Macromodel for micromechanical, multi-electrode structures in force feedback control systems

Electrostatic force feedback loops are commonly used for measurement and position control in micromechanical sensors and actuators. They are widely implemented with switched capacitor, sigma-delta (Σ/Δ) architectures as they provide perfect compatibility with capacitive sensing methods. The electromechanical subsystem, as a vital part of a Σ/Δ-loop, strongly influences the loop characteristics. In order to characterize a Σ/Δ-loop, however, transient simulations over a long time period are inevitable. Full 3D physical simulations (FEM/BEM) at system level are numerically impractical. Therefore, effective macromodels of the electro-mechanical subsystem are required. Established macromodels can handle one-dimensional systems and weakly coupled multi-dimensional systems but are not applicable to multidimensional, multi-electrode systems. A novel modeling technique is presented for micromechanical, multi-electrode structures which is based on Lagrange´s equations. It considers the coupling of the structure kinematics with the electrical field between the electrodes. It further reflects parasitic excitation of the mechanical fundamental eigenmodes of the structure. The technique is applied to a new electrostatic levitation system controlling a micromechanical plate