This paper develops a magnetic equivalent circuit model suitable to the design and optimization of the synchronous ac homopolar machine. The ac homopolar machine is of particular interest in the application of grid-based flywheel energy storage, where it has the potential to significantly reduce self-discharge associated with magnetic losses. The ac homopolar machine features both axial and radial magnetizing flux paths, which requires finite element analysis to be conducted in 3-D. The computation time associated with 3-D finite element modeling is highly prohibitive in the design process. The magnetic equivalent circuit model developed in this paper is shown to be a viable alternative for calculating several design performance parameters and has a computation time which is orders of magnitude less than that of 3-D finite element analysis. Results obtained from the developed model are shown to be in good agreement with finite element and experimental results for varying levels of saturation.Index Terms-Energy storage, finite element analysis, flywheels, magnetic equivalent circuit (MEC), renewable generation. ;).R. Nilssen and T. Undeland are with the Department of Electric Power Engineering,

Flywheel energy storage technology has been successfully commercialized for applications requiring high power, high cycle-life, and short storage intervals. High idling losses have prevented the use of flywheel technology in applications that require longer storage intervals, such as grid-based, load-following energy storage. This paper proposes the use of an outer-rotor ac homopolar motor to significantly decrease idling losses, increase energy density, and decrease cost. Motor sizing equations, a comparison to the typically-used permanent magnet motor, and 3D finite element analysis of an example design are presented. It is shown that for high-performance flywheel designs, the ac homopolar motor can have a torque density comparable to that of a permanent magnet motor.

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Bearingless ac homopolar machines combine magnetic bearing and motor/generator functionality into a single electric machine which features variable excitation, high power density at high rotational speed, a simple and robust rotor structure, and magnet-less excitation. These features make the bearingless ac homopolar machine a promising machine for highspeed flywheel energy storage systems (FESS). The variable excitation of the bearingless ac homopolar machine has the potential to increase the FESS's efficiency by allowing for low excitation during periods of free-wheeling and high-speed operation. However, the magnetic suspension's position stiffness and current stiffness depend upon the excitation level. This dependency must be taken into account in the suspension controller or the magnetic suspension may become unstable at certain excitation levels. A technique for modeling this dependence is presented in this paper and explored through 3D finite element simulation. A prototype design is analyzed for two rotor structures: one with a square airgap length profile and one with an inverted sinusoidal airgap length profile.Index Terms-ac homopolar machine, active magnetic bearing, flywheel energy storage