An　integrated　third　kind　boundary　condition　for　2D　eddy　current　problems　with　multi-connected　region　and　corresponding　FE　equations　for　a　formulation　of　magnetic　intensity　H　are　presented.　In　this　method,　there　is　only　one　unknown　per　node　in　the　problem,　and　only　the　conducting　region　needs　to　be　considered.　The　new　method　is　applied　to　the　eddy　current　non-destructive　test　of　a　tube,　which　reveals　some　of　the　numerical　properties　in　remote-field　eddy　current　NDT.