## 2 Transition Design
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Description** & **Value (mm)** \\ \hline \(a\) & SIW width & 12.500 \\ \(ae\) & Equivalent RWG width & 11.865 \\ \(d\) & Via hole diameter & 1.000 \\ \(p\) & Via hole pitch & 2.000 \\ \(s\) & Short distance & 2.500 \\ \(b\) & Substrate height & 1.575 \\ \(t\) & Metal thickness & 0.045 \\ \(sl\) & Slot length (vertical) & 11.000 \\ \(sl\) & Slot length (horizontal) & 3.500 \\ \(sw\) & Slot width & 0.500 \\ \(wga\) & RWG width & 19.050 \\ \(wgb\) & RWG height & 9.525 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: RWG-to-SIW transition geometrical parameters. Figure 3: RWG symmetric E-plane iris (side view, air inside metallic background). Figure 2: RWG-to-SIW transition simulated scattering parameters (no matching elements). increased computational cost. Since a complete full-wave design of a matching network with more than one iris would be extremely time-consuming and convergence could not be ensured, the idea was to perform a hybrid synthesis using both simulators. In particular, a proper equivalent circuit associated to the iris geometrical parameters was extracted and then employed in the design of a wideband matching network at a circuit level, and finally the output geometrical parameters were used in the full-wave design. Note that a circuit simulation does not take into account potential higher order modes interactions, thus a final full-wave optimization may be necessary. Anyway, given this first estimate, this process would lead to a very good return loss performance quite rapidly. It is well known from waveguide theory that a symmetric E-plane iris can be schematized as an equivalent shunt capacitance [24; 25], but this statement holds true only in a neighborhood of the center frequency. As a consequence, such circuits cannot accurately predict the iris behavior in terms of scattering parameters over the large bandwidth required in many applications and, for this reason, another model must be derived. In this design the iris is to be characterized when its height \(h\), which is the only varying parameter, is swept from 0.5 mm to 4.5 mm with a step of 0.5 mm. A total of nine scattering matrices are extracted from full-wave simulations and, for each geometrical configuration, it can be verified from circuit simulations that a suitable broadband modeling of the iris can be achieved by introducing a lumped shunt inductance. The equivalent circuit is shown in Figure 4.  
It consists of one capacitance \(C\) and one inductance \(L\) connected so as to form a shunt \(LC\) resonator. In addition to this, two identical transmission lines with normalized impedance \(Z_{0}=1\) and electrical length \(\theta\) are introduced to account for second order contributions to the phase of the electric field. For each sampled iris configuration, defined by the parameter \(h\), an optimization of the three circuit element values \(C\), \(L\), and \(\theta\), is to be performed so that the scattering matrix of the equivalent circuit fits the corresponding one from the full-wave simulation over a wide bandwidth. The fitting procedure results are listed in Table 2.  
The proposed equivalent circuit performance for a selected iris geometry are reported in Figure 5, both in magnitude and phase, and compared with the standard shunt capacitance model. A very good agreement was verified on a 40% fractional bandwidth. It is worth mentioning that this wideband fitting cannot be obtained with the standard single-element model. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Iris Height (mm)** & \(C\) **(pF)** & \(L\) **(pH)** & \(\theta\) **(deg)** \\ \hline
0.500 & 1.776 & 314.162 & \(-\)0.512 \\
1.000 & 3.838 & 165.103 & \(-\)1.526 \\
1.500 & 6.438 & 100.696 & \(-\)2.192 \\
2.000 & 10.131 & 63.611 & \(-\)2.765 \\
2.500 & 14.894 & 44.643 & \(-\)3.374 \\
3.000 & 22.089 & 29.057 & \(-\)4.039 \\
3.500 & 31.240 & 22.417 & \(-\)4.537 \\
4.000 & 49.427 & 14.315 & \(-\)5.198 \\
4.500 & 110.015 & 6.473 & \(-\)5.883 \\ \hline \hline \end{tabular}
\end{table}