# A Phased Array Antenna with New Elements Designed Using Source Transformations
## 3 Full-Wave Simulation Results and Discussion  
To demonstrate the accuracy of the proposed theoretical model of the linear \"pinwheel\" antenna array, numerical simulations were performed in the commercially available finite-element analysis (FEA) software COMSOL Multi- physics(r). The transformation electromagnetics/optics (TE/TO) approach often results in anisotropic, non-homogeneous, and complex material parameters in matrix form. COMSOL Multiphysics has the capability to validate works related to transformation electromagnetics/optics, as it allows the specification of material anisotropy and continuous inhomogeneity, as found in [13; 14; 15; 16; 17; 18]. The work presented here adopts the validation process presented in [13; 14; 15; 16; 17; 18] by comparing theory with COMSOL results. In the \"pinwheel\" transformation, along with the transformed current source from Equation (6) and transformed material parameters from Equation (7), the \"pinwheel\" shaped antenna geometry is also a function of the proposed transformation. COMSOL Multiphysics has a unique functionality, which allows the definition of the complex \"pinwheel\" shaped geometry and material, as opposed to other commercially available full-wave tools. First, the single element, as shown in Figure 1a, was simulated to verify the transformed current from (6) and the transformed media from (7). Simulation results from a half-wave (\(\lambda/2\)) dipole antenna in free-space are shown in Figure 3a. A frequency of 10 GHz was chosen. If the dipole is twisted by an angle of 180\({}^{\circ}\) without proper material compensation from (7), the field pattern changes significantly, which is demonstrated in Figure 3b. The field pattern is recovered outside the transformation media once the correct material is used from (7), which is illustrated in Figure 3c. The fields from the dipole in Figure 3a and the transformed \"pinwheel\" antenna in Figure 3c outside the material shell are the same. This is emphasized in Figure 3d, which shows almost no field distribution outside the transformation media when the difference between the two fields is taken. This also confirms that the current distribution in (1) is conserved under the \"pinwheel\" coordinate transformation in (6). It is also important to note that the meshing method used for the simulation is free-triangular. The mesh used in the simulation has over 45,000 elements to ensure convergence and accurate results and also to effectively approximate the continuous profiles given by Equation (7). The minimum meshing element quality is 0.2694, the average element quality is 0.9154, and the element area ratio is 0.001049. The total mesh vertices are 22,555. Figure 4 demonstrates the verification of the proposed transformed antenna array design. Figure 4a represents the electric field of the reference dipole antenna array as described in Figure 1b. The phase difference between any two adjacent elements is set to 90\({}^{\circ}\) to scan the beam at an angle \(\theta_{s}=22.5^{\circ}\). For a fair comparison, Figure 4b demonstrates the electric field distribution radiated by the transformed \"pinwheel\" linear array, where the \"pinwheel\" antenna elements are not enclosed by the material parameters defined by (7). The fields in Figure 4b are significantly different from that radiated by the original linear dipole array in Figure 4a. Figure 4c shows the electric field distribution radiated by the proposed \"pinwheel\" antenna array, where each antenna element is enclosed by the transformation medium from (7) and the current distribution is given by (6). The phase difference between any two adjacent \"pinwheel\" elements is kept the same as the \"reference array\". The simulation results show that the radiated fields by the transformed  
\"pinwheel\" linear array (Figure 4c) and the reference dipole array (Figure 4a) are virtually identical.