## 2 Mathematical Foundation of Generalized Algorithm
### Internal Filling of Enclosures  
Internal fillings, such as printed circuit boards, electronic components, cable assemblies, etc., can have a significant effect on the shielding and resonance performances of the enclosure. Several modifications of the equivalent circuit model have been developed to analyze filled enclosures. For example, in [38], a model was proposed to evaluate the SE of a rectangular enclosure populated with dielectric plates replacing the dielectric layers of printed circuit boards. In this equivalent circuit, the enclosure is represented as a set of rectangular waveguide sections with the same dimensions, but different dielectric properties of the materials filling their cross-sections (see Figure 7). Figure 6: Cylindrical enclosure with the aperture at the base. In waveguide sections populated with a dielectric, the characteristic impedance and propagation constant are calculated as:  
\[Z^{\prime}{}_{g}=\frac{Z_{0}}{\sqrt{\hat{\varepsilon}_{r}}\sqrt{1-\left(\frac{ \lambda^{\prime}}{2a}\right)^{2}}}, \tag{31}\]  
\[k^{\prime}{}_{g}=\frac{2\pi}{\lambda^{\prime}}\sqrt{1-\left(\frac{\lambda^{ \prime}}{2a}\right)^{2}}, \tag{32}\]  
where; \(\hat{\varepsilon}_{r}\)= \(\varepsilon^{\prime}-j\varepsilon^{\prime\prime}\) is the complex relative permittivity of this dielectric, and \(\lambda^{\prime}=\lambda/\sqrt{\hat{\varepsilon}_{r}}\). A similar approach was used in [39] to analyze the SE of cylindrical enclosures filled with dielectrics. To calculate \(Z^{\prime}{}_{g}\) and \(k^{\prime}{}_{g}\), the same expressions, (27)-(28), were used, but they had component \(2a\) replaced by \(\lambda_{c\ mm}\), calculated using (26). Another modification of the model from [20] is proposed in [40], in which the value of \(\hat{\varepsilon}_{r}\) in (27)-(28) is replaced by the effective relative permittivity \(\varepsilon_{\mathit{eff}}\). This version allows one to calculate the SE of an enclosure with dielectric structures of an arbitrary shape (but only when the \(TE_{10}\) dominant mode is excited). In [41], the authors proposed a model to evaluate the SE of a rectangular enclosure filled with conductive posts, which can be used to approximate dipole antennas, wires, cable assemblies, etc. In the equivalent circuit, the post is replaced by a T-section formed by impedances \(jx_{1}\) and \(jx_{2}\) (see Figure 8). The values of \(jx_{1}\) and \(jx_{2}\) are calculated as:  
\[x_{2}=-\big{(}a/\lambda_{g}\big{)}\bigg{(}\frac{2\pi r_{0}}{a}\bigg{)}^{2}(1+F _{1})^{-1}, \tag{33}\]  
\[x_{1}=\bigg{(}\frac{a}{2\lambda_{g}}\bigg{)}\big{(}S^{\prime}-F^{\prime}{}_{0 }-F^{\prime\prime}{}_{0}\big{)}-\frac{x_{2}}{2}, \tag{34}\]  
Figure 8: Enclosure filled with a conductive post (**a**), equivalent circuit for calculating SE (**b**). Figure 7: Enclosure populated with dielectric plate (**a**), equivalent circuit for calculating SE (**b**). where; \(r_{0}\) is the radius of the post, \(\lambda_{g}=\lambda/\sqrt{1-\left(\lambda/2a\right)^{2}}\) is the wavelength in the waveguide, and the expressions for calculating \(F_{1}\), \(S^{\prime}\), \(F^{\prime}\), and F\({}^{\prime\prime}\) are not given in this paper to save space and to preserve narrative integrity. In [42], the model was developed to evaluate the SE of an enclosure filled with conductive plates. This model can be used to approximate the conductive layers of printed circuit boards. The plate is considered as a capacitive waveguide diaphragm that fills the entire width of the enclosure (Figure 9a). In the equivalent circuit, the plate is replaced by the impedance \(Z_{cp}\) (see Figure 9b), which is defined as:  
\[Z_{cp}=jB^{-1}, \tag{35}\]  
where; \(B\) is the susceptance of a capacitive diaphragm. Several expressions for calculating the \(B\) values, depending on the plate height \(h\), are given in [42].