Planar Coil Optimization in a Magnetically Shielded Cylinder

Hybrid magnetic shields with both active field generating components and high-permeability magnetic shielding are increasingly needed for various technologies and experiments that require precision-controlled magnetic field environments. However, the fields generated by the active components interact with the passive magnetic shield, distorting the desired field profiles. Consequently, optimization of the active components needed to generate user-specified target fields must include coupling to the high-permeability passive components. Here, we consider the optimization of planar active systems, on which an arbitrary static current flows, coupled to a closed high-permeability cylindrical shield. We modify the Green’s function for the magnetic vector potential to match boundary conditions on the shield’s interior surface, enabling us to construct an inverse optimization problem to design planar coils that generate user-specified magnetic fields inside high-permeability shields. We validate our methodology by designing two biplanar hybrid active-passive systems, which generate a constant transverse field, $\mathbf{B}=\hat{\mathbf{x}}$, and a linear field gradient, $\mathbf{B}=(-x\hat{\mathbf{x}}-y\hat{\mathbf{y}}+2z\hat{\mathbf{z}})$, respectively. For both systems, the inverse-optimized magnetic field profiles agree well with forward numerical simulations. Our design methodology is accurate and flexible, facilitating the miniaturization of high-performance hybrid magnetic field generating technologies with strict design constraints and spatial limitations.

Figure 1

A cylindrical hollow high-permeability magnetic shield of length $L_s$, radius ${\rho }_s$, and with planar end caps located at $z=\pm L_s/2$ encloses an interior circular current source (bounded by the upper dashed curve) of radius ${\rho }_c<{\rho }_s$, which lies in the $z=z^{\prime }$ plane.

Figure 2

Schematic diagram showing the areas occupied by the biplanar coils (light gray) inside a cylindrical closed high-permeability magnetic shield (black outline). The high-permeability shield is of length $L_s=1$ m and radius ${\rho }_s=0.5$ m, with planar end caps located at $z=\pm L_s/2$. The shield encloses interior conducting planes of radius ${\rho }_c=0.45$ m that are located at $z^{\prime }=\pm 0.45$ m. The optimization region (red) is bounded along the $z$ axis between the coil planes with top and bottom positions $z=\pm z^{\prime }/2$, respectively, and extends radially between $\rho =[0,{\rho }_c/4]$.

Figure 3

Wire layouts (a) and performance (b)–(d) of a hybrid active-passive system optimized to generate a constant transverse magnetic field, $B_x$. Current flows on the surface of two disks of radius ${\rho }_c=0.45$ m, which are separated symmetrically from the origin and lie at $z^{\prime }=\pm 0.45$ m. The wire layouts are optimized to generate a constant transverse field, $B_0=1\mu \mathrm{T}$, across the cylinder and normal to its axis of symmetry. The current-carrying planes are placed symmetrically inside a perfect closed magnetic shield of radius ${\rho }_s=0.5$ m and length $L_s=1$ m and the magnetic field is optimized between $\rho =[0,{\rho }_c/4]$ and $z=\pm z^{\prime }/2$, as shown in Fig. 2. The least squares optimization is performed with parameters $N=50$, $M=1$, $\beta =1.77\times {10}^{-9}{\mathrm{T}}^2/\mathrm{W}$, $t=0.5$ mm, and $\rho =1.68\times {10}^{-8}\mathrm{\Omega }\mathrm{m}$. (a) Color map of the optimal current streamfunction calculated for the upper current-carrying plane in Fig. 2. Blue and red shaded regions correspond to current counterflows 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 $N_{\phi }=12$ global contour levels. Streamfunction on the lower light gray plane in Fig. 2 is geometrically identical but the current direction is reversed. (b) Transverse magnetic field, $B_x$, calculated versus transverse position, $x$, for $y=z=0$, from the current continuum in (a) in three ways: analytically using Eqs. (26)–(28) (solid red curve); numerically using comsol Multiphysics version 5.5a, 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 curve). Black lines enclose the regions where the calculated field deviates from the target field by $5\mathrm{\%}$ (dashed) and $1\mathrm{\%}$ (dash-dot). (c) Transverse magnetic field, $B_x$, calculated versus axial position, $z$, for $x=y=0$, from the current continuum in the same three ways as for (b). (d) Enlarged section of (c) showing good agreement between the numerical and analytical results throughout the optimization region.

Figure 4

Color maps showing the magnitude of the magnetic field, in the $x$-$z$ plane inside a closed finite-length perfect magnetic conductor generated by the active-passive system depicted in Fig. 3: (a) transverse component, $B_x$, and (b) axial component, $B_z$. The field profiles are calculated numerically using comsol Multiphysics version 5.5a. The magnetic field is optimized between $\rho =[0,{\rho }_c/4]$ and $z=\pm z^{\prime }/2$; dashed red lines in (a),(b). Contours, in (a) only, show where the field deviates from the target field by $5\mathrm{\%}$ (dashed curves) and $1\mathrm{\%}$ (dash-dot curves).

