Mixed-Level Model for Highly Perforated Torsional Actuators Coupling the Mechanical, the Electrostatic and the Fluidic Domain
Robert Sattler, Gabriele Schrag and Gerhard Wachutka
Institute for Physics of Electrotechnology, DE
Keywords: compact modeling, mixed-level modeling, squeeze-film damping, reduced order modeling, coupled simulation
Electrostatic attraction between neighboring electrodes is a widely used actuation principle employed in tiltable micromirrors or microrelays. Electrostatic actuation is especially favored for switching applications in mobile telecommunication because of its low power consumption. State-of-the-art commercial software tools are able to analyze the static equilibrium between the electrostatic and the mechanical forces by solving the underlying partial differential equations sequentially in each of the two energy domains, which is a computationally expensive approach for complex devices. But for switching applications it is primarily the transient behavior that needs to be analyzed. Even if fluidic damping effects are neglected it is prohibitive to simulate the transient behavior of an industrial microrelay (Fig. 1) on the continuous field level in view of the computational cost. On the other hand, since damping can only be neglected under vacuum conditions, which is only achievable with an expensive packaging, it is unavoidable to take squeeze-film damping effects into account in the analysis and design optimization of most MEMS devices.
In order to tackle this problem, we follow a new mixed-level approach, which makes it possible to reduce the model complexity by incorporating the fluidic damping and the electrostatic actuation in system-level models in a physically correct and accurate manner. Besides our method offers high flexibility with respect to variations of the device geometry and linking to additional other energy and signal domains. The damping model is based on the Reynolds equation, a simplification of the Navier-Stokes equation (NSE), which is solved using the Finite Network method. The basic ideas of this approach have been presented in [1,2]. In the terminology of Kirchhoffian network variables, we consider the fluidic massflow (and the mechanical torque, respectively) as generalized flux ("through" variable), which is driven by the pressure gradient in the fluid (or the angular velocity of the mechanical parts, respectively) acting as generalized force ("across" variable). The mechanical and the electrostatic model have been reported in . Physically based, scalable compact models are integrated in the network to correct for finite size effects and perforations in the structure.
One of the objectives of the present work has been to extend the compact models, which account for the fluidic damping caused by perforations of the mechanical structure, to the case of large displacements. Fig. 2 displays the fluidic resistance as extracted from a FEM analysis of a perforation based on the NSE; evidently the fluidic resistance increases with decreasing squeeze film thickness. Hence, assuming the fluidic resistance to be independent of the fluid film thickness (as reported in ) is a good approximation only, if the fluid-film thickness is not too small. For modeling the pull-in behavior, however, this effect must not be neglected.
Another objective has been to properly include electrostatic fringing fields in the mixed-level scheme, which arise at the edges and near the holes of a highly perforated plate and contribute to the electrostatic torque acting on the mechanical structure. Compact models for the fringing fields around a single hole (Fig. 3) and near the edges of the plate have been extracted from FEM simulations. The previous analytical model  describing the electrostatic torque on the actuator without fringing fields has been extended accordingly, and it also accounts for the shape of the fixed ground electrode. The influence of fringing fields and the effect of reducing the size of the ground electrode are depicted in Fig. 4 (with the scale normalized to the electrostatic torque of an actuator where the ground electrode has the same size as the movable electrode).
Equipped with this fluid-electro-mechanical macromodel, it is now possible to study the dynamic behavior of electrostatically actuated microdevices with complex geometry in a cost-effective but still very accurate manner. This is demonstrated, for instance, in Fig. 5 and 6 where the transient pull-out and pull-in behavior of the microswitch shown in Fig. 1 is compared with measurements for three different degrees of perforation (ratio of the hole area to the plate area in percent).
NSTI Nanotech 2003 Conference Technical Program Abstract