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A rectangular metallic box loaded by dielectric pucks and excited by rectangular waveguides can be used as a building block for the realization of a wide class of resonators and filters for microwave and millimeter wave applications. Commercial software can perform the analysis of this structure by using numerical methods such as the finite difference time domain or the finite elements frequency domain. A computation time of many hours is required for a single analysis making the automatic design of a complex structure a practically unaffordable task. A rigorous formulation to obtain the generalized admittance matrix of the basic building block is presented here. This formulation based on the Green's function of the empty rectangular box and modal field expansions avoids any problem related to singularities and a fast convergence and a high accuracy have been obtained.