## 2 Formulation of the Problem  
The reference scenario is a reverberation chamber made by a rectangular cavity having dimensions \(a\) along x direction, \(b\) along y direction, and \(c\) along z direction, as shown in Figure 1. The walls of the chamber have a finite conductivity that accounts for all the loss mechanisms, and its value is chosen to have the same quality factor obtained by measurements [13]. A set of \(N_{S}\) monopoles is placed on the RC's walls to achieve the field sampling. The EUT is placed into the working volume of the RC. The application of the equivalence theorem involves the use of a closed geometrical surface (equivalence surface) surrounding the radiating object. In our case, it is a rectangular surface whose faces are parallel to the RC's walls, and it is the geometrical support where the electric (\(\overset{\rightarrow}{Ieq}\)) and magnetic (\(\overset{\rightarrow}{Meq}\)) equivalent currents, which characterize the EUT radiation, flow. No particular constraint limits the choice of the equivalent surface, whose shape is due to practical reasons. The equivalent current distribution is discretized by introducing a proper discretization of the equivalence surface: each face is divided into a set of small rectangular sub-areas. If each sub-area is small enough, with respect to the wavelength, the current distribution on each sub-area can be considered constant. For example, Figure 2 shows a rectangular face of the equivalence surface orthogonal to the x axis, and the i-th sub-area (S) having dimensions \((2\Delta_{y},2\Delta_{z})\), centered in \((x_{H},y_{H},z_{H})\). In this case, the equivalent magnetic and electric currents are Equation (1)  
\[\left\{ \begin{array}{ll}\overline{M_{eq}^{i}}=M_{eqy}^{i}\hat{y}+M_{eqz}^{i} \hat{z}&\\ \overline{j_{eq}^{i}}=J_{eqy}^{i}\hat{y}+J_{eqz}^{i}\hat{z}&\end{array} \right. i=1,\ldots,N_{eq}^{(x)} \tag{1}\]  
where \(N_{eq}^{(x)}\) is the total number of sub-areas after the discretization of the considered face. Equation (1) clearly shows that, for each sub-area, four current components must be determined to characterize the EUT emission. Similar expressions can be easily obtained for the other faces. Figure 1: The scenario: an RC, its dimensions, the working volume, and the equivalence surface surrounding the EUT. Figure 2: Sub-surface element in the face orthogonal to x axis and its coordinates and dimensions. The electric field radiated by the equivalent currents in a rectangular cavity can be calculated using a modal expansion [18], including irrotational and TE and TM solenoidal fields (2). \[\overline{E}=\overline{E}_{IRR}+\overline{E}_{TE}+\overline{E}_{TM} \tag{2}\]  
The irrotational component of the electric field is (3)  
\[\overline{E}_{IRR}=-\frac{1}{j\omega\epsilon}\sum_{m,n,p}\biggl{(}\int_{S} \overline{J}_{\alpha\eta}\cdot\overline{J}_{m,n,p}dS\biggr{)}\overline{J}_{m,n,p} \tag{3}\]  
and the solenoidal components are (4):  
\[\overline{E}_{TE,TM}=\sum_{m,n,p}\left[\frac{\int_{S}\overline{M}_{\alpha\eta} \cdot\overline{h}_{m,n,p}^{TE,TM}dS-j\omega\mu\int_{S}\overline{J}_{\alpha\eta }\cdot\overline{g}_{m,n,p}^{TE,TM}dS}{k_{m,n,p}^{2}-\beta_{TE,TM}^{2}}\right] \overline{e}_{m,n,p}^{TE,TM}, \tag{4}\]  
where  
\[\beta_{TE,TM}^{2}=\beta^{2}\Biggl{[}1-(-1+j)\frac{\omega_{n,m,p}}{\omega Q_{m,n,p}^{TE,TM}}\Biggr{]}, \tag{5}\]  
\(\overline{J}_{m,n,p}\) and \(\overline{e}_{m,n,p}^{TE,TM}\) are irrotational and TE and TM are divergenceless electric eigenvectors, respectively, whereas \(\overline{h}_{m,n,p}^{TE,TM}\) are the TE and TM divergenceless magnetic eigenvectors. \(k_{m,n,p}\), \(Q_{m,n,p}^{TE,TM}\), and \(\omega_{n,m,p}\) are the corresponding eigenvalues, quality factors, and resonant angular frequencies, respectively, and \(\beta^{2}=\omega^{2}\mu\epsilon\). The \(Q_{m,n,p}^{TE,TM}\) is the quality factor of each mode, and it is calculated considering an equivalent wall conductivity value retrieved from real chamber quality factor measurements [13; 15], thus accounting for all loss mechanisms inside the real RC.