# Ultra-Wideband Bandpass Filters Using Tapered Resonators
## 4 UWB Bandpass Filter Design  
In this section, UWB bandpass filters are designed. The UWB filter configuration adopted in this paper is the conventional UWB structure based on optimum high-pass filter structure [19]. This filter consists of short-circuited uniform stubs which can be designed to cover the UWB spectrum. For n = 4 the filter consists of four short-circuited transmission line stubs with characteristic impedances of \(Z_{01}\), \(Z_{02}\), \(Z_{03}\), and \(Z_{04}\) with electrical lengths of \(\theta_{\rm s}\) separated by connecting lines of electrical length \(\theta_{\rm c}=2\theta_{\rm s}\), as shown in Figure 4. To simplify the design, we selected the impedance values as \(Z_{01}=Z_{04}\), \(Z_{02}=Z_{03}\), and \(Z_{01}=2Z_{02}\). In the proposed design, tapered transmission line stubs are used to replace the uniform short-circuited stub in the classical design, as shown in Figure 5. The characteristic impedances of the tapered transmission line stubs vary from \(Z_{0i1}\) to \(Z_{0i2}\) (\(i\) = 1,2,3,4) where \(Z_{011}=Z_{041}\), \(Z_{021}=Z_{031}\), and \(Z_{011}=2Z_{021}\). The electrical lengths of the tapered short-circ  
ETTL and LTTL, respectively. For \(k/\beta_{0}=0.25\) exponential tapered transmission line short-circuited stubs, the resulting impedances are \(Z_{012}=Z_{042}=161.3\)\(\Omega\), and \(Z_{022}=Z_{032}=80.6\)\(\Omega\). For \(k/\beta_{0}=0.375\) stubs, the impedances are \(Z_{012}=Z_{042}=189.8\)\(\Omega\), and \(Z_{022}=Z_{032}=94.9\)\(\Omega\). For \(k/\beta_{0}=0.5\) stubs, the impedances are \(Z_{012}=Z_{042}=221.8\)\(\Omega\), and \(Z_{022}=Z_{032}=110.9\)\(\Omega\). The layout of the proposed UWB bandpass filter is shown in Figure 6. The filter is simulated and fabricated on a Roger RT5880 substrate with a relative dielectric constant equal to 2.2 and a thickness of 0.78 mm. The parameters of the proposed microstrip filters are given in Table 2.  
The semi-lumped equivalent circuit of the tapered short-circuited stubs filter can be determined by replacing the short-circuited tapered stubs with its lumped LC equivalent resonator, as shown in Figure 7. The equivalent inductances and capacitances can be calculated from (9) and (10) for the ETTL and from (13) and (14) for the LTTL. The values of the equivalent inductances and capacitances for the ETTL shunt stubs at \(f_{0}=6.85\) GHz are given in Table 3. Figure 8 shows the response of the semi-lumped equivalent filter of \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \(k/\beta_{0}\) & 0.25 & 0.375 & 0.50 \\ \hline \(l_{1}\) & & 14.92 & 14.92 & 14.92 \\ \hline \(l_{2}\) & & 14.17 & 14.17 & 14.17 \\ \hline \(l_{3}\) & & 14.92 & 14.92 & 14.92 \\ \hline \(\begin{array}{c}Terminal\\ stub widths\end{array}\) & \(W_{T1}\) & 0.51 & 0.51 & 0.51 \\ \cline{2-5} \(\begin{array}{c}Triminal\\ stub widths\end{array}\) & \(W_{T2}\) & 0.16 & 0.08 & 0.03 \\ \hline \(\begin{array}{c}Middle\\ stub widths\end{array}\) & \(W_{M1}\) & 2.00 & 2.00 & 2.00 \\ \cline{2-5} \(\begin{array}{c}Triminal\\ stubs\end{array}\) & \(W_{M2}\) & 1.10 & 0.80 & 0.53 \\ \hline \(\begin{array}{c}Lengths of the terminal\\ stubs\text{ (L}_{sT}\text{)}\end{array}\) & & 7.27 & 6.86 & 6.51 \\ \hline \(\begin{array}{c}Lengths of the middle\\ stubs\text{ (L}_{sM}\text{)}\end{array}\) & & 7.03 & 6.63 & 6.28 \\ \hline \(\begin{array}{c}W_{c}\end{array}\) & & & 2.07 & \\ \hline \(\begin{array}{c}W_{v}\end{array}\) & & & 0.7 & \\ \hline \(\begin{array}{c}W_{f}\end{array}\) & & & 2.4 & \\ \cline{2-5} \(\begin{array}{c}Radius\text{ of }vis\text{)}\end{array}\) & & 0.25 & \\ \hline \(\begin{array}{c}W_{s}\end{array}\) & & 20 & \\ \hline \(\begin{array}{c}\text{L}_{s}\end{array}\) & & 59.03 & \\ \hline \(\begin{array}{c}\text{L}_{f}\end{array}\) & & 5 & \\ \hline \end{tabular}
\end{table} 
Table 2: Dimensions of the elements of the designed filters (millimeters). Figure 6: Layout of the proposed microstrip UWB bandpass filters with tapered resonators. ETTL stubs where \(k/\beta_{0}=0.25\).