Efficient Pre-stressed Harmonic Analysis of RF-Microresonators by Means of Model Order Reduction L. Del Tin 1 , R. Gaddi 1 , A.Gnudi 1 , E. Rudnyi 2 , A. Greiner 2 , J. G. Korvink 2 1 ARCES-DEIS, University of Bologna, Viale Risorgimento 2, 40136 Bologna, Italy. Email: ldeltin@arces.unibo.it 2 IMTEK, Laboratory for simulation, Department of Microsystem Engineering, University of Freiburg, Georges- Koehler-Allee 103, 79110, Freiburg, Germany Abstract A simulation methodology to reduce computational time of pre-stressed 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 spatial discretization 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 allows to perform rapid device design and optimization. Once the device design and working conditions have been defined its reduced model can also be used to implement a behavioural model that can be employed in system level simulations. 1. Introduction Vibrating micromechanical structures with electrostatic actuation are good candidates for demanding frequency-selective applications, especially in the wireless telecommunication field. Recent years have therefore witnessed continuous efforts focused on their design and modeling. Good performance in terms of both quality factor and power consumption together with small dimensions, make micromechanical resonator a valid alternative to previously adopted devices, such as ceramic filters, SAW filters and quartz crystals. Moreover, RF- MEMS devices offer the possibility for on-chip fabrication, which implies an overall cost reduction [1]. Microelectromechanical resonators are usually operated with a harmonic voltage of small amplitude, superimposed on a large bias voltage. The characterization of their frequency response requires therefore harmonic simulation in which deformation of the structure and nonlinear effects due to static loading are considered. This analysis is generally referred to as harmonic pre-stressed analysis. Pre-stress effects have significant influence on the total stiffness of the structure, and have therefore to be taken into account for correct harmonic simulation. For devices with simple geometries, analytical formulas are used to approximately compute the device resonance frequency and equivalent circuit models are used to simulate the frequency behaviour [2]. With the increase of geometrical complexity, the finite element method (FEM) is employed to improve modeling accuracy [3]. However, harmonic simulation of electromechanical problems using FEM is computationally expensive because two coupled energy domains are involved. FEM discretizes the device in space and leads to a system of ordinary differential equations (ODE) whose dimension is generally very large. Harmonic analysis requires the solution of such a system in the frequency domain for each frequency step. This is a major drawback when simulation is applied to device design optimization with respect to model parameters. Because of high computational complexity, finite element simulation in practice is often limited to a static or modal analysis. In this paper, a simulation approach is presented that enables a significant speed-up of the harmonic pre- stressed analysis of electrostatically driven micromechanical devices. The approach is based on the application of model order reduction (MOR). Since harmonic analysis is a small signal linear analysis around a bias point, a technique for linear MOR can be applied. System matrices describing the electromechanical problem made by the FEM software ANSYS ® are read directly from ANSYS binary files and a low-order model of the device is automatically extracted. This is done with the tool mor4ansys, which performs moment-matching model order reduction via the Arnoldi algorithm [4]. The reduced model, which takes into account both electrical and mechanical pre-stress effects, is then used to perform linear harmonic analysis. Simulation results for a lateral clamped-clamped beam resonator are presented and compared with the full model. 2. Finite element modeling and simulation The finite element tool ANSYS is used for modeling of the microelectromechanical device. During the analysis electrical and mechanical domains are directly coupled. For this purpose, electromechanical coupling is modeled using transducer elements (TRANS126) implemented in ANSYS. These lumped elements model the capacitive response of a device to a motion in one direction. Each element has two nodes, with an electrical and a mechanical d.o.f. for each node, representing voltage and displacement along one of the coordinate directions. The electrostatic behaviour of the electromechanical device is then completely described in terms of capacitance between the conductive parts that are connected by nodes of the TRANS126 element. Therefore, only movable structures have to be represented and meshed. The relation between the capacitance and the distance (in the chosen direction) for each element can be computed automatically by ANSYS using the analytical formula for a parallel plate capacitor. Alternatively the relation can be specified on the basis of a separate series of electrostatic simulations for various distance values. This second approach allows to achieve a good degree of approximation for devices with complex geometry [5]. Electrostatic forces are then computed using energy principles. TRANS126 modeling capabilities are limited to devices having electrostatic forces between conductors