Coil-dominated combined function magnet design

Most coil-dominated combined function magnets are built by stacking multipole coils around the beam aperture, which result in unnecessary cancellations of currents. This article suggests that conductor positions can be computed analytically based upon the combined potentials, so that significant ampere-turns can be reduced. The two dimensional complex potential theory is the base of this method. Examples are demonstrated. Detailed formulas are laid out and ready for various applications.