# Quad-Beam 4 \(\times\) 2 Array Antenna for Millimeter-Wave 5G Applications
## 2 Single-Element Antenna
### Antenna Modeling Using Transmission Line  
In the previous section, the antenna structure is designed and tuned using the finite-element method (FEM), which provides the overall response of the antenna structure. However, it does not provide insight into the part-by-part structure behavior, and this can be comprehended by transforming the antenna geometry to a transmission line model. Figure 4: (a) The 3D radiation pattern and peak gain of Ant1, and (b) a 2D representation with co- and X-polarization in the xz- and yz-planes. Figure 5: A Poynting vector representation of Ant1. The transmission line model offers detailed insight into the changes in the impedances in figure-eights at distinct points, as well as the effects of feed line and ground. Thus, this section discusses the above antenna transmission line model in detail. At mmWave, the dimension of the proposed antenna is comparable with the wavelength; as a result, the voltage and current on the structure at different points vary, and consequently, the phase changes. Thus, the modeling of the antenna as a parallel RLC circuit does not fully comply. As a result, we split the entire structure into small lengths of transmission line, as shown in Figure 6, and we apply the transmission line theory to find the total impedance \((Z_{T})\) at the input. The conducting area of the patch is transformed to multiple series inductances at different points due to varying conductive widths. Likewise, the slot in the patch is transformed into the shunt capacitance connected between the series inductive impedances. The same principle is applied to the feed and ground layer. This impedance circuit is realized in the circuit simulator and tuned to achieve the desired response. Further, the total input impedance is computed using the transmission line equations. \(Z_{IT}\) is the intermediate impedance at the input of the quarter-wave transformer, with a quarter-wave transform impedance of 75 \(\Omega\). The load impedance \((Z_{L})\) is the impedance of two elliptical rings, which are further split as impedances \(Z_{1}\), \(Z_{2}\), and \(Z_{c1}\). The elliptical ring has varied widths; consequently, its impedance also varies. Thus, it is modeled with two impedances, \(Z_{1}\) and \(Z_{2}\). The capacitance effect in the ring is represented by the shunt capacitive impedance \(Z_{c1}\). \(Z_{c2}\) and \(Z_{cg}\) are the coupling capacitors between the patch and the ground plane. Therefore, the load impedance \(Z_{L}\) and ground impedance \(Z_{g}\) are obtained using Equations (2)-(10), as follows:  
\[Z_{L}=\frac{Z_{1c2}\,Z_{1c2}}{Z_{1c2}+Z_{1c2}}, \tag{2}\]  
where  
\[Z_{c12}=Z_{c11}=\frac{Z_{1}Z_{c1}}{2Z_{1}+Z_{c1}}, \tag{3}\]  
\[Z_{11}=\frac{\left(Z_{1}\right)^{2}}{2Z_{1}+Z_{c1}}, \tag{4}\]  
\[Z_{c112}=Z_{c113}=Z_{c11}+Z_{2}\;, \tag{5}\]  
Figure 6: Modeling of the single-element antenna with the transmission line theory. \[Z_{p}=\frac{Z_{\rm c112}\ Z_{\rm c113}}{Z_{\rm c112}+Z_{\rm c113}}, \tag{6}\]  
\[Z_{\rm Lc2}=Z_{\rm 11}+Z_{p}+Z_{\rm c2}, \tag{7}\]  
\[Z_{L}=\frac{Z_{\rm Lc2}\ Z_{\rm Lc2}}{Z_{\rm Lc2}+Z_{\rm Lc2}},\rm{and} \tag{8}\]  
Figure 5:0  
Figure 5:1  
The load and ground impedance together are obtained by \(Z_{\rm Lgd}\), as follows:  
Figure 5:2  
Due to the quarter-wave transform, the input impedance \(Z_{\rm IT}\) is obtained as follows:  
Figure 5:3  
Further applying the transmission line equation, the total input impedance is obtained as follows:  
Figure 5:4  
where \(\beta_{1}\) and \(l_{1}\) are the phase constant and length, respectively, of the impedance \(Z_{\rm FI1}\). The Figure 6 is modeled in the circuit simulator to find its impedances. The reflection coefficient graph of the simulated and modeled antenna is compared in Figure 7a. From the circuit simulator, the obtained impedances are \(Z_{1}=27.1+77.28j\), \(Z_{\rm 2}=8.2+50.89j\), \(Z_{\rm c1}=-53.05j\), \(Z_{\rm c2}=-15.6j\), \(Z_{\rm cg}=-84.2j\), and \(Z_{lg}=111.21j\).