The complementary energy bounds of dual formulations on dual meshes for eddy current problems are investigated. Based on the algebraic discretization of Maxwell equations on the dual meshes using different working variables, two sets of dual formulations, both for the finite-element method (FEM) and the finite integral technique (FIT), can be derived. A rational comparison between these formulations and the circuit networks is performed. The complementary bounds of magnetic energy and power losses are studied through an example. The study shows that the energy bounds are due to the numerical discretization of the Hodge operators on dual meshes. The dependency of the upper and lower energy bounds on the field excitation is also illustrated. The relation of the numerical bounds of the global circuit elements to the dual mesh discretization is given.