## 3 Operational Principle of the SUWB-BPF with In-Band Frequency Suppression  
A novel T-shaped shorted stub-loaded resonator with a central square ring, also known as a quad-mode stub-loaded resonator, was utilized for the realization of in-band frequency suppression in the SUWB-BPF. The operating principle for the formation of notch bands was completed in two steps. Initially, the folded C-shaped resonator, as shown in Figure 6, was coupled to the basic wideband filter, and after optimizing the length of the proposed resonator, a single stop band at 8.3 GHz was obtained, as depicted in Figure 7 with its control shown in Figure 8. In the second step, a centrally shorted square ring was  
Figure 4: S\({}_{11}\) vs. S\({}_{21}\) frequency plot by loading conventional stub. Figure 5: Simulated frequency plots of the SUWB-BPF with proposed rectangular pads. loaded to the C-shaped structure for the realization of two more stopbands at 4.9 GHz and 11.5 GHz. In this way, a combination of the C-shaped resonator and square ring resonator was obtained, which is also called a T-shaped shorted stub-loaded resonator with a central square ring or a quad-mode stub-loaded resonator. The name quad-mode stub-loaded resonator is given by the authors of this work as it generates four resonant modes, which will be explained in the section below. The complete configuration of the proposed resonator for the triband responses is shown in Figure 9a, which is composed of a ring resonator, a common half wavelength (\(\lambda/2\)) uniform-impedance resonator (UIR), and a short stub. It can be seen that the proposed resonator is symmetrical with respect to the central vertical plane XX'; therefore, a well-known classical method called the even-odd-mode is performed to characterize the resonance behavior of the proposed architecture discussed in [22; 23]. This method will simplify the mathematical calculation by dividing the resonator into two sections along the XX' plane; one section will behave as a magnetic wall (M.W. ), called even mode excitation, while the other one behaves as an electric wall (E.W. ), called odd mode excitation, as shown in Figure 9b,c, respectively. According to the conventional theory of the even-odd-mode analysis, the resonator on the symmetrical plane XX' is shorted under odd mode excitation and open-circuited under even mode excitation, but in this topology, the plane XX' for the even mode excitation is shorted with one end due to it already being present in the topology, as shown in Figure 9a. Figure 6: The proposed C-shaped resonator topology for the 8.3 GHz stopband. Figure 7: Response of the proposed C-shaped resonator at 8.3 GHz. Figure 8: Control of the C-shaped resonator for the 8.3 GHz stopband. Figure 9: (**a**) The presented geometry of the quad-mode stub-loaded resonator. (**b**) Even-mode excitation circuit. (**c**) Odd-mode excitation circuit. The odd-mode equivalent circuit in Figure 9c consists of two quarter wavelength paths with one end grounded, as shown in Figure 10a,b. Thus, two resonance frequencies \(f_{odd1}\) and \(f_{odd2}\) can be realized by setting the resonance condition, i.e., \(Y_{in\ odd1}=0\) and \(Y_{in\ odd2}=0\). According to the basic microwave network theory, the input admittance of the odd modes can be calculated using the mathematical equation below [24]:  
\[Y_{in}=Y_{0}\frac{Y_{L}+jY_{0}\tan\theta}{Y_{0}+jY_{L}\tan\theta} \tag{13}\]  
\[Y_{in}=Y_{0}\frac{Y_{L}+jY_{0}\tan(L2\pi/\lambda_{\rm g})}{Y_{0}+jY_{L}\tan(L 2\pi/\lambda_{\rm g})} \tag{14}\]  
where \(\lambda_{g}=\frac{C}{f\sqrt{e_{ff}}}\) and  
\[\varepsilon_{eff}=\frac{1+\varepsilon_{r}}{2}+\frac{\varepsilon_{r}-1}{2} \times\frac{1}{\sqrt{1+12\frac{b}{w}}}\]  
Figure 4:0  
Figure 4:1  
Figure 4:2  
Figure 4:3  
For even-mode excitation, the circuit in Figure 9b also contains two paths: a \(\lambda/2\) resonator with one end open constitutes path-I, as shown in Figure 11a, while a \(\lambda/4\) resonator with one end grounded constitutes path-II, as illustrated in Figure 11b, respectively.