In pulsed power supply for electromagnetic launchers, air-core pulsed alternators have obvious advantages to enhance the power density and energy density. Compared with conventional machines, the large excitation currents are required to establish the high flux densities needed in the air-core magnetic circuits, so the self-excitation mode is adopted in these systems. Therefore, we make great efforts on the conditions of self-excitation mode. This paper presents a detailed description about the transient process of the air-core pulsed alternator operating at self-excitation mode. First, it's necessary to supply a seed current rapidly prompting the alternator running at self-excitation mode. So the capacitance and pre-charged voltage on the capacitor in excitation initialization module determining the seed current is analyzed, a reasonable value range of the capacitance and the voltage can be decided. Then, when the machine is in self-excitation mode, a detailed study about the current and the voltage through the excitation winding and the armature winding are developed. Also some new concepts, such as 'R-feq' meaning the equivalent impedance of the excitation winding and 'K-fi' applied to evaluate the growth rate of the excitation current, are proposed to simplify the transient analysis. They also play a guiding role to improve our systems. Finally, the results of the SABER simulation are presented to validate our analysis.