An Estimate of the Maximum Gradients in Superconducting Quadrupoles

In this paper we show that the electromagnetic design of several superconducting quadrupoles, built for particle accelerators, can be rather well approximated by a 36 degree sector coil with a wedge, canceling the first two field harmonics. We therefore carry out a complete analysis of this lay-out, obtaining an approximated equation for the critical gradient as a function of the coil area, magnet aperture, and of the superconducting properties of the cable. Using this model, we estimate through numerical methods the maximum critical gradient that can be obtained in quadrupole of a given aperture for Nb-Ti, Nb-Ti-Ta and Nb$_{3}$Sn.