J Comput Electron (2006) 5:205–210 DOI 10.1007/s10825-006-8845-y Equivalent circuit for RF flexural free-free MEMS resonators M. Mastrangeli · A. Nannini · D. Paci · F. Pieri C  Springer Science + Business Media, LLC 2006 Abstract A method to extract a lumped-parameter equiva- lent circuit for a free-free flexural MEMS resonator, based on the Euler-Bernoulli beam equation and exploiting a modal analysis approach, is presented. The dynamic behaviour pre- dicted by the equivalent circuit is compared with FEM sim- ulations, and the effect of a geometrical mismatch is investi- gated as well. The resonance frequency and the quality factor are correctly predicted. The method could be used for more complex systems of interconnected beams. The circuit can be used as a quick and intuitive analysis tool for the system- level designer and to allow the simulation of the device in a system-level design environment. Keywords Equivalent circuit . Lumped-parameters . MEMS flexural resonators . FEM simulations . Euler-Bernoulli beam Introduction Compact models for MEMS components, which operate on multiple physical domains, are required for efficient simula- tion of complex systems, including MEMS devices as well as analog and digital circuits. Consequently, a large amount of MEMS literature has been devoted to the development of such models (often based on a lumped-parameter approach) and to their integration in the flow of system simulation and design [1–3]. In this respect, electrical lumped-parameter M. Mastrangeli · A. Nannini · D. Paci () · F. Pieri Dipartimento di Ingegneria dell’Informazione, Universit` a di Pisa, Via Caruso 16, 56122 Pisa Italy e-mail: dario.paci@iet.unipi.it D. Paci CNR-IEIIT Sezione di Pisa, via Caruso 16, 56122 Pisa Italy equivalent circuits are especially useful. They reduce the number of state variables of the model and they can be im- plemented in a language for behavioural modelling [3], but also on more established and conventional (from the point of view of the electronic design) circuit simulators. This last aspect is relevant not only because circuit simulators are read- ily available in a design environment, but also because the equivalent circuit gives to the designer a quick insight into the component behaviour regardless of the underlying physical working principle, leading to a tighter interaction between the system-level and the device designer. Among MEMS devices, electromechanical resonators have been proposed as fundamental components for RF fil- ters and mixers [4], as well as frequency references for RF oscillators [4–6]. Several different designs, based on tor- sional, flexural, and bulk vibrational modes, have been pre- sented, and equivalent circuits for a number of different designs have been proposed [5, 7, 8], although a general approach to equivalent circuit extraction is not established yet. In this paper we present a method to extract a lumped- parameter equivalent circuit for laterally actuated free-free flexural resonators. FEM simulations of the frequency re- sponse of devices with different dimensions were performed to validate the model and investigate the effect of design fabrication error. Resonator structure and operation The free-free flexural resonator is shown in Fig. 1: the me- chanical structure is composed by a main beam with free ends, suspended at the nodal points of its first resonance mode by two support clamped-clamped beams. The resonator is driven into motion by an electrostatic force generated by the Springer