## 2 Design Goals and Procedure  
The proposed antenna radiates at 2.4 GHz, with a desired 10% minimum bandwidth and 95% minimum radiation efficiency, through a floating metallic thin patch with \(h_{air}\) mm of an air dielectric (\(\epsilon_{r_{air}}=1\)) between the patch and the ground plane. The patch is implemented with a fully metalized layer of a substrate of \(h_{patch}\) mm total thickness (considering the 17 \(\upmu\)m of cladding and 9 \(\upmu\)m of copper electrodeposition to completely metalize the whole patch); in fact, the patch can also be completed with just a copper sheet of \(h_{patch}\) thickness and sufficient stiffness. For the feeding section, a microstrip line of a \(w_{ms}\) mm width is used; it is set in another layer of substrate (Rogers-4003C, \(\epsilon_{r_{sub}}=3.55\), \(\tan\delta=0.0027\), and 17 \(\upmu\)m of cladding for each side) of an \(h_{ms}\) mm thickness. A proper fastener to hold and stabilize the floating patch is needed, which is the feeding pin rod that injects the power into the patch from the microstrip line; the design and adjustment of this element are critical for achieving a good impedance matching and a successful device with high values of efficiency. The well-known design equations for a microstrip patch included in Chapter 14 of [10] and summarized in [25] are used to calculate the initial values of the patch size; the used equations give only an initial approximation of the dimensions, so the device is modeled and numerically evaluated through optimization algorithms using CST Studio Suite to obtain the final dimensions and the adjustment of the pin rod's position in the patch. The appearance and design parameters of the prototype are illustrated in Figure 1. The three parts composing the prototype are shown: the lower layer with a microstrip line whose ground plane is also used as the ground plane for the patch; the metallic floating patch optimized to radiate at 2.4 GHz; and the pin rod holding the patch and feeding it from the microstrip, whose insertion position and isolation from the ground plane are optimized to maximize the impedance matching. The patch is just a completely metalized piece of Rogers-4003C of a thickness of 0.813 mm, giving a total thickness of \(h_{p}=0.866\) mm; its shape is a rectangle of the size \(L_{p}\times W_{p}\) at the distance \(h_{air}\) over the ground of the microstrip layer. This distance is chosen to be \(h_{air}=8.13\) mm to ease the manufacture (it is the piling of ten substrates of a height of 0.813 mm). For a chosen radiation frequency of \(f_{r}=2.4\) GHz and the air as the patch dielectric (\(\epsilon_{r_{eff}}=\epsilon_{r_{air}}=1\)), the design equations for the patch are simplified as follows:  
\[W_{p}=\frac{c}{2f_{r}} \tag{1}\]  
\[\Delta L=0.722h_{air}\frac{0.264+\frac{W_{p}}{h_{air}}}{0.8+\frac{W_{p}}{h_{air}}} \tag{2}\]  
\[L_{p}=\frac{c}{2f_{r}}-2\Delta L \tag{3}\]  
where \(W_{p}=62.457\) mm is the size of the radiating edges (y axis), \(L_{p}=51.459\) mm is the size of the non-radiating edges (x axis), \(\epsilon_{r_{eff}}\) is the effective dielectric constant (that matches \(\epsilon_{r_{air}}=1\) in this case), and \(\Delta L=5.499\) mm is the correction in size due to the fringing effects on the patch. The size of the patch reference plane (\(L_{s}\times W_{s}\)) must be greater than the patch; it is recommended to be at least approximately the size of the patch incremented six times by the value of the dielectric (\(L_{s}=L_{p}+6h_{air}\), \(W_{s}=W_{p}+6h_{air}\)). The microstrip layer is integrated into an \(h_{ms}=0.813\) mm thick substrate of the size \(L_{s}=120\) and \(W_{s}=110\) mm with a 50 \(\Omega\) impedance microstrip line on the bottom side of the length \(l_{ms}=40\) mm and width \(w_{ms}=1.784\) mm. The top side of this layer is the common ground plane for the microstrip and patch. Figure 1: Appearance and design parameters of the floating patch prototype.