  This paper reviews magnetic flux signal calculations through pick-up loops
using vector spherical harmonic expansion under the quasi-static approximation,
and presents a near-analytical method of evaluating the flux through arbitrary
parametrizable pick-up loops for each expansion degree. This is done by
simplifying the surface flux integral (2D) into a line integral (1D). For the
special case of tangential circular sensors, we present a fully analytical
recursion calculation. We then compare commonly-used cubature approximations to
our (near-)analytical forms, and show that cubature approximations suffer from
increasing errors for higher spatial frequency components. This suggests the
need for more accurate evaluations for increasingly sensitive sensors that are
being developed, and our (near-)analytical forms themselves are a solution to
this problem.
