Recent progress in the development of methods used to predict AC loss in superconducting conductors is summarized. It is underlined that the loss is just one of the electromagnetic characteristics controlled by the time evolution of magnetic field and current distribution inside the conductor. Powerful methods for the simulation of magnetic flux penetration, like Brandt's method and the method of minimal magnetic energy variation, allow us to model the interaction of the conductor with an external magnetic field or a transport current, or with both of them. The case of a coincident action of AC field and AC transport current is of prime importance for practical applications. Numerical simulation methods allow us to expand the prediction range from simplified shapes like a (infinitely high) slab or (infinitely thin) strip to more realistic forms like strips with finite rectangular or elliptic cross-section. Another substantial feature of these methods is that the real composite structure containing an array of superconducting filaments can be taken into account. Also, the case of a ferromagnetic matrix can be considered, with the simulations showing a dramatic impact on the local field. In all these circumstances, it is possible to indicate how the AC loss can be reduced by a proper architecture of the composite. On the other hand, the multifilamentary arrangement brings about a presence of coupling currents and coupling loss. Simulation of this phenomenon requires 3D formulation with corresponding growth of the problem complexity and computation time.

Investigation of ac loss under the simultaneous action of the transport ac current and the external ac magnetic field is of prime importance for the reliable prediction of dissipation in electric power devices such as motors/generators, transformers and transmission cables. An experimental rig allowing us to perform ac loss measurements in such conditions, on short (10 cm) tape samples of high-temperature superconductor Bi-2223/Ag, was designed and tested. Both the electromagnetic and thermal methods were incorporated, allowing us to combine the better sensitivity of the former and the higher reliability of the latter. Our main aim was to see how the ac loss depends on the phase shift between the transport current and the external magnetic field. Such a shift could have different values in various applications. While in a transformer winding, the maximum phase shift at full load will probably not exceed a few degrees, in a three phase transmission cable in tri-axial configuration it is around 120 • . Therefore, we explored the whole range of phase shifts from 0 to 360 • . Surprisingly, the maxima of dissipation did not coincide with zero shift as expected from qualitative considerations.

Numerical calculations using the commercial finite-element code FEMLAB have been carried out to find the distribution of current density in a wire of elliptical cross-section from a hard superconductor carrying the critical current. Dependence of the critical current density on local magnetic field and its orientation has been taken into consideration in these calculations. The results of critical current calculations for wires of different shapes and dimensions clearly revealed how this quantity can be influenced by the distribution of the self magnetic field. The outcome found for wires with cross-sections resembling tapes made from a high-temperature superconductor Bi-2223 have been approximated by two engineering formulae: one can be used to cancel out the influence of the tape width, and the other allows us to perform a correction on the aspect ratio. These expressions would allow the quality of superconducting properties expressed by the self-field critical current density to be compared also for wires of different cross-sections.