# Moment-Based Stochastic Analysis of a Bistable Energy Harvester with Matching Network
###### Abstract
Vibrations of the support induce oscillations of the beam that in turn are transformed into electrical power by means of layers of a piezoelectric material deposed on the cantilever. Nonlinearity is introduced in the design by the tip magnet, that is fixed to a support in front of the inertial magnet but with opposed polarities, to create a biased inverted pendulum with magnetic repulsive force. When the two magnets are close enough, magnetic repulsion forces the beam to the left or to the right of the vertical position. The unforced system exhibits bi-stability, with two stable equilibrium points, each with its own basin of attraction, separated by an unstable saddle, corresponding to the vertical position. We show novel results derived through stochastic analysis complemented by numerical approaches. The energy harvester description derives from modeling the mechanical part, the linear piezoelectric materials, and the circuit representation of the electrical load. Nonlinearities are included by means of a proper model for the mechanical elastic potential. The result is a description in the form of a stochastic differential equation (SDE) system. For the linear energy harvester, we use stochastic calculus to analytically evaluate the output average power and the power efficiency. For the nonlinear system, the same quantities cannot be computed analytically, because the resulting stochastic differential equations are nonlinear and cannot be solved. To overcome this issue, we resort to a moment closure technique that permits to calculate the moments of the probability distribution for the electro-mechanical variables in the weak noise limit. In the medium-strong noise regime, we use stochastic numerical integration schemes to calculate the output average power and power efficiency. Inspired by our recent work on the application of circuit theory to improve the efficiency of energy harvesting systems, we interpose an \(LC\) matching network between the transducer and the load [18; 19], whose advantages are assessed in terms of the output (i.e., on the load) average voltage, output average power and conversion power efficiency. The paper is organized as follows: In Section 2, we derive the stochastic differential equations that describe the energy harvester subject to random vibrations, modeled as a white Gaussian noise. As a first contribution, in Section 3, we derive the equation governing the power balance of the electro-mechanical system, showing that determination of the output average power and of the power efficiency requires to calculate the variance of the output voltage. Section 4 is dedicated to the second main contribution: we introduce a novel, systematic procedure to derive dimensionless stochastic differential equations describing the system. In particular, we show how dimensionless time must be introduced to correctly take into account the stochastic nature of the governing equations. Section 5 is devoted to the analysis of a linear energy harvester, showing how stochastic calculus can be used for the design and optimization of the matching network. Section 6 represents the main contribution of the work: we develop a methodology based on the Gaussian moment closure technique for the design and optimization of the matching network for nonlinear energy harvesters, a quite important result, as few techniques exist for the design of nonlinear systems. We show that the optimized matching network increases the average harvested power and power conversion efficiency by more than 30 times in the weak noise limit, and by more than 5 times for medium-strong noise intensity. Finally, Section 7 is devoted to conclusions.