## 2 Common-Mode Conducted Emissions of Power Switching Circuits  
The source and the propagation path of the CM EMI delivered by a switching circuit are discussed in this Section. The analysis is carried out referring to the circuit shown in Figure 2. The EUT comprises input capacitors represented by the \(Z_{\mathrm{in}}\) impedance, ideal high-side (\(S_{\mathrm{HS}}\)) and low-side (\(S_{\mathrm{LS}}\)) switches, which are connected to the output load impedance \(Z_{\mathrm{load}}\) through the inductor \(L_{\mathrm{pow}}\). In order to simplify the analysis of the CM EMI, the EUT terminals are loaded with the LISNs impedance, which can be approximated by a \(R_{\mathrm{LISN}}=50\,\Omega\) resistance in the frequency range of interest. Finally, the parasitic capacitances that load the EUT nodes toward the reference ground plane are indicated with dashed lines to be distinguished from the other components. More precisely, \(C_{\mathrm{p,lg}}\) connects the negative input terminal (node LG) of the EUT, \(C_{\mathrm{p,ps}}\) the positive input terminal (PS), \(C_{\mathrm{p,sw}}\) the switching node sw and \(C_{\mathrm{p,l}}\) the load to the ground plane. Moreover, the CM current (\(i_{\mathrm{CM}}\)), the DM current (\(i_{\mathrm{DM}}\)), the switching current (\(i_{\mathrm{sw}}\)), and the voltage across the low-side switch (\(v_{\mathrm{sw}}\)), which have been previously defined on Figure 1, are depicted too. As previously mentioned, the DM current is mainly due to the switching current \(i_{\text{sw}}\), whereas \(i_{\text{CM}}\) to the switching voltage \(v_{\text{sw}}\), resulting in a current to flow in \(\text{C}_{\text{p,sw}}\). In order to simplify further the model shown in Figure 2 for the analysis of the CM EMI, the DM current is not taken into account, i.e., \(i_{\text{DM}}=0\). This condition is valid only if \(Z_{\text{in}}=0\), meaning that the switching current flows completely in the input capacitors. As far as the CM EMI is concerned, the node PS is short-circuited to the node LG and the impedance shown by the two LISNs is equal to \(R_{\text{CM}}=25\,\Omega\). Under these assumptions, the only current loop involved in the CM EMI is the one that comprises the LISN impedances, the low-side switch and the capacitances \(\text{C}_{\text{p,sw}},\text{C}_{\text{p,ps}},\text{C}_{\text{p,lg}}\). Such a loop is redrawn in Figure 3, where the noise source \(v_{\text{sw}}\), the capacitive coupling \(\text{C}_{\text{p,sw}}\) and the EMI receiver are clearly identified. By definition, the CM EMI voltage \(v_{\text{CM}}\) is equal to  
\[v_{\text{CM}}=\frac{v_{\text{LISN,p}}+v_{\text{LISN,n}}}{2}, \tag{1}\]  
where \(v_{\text{LISN,p}}\) and \(v_{\text{LISN,n}}\) are the voltages across the measurement ports of the two LISNs, as shown in Figure 2. The frequency spectrum of \(v_{\text{CM}}\), denoted as \(V_{\text{CM}}(j\omega)\) in what follows, is related to the frequency spectrum of the switching voltage \(V_{\text{sw}}(j\omega)\) according to  
\[V_{\text{CM}}(j\omega)=\frac{j\omega\text{C}_{\text{p,sw}}R_{\text{CM}}}{1+j \omega R_{\text{CM}}(\text{C}_{\text{p,sw}}+\text{C}_{\text{p,lg}}+\text{C}_ {\text{p,ps}})}V_{\text{sw}}(j\omega). \tag{2}\]  
Typically it is \(\text{C}_{\text{p,sw}}<<\text{C}_{\text{p,lg}}+\text{C}_{\text{p,ps}}\), thus the magnitude of the CM transfer function (\(|TF|\)) can be approximated by its numerator for \(\omega<(R_{\text{CM}}(\text{C}_{\text{p,lg}}+\text{C}_{\text{p,ps}}))^{-1}\), meaning that \(\text{C}_{\text{p,sw}}\) is the only capacitance that contributes to the CM EMI in the frequency range of interest. Equation (2) ideally allows one to predict the CM EMI provided that the frequency spectrum of \(V_{\text{sw}}(j\omega)\) and the values of the parasitic capacitances are known. An upper bound for the \(v_{\text{sw}}\) frequency spectrum is provided by the trapezoidal approximation, whereas, the value of \(\text{C}_{\text{p,sw}}\) can be either extracted by analytical formula if the geometry is simple, as in [1], or by using finite element simulators [18].