# Optimized Multiport DC/DC Converter for Vehicle Drivetrains: Topology and Design Optimization
## 2 Analysis of Non-Isolated Interleaved DC-DC Converter
### Weight of Inductors.  
The weight of an inductor mainly consists of the weight of copper coil (or winding) and the weight of core. An inductor design is illustrated in Figure 5. The weight of a coil \(W_{coil}\) is dependent on the physical structure of winding (i.e., length, diameter, number of turns, and number of layers) while the weight of the core \(W_{core}\) is dependent on the type of core (i.e., material, shape). The air gap \(g\) is added to prevent saturation in the inductor.  
To determine the weight of a coil \(W_{coil}\) (kg), it is first necessary to calculate the number of turns \(N_{t}\) (turn) as in Equation (6); \(K_{u}(-)\) is utilization factor, \(W_{a}\) (mm\({}^{2}\)) is window area, \(A_{cu}\) (mm\({}^{2}\)) is cross section of the wire:  
\[N_{t}\leq\frac{K_{u}W_{a}}{A_{cu}} \tag{6}\]  
The diameter of the wire conductor \(D_{wire}\) (mm) is calculated as Equation (7), where \(J_{rms}\) (\(\frac{\Lambda}{\text{mm}^{2}}\)) is current density:  
\[D_{wire}=\sqrt{\frac{4}{\pi}\frac{\hat{I_{L}}}{J_{rms}}} \tag{7}\]  
which gives the number of layers \(N_{layer}\) as in Equation (8):  
\[N_{layer}=floor\left(\frac{D_{wire}N_{t}}{c}\right)+1 \tag{8}\]  
where the function \(floor\) (\(X\)) returns the nearest integer less than or equal to \(X\). And \(c\) (mm) is the height window of the core as shown in Figure 5b.  
The number of turns on each layer \(N_{t,\ layer}\) (turn) is then calculated as Equation (9):  
\[N_{t,\ layer}=round\left(\frac{N_{t}}{N_{layer}}\right) \tag{9}\]  
Figure 5: (**a**) Structure of an inductor; (**b**) Geometry parameters of a core.  
where the function \(round(X)\) returns to the nearest integer with \(X\).  
TThe length of the coil \(l_{wire}\) (mm) as in Equation (10) is then based on the number of turns \(N_{t}\), the number of layers \(N_{layer}\) and the geometric parameters of the core (\(a\) and \(d\) in Figure 5b).  
\[l_{wire}=\sum_{m=1}^{N_{layer}}2N_{t,layer}[a+d+D_{wire}(2m-1)] \tag{10}\]  
The weight of the coil \(W_{coll}\) (kg) is finally calculated as in Equation (11), where \(\rho_{cu}\) (8940 \(\frac{\text{kg}}{m^{3}}\)) is the mass density of copper:  
\[W_{coil}=\rho_{cul}l_{wire}A_{cu} \tag{11}\]  
For one port, the total weight of inductors \(W_{\Sigma ind}\) (kg) is based on the number of inductors according to the number of phases \(N_{ph}\), the weight of core \(W_{core}\), the weight of coil \(W_{coil}\). (assumed that the weight of bobbin \(W_{bobbin}\) is constant):  
\[W_{\Sigma ind}=N_{ph}(W_{coil}+W_{core}+W_{bobbin}) \tag{12}\]  
The inductor design is not straightforward. In practical design, the core is selected from available commercial products and the wire is decided by the amplitude of the inductor current. For the sake of optimization, it is necessary to derive the dependence of the inductance value on the design specifications (i.e., input current, switching frequency, output voltage). The main idea is that the designed inductor should guarantee the pre-defined current ripple which is also dependent on the duty cycle and the switching frequency \(f_{sw}\). In addition, the core of the inductor is not saturated. The inductance value \(L\) (\(\mu\)H) can be derived from a second-degree polynomial equation as Equation (13). The detailed derivation steps are explained in the Appendix A.  
\[L^{2}I_{L}^{2}+L\left[\frac{I_{L}(1-D_{max})D_{max}}{f_{sw}}-K_{u}W_{a}I_{w}A_ {c}B_{max}\right]+\left[\frac{V_{o}(1-D_{max})D_{max}}{2f_{sw}}\right]^{2}=0 \tag{13}\]  
From Equation (13), it is possible to calculate the maximum inductance that can be achieved by a given core. It is obvious that out of the two roots of (13), only the real root has a physical meaning.