Ring 411 exemplifies an MPMR having cuboid-shaped permanent magnet elements (i.e. phase 1) separated by relatively small non-magnetic gaps (i.e. phase 2). Ring 413 exemplifies an MPMR having cuboid shaped permanent magnet elements (i.e. phase 1) separated by relatively large non-magnetic gaps (i.e. phase 2). Clearly, the fraction of the total ring volume occupied by non-magnetic gaps is small in the case of ring 411 and relatively large in the case of ring 413. Thus, rings 411 and 413 are MPMRs that are phase-dissimilar and array 400 may contain many phase-dissimilar MPMRs.

Furthermore, ring 411 may also have a magnetization vector in a direction substantially different (e.g., by more than 45 degrees) from ring 413. For instance, the magnetization vectors of the permanent magnet segments may point in the ?Z direction in ring 411, and ?45 degrees in the r-Z plane in ring 413. In an embodiment, the two or more mixed-phase magnet rings contain only one magnetic phase with a magnetization vector in a direction different by more than 45 degrees from one another. Each MPMR ring possesses rotational symmetry with an azimuthal period equal to 360/N degrees where N is the number of segments in the ring. (For a continuous ring, i.e., for N→∞, the rotational symmetry is continuous). In some embodiments, the disclosed MPMR rings have discrete rotational symmetry of an order N≥8.