Optimal Inverse Design of Magnetic Field Profiles in a Magnetically Shielded Cylinder

Magnetic shields that use both active and passive components to enable the generation of a tailored low-field environment are required for many applications in science, engineering, and medical imaging. Until now, accurate field nulling, or field generation, has only been possible over a small fraction of the overall volume of the shield. This is due to the interaction between the active field-generating components and the surrounding high-permeability passive shielding material. In this paper, we formulate the interaction between an arbitrary static current flow on a cylinder and an exterior closed high-permeability cylinder. We modify the Green’s function for the magnetic vector potential and match boundary conditions on the shield’s interior surface to calculate the total magnetic field generated by the system. We cast this formulation into an inverse optimization problem to design active–passive magnetic field shaping systems that accurately generate any physical static magnetic field in the interior of a closed cylindrical passive shield. We illustrate this method by designing hybrid systems that generate a range of magnetic field profiles to high accuracy over large interior volumes, and simulate them in real-world shields whose passive components have finite permeability, thickness, and axial entry holes. Our optimization procedure can be adapted to design active–passive magnetic field shaping systems that accurately generate any physical user-specified static magnetic field in the interior of a closed cylindrical shield of any length, enabling the development and miniaturization of systems that require accurate magnetic shielding and control.

Figure 1

Schematic diagram showing a high magnetic permeability cylinder of length $L_s$ and radius ${\rho }_s$ with planar end caps located at $z=\pm L_s/2$. This cylinder encloses an interior conducting open cylindrical shell of length $L_c=L_1-L_2$ and radius ${\rho }_c$.

Figure 2

Schematic diagram showing how azimuthal (left) and axial (right) currents are reflected by two infinite parallel planes (hatched) of infinite extension and permeability (i.e., perfect magnetic conductors) located at $z=\pm L_s/2$. Left: Azimuthal currents located in the $z^{\prime }=d$ plane are translated, by the reflection process, to the $z^{\prime }=(-1{)}^{p}d+p{L}_s$ planes. Right: Axial currents flowing from $z^{\prime }=d_1$ to $z^{\prime }=d_2$ are reflected to positions $z^{\prime }=(-1{)}^p{d}_1+p{L}_s$ and $z^{\prime }=(-1{)}^p{d}_2+p{L}_s$, respectively, with odd reflections reversing the initial current flow direction. Black circles show cross sections of wires carrying current into (red crosses) and out of (red circles) the page.

Figure 3

Wire layouts (a) and performance (b),(c) of an optimized hybrid active–passive constant transverse field-generating system in which current flows on a cylinder of length $L_c=0.95$ m and radius ${\rho }_c=0.245$ m. The wire layouts are optimized to generate a constant transverse field, $B_x=1\mu \mathrm{T}$, across the cylinder and normal to its axis of symmetry. The current-carrying cylinder is placed symmetrically inside a perfect closed magnetic shield of length $L_s=1$ m and radius ${\rho }_s=0.25$ m and the magnetic field is optimized between $\rho =[0,{\rho }_c/2]$ and $z=\pm L_c/4$; see the dashed black lines in (b),(c). The least-squares optimization is performed with parameters $N=200$, $M=1$, and $\beta =5.95\times {10}^{-12}$. (a) Color map of the optimal current streamfunction on the cylinder [blue and red shaded regions correspond to the flow of current in opposite senses and their intensity shows the streamfunction magnitude from low (white) to high (intense color)]. Solid and dashed black curves represent discrete wires with opposite senses of current flow, approximating the current continuum with $N_{\phi }=24$ contour levels. (b) Transverse magnetic field $B_x$ versus axial position $z$ calculated from the current continuum in (a) in three ways: analytically using Eqs. (37)–(39) (solid red curve); numerically using comsol Multiphysics version 5.3a and modeling the high-permeability cylinder as a perfect magnetic conductor (blue dotted curve); numerically without the high-permeability cylinder and using the Biot-Savart law with $N_{\phi }=100$ contour levels (dashed green). (c) Enlarged section of (b) emphasizing the high level of field uniformity and the agreement between the numerical and analytical results over the optimization region.

