In some embodiments, the asymmetric magnet array is provided with at least two mixed-phase permanent magnet rings that are phase-dissimilar. In the context of the present invention, a mixed-phase magnet ring (MPMR) is defined as a magnet ring comprising multiple, repeating segments, each of which consisting of two or more phases, at least one of which is comprised of a permanent magnetic material.

A phase is defined as an element characterized by a particular combination of (i) material composition, (ii) geometric shape and relative position within the segment, and (iii) magnetization state. The magnetization state is represented by three components of magnetic moment, M=(M_r,M_θ,M_z), which are shared by corresponding phases in different segments, in the aforementioned cylindrical reference frame of coordinates. The materials of the various phases may be (but not limited to) permanent magnets, ferromagnetic, ferrimagnetic, paramagnetic, diamagnetic, antiferromagnetic or non-magnetic. The total magnetic field of an MPMR at any point is calculated by superposing the contributions of all phases in the ring which have nonzero values of M.

The phases fill the entire MPMR effective volume, which is defined as the volume of a polygonal annular ring of a minimum cross-sectional area, which just encloses all magnetic phases in the ring. The volumetric ratio of a phase is defined as the ratio of the phase volume to the effective volume of the MPMR.