## 2 Filter Design
The notch band characteristics are formed by using defect microstrip structure similar to using defect ground structure [19], and the resonant unit cell are modeled by L-shaped defect microstrip structure and the T-shaped resonator, i.e., Figure 4b, only considers the first and second resonant modes, which shows the equivalent lumped circuit of the resonant elements composed of the T-shaped resonator and L-shaped defect microstrip structure. According to the theory of transmission line, each resonant element can be equivalent to a simple LC parallel resonant circuit with a characteristic that is similar to the first order Butterworth low-pass filter. Therefore, the equivalent circuit model Equations (6)-(10) of the two resonant elements composed of the T-shaped resonator and the L-shaped defect microstrip structure can be deduced from the Butterworth low-pass filter circuit model, where \(f_{T}\) means a transit frequency, while \(f_{01}\) and \(f_{02}\) describe the lower and the higher resonant frequencies. The 3-dB bandwidth at \(f_{01}\) and \(f_{02}\) are  
Figure 4: (**a**) The structure of T-shaped resonator; (**b**) even-mode equivalent circuit; (**c**) odd-mode equivalent circuit (\(Z_{5}\) and \(Z_{6}\) are the characteristic impedance. \(L_{5}\) and \(L_{6}/2\) are length of the open-circuited stub and the short-circuited stub). Figure 5: (**a**) The element structure of L-shaped defect microstrip structure; (**b**) the equivalent lumped circuit of T-shaped resonator and L-shaped defect microstrip structure. denoted by \(\Delta_{3\text{dB}_{-f_{01}}}\) and \(\Delta_{3\text{dB}_{-f_{02}}}\). The imaginary parts of three \(Z\) parameters at \(f_{T}\) are \(X_{11}\), \(X_{22}\) and \(X_{21}\). The following equivalent formulas regarding resonant elements composed of the T-shaped resonator and L-shaped defect microstrip structure can be derived from Reference [3]. The equivalent equations of resonant elements composed of the T-shaped resonator and L-shaped defect microstrip structure are shown in Reference [19]. \[Z_{\text{ino}}=jZ_{6}\tan\theta_{6} \tag{2}\]0  
\[Z_{\text{ino}}=jZ_{6}\tan\theta_{6} \tag{2}\]1  
\[Z_{\text{ino}}=jZ_{6}\tan\theta_{6} \tag{2}\]2  
\[Z_{\text{ino}}=jZ_{6}\tan\theta_{6} \tag{2}\]3  
Defect microstrip structure has the same slow wave characteristics as DGS, so the resonant frequency can be derived from the method of DGS [20]. The L-shaped defect microstrip structure can be regarded as a half guided wavelength crooked slot line resonator. Therefore, the resonant frequency \(f\) may be achieved as:  
\[Z_{\text{ino}}=jZ_{6}\tan\theta_{6} \tag{2}\]4  
where \(c\) is the free-space speed of light, \(L=L_{7}+L_{8}\) the total length of the L-shaped defect microstrip structure, and \(\varepsilon_{\text{slot}}\) is the effective dielectric constant of the slot acquired by the closed-form equations in Reference [21]. The circuit parameters extracted from the ADS is shown in \[Z_{\text{ino}}=jZ_{6}\tan\theta_{6} \tag{2}\]5 Figure 6 shows the comparison of HFSS and ADS simulation results regarding the equivalent equation of resonant elements composed with the T-shaped resonator and L-shaped defect microstrip structure. It can be seen from Figure 6 that the ADS simulation results and the HFSS simulation results have the same resonant frequencies at 3.5 GHz and 7.5 GHz with the same insertion loss 28.6 dB, which proves the validity and correctness of the equivalent circuit. Figure 7 shows the photograph of the fabricated filter which selects RT/Duorid5880 (Rogers Corporation, Chandler, AZ, USA) as a substrate with the size of 32 mm \(\times\) 10 mm \(\times\) 1 mm. \begin{table}
\begin{tabular}{c c} \hline \hline \(C\) (pF) & \(L\) (nH) \\ \hline \(C_{\text{p1}}=2.415\) & \(L_{\text{p1}}=0.85\) \\ \(C_{\text{p}}=2.489\) & \(L_{\text{s1}}=0.18\) \\ \(C_{\text{p2}}=0.1216\) & \(L_{\text{s2}}=-0.105\) \\ – & \(L_{\text{p2}}=0.47\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The circuit parameters in Figure 5b extracted from the advanced design system (ADS).