Magnetic Analysis of the Nb$_3$Sn low-beta Quadrupole for the High Luminosity LHC

As part of the Large Hadron Collider Luminosity upgrade (HiLumi-LHC) program, the US LARP collaboration and CERN are working together to design and build 150 mm aperture $Nb_3Sn$ quadrupoles for the LHC interaction regions. A first series of 1.5 m long coils were fabricated, assembled and tested in the first short model. This paper presents the magnetic analysis, comparing magnetic field measurements with the expectations and the field quality requirements. The analysis is focused on the geometrical harmonics, iron saturation effect and cold-warm correlation. Three dimensional effects such as the variability of the field harmonics along the magnet axis and the contribution of the coil ends are also discussed. Moreover, we present the influence of the conductor magnetization and the dynamic effects.