## 2 Materials and Methods
### Proposed DC/DC Converter
#### 2.2.1 Coupled Inductor Analysis and Implementation  
A power inductor is composed of a winding and a magnetic core [19; 20], as shown in Figure 5. The winding is typically made up of round Litz copper wire, which can pass a high current at a given voltage. The role of the magnetic core is to channel the magnetic field lines in order to transfer or to store energy. Materials with high permeability, supporting a significant induction field without saturation and with low losses at the target switching frequency are required. In our application, the coupled inductor is made of the magnetic core ETD 49 [21] and a copper winding. Inductor losses occur from two sources, namely the losses in the core and the losses generated in the windings. Winding losses are due to the wire resistance, \(R\), and are evaluated by [PERSON]'s law \(P_{loss\_avg}=Ri^{2}\). In this study ohmic winding losses are not considered and the focus is mainly on the core losses. Based on theory and empirical tests [16; 17; 18; 19; 20] core losses depend directly on the hysteresis loop and to the maximum flux, \(\mathrm{B_{max}}\). Consequently, we must avoid flux saturation on the core to avoid losses. The hysteresis characteristic of an inductor is presented in Figure 6. Depending on the inductor core size, it is known that in \(B=f(H)\) characteristics, the linear region can be long or short. When the core size increases, the achieved maximum current in the linear region increases as well. To avoid a complicated theoretical approach, we reduce the hysteresis loop to a linear relationship between flux, \(B\), and magnetic field, \(H\). In the conventional power converters, under all operating conditions of the PVG, the inductor operation point needs to meet the linear region of the \(B\)-\(H\) curve [17; 18; 19; 20]. Figure 5: Basic ferromagnetic circuit. The sensing of the PV current in Figure 6 allows us to know the operating point position of the inductor on the hysteresis characteristic. Indeed, since the total flux is given by [17; 18; 19; 20]  
\[\varphi_{T}=\frac{N^{2}i}{\mathcal{R}} \tag{1}\]  
where \(\mathcal{R}\) is the reluctance of the magnetic core, \(N\) is the coil number of turns, and the inductance, \(L\), is defined as the ratio between the total flux, \(\varphi_{T}\), and the current, \(i\) flowing in the winding  
\[L=\frac{\varphi_{T}}{i} \tag{2}\]  
The flux is therefore proportional to the current that flows into the inductor. Consequently, the inductor core is sized to function in the linear region where the maximum dissipated current into the inductor does not exceed one third (1/3) of the nominal PVG current. This means that the inductor could operate in the saturation zone when the PVG current increases, as shown in Figure 6.  
To avoid saturation, we propose to add a second coil in order to produce an additional flux in the magnetic core. The latter can be added or subtracted from the first one, depending on the direction of the second flux. The basic model of a coupled inductor is shown in Figure 7. The coil \(L_{1}\) traversed by the current \(i_{1}\) produces the flux \(\varphi_{11}\) self-induction through each turn of \(L_{1}\) and the mutual flux \(\varphi_{21}\) through each turn of \(L_{2}\). Similarly, the coil \(L_{2}\) traversed by the current \(i_{2}\) produces the self-induction flux \(\varphi_{22}\) through each turn of \(L_{2}\) and the mutual flux \(\varphi_{12}\) through each turn of \(L_{1}\). Fluxes \(\varphi_{11}\), \(\varphi_{21}\) and fluxes \(\varphi_{21}\), \(\varphi_{22}\) are added to each part (Figure 7b), or subtracted (Figure 7c), according to the positive direction of currents chosen arbitrarily, and the winding of the coils direction [17; 18; 19; 20]. Figure 6: Conventional boost inductor current and corresponding operating point on the \(B\)-\(H\) curve. Assuming that the flux coil is the same for each turn, the total flux is given by  
\[\left\{