Study of Superconducting Magnetization Effects and 3D Electromagnetic Analysis of the Nb$_3$Sn cos$\theta$ Short Model for FCC

The Falcon Dipole (Future Accelerator post-LHC Cos-theta Optimized <inline-formula><tex-math notation="LaTeX">$\rm {Nb_3Sn}$</tex-math></inline-formula> Dipole) is a single aperture <inline-formula><tex-math notation="LaTeX">$\rm {Nb_3Sn}$</tex-math></inline-formula> cos-theta dipole short model in the framework of the Future Circular Collider (FCC) project. The Italian Institute of Nuclear Physics (INFN), in collaboration with CERN, is in charge of designing and constructing the magnet, which is a crucial step towards the construction of High Field <inline-formula><tex-math notation="LaTeX">$\rm {Nb_3Sn}$</tex-math></inline-formula> magnets suitable for a post LHC collider. This paper recalls the electromagnetic design, the field quality and performances of the Falcon Dipole. The coil ends design has been implemented in a 3D FEM to study the peak field distribution on the magnet and influence on field quality. A special focus is given to the 2D analysis to study the effect of superconductor magnetization on field quality from the injection to the final energy and then to investigate the effectiveness of compensation methods.