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Korea Institute of Radiological & Medical Sciences (KIRAMS) has been developing a superconducting cyclotron since 2007. This paper is concentrated on the design of the superconducting magnet system. The superconducting magnet for a cyclotron consists of an iron yoke and a superconducting coil system. This paper presents the designed spiral sector for beam stability and the design results of the superconducting coil system. Additional test results of the superconducting coils are described.