## 2 Materials and Methods  
Phased-array antenna systems, commonly referred to as multi-beam antennas, utilize multiple beam antennas to determine the direction of the beam based on the phase and its variation [52]. These antennas offer extensive coverage at wide angles and improved connectivity. The critical component of the antenna system is the beamforming network (BFN), which adjusts the phase and amplitude of the radiating elements to steer the main beam [53,54]. BFNs can be classified into two categories: digital BFNs, and RF BFNs [55]. Digital BFNs are typically employed in adaptive array systems, while RF BFNs are used in multi-beam antenna systems.  
Digital BFNs often encounter challenges such as analog-to-digital offset errors, receiver imbalances, and frequency-dependent errors [56]. On the other hand, RF BFNs can be further divided into lens-based beamformers and circuit-based beamformers. Circuit-based BFNs are constructed using directional couplers, phase shifters, and crossovers as building blocks. The circuit-based BFNs are commonly used due to their simple fabrication, cost-effectiveness, lack of external feed requirement, and compact design. The circuit based BFNs can be divided into two categories based on a variable phase shifter and a fixed passive beamformer such as a Butler Matrix (BM). The variable phase shifter has high power consumption and high insertion loss, whereas the BM provides high isolation between the input and output ports [57].  
The BM network is used to feed the multiple antenna array element of \(N\times N\) configuration and consists of \(N\) input and \(N\) output ports. In the Butler matrix, \(\frac{N}{2}log_{2}(N)\)\(90^{\circ}\) hybrid coupler, \(\frac{N}{2}(log_{2}(N)-1)\) fixed-phase shifters, and 0 dB crossovers are used to form the beam pattern. The signal from the transceiver is fed to the antenna elements by the beamforming module with a gradually changing phase difference between the output ports. This enables the beamforming operation and facilitates the transmission of the signal in various directions [52]. The block diagram of the BM with 4 inputs and outputs is shown in Figure 2. For the 4 \(\times\) 4 configuration, it consists of four branch line couplers, two phase shifters, and two crossovers. Ports 1-4 are considered as input ports and 5-8 are considered as output ports connected to the antenna array.  
The BM was specifically designed for operation at 5.5 GHz, utilizing a loss-free substrate Rogers RT5880 using CST 2018 software (Dassault Systemes, Velizy-Villacoublay, France). This substrate possesses a relative permittivity of 2.2 and has a thickness of 0.787 mm, with a copper cladding of 0.035 mm [58,59]. The BM network, designed for four input and output ports, incorporates four hybrid couplers, two phase shifters, and two crossovers to ensure isolation between the ports. Its purpose is to generate four distinct beams located at different positions. Further explanation and the design details of each component within the Butler matrix are discussed below.  
The following assumptions were made during the modeling process to ensure focus and manageability:  
Figure 2: Block diagram of the beamforming Butler matrix network.  
* The environment was considered to be homogeneous, assuming uniformity in the tunnel geometry and ignoring geological variations such as rock type differences or irregularities.
* Interference from nearby electronic devices, mining equipment, or environmental noise was not modeled, as the focus was on the antenna's beam steering performance.
* The materials used (such as the substrate Rogers RT5880) were assumed to have ideal and constant properties across all simulations.