Optimizing magnetization orientation of permanent magnets for maximal gradient force

The force exercised on a permanent magnet (PM) in a nonuniform field (gradient force) is dependent on the magnetization orientation of the magnet. In this paper, it is shown theoretically that the gradient force is greatest when the magnetization through the magnet, or at least at its surface, is collinear with the external field. The formulae for calculating the force between an axis-symmetric optimal magnet and a coaxial axis-symmetric coil are presented. Using the finite element method (FEM), calculations of the magnetic field distribution of an optimal cylindrical magnet and some its approximations are performed. The forces between these magnets and a pancake coil are computed and compared. For a system consisting of a magnet with a height of 1 unit and a diameter of 2 units and magnetization invariant in field and an annular pancake coil with a diameter of 2.4 units, a thickness of 0.2 units, an inner diameter of 0.4 units and a distance from the magnet of 0.2 units, the force on the optimal magnet was 1.44 times greater than the force on an axially magnetized magnet of the same size and magnetization magnitude. The optimal magnetization may be approximated by magnetization inclined at a constant angle to the axis and by a combination of axially and radially magnetized sections. With magnetization at a constant angle to the axis in the axis plane, the force was greatest when the angle was about 45°, being 1.38-fold compared to the force on an axially magnetized magnet. When the magnet was composed of an axially magnetized cylindrical core and a radially magnetized outer ring, the force was greatest when the volume of the core was approximately equal to the volume of the ring, being 1.26-fold compared to the force on an axially magnetized magnet. The optimal magnet and its approximations also provided a reduced stray field. A short review of methods of the fabrication of permanent magnets (PMs) with a continuous variation of the magnetization orientation and with radial magnetization orientation is given.The results of this study can be used to design linear electromagnetic (micro)actuators.