The closed-form expression of multi-conductor lossy transmission lines is presented. The model is analytically derived from a set of telegrapher's equations. Previous works for modeling the transmission lines have relied on some approximation procedures, for example, Pade approximation, Krylov subspace method, and so on. Thus, these methods never give the exact poles and residues. In this paper, a macromodel consisting of the exact poles and residues is presented. Further, it is proven that the passivity of the macromodel is preserved.