Force formulation of a three-phase tubular linear machine with dual Halbach array

A novel dual Halbach array is proposed in this paper to increase the output performance of tubular linear machines. The magnetic field distribution in three-dimensional (3D) space is formulated with Bessel functions analytically. Following that, the modeling of force output for linear machines with three-phase windings is carried out based on Lorentz force law. The formula component of force ripple is separated from the analytical expression of total force. It is found that for multi-phase tubular linear machines, the force output and force ripple are not only closely related to the instantaneous position of the mover, but also the starting position. Therefore, the force output of the linear machine is formulated for three typical starting positions as a function of mover positions. The derived models are simulated with respect to the mover´s motion. They are also validated with numerical results from finite element calculation. The study shows that an appropriate starting position of the mover helps to improve the output performance of the tubular linear machines. The proposed analytical force model can also be employed to analyze the influence of the structure parameters on the force output of linear machines with similar structures.