  A yoke-based permanent magnet, which has been employed in many watt balances
at national metrology institutes, is supposed to generate strong and uniform
magnetic field in an air gap in the radial direction. However, in reality the
fringe effect due to the finite height of the air gap will introduce an
undesired vertical magnetic component to the air gap, which should either be
measured or modeled towards some optimizations of the watt balance. A recent
publication, i.e., {\it Metrologia} 52(4) 445 [1], presented a full field
mapping method, which in theory will supply useful information for profile
characterization and misalignment analysis. This article is an additional
material of [1], which develops a different analytical algorithm to represent
the 3D magnetic field of a watt balance magnet based on only one measurement
for the radial magnetic flux density along the vertical direction, $B_r(z)$.
The new algorithm is based on the electromagnetic nature of the magnet, which
has a much better accuracy.

Instrumentation and Detectors