Optimization of SMES coil by using virial theorem

The coil for the superconducting magnetic energy storage (SMES) is optimized by use of the virial theorem with stored energy and stress. In this work, we show the theoretical limit of stored energy with the maximum stress. To achieve the ideal limit, we propose the toroidal coil with helical winding. It is a hybrid coil of a toroidal field (TF) coil and a solenoidal coil helically wound on a torus. The winding is modulated in such a way that the toroidal field is created in the torus whereas the poloidal field is only out of the torus. In this case, the electromagnetic force is represented by the difference in the poloidal and the toroidal magnetic pressure. The virial theorem in the magnet is the relation of the magnetic energy and the averaged stress, and shows that the best coil to store the magnetic energy under the weakest averaged stress requires equal averaged principal stresses in all directions, which determines the ratio of the poloidal and toroidal current of our toroidal coil. The coil increases the magnetic energy to 4 times the conventional TF coil with the same maximum stress.