# Miniaturized Compact Reconfigurable Half-Mode SIW Phase Shifter with PIN Diodes
## 2 Design of the HM-SIW Phase Shifter
### Transmission Line Topology  
Figure 1 illustrates the geometric parameters of a HM-SIW structure that transfers a desired signal. The width of the transmission line can be reduced when using a half-mode SIW, but on the other hand, most of the SIW features are still maintained. Instead of vias in both corners, one edge will be an open aperture.  
Dividing a SIW transmission line in half creates the HM-SIW. On the open aperture plane, the magnetic field is nearly parallel, which creates an equivalent perfect magnetic conductor. This topology can handle a lower amount of signal transmission power and requires a smaller frequency wavelength compared to the thickness of the substrate. Both of the aforementioned facts are acceptable for electronically controlled phase shifter designs and the desired operation frequency of 4-8 GHz.  
In addition, the port widths are configured to transmit 50 \(\Omega\) from microstrip line theory [40]. This configuration is being continued to prepare a constant matching feature to be used with the HM-SIW structure. A typical microstrip line has been used to feed the proposed structure. For adequate impedance matching, the microstrip line segment connecting the SIW has been tapered. We refer to the employed structure as the microstrip to SIW transition [41].  
Starting from the cutoff frequency and exploiting the operating conditions of a SIW line, width (\(W_{g}\)), pitch, or distance between the vias (\(p\)), and via diameter (\(d\)) can be formed. Equations (1)-(9) describe the conditions that govern SIW operation in dominant mode [42]. Then, by exploiting the phenomenon of the magnetic wall, we can define HM-SIW [43].  
\[\lambda_{g}=\frac{c_{0}}{\sqrt{\varepsilon_{r}}}\;, \tag{1}\]  
\[\frac{\lambda}{2}\;<\;W_{\mathit{eff}}\;<\;\lambda, \tag{2}\]  
\[p\;<\;d(\text{Physically realizable condition of vias.}), \tag{3}\]  
\[\frac{p}{\lambda_{g}}\;<\;0.25(\text{avoiding bandgap}), \tag{4}\]  
\[d<\frac{\lambda_{g}}{5},\;\text{and}\;p\leq 2d\;(\text{reducing radiation loss}), \tag{5}\]  
\[\frac{p}{\lambda_{c}}>0.05\;\;(\text{avoiding over perforation}), \tag{6}\]  
\[W_{\mathit{eff}}=W_{g}-1.08\frac{d^{2}}{p}+0.1\frac{d^{2}}{W_{g}}\;, \tag{7}\]  
Figure 1: Schematic of the proposed HM-SIW structure from simulation software.  
\[\frac{p}{d}<3,\text{ and }\frac{d}{W_{g}}<0.2\text{ (accurate range)}, \tag{8}\]  
\[L_{\textit{eff}}=L-1.08\frac{d^{2}}{p}+0.1\frac{d^{2}}{L}\text{,} \tag{9}\]  
where \(c_{0}\), \(\lambda_{g}\), \(L_{\textit{eff}}\), and \(W_{\textit{eff}}\) are, respectively, the speed of light, guided wavelength, effective length, and width [44]. Equations (1) and (2) are applied to calculate the transmission line geometry for 4 GHz when modeling SIW with a dielectric-filled rectangular waveguide. Then, the \(W_{\textit{eff}}\) is split in half for a half-mode SIW. Furthermore, the diameter of the via (d) and the distance between the vias (\(p\)) are set up with Equations (3)-(9) and the fabrication limits. The diameter of the vias holes is \(d\) = 1.016 mm, and the distance between two vias centers is \(p\) = 1.5 mm. Each vias is 0.492 mm away from the metal edge, and \(W_{g}\) = 15 mm.