# Phased Antenna-Array Synthesis Using Taylor-Series Expansion and Neural Networks
## 4 Validation Using CST Microwave Studio for 5G Antenna Arrays
### Antenna Design
This detailed design methodology ensures that the proposed antenna meets the stringent performance requirements of 5G millimeter-wave communications while achieving optimal impedance matching and enhanced radiation efficiency. Equations (10)-(15) present standard design formulas for microstrip patch antennas. They are included here for completeness and clarity and are derived from the classical reference [25]. To enhance the analytical understanding of the antenna's dual-band behavior, an equivalent circuit model (ECM) is proposed, as shown in Figure 7. The ECM consists of two parallel RLC resonators, each representing one of the antenna's resonant frequencies at 28.14 GHz and 37.88 GHz. These branches are connected to a 50 \(\Omega\) transmission line that models the microstrip feed. Each RLC cell captures the resonant characteristics, impedance response, and loss mechanisms of its respective frequency band. This circuit-level abstraction provides an analytical perspective on the impedance matching and coupling effects observed in the electromagnetic simulations and can serve as a useful tool for future parametric tuning and rapid prototyping. Although a detailed parameter extraction is outside the scope of this work, the conceptual ECM reinforces the physical interpretation of the antenna design. Such equivalent circuit models have been widely adopted in the literature  
Figure 6: Geometry of the studied antenna. to interpret the impedance behavior and mode separation in dual-band or multi-band antennas, especially in the millimeter-wave range [48; 49]. The analysis of the results presented in Figure 8 highlights the dual-band behavior of the proposed antenna. It shows distinct resonances at 28.14 GHz and 37.88 GHz, with respective reflection coefficients of S\({}_{11}\) = \(-\)19 dB and S\({}_{11}\) = \(-\)19.33 dB. These values indicate optimal impedance matching, significantly reducing reflection losses and ensuring efficient signal transmission in the two targeted frequency bands. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value (mm)** & **Parameter** & **Value (mm)** \\ \hline WS & 4.5 & Wb & 0.4 \\ LS & 4.5 & Lb & 0.3 \\ Lp & 2.6 & LC2 & 1.8 \\ Lu & 0.65 & L3 & 1.4 \\ Wu & 0.2 & WK & 0.8 \\ Le & 0.2 & Lt & 0.2 \\ We & 0.44 & Wt & 0.2 \\ LC1 & 1.1 & Lf & 4.34 \\ WC1 & 1.4 & Wf & 0.86 \\ R & 0.3 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Geometric parameters of the antenna. Figure 8: Simulated reflection coefficient with CST MICROWAVE STUDIO. Figure 7: Equivalent circuit model of the dual-band antenna. Figure 9, which represents the antenna's directivity, highlights the antenna's ability to focus its radiation in a specific direction. The directivity of the antenna is 5.98 dBi, indicating that the antenna emits in a narrow beam, concentrating energy in a particular area. This high directivity is essential for applications like point-to-point communications and radar systems. Conversely, low directivity would indicate broader coverage but less targeted radiation. Figure 10 shows the antenna's gain, which measures the antenna's efficiency in transmitting or receiving signals in a given direction compared to an isotropic antenna. The gain of the antenna is 4.29 dBi, meaning the antenna is quite efficient at focusing energy in a specific direction. A high gain enhances the range and performance in applications for which signal focus is crucial, such as long-distance communication systems, satellites, and radar. Together, these two parameters (directivity of 5.98 dBi and gain of 4.29 dBi) characterize the antenna's efficiency and performance in various use cases. The 3D radiation pattern of the antenna, presented in Figure 11, provides valuable insights into the antenna's performance at two distinct frequencies: (a) at 28.14 GHz and (b) at 37.88 GHz. (a) At 28.14 GHz: The radiation pattern at this frequency shows a specific distribution of the radiated energy, with the main lobe directed at a particular angle, indicating the primary direction of maximum radiation.