Available online at www.sciencedirect.com Sensors and Actuators A 142 (2008) 203–210 Nonlinear behavior of SOI free-free micromechanical beam resonator Moorthi Palaniapan ∗ , Lynn Khine Signal Processing and VLSI Lab, Department of Electrical and Computer Engineering, National University of Singapore, Singapore Received 30 September 2006; received in revised form 17 July 2007; accepted 7 August 2007 Available online 17 August 2007 Abstract Measured nonlinear behavior of a capacitively driven free-free micromechanical beam resonator at different driving conditions is presented. The resonator, fabricated in SOIMUMPs process, has a measured resonant frequency of 654 kHz with an average quality factor, Q value of 12,000 operating at a pressure of 37.5 Torr. The overall nonlinearity (including mechanical and electrical) in the resonator was found to be triggered after critical ac drive voltage amplitude of about 120mVpp was exceeded. The observed nonlinearity was relatively independent of the proof-mass dc voltage, V P , as long as the critical ac drive voltage is not exceeded. Furthermore, partial compensation of spring hardening effect (arising from mechanical nonlinearity) with spring softening effect (from capacitive force used to actuate the resonator) is observed in this work. This feature is particularly useful for MEMS oscillator applications where frequency tuning can be done by varying V P without inducing nonlinearity in the resonator. © 2007 Elsevier B.V. All rights reserved. Keywords: Micromechanical; MEMS; Nonlinearity; Resonator; Oscillator 1. Introduction Micromechanical resonators [1–7] have been studied exten- sively and their popularity is growing due to great promise in diverse sensing applications in inertial sensors, chemical sen- sors, and in RF communications such as oscillators and filters. In most cases, mechanical nonlinearities [2] arising from large mechanical displacements are highly undesirable for optimal performance of device especially for high quality factor Q appli- cations, such as degradation in phase noise performance of micromechanical resonator-based oscillator [3]. A larger nonlin- ear mechanical displacement of the resonator not only reduces the effective Q of the resonator due to higher energy losses but also introduces long-term reliability issues. However, there are applications where nonlinear behavior is desired and can be exploited to improve overall device performance, such as the suppression of feedback amplifier noise of an oscillator when nonlinear mechanical resonator is used as a frequency- controlling component [4]. Hence, it is crucial for designer to know the necessary driving conditions for resonator to oper- ∗ Corresponding author. Tel.: +65 65168723; fax: +65 67791103. E-mail address: elemp@nus.edu.sg (M. Palaniapan). ate in the linear or nonlinear mode of operation, as well as the knowledge on how to control the amount of nonlinearity. For a capacitively driven-and-sensed micromechanical res- onator, the interactions of applied capacitive force and mechanical restoring forces govern the overall resonance and nonlinear behavior. The capacitive force results from the com- bined influence of dc proof-mass voltage, V P and ac drive voltage, v ac . On the other hand, due to restrained structural boundaries, there exists mechanical restoring force that is composed of axial force, bending, and mid-plane stretching components [5], which are unique for particular design archi- tecture of the resonator. Although the capacitive force and the restoring mechani- cal force have nonlinear dependence on the displacement of the resonator structure, these forces vary linearly for rela- tively small displacements relative to critical dimensions of the resonator. For example, the critical dimension for capac- itive force nonlinearity will be the electrode-to-resonator gap used for actuation and sensing while that for mechanical non- linearity will be the width and width-to-length ratio in the direction of motion for flexural mode resonators. Nevertheless, a large capacitive driving force could induce bifurcation in res- onator response, a well-known “Duffing” behavior, resulting in electrical nonlinearities. The responding mechanical restoring 0924-4247/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.sna.2007.08.016