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In an induction heater for heating of aluminum billets, the main reason for using high-temperature superconducting (HTS) tapes instead of conventional copper conductors is to reduce the large power losses in the induction coil. In this work we present how to calculate the different loss contributions of an induction heater based on HTS tapes. Calculations of these losses are used in the design of an HTS induction heater with a rated power of about 10 kW operating at liquid nitrogen temperature. The calculations predict an optimal current that yields the highest efficiency of the induction heater.