## 2 Case of Study: Effect of T-Shaped Stub-Loading  
When two radiating structures are placed in close proximity, even if they have different resonant frequencies, mutual electromagnetic coupling occurs. This coupling can be direct through space, through scattering from nearby objects, or even through the excitation network feeding the elements, such as the feed port and ground plane. In a structure like the one proposed in Figure 1b, which can be considered as an array of two antennas separated by a distance d, it can be treated as a two-port network. The mutual impedance between both antennas can be determined by (1). \[Z_{12}=\left.\frac{V_{1}}{I_{2}}\right|_{I_{1}=0} \tag{1}\]where \(V_{1}\) is the voltage at antenna 1 and \(I_{2}\) is the current on radiator 2. However, mutual impedance is difficult to calculate numerically [10] but can be determined using numerical methods or electromagnetic simulation software. Due to mutual coupling, antenna impedance varies depending on several factors, such as element spacing, excitation point, radiator geometry, and operating frequency. Assuming an isolated antenna, as shown in Figure 1a, its input impedance is given by (2)  
\[Z_{11}=\frac{V_{1}}{I_{1}} \tag{2}\]  
when a second radiating structure is positioned at a distance \(h\) from the first, the radiation of this induces a field in the second one, which in turn generates its own radiated field. This mutual interaction influences both structures, resulting in a change in the impedance of the first radiator, as expressed in Equation (3a). \[V_{1}=Z_{11}I_{1}+Z_{12}I_{2} \tag{3a}\]  
Similarly, the voltage in radiator 2 is influenced by the interaction with current \(I_{1}\), as shown in (3b). \[V_{2}=Z_{22}I_{2}+Z_{21}I_{1} \tag{3b}\]  
Assuming that antenna 2 has a load impedance determined by the interaction with antenna 1, which can be expressed as \(Z_{L2}\), then the voltage at its terminals can be expressed as \(V_{2}=-Z_{L2}I_{2}\). Therefore, Equation (3b) can be rewritten as (4). \[-Z_{L2}I_{2}=Z_{22}I_{2}+Z_{21}I_{1} \tag{4}\]  
From Equation (4), \(I_{2}\) is obtained and substituted into Equation (3a), yielding  
\[V_{1}=Z_{11}I_{1}+Z_{12}\bigg{(}\frac{-Z_{12}I_{1}}{Z_{L2}+Z_{22}}\bigg{)}\] (5a) (5a) can be rewritten as (5b) \[\frac{V_{1}}{I_{1}}=Z_{11}-\frac{(Z_{12})^{2}}{Z_{L2}+Z_{22}}=Z_{L1} \tag{5b}\]  
where \(Z_{L1}\) represents the loaded impedance of the primary radiator. Notably, \(Z_{L1}\) is a function of the mutual impedance and loaded impedance of the secondary radiator. Therefore, the positional relationship, geometric configuration, and dimensional properties  
Figure 1: (**a**) conventional strip monopole, scale in mm; (**b**) strip monopole with T-stub, scale in mm. of the secondary radiator exert a direct influence on the characteristic impedance of the primary radiator. As mentioned above, the calculation of mutual impedance is not easy to achieve; then, a simulation using Finite Element Method (FEM) was performed to investigate the impact of the stub-loading. To initiate, assuming a substrate with a relative permittivity of 10 and a thickness of 0.635 mm, a conventional strip monopole 10 mm in length from the ground plane and fed by a 50 Ohm coplanar line, was considered as a baseline, as shown in Figure 1a. The ground plane's (GND) dimensions are 5 mm by 3 mm. To introduce the reactive load, a T-shaped stub was added to the monopole, as depicted in Figure 1b. The black-colored geometries in Figure 1b are placed on one side of the substrate, and the T-stub, colored gray, is placed on the backside of the substrate. The monopole's length and width are kept for this configuration. To investigate the impact of these relationships and to achieve impedance stabilization over an extended bandwidth, a parametric analysis of the thinner part of the stub's position relative to the monopole's GND was undertaken utilizing Ansys(r) Electronics simulation software 2024.