Butler matrix (BM) is a widely used beamforming technique in array signal processing that enables efficient and precise signal distribution to multiple antenna elements. However, the frequency-dependent nature of their design restricts the effectiveness of the conventional Butler matrix to a specific frequency band only, thus leading to the narrow band operation. This paper presents the analysis of the frequency-dependency behaviour through a 4-by-4 BM based on its first-stage operation. Through this analysis, a mathematical correlation was developed between the BM parameters, specifically the electrical length and phase components, by utilizing the phase–to–frequency gradient. This correlation serves as a foundation for subsequent design stages. The 4x4 BM is developed for a sub-6 GHz 5G operation at 3.5 GHz.