Currently, three types of superconducting quadrupole magnets are used in particle accelerators: \(\cos {2\theta }\) , CCT, and serpentine. However, all three coil configurations have complex spatial geometries, which make magnet manufacturing and strain-sensitive superconductor applications difficult. Compared with the three existing quadrupole coils, the racetrack quadrupole coil has a simple shape and manufacturing process, but there have been few theoretical studies. In this paper, the two-dimensional and three-dimensional analytical expressions for the magnetic field in coil-dominated racetrack superconducting quadrupole magnets are presented. The analytical expressions of the field harmonics and gradient are fully resolved and depend only on the geometric parameters of the coil and current density. Then, a genetic algorithm is applied to obtain a solution for the coil geometry parameters with field harmonics on the order of \(10^{-4}\) . Finally, considering the practical engineering needs of the accelerator interaction region, electromagnetic design examples of racetrack quadrupole magnets with high gradients, large apertures, and small apertures are described, and the application prospects of racetrack quadrupole coils are analyzed.
