## 4 Wideband Energy-Management Circuit
### Proposed Energy-Management Circuit
\[V_{multiplier}=2 nV_{in}-\Delta(n), \tag{1}\]  
\[\Delta(n)=V_{dcap,L}(2n)+V_{diode,L}(2n)+\delta_{o}(n), \tag{2}\]  
\[V_{ripple}=\frac{I_{o}(n^{2}+\frac{n}{2})}{8 fC_{VM}} \tag{3}\]  
As shown in Figure 9, the output voltage of the voltage multiplier presents some ripples in the first stages. These ripples are related mainly to the selected VM capacitors (\(C_{VM}\)). Typically, the VM capacitors should be selected according to the electromagnetic converter output current (\(I_{o}\)), the number of the VM stages, as well as the working frequency (\(f\)), as shown in Equation (3). By increasing the VM stages, the VM capacitors' values are required to be larger in terms of capacity (Equation (3)). In reality, increasing the value of  
Figure 8: Schematic of the proposed energy-management circuit for the electromagnetic converter. the VM capacitors does not only reduce the voltage ripple (\(V_{ripple}\)) in the output voltage, but also increases their time of charge and the super capacitor. For this, in this paper, smaller capacitor values have been adopted in the first two VM stages of about 1 uF, while stage numbers 3, 4, and 5 use capacitors with 10 uF of capacity. The last voltage multiplier stage uses a capacitor of 20 uF, and the capacitor connected to the DC-DC input (\(C_{Vit}\)) has 1000 uF to generate a current pick to charge the super capacitor for a longer time (Figure 8). With these values, the activation of the WSN in a proper way is guaranteed. When the WSN is connected to the super capacitor, some current losses flow to supply the node properly, which reduce the time to charge of the super capacitor. For this, disconnecting the load from the super capacitor during the charging process is required for fast charging. This can be achieved through the utilization of Under-Voltage Lock-Out (UVLO) circuits [29; 30] designed to use hysteresis comparator-based circuits and MOSFET-based circuits [31]. By using active components, such as comparators, switches, or even detectors, they increase the circuit cost and consumption, which makes the circuit with a singular MOSFET more recommended. Figure 8 shows the proposed circuit to connect the WSN and the super capacitor when their voltage is between 3 V and 3.3 V. The circuit is composed of an N-channel MOSFET connected between the supply circuit ground and the wireless sensor node ground. The proposed architecture does not cause a voltage drop of the super capacitor voltage that supplies the WSN. The MOSFET becomes conductive by a gate voltage higher than 1 V delivered by the \(P_{Good}\) pin of the used DC-DC converter. However, the activation time of the WSN by the \(P_{Good}\) signal becomes high when the super capacitor voltage is about 3 V, which is sometimes not sufficient to transmit the message by the WSN. For this, a higher capacitor in the final stage of the VM circuit is connected to the input of the DC-DC converter that injects a higher current to supply the WSN. One of the major aspects to follow is the sizing of the circuit capacitors, which should not be too big to reduce the time to charge and not too small that they are not able to properly supply the WSN. The size of the super capacitor (\(C_{super-Cap}\)) can be defined analytically based on the explored energy of the used WSN (\(E_{explored}\)), as well as the minimal (\(V_{t}\)) and the maximal (\(V_{i}\)) required input voltage of the WSN, as presented in Equation (4). The explored energy can be defined based on the power consumption (\(P_{consum}\)), the time of activation, and the associated DC-DC converter energy-management circuit power consumption (\(P_{EMcircuit}\)). Besides, it considers the power dissipation due to the super  
Figure 9: Output voltage at different stages: red, EM converter signal; blue, first stage of the VM; green, second stage of the VM; and purple, third stage of the VM outcomes. capacitor discharge (\(P_{self-dis}\)) when the WSN is not activated. For this, the accurate super capacitor size can be explored, as shown in Equation (5).