# Wide-Band Wide-Beam Circularly-Polarized Slot-Coupled Antenna for Wide-Angle Beam Scanning Arrays
## 3 Design Methodology  
In order to address in a computationally-effective way the synthesis problem at hand, a customized implementation of the system-by-design (_SbD_) paradigm [32] is exploited and briefly summarized in the following. More specifically, the \"_Problem Formulation_\" _SbD_ functional block [32] is customized to (_i_) define a proper set of geometric descriptors of the _WWCS_ layout and (_ii_) formulate a suitable multi-objective cost function accounting for several user-defined requirements on both impedance matching and radiation features. Concerning (_i_), once the characteristics of the substrates (i.e., material/thicknesses) of the layers \(\ell_{l},l=1,\dots,\mathcal{L}\), and the width of the microstrip feeding line \(W_{f}\) are determined as detailed in [8] (p. 148, Equation 3.197) to yield the desired characteristic impedance \(Z_{0}\) (e.g., \(Z_{0}=50\) [\(\Omega\)]), the set \(\underline{\Omega}=\{\Omega_{k};\,k=1,\dots,K\}\) of geometric descriptors (Figures 1 and 2) is defined as \(\underline{\Omega}=\left\{R_{p},\,L_{f},\,L_{1},\,W_{1},\,L_{2},\,W_{2},\,D,\, \alpha,\,\beta\right\}\), \(\alpha\) [\(\alpha\triangleq 2\times\left(\frac{R_{r}}{L_{a}}\right)\)] and \(\beta\) [\(\beta\triangleq\frac{W_{r}}{\left(\frac{L_{a}}{L_{a}}-R_{r}\right)}\)]being auxiliary parameters (\(0<\alpha<1\); \(0<\beta<1\)) that avoid the generation of physically-unfeasible geometries for the secondary element by enforcing the constraints \(R_{r}<\left(\frac{L_{q}}{2}\right)\) and \(W_{r}<\left(\frac{L_{q}}{2}-R_{r}\right)\), respectively [Figure 2c]. The synthesis problem at hand can then be stated as follows:  
_WWCS Antenna Design Problem--_Determine the optimal setup of the degrees-of-freedoms (\(DoF\)s), \(\underline{\Omega}^{(opt)}\), such that the corresponding _WWCS_ radiator (_i_) exhibits a suitable impedance matching within the user-defined wide frequency range \(f_{\min}\leq f\leq f_{\max}\), (_ii_) radiates an azimuth-invariant wide-beam pattern suitable for _WASAs_, and (_iii_) implements a _LHCP/RHCP_ with high polarization purity within the half-space region (\(0\leq\theta\leq 90\)) \(\bigcup\) (\(0\leq\varphi\leq 360\)) [Figure 1a]. As for (_ii_), because of the conflicting requirements on the bandwidth and the radiation features as well as the non-linear dependence of these latter on \(\underline{\Omega}\), the original synthesis problem is recast into a global optimization one, where  
\[\underline{\Omega}^{(opt)}=\arg\biggl{\{}\min_{\underline{\Omega}}\Phi( \underline{\Omega})\biggr{\}} \tag{2}\]  
\(\Phi(\underline{\Omega})\) being the cost function, which quantifies the mismatch with the synthesis targets, given by  
\[\Phi(\underline{\Omega})=\sum_{\gamma\in\underline{\Gamma}}w_{\gamma}\Phi_{ \gamma}(\underline{\Omega}) \tag{3}\]  
where \(\underline{\Gamma}=\{S_{11},\,HPBW,\,AR,\,XPD\}\), and \(w_{\gamma}\) is a real weight associated with the \(\gamma\)-th cost function term \(\Phi_{\gamma}(\underline{\Omega})\). More in detail, the impedance bandwidth term of the cost function \(\Phi\) (\(\gamma=S_{11}\)) is defined as follows  
\[ \begin{array}{rl}\Phi_{S_{11}}(\underline{\Omega})=\frac{1}{Q}\sum_{q=1}^{Q }\frac{\left\{\left|S_{11}^{(dB)}(f_{q}\big{|}\underline{\Omega})\right|-S_{1 1}^{th}\right\}}{\left|S_{11}^{th}\right|}\times\\ \times\mathcal{H}\biggl{\{}\left|S_{11}^{(dB)}(f_{q}\big{|}\underline{\Omega} )\right|-S_{11}^{th}\biggr{\}}\end{array}  \tag{4}\]  
where  
\[S_{11}^{(dB)}(f_{q}\big{|}\underline{\Omega})=20\times\log_{10}\biggl{(}\frac {Z_{in}\big{(}f_{q}\big{|}\underline{\Omega})-Z_{0}}{Z_{in}\big{(}f_{q} \big{|}\underline{\Omega})+Z_{0}}\biggr{)} \tag{5}\]  
is the reflection coefficient at the antenna input port, \(Z_{in}\big{(}f_{q}\big{|}\underline{\Omega}\big{)}\) being the input impedance.