## 2 Design of the Proposed Antenna Array
### CSC Feeding Network Layer  
The design processes of the driven layer and parasitic layer are described in detail above. In this paper, the proposed multilayer antenna array is obtained with a compact size of 4 \(\times\) 4 \(\times\) 3 (row \(\times\) column \(\times\) layer); therefore, it is necessary to design a feeding network for the 2 \(\times\) 2 proposed multilayer antenna subarray (as shown in Figure 4). If a larger array of antennas is required, it can be expanded on this basis.  
A CPW-Slot-Couple (CSC) feeding network is implemented in the bottom layer, which is a 1 to4 series-parallel feeding network. Figure 7 shows the schematic diagram and equivalent circuit diagram. Port 1 is the input port, and ports 2-5 are the output ports, which are, respectively, connected to the driven slot antenna elements. \(Z_{1}\) and \(Z_{f}\) are the characteristic impedance of the input and output ports, respectively, which are both 50 ohms. Among other things, it also contains two quarter-wavelength impedance matching lines, \(Z_{2}\) and \(Z_{4}\), and a two-wavelength transmission line, \(Z_{3}\); all lines are enclosed by metal vias. According to the equivalent circuit impedance matching principle, \(P_{1}\) represents the input power, \(\theta\) represents the electrical length of each section of the transmission line, and \(Z_{in1}\), \(Z_{in2}\), \(Z_{in1p}\), and \(Z_{in2p}\) represent the equivalent input impedance; the impedance equation of each section of the transmission line is expressed as follows:  
\[Z_{3}=Z_{in2}=Z_{in2p}\parallel Z_{in2p};\ \ \ Z_{4}^{2}=Z_{in2p}Z_{f} \tag{2}\]  
\[Z_{1}=Z_{in}=Z_{3}\parallel Z_{in1p}\parallel Z_{in1p};\ \ \ Z_{2}^{2}=Z_{in1p}Z_{f} \tag{3}\]  
\[Z_{2}=Z_{4}=2Z_{f} \tag{4}\]  
The characteristic impedance of each transmission line can be obtained by the above equation, and the corresponding line width and length can be calculated. The impedance and size values of each transmission line are shown in Table 2.  
According to the calculated size of the feeding network, a 3D model is established in the electromagnetic simulation software HFSS. The simulated S-parameter curve of the feeding network is shown in Figure 8. It can be seen from the figure that in the frequency range of 26 GHz to 31 GHz, the reflection coefficient of port 1 \(|\)S11\(|\) > 10 dB, and the  
\begin{table}
\begin{tabular}{c c c c c c} \hline  & \((Z_{1},\theta_{1})\) & \((Z_{2},\theta_{2})\) & \((Z_{3},\theta_{3})\) & \((Z_{4},\theta_{4})\) & \((Z_{f},\theta_{f})\) \\ \hline Impedance & 50 \(\Omega\) & 50 \(\Omega\) & 100 \(\Omega\) & 100 \(\Omega\) & 50 \(\Omega\) \\ Elec. Length & 360\({}^{\circ}\) & 90\({}^{\circ}\) & 360\({}^{\circ}\) & 90\({}^{\circ}\) & 654.5\({}^{\circ}\) \\ Width & 1.578 mm & 0.484 mm & 0.484 mm & 0.484 mm & 1.578 mm \\ Length & 7.2 mm & 1.87 mm & 14.96 mm & 1.87 mm & 13.09 mm \\ \hline \end{tabular}
\end{table}
Table 2: Physical dimensions of the CSC feeding network.  
Figure 7: CSC feeding network: (**a**) schematic diagram; (**b**) equivalent circuit diagram.  
reflection coefficients of port 1 to 2, 3, 4, and 5 ( |S12-51 ) are in the range of 6-8 dB, which meets the design requirements of the 1 to4 feeding network. The simulated S11 curve of the feeding network without vias is also shown in Figure 8a. It can be seen that the S11 of the feeding network without vias deteriorates rapidly at frequencies >30 GHz. As shown in Figure 8b, it can be seen from the surface current distribution diagram that at 30 GHz, the metal vias can effectively reduce the surface current distribution, reduce energy loss, and improve the isolation between each feed port.