The wide band high power traveling wave tubes (TWTs) employed in radar, communication systems, etc. are always facing the backward wave oscillation (BWO) problem. However, it takes much time and computer resource to simulate BWO by the large electromagnetic software. Thus, several parametric models are developed to solve the problem faster. Most of those models do not discuss the saturated oscillation power. In this paper, a three-dimensional (3D) nonlinear backward-wave interaction model is presented, by which the BWO phenomenon can be accurately studied in TWTs and the oscillation power is also analyzed. This model is established with the equation of 3D excitation fields combined with 3D motion equations and 3D space charge force. The oscillation frequencies and the start-oscillation lengths are calculated by one-dimensional (1D) and 3D models, respectively, and they are carefully compared in the cases of with and without the space charge force, indicating that the space charge force in 1D model is much weaker than in 3D model. The reason for that is the model of current density for space charge model in 1D model is supposed to be proportional to particle radius, but the one in 3D model is almost uniform, which is indicated by 3D beam trace distribution analysis. The BWO saturated powers and the oscillation frequencies are studied by this nonlinear 3D backward-wave interaction model. The simulation results show that the BWO saturated power increases as the beam-wave interaction length extends before many trajectories intercept the helix. While the oscillation frequencies decrease, the large saturated power supplies more energy to the beam at the very beginning in beam-wave interaction starting region. Then the BWO suppression induced by the magnetic field effect of the beam ripple is also under consideration. As the magnetic force increases, not only some cross area of interaction beam is suppressed, but also the interaction impedance of -1 space harmonic decreases. So increasing magnetic field strength can obviously reduce BWO, while the effect on forward wave interaction should be balanced. Finally, a Ka-band tube is used to validate the 1D and 3D nonlinear backward-wave interaction models. The BWO frequencies at different voltages are compared among the experimental results and the calculations by 1D and 3D models. The results from the 3D model in the test voltage range are 4.8% lower than the experimental data, while the difference from the results of the 1D model is 6.7%. The 3D model seems to be more accurate than the 1D model.