An advanced superconducting fault current limiter (SFCL) using $high-T_c$ superconducting (HTS) wire has been developed. The SFCL has a non-inductively wound magnet for reducing loss in normal state. Two types of non-inductively wound magnets, the solenoid type and the pancake type, were designed and manufactured by using Bi-2223 wire in this research. Short-circuit tests of the magnets were performed in sub-cooled $LN_2$ cooling system of 65 K. The magnets are thermally more stable and have a higher critical current in 65 K sub-cooled $LN_2$ cooling system than in 77 K saturated one. Because the resistivity of matrix at 65 K is lower than the resistivity at 77 K, the magnets generate a small resistance to reduce the fault current when the quench occurs. The magnets could limit the fault current to low current level with such a small resistance. The current limiting characteristic of the magnets was analyzed from the test result. The solenoid type was wound in parallel to make it non-inductive. The pancake type was also connected in parallel to be compared with the solenoid type in the same condition. The solenoid type was found to have a good thermal stability compared with the pancake type. It also had as large resistance as the pancake type to limit the fault current in sub-cooled $LN_2$ cooling system.