# A Reconfigurable Single-/Dual-Bandstop Filter with Controllable Equal-Ripple Performance
## 2 Design and Analysis of the Proposed Reconfigurable BSF
### State II: D-Bsf
#### 2.2.1 Design Equation of State II  
The topologies of the even-mode and odd-mode equivalent circuits for State II are shown in Figure 6.  
For the even mode, the input impedances of three step-impedance open stubs are represented by \(Z_{ine(3)}\), \(Z_{ine(4)}\), and \(Z_{ine(5)}\), where \(Z_{ine(5)}\) is consistent with \(Z_{ine(1)}\) and can be \begin{table}
\begin{tabular}{c c c c c} \hline  & \(Z_{1}\) (\(\Omega\)) & \(Z_{2}\) (\(\Omega\)) & \(Z_{3}\) (\(\Omega\)) & \(Z_{4}\) (\(\Omega\)) \\ \hline Example I & 2.9314 & 1.9270 & 0.6000 & 0.4564 \\ \hline Example II & 2.1529 & 0.8939 & 1.2000 & 1.4450 \\ \hline Example III & 2.0803 & 0.6733 & 1.8000 & 2.7807 \\ \hline \end{tabular}
\end{table} 
Table 1: The design parameters for three design examples of State I. (\(SR_{S}\) = 20 dB and \(RL_{S}\) = 25 dB). Figure 5: \(S\)-parameters of Example I, Example II, and Example III. (\(SR_{S}\) = 20 dB and \(RL_{S}\) = 25 dB). Figure 6: Topologies of (a) even-mode and (b) odd-mode equivalent circuit for State II. obtained by (1). \(Z_{inc2}\) is defined as the total input impedance of the even-mode equivalent circuits. These input impedances can be calculated by the following:  
\[Z_{inc(3)}=-jZ_{5}\cot\theta \tag{24}\]  
\[Z_{inc(4)}=\frac{-2 jZ_{1}Z_{4}+jZ_{1}(Z_{1}+2Z_{4})\sin^{2}\theta}{\cos\theta \sin\theta(Z_{1}+2Z_{4})} \tag{25}\]  
\[Z_{inc(5)}=Z_{inc(1)} \tag{26}\]  
\[Z_{inc2}=\frac{1}{j\frac{\sin\theta}{Z_{5}\cos\theta}+j\frac{\cos\theta\sin \theta(Z_{1}+2Z_{4})}{2Z_{1}Z_{4}-Z_{1}(Z_{1}+2Z_{4})\sin^{2}\theta}+j\frac{ \cos\theta\sin(Z_{2}+Z_{3})}{Z_{2}Z_{3}-Z_{2}(Z_{2}+Z_{3})\sin^{2}\theta}} \tag{27}\]  
Similarly, for the odd mode, \(Z_{inc(3)}\) and \(Z_{inc(4)}\) are defined as the input impedances of the step-impedance short stub, while \(Z_{inc(5)}\) is defined as the input impedance of the step impedance open stub. The total input impedance of the odd-mode equivalent circuits is represented by \(Z_{inc2}\). These input impedances are expressed by the following:  
\[Z_{inc(3)}=jZ_{5}\tan\theta=jZ_{5}\frac{\sin\theta}{\cos\theta} \tag{28}\]  
\[Z_{inc(4)}=jZ_{1}\tan\theta=jZ_{1}\frac{\sin\theta}{\cos\theta} \tag{29}\]  
\[Z_{inc(5)}=\frac{-jZ_{2}Z_{3}+jZ_{2}(Z_{2}+Z_{3})\sin^{2}\theta}{\cos\theta \sin\theta(Z_{2}+Z_{3})} \tag{30}\]  
Figure 5:0  
The reflection coefficients for the odd-mode equivalent circuit \(\Gamma_{o}^{D}\) and even-mode equivalent circuit \(\Gamma_{e}^{D}\) can be obtained as follows:  
Figure 5:1  
Figure 5:2  
The reflection and the transmission coefficients of the DBSF mode \(S_{11}^{D}\) and \(S_{21}^{D}\) can be obtained as follows:  
Figure 5:3  
Figure 5:4  
In order to analyze conveniently, the ratio of \(S_{11}^{D}\) to \(S_{21}^{D}\) is defined as the transfer function for State II \(F_{D}\), which can be expressed as follows:  
Figure 5:5  
where \(B_{j}\) (\(j=0,1,\ldots,7\)) are the coefficients for the power terms of \(F_{D}\), and can be deduced by the following:  
Figure 5:6  
Figure 5:7\[B_{2} =\left( \begin{array}{c}3Z_{2}^{4}Z_{2}^{2}-3Z_{2}^{4}+2Z_{2}^{3}Z_{ 2}Y_{2}^{2}-2Z_{2}^{3}Z_{3}-4Z_{2}^{3}Z_{5}-5Z_{2}^{2}Z_{3}Z_{5}\\ -Z_{2}^{2}Z_{2}^{2}-Z_{2}Z_{2}^{3}Z_{5}-2Z_{2}Z_{3}Z_{5}^{2}-2Z_{3}^{2}Z_{5}^{2} -2Z_{2}^{4}Z_{4}-2Z_{3}^{2}Z_{5}\\ -2Z_{2}^{3}Z_{4}Z_{5}-2Z_{2}^{2}Z_{3}Z_{4}Z_{5}\\ +(-Z_{2}^{4}Z_{5}^{2}-4Z_{2}^{4}Z_{4}Z_{5}-2Z_{2}^{3}Z_{4}Z_{5}\\ +(-Z_{2}^{4}Z_{5}^{2}-4Z_{2}^{4}Z_{4}Z_{5}-2Z_{2}^{3}Z_{4}Z_{5}^{2}-2Z_{2}^{2}Z_ {3}Z_{4}Z_{5}^{2})Z_{1}-2Z_{2}^{4}Z_{4}Z_{5}^{2}\\ \end{array} \right)Z_{1}^{3}\] \[B_{3}=Z_{1}Z_{2}^{4}Z_{5}\left(Z_{1}^{2}+2Z_{1}Z_{4}+2Z_{4}Z_{5} \right)+2Z_{1}^{3}Z_{2}^{3}Z_{5}(Z_{2}+Z_{3}) \tag{40}\] \[B_{4}=(Z_{2}+Z_{3})\left(