Plane elliptic or toroidal multipole expansions for static fields: Applications within the gap of straight and curved accelerator magnets

The purpose of this paper is to present new basis functions suitable to parameterize two‐dimensional static potentials or (magnetic) fields and to show their application in practical cases.

Regular multipole solutions of the potential equation in plane elliptic coordinates are found by separation. The resulting set of functions is reduced to complete subsets suitable for expanding regular potentials or irrotational source‐free fields. Approximate regular plane solutions of the potential equation in local toroidal coordinates are computed by R‐separation and power series expansions in the inverse aspect ratio. The harmonic signals induced in a coil rotating in such a toroidal multipole field are computed from the induction law by similar expansions.

The elliptic expansions are useful in a larger area than circular multipole expansions and give better fits. The toroidal expansions permit one to estimate the effect of the curvature of magnets on the field and give better adapted expansions. However, while the scalar multipoles for the potential are orthogonal the vector fields derived for the two‐dimensional field are not.

Derivations presuppose analytical fields.

Field data obtained from numeric field calculations or measurements do not represent exactly analytic fields. Application of the expansion requires care and checks.

The paper presents novel approaches for parameterizing longitudinally uniform cylindrical or toroidally uniform potentials and fields.