## 3 Characteristic Mode Analysis of the MTS  
Recently, CMA has been utilized as a novel and effective approach for analyzing the internal working mechanisms and for designing the MTS antennas [25; 26; 27]. Characteristic modes are defined as a set of surface currents and radiation fields that depend only on the shape of the conducting object and are independent of any external sources. In antenna design, CMA provides an important theoretical basis and design guidance for optimiz \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Parameters** & **Value (mm)** & **Parameters** & **Value (mm)** \\ \hline \(W\) & 50 & \(h_{2}\) & 0.8 \\ \(g\) & 0.55 & \(l_{1}\) & 12.9 \\ \(p\) & 7.3 & \(l_{2}\) & 6.8 \\ \(p_{1}\) & 3.5 & \(l_{3}\) & 6 \\ \(c\) & 3 & \(l_{4}\) & 9.3 \\ \(r_{1}\) & 3.3 & \(w_{1}\) & 1.5 \\ \(r_{2}\) & 5.3 & \(w_{2}\) & 0.5 \\ \(h_{1}\) & 2 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Optimized antenna geometric dimension parameters. Figure 1: Antenna structure schematic. (**a**) Top view of a MTS that includes parasitic patches and mode-suppressing patches; (**b**) middle layer view of a metal ground plane with a circular ring slot; (**c**) bottom view of an L-shaped microstrip feed line; (**d**) side view; (**e**) view of ring slot and the feed line overlapped. ing antenna performance and facilitating a deeper understanding of antenna radiation mechanisms. The eigenvalue equations are defined as follows [28]:  
\[XJ_{n}=\lambda_{n}RJ_{n} \tag{1}\]  
where X and R are the imaginary and real parts of the impedance matrix, respectively, which is used to solve the electric field integral equation. \(\lambda_{n}\) is the eigenvalue, and \(J_{n}\) is the eigenvector, which represents the mode current of the n-th (\(n=0,1,2,3,\ldots\)) characteristic mode. The characteristic angle represents the phase disparity between the tangential components of the mode current and the mode electric field, and it is mathematically defined as follows [28]:  
\[\alpha_{n}=180^{\circ}-\tan^{-1}(\lambda_{n}) \tag{2}\]  
Modal significance (MS) is only related to the conductor itself, and its characteristic curve reflects the resonance potential at different frequencies. It is represented as follows [28]:  
\[\text{MS}=\frac{1}{1+\lambda_{n}} \tag{3}\]  
Generally, when MS > 0.707, this mode is in the resonance frequency band. To achieve CP radiation, at least two orthogonal modes need to be excited simultaneously, with these two modes having equal (or similar) MS values, a 90\({}^{\circ}\) difference in characteristic angle, and similar radiation patterns. To understand the working mechanism of CP, the radiation patterns and mode distributions of the antenna's MTS are studied using the CST 2022 simulation software. The structure of the MTS in the electromagnetic simulation software CST is shown in Figure 2. The region surrounding the antenna is set as a radiation boundary condition, and the dielectric layer of the antenna is set as a loss-free substrate material. As illustrated in Figure 3, the mode currents for the first four modes are depicted under the conditions of a frequency of 6.9 GHz and an initial phase of 0\({}^{\circ}\). The red arrows in Figure 3 specifically mark the regions of peak current density, the blue arrows denote the areas with minimum current density, and the black arrows are employed to indicate the direction of current for each mode. Additionally, the scale of current amplitude, which provides a quantitative reference for the magnitude of the currents, is conveniently placed on the right side of Figure 3.  
Figure 2: Simulation of CMA for the MTS antenna. Owing to the symmetry of the MTS, \(\mathbf{J}_{1}\) (abbreviation for the current of Mode 1) which is polarized in the Y-direction and \(\mathbf{J}_{2}\) which is polarized in the -X-direction are mutually orthogonal, thereby constituting a pair of orthogonal modes. Given that all currents within these modes are in phase, both modes generate broadside radiation patterns, as illustrated in Figure 4a,b.