## 6 Filter Volume Optimization  
In this section, the primary purpose is to optimize the EMI filter size considering the selected proper phase shift. To investigate the efficiency of the proposed method, EMI filter volume can be calculated based on [15]. As mentioned above, the symmetrical two-stage EMI filter [18], shown in Figure 2, has been considered for this paper's analysis. Hence, the EMI filter capacitor size can be obtained from (19) [15]:  
\[V_{\text{C}}=k_{\text{C1}}C_{DM}u_{g}^{2}+k_{\text{C2}} \tag{19}\]  
where the factor \(k_{\text{C1}}\) denotes the capacitor volume proportionality to the stored energy and \(k_{\text{C2}}\) denotes a voltage-dependent factor. Furthermore, the inductor size is obtained by (20) [15]:  
\[V_{L}=k_{L1}L_{DM}l_{g}^{2}+k_{L2}L_{DM}+k_{L3}l_{g} \tag{20}\]  
\(k_{L1}\) is a constant factor describing the proportionality between the stored energy \(E_{L}=1/2\cdot L_{DM}\cdot{I_{g}}^{2}\) and the inductor volume. These factors can be derived analogously to \(k_{L1}\), \(k_{L2}\), and \(k_{L3}\) from the manufacturer's data using Magnetics toroid cores [10], which is present in Table 1. So, the total volume of the two-stage symmetrical EMI filter is calculated from (21):  
\[V_{tot}=2(n_{f}+1)V_{\text{L}}+n_{f}V_{\text{C}}\rightarrow\text{min} \tag{21}\]  
where \(n_{f}\) is the number of filter stages [9; 15; 18]. Solving (19)-(21) and (2) results in optimum filter component parameters for a certain number of filter stages \(n_{f}\). To simplify the calculation analysis, (2) is simplified for two-stage EMI filter as  
\[A_{tt_{\text{avg}}}(f)=(j2\pi f)^{2n_{f}}.(2L_{DM})^{n_{f}}.C_{DM}^{n_{f}} \tag{22}\]  
Figure 9: Input ripple current of the DC capacitor on the basis of the number of the interleaved converters with \(f_{\text{avg}}=37.5\) kHz, \(P_{0}=1\) kW, and \(L_{\text{boost}}=1\) mH based on PLECS software simulations. The conventional phase shift is considered \(360^{\circ}\)/N and unconventional is selected based on the (17) and (18). Finally, EMI filter components are calculated by  
\[C_{DM}=\sqrt{\frac{(n_{f}+1)(k_{L1}.I_{g}^{2}+k_{L2}).\ ``\sqrt{Att_{req,DM}}}{2n_{f}.k_{C1}.u_{g}^{2}.(2\pi f_{D})^{2}}} \tag{23}\]  
\[L_{DM}=\sqrt{\frac{n_{f}k_{C1}.u_{g}^{2}.\sqrt[2]{Att_{req,DM}}}{2(n_{f}+1).(k_{ L1}.I_{g}^{2}+k_{L2}).(2\pi f_{D})^{2}}} \tag{24}\]  
Finally, Figure 10 shows a flowchart demonstrating the steps to design the optimal filter volume based on the analytical EMI noise estimation approach. Further, as it is clear from Figure 8, the relation (17)-(18) does cover some unconventional phase angles that filter required attenuation minimized from the analytical method. On the other hand, there are other points for minimization filter required attenuation. The green dash line in Figure 10 shows the general method for phase shift selection based on the analytical estimation. Moreover, (17)-(18) can use instead of the general unconventional phase-shifting method to improve computation time. This flowchart is mainly applied to calculate the EMI filter component with only a few equations considering proper phase shifts with unconventional phase shift or analytical estimation. Hence, Figure 11 shows optimal DM EMI filter boxed-volume approximation, including conventional and unconventional phase shifts in Band-B based on Table 3 and Figure 10. As previously mentioned, the unconventional phase shift efficiency in a decreased EMI filter size is obtained from Figure 11. Here, two cases are presented with different phase shifts, including conventional as 180\({}^{\circ}\) and unconventional as 45\({}^{\circ}\) selected from Table 3 based on the \(f_{sw}\) = 37.5 kHz for two units interleaved. \[V_{L}=k_{L1}L_{DM}l_{g}^{2}+k_{L2}L_{DM}+k_{L3}l_{g} \tag{20}\]0 Flow chart of DM EMI filter boxed-volume optimization. Table 4 provides the outcome of the two case studies, including the attenuation requirement and corner frequency.