## 3 Materials and Methods  
### Transition Design Equations and Parameters  
A two-port rectangular waveguide WR-28 with a broadside dimension (a) of 7.112 mm and width (b) of 3.556 mm to microstrip line transition was designed for full Ka-band operation, using ANSYS HFSS software. The waveguide length of the design was chosen as an integer multiple of \(\lambda_{\mathrm{g}}/2\). The basic analytical calculations were made using standard mathematical relationships and the principle of an air-filled 500 \(\Omega\) WR-28 (RWG) and 50 \(\Omega\) microstrip line [1; 3; 4; 5; 6]. A window or slot was cut at a distance \(\lambda_{\mathrm{g}}/4\) from the shorting (back short) end of the RWG. The slot height and width were optimized using thumb rules for better transmission between the MSL and RWG, as depicted in Figure 1a,b. A 50 \(\Omega\) microstrip was fabricated on double-layered RT Duroid 5880 substrates using photolithography and etching. The optimized dimensions of the 3D-transition model are shown in Table 1 and illustrated in Figure 1a,b. The substrate used at 30 GHz was a soft substrate Roger RT 5880 (Duroid) with a loss tangent of 0.0009. The microstrip line comprised a partial ground plane that just touched the slot/window placed at a back-short distance (d = \(\lambda_{\mathrm{g}}/4\)) from the guided wavelength of the WR-28 at the bottom surface, and a patch with a fishtail-shaped probe was printed on the top surface of the substrate, as shown in Figure 2a-c. At a Ka-band frequency of 30 GHz, the following design equations were used for the microstrip line and waveguide analysis [14; 15; 16]:  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Part Name** & **Dimension Designation** & **Dimension Values (mm)** \\ \hline \multirow{8}{*}{Rectangular Waveguide (RWG)} & Waveguide type & WR-28 \\  & Dielectric & Air-filled PEC (AI) \\  & Broadside & a = 7.112 \\  & Narrow side & b = 3.556 \\  & Length & L\({}_{\mathrm{WG}}\) = 19.9 \\ \hline \multirow{8}{*}{Microstrip Line (MSL)} & Material & Copper (Cu) \\  & Width & w = 0.37 \\ \cline{1-1}  & Length & L\({}_{\mathrm{strip}}\) = 10.4 \\ \cline{1-1}  & Thickness & t = 0.035 \\ \cline{1-1}  & Fish-tail lengths (upper) & L\({}_{\mathrm{FT1}}\) = L\({}_{\mathrm{FT4}}\) = 2.414 \\ \cline{1-1}  & Fish-tail lengths (lower) & L\({}_{\mathrm{FT2}}\) = L\({}_{\mathrm{FT3}}\) = 1.9416 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Dimensions of the Ka-band MS Line-to-RWG Transition.  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Part Name** & **Dimension Designation** & **Dimension Values (mm)** \\ \hline \multirow{4}{*}{E-plane 90\({}^{\circ}\) broadside Waveguide Slot} & Dielectric & Air-filled \\  & Width & W\({}_{\text{Slot}}\) = 7.0 \\  & Length & L\({}_{\text{Slot}}\) = 1.015 \\  & Height & H\({}_{\text{Slot}}\) = 0.381 \\ \hline \multirow{4}{*}{Ground} & Material & Copper (Cu) \\  & Width & W\({}_{\text{gnd.}}\) = 7.0 \\  & Length & L\({}_{\text{gnd.}}\) = 9.629 \\  & Thickness & t = 0.035 \\ \hline \multirow{4}{*}{Substrate} & Material & Roger RT Duroid 5880 (tm) \\  & Width & W\({}_{\text{sub.}}\) = 7.0 \\  & Length & L\({}_{\text{sub.}}\) = 14.2 \\  & Height & h = 0.127 \\ \hline \multirow{4}{*}{Back Short} & Back-short distance & d = \(\lambda_{\text{g}}/4\) = 1.9 \\  & Broadside & a = 7.112 \\ \cline{1-1}  & Narrow side & b = 3.556 \\ \hline \hline \end{tabular}
\end{table}
Table 1: _Cont._  
Figure 1: (a) Rectangular – waveguide to fishtail-shaped microstrip line transition at 30 GHz. (b) Side view of the rectangular waveguide to fish-tail-shaped microstrip line transition at 30 GHz.  
Figure 2: _Cont._