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Superconducting magnets consisting of conductor having a low n-value can be operated shortly at supercritical currents without quenching the magnet. This applies also to epoxy impregnated coils. However, when sufficiently high disturbance occurs at a supercritical current a magnet quenches. Traditional approaches on minimum quench energy (MQE) do not consider the supercritical region. In this study we measured MQE at sub- and supercritical currents for MgB2 and Bi-2223/Ag conductors having low n -values, i.e. below 15. Simulation results from a Finite Element Method model were compared with the measurement results and the differences in MQE between the sub- and supercritical regime were scrutinized.