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A large-signal field analysis of an O-type travelling-wave amplifier employing a practical focusing-field arrangement is presented, based on a coupled Eulerian-Lagrangian formulation of the problem. The resulting three-dimensional electron trajectories give rise to a beam that varies with the axial co-ordinate and time. It is shown that the boundary conditions on the moving (a priori unknown) beam boundary can be automatically satisfied by redefining the charge and current densities for all values of the radial co-ordinate. The key step in the analysis is a representation of the field components as functions of the electron-arrival time and the electronradial position, through a sequence of Green's functionals for the slow-wave circuit. Substitution of this representation into the electron ballistic equations reduces the latter to a fixed-point problem for a pair of operators in an appropriate function space. The fixed point, and therefore the solution for the electromagnetic field components, can be obtained by standard successive approximation techniques. Finally, it is shown that when the radial component of the focusing magnetic field vanishes and its axial component becomes constant, the electron trajectories degenerate to straight lines parallel to the tube axis.