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In the current energy scenario, due to the increment in power generation from renewable sources, the importance of electrical storage systems has increased significantly, and as a consequence, the study of the improvement of its efficiency and the design of new storage systems also increased. Superconducting material permits the design of Superconducting Magnetic Energy Storage (SMES). The main problem of SMEs is the low energy density they have, what make the optimization of design to be one of the keys for inclusion of this elements in the power grid and other specific applications as, for instance, flux pumps. As the only basic forms for SMES, and with the objective of its mathematical optimization, this work (i) evaluates the mathematical equations for a real solenoidal winding, and (ii) develops the equivalent equations for a toroidal winding, from the electromagnetic laws. Then, (iii) a practical structure formed by short solenoids connected in series along a circular axis (quasi-toroidal structure) is studied. Due to the large number of equations involved in this case, the finite-element method in used here. Finally (iv), in order to validate the results without building the complete solenoid (impossible at the time), one of the magnetic coupling between two solenoids in the quasitoroidal winding was developed according with the theoretical method, and experimentally tested. The study was carried out by programming different dimensions in order to make conclusions for a further development of an optimization algorithm. These conclusions are presented. This work is the first stage for the optimized design of a SMES, and presents the complete equations of the real toroidal winding as the base of the outline dimensions of a practical quasi-toroidal SMES.

In the field of the application of HTS in electrical motors, most prototypes are made using superconducting coils based on tape and located in the position where copper coils work in a similar conventional motor. Other prototypes use superconducting bulks (usually disk-shaped) in those positions where normal magnets should work in similar conventional motors. But it is very unusual to find designs using electrical coils based on bulks. This is a challenge whose main problem is the difficulty in machining the superconductor bulks to get the proper shape because of the impossibility of bending the material to wind coils. The design of a bipolar single-turn coil made from a superconducting YBCO disk was proposed by the group of Electrical Application of Superconductors, at the University of Extremadura, several years ago to be an element for the design of a modular two-phase inductor for an air core axial-flux motor. The shape of each coil looks like an 'S'. When a current flows through the circuit, two opposite magnetic fields appear in the upper and lower halves of the piece. Until now, attempts to get a good superconducting circuit by cutting a YBCO disk into the required shape have failed because of cracks appearing in the crystal during the process. Last year, our group at the University of Extremadura began to work with ATZ GmbH who have improved the machining process and made the coils. In this paper we present the coil and the first tests carried out.