Figure 4

Wire layouts (a) and performance (b) of an optimized hybrid active–passive linear transverse gradient field-generating system in which current flows on a cylinder of length $L_c=0.95$ m and radius ${\rho }_c=0.245$ m. The wire layouts are optimized to generate a linear transverse field gradient, $d{B}_x/dz=1\mu \mathrm{T}/$m, along the $z$ axis of the cylinder. The current-carrying cylinder is placed symmetrically inside a perfect closed magnetic shield of length $L_s=1$ m and radius ${\rho }_s=0.25$ m and the magnetic field is optimized between $\rho =[0,{\rho }_c/2]$ and $z=\pm L_c/4$; see the dashed black lines in (b). The least-squares optimization is performed with parameters $N=200$, $M=1$, and $\beta =5.95\times {10}^{-12}$. (a) Color map of the optimal current streamfunction on the cylinder [blue and red shaded regions correspond to the flow of current in opposite senses and their intensity shows the streamfunction magnitude from low (white) to high (intense color)]. Solid and dashed black curves represent discrete wires with opposite senses of current flow, approximating the current continuum with $N_{\phi }=24$ contour levels. (b) Transverse magnetic field $B_x$ versus axial position $z$ calculated from the current continuum in (a) in three ways: analytically using Eqs. (37)–(39) (solid red curve); numerically using comsol Multiphysics version 5.3a and modeling the high-permeability cylinder as a perfect magnetic conductor (blue dotted curve); numerically without the high-permeability cylinder and using the Biot-Savart law with $N_{\phi }=100$ contour levels (dashed green).

Figure 5

Color maps showing the magnitude of the transverse magnetic field $B_x$ in the $y$-$z$ plane inside a closed finite length perfect magnetic conductor generated by two active–passive systems: (a) the constant transverse field-generating system depicted in Fig. 3; (b) the linear transverse gradient field-generating system depicted in Fig. 4. The field profiles are calculated numerically using comsol Multiphysics version 5.3a. Contours show where the field deviates from the target field by $1$% (solid curves), $0.1$% (dashed curves), $0.01$% (dash–dot curves), and $0.001$% (dotted curves; in (a) only).

Figure 6

Plots showing how the performance of the optimized hybrid active–passive constant transverse field-generating system depicted in Fig. 3 depends (a) on the thickness $d$ of the high-permeability cylinder (inset), and (b) on the radius ${\rho }_h$ of the circular axial entry hole in the two end caps (inset), which allow access to the interior of the system. (a) Root-mean-square deviation $\mathrm{\Delta }{B}_x^{\mathrm{rms}}$ evaluated along the axis of the optimized field region, as a function of the high-permeability shield thickness $d$, taking ${\mu }_r=20000$ and ${\rho }_h=0$. Light blue crosses show $\mathrm{\Delta }{B}_x^{\mathrm{rms}}$ values calculated numerically from the current continuum in Fig. 3 using comsol Multiphysics version 5.3a. Horizontal dashed red line shows the analytical value of $\mathrm{\Delta }{B}_x^{\mathrm{rms}}=0.0232\mathrm{\%}$ calculated using Eqs. (37)–(39). (b) Root-mean-square deviation $\mathrm{\Delta }{B}_x^{\mathrm{rms}}$ evaluated along the axis of the optimized field region, as a function of the normalized axial hole radius ${\rho }_h/{\rho }_s$, taking ${\mu }_r=20000$ and $d=1$ mm. Purple crosses show $\mathrm{\Delta }{B}_x^{\mathrm{rms}}$ values calculated numerically from the current continuum in Fig. 3 using comsol Multiphysics version 5.3a. For comparison, the horizontal dashed light blue line shows the numerical rms deviation $\mathrm{\Delta }{B}_x^{\mathrm{rms}}=0.0190\mathrm{\%}$ calculated when $d=1$ mm and ${\rho }_h=0$.

