## 2 Antenna Design
The table also shows that the radiation efficiency increased as the electrical size of the antenna increased with a thicker h.  
The final design parameters are presented in Table 2. The antenna resonated at 2.45 GHz with an impedance bandwidth of 1.2% with a \(-\)10 dB criterion, as shown in Figure 4. It was electrically small with its electrical size (\(ka\)) of 0.47 at the resonant frequency (\(k\), free-space wave propagation constant; \(a\), radius of the imaginary sphere enclosing the antenna). The simulated radiation pattern at the elevation and azimuth planes will be shown in a later section with the measurement results. The designed antenna showed a quasi-isotropic radiation pattern with a maximum gain deviation of 3.01 dB as shown in Figure 5. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter (mm)** & **h = 0.32** & **h = 0.72** & **h = 1.12** \\ \hline Frequency (GHz) & 2.30 & 2.45 & 2.51 \\ \hline Gain deviation (dB) & 2.66 & 3.01 & 3.09 \\ \hline Radiation Efficiency (\%) & 81.3 & 82.6 & 88.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter study results on h.  
\begin{table} \begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline M1 & 8.4 & W3 & 2.4 \\ \hline M2 & 13.8 & W4 & 2 \\ \hline M3 & 12.1 & W5 & 3.15 \\ \hline M4 & 6.7 & r & 9.25 \\ \hline W1 & 2.2 & h & 0.72 \\ \hline W2 & 2 & \(\alpha\) & \(271.06^{\circ}\) \\ \hline \hline \end{tabular}  Note that all the units are in millimeters except \(\alpha\), which is in degrees. Figure 3: Parameter study results on h: (**a**) Reflection coefficient. (**b**) Input impedance. Lastly, the axial ratio (AR) of the antenna at the resonant frequency was calculated in a spherical space and is plotted in Figure 6 to show the spatial polarization dependency. We found that a circular polarization with AR \(<\) 3 dB was created at certain points but overall linear polarization was formed. This can be explained by the antenna configuration with the electric dipole and magnetic dipole being along the \(x\)-axis and the \(z\)-axis, respectively. The antenna was linearly polarized when the field components of the two dipoles were in the same magnitude and phase in the xy-plane (i.e., \(\theta\) = 90\({}^{\circ}\)) and  
Figure 4: Simulation and measurement results of the reflection coefficients of the proposed antenna. Figure 5: Simulation result of a gain deviation over the entire space at the resonant frequency. Figure 6: Simulation results of the axial ratio over the entire space. the yz-plane (i.e., \(\varphi\) = 90\({}^{\circ}\)). This showed a circular polarization at the points in the xz-plane, such as when \(\theta\) is 45\({}^{\circ}\) and 135\({}^{\circ}\) and \(\varphi\) is 0\({}^{\circ}\) and 180\({}^{\circ}\). These occurred where the electric field components of the electric dipole E\({}_{\theta}\) and magnetic dipole E\({}_{\varphi}\) met with same magnitude and 90\({}^{\circ}\) phase shift. This showed that the proposed antenna can generate the quasi-isotropic radiation pattern from the sum of two different polarizations from the two different dipole configurations. This also can be verified from the pattern measurement (See Figure 9c). Thus, the proposed antenna shows a quasi-isotropic radiation pattern not only in terms of its magnitude but also in terms of polarization.