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A yoke-based permanent magnet, which has been employed in many watt balancesat national metrology institutes, is supposed to generate strong and uniformmagnetic field in an air gap in the radial direction. However, in reality thefringe effect due to the finite height of the air gap will introduce anundesired vertical magnetic component to the air gap, which should either bemeasured or modeled towards some optimizations of the watt balance. A recentpublication, i.e., {\it Metrologia} 52(4) 445 [1], presented a full fieldmapping method, which in theory will supply useful information for profilecharacterization and misalignment analysis. This article is an additionalmaterial of [1], which develops a different analytical algorithm to representthe 3D magnetic field of a watt balance magnet based on only one measurementfor the radial magnetic flux density along the vertical direction, $B_r(z)$.The new algorithm is based on the electromagnetic nature of the magnet, whichhas a much better accuracy.

An analytical algorithm for 3D magnetic field mapping of a watt balance magnet