Coulomb actuated microbeams: A Chebyshev Edgeworth approach to highly efficient lumped parameter models

Published 05.01.2022 in arXivLabs

Coulomb-actuated microbeams play a crucial role in many MEMS (Micro-Electro-Mechanical Systems) applications.
This publication describes the stable and unstable equilibrium states of prismatic Coulomb-actuated Euler-Bernoulli microbeams using the lumped parameter model. The observations of the research paper “Coulomb-actuated microbeams revisited: experimental and numerical modal decomposition of the saddle-node bifurcation” allows us here to develop a model with a single degree of freedom and lumped parameters. The zero-mode LPM based on the Chebyshev-Edgeworth approach proves the physics of these microbeams.  

Coulomb actuated microbeams
Large-Signal Equivalent-Circuit Model of Asymmetric Electrostatic Transducers

Published 07.10.2021 in IEEE/ASME Transactions on Mechatronics

This article presents a circuit model that is able to capture the full nonlinear behavior of an asymmetric electrostatic transducer whose dynamics are governed by a single degree of freedom. Effects such as stressstiffening and pull-in are accounted for. The simulation of a displacement-dependent capacitor and a nonlinear spring is accomplished with arbitrary behavioral sources, which are a standard component of circuit simulators. The approach demonstrates the behaviour of a electrostatic MEMS microspeaker.  

Overview of the large-signal equivalent-circuit model of an electrostatic transducer. Measured and simulated THD and sound pressure level of the ANED microspeaker.
Coulomb-actuated microbeams revisited: experimental and numerical modal decomposition of the saddle-node bifurcation

Published 28.05.2021 in Microsystems & Nanoengineering

Stable and unstable states of Coulomb-actuated microbeams can be simulated successfully by combining FEM with arc-length solvers. Both approaches confirm that the shape of the bending profile of a Coulomb-activated prismatic microbeam, clamped at both ends, is almost identical to the shape of the Euler-Bernoulli zero-mode.
The observation paves the way for lumped-parameter models of high accuracy that only use a single spatial degree of freedom, amenable to direct physical interpretation. The fruit of this research work is a surprisingly accurate zero-mode approximation that has numerous applications in the modeling of the physics of electrostatic MEMS actuators.  

Concept and proof for an all-silicon MEMS micro speaker utilizing air chambers 

Published 07.10.2019 in Microsystems & Nanoengineering

MEMS-based microspeakers are used as sound transducers for smart devices such as hearables. Such devices require high sound pressure levels, low harmonic distortion and low power consumption for industrial, consumer and medical applications.
The novel microspeaker concept based on nanoscopic electrostatic actuation presented in this paper essentially involves in-plane electrostatic bending actuators and uses the chip volume rather than its surface for sound generation. Key advantages of the concept are CMOS compatibility including integration capabilities with microelectronic circuitry, scalable mass production, low unit cost, small foot print for high sound quality and very low power consumption. The research paper describes the principle, design, fabrication and initial characterisation results.  

Geometric setup of an all-silicon MEMS microspeaker
A small-gap electrostatic micro-actuator for large deflections

Published 11.12.2015 in nature communications

In this research paper, an electrostatic actuator principle is presented, fabricated in a CMOS-compatible process, which allows high deflections with small electrode gaps. The concept makes the huge electrostatic forces within nanometre small electrode separation accessible for large deflections. Electrostatic actuations that are larger than the electrode separation were measured. The driving concept enables the investigation and development of micro systems with a high potential for improved device and system performance. A solution is demonstrated to prevent or reduce the occurrence of pull-in related failure mechanisms of MEMS actuators and sensors.  

A small-gap electrostatic micro-actuator for large deflections


Large deflection of electrostatically actuated microbeams – How do the bending modes contribute beyond pull-in?

Published 8.-10.11. 2021 at MikroSystemTechnik Congress 2021 in Stuttgart-Ludwigsburg

In our research paper we addressed the question of how the various Euler-Bernoulli bending modes contribute to the physics of a Coulomb-actuated prismatic clamped-clamped microbeam.
To test our hypothesis, we analyzed numerically and experimentally the contribution of the Euler-Bernoulli zero mode to the nonlinear bending profile of a suitable microbeam. For this purpose we performed a FEM simulation, using an arc length solver, and compared the simulation with a MEMS experiment. The experiment and FEM simulations match almost perfect. This proves the validity of our zero-mode hypothesis. As a result of this modeling, we developed a single degree of freedom lumped parameter model.
Our analysis shows that it is possible to accurately model Coulomb-actuated microbeams using the Euler-Bernoulli equation.  

Prismatic coulomb-actuated clamped-clamped microbeam
Balanced MEMS loudspeaker: Coulomb forces vs. viscosity of air and squeeze film damping

Published 15. – 18.08.2021 at DAGA 2021 in Vienna

In our presentation at DAGA 2021, we discussed the influence of fluid dynamics on the damping of a balanced nanoscopic electrostatic drive (NED) in a test loudspeaker. We reported on the FEM-based determination of the natural frequency and quality factor of an actuator designed to drive a balanced MEMS speaker.
On the one hand, an electrostatic MEMS microspeaker benefits from small electrode spacing leading to high driving forces. At the same time, the fluid dynamics of air can cause strong damping forces in the microscopic cavities of a µSpeaker chip. 

Exploded view of the FEM model of the actuator in the state without a cover wafer. (a) Air domain above the chip. (b) Structural domain including the actuator. (c) Air domain within the microcavity, modeled with thermoviscous acoustics.
Nonlinearity of balanced MEMS loudspeakers: optical experiments and numerical modeling using time-harmonic signals

Published 11. – 16.07.2021 at the 27th International Congress on Sound and Vibration

The recently developed NED technology offers a high potential for electrostatic transducers for audio and ultrasound applications. MEMS audio transducers cover the full frequency range required for high fidelity audio applications. Such audio reproduction also demands minimizing harmonic distortions substantially below 1 %. A major advance in this direction is combining the NED principle with a push-pull driving scheme in a balanced design (BNED).
The nature of the Coulomb force, the impact of stress stiffening and the large deformations required for generating high sound pressures potentially contribute to the harmonic distortion and therefore need advanced experimental methods and simulation models to allow for an apt design.  

(a) The measurement setup: The Photron Fastcam SA1.1 high-speed camera on top of the Leica DM8000 microscope. (b) A micrograph of the free end of the BNED structure. The red marker shows the pattern tracking location used for the extraction of the movement.
Electrostatic all-silicon MEMS speakers for in-ear audio applications

Presented 09. – 13.09.2019 at the 23rd International Congress on Acoustics

This paper for the 23rd International Congress on Acoustics presents the acoustic properties of an electroacoustic MEMS transducer. The transducer is based on electrostatic bending actuators with lateral deflection. The actuators are arranged between a vented handle and cover wafer and are designed to emit sound in the audible frequency range. The results of acoustic measurements are discussed demonstrating audio reproduction capabilities for in-ear audio applications. 

Electrical circuit of the model for our electrostatic transducer
CMOS-compatible MEMS loudspeaker for in-ear applications

Presented 19. – 22.03.2018 at DAGA 2018 in Munich

At DAGA 2018, we introduced our electrostatic MEMS micro speakers for the first time, whereby no membrane but the wafer volume is used to generate sound. The article explains the groupwise counter-phase movement of the bending transducers, whereby vibration-free sound transmitters are possible. With the lateral arrangement of air inlets and outlets, several chips can be stacked and electronics for control as well as further MEMS sensor technology can be monolithically integrated.  

CMOS-compatible MEMS loudspeaker for in-ear applications
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