## 2 Case of Study: Effect of T-Shaped Stub-Loading
C1 represents the equivalent capacitance between the ground plane of the coplanar line and the thick part of the stub, while L1 emulates the behavior of the inductance associated with the section of the stub closest to the ground plane. L2 models the extra inductance associated with the magnetic coupling between the monopole and the thick part of the stub near the ground plane. C2 is the associated capacitance formed between the monopole and the thick part of the stub near the ground plane. For the second part, L3 represents the inductance associated with the section of the stub in the middle part until the junction with the narrow part, while C3 emulates the behavior of the capacitance generated depending on the width of the stub line. While L4 and C4 represent the extra inductance due to the magnetic coupling between the monopole and the stub and the associated capacitance between both structures in this part of the arrangement, respectively. Finally, for the third part, L5 emulates the behavior of the inductive nature of the thinner section of the stub, orthogonal to the thicker section, and C5 is the capacitance presented between the monopole and the thin section of the stub. The values of the elements were determined based on the wideband coupling response obtained from the electromagnetic simulation, with initial values based on the reflection coefficient of a Butterworth filter prototype, and the final values were obtained through a parameterization process using Advanced Design System(r) (ADS 2025 Update 2.0). The comparison of the response obtained using electromagnetic simulation (Ansys) and the circuit model (ADS) is presented in Figure 3b. In order to expand, even more, the bandwidth of the antenna loaded with the T-shaped stub, an additional monopole with a shorter length is introduced. This shorter monopole creates a higher resonance, which overlaps with the resonance of the longer monopole, leading to a wider operating bandwidth. This approach is commonly used in antenna design and has been implemented in various structures, including logarithmic antennas [10] and fractal antennas [12]. The proposed configuration is displayed in Figure 4a, and its frequency response is presented in Figure 4b. In Figure 4a, the second arm of the monopole was sized to have a resonance close to that of the main arm, resulting in overlapping resonances. For this configuration, the ground plane dimensions were maintained, as shown in Figure 1b. As seen in Figure 4b, the two-arm monopole exhibits two resonances, one near 4.3 GHz and another at 6.3 GHz. As expected, the resonance of the larger monopole was shifted due to the loading generated by the second arm. At the first resonance, the radiator is mismatched and exhibits a high \(Q\)  
\[Z_{11}=\frac{V_{1}}{I_{1}} \tag{2}\]0 (**a**) Two-branches monopole; (**b**) Frequency response. \[Z_{11}=\frac{V_{1}}{I_{1}} \tag{2}\]1 (**a**) Electric circuit model; (**b**) comparison of simulated \(S_{11}\) parameters. while at the second resonance, a bandwidth of approximately 2.5 GHz is obtained with a port coupling of around \(-\)23 dB. This response was left unoptimized to allow for better performance when integrating the tuning with the stub. The results of this integration are presented in the following section.