Edgewise Bending Strain in Helical Coils With Geodesic Windings Based on Virial Theorem

Distributions of edgewise bending strain in helical coils with the geodesic winding based on virial theorem are analyzed theoretically and numerically. A force-balanced coil (FBC) is a multipole helical coil combining toroidal field (TF) coils and a solenoid helically wound on a torus. The combination reduces the net electromagnetic force in the direction of the major radius by canceling out the centering force due to the TF coil current and the hoop force due to the solenoid current. The FBC concept was extended using the virial theorem, which shows the theoretical lower limit of stress in the coils and their supporting structure. High-field coils should accordingly have the same averaged principal stresses in all directions, which is named the virial-limit condition. Since FBC winding is modulated to reduce the tilting force, the winding is slightly similar to but different from the shortest geodesic trajectory and has no tensile load. To apply FBC to high-temperature superconducting tapes, the degradation of superconducting properties originating from edgewise bending strain is an important problem. Since the geodesic trajectory is a kind of a straight line on a curved surface and curves only to the normal direction of the surface, it is expected that the tape with geodesic trajectories has a small residual stress. In this paper, we analyze the effect of the winding modulations including the geodesic modulation for the optimization of residual stress in helical windings.