# Extremely Low-Profile Monopolar Microstrip Antenna with Wide Bandwidth
## 2 Antenna Design and Analysis
### Parametric Analysis  
The parametric results of the proposed antenna are shown in Figure 4. A relative dielectric constant of \(\varepsilon r\), a feed patch size of \(p\), a coupling gap size of \(g\), a ground pad under the feed patch of \(l\)\(f\) and a width occurring from ground slots of \(w\) were investigated. For the parametric study, the following initial values of the dimension were chosen for clear degenerated modes: \(\varepsilon r=2.2\), \(p=10.0\) mm, \(g=0.2\) mm, \(l\)\(f=3.0\) mm and \(w=0.2\) mm. In Figure 4a, the relative permittivity of the substrate, \(\varepsilon r\), changed from 2.2 to 6.15 with the fixed \(t\) of 0.254 mm. Overall, since the dielectric constant of the substrate forms a guided wavelength in which the wavelength in free space is inversely proportional to the square root of the dielectric constant, it can be seen that the resonant frequencies decreased from 5.5 GHz to 3.5 GHz as the dielectric constant increased. This is due to the fact that, compared to the length of the wavelength in the free space, the length of the wavelength in the guide wavelength inside the dielectric substrate was reduced. In particular, when the coupling gap of 0.2 mm which determines the effective dielectric constant computed from the air upper side of 1.0 and the substrate lower side of 2.2 are mathematically equal to the substrate thickness of 0.254 mm with a dielectric constant of 2.2, the degenerated modes were strongly coupled. Thus, it can be seen that all of the degenerated even and odd resonant frequencies were affected from the top front patch configuration for the lower even TM\({}_{02}\) mode and substrate cavity resonances between the top patch and the bottom ground metal capacitance for the upper odd TM\({}_{02}\) mode. Figure 4: Simulated results of the proposed antenna: (**a**) relative dielectric constant, (**b**) feed size, (**c**) gap size and (**d**) ground soldering pad. As can be seen in Figure 4b, as the length of six isosceles trapezoid radiators increased, which caused the one-centered hexagonal patch size to increase, both of the even and odd mode resonant frequencies of the proposed antenna decreased. The main reason for this result is that the physical patch size determines the electrical wavelength of the resonant frequencies for the proposed antenna. In other words, as the size of the trapezoidal radiator increases, the guided wavelength increases electrically due to the TM\({}_{02}\) mode of the boundary condition determined by the trapezoidal radiator patch. However, the matching effect was different between even and odd TM\({}_{02}\) modes, and the impedance matching results are shown in different amounts. Therefore, 11.0 mm-long patch length showed a lower resonant frequency of 4.8 GHz, and 9.0 mm-long patch length produced the highest resonant frequency of 6.8 GHz. Figure 4c exhibits the frequency response when the gap of \(g\) between each patch was varied from 0.1 mm to 0.5 mm. In the case of \(g=0.5\) mm, due to the asymmetric cavity geometry that was shown in the cuboid, the \(g\) was 0.5 mm, the \(t\) was 0.254 mm and the degenerated modes merged into a single dominant frequency. However, when the \(g\) and the \(t\), which was also affected by the effective dielectric constant, were almost identical, the identifiable even TM\({}_{02}\) and odd TM\({}_{02}\) modes could be separated. Additionally, as the other parameters were not changed, the couplings were kept. It can be seen that the amount of electrical coupling of the even TM\({}_{02}\) mode and odd TM\({}_{02}\) mode varies with the \(g\), since the \(g\) generates coupling between top radiating patches and also controls the coupling cavity between the top patch and bottom ground. Therefore, if the operating frequencies of the degenerated TM\({}_{02}\) modes are determined through the proposed antenna radiator and feeding patch, the \(g\) should be maximized to obtain broad operating bandwidth.