# Intuition and Symmetries in Electromagnetism: Eigenstates of Four Antennas
## 3 Results  
To distinguish the different cases of symmetry, it is necessary to use the phases of the physical quantities J, E, H and the coupling parameters S, Z, Y,...etc. In our case, we used the MoM-GEC and HFSS as simulation tools (see Figure 3). In this context, several results were shown to validate this approach. First, we validated the MoM GEC and HFSS on the input impedance of the planar dipole antenna used in the four antennas configuration, as indicated in Figure 4[18]. After that, a validation based on the boundary conditions of four antennas was proven by the surface currents described by the guide modes and test functions, solved with MoM GEC, as given in Figures 5 and 6[18].  
A limitation with the MoM-GEC is shown to differentiate the different cases of symmetries. According to the MoM GEC formulation, the coupling parameters Z, Y and S are independent of the phase shifts of the excitation sources (From the coupling expression \(Z=\frac{1}{A^{\dagger}[Z_{di}]^{-1}A}\)) [18]. These phase shifts will be caused only by the amplitudes of the surface  
Figure 3: Example of a four symmetric antennas configuration with amplitudes of 11\(-\)1\(-\)1 (under HFSS): (ox) electrical walls, (oy) magnetic walls.  
currents \(J_{S}\) and the surface electric field \(E_{S}\) (\(J_{S}\) and \(E_{S}\) are complex numbers). We were able to differentiate these different symmetries with the MoM-GEC using the surface current phase of the four-antennas structure, as given in Figures 7 and 8. It is an advantage for HFSS to show and distinguish these symmetries with the coupling parameters Z, Y and S by using the commands Arg(S), Arg(Y) and Arg(Z) (or the HFSS Phase commands).  
Figure 4: Input impedance seen at the source of one of four antennas given by the MoM GEC method and the HFSS tool (Validation) [18].  
Figure 5: Distribution of the current density for (2 × 2) phased half-wavelength planar dipoles described with the guide’s modes functions at f = 5.4 Ghz, \(\alpha_{0}\) = 0 rad m\({}^{-1}\) and \(\beta_{0}\) = 0 rad m\({}^{-1}\) (MoM-GEC method) [18].  
According to Figure 9, we considered Arg(\(S_{12}\)) between the interaction of two sources of the configuration 1111 (two symmetrical sources in-phase) and Arg(\(S_{12}\)) between the interaction of two sources of the configuration 11-1-1 (two anti-symmetrical sources in phase opposition). We found a phase shift of angle \(\pi\) is established between Arg(\(S_{12}^{\text{symmetric sources}}\)) and Arg(\(S_{12}^{\text{anti symmetric sources}}\)), at any point of the frequency band [0 20] GHz, as depicted in Figure 9.  
This verifies, Arg(\(S_{12}^{\text{symmetric sources}}\)) - Arg(\(S_{12}^{\text{anti symmetric sources}}\)) = \(\pi\) (or = 180\({}^{\circ}\)) (also, Arg(\(Z_{12}^{\text{symmetric sources}}\)) - Arg(\(Z_{12}^{\text{anti symmetric sources}}\)) = \(\pi\)) which seems to follow the equation: \(\Phi_{1}-\Phi_{2}=\pi\) and the reasoning which follows Equation (6). In our case, we used two frequencies, 3.9948 Ghz and 11.8029 Ghz, on the whole frequency band. At each frequency, if we add or subtract their values (between Arg (S21) of two symmetrical sources and Arg (S21) of two antisymmetrical sources), we find a phase shift of 180 degrees. Note that we checked this difference on the whole frequency band. Finally, we can distinguish the cases of symmetries by the introduction of phase shift in the matrix of [5] parameters, as explained in the previous section.  
Figure 6: Distribution of the current density for (\(2\times 2\)) phased half-wavelength planar dipoles described with the trial functions (test functions) at \(f\) = 5.4 Ghz, \(a_{0}\) = 0 rad m\({}^{-1}\) and \(\beta_{0}\) = 0 rad m\({}^{-1}\) (MoM-GEC method) [18].  
Figure 7: Current \(\dagger_{c}\)’s phase (or Angle) of a symmetry configuration with an electric wall along (ox) and a magnetic wall along (oy): the amplitudes of the excitations are 1111 (MoM-GEC method).