## 2 Circuit Modeling of the BPF and the WG ZOR, and Its Geometry  
### Required Specifications of the BPF for Satellite Wireless Communication  
A bandpass filter is needed in the transponder as the communication system for the satellites with the following assessment items and values. The mathematical approach discovers the function of Chebyshev-type with \(N_{\text{order}}\) = 8 is quite close to the specifications in Table 1 for the amplitude of the bandpass filter as in [13]. It is expressed with the circuit network. \[ \begin{array}{l}t(s)=\frac{5.356\times 10^{8}s-8.87\times 10^{19}}{s^{2}-2.1185\times 10^{9}s-2.193\times 10^{12}}+\frac{5.896\times 10^{8}s+9.092\times 10^{19 }}{s^{2}+1.172\times 10^{9}s+2.288\times 10^{12}}\\ +\frac{-1.08\times 10^{8}s+4.25\times 10^{15}}{s^{2}+8.982\times 10^{8}s+2.138 \times 10^{12}}+\frac{-1.14\times 10^{8}s-4.552\times 10^{19}}{s^{2}+9.329\times 10^{8}s +2.306\times 10^{124}}\\ +\frac{7.573\times 10^{8}s-5.973\times 10^{8}}{s^{2}+5.807\times 10^{8}s+2.095 \times 10^{124}}+\frac{8.043\times 10^{8}s+6.506\times 10^{18}}{s^{2}+4.6 153\times 10^{8}s+2.353\times 10^{27}}\\ +\frac{-2.253\times 10^{8}s-3.449\times 10^{18}}{s^{2}+2.015\times 10^{5}s+2.073 \times 10^{124}}+\frac{8.043\times 10^{8}s+5.066\times 10^{18}}{s^{2}+4.6 153\times 10^{8}s+2.353\times 10^{27}}\\ +\frac{-2.253\times 10^{8}s-3.449\times 10^{18}}{s^{2}+2.015\times 10^{5}s+2.07 3\times 10^{124}}+\frac{8.043\times 10^{8}s+5.066\times 10^{18}}{s^{2}+4.6 153\times 10^{8}s+2.353\times 10^{27}}\\ \end{array}  \tag{1}\]  
Figure 1 has eight parallel LC resonators cascaded through transmission-line segments as coupling elements. Equation (1) is the transfer function of the curve fitting the amplitude. The long expression is decomposed into a few terms. The maximum order of the denominator of \(t(s)\) is the same as the number of the resonators. As is done for filter designs, the transmission-line segments or waveguide sections of nearly quarter-wavelength work as an inductor, as mentioned in [14], and put between the resonators for the bandwidth. In the PCB filters and waveguide filters, the LC resonators seen in Figure 1 are changed to the distributed elements, the lengths of which are half-wavelength and odd multiple half-wavelengths. This is the conventional rule which is mostly used owing to simplicity and convenience. If, and only if, this rule is kept, the bandpass filter ends up with a long structure when a higher-order filtering is required, like the specifications in Table 1. The configuration of Figure 1 with the parallel LC resonators leading to conventional resonators is modified into something with new resonators as follows. \begin{table}
\begin{tabular}{c c} \hline \hline
**Item** & **Value** \\ \hline Insertion Loss & \(\leq\)1 dB \\ Center Frequency & 7.5 GHz \\ Bandwidth & 500 MHz \\ Reflection coefficient & \(\leq\)\(-\)15 dB \\ Out of Band Rejection (or Skirt) & \(\leq\)\(-\)40 dB(\(\mathrm{\{c}}\pm\) 500 MHz) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Specifications the design will meet. Figure 1: Equivalent circuit network of the bandpass filter. The resonators of Figure 2a will be filled with new ones. By satisfying the requirement, circuit calculation gives the unknowns of Figures 1 and 2 the values in Table 2. Figure 2b presents S\({}_{11}\) as the reflection coefficient and S\({}_{21}\) as the transmission coefficient of the device. This ideal circuit results in the satisfactory frequency response. The circuit calculation is operated based on the following mathematical procedures. \[ \begin{bmatrix}ABCD_{\text{Tot.}}^{\text{Circ.}}\end{bmatrix}  = \begin{bmatrix}ABCD_{\text{Res.1}}^{\text{Circ.}}\end{bmatrix} \begin{bmatrix}ABCD_{\text{Im}_{12}}^{\text{Circ.}}\end{bmatrix} \begin{bmatrix}ABCD_{\text{Res.2}}^{\text{Circ.}}\end{bmatrix} \cdots \begin{bmatrix}ABCD_{\text{Im}_{\text{Im}_{\text{Im}}}}^{\text{Circ.}} \end{bmatrix}\begin{bmatrix}ABCD_{\text{Res.8}}^{\text{Circ.}}\end{bmatrix}  \\ABCD_{\text{Res.1}}^{\text{Circ. }}\end{bmatrix}\] \[