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In this paper, we propose and demonstrate an original exploitation of relaxation oscillations of erbium-doped-fiber laser ring resonators based on a general model for Er cross-sectional characterization. Its main advantage as compared to other existing exploitations relies on the fact that only weak approximations relative to spatial light distribution in the laser are imposed. It is therefore applicable to various resonator configurations, i.e., independent of the active-media length and losses encountered during the single-pass loop propagation. An experimental demonstration is finally presented by comparing the extracted erbium parameters (intrinsic saturation power and absorption) to those obtained with other methods.