## 2 Antenna Design and Theoretical Analysis
However, the reflection amplitude of \(f_{21}\) gradually increases, and the reflection coefficient of \(f_{22}\) exhibits the characteristics of improvement and then deterioration with the increase in \(H2\). Figure 5d shows the graph of \(f_{3}\) varying with \(H2\). It can be seen that, when \(H2\) increases from 0 mm to 0.25 mm, \(f_{3}\) changes from a single-frequency point resonance of 5.06 GHz to a multi-frequency point resonance. As \(H2\) continues to increase,\(f_{3}\) gradually moves to a higher frequency. It can also be observed from [17] that the equivalent permittivity \(\varepsilon_{efq}\) of the substrates with an air layer can be calculated using the following formula:  
\[\varepsilon_{efq}=\frac{\varepsilon_{r}(H2+H3)}{H2\cdot\varepsilon_{r}+H3} \tag{1}\]  
Figure 3.0 The effect of different parameters on simulated S\({}_{11}\) of Ant. 3: (**a**) the effect of different R5 on frequency bands \(f_{1}\) and \(f_{2}\); (**b**) the effect of different R5 on frequency band \(f_{3}\); (**c**) the effect of different \(H2\) on frequency bands \(f_{1}\) and \(f_{2}\); (**d**) the effect of different \(H2\) on frequency band \(f_{3}\). where \(\epsilon_{r}\) and \(H3\) represent the relative permittivity and thickness of Substrate 2. When the thickness \(H2\) of the air layer increases, it can be seen from (1) that \(\epsilon_{eff}\) gradually decreases, causing the three frequency bands of Ant. 3 to shift to a higher frequency. Finally, 0.5 mm is selected as the optimal value of \(H2\) to ensure the broadband characteristics of Ant. 3. The vector current distribution on the surface of both the circular patch and the parasitic patch is given in Figure 6 to vividly explain the broadband characteristics of Ant. 3. It can be seen from Figure 6a that the current on the parasitic patch at 1.87 GHz mainly flows along the \(x\)-axis direction with the characteristics of in-phase. Taking into account the current distribution of the circular patch, it can be concluded that Ant. 3 at 1.87 GHz is working in TM\({}_{01}\) mode with a broadside radiation pattern. It should be noted that, after the parasitic patch is loaded, the influence of R5 on \(f_{1}\) is greater than that of R4. Due to the coupling effect between the parasitic patch and the circular patch, the original resonance points of Ant. 2 at 3.64 GHz and 4.43 GHz are close to each other, and finally, \(f_{21}\) and \(f_{22}\) of Ant. 3 are formed. Figure 6b shows the current distribution on the two patches when Ant. 3 operates at 3.84 GHz. The vector current on the circular patch flows from the center of the patch in two opposite directions along the \(x\)-axis, with out-of-phase characteristics. Moreover, the surface current on the parasitic patch also shows out-of-phase flowing from the center of the parasitic patch along the \(y\)-axis in two opposite directions. Attributed to the out-of-phase feature, Ant. 3 works at \(f_{2}\) with an omnidirectional radiation pattern, which is similar to the center-fed monopole patch antenna that resonates in the TM\({}_{02}\) mode [2]. The benefit of the weak coupling between the circular radiating patch and the parasitic patch, when Ant. 3 works at 5.38 GHz, is that the current distribution of the circular patch is diametrically opposite to that of the parasitic patch, as shown in Figure 6c. Meanwhile, Figure 7a shows the electric field distribution of Ant. 3 at 5.38 GHz, which is the same as the working mechanism of the TM\({}_{03}\) mode in Figure 7b. Combining Figures 6c and 7, it can be concluded that at 5.83 GHz the TM\({}_{03}\) mode dominates the working mode of Ant. 3 and the resonance bandwidth is 450 MHz. Figure 3.1 The simulated surface current distribution on the circular patch and parasitic patch of Ant. 3: (**a**) 1.87 GHz; (**b**) 3.84 GHz; (**c**) 5.38 GHz.