# Modeling a Wave Energy Harvester for Supplying Data Buoys
## 3 Results
### The Harvester Prototype  
A prototype of the wave energy harvester was built and tested, as shown in Figure 6. The track, made by means of a 3D printer, has an effective length \(L=200\) mm. The rolling cylinder is made up of a set of five magnets having total height \(L_{t}=2L_{1}+2L_{2}+L_{3}=32\) mm with \(L_{1}=4\) mm, \(L_{2}=2\) mm, \(L_{3}=20\) mm, radius \(R=4\) mm, radius \(R_{e}=5\) mm and total mass \(m=17.9\) g. The resulting parameter \(k\) in Equation (4) takes value \(k=1/0.569\). Each coil placed along the track has 1650 turns, internal resistance \(R_{c}=150\)\(\Omega\), internal inductance \(L_{c}=10\) mH, internal diameter \(5\) mm and external diameter \(9\) mm. To experimentally test the harvester, it was mounted on a plexiglass sheet hinged on two aligned pivots and controlled on one edge by a vertical linear actuator, which determines its inclination over time. Figure 5: Boundary line between the region of bang-bang operation and other operating conditions, as analytically predicted by Equation (21) for different values of the track length \(L\). The electromechanical coupling coefficient, \(\theta(x)\), was determined by experimental tests. In particular, the track was placed in a fixed position with an inclination angle \(\alpha=5^{\circ}\) and the magnetic cylinder was allowed to roll from the upper end of the track downwards. A single pair of series-connected coils was placed 30 mm from the bottom end and its open-circuit voltage was measured. Since, under these conditions, the current is zero, the motion of the cylinder \(x(t)\) and its velocity \(\dot{x}(t)\) can be easily calculated from Equation (4). Moreover, the measured voltage is equal to the electromotive force, \(e(t)=v_{L}(t)\), according to Equation (7). Once \(\dot{x}(t)\) and \(e(t)\) are known, the coupling coefficient can be calculated from Equation (6). The so-determined electromechanical coupling coefficient, \(\theta(x)\), is reported in Figure 7a. By using the electromechanical coupling coefficient, the time evolution of the cylinder position, in the case in which the current in the coils is zero, was compared with the case in which the current flows in two pair of coils was placed 30 mm from the track ends and connected to the optimal resistive load. The comparison reported in Figure 7b shows  
Figure 6: Photo of the experimental setup, made up of an oscillating plane, and of the wave energy harvester based on a magnetic cylinder rolling on a track. Figure 7: (**a**) Electromechanical coupling coefficient determined by measurements on the prototype of the wave energy harvester. (**b**) Comparison of the time evolution of the cylinder position in case the current into the coils is null and in case the current is that of the optimal load. Three different initial values of \(\phi\) were considered. For each considered case, the relationship between \(L\), \(L_{T/2}\) and \(L_{inv}\) is reported in Figure 8a,c,e, whereas the corresponding time evolution of the cylinder position \(x(t)\) and the time evolution of the phase \(\phi\) at the detachment instants from the track end \(x_{1}\) are reported in Figure 8b,d,f, respectively. It should be noted that in all the three cases considered, despite the different initial conditions, the phase \(\phi\) converges toward the stable equilibrium value \(\phi_{EQ}=12^{\circ}\), as analytically predicted by Equation (18). Different from the cases considered above, in Figure 9a,b, the parameters \(\alpha_{M}=20^{\circ}\) and \(T=1.2\) s were set, leading to \(L<L_{T/2,0}\). According to the theoretical investigation, the phase \(\phi\) converges to \(\phi_{EQ}=0^{\circ}\). Finally, in Figure 9c,d, it was considered the case \(\alpha_{M}=20^{\circ}\) and \(T=0.95\) s, leading to \(L_{T/2,M}<L\). In this case, as analytically predicted, the system is not able to reach the bang-bang operation. Experimental tests were also carried out to validate the theoretical findings regarding the boundary of the region of bang-bang operation.