## 4 Wideband Energy-Management Circuit
### Proposed Energy-Management Circuit  
The structure of the adopted energy-management circuit for an electromagnetic converter is shown in Figure 8. It consists of a VM associated with a wideband DC-DC converter that charges the super capacitor. Besides, an Under-Voltage Lock-Out (UVLO) circuit is associated with the energy-management circuit to activate the WSN when the super capacitor is full of charge. The rectification stage aims to transform the AC signal of the EM to a DC signal. The rectification stage reduces the output voltage according to the properties and numbers of the used diode. Generally, the rectification stage is associated with a DC-DC converter to ensure a constant output voltage to the super capacitor. The used EM converter generates a maximum open voltage of around 4 V peak-to-peak, with an acceleration of 0.2 g and a frequency of 13 Hz (Figure 7). The main requirement of the proposed energy-management circuit is to charge the super capacitor for different input voltages with a range between 2 V peak-to-peak to 4 V peak-to-peak, where three main possible approaches can be suggested. In the first approach, the use of a bridge rectifier is promoted to reduce the DC-DC converter input voltage of 2V drop related to the used diodes. In this case, the DC-DC converter should be able to provide a low input voltage, where their efficiency and time to start-up are critical. In the second approach, wide-band converters are used, which are composed of a combination of dual DC-DC converters with specific control circuits, resulting in high-costand complex circuits. Besides, the use of switches with a higher inner resistance causes energy losses. Another approach that relies on this is to use VM circuits in conjunction with AC signals to achieve higher DC-DC converter input voltages. Through this approach, a DC-DC converter based on buck architecture can reduce the cold-start phase and charge the super capacitor within a wide band of acceleration. Various commercial DC-DC converters have been developed and used for energy harvesting with vibration converters; LTC3588-1 was selected, with an efficiency of 82% [28] for energy storage in the super capacitor. It presents a wideband hysteresis voltage supervisor like that of buck converters. As the super capacitor is being charged, the LTC3588-1 maintains a high impedance state to minimize leakage current during the energy-conversion phase. The output voltage of the used DC-DC converter is defined as 3.3 V, which presents the maximum possible supply voltage of the WSN. Various VM architectures have been presented and compared in the literature [28], including Dickson and Cockcroft-Walton architectures. Both architectures can reach the main required output voltage. However, the Dickson architecture is more recommended for low-power applications due to its small capacitor stage. The Dickson VM is composed of dual diodes and capacitors on each stage, as shown in Figure 8. Typically, with an input voltage (\(V_{in}\)) of 0.65 V, the VM output voltage (\(V_{multiplier}\)) based on three stages becomes around 4 V. However, as shown in Figure 9, in this case with real implementation, the output voltage of the three stages of the voltage multiplier is around 2.1 V. In fact, in Figure 9, for an acceleration of 0.15 g, the input signal presents a maximal amplitude of 0.65 V, expressed by the red color, where stages 1, 2, and 3 of the VM are expressed by blue, green, and purple colors, respectively. The VM voltage amplitudes are 0.75 V, 1.3 V, and 2.1 V for stages 1, 2, and 3, respectively. This voltage drop (\(\Delta(n)\)) is related to various parameters, as shown in Equations (1) and (2), such as the losses due to the used capacitors (\(V_{dcap,L}\)) and diodes (\(V_{diode,L}\)), where the error related to the working frequency as well as the number of VM stages (\(n\)).