Korea Institute of Radiological & Medical Sciences (KIRAMS) has been developing a K120 superconducting cyclotron. We have designed the magnetic field of a K120 superconducting cyclotron. Two pairs of superconducting NbTi coils have been introduced to accelerate variable ions ($0.125{leq}Z/A{leq}0.5$). The spiral sector has been designed for stable beam acceleration. The basic design parameters of the magnet geometries are calculated by analytic equations. The magnetic fields and the geometries are simulated by TOSCA and are verified by GENSPEO. We use Carbon ions during the design process as the criteria. The main parameters and properties of the K120 superconducting magnet are presented.