We　implement　a　three-dimensional　formulation　for　eddy　current　problems　based　on　the　reduced　magnetic　scalar　potential　enforcing　a　high　order　surface　impedance　boundary　condition　(SIBC)　which　takes　into　account　the　curvatures　of　the　conductor　surface.　Based　on　perturbation　theory,　the　formulation　reduces　to　three　Laplace　boundary　value　problems　with　Neumann　boundary　conditions　and　hence　can　be　easily　implemented　in　any　existing　Finite　Element　Method　(FEM)　or　Boundary　Element　Method　(BEM)　code　for　the　Laplace　equation.　The　appropriate　choice　of　the　small　parameter　in　the　perturbation　approach　correctly　represents　the　order　of　accuracy　of　the　SIBC.　The　validation　is　carried　out　by　comparison　with　full　FEM　solutions　of　a　canonical　test　problem　and　of　a　more　realistic　example　of　a　non-destructive　testing　probe.　The　validity　of　the　extension　of　a　high　order　SIBC　to　lower　frequencies　is　verified　and　the　fields　can　be　obtained　at　any　frequency　in　the　range　of　interest　once　the　formulation　is　solved　only　once.　©　2021　American　Institute　of　Physics　Inc..　All　rights　reserved.