Design and analysis of vibration energy harvesters based on peak response statistics

Energy harvesting using cantilever piezoelectric vibration energy harvesters excited by Gaussian broadband random base excitation is considered. The optimal design and analysis of energy harvesters under random excitation is normally performed using the mean and standard deviation of a response quantity of interest, such as the voltage. An alternative approach based on the statistics of the peak voltage is developed in this paper. Three extreme response characteristics, namely (a) level crossing, (b) response peaks above certain level, and (c) fractional time spend above a certain level, have been employed. Two cases, namely the harvesting circuit with and without an inductor, have been considered. Exact closed-form expressions have been derived for number of level crossings, statistics of response peaks and fractional time spend above a certain level for the output voltage. It is shown that these quantities can be related to the standard deviation of the voltage and its derivative with respect to time. Direct numerical simulation has been used to validate the analytical expressions. Based on the analytical results, closed-form expressions for optimal system parameters have been proposed. Numerical examples are given to illustrate the applicability of the analytical results.