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There are lots of sphere–plane air gaps in valve halls in extra-high-voltage and ultra-high-voltage converter stations. Accurate prediction for discharge characteristics of sphere–plane gaps is of great significance for the selection of shielding structure and determining the dielectric strength of the valve hall. In this paper, based on the physical process of corona inception and continuous leader inception, a computational model for calculating the discharge voltage of a sphere–plane air gap under positive switching impulse voltage is proposed, and the leader characteristics and discharge voltage are analyzed. Then, the switching impulse discharge test of the sphere–plane gap with 0.15, 2, and 0.3 m radius spherical electrodes is carried out to verify the correctness of the model. The results show that the discharge voltage calculated by the proposed method is consistent with the test results, and the error is within 7.3%. The leader inception voltage and inception time increase with the increasing spherical electrode radius at the same gap distance, and an identical spherical electrode require higher leader inception voltage and faster leader inception time with the increasing gap distance.