# On Modeling Antennas Using MoM-Based Algorithms: Wire-Grid versus Surface Triangulation
## 2 Materials and Methods
### Wire-Grid
#### 2.2.1 Wire-Grid: Overview
The errors produced by both methods were located near the surfaces being modeled, a consequence of the fact that wire-grid is a simplified representation of a solid real object. All this supported the necessity to use a finer mesh when modeling via wire-grid and EAR methods to improve the results, but of course this will be at the expense of computational costs. More recently, the authors of [162] presented a novel modeling approach that uses wire-grid that is not sensitive to the choice of the wire radius or to the EAR rules. They established a theoretical basis for partitioning the surface into thin flat strips that can be then replaced by wires having an equivalent radius. These radii can be calculated from well-known formulas for planar dipoles. They proved on the example of a cube, a cylinder and a spherical closed surfaces that the wire-grid obtained using their way is equivalent to the original conducting surface since they both have the same electromagnetic response. They also declared that their equivalent wire-grid modeling frame can be extended to surfaces that are not in free space, as in the case of planar antennas on dielectric substrates, due to its generality. The researchers in [163] tried to summarize guidelines for the design of wire-grid models of complex surfaces like an ellipsoid, a sphere, strips, cubes, plates, and aircraft. They also demonstrated also one of the wire-grid limitations when modeling complex objects, where wire grids of such objects tend to be irregular in segment length, cell size, and mesh area, and \"square\" cells cannot be fitted in all regions. They proposed the use of triangular cells in such regions to achieve \"elegant transitions\". These guidelines helped to identify wire-grid model limiting factors: individual long segments, individual large meshes, match point errors, spacing errors, crossed wires errors, and so forth. In addition, the work [34] summarized rules and guidelines to modeling using the NEC and ESP moment method codes. On their base, an interface computer program GEOM was developed to reduce the human effort and modeling errors by using these modeling rules and the ability to discretize complex structures. It was also shown that segmentation near a source in NEC has significant effect on numerical results since the geometry modeling plays an important role in the applications of these different codes. Moreover, this work demonstrated the fact that the scheme of length segmentations and the resolution of ground meshes can noticeably change the predicted characteristics, such as antenna input impedances. The authors of this work also proved that due to the nature of the NEC code, finer segmentation does not necessarily provide better results when segment lengths become very small, on the contrary numerical errors are no longer negligible. On the other hand, they confirmed the fact that the ESP code can get around some of NEC's problems, however, it has its own limitations like dealing with symmetric structures and nonplanar surfaces. The authors of [164] chose to theoretically analyze the relationship between the random positioning errors upon grid of parallel infinitely long perfectly conducting thin wires impedance and the reflection and transmission characteristics of the grid. They found that increasing these errors increases the magnitude of the wire-grid impedance and the transmission through it, and also reduces the reflection from it. On the other hand, when the wires are not analyzed using the thin-wire theory, the problem of the circumferential current distribution on closely spaced electrically thin wires may occur, and it can affect the determination of the antenna efficiency. This problem was discussed in [165] and solved by modifying the formulation of piecewise-sinusoidal reaction for thin-wire structures. The authors showed that this problem can be faced even when the wire-to-wire separation exceeds several wire diameters.