Adjoint Variable Method for the Study of Combined Active and Passive Magnetic Shielding

<table>Domain <svg height="10.6875" id="M29" style="vertical-align:-0.0pt" version="1.1" viewbox="0 0 11.75 10.6875" width="11.75" xmlns="http://www.w3.org/2000/svg">
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</svg>. <svg height="14.375" id="M30" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 15.3875 14.375" width="15.3875" xmlns="http://www.w3.org/2000/svg">
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<g transform="translate(72,-60.5)">
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<tspan style="font-size: 12.50px; " x="0" y="0">Γ</tspan>
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<text transform="matrix(1,0,0,-1,-64.61,60.55)">
<tspan style="font-size: 8.75px; " x="0" y="0">2</tspan>
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</svg> is the bottom
horizontal border of <svg height="10.6875" id="M31" style="vertical-align:-0.0pt" version="1.1" viewbox="0 0 11.75 10.6875" width="11.75" xmlns="http://www.w3.org/2000/svg">
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<g transform="translate(72,-63.45)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">Ω</tspan>
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</g>
</svg> and <svg height="14.375" id="M32" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 15.3875 14.375" width="15.3875" xmlns="http://www.w3.org/2000/svg">
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<g transform="translate(72,-60.5)">
<text transform="matrix(1,0,0,-1,-71.95,63.67)">
<tspan style="font-size: 12.50px; " x="0" y="0">Γ</tspan>
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<tspan style="font-size: 8.75px; " x="0" y="0">1</tspan>
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</svg> is the rest of
the border of <svg height="10.6875" id="M33" style="vertical-align:-0.0pt" version="1.1" viewbox="0 0 11.75 10.6875" width="11.75" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,10.6875)">
<g transform="translate(72,-63.45)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">Ω</tspan>
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</g>
</svg>; <svg height="14.7125" id="M34" style="vertical-align:-3.23158pt" version="1.1" viewbox="0 0 22.700001 14.7125" width="22.700001" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.7125)">
<g transform="translate(72,-60.23)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">Ω</tspan>
</text>
<text transform="matrix(1,0,0,-1,-62.65,60.37)">
<tspan style="font-size: 8.75px; " x="0" y="0">W</tspan>
</text>
</g>
</g>
</svg> is the
workpiece; <svg height="14.6" id="M35" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 18.475 14.6" width="18.475" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.6)">
<g transform="translate(72,-60.32)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">Ω</tspan>
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<text transform="matrix(1,0,0,-1,-62.65,60.37)">
<tspan style="font-size: 8.75px; " x="0" y="0">P</tspan>
</text>
</g>
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</svg> is the passive
shield; <svg height="14.6" id="M36" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 26.375 14.6" width="26.375" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.6)">
<g transform="translate(72,-60.32)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">Ω</tspan>
</text>
<text transform="matrix(1,0,0,-1,-62.65,60.37)">
<tspan style="font-size: 8.75px; " x="0" y="0">T</tspan>
<tspan style="font-size: 8.75px; " x="4.8923678" y="0">A</tspan>
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</g>
</svg> is the target
area; <svg height="17.3375" id="M37" style="vertical-align:-5.32956pt" version="1.1" viewbox="0 0 22.4625 17.3375" width="22.4625" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,17.3375)">
<g transform="translate(72,-58.13)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">Ω</tspan>
</text>
<text transform="matrix(1,0,0,-1,-62.65,60.37)">
<tspan style="font-size: 8.75px; " x="0" y="0">C</tspan>
</text>
<text transform="matrix(1,0,0,-1,-56.81,58.18)">
<tspan style="font-size: 6.25px; " x="0" y="0">i</tspan>
</text>
</g>
</g>
</svg> are the
compensation coils of the active shield.</table>

Mathematical Problems in Engineering

fig2

Figure 2

Figure 2: Adjoint Variable Method for the Study of Combined Active and Passive Magnetic Shielding 