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The air-core pulsed alternator has advantages in the application of pulsed power region, such as in electromagnetic launchers. The self-excitation and discharge processes in an air-core alternator are more complicated than traditional iron-core generators. In this paper, a mathematical model of an air-core pulsed alternator is built and several critical problems are analyzed, such as the conditions for successful self-excitation, the time needed for self-excitation, the waveforms of electromagnetic torque, discharge current and load voltage, and the compensation action during discharge, the effect of triggered angles, the rotational speed curve, and the energy reclaiming. Investigation of self-excitation and discharge processes in an air-core pulsed alternator is helpful for understanding the operation principle, predicting the performance, and analyzing and designing the system.