Figure 7

Color maps showing the magnitude of the transverse magnetic field $B_x$ in the $y$-$z$ plane inside a magnetic shield with permeability ${\mu }_r=20000$, thickness $d=1$ mm, and a circular axial entry hole of normalized radius ${\rho }_h=0.25{\rho }_s$ around the center of each end cap generated by two active–passive systems: (a) the constant transverse field-generating system depicted in Fig. 3; (b) the linear transverse gradient field-generating system depicted in Fig. 4. The field profiles are calculated numerically using comsol Multiphysics version 5.3a. Contours show where the field deviates from the target field by $1$% (solid curves), $0.1$% (dashed curves), $0.01$% (dash–dot curves), and $0.001$% (dotted curves; in (a) only).

Figure 8

Wire layouts (a) and performance (b) of an optimized hybrid active–passive uniform axial field-generating system in which current flows on a cylinder of length $L_c=0.95$ m and radius ${\rho }_c=0.245$ m. The wire layouts are optimized to generate axial constant fields $B_z$ of two different magnitudes along the $z$ axis of the cylinder in two separate regions. The current-carrying cylinder is placed symmetrically inside a perfect closed magnetic shield of length $L_s=1$ m and radius ${\rho }_s=0.25$ m and target magnetic field values of $B_z=1\mu \mathrm{T}$ and $B_z=2\mu \mathrm{T}$ are optimized between $\rho =[0,{\rho }_c/2]$ and either $z=[-0.45L_c,0.15L_c]$ [dashed black lines in (b) where target $B_z=1\mu \mathrm{T}$] and $z=[0.15L_c,0.45L_c]$ [dash–dot black lines in (b) where target $B_z=2\mu \mathrm{T}$]. The least-squares optimization is performed with parameters $N=200$, $M=0$, and $\beta =2.97\times {10}^{-11}$. (a) Color map of the optimal current streamfunction on the cylinder [blue and red shaded regions correspond to the flow of current in opposite senses and their intensity shows the streamfunction magnitude from low (white) to high (intense color)]. Solid and dashed black curves represent discrete wires with opposite senses of current flow, approximating the current continuum with $N_{\phi }=8$ contour levels. (b) Axial magnetic field $B_z$ versus axial position $z$ calculated from the current continuum in (a) in three ways: analytically using Eqs. (37)–(39) (solid red curve); numerically using comsol Multiphysics version 5.5a and modeling the high-permeability cylinder as a perfect magnetic conductor (blue dotted curve, which essentially overlays the red curve); numerically without the high-permeability cylinder and using the Biot-Savart law with $N_{\phi }=250$ contour levels (dashed green curve).

Figure 9

Wire layouts (a) and performance (b),(c) of an optimized hybrid active–passive quadratic gradient field-generating system in which current flows on a cylinder of length $L_c=0.95$ m and radius ${\rho }_c=0.245$ m. The wire layouts are optimized to generate a quadratic gradient field $\mathbf{B}=(2xz\hat{\mathit{x}}-2yz\hat{\mathit{y}}+(x^2-y^2)\hat{\mathit{z}})\mu \mathrm{T}$. The current-carrying cylinder is placed symmetrically inside a perfect closed magnetic shield of length $L_s=1$ m and radius ${\rho }_s=0.25$ m and the magnetic field is optimized between $\rho =[0,{\rho }_c/2]$ and $z=\pm L_c/4$; see the dashed black lines in (b) and (c), respectively. The least-squares optimization is performed with parameters $N=200$, $M=2$, and $\beta =5.95\times {10}^{-12}$. (a) Color map of the optimal current streamfunction on the cylinder [blue and red shaded regions correspond to the flow of current in opposite senses and their intensity shows the streamfunction magnitude from low (white) to high (intense color)]. Solid and dashed black curves represent discrete wires with opposite senses of current flow, approximating the current continuum with $N_{\phi }=24$ contour levels. (b) Axial magnetic field $B_z$ versus transverse position $x$ at $z=0$ calculated from the current continuum in (a) in three ways: analytically using Eqs. (37)–(39) (solid red curve); numerically using comsol Multiphysics version 5.5a and modeling the high-permeability cylinder as a perfect magnetic conductor (blue dotted curve, which essentially overlays the red curve); numerically without the high-permeability cylinder and using the Biot-Savart law with $N_{\phi }=250$ contour levels (dashed green curve). (c) Transverse magnetic field $B_x$ versus axial position $z$ at $x=0.05$ m calculated from the current continuum in (a) in the same three ways as (b); labeled as in (b).