In this paper, we analyze the effective behaviour of the solution of an elliptic problem in a two-phase composite material with non-standard imperfect contact conditions between its constituents. More specifically, we consider on the interface an equi-valued surface condition and a non-local flux condition involving a scaling parameter $α.$ We perform a homogenization procedure by using the periodic unfolding technique. As a result, we obtain two different effective models, depending on the scaling parameter $α.$ More precisely, in the case $α > −1,$ we are led to a standard Dirichlet problem for an elliptic equation, while in the case $α = −1,$ we get a bidomain system, consisting in the coupling of an elliptic equation with an algebraic one.