## 5 Conclusions
### Electric Field Generated by a Sub-Surface Orthogonal to z Axis  
Let us consider the rectangular faces of the equivalence surface orthogonal to the z axis, and one of its sub-surfaces (S) having dimensions (\(2\Delta_{x},2\Delta_{y}\)), and centered in (\(x_{H}\),\(y_{H}\),\(z_{H}\)).  
In this case, the equivalent magnetic and electric currents are (A5):  
\[\left\{\begin{array}{ll}\overline{M_{eq}^{i}}=M_{eqx}^{i}\hat{x}+M_{eqy}^{i }\hat{y}&\\ \overline{j_{eq}^{i}}=J_{eqx}^{i}\hat{x}+J_{eqy}^{i}\hat{y}&i=1,\ldots,N_{eq}^ {(z)}\end{array}\right. i=1,\ldots,N_{eq}^{(z)} \tag{12}\]where \(N_{eq}^{(z)}\) is the total number of sub-areas after the discretization of the considered face.  
The electric field radiated by the equivalent currents in a generic point \(P(x,y,z)\) is reported in (A6)-(A8).  
\[\overline{E}_{IRR}(x,y,z)= -\frac{1}{\mu\omega}\frac{32}{abc}\sum\limits_{m,n,p}\frac{1}{p^{k_{ n},\mu,p}}\sin(k_{x}\Delta_{x})\sin(k_{y}\Delta_{y})\sin(k_{z}z_{H})\left[\frac{i _{\text{erg}}^{i}\sin(k_{x}x_{H})\cos(k_{y}y_{H})}{k_{x}}\right.\] \[\left.-\frac{i_{\text{erg}}^{i}\cos(k_{x}x_{H})\sin(k_{y}y_{H})}{k _{x}}\right]\left\{\begin{array}{l}k_{x}\cos(k_{x}x)\sin(k_{y}y)\sin(k_{z}z) \hat{x}\\ k_{y}\sin(k_{x}x)\cos(k_{y}y)\sin(k_{z}z)\hat{y}\\ k_{z}\sin(k_{x}x)\sin(k_{y}y)\cos(k_{z}z)\hat{z}\end{array}\right.,\] (A6) \[\overline{E}_{TE}(x,y,z)=\frac{32}{abc_{m}\omega_{n}}\sum \limits_{m,p}\frac{1}{k_{x}\omega_{n}-\beta_{TE}^{-1}}\sin(k_{x}\Delta_{x}) \sin(k_{y}\Delta_{y})\] \[\left\{\cos(k_{z}z_{H})\left[\frac{M_{eqx}^{i}\cos(k_{x}x_{H}) \sin(k_{y}y_{H})}{k_{x}}-\frac{M_{eqy}^{i}\sin(k_{x}x_{H})\cos(k_{y}y_{H})}{ k_{y}}\right]\right.\] \[\left.-j\omega\mu\sin(k_{z}z_{H})\left[\frac{i_{\text{erg}}^{i} \sin(k_{x}x_{H})\cos(k_{y}y_{H})}{k_{y}}\right.\right.\] (A7) \[\left.\left.+\frac{i_{\text{erg}}^{i}\cos(k_{x}x_{H})\sin(k_{y}y_{ H})}{k_{x}}\right]\right\}\left\{\begin{array}{l}-k_{y}\cos(k_{x}x)\sin(k_{y}y) \sin(k_{z}z)\hat{x}\\ k_{x}\sin(k_{x}x)\cos(k_{y}y)\sin(k_{z}z)\hat{y}\end{array}\right.\] \[\overline{E}_{TM}(x,y,z)= \frac{32}{abc_{m}}\sum\limits_{m,n,p}\frac{1}{k_{x}^{2}\alpha_{n,p}-\beta_{TM}^{-1}}\sin(k_{x}\Delta_{x})\sin(k_{y}\Delta_{y})\left\{-\cos(k_{ z}z_{H})\left[\frac{M_{eqx}^{i}\cos(k_{x}x_{H})\sin(k_{y}y_{H})}{k_{y}}\right.\right.\] (A8) \[\left.+\frac{M_{eqy}^{i}\sin(k_{x}x_{H})\cos(k_{y}y_{H})}{k_{y}} \right]\] \[-j\omega\mu\frac{k_{x}}{k_{x,n,p}}\sin(k_{y}y_{H})\left[-\frac{i_{ \text{erg}}^{i}\sin(k_{x}x_{H})\cos(k_{z}z_{H})}{k_{x}}\right.\] \[\left.\left.+\frac{i_{\text{erg}}^{i}\cos(k_{x}x_{H})\sin(k_{z}z_{H })}{k_{y}}\right]\right\}\left\{\begin{array}{l}-\frac{k_{x}k_{x}}{k_{x}^{2}} \cos(k_{x}x)\sin(k_{y}y)\sin(k_{z}z)\hat{x}\\ -\frac{k_{y}k_{x}}{k_{x}^{2}}\sin(k_{x}x)\cos(k_{y}y)\sin(k_{z}z)\hat{y}\\ \sin(k_{x}x)\sin(k_{y}y)\cos(k_{z}z)\hat{z}\end{array}\right.,\]  
Where \(\beta_{TE,TM}^{2}\)= \(\beta^{2}[1-1+j)\frac{\omega_{n,m,p}}{\omega Q_{m,n,p}^{TE,TM}}\hat{\beta}_{ k}=\left\{\begin{array}{ll}1,&k\
eq 0\\ 2,&k=0\end{array}\right.,\)\(k_{x}=\frac{m\pi}{a},k_{y}=\frac{n\pi}{b},k_{x}=\frac{p\pi}{c}.\)  
\(\overline{f}_{m,n,p}\) and \(\overline{g}_{m,n,p}^{TE,TM}\) 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 \(\beta^{2}=\omega^{2}\mu e.\) (_Rev_ 2.10).