Scaling laws of electromagnetic and piezoelectric seismic vibration energy harvesters built from discrete components

This paper presents a theoretical study on the scaling laws of electromagnetic and piezoelectric seismic vibration energy harvesters, which are assembled from discrete components. The scaling laws are therefore derived for the so called meso-scale range, which is typical of devices built from distinct elements. Isotropic scaling is considered for both harvesters such that the shape of the components and of the whole transducers do not change with scaling. The scaling analyses are restricted to the case of linearly elastic seismic transducers subject to tonal ambient vibrations at their fundamental natural frequency, where the energy harvesting is particularly effective. Both resistive-reactive and resistive optimal electric harvesting loads are considered. The study is based on equivalent formulations for the response and power harvesting of the two transducers, which employ the so called electromagnetic and piezoelectric power transduction factors, Πcm2 and Πpe2. The scaling laws of the transduction coefficients and electrical and mechanical parameters for the two transducers are first provided. A comprehensive comparative scaling analysis is then presented for the harvested power, for the power harvesting efficiency and for the stroke of the two harvesters. Particular attention is dedicated to the scaling laws for the dissipative effects in the two harvesters, that is the Couette air losses and eddy currents losses that develop in the electromagnetic harvester and the material, air and dielectric losses that arise in the piezoelectric harvester. The scaling laws emerged from the study, are thoroughly examined and interpreted with respect to equivalent mechanical effects produced by the harvesting loads.