## 1 Introduction  
Energy Harvesting (EH) techniques mainly consist of recovering a bit of environmental energy, either due to natural sources or human activities, that would normally be wasted. While renewable energies are being exploited everywhere at large scale, EH is usually performed in remote places where a power grid is unavailable or expensive to connect to but some strategic electronic circuits need to be supplied. In particular, EH could be used to supply relevant low-power consumption electronics, such as Wireless Sensor Networks (WSN), that can be used, for example, in Structural Health Monitoring (SHM) of bridges or viaducts [1, 2]. More generally, these techniques are valid candidates to supply power to electronic devices or sensors that strongly interconnected and which constitute the Internet of Things (IoT) [3, 4] in such a way that avoids the need for batteries, which require recharging or substitution and are not eco-friendly. Indeed, the main advantages of EH techniques is the capability to harvest and convert ambient energy into electricity in a location that is very close to the end-user [5]. Among all the different EH devices, we will focus on the Vibration Energy Harvesters (VEHs) which convert environmental vibrations into electric energy by using electromagnetic generators [6] or _smart_ materials, such as piezoelectrics or magnetostrictives [7, 8]. VEHs can be installed as active dampers for internal combustion engines or, more effectively, in common remote places such as roads and bridges. Vibrations and stresses causedby the ongoing vehicle traffic on such roadways certainly represent one of the less exploited sources of anthropic energy [9]. VEHs based on magnetostrictive materials mainly exploit the so-called \"Villari effect\" (or inverse magnetostrictive effect) and the Faraday's law [10]. Magnetostrictive materials, such as _Galfenol_, show good mechanical characteristics, high energy density and low temperature response dependence [11; 12], without being affected by cracking and depolarization phenomena as piezoelectrics are [13]. Nevertheless, they have peculiar characteristics such as strong non-linearity and hysteretic behavior, and the harvested energy depends on magnetic bias and mechanical pre-stress, so suitable modeling must be adopted [14; 15; 16]. These characteristics lead to the need for a careful modeling in order to exploit them to the fullest, as reported in [17]. Figure 1 shows a 2D sketch of a force-driven harvester device based on magnetostrictive material. By means of the Villari effect, time-varying mechanical stress applied to the top creates a flux density variation into the material. Then, by exploiting [PERSON]'s law, the device produces an output voltage (\(V_{1}\)) across an electric load through a coil wounded around the magnetostrictive material [18]. Permanent magnets and an iron path provide the magnetic bias to increase the converted energy. Moreover, the iron is exploited also as a structural frame. PEIs play a key role in EH systems, in particular for Vibration Energy Harvesting ones. Indeed, the total power amounts recovered in energy harvesting conversion mechanisms are typically low, even below mW, while output voltages can be far below 1 V [19; 20; 21]. Furthermore, the energy source for the EH (vibrations in this case) is typically quite random in nature, and therefore, the need for a custom PEI is often compulsory [22]. The latter aims to increase the harvested energy and the available voltage output [23], with a consequent efficiency improvement. Indeed, depending on the nature of the energy source, the output voltage of the magnetostrictive EH device could be impulsive or, more generally, non-periodic. Consequently, the purpose of any PEI should not only be the rectification and boosting of the voltage, but also self-adaptability with respect to the input voltage, aiming at an optimal conversion efficiency. Then, the system should be able to measure the positive and negative pulses independently and suitably trigger the circuit.