A Predictive Model and Design Study of Frequency-Up-Conversion-Based Low-Frequency Vibrational Energy Harvesters

This paper reports a predictive model for impact-based frequency-up conversion in MEMS electrostatic vibration energy harvesters (e-VEH) with soft stoppers and tilt sidewalls. Experimental studies demonstrated that these converters exhibit a dimensional figure-of-merit (normalized power) an order of magnitude higher than the state-of-the-art due to frequency up-conversion. However, no model that can explain the observed results has yet been reported. This work shows that such converters can be modeled as coupled oscillators, a concept used for the first time in the analysis of e-VEHs. While the shuttle mass is modeled as a point mass supported by a spring, a second oscillator is added to represent the vibration of the electrodes during and following the impact. Another key aspect is that the effect of non-parallel electrodes is considered for both the squeeze film damping and the electrostatic forces. The results predicted by the model compare favorably to previously measured data and are a significant advancement to the state-of-the-art models, since the sub-resonant decaying oscillation is first predicted by a numeric model without data-fitting.

The device considered here consists of a shuttle supported by suspension beams with the parylene-covered, slanted, comb-like electrodes on two sides. Additionally, the device employs cantilever beams as soft stoppers. The device’s resonant behavior is tested with the literature standard DC bias circuit[3]. The model predicts the output voltage by simultaneously solving the equations for Newton’s second law and Kirchhoff’s law associated with the device and testing circuit. In the lumped element model, the plate is represented as the first resonator of mass m₁, while the high aspect ratio electrode, which deforms easily, is represented by a second oscillator of mass, m₂, coupled to the first mass by a spring. Finally, a spring that becomes active at a certain displacement of the first mass is added to represent the effect of the soft-stopper. The forces acting on the oscillators and the corresponding displacement are marked. During each vibration period, the second resonator impacts the insulated stationary electrodes, entering an oscillation of higher frequency.

Both measured and analytical results show the same behavior with two groups of decaying sub-resonant waves appearing in each vibration period. Each group corresponds to the impact of the mobile electrode with the left and right stationary electrodes. Also, the peak values compare favorably to the experimental data. For the device’s resonant frequency, no difference is observed between up sweeps and down sweeps due to the small vibration amplitudes. The resonant peak shift to left under increasing bias voltage, namely stronger spring softening. Nonlinear behaviors become obvious when the device is operated under larger vibration amplitudes, where the spring hardening and hysteresis are both captured by the model and numerical results are in good agreement with the experimental data.