Figure 5

Wire layouts (a) and performance (b),(c) of a hybrid active-passive system optimized to generate a linear variation of $B_z$ with $z$ position. Current flows on the surface of two disks of radius ${\rho }_c=0.45$ m, which are separated symmetrically from the origin and lie at $z^{\prime }=\pm 0.45$ m. The wire layouts are optimized to generate a linear axial field gradient, $d{B}_z/dz=2\mu \mathrm{T}/\mathrm{m}$, along the $z$ axis of the cylinder through the harmonic $\mathbf{B}=(-x\hat{\mathbf{x}}-y\hat{\mathbf{y}}+2z\hat{\mathbf{z}})$. The current-carrying planes are placed symmetrically inside a perfect closed magnetic shield of radius ${\rho }_s=0.5$ m and length $L_s=1$ m and the magnetic field is optimized between $\rho =[0,{\rho }_c/4]$ and $z=\pm z^{\prime }/2$, as shown in Fig. 2. The least squares optimization is performed with parameters $N=50$, $M=0$, $\beta =1.77\times {10}^{-9}{\mathrm{T}}^2/\mathrm{W}$, $t=0.5$ mm, and $\rho =1.68\times {10}^{-8}\mathrm{\Omega }\mathrm{m}$. (a) Color map of the optimal current streamfunction calculated for the upper current-carrying plane in Fig. 2. Intensity of red shaded regions 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 $N_{\phi }=12$ global contour levels. Streamfunction on the lower light gray plane in Fig. 2 is geometrically identical but the current direction is reversed. (b) Transverse magnetic field, $B_x$, calculated versus transverse position, $x$, for $y=z=0$, from the current continuum in (a) in three ways: analytically using Eqs. (26)–(28) (solid red curve); numerically using comsol Multiphysics version 5.5a, 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 curve). Black lines enclose the regions where the calculated field deviates from the target field by $5\mathrm{\%}$ (dashed) and $1\mathrm{\%}$ (dash-dot). (c) Axial magnetic field, $B_z$, calculated versus axial position, $z$, for $x=y=0$, from the current continuum in (a) in the same three ways as for (b).

Figure 6

Color maps showing the magnitude of the magnetic field, in the $x$-$z$ plane inside a closed finite-length perfect magnetic conductor generated by the active-passive system depicted in Fig. 5: (a) transverse component, $B_x$, and (b) axial component, $B_z$. The field profiles are calculated numerically using comsol Multiphysics version 5.5a. The magnetic field is optimized between $\rho =[0,{\rho }_c/4]$ and $z=\pm z^{\prime }/2$; dashed red lines in (a),(b). Contours show where the field deviates from the target field by $5$% (dashed curves) and $1$% (dash-dot curves).

Figure 7

Schematic diagram showing how the approximate zonal (a) and tesseral (b) harmonic responses are generated by simple current loops with cylindrical and $m$-fold azimuthal symmetries, respectively. The discrete current loops (black) are located at $z=z_1^{\prime }$ and $z=z_2^{\prime }$ with their $p\mathrm{th}$ reflected pseudoimage currents (red) generated by the surface of the cylindrical shield at $\rho ={\rho }_s$ and by the planar end caps at $z=\pm L_s/2$, in accordance with the method of images described by Eq. (24). Images of the two planar (black) coils resulting from the end caps are located at $z=(-1{)}^p{z}_1^{\prime }+p{L}_s$ and $z=(-1{)}^p{z}_2^{\prime }+p{L}_s$, respectively, where $p\in \mathbb{Z}$ (two such image coils are shown blue). For all odd reflections, the axial current direction is reversed.

Figure 8

Improvement in the performance of a hybrid active-passive system, optimized to generate a constant transverse field $B_0=1\mu \mathrm{T}$, upon increasing the radius of the passive shield. Current flows on the surface of two disks of radius ${\rho }_c=0.45$ m, which are separated symmetrically from the origin and lie at $z^{\prime }=\pm 0.45$ m. The wire layouts are optimized to generate constant $B_0=1\mu \mathrm{T}$ between $\rho =[0,{\rho }_c/4]$ and $z=\pm z^{\prime }/2$, as shown in Fig. 2. The current-carrying planes lie within the bore of a perfect closed magnetic shield of radius ${\rho }_s=[{\rho }_c,3{\rho }_c]$ and length $L_s=1$ m. Each least squares optimization is performed with parameters $N=50$, $M=1$, $\beta =1.77\times {10}^{-9}{\mathrm{T}}^2/\mathrm{W}$, $t=0.5$ mm, and $\rho =1.68\times {10}^{-8}\mathrm{\Omega }\mathrm{m}$. (a) Transverse magnetic field, $B_x$, calculated analytically versus transverse position, $x$, using Eqs. (26)–(28) (red curves), where ${\rho }_s/{\rho }_c=1.01$ (dotted curve), ${\rho }_s/{\rho }_c=1.25$ (dash-dot curve), ${\rho }_s/{\rho }_c=1.5$ (dashed curve), and ${\rho }_s/{\rho }_c=2.5$ (solid curve). (b) Root mean square (rms) deviation, $\mathrm{\Delta }{B}_x^{\mathrm{rms}}$, between the calculated and target fields evaluated along the axis of the optimized field region and plotted as a function of ${\rho }_s/{\rho }_c$. Light blue crosses show $\mathrm{\Delta }{B}_x^{\mathrm{rms}}$ values calculated analytically using Eqs. (26)–(28). Horizontal dashed red line shows the analytical value of $\mathrm{\Delta }{B}_x^{\mathrm{rms}}=1.18\mathrm{\%}$ obtained in the wide shield limit (${\rho }_s/{\rho }_c\gg 1$).

Figure 9

Wire layouts (a) and performance (b) of a hybrid active-passive system optimized to generate a linear variation of $B_x$ with $z$ position. Current flows on the surface of two disks of radius ${\rho }_c=0.45$ m, which are separated symmetrically from the origin and lie at $z^{\prime }=\pm 0.45$ 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 through the harmonic field profile $\mathbf{B}=(z\hat{\mathbf{x}}+x\hat{\mathbf{z}})$. The current-carrying planes are placed symmetrically inside a perfect closed magnetic shield of radius ${\rho }_s=0.5$ m and length $L_s=1$ m and the magnetic field is optimized along the $z$ axis between $z=\pm z^{\prime }/2$. The least squares optimization is performed with parameters $N=100$, $M=1$, $\beta =1.77\times {10}^{-8}{\mathrm{T}}^2/\mathrm{W}$, $t=0.5\mathrm{mm}$, and $\rho =1.68\times {10}^{-8}\mathrm{\Omega }\mathrm{m}$. (a) Color map of the optimal current streamfunction calculated for the upper current-carrying plane in Fig. 2 [blue and red shaded regions correspond to current counterflows 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 $N_{\phi }=12$ global contour levels. Streamfunction on the lower light brown plane in Fig. 2 is geometrically identical but the current direction is reversed. (b) Transverse magnetic field, $B_x$, calculated versus axial position, $z$, from the current continuum in (a) in three ways: analytically using Eqs. (26)–(28) (solid red curve); numerically using comsol Multiphysics version 5.5a, 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 curve). Black lines enclose the regions where the axial field gradient deviates by $5\mathrm{\%}$ (dashed) and $1\mathrm{\%}$ (dash-dot).

Figure 10

Wire layouts (a),(b) and performance (c) of a hybrid active-passive linear system optimized to generate a constant axial field, $B_z$. Current flows on the surface of two disks of radii ${\rho }_{c1}=0.95$ m (on the upper current-carrying plane in Fig. 2) and ${\rho }_{c2}=0.35\mathrm{m}$ (on the lower current-carrying plane in Fig. 2), respectively, which are separated symmetrically from the origin and lie at $z^{\prime }=\pm 0.45$ m. The wire layouts are optimized to generate a constant axial field, $B_0=1\mu \mathrm{T}$, along the cylinder and parallel to its axis of symmetry. The current-carrying planes are placed symmetrically inside a perfect closed magnetic shield of radius ${\rho }_s=1$ m and length $L_s=1$ m and the magnetic field is optimized along the $z$ axis between $z=\pm z^{\prime }/2$. The least squares optimization is performed with parameters $N=100$, $M=0$, $\beta =1.77\times {10}^{-8}$ , $T^2/W,t=0.5$ mm, and $\rho =1.68\times {10}^{-8}\mathrm{\Omega }\mathrm{m}$. (a),(b) Color maps of the optimal current streamfunction on the upper and lower planar disks, respectively [blue and red shaded regions correspond to current counterflows 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 $N_{\phi }=6$ global contour levels. (c) Axial magnetic field, $B_z$, versus axial position, $z$, calculated from the current continuum in (a),(b) in three ways: analytically using Eqs. (26)–(28) (solid red curve); numerically using comsol Multiphysics version 5.5a, 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 curve). Black lines enclose the regions where the axial field gradient deviates by 5% (dashed) and 1% (dash-dot).