# Modeling a Wave Energy Harvester for Supplying Data Buoys
## 2 Materials and Methods
### Bang-Bang Mode
Therefore, cycle after cycle, the instantaneous phase of each cycle becomes less and less until the phase reaches the value \(\phi_{EQ}\). If the initial phase \(\phi\) belongs to the interval \(\phi_{L}<\phi<\phi_{N}\), the curves in Figure 4a show that the track length \(L\) is greater than the half-period length \(L_{T/2}\) and less than the inversion length \(L_{inv}\). Thus, like in the case \(0\leq\phi<\phi_{EQ}\), there is an increase of \(\phi\) toward values greater than \(\phi_{N}\). Finally, if the initial phase \(\phi\) is greater than \(\phi_{N}\), the track length \(L\) is greater than the inversion length \(L_{inv}\). Thus, before reaching the other end of the track, the cylinder changes the direction. It is not easy to predict the behavior in this case, but it is presumable that sooner or later the cylinder will end up at one end of the track with an initial phase in the interval \(0<\phi<\phi_{L}\). From that moment on, the phase will converge towards \(\phi_{EQ}\). From the above considerations, it is possible to deduce that \(\phi_{EQ}\) represents an equilibrium condition that is stable. Let us now assume that \(L<L_{T/2,0}\), as shown in Figure 4b. In this case, there is no intersection between the increasing part of the curve \(L_{T/2}\) and the straight line \(L\). Repeating considerations like those made above, it is easy to deduce that the stable equilibrium phase  
\[\dot{x}=\frac{g\text{sin}\,\alpha_{M}}{k\,\omega}[\text{cos}\,(\omega\ t+\phi _{0})-\text{cos}\,\phi] \tag{10}\]3 Representation of the half-period length, \(L_{T/2}\), and of the inversion length, \(L_{inv}\), as a function of the instantaneous phase at the detachment instant, \(\phi\), in three case studies. (**a**) The track length is in the interval \(L_{T/2,0}\leq L\leq L_{T/2,M}\). (**b**) The track length is assumed to be \(L<L_{T/2,0}\). (**c**) The track length is assumed to be \(L_{T/2,M}<L\). is \(\phi_{EQ}=0\) and the interval of the initial phase \(\phi\) that directly takes to the equilibrium phase is \(0<\phi<\phi_{L}\). On the other hand, if \(L>L_{T/2\_M}\), as shown in Figure 4c, there is no intersection between the curve \(L_{T/2}\) and the straight line \(L\). In this case, for no value of the initial phase, the cylinder is able to reach the other end of the track in \(T/2\), making bang-bang operation impossible. Concluding, only if \(L_{T/2\_0}\leq L\leq L_{T/2\_M}\) and if \(L<L_{T/2\_0}\), the system reaches a bang-bang mode of operation. Thus, a stable bang-bang mode of operation is possible only if \(L<L_{T/2\_M}\), which according to Equation (17), leads to the condition  
\[\dot{x}=\frac{g\text{sin}\,\alpha_{M}}{k\,\omega}[\text{cos}\,(\omega\ t+\phi _{0})-\text{cos}\,\phi] \tag{10}\]4  
Equation (21), which is graphically shown in Figure 5, establishes a relationship between the track length \(L\), the maximum tilt angle \(\alpha_{M}\) and the oscillation period \(T\) in order to ensure a bang-bang operation. Therefore, Equation (21) can be employed to design the wave energy harvester to operate in the more efficient way, which is the bang-bang mode of operation.