# Moment-Based Stochastic Analysis of a Bistable Energy Harvester with Matching Network
###### Abstract  
We discuss the analysis of a piezoelectric energy harvester for random mechanical vibrations, and we assess the performance improvement guaranteed by interposing a matching network between the transducer and the electrical load, in terms of average output power and power efficiency. The mathematical model describing the harvester is a system of stochastic differential equations, where both cases of linear and nonlinear devices are considered. In the linear case, the power delivered to the load is increased by a factor of about 20 with respect to the direct connection, with a similar increase in the conversion efficiency. In the nonlinear case, we use a moment closure technique to calculate the first- and second-order moments of the electro-mechanical variables in the weak noise limit. Moment calculation is used to determine the optimal values of the matching network components that maximize the performance. In the strong noise limit, the state equations are integrated numerically to determine the same performance metrics. Our analysis shows that a properly designed matching network improves the performance by a significant amount, especially at low noise intensity. energy harvesting; piezoelectric energy harvester; nonlinear dynamical systems; equivalent circuits; impedance matching; power efficiency; nonlinear resonance +
Footnote †: journal: Applied  
1  
MCDI 2023338  
of the combined machinery of stochastic analysis [30; 31] and nonlinear dynamics [32] is required. A second major challenge is represented by the obvious requirement to design harvesting systems with a very high efficiency. In fact, energy harvesters (or, more in general, electrical power supplies), are often working far from their optimal condition due to a non-ideal energy transfer from the (mechanical) source to the (electrical) load. Such an operating condition can be conveniently represented exploiting an electro-mechanical analogy, allowing for the construction of an equivalent circuit describing the entire harvester [28; 33]: this representation, in turn, suggests the introduction of a matching network between the harvesting device and the electrical load to eliminate, or at least mitigate, the mismatch [18; 19; 34]. Impedance matching is a classical design technique that has been exploited in RF and microwave electronics for about 70 years [35; 36; 37]; however, in other electronics areas, the energy transfer improvement is mostly limited to power factor correction schemes. A solution inspired by power factor correction is the connection of a shunted reactive element (capacitor or inductor) in parallel to the resistive load [18; 34; 38; 39]. The role of the shunted element is to reduce the time lag between the voltage and current in the load, thus reducing the amount of power reflected from the load to the source. An alternative solution, based on impedance matching theory, consists in interposing a matching network between the harvester and the electrical load. The matching network must be designed to eliminate the impedance mismatch between the mechanical and the electrical parts. The advantage offered by the matching network solution is that it not only reduces the power reflected from the load to the source but also maximizes the average power absorbed by the load, in general outperforming power factor corrected solutions. Obviously, perfect matching is achievable only at a specific frequency. Still, the matching network can be optimized to offer partial matching over a relatively wide frequency interval, maximizing the average power absorbed by the load and the power conversion efficiency, even for multi-frequency or broadband inputs. In this work, we study a bistable piezoelectric energy harvester, as schematized in Figure 1, subject to random mechanical vibrations. In this device, a cantilever beam is fixed at one end to a vibrating support, with a magnet of mass \(m\) at the opposite end to amplify oscillations.