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Analytical expressions for the multiparametric sensitivity analysis of coupled-via S -parameters in board and package structures are derived in this paper. These expressions are validated with finite difference approximations of S-parameters obtained from three-dimensional field solvers and of the analytical expressions themselves. Sensitivity analysis with respect to multiple geometric and material variations provides quick early-design insight without resorting to complete three-dimensional field simulation. Sensitivities for eccentric effect of via drilling and exterior problem are also studied. First derivative data is also critical for gradient-based optimization of system-level performance which includes via-via coupling. The proposed approach is a stepping stone towards early design and optimization for large-scale via structures in microelectronics systems.