## 3 Design Examples for the Proposed Synthetization Approach
### Design Example for the Conventional 5-Section IT  
The topology of 5-section IT is shown in Figure 5. From Equation (4), \(A_{T}\), \(B_{T}\), \(C_{T}\) and \(D_{T}\) can be calculated as follows:  
\[A_{T}=a_{5}\cos^{5}\theta+a_{3}\cos^{3}\theta+a_{1}\cos\theta \tag{19}\]  
\[B_{T}=\frac{j}{\sin\theta}(b_{6}\cos^{6}\theta+b_{4}\cos^{4}\theta+b_{2}\cos^ {2}\theta+b_{0}) \tag{20}\]  
\[C_{T}=\frac{j}{\sin\theta}(c_{6}\cos^{6}\theta+c_{4}\cos^{4}\theta+c_{2}\cos^ {2}\theta+c_{0}) \tag{21}\]  
\[D_{T}=d_{5}cos^{5}\theta+d_{3}cos^{3}\theta+d_{1}cos\theta \tag{22}\]  
\(a_{p}\), \(d_{p}\) (\(p=1,3,5\)), \(b_{q}\) and \(c_{q}\) (\(q=0,2,4,6\)) are the coefficients of the polynomial functions which are determined by characteristic impedances of \(Z_{e\tau 1}\), \(Z_{o41}\), \(Z_{e\tau 2}\), \(Z_{o42}\), \(Z_{e\tau 3}\), \(Z_{o43}\), \(Z_{e\tau 4}\), \(Z_{e\tau 5}\) and \(Z_{o45}\) (the detailed equations are listed in Table 1 in Appendix A).  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Number of Substituted CLSs by TLs** & **Number of Substitutable Combinations** \\ \hline
1 & \(C_{N}^{1}\) \\
2 & \(C_{N}^{2}\) \\... &... \\ \(N-3\) & \(C_{N}^{N-3}\) \\ \(N-2\) & \(C_{N}^{N-2}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: The number of substitutable cases for \(N\)-section IT.  
Figure 5: Topology of \(N\)-section IT [28] under the condition of \(N=5\).  
After the complicated arithmetical operation from Equations (10) to (13), \(F_{circuit}\) can be written as  
\[\begin{array}{rcl}F_{circuit}&=&\frac{1}{T_{1}T_{2}T_{3}T_{4}T_{5}}(X_{5}\cos^ {5}\theta+X_{3}\cos^{3}\theta+X_{1}\cos\theta)\\ &+&\frac{j}{T_{1}T_{2}T_{3}T_{4}T_{5}\sin\theta}(Y_{6}\cos^{6}\theta+Y_{4} \cos^{4}\theta+Y_{2}\cos^{2}\theta+Y_{0})\end{array} \tag{23}\]  
where  
\[X_{p}=(a_{p}-k\cdot d_{p})/2\sqrt{k} \tag{24}\]  
\[Y_{q}=(b_{q}-k\cdot c_{q})/2\sqrt{k} \tag{25}\]  
Similarly, from Equations (10) to (13), \(F_{formula}\) can be written as  
\[F_{formula}=\varepsilon\cos(5\varphi+\xi)=\frac{\varepsilon}{\sin\theta}[u_ {6}\cos^{6}\theta+u_{4}\cos^{4}\theta+u_{2}\cos^{2}\theta+u_{0}] \tag{26}\]  
where  
\[u_{6}=\frac{16}{\cos^{6}\theta_{c}}(1+\sqrt{1-\cos^{2}\theta_{c}}) \tag{27}\]  
\[u_{4}=-\frac{4}{\cos^{4}\theta_{c}}(5+7\sqrt{1-\cos^{2}\theta_{c}}) \tag{28}\]  
\[u_{2}=\frac{1}{\cos^{2}\theta_{c}}(5+13\sqrt{1-\cos^{2}\theta_{c}}) \tag{29}\]  
\[u_{0}=-1 \tag{30}\]  
From \(F_{circuit}=F_{formula}\), the seven design conditions can be summarized as follows to maintain the Chebyshev response.  
\[\left\{\begin{array}{l}a_{5}-k\cdot d_{5}=0\\ a_{3}-k\cdot d_{3}=0\\ a_{1}-k\cdot d_{1}=0\end{array}\right.\text{and}\left\{\begin{array}{l}(b_ {6}-k\cdot c_{6})/2\sqrt{k}=\varepsilon u_{6}\\ (b_{4}-k\cdot c_{4})/2\sqrt{k}=\varepsilon u_{4}\\ (b_{2}-k\cdot c_{2})/2\sqrt{k}=\varepsilon u_{2}\\ (b_{0}-k\cdot c_{0})/2\sqrt{k}=\varepsilon u_{0}\end{array}\right. \tag{31}\]  
For the section number \(N=5\), there are 10 parameters (\(Z_{e\sigma 1}\), \(Z_{\alpha 1}\), \(Z_{e\sigma 2}\), \(Z_{\alpha 2}\), \(Z_{e\sigma 3}\), \(Z_{\omega 3}\), \(Z_{e\sigma 4}\), \(Z_{\omega 4}\), \(Z_{e\sigma 5}\) and \(Z_{\omega 4}\)5) corresponding to seven design conditions for the Chebyshev response; thus, three variable parameters can be utilized to simplify the circuit topology.