Scaling Laws for Magnetic Energy in Superconducting Quadrupoles

The stored energy in superconducting magnets is one of the main ingredients needed for the quench calculation and for designing quench protections. Here we propose an analytical formula based on the Fourier transformation of the current density flowing within the winding to determine the magnetic energy stored in superconducting quadrupoles made of sector coils. Two corrective coefficients allowing to estimate the energy enhancement produced either by current grading or by the presence of an unsaturated iron yoke are respectively derived from a numerical and an analytical study. This approach is applied to a set of real quadrupoles to test the validity limits of the scaling law, which are shown to be of ~10%.