## 3 Required EMI filter Attenuation in Interleaving Units Using Conventional Phase Shift  
The interleaved boost PFC and the related advantages have been reported in the past literature [1]. Here, Figure 1 illustrates the block-diagram of the test system, including two interleaved boost PFCs, EMI receiver, LISN and EMI filter, and the boost-inductor design for continuous conduction mode (CCM) [16; 17] operation has already provided in the literature. So, for the simplicity, only the widely used equations are specified in this subsection, and the boost inductor size for CCM operations is obtained by (5) assuming 22% input ripple current according to [18]:  
\[L_{boost}=\quad U_{dc}/4.\Delta i_{L,\text{max}}.f_{sw} \tag{5}\]  
The value of parameters for a single-phase PFC are summarized in Table 1. The parameters are used to define the filter's attenuation requirements with respect to various switching frequencies in CCM. So, inductor size, as presented in Table 2, can be obtained via (5) for the different case studies. Thus, the maximum peak values of the spectrum (\(U_{max}\)) for various frequency switchings are achieved based on PLECS simulations and Equation (4). Figure 4 illustrates the phase shift implementation between the two units with phase shift 180\({}^{\circ}\) in order to decrease input ripple currents on the boost stages. Notably, as shown in Figure 4, selecting the proper phase shift may affect the ripple of input current. Providing numerous simulation case studies with different phase shifts and switching frequencies is a time-consuming and complicated task at the system-level analysis. In order to alleviate the computational burden/time, a new analytical estimation is proposed in Section 5. In this section, the interleaving technique is evaluated to get the optimal design of DM EMI filter. So, up to four interleaved units have been working at different switching frequencies in Band A and Band B to get the connection between the attenuation requirement considering the interleaving and phase shifting. So, the conventional phase shift of 360\({}^{\circ}\)/N (N is the number of the interleaved converters) is employed among the interleaved units. Interestingly, the first noise peak value erects at the switching frequency of \(f_{sw}\) in Band B. By interleaving task, the equivalent switching frequency is equal to N \(f_{sw}\), where N denotes the number of the interleaved units. More details about the impact of the interleaving task on ripple current has been reported in [1]. Figure 5a depicts the connection between the attenuation requirement and the switching frequency up to four units for CCM mode in Band A. As the equivalent switching frequency appears in \(N\,f_{sw}\), thus there is no need to employ a filter if the switching frequency is considered higher than 75 kHz in two units, 50 kHz in three units, and 37.5 kHz in four units. Further, Figure 5b depicts the filter corner frequency (\(f_{c}\)) via different switching frequencies for a two-stage DM EMI filter in Band A subjected to (1)-(2). Notably, as depicted in Figure 5c,d, the filter corner frequency increases while the filter attenuation requirement decreases at a specific switching frequency range (30-37.5 kHz, 50-75 kHz,  
Figure 3: LISN per-phase circuit diagram recommendation by CISPR-16 for 9–500 kHz [9; 10]. and >150 kHz) in two units interleaved. For example, in order to select switching frequencies of 35 kHz and 30 kHz in Band B, filter design frequencies appear at 175 kHz and 150 kHz, respectively (fifth carrier harmonics occur above 150 kHz). Hence, a case study with 175 kHz compared with 150 kHz obtains a smaller component size and higher filter corner frequency in the same filter attenuation requirement. Thus, switching at the aforementioned critical frequencies and utilizing a switching frequency lower than them is not highly efficient. Because this increases the filter corner frequency without affecting the boost inductor size. In addition, using the interleaving technique leads to decreased input ripple currents.