Analysis of Nb$_3$Sn Accelerator Magnet Training

Nb$_3$Sn accelerator magnet technology has made significant progress during the past decades. For the first time, it is planned to be used in a real accelerator. A relatively small number of Nb$_3$Sn quadrupoles and dipoles will be installed in the LHC to increase machine luminosity. Although it will prove the possibility of using Nb$_3$Sn magnets in real machines, many questions of scaling this technology up remain. One of them is related to slow training of Nb$_3$Sn magnets compared to the traditional Nb-Ti accelerator magnets. Since the goal is to operate thousands of Nb$_3$Sn magnets in a future post-LHC accelerator, the slow training will affect both the practical design margin and the nominal operation field. Consequently, the cost of the project to reach the design field level is also increased. To improve our understanding of slow magnet training the existing Fermilab data from Nb$_3$Sn magnet tests we reanalyzed. A summary of coil training features and correlations with fabrication parameters observed is presented in this paper.