Abstract: This paper studies the smooth cusp magnetic field for an axis-encircling large-orbit electron gun, which is realized by three different coils. Theoretical analysis and computer simulation of the field distribution agree well with each other. The Lagrangian for the electron is used to deduce the electrons equations of motion. The trajectories of electrons are then numerically calculated with the equations and the result is consistent with that of the three-dimensional particle-in-cell simulation. Through optimization of the ampere turns of the three coils, the velocity ratio from 1.4 to 2.5 and the axial velocity spread below 8% (at the velocity ratio equal to 1.9) are obtained with a 40 kV, 1 A electron beam.