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This paper presents a 3-D analytical model of an axial flux permanent-magnet synchronous machine, based on formal resolution of Maxwell equations. This method requires much less computation time than conventional 3-D finite elements, and is therefore suitable for optimization purposes. In a first part, the mathematical procedure used to compute the machine no-load flux is described in detail. This method is 3-D, and then takes into account the radial edge effects of the machine, as well as the curvature effects by a resolution in cylindrical coordinates. Moreover, the originality of this method lies in the fact that it is totally analytical. The obtained results are verified using 3-D finite elements, and compared with simpler analytical models of axial flux machines, taken from the literature. This work puts in evidence the advantages of the proposed model. In particular, it is shown that the radial edge effects are important for a correct estimation of the no-load flux. On the contrary, the curvature effects are a second-order phenomenon.