Low-frequency magnetic field shielding effect of artificial joint-free REBCO coils

Below, we will address the details for the artificial design. Two coils were placed concentrically using REBCO tape without any joint resistance, built with a special cutting method. The joint-free coil consists of an inner coil and outer coil, with the inner radius r₁ and turn number n₁ for the inner coil and the outer radius r₂ and turn number n₂ for the outer coil. The inner coil is closer to the center shielding position, and contributes more induced magnetic field to the center reference point. However, the induced current of the inner coil is not high enough, because its induction area of magnetic flux is relatively small.

To solve the problem, a winding method was designed, i.e. connecting the inner coil and the outer coil using single slitting REBCO tape without joint resistance. Since the outer coil exhibits a larger induction area, in the shielding process, it could provide extra electric potential to drive the inner coil, so that the inner coil obtains larger current and reversing magnetic field, as well as the enhanced shielding effect, which is significantly practical for precision sensors.

The ring shaped coils, as shown in figure 1, are prepared by cutting and splitting a piece of 4.5 mm wide REBCO long tape from the center, the width of the gap is 0.5 mm. It is possible to obtain a multi-turn coil by stacking these unfolded tapes concentrically as reported in [23]. We firstly calculated a two coil system as shown in figure 1(a). The radius is 70 mm for the inner coil and 180 mm for the outer coil made from REBCO tapes of 250 mm and 600 mm in length, respectively. The shielding area is inside the inner coil. We name this topology as Coil 1/1 where two coils are independent.

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Figure 1(b) illustrates an improved topology by cutting the tape in a designed pattern. In this case, the inner and outer coil, when unfolded following the way shown in the figure, are connected in series without any joint as well. The resultant artificial coil is named as Coil 1–1 to distinguish it from coil 1/1 mentioned above. The radius of the inner (r₁) and outer coils (r₂) are 70 mm and 180 mm, respectively as well. This coil can be further extended to a 4-turn (Coil 2–2) or 6-turn coil (Coil 3–3) as shown in figure 2, if the REBCO tape is wide enough to enhance the shielding effect.

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The shielding effect was simulated numerically using the H-formulation combined with the E-J nonlinear relationship of REBCO coated conductors. The simulation was performed using COMSOL Multiphysics FEM software. We use the field dependant critical current density, J_c(${B_\parallel }$,${B_ \bot }$) formula reported in [25–28].

Faraday's Law (1) and Ampère's Law (2) are used as the governing equations:

$$\begin{equation}\nabla \times {\mathbf{E}} = - {\mu _0}\frac{{\partial {\mathbf{H}}}}{{\partial t}}\end{equation} \tag{ 1 }$$

$$\begin{equation}{\mathbf{J}} = \nabla \times {\mathbf{H}}\end{equation} \tag{ 2 }$$

The E–J power law is used to describe the nonlinear resistivity of the high temperature superconductor:

$$\begin{equation}{\mathbf{E}} = {E_c}{\left( {\frac{{\mathbf{J}}}{{Jc\left( B \right)}}} \right)^n}\end{equation} \tag{ 3 }$$

The critical electrical field E_c is set equal to 1 µV cm⁻¹, the exponent n = 38 in the power law is used to describe how abrupt the transition from the superconducting to the normal state is. The in-field critical current density, J_c(B) use the following.

$$\begin{equation}Jc\left( B \right) = \frac{{{J_{{C_0}}}}}{{{{\left(1 + \frac{{\sqrt {{k^2}\left| {{B_\parallel }{|^2} + } \right|{B_ \bot }{|^2}} }}{{{B_0}}}\right)}^\alpha }}}\end{equation} \tag{ 4 }$$

The parameters B₀, k, and α are used to describe the relation between magnetic field and in-field critical current density J_c(B), and have the respective values of 42.6 mT, 0.295 and 0.7, where ${B_\parallel }$ and ${B_ \perp }$ are the parallel component and perpendicular component of the magnetic field to the surface of REBCO tape, and J_c0 is the critical current density of REBCO in self-fields, as the self-field critical current I_c0 of the REBCO tape is 40 A per millimeter in width, at 77 K.

The models were built using COMSOL Multiphysics. In geometric modeling, the length and width of REBCO tape is in the millimeter range, however the thickness of REBCO superconducting layer is as low as 1 μm, which is much smaller than the length and width. The large ratio between length (width) and thickness will increase the difficulty of calculation. To solve this problem, the REBCO layer with 1 μm thickness can be equivalently transformed to REBCO tape with 0.3 mm thickness including both substrate and covering layer.

The self-field critical current I_c0 of the REBCO tape is fixed at 40 A per millimeter in width, so the critical current density of the 1 μm thick REBCO layer is 4 × 10¹⁰ A m⁻². Since the thickness of the total REBCO tape is 0.3 mm, in modeling, the tape is assumed to be made of 0.3 mm thick REBCO layer only. According to the thickness ratio of 0.3 mm to 1 μm, the critical current density J_c0 of the total tape can be set as 1.33 × 10⁸ A m⁻², the current density J and in-field critical current density J_c(B) are equivalently transformed as well.

Meanwhile, considering the thin thickness of REBCO film, the mesh of tape should be built in only a single layer in the REBCO c axial direction (perpendicular to the surface of tape), as figure 3 shows. In the thickness direction, the size of each mesh is 0.3 mm, the same as the thickness of the tape. In the width direction of the tape, the 4.5 mm width was divided into 45 parts, the size of each mesh is 0.1 mm. In the length direction of tape, the size of each mesh is 