The proposed method allows to calculate explicitly, using mathematical notions of barycenter and mixed product, the conduction duration switches of a n-leg 2-level Voltage Source Inverter (VSI) supplying a n-wire load. This method depends on a new geometrical and vectorial VSI characterization. Its geometrical feature enables optimization and graphic representation such as the Space Vector Method (SVM) for a 3-leg VSI. Its vectorial feature enables to generalize to n phase systems the properties found out with three phase systems.