MEMS electrostatic energy harvesters with end-stop effects

In micro scale energy harvesting devices, end-stops that limit the proof mass motion are inevitable from reliability concerns and can even be exploited as a functional element to achieve a broadband response. To investigate how these can be modelled, both characterization and modelling of vibration energy harvesters with end-stop effects are presented in this paper. A Hertz contact model of the impact force between the proof mass and the end-stops is analysed and compared to a linear stiffness model. The resulting impact force model is then included into a SPICE model of an electrostatic harvester. The performance prediction of the model is validated by comparing simulations and measurements on two different prototypes, one with mechanical quality factor Q(m) = 5.7 and one with Q(m) = 203.5. The electromechanical coupling factors of the two devices are respectively k(2) = 1.44% and 2.52%. Both devices display the well-known jump phenomenon and output voltage saturation during, respectively, frequency and amplitude sweeps. Under low-level broadband excitations, the high-Q(m) device performs in agreement with linear theory at an efficiency of 71.8%. For sufficiently high acceleration power spectral density (PSD), it displays a soft limit on the output power and a bandwidth increase, e. g. a factor 3.7 increase of 3 dB bandwidth when increasing the acceleration PSD from 0.34 x 10(-3) to 0.55 x 10(-3) g(2) Hz(-1). The end-stop effects reduce the device efficiency down to 35.4% at 1.69 x 10(-3) g(2) Hz(-1). A comparison between model and experiment shows that a model with end-stop stiffness extracted from the contact analysis can adequately model the nonlinear end-stop effects both for narrow-and broadband accelerations.