Distributions of stress in helical coils with the geodesic winding based on virial theorem are analyzed theoretically and numerically. A force-balanced coil (FBC) is a multi-pole helical hybrid 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 major radius by canceling 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 that strength of magnetic field is restricted by working stress in the coils and their supporting structure. High-field coils should accordingly have same averaged principal stresses in all directions which is named the virial-limit condition. Recently, we made a model FBC which were neither impregnated with epoxy resin nor reinforced with stainless steel wires. Using acoustic emission measurements, we found that the wires vibrated in response to electromagnetic force changes. 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. In order to reduce the vibration, the geodesic winding is expected to be effective. In this work, we analyze the effect of the winding modulations including the geodesic modulation for the stress distribution of helical windings.

Stability and force equilibrium of toroidal helical coil with circular cross section were investigated. In this paper, we derived and calculated equations for equilibrium of toroidal helical coil based on an axisymmetric surface current model. The helical coil, which is modulated in such a way that a magnetic surface coincides with the coil surface, can reduce overturning force generated by electromagnetic forces. We analyzed the stability of the toroidally modulated helical coil and confirmed that this modulated configuration is held in equilibrium while it is unstable. This condition means the minimum inductance condition. Moreover, we formulated the relationships between electromagnetic forces and stresses acting on toroidal helical coil with circular cross section, and we compared the stress distributions calculated by the equations with those from finite element method (FEM) analysis.

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Virial-limit coils for application to magnetic confinement nuclear fusion devices are hybrid helical coils that level electromagnetic stresses. In past research, a possibility that a large bending stress is caused in helical coils made of a cable-in-conduit configuration (CICC) was pointed out. In this paper, we analytically evaluated the stresses of helical coils made of CICC by modeling them as springs and calculated a detailed stress distribution by finite-element method (FEM) analysis. FEM results show that the bending and torsional stresses, which are ten times or more as large as the membrane stress, are caused. These stresses are caused by the moment of electromagnetic forces, which can be explained by modeling helical coils as coil springs. We propose a helical coil that lessens the moment and confirms the effect of the reduced moment, and the coil under this optimal condition of moment reduction is termed a moment-free coil.Index Terms-Bending and torsional stresses, electromagnetic force, finite-element method (FEM), magnet confinement device, virial-limit coil (VLC).