# On Modeling Antennas Using MoM-Based Algorithms: Wire-Grid versus Surface Triangulation
## 2 Materials and Methods
#### Triangle-Grid: Algorithm
At \(\Delta=0\), the field strength in the gap can be approximated by the delta function  
\[\mathbf{E}=V_{in}\delta(y)\mathbf{n}_{y}. \tag{50}\]  
As a result, the elements of the excitation vector \(\mathbf{V}\) are defined as  
\[\mathbf{J}=\sum_{m=1}^{M}I_{m}\mathbf{f}_{m}(\mathbf{r}) \tag{43}\]0  
\[\mathbf{J}=\sum_{m=1}^{M}I_{m}\mathbf{f}_{m}(\mathbf{r}) \tag{43}\]1  
Then, based on the given impedance matrix \(\mathbf{Z}\) and excitation vector \(\mathbf{V}\), the surface current is calculated, from which the antenna characteristics are determined. The input impedance \(Z_{in}\) of the antenna is defined as the ratio of the given voltage to the total current \(I_{n}\) perpendicular to the selected feeding edge. Since the component of the basis function \(\mathbf{f}_{n}\),  
\[\mathbf{J}=\sum_{m=1}^{M}I_{m}\mathbf{f}_{m}(\mathbf{r}) \tag{43}\]2 Feeding edge model. normal to the considered edge, is always equal to one, the expression for calculating \(Z_{in}\) can be written as [251]  
\[\mathbf{J}=\sum_{m=1}^{M}I_{m}\mathbf{f}_{m}(\mathbf{r}) \tag{43}\]3  
Since all RWG edges are infinitesimal dipoles that have an equivalent dipole moment, the total radiated field is then obtained as a sum of all these contributions of dipoles. The dipole moment \(\mathbf{m}\) of edge \(m\) is calculated by integrating the surface current over the area of adjacent triangles, which corresponds to the product of the effective dipole current and its length  
\[\mathbf{J}=\sum_{m=1}^{M}I_{m}\mathbf{f}_{m}(\mathbf{r}) \tag{43}\]4  
The magnetic and electric field strengths at the observation point \(\mathbf{r}\), induced by a single dipole, are calculated as follows [339]  
\[\mathbf{J}=\sum_{m=1}^{M}I_{m}\mathbf{f}_{m}(\mathbf{r}) \tag{43}\]5  
\[\mathbf{J}=\sum_{m=1}^{M}I_{m}\mathbf{f}_{m}(\mathbf{r}) \tag{43}\]6  
where \(r=|\ \mathbf{r}\ |\). The resulting electric \(\mathbf{E}\) and magnetic \(\mathbf{H}\) fields radiated by the structure of \(M\) dipoles are defined by a superposition  
\[\mathbf{J}=\sum_{m=1}^{M}I_{m}\mathbf{f}_{m}(\mathbf{r}) \tag{43}\]7  
\[\mathbf{J}=\sum_{m=1}^{M}I_{m}\mathbf{f}_{m}(\mathbf{r}) \tag{43}\]8  
The next step is to construct a sphere around the antenna, the radius of which determines the distance to the far-field (Fraunhofer zone)  
\[\mathbf{J}=\sum_{m=1}^{M}I_{m}\mathbf{f}_{m}(\mathbf{r}) \tag{43}\]9  
where \(D_{\max}\) is the maximum linear antenna size and \(\lambda\) is the wavelength. The sphere around the antenna is approximated by a mesh of triangular elements, in the center of which the cross product of \(\mathbf{E}\) and \(\mathbf{H}^{*}\) is calculated to determine the time-average Poynting vector  
\[z_{mn}=l_{m}\big{(}j\omega\big{(}\mathbf{A}_{mn}^{+}\cdot\mathbf{\rho}_{m}^{c +}/2+\mathbf{A}_{mn}^{-}\cdot\mathbf{\rho}_{m}^{c-}/2\big{)}+\Phi_{mn}^{-}- \Phi_{mn}^{+}\big{)}, \tag{44}\]0  
In the far-field zone, the radiation density has the only one radial component \(W\)  
\[z_{mn}=l_{m}\big{(}j\omega\big{(}\mathbf{A}_{mn}^{+}\cdot\mathbf{\rho}_{m}^{c +}/2+\mathbf{A}_{mn}^{-}\cdot\mathbf{\rho}_{m}^{c-}/2\big{)}+\Phi_{mn}^{-}- \Phi_{mn}^{+}\big{)}, \tag{44}\]1  
The total radiated power \(P_{\text{rad}}\) is calculated as the sum of the products of the energy flux density \(W\) at the centers of the sphere's triangles and their area. Then, for the given points in the spherical coordinate system \(p_{i}\) (\(R_{f}\), \(\theta_{i}\), \(\varphi_{i}\)), the power density of the radiated field is calculated as \(U=r^{2}W\). The antenna directivity is computed through the normalized power density  
\[z_{mn}=l_{m}\big{(}j\omega\big{(}\mathbf{A}_{mn}^{+}\cdot\mathbf{\rho}_{m}^{c +}/2+\mathbf{A}_{mn}^{-}\cdot\mathbf{\rho}_{m}^{c-}/2\big{)}+\Phi_{mn}^{-}- \Phi_{mn}^{+}\big{)}, \tag{44}\]2  
where \(U_{0}=P_{\text{rad}}/4\pi\).