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Electromagnetic quasistatic (EMQS) fields, where radiation effects are neglected, while Ohmic losses and electric and magnetic field energies are considered, can be modeled using Darwin-type field models as an approximation to the full Maxwell equations. Commonly formulated in terms of magnetic vector and electric scalar potentials, these EMQS formulations are not gauge invariant. Several EMQS formulations resulting from different gauge equations are considered and analyzed in terms of their structural properties and their modeling capabilities and limitations. Associated discrete field formulations in the context of the Maxwell-grid equations of the Finite Integration Technique are considered in frequency and time domain and are studied with respect to their algebraic properties. A comparison of numerical simulation results w.r.t. reference solutions obtained with established formulations for the full Maxwell equations are presented.