Homogenization of Periodic Electrical Networks Including Voltage to Current Amplifiers

Abstract

We derive the homogenized model of a periodic electrical network that includes resistive devices, voltage to current amplifiers, sources of tension, and sources of current. First, a mixed variational formulation is associated with the classical equations of such electrical networks. In an abstract framework, inf-sup conditions are given for the existence and uniqueness of its solution. Second, optimal conditions, based on the network topology, are stated so that the inf-sup conditions are satisfied. Third, the homogenized model of such a periodic network is derived using the two-scale convergence developed for circuits by Lenczner [C. R. Acad. Sci. Paris Ser. II B, 324 (1997), pp. 537--542]. Finally, numerical comparisons of the solutions of the homogenized model and of the complete one are detailed. It underlines clearly, if necessary, the strong interest of using the homogenized model when the number of periodic cells is large enough.