Thick Cylindrical Coil Systems for Strong Magnetic Fields with Field or Gradient Homogeneities of the 6th to 20th Order

For strong homogeneous fields the ideal loop and solenoid elements of a prototype system may be expanded into thick cylindrical coils whose current density is (a) uniform or (b) inversely proportional to the cylindrical radius. Error coefficients of the zonal harmonic series for field or gradient that are cancelled in any given prototype continue to vanish as the elements are expanded, even to fusion, if at each stage the coil boundaries are adjusted by an iterative method. This involves the use of (a) the harmonic source functions Un* or (b) the Legendre functions Pn; either set is best computed by a recursion formula. An alternative method based on Lyle's principle requires no iterations and only trivial calculations. Though it also is valid for systems of any order, it can only expand a set of loops into coils with square cross sections of limited area. Also, the true null coefficients are replaced by small residues proportional to 4th and higher powers of the section dimensions when the second method ...