Force characterization and coordinate transformation of a fixed-bias flux three-phase magnetic bearing

Conventional hybrid magnetic bearings employ permanent magnets (PMs) for bias flux together with actively controlled electromagnets; however, the control forces are typically obtained by modulating excitation currents on top of the bias field. This approach only partially utilizes the PM flux and requires relatively complex power electronics, limiting its efficiency and compactness. This paper presents a fixed-bias flux three-phase magnetic bearing concept that employs permanent magnets to establish a constant bias flux and three-phase electromagnet assembly to generate control forces. Compared with conventional magnetic bearing structures, the proposed configuration produces forces with an angular separation of 120°, enabling a noticeable relationship between the coil currents and resulting force vectors. The force–current characteristics were derived via a vector analysis of 12 electromagnet poles and expressed in Cartesian components, showing that each three-phase group produces a unidirectional resultant force aligned with its phase axis. To further clarify the force–current relationship, we introduced a two-step coordinate transformation: first, a Clarke transform from the three-phase to the stationary α–β frame, and second, a rotation into a force–current aligned frame. These transformation ensures alignment between the current and force vectors, thereby providing a convenient basis for control design. The results verify a theoretical framework for analyzing and controlling fixed-bias flux three-phase magnetic bearings with potential applications in high-stability and energy-efficient rotor systems.