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A general equivalent circuit for two-port lossy non-symmetric reciprocal networks is proposed. The equivalent circuit is based on the eigenstate decomposition. It has three complex parameters that are obtained from the eigenvalues and eigenvectors of the admittance or impedance matrix of the network in a straightforward way. The interconnection of the circuit elements is simple and compact. The applicability of the equivalent circuit is general, and the real parts of its immittances are always positive. To verify its behavior, the equivalent circuit of three different structures is extracted: an artificial composite right/left-handed transmission line unit cell, a series-fed coupled patch radiating element, and the complementary strip-slot. These cases offer a wide array of results, obtained from theoretical, simulated, and measured data. Nevertheless, the parameters of the equivalent circuit behave as expected in every case study in a very broad bandwidth. The equivalent circuit can be used advantageously in the analysis and design of non-symmetric two-ports, like unit cells of leaky-wave antennas.