- The recently developed manufacturing technologies use high pressure and various doping additions to prepare bulk MgB2-based materials with high critical current density. We use a contactless method transformer to measure AC losses and voltage-current characteristic of MgB2 rings synthesized under high pressure. The method is based on the transformer configuration where the superconducting ring forms the secondary coil of a transformer in which the primary coil is connected with a voltage source. Using the Hall-probe technique, the magnetic flux density along the ring axis was measured as a function of the instantaneous current in the primary coil with following calculation of the electric field, current, and AC losses in the superconductor. The obtained dependence of the losses on the primary current (applied magnetic field) is fitted by a power law with the exponent ~2.1 instead of the cubic dependence predicted by Bean's model and power-law electric field-current density (E-J) characteristics with a large value of the exponent n. It was shown that the obtained E-J characteristic is well fitted by the dependence used in the extended critical state model based on the account of the viscous vortex motion in the flux flow regime. The critical current value measured by the transformer method (transport current) is about an order less than the value obtained from the magnetization measurements. A possible explanation of the obtained peculiarities lies in a specific non-uniform structure of the investigated MgB2 samples where the non-superconducting and superconducting regions were formed.