Second Generation Coil Design 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 a 150 mm aperture Nb$_{3}$Sn quadrupole for the LHC interaction regions. A first series of 1.5 m long coils were fabricated and assembled in a first short model. A detailed visual inspection of the coils was carried out to investigate cable dimensional changes during heat treatment and the position of the windings in the coil straight section and in the end region. The analyses allow identifying a set of design changes which, combined with a fine tune of the cable geometry and a field quality optimization, were implemented in a new, second-generation, coil design. In this paper we review the main characteristics of the first generation coils, describe the modification in coil lay-out, and discuss their impact on parts design and magnet analysis.