## 3 Design of a Resonator and Experimental Results
\[C_{ex}=[(l_{TH}+l_{TV})+2\ (t+g_{ex})]C_{op}\bigg{[}\frac{\varepsilon_{0}(1+ \varepsilon_{r})}{2}\bigg{]} \tag{1}\]  
\[C_{in,m,n}=\sum_{n=1}^{N-1}\frac{C_{ex}}{4}=\frac{N-1}{2}[2l_{in,\ m,n}-(2N-1) \ (t+g_{in})]C_{m}\bigg{[}\frac{\varepsilon_{0}(1+\varepsilon_{r})}{2} \bigg{]}\oplus C_{m}=\frac{\varepsilon_{r}A}{d} \tag{2}\]  
\[C_{p1}=(l_{TH}+l_{TV})\ g_{ex}+g_{in}\bigg{[}\frac{\varepsilon_{0}(1+ \varepsilon_{r})}{2}\bigg{]} \tag{3}\]  
\[C_{p2}=(l_{in,m,n}t)\ g_{ex}+g_{in}\bigg{[}\frac{\varepsilon_{0}(1+\varepsilon _{r})}{2}\bigg{]} \tag{4}\]  
\[C_{s}=3.937\times 10^{-5}l_{T}(\varepsilon_{r}+1)[0.11(N-3)+0.252] \tag{5}\]  
\[C_{op}=3.937\times 10^{-5}l_{s}(\varepsilon_{r}+1) \tag{6}\]  
The inductance used in designing the resonator is given by Equations (7)-(10), where \(\lambda_{g}\), the guided wavelength, was 18.8 mm [22]. \[L_{ex}=[2(l_{TH}+l_{TV})]-g_{ex}-2w_{ex} \tag{7}\]  
Figure 2.0  
Figure 2.1  
Figure 2.2 Proposed resonator. Figure 2.3  
In the resonator design, the characteristic impedance was given by Equations (11)-(17), where \(\omega_{r}\) is the resonant frequency, \(\beta\) is the phase constant, and \(Z_{0}\) is the characteristic impedance (\(Z_{0}=50\)\(\Omega\)) [22]. Figure 2.4  
Figure 2.5  
Figure 2.6  
Figure 2.7  
Figure 2.8  
Figure 2.9  
Figure 1:0  
Split\(-\)ring resonators generally have high-Q\({}_{\text{L}}\) values [23]. In addition, the open-loaded T-type stub (\(Z_{T}\) and \(Z_{S}\)) operates similar to a band-stop-type resonator with a high Q\({}_{\text{L}}\), as shown in Figure 1:1 From the figure, \(\theta_{T}\) and \(\theta_{S}\) are 30\({}^{\circ}\) and 45\({}^{\circ}\), respectively, at the resonant frequency [24]. Then, \(Z_{T}\) and \(Z_{S}\) can be solved using Equations (18) and (19) for \(\theta_{T}\). Figure 1:2  
Figure 1:3  
Figure 1:4 Change in electrical length according to the impedance of the stub. where \(Z_{0}\) is characteristic impedance (50 \(\Omega\)). Then, \(\theta_{T}\) must be greater than 0\({}^{\circ}\) and less than 45\({}^{\circ}\) (2\(\theta_{T}\) < \(\theta_{T}\) = 90\({}^{\circ}\)). If \(\theta_{T}\) is greater than 45\({}^{\circ}\) (\(\theta_{T}\) < 90\({}^{\circ}\)), \(Z_{S}\) reaches infinity, as shown in Figure 1:5 For the shunt stub, \(-\)jZ\({}_{S}\) cot \(\theta_{S}\) = \(\infty\); thus, the open-loaded T-type stub behaves similar to an open stub. \(Z_{T}\) and \(Z_{S}\) are 86.6 \(\Omega\) and 75 \(\Omega\), and \(\theta_{T}\) and \(\theta_{S}\) are 30\({}^{\circ}\) and 45\({}^{\circ}\), respectively. The calculated values of the characteristic impedance, electrical, length, and physical length of the proposed resonator are listed in Figure 1:6  
In the open-loaded T-type stub of the proposed resonator, the Q\({}_{\rm L}\) started to change according to the tuning of the stub length. When the proposed resonator was designed by integrating the open-loaded T-type stub into the split-ring resonator, the Q\({}_{\rm L}\) was 333, as shown in Figure 5, and the length of the stub was 0.82 mm. However, if the stub length is tuned, Q\({}_{\rm L}\) varies. For example, the Q\({}_{\rm L}\) increases (from 83.0 to 583) in the stub length range of 0.78 mm to 0.86 mm, and it gradually decreases (from 500 to 98.7) thereafter. When the open-loaded T-type stub is directly connected to the split-ring resonator, the Q\({}_{\rm L}\) of the resonator is greatly increased. Figure 4 shows the results of the simulation of the Q\({}_{\rm L}\) of the split-ring resonator (single), open-loaded T-type stub, and integrated splitting resonator (single) with an open-loaded T-type stub. From the figure, the Q\({}_{\rm L}\) values of the split-ring resonator (single), open-loaded T-type stub, and integrated split-ring resonator (single) with the open-loaded T-type stub were 206 (S\({}_{21}\): 24.8 dB), 322 (S\({}_{21}\): 37.4 dB), and 323 (S\({}_{21}\): 56.6 dB), respectively.