# On Modeling Antennas Using MoM-Based Algorithms: Wire-Grid versus Surface Triangulation
## 2 Materials and Methods
### General Mathematical Formulation  
Analysis of antenna characteristics based on MoM is carried out by reducing the surface integral equations to system of linear algebraic equations (SLAE). Thus, when an electromagnetic wave incident on a conducting surface, surface electric currents are induced on it, creating a scattered electromagnetic field. Considering that [PERSON]'s equations are linear, the total \(\mathbf{E}^{\mathrm{tot}}\) and \(\mathbf{H}^{\mathrm{tot}}\) fields can be represented as a superposition of incident (\(\mathbf{E}^{\mathrm{inc}}\), \(\mathbf{H}^{\mathrm{inc}}\)) and scattered (\(\mathbf{E}^{\mathrm{scat}}\), \(\mathbf{H}^{\mathrm{scat}}\)) waves. Under the boundary conditions on the surface \(S\) of a perfect electric conductor (PEC), the total tangential components of the electric and magnetic fields are described by the expressions  
\[ \begin{array}{l}\mathbf{n}\times\mathbf{E}^{\mathrm{tot}}(\mathbf{r})\,|_{S }=\mathbf{n}\times(\mathbf{E}^{\mathrm{inc}}(\mathbf{r})+\mathbf{E}^{\mathrm{ scat}}(\mathbf{r}))\,|_{S}=0,\\ \mathbf{n}\times\mathbf{H}^{\mathrm{tot}}(\mathbf{r})\,|_{S}=\mathbf{n}\times (\mathbf{H}^{\mathrm{inc}}(\mathbf{r})+\mathbf{H}^{\mathrm{scat}}(\mathbf{r})) \,|_{S}=\mathbf{J}(\mathbf{r}),\end{array}  \tag{1}\]  
where \(\mathbf{n}\) is the vector of the external normal from the observation point \(\mathbf{r}\). Here, the decomposition into the components of the external incident (\(\mathbf{E}^{\mathrm{inc}}\), \(\mathbf{H}^{\mathrm{inc}}\)) and scattered (\(\mathbf{E}^{\mathrm{scat}}\), \(\mathbf{H}^{\mathrm{scat}}\)) fields is used. The first two are the components that exist in the absence of the object under consideration, and the latter are the components generated by the induced current \(\mathbf{J}(\mathbf{r})\) on the surface of this object. Since the induced magnetic currents \(\mathbf{M}(\mathbf{r})\) on the surface of perfect electric conductors are zero, the scattered electric and magnetic fields can be expressed through the electric vector \(\mathbf{A}(\mathbf{r})\) and scalar \(\mathbf{\Phi}(\mathbf{r})\) potentials as  
\[\mathbf{E}^{\mathrm{scat}}(\mathbf{r})=-j\omega\mathbf{A}(\mathbf{r})-\
abla \Phi(\mathbf{r}), \tag{2}\]  
\[\mathbf{H}^{\mathrm{scat}}(\mathbf{r})=\frac{1}{\mu}\
abla\times\mathbf{A}( \mathbf{r}). \tag{3}\]  
The Green's function of free space (homogeneous isotropic medium) G(\(\mathbf{r}\), \(\mathbf{r}^{\prime}\)), which determines the potential at point \(\mathbf{r}\) generated by a unit point charge at source point \(\mathbf{r}^{\prime}\), is used to later calculate the vector and scalar potentials [40]  
\[\mathrm{G}\big{(}\mathbf{r},\mathbf{r}^{\prime}\big{)}=\frac{e^{-jk|\mathbf{r }-\mathbf{r}^{\prime}|}}{4\pi|\mathbf{r}-\mathbf{r}^{\prime}|}, \tag{4}\]  
where  
\[\mathbf{H}^{\text{scat}}(\mathbf{r})=\int\limits_{S^{\prime}}\
abla G\left( \mathbf{r},\mathbf{r}^{\prime}\right)\
abla^{\prime}\times\mathbf{J}\left( \mathbf{r}^{\prime}\right)dS^{\prime}. \tag{8}\]  
Further substitution of (7) into (1) leads to the Electric Field Integral Equation (EFIE)  
\[\frac{1}{j\omega z_{0}}\mathbf{n}\times\mathbf{E}^{\text{inc}}(\mathbf{r})= \mathbf{n}\times\int\limits_{S^{\prime}}G\left(\mathbf{r},\mathbf{r}^{\prime} \right)\left[\mathbf{J}\left(\mathbf{r}^{\prime}\right)+\frac{1}{k^{2}}\
abla ^{\prime}\left(\
abla^{\prime}\cdot\mathbf{J}\left(\mathbf{r}^{\prime}\right) \right)\right]dS^{\prime}, \tag{9}\]  
where \(z_{0}\) is the free space characteristic impedance. The EFIE relates the distribution of currents on the PEC surface to the distribution of electric fields in the surrounding space. To solve the EFIE, MoM is widely applied [9], which is based on approximating the current distributions in the modeled structure using a set of basis functions. For this purpose, the equation to be solved is represented in the following form  
\[Lf=g, \tag{10}\]  
where \(L\) is a linear integral operator, \(g\) is a known function (induced field), and \(f\) is the unknown function (currents).