## 2 Transition Design

Table 2: Equivalent circuit parameters for the iris configurations. Figure 4: Symmetric E-plane iris equivalent circuit. Figure 5: _Cont._  
A least-squares polynomial interpolation of the computed circuit element values was to be performed to have an estimate of the iris scattering matrices for any height value in the above mentioned range. The fitting polynomials grades were selected to be \(n=3\) for each parameter as a trade-off between interpolation accuracy and computational cost in the following design stage. Fitting polynomials for \(C\), \(L\), and \(\theta\) were:  
\[C(h)=\frac{c_{3}h^{3}+c_{2}h^{2}+c_{1}h+c_{0}}{h-4.7625} \tag{1}\]  
\[L(h)=l_{3}h^{3}+l_{2}h^{2}+l_{1}h+l_{0} \tag{2}\]  
\[\theta(h)=t_{3}h^{3}+t_{2}h^{2}+t_{1}h+t_{0} \tag{3}\]  
Note that (1) differs from the cubic curves (2) and (3) because of a vertical asymptote given by the geometrical upper limit for the iris height, which completely shorts the waveguide when \(h=wgb/2\). Once the interpolation coefficients \(c_{i}\), \(l_{i}\), and \(t_{i}\) are calculated, Equations (1)-(3) can be easily implemented in any circuit simulation software and scattering matrices with an accuracy close to that of a full-wave simulator can be obtained. The polynomial coefficients and interpolation results are respectively shown in Table 3 and Figure 6 for completeness sake. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{4}{c}{**Interpolation Coefficients**} \\ \hline \(c_{3}\) & \(c_{2}\) & \(c_{1}\) & \(c_{0}\) \\ \(1.364\) & \(-6.294\) & \(-4.894\) & \(-4.044\) \\ \(l_{3}\) & \(l_{2}\) & \(l_{1}\) & \(l_{0}\) \\ \(-12.950\) & \(126.500\) & \(-413.900\) & \(482.100\) \\ \(t_{3}\) & \(t_{2}\) & \(t_{1}\) & \(t_{0}\) \\ \(-0.051\) & \(0.428\) & \(-2.319\) & \(0.503\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Polynomial interpolation coefficients for the circuit element values \(C\), \(L\), and \(\theta\). Figure 6: Fitting values (red dots) and polynomial interpolation (solid line) for the circuit element parameters. (**a**) C; (**b**) \(L\); (**c**) \(\theta\). Figure 9: Full-wave simulated scattering parameters for three optimized matching networks. **(a)** One iris; **(b)** Two irises; **(c)** Three irises. The 20-dB return loss bandwidth improves with an increasing number of irises. Fractional bandwidths of 3.28%, 17.59%, and 24.51% were obtained in the three cases. Insertion loss was better than 0.2 dB and flatter in the frequency ranges corresponding to good input port matching. A comparison in terms of bandwidth and dimensions between this device and right-angle RWG-to-SIW transitions in referenced works [12; 13; 14; 15; 16; 17; 18; 19; 20; 21] is presented in Table 6. All geometrical parameters refer to the SIW. In particular, the SIW height (i.e. the substrate thickness) is defined as fractions of the cut-off wavelength \(\lambda_{c}=2a_{e}\sqrt{\varepsilon_{r}}\) in the ratio \(r\). As the substrate thickness decreases, the SIW broad walls get closer to one another. The lower the ratio \(r\) the lower the SIW section input impedance, and as a consequence the harder is to obtain good matching over a wide band. To perform a fair comparison between the various solutions reported in the literature [12; 13; 14; 15; 16; 17; 18; 19; 20; 21] the bandwidth performance are therefore normalized with the introduction of the merit factor \(F=BW_{20 dB}/r\). It is worth mentioning that simulated results for standalone transitions are reported, in order to provide a comparison among data not affected by manufacturing errors in the fabricated prototypes. Though the fractional bandwidth of the present work is not the largest among the list of referenced papers, the high performance of the proposed solution becomes evident when considering also the ratio \(r\) and the merit factor \(F\).