Quantitative　characterization　of　tensile　stress　in　electroplated　nickel　coatings　on　the　copper　substrate　with　the　thickness　range　of　2.809　µm　to　35.654　µm　in　the　elastic　stage　was　experimentally　performed　with　a　magnetic　incremental　permeability　(MIP)　sensor　in　the　study.　Firstly,　a　differential　mutual-inductance　sensor　based　on　the　parallel　excitation　of　bifilar　co-directional　coil　was　proposed.　With　the　proposed　sensor,　multiple　signals　including　high-frequency　eddy　current　(HFEC),　tangential　magnetic　field　strength,　and　magnetic　incremental　permeability　butterfly　curve　(MIPBC)　could　be　measured　synchronously　and　eight　typical　magnetic　parameters　were　obtained　from　experimental　signals.　Then,　four　parameters　including　the　peak　height　of　MIPBC,　amplitude　at　cross　point　of　MIPBC,　difference　frequency　of　HFEC,　and　sum　frequency　component　of　HFEC　were　selected　to　quantitatively　characterize　tensile　stress.　Experimental　results　showed　that　the　four　parameters　depended　on　tensile　stress　and　showed　a　parabolic　trend.　The　parameters　of　the　parabolic　equation　varied　with　nickel　coating　thickness.　Finally,　a　polynomial　surface　fitting　method　based　on　the　combination　of　the　factors　of　thickness　and　tensile　stress　were　derived　with　the　selected　four　parameters　and　the　coefficient　of　determination　R2　was　higher　than　0.9.　The　further　comparison　between　theoretical　results　and　predicted　stress　confirmed　that　the　proposed　MIP　sensor　could　realize　the　quantitative　detection　of　tensile　stress　in　different　nickel　coatings.　©　2024　Elsevier　B.V.