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Modeling of a microelectromechanical system (MEMS) requires coupling of structural mechanics with electrostatics. The simplest approach is to describe the electrostatic force by means of lumped capacitors (TRANS126 element in ANSYS). Yet, even in this case the model is already nonlinear.

Small signal analysis, that is, linearization of a nonlinear model around an operation point, is an evident option to apply linear model reduction. Common practice in the area of RF-resonators is to use a bias voltage and then a small harmonic signal to operate the device. Pre-stressed simulation in ANSYS is designed to model this behavior. During nonlinear static analysis, the effect of the bias voltage is included into the stress-stiffening matrix and the latter is used during harmonic response simulation. The methodology to incorporate model reduction into this process for RF-resonators is presented in papers listed below.

*L. Del Tin, R. Gaddi, A. Gnudi, E. B. Rudnyi, A. Greiner, J. G. Korvink.*

Efficient pre-stressed harmonic analysis of RF-microresonators by means of model order reduction.

Thermal, Mechanical and Multiphysics Simulation and Experiments in
Micro-Electronics and Micro-Systems, Proceedings of EuroSimE 2006, Como (Milano), Italy, 24-26 April, p. 226-229, 2006.

A simulation methodology to reduce computational time of pre-stress harmonic analysis of radio frequency (RF) microresonators is demonstrated. The methodology is based on the application of model order reduction to a system of ordinary differential equations obtained after discretization in space by finite element software. Model order reduction produces a low dimensional approximation of the original system and hence enables a substantial reduction of simulation time while maintaining a very small approximation error. The approach is thus allows us to perform rapid device design and optimization. Reduced models can be also used to implement hardware description language model to employ in system level simulation.

*L. Del Tin, R. Gaddi, E. B. Rudnyi, A. Greiner, J. G. Korvink.*

Model Order Reduction for the Extraction of Small Signal Equivalent Circuit Models of RF-MEMS.

SCEE 2006, Scientific Computing in Electrical Engineering, Sinaia - Romania, 17-22 September 2006.

A reduced order model for the small signal analysis of micromechanical structures (MEMS) has been extracted by applying model order reduction (MOR) to their finite element models. The low-order model conserves the accuracy belonging to the finite element method, while drastically reducing the computational time. Moreover, it gives a description of the device terminal behaviour and can therefore be employed for circuit and system level simulations of MEMS devices.

*Laura Del Tin.*

Reduced-order Modelling, Circuit-level Design and SOI Fabrication of Microelectromechanical Resonators.

PhD Thesis, Universita degli Studi di Bologna, Facolta di Ingegneria, 2007.

*L. Del Tin, A. Greiner, J. G. Korvink.*

**
Model Order Reduction for Circuit Level Simulation of RF MEMS Frequency Selective Devices**.

Sensors Letters, Volume 6, Number 1, February 2008, 2008, pp. 1-8.

Final Paper at IngentaConnect.

The development of complex radio frequency circuits with integrated micromechanical devices requires the availability of tools for predictive design, optimization and verification of the complete system. In this paper, a methodology towards this goal is proposed. It enables the extraction of non-linear low order models of vibrating micromechanical devices suitable for use in a standard circuit level simulator. The reduction of the complexity of the model is achieved by using moment matching model order reduction, together with an approximated lumped description of the electrostatic forces. In this way, both high simulation accuracy and low computational complexity are obtained.

Evgenii B. Rudnyi

Designed by

Masha Rudnaya