# Phased Antenna-Array Synthesis Using Taylor-Series Expansion and Neural Networks
## 4 Validation Using CST Microwave Studio for 5G Antenna Arrays  
### Antenna Design  
The basic structure envisaged for validation is a dual-band rectangular microstrip patch antenna fabricated on a Rogers RT4003C substrate. It exhibits two resonances at 28.14 GHz and 37.88 GHz, tailored for 5G applications. The substrate features a thickness of 0.41 mm, a relative permittivity of \(\varepsilon_{r}=3.55\), and a low loss tangent of 0.0027. The patch is fed via a 50 \(\Omega\) microstrip line to ensure proper impedance matching. Dual-band operation is achieved through slot insertion and careful tuning of the patch geometry. All geometric parameters--including length, width, slot configuration, and feed position--were first derived analytically and then refined through full-wave CST simulations. Millimeter-wave (mm-wave) frequency bands have attracted considerable attention in wireless communications, particularly for 5G networks, due to their wide spectral bandwidth and ability to support exceptionally high data rates. To harness these frequency bands effectively, a meticulous antenna design process is essential, with the selection of substrate material playing a pivotal role. Parameters such as thickness, relative permittivity, and loss tangent directly impact the antenna's electromagnetic performance, influencing radiation efficiency, impedance matching, and dielectric losses. In this section, we will design and simulate antenna arrays to validate our method using CST Microwave Studio. We also present the design and optimization of a dual-band antenna operating at the critical 5G frequencies of 28 GHz and 38 GHz, specifically tailored to next-generation communication systems. The proposed antenna is fabricated using the Rogers RT4003C substrate, selected for its excellent electromagnetic properties, which are characterized by the following:  
* Thickness: 0.41 mm. * Relative permittivity: \(\varepsilon_{r}=3.55\). * Loss tangent: \(\tan\delta=0.0027\). The antenna is fed via a 50 \(\Omega\) microstrip transmission line to ensure optimal impedance matching with high-frequency circuitry. Its structural design features a rectangular patch geometry with a radiating element of width \(W\) and length \(L\), carefully determined using classical analytical models for microstrip patch antennas. The slot dimensions were empirically adjusted to support the dual-band response and ensure proper impedance matching. Geometric parameter optimization aimed at minimizing reflection losses, enhancing radiation efficiency, and meeting the stringent requirements of mm-Wave communications. The final antenna dimensions are summarized in Figure 6 and Table 2, illustrating the necessary design refinements for stable operation at the target frequencies. The patch width \(W\) is calculated using the following equation:  
\[W=\frac{c}{2f_{c}\sqrt{(\varepsilon_{r}+1)/2}} \tag{10}\]  
The central frequency, \(f_{c}\), and the effective permittivity, \(\varepsilon_{eff}\), are crucial in determining the microstrip patch antenna dimensions. The effective permittivity, accounting for the dielectric substrate's influence, is given by the following:  
\[\varepsilon_{eff}=\frac{\varepsilon_{r}+1}{2}+\frac{\varepsilon_{r}-1}{2} \bigg{(}1+12\frac{h}{W}\bigg{)}^{-0.5} \tag{11}\]  
where \(h\) is the substrate thickness. The extended incremental length, \(\Delta L\), caused by fringing fields, is approximated as follows:  
\[\Delta L=0.412h\frac{(\varepsilon_{eff}+0.3)(W/h+0.264)}{(\varepsilon_{eff}-0.258 )(W/h+0.8)} \tag{12}\]  
The effective length \(L_{eff}\) is calculated as follows:  
\[L_{eff}=\frac{c}{2f_{c}\sqrt{\varepsilon_{eff}}} \tag{13}\]  
Finally, the actual physical length \(L\), accounting for fringing effects, is determined by the following:  
\[L=L_{eff}-2\Delta L \tag{14}\]  
where \(c\) is the speed of light in free space (\(3\times 10^{8}\) m/s), and \(f_{c}\) is the resonant frequency.