## 2 2-D Phased Array Design
### Design Model for the Configurations  
Two new 2-D configurations are proposed using 4 \(\times\) 9 CORPSs blocks, as shown in Figures 1 and 2. The array factor, considering the 2-D phased array design, is given by [28]:  
\[AF(\theta,\phi)=\sum_{n=1}^{N}\sum_{m=1}^{M}I_{n,m}e^{i[kd(n-1)\psi_{x}+kd(m-1 )\psi_{y}+\beta_{x}+\beta_{y}]} \tag{1}\]  
where  
\[\psi_{x}=sin(\theta)\text{cos}(\phi) \tag{2}\]  
\[\psi_{y}=sin(\theta)\text{sin}(\phi) \tag{3}\]  
\[\beta_{x}=-kd(n-1)\text{sin}(\theta_{0})\text{cos}(\phi_{0}) \tag{4}\]  
\[\beta_{y}=-kd(m-1)\text{sin}(\theta_{0})\text{sin}(\phi_{0}) \tag{5}\]Figure 1: Design case 1 fed by 9 CORPSs blocks of 4 inputs and 9 outputs. Figure 2: Phase plane generated by the design or 2-D phased array system 1. The number of elements is defined as \(N\) over the \(x\) axis and \(M\) over the \(y\) axis, the current excitation for each \(nm\)-th antenna element is set by \(I_{nm}\), and the phase excitation values are set by \(\beta_{x}\) and \(\beta_{y}\) at the values of (\(\theta_{0}\), \(\phi_{0}\)) as the scanning angle. The design case 1 (illustrated in Figure 1) utilizes 81 antennas in the 2-D phased array system. This antenna system is fed by nine CORPSs blocks of four inputs and nine outputs. This system design feeds 81 antenna elements by using 35 control ports (or phase shifters). The antenna system reduces to 57% the number of control ports or phase shifters employed in the antenna system providing beam-scanning for a range of \(\pm\)40\({}^{\circ}\) in azimuth and for a range of \(\pm\)30\({}^{\circ}\) in elevation. The amplifiers layer is connected to the output of the feeding network. The amplitude level is controlled in each antenna element. So, the raised cosine distribution can be applied to the planar array by compensating for the amplitude value generated at every output of the feeding network. The phase shifters layer is set as the first stage to control the phase flux into the network and the antenna elements. The idea is to generate the required phase plane for adequate beam-scanning. Figure 2 shows the phase plane generated by this 2-D phased array system. It is interesting to observe that by setting the input phases (blue color circles) and 4 \(\times\) 9 CORPSs blocks, it is possible to generate the rest of the output phases (red color circles) to build all the phase planes for beam-scanning. Every input port of the feeding system sets a phase value that permits the adequate recombination in the 4 \(\times\) 9 CORPSs blocks to generate the progressive phase excitation or phase plane shown in Figure 2.  
The design case 2 (illustrated in Figure 3) uses two stages of 4 \(\times\) 9 CORPSs blocks (13 CORPSs blocks). The first stage utilizes four CORPSs blocks in order to generate 36 outputs to feed nine CORPSs blocks connected to the 81 antenna elements of the array system, i.e., 81 antenna elements are controlled by 15 phase shifters. The 2-D phased antenna system can achieve a reduction of 81% in the number of phase shifters used for a beam-scanning of \(\pm\)30\({}^{\circ}\) in the azimuth and \(\pm\)30\({}^{\circ}\) in the elevation plane. As with the previous configuration, the amplifiers layer is connected to the output of the feeding network. So, the amplitude control is operated in the same way. The phase shifters layer or phase control unit provides 15 phase values (blue color circles) to feed the first stage and to generate the rest of the output phases (red color circles) of the phase plane (81 phase values for the planar array) (see Figure 4) for beam-scanning. Please note that less control ports are required to feed the 81 antenna elements of the phased array system. As illustrated, the 2-D phased array system uses the 4 \(\times\) 9 CORPSs blocks to generate the required phase plane for beam-scanning using less control ports or phase shifters. As shown previously, the outputs of the 4 \(\times\) 9 CORPSs blocks connect to a set of amplifiers.