# Simulation and Design of Three 5G Antennas
## 2 Materials and Methods
### Theoretical Analysis of Microstrip Antenna Arrays
#### 2.1.2 Planar Array  
On the basis of a linear array, we can arrange several linear arrays in a planar configuration. This is known as a planar array antenna. Here, we will introduce the rectangular planar array as a classic arrangement, where the elements consist of identical antenna units. This is illustrated in Figure 2, which represents a schematic of a rectangular planar array.  
In Figure 2, a three-dimensional coordinate system is established, with the (0, 0) antenna unit placed at the origin, and a planar array set in the xoy plane. Assuming the direction parallel to the x-axis is termed 'columns', each column contains N\({}_{x}\) antenna units, and the distance between adjacent units in a column is denoted as 'd\({}_{x}\)'. Similarly, the direction parallel to the y-axis is termed 'rows', with each row containing N\({}_{y}\) antenna units, and the distance between adjacent units in a row is 'd\({}_{y}\)'. Therefore, the coordinates of a particular antenna unit (m, n) can be expressed using Equation (6).  
Figure 1: A uniform linear array antenna arranged along the x-axis.  
Based on the earlier assumptions, the planar array antenna is composed of \(\rm N_{x}\times N_{y}\) individual antenna units. Under ideal conditions, the current flowing through the antenna unit located at position (m, n) is denoted as \(\dot{I}_{mn}\). The expression for the array factor \(S(\theta,\phi)\) of the planar array antenna can be derived as shown in Equation (7).  
\[S(\theta,\phi)=\sum_{m=0}^{N_{x}-1}\sum_{n=0}^{N_{y}-1}\dot{I}_{mn}exp\left[jk \big{(}md_{x}cos\phi+nd_{y}sin\phi\big{)}sin\phi\right], \tag{7}\]  
Assuming the current flowing through each antenna unit parallel to the x-axis varies with a phase difference of \(\alpha_{x}\) and has equal magnitudes, the current in the x-axis direction is described as in Equation (8).  
\[\dot{I}_{\rm xm}=I_{\rm xm}\mathrm{exp}(-\mathrm{j}\mathrm{m}\alpha_{x}), \tag{8}\]  
Similarly, assuming that the current passing through each antenna unit aligned parallel to the y-axis varies with a phase difference of \(\alpha_{y}\) and has the same magnitude, the current in the y-axis direction is as represented in Equation (9).  
\[\dot{I}_{\rm ym}=I_{\rm ym}\mathrm{exp}\big{(}-\mathrm{j}\mathrm{m}\alpha_{y} \big{)}, \tag{9}\]  
Consequently, the current passing through the planar array antenna is expressed as in Equation (10).  
\[\dot{I}=\dot{I}_{\rm xm}\cdot\dot{I}_{\rm ym}=I_{\rm xm}I_{\rm ym}\mathrm{exp} \left[-\mathrm{j}\big{(}m\alpha_{x}+n\alpha_{y}\big{)}\right], \tag{10}\]  
Another expression for the array factor \(S(\theta,\phi)\) can be derived as shown in Equation (11).  
\[\left\{\begin{aligned} S(\theta,\phi)=S_{\rm x}(\theta,\phi)\cdot S _{\rm y}(\theta,\phi)\\ S_{\rm x}(\theta,\phi)=\sum_{m=0}^{N_{\rm x}-1}I_{\rm xm}\mathrm{ exp}[\mathrm{j}\mathrm{m}(kd_{x}cos\phi\mathrm{s}\mathrm{i}\mathrm{n} \theta-\alpha_{x})\\ S_{\rm y}(\theta,\phi)=\sum_{n=0}^{N_{\rm y}-1}I_{\rm ym}\mathrm{ exp}\left[\mathrm{j}\mathrm{n}\big{(}kd_{y}\mathrm{s}\mathrm{i}\mathrm{n} \theta-\alpha_{y}\big{)}\right.\end{aligned}\right. \tag{11}\]  
Hence, it is apparent that the array factor \(S(\theta,\varphi)\) of the planar array antenna is the product of the array factors in the x-axis and y-axis directions, in line with the principles of the array antenna radiation pattern theory [18].  
Figure 2: Schematic of a rectangular planar array.