Dynamic response modelling and characterization of a vertical electrothermal actuator

Mathematical modelling and characterization of the dynamic response of a microelectromechanical system (MEMS) electrothermal actuator are presented in this paper. The mathematical model is based on a second-order partial differential equation (one-dimensional heat transfer) and a second-order ordinary differential equation ( mechanical dynamic equation). The simulations are implemented using the piecewise finite difference method and the Runge-Kutta algorithm. The electrothermal modelling includes thermal conduction, convective thermal loss and radiation effects. The temperature dependence of resistivity and thermal conductivity of single crystal silicon have also been taken into consideration in the electrothermal modelling. It is calculated from the simulation results that the 'cold' beam of the electrothermal actuator is not only a mechanical constraint but also a thermal response compensation structure. The 0-90% electrothermal rise times for the individual 'hot' and 'cold' beams are calculated to be 32.9 ms and 42.8 ms, respectively, while the 0-90% electrothermal rise time for the whole actuator is calculated to be 17.3 ms. Nonlinear cubic stiffness has been considered in the thermal-mechanical modelling. Dynamic performances of the device have been characterized using a laser vibrometer, and the 0-90% thermal response time of the whole structure has been measured to be 16.8 ms, which matches well with the modelling results. The displacements of the device under different driving conditions and at resonant frequency have been modelled and measured, and the results from both modelling and experiment agree reasonably well. This work provides a comprehensive understanding of the dynamic behaviour of the electrothermal actuation mechanism. The model will be useful for designing control systems for microelectrothermal actuated devices.