This paper presents a comprehensive modeling and analysis of spin-valve Giant Magnetoresistance (GMR) elements and their application in bridge sensor configurations. The resistance-magnetic field (R-H) characteristics of spin-valve GMR elements are first modeled using both traditional linear approximation and a hyperbolic tangent (tanh)-based phenomenological model. Experimental data are utilized to validate the models, highlighting the limitations of linear approximation in representing the gradual saturation behavior observed at higher magnetic fields. The tanh-based model demonstrates superior accuracy in fitting the entire R-H curve, including the saturation regions. Furthermore, analytical expressions for output voltage in half-bridge and full-bridge Wheatstone configurations are derived using the tanh-based model, providing insights into the influence of GMR element characteristics on bridge sensor performance. Sensitivity analysis indicates that bridge configurations employing tanh-based modeling offer enhanced linearity and noise immunity, enabling the design of high-performance magnetic sensors for industrial and biomedical applications. The proposed modeling approach bridges the gap between individual GMR element characterization and practical bridge sensor design, offering a robust framework for accurate prediction and optimization in advanced magnetic sensing systems.