## 2 Proposed Boost Architecture for Magnetostrictive Energy Harvesters
### Circuit Implementation  
Figure 3 shows the electrical circuit of the full-wave AC-DC boost converter considered in this work, while Figure 4 shows qualitative trends of the most interesting electrical quantities involved. V\({}_{\text{in}}\) is an AC voltage source (T is its period), and it represents the generic output voltage of a magnetostrictive EH device, while L and R\({}_{\text{coil}}\) represent the inductor and its internal resistance. The two MOSFETs behave as controlled switches where the opening and closing times depend on the voltages applied to the gates. In particular, when the switches are both closed, the AC voltage source is directly connected to the inductor. The latter charges, as represented in magenta color in Figure 4, and their energy \(W_{\text{L}}\) can be expressed as:  
\[W_{\text{L}}=\frac{1}{2}\text{ L i}_{\text{L}}^{2} \tag{1}\]  
where i\({}_{\text{L}}\) is the inductor current. The energy is then released through the diodes D1 and D2, respectively, when each MOSFET alternately opens, as represented in green color in Figure 4. Here, the output load is simplified to a capacitor C and a load resistor R\({}_{\text{out}}\). In more detail, for the circuit of Figure 3, three operating conditions can be identified with respect to the input signal period:  
* Both MOSFETs are closed: the inductor L is positively or negatively charged by the AC source and capacitor C discharges over the output resistor R\({}_{\text{out}}\);
* MOSFET1 closed and MOSFET2 open: the positively charged inductor L discharges over the output capacitor-resistor loop through diode D1;
* MOSFET1 open and MOSFET2 closed: the negatively charged inductor L discharges over the output capacitor-resistor loop through diode D2. The output voltage (V\({}_{\text{out}}\)) then increases at each inductor current commutation until a steady-state condition is reached. With respect to the inductor current trend, three operating modes can be considered: the Discontinuous current Conduction Mode (DCM), the Continuous current Conduction Mode (CCM) and the \"Critical\" Boundary current Conduction Mode (BCM). The DCM and CCM operating modes refer to whether the inductor fully discharges and the current goes to zero or not during each switching period [40; 41]. The DCM mode is the critical condition where the inductor current goes to zero right at the end of each switching period. With the aim to reduce the electromagnetic interference and to increase the efficiency, it is convenient to use the Boost converter in the DCM operating mode. Indeed, in DCM mode, the dissipation switching transition decreases because the switch current always  
Figure 3: Electrical schematic of the proposed AC–DC boost Converter. starts from zero [42, 43]. Moreover, at each cycle, the whole energy stored in the inductor is delivered to the output, contrary to what happens in CCM. In this paper, the MOSFET's gates are driven by two different Pulse Width Modulation (PWM) signals that are defined by a Time Delay (T\({}_{\text{D}}\)) and a Duty Cycle (D) with respect to the sign changes of the input voltage, as sketched in Figure 4. In the following, the Time Delay and the Duty Cycle are defined in percentages with respect to the input voltage period (T), and the two PWM signals are simply T/2 out of phase with each other. Nevertheless, the choice of T\({}_{\text{D}}\) and D can be regulated and optimized in order to maximize the output voltage and then the harvested power. The circuit parameters and the model of the electronic components adopted are reported in Table 1.  
The proposed AC-DC boost scheme is shown in Figure 2. Two switching elements as MOSFETs are controlled by a block reading zero crossing of the voltage of the energy harvester. As for the classic AC-DC boost, two diodes are connected to the next energy management stage, which can be as simple as a capacitor. The control block, described in the next subsection, has the task to drive the MOSFETs in order to boost the voltage and maximize the power transfer.