectively. Local and global well-posedness results are obtained for the corresponding hyperbolic and parabolic evolution problems as well as a criterion for global existence excluding the occurrence of finite time singularities which are not physically relevant. Existence of a stable steady state is shown for sufficiently small $$\lambda $$ . Non-existence of steady states is also established when $$\varepsilon $$ is small enough and $$\lambda $$ is large enough (depending on $$\varepsilon $$ ).

Optimal control of a singular PDE modeling transient MEMS with control or state constraints

A particular feature of certain microelectromechanical systems (MEMS) is the appearance of a so-called “pull-in” instability, corresponding to a singularity in the underlying PDE model. We here consider a transient MEMS model and its optimal control via the dielectric properties of the membrane and/or the applied voltage. In contrast to the static case, the control problem suffers from low dimensionality of the control compared to the state and hence requires different techniques for establishing first order optimality conditions. For this purpose, we here use a relaxation approach combined with a localization technique.

Modeling and Simulation of the Ferroelectric Based Micro Gyroscope: FEM Analysis

This paper presents the design and modeling of micro-electromechanical systems (MEMS) on the ternary ferroelectric compounds (PZT) based by using finite element model (FEM) simulation. The conceptual framework establishes five steps to perform the critical analysis: design a novel structure, define the failure mechanisms under the given conditions, analyze different vibrations, analyze the operation principle, and detect resonance modes. In addition, MEMS failure modes were analyzed under different scenarios and the obtained results discussed.

Thermal and Electrical Conductivities of Nanocrystalline Nickel Microbridges

DC electrical self-heating (Joule heating) is exploited to characterize the thermal behavior of Ni microbridges. The temperature rise of the devices due to self-heating is monitored using an infrared microscope for current densities up to 105 A/cm2. The obtained temperature profiles reveal significant heating at the bases of the microbridges. Simulations are performed in order to extract the thermal conductivity of the electroplated Ni thin film from the experimental data. The thermal conductivity is found to be 78.8 W/m · K or 13% less than that of bulk Ni. As current flows through the microbridges, they deflect upward, significantly changing the system response and pull-in voltage required for actuation. Additionally, the electrical resistivity and specific electrical contact resistances between the microbridges and the anchor points are reported. The electroplated Ni is found to have an electrical resistivity of 9.7 μΩ · cm which agrees with other values in the literature for thin-film Ni. By combining the electrical and thermal measurements, it is possible to determine the phonon and electron contributions to thermal conductivity. Although demonstrated on Ni films, this technique can be applied to any metallic film without modification. Such characterization of transport properties of constituent materials is important in the modeling of microelectromechanical systems and enables device performance to be predicted with improved accuracy.