## 3 Complementary Model Description
### Setting the EMI Constraints  
As mentioned above, a clear design frame must be defined in order to take into account the impact of frequency and inductor ripple magnitude on conducted EMI. For this purpose, it is defined that the output ripple current going out from the converter must not exceed some limits. Taking into consideration conducted EMI regulations [26; 27], in a tentative to make it clear and simple to use in the inductor value selection process, it has been defined that the current harmonics are going out of the power converter, flowing into the two 50\(\Omega\) resistors of a conducted EMI testing equipment, should not exceed 60 dBuV no matter the frequency considered. These choices are slightly different from the regulations. Nonetheless, they are offering a realistic reference that eases the design and optimization process. Figure 12 below presents the schematic of a typical differential mode (DM) input filter circuit inserted between the converter itself and the LISN (Line Impedance Stabilization Network) testing equipment considered for conducted EMI compliance.  
In this schematic, a traditional differential mode filter topology is considered, with symmetric input inductors \(L\) (one on the phase and one on the neutral) together with Cx1 and Cx2 capacitors for DM current ripple filtering, between the converter and the LISN, used for normative tests. In the upper-frequency range, above 100 kHz, the LISN looks similar to a set of two 50 \(\Omega\) resistors, from which, one can measure the current flowing through. Setting the limit at 60 dBuV across each LISN resistor means that the maximum current ripple harmonic must not exceed 2 uA. If the input inductor current ripple magnitude is higher, the \(C_{x1}\) and \(C_{x2}\) capacitors will have to be adjusted to filter the current harmonics. If the \(C_{x}\) capacitor value exceeds 20 uF, extra \(L_{filter}\) filtering inductors will have to be inserted between \(C_{x1}\) and \(C_{x2}\) in order to increase the global filter order as shown in Figure 13. A limit for \(C_{x}\) has been introduced because large \(C_{x}\) capacitors have low resonant frequencies, preventing them from a satisfactory filtering over a large frequency range.  
The Equation (13) below presents the ideal attenuation with respect to the frequency offered by a \(C_{x}\) capacitor filter topology, considering \(C_{x}\) equals \(C_{x1}+C_{x2}\).  
\[\frac{V_{Isin}}{I_{L}}=\frac{(R_{Isin_{100}}/2)}{1+Cx\times R_{Isin_{100}} \times j\omega} \tag{13}\]  
Figure 12: Schematic of a VSI + its differential mode input filter for conducted EMI compliance + the LISN resistors (LISN simplified representation in high frequency range).  
Please note that the LISN total resistor is 100 \(\Omega\), the sum of two 50 \(\Omega\) resistors. However, the probe is inserted across one of the two resistors, justifying the factor 2 in Equation (13) above. Also, in Equations (13) and (14), \(I_{L}\) is the current flowing in the inductors of the chopper.  
Equation (14) below presents the ideal attenuation with respect to the frequency offered by a \(C_{x1}-L_{filter}-C_{x2}\) filter topology. The operation to select the minimum inductor value for this filter is described in the following Section 3.3.2.  
\[\frac{V_{isin}}{I_{L}}=\frac{(R_{lsin_{100}}/2)}{1+j\omega\big{(}R_{lsin_{100} }C_{x1}+R_{lsin_{100}}C_{x2}\big{)}-\omega^{2}\big{(}L_{filter}C_{x2}\big{)}-j \omega^{3}\big{(}R_{lsin_{100}}L_{filter}C_{x1}C_{x2}\big{)}} \tag{14}\]