## 3 Quarter-Mode SIW Antenna Design  
Having a properly designed SIW that guides the TE\({}_{10}\) mode, an SIW-based antenna can be designed and then further miniaturized by taking advantage of the symmetry of the TE\({}_{10}\) mode [10; 13; 14]. Figure 4a shows the E-field amplitude distribution inside a section of an SIW. As can be seen, the E-field is symmetric along the transverse and longitudinal axes. Next, Figure 4b shows the E-field distribution when the SIW section is cut in half. The open boundary condition along the transverse axis forms a pseudo-magnetic wall due to the large ratio of the SIW's width to height. With this virtual wall, the E-field distribution hardly changes, providing a half-mode SIW (HMSIW) [15; 16]. In addition, a quarter-mode SIW (QMSIW) can be realized by cutting the HMSIW along the longitudinal axis where the E-field is symmetric. This is described in Figure 4c. Figure 4: E-field amplitude plots of (**a**) a full-section of SIW, (**b**) a half-section cut in the transverse symmetric axis, and (**c**) a quarter-section cut in the longitudinal symmetric axis. Figure 3: Geometry of a section of SIW. Based on the QMSIW structure, the proposed antenna was designed, and its geometry is shown in Figure 5. Two copper plates (patch and ground) were separated by air. The size of the patch corresponded to the size of QMSIW at 88 MHz determined by Equation (1). There were three different kinds of cylindrical metallic posts in between the patch and ground: feeding probe, shorting pins, and via fence. The feeding probe was the inner conductor of a coaxial cable brought to the radio frequency (RF) source, while the shorting pins were directly connecting the patch and ground. The outer and inner conductors of the coaxial probe were electrically connected to the lower ground and upper patch, respectively. The diameter of the shorting pins was 1 mm, which corresponded to a gauge size of 18. Their locations were carefully chosen relative to the feeding probe location to induce in-phase currents, thus promoting an omnidirectional radiation pattern [17]. By adding the shorting pins, the antenna resonant frequency was shifted lower, thus the antenna size parameters \((W,L)\) were adjusted to shift it back. The QMSIW structure was formed by the via fence and two magnetic walls as depicted in Figure 5b. The via's spacing and diameter were determined by Equation (2). Table 1 lists the geometrical parameters and their values, which were finalized after optimizing the initial model using the full-wave EM simulation tool, Ansys HFSS. Several key parameters such as the ground and patch size, feeding probe and shorting pin locations, numbers of vias and its diameter were optimized one by one to reach the dimensional values in Table 1. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value (cm)** & **Parameter** & **Value (cm)** \\ \hline \(L\) & 125 & \(x_{\text{f}}\) & 62 \\ \(W\) & 125 & \(y_{\text{f}}\) & 73 \\ \(h\) & 9 & \(x_{\text{p}}\) & 6.2 \\ \(s\) & 5 & \(y_{\text{p}}\) & 33.3 \\ \(d\) & 2.5 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: QMSIW antenna parameters. Figure 5: Geometry of the proposed QMSIW antenna: (**a**) a bird’s-eye view, (**b**) a top view. Figure 6 shows a parametric simulation study on the reflection coefficient (S11) by varying the height, \(h\). The S11 was calculated in the frequency range from 60 to 120 MHz. One can observe that the S11 is getting lower at the frequency of interest, 88 MHz, as \(h\) increases from 6 cm to 9 cm, while other parameters remain the same. This implies a better impedance matching condition is achieved due to an increase of the shunt inductance and a decrease of the shunt capacitance by increasing \(h\)[18]. We set the height \(h\) = 9 cm, which is approximately 0.025 wavelength at 88 MHz to achieve a low-profile form factor. Figure 7 shows the effect of the shorting pins on the antenna's radiation pattern. There are 2D radiation patterns of the antenna along the azimuth plane with no, one, and two shorting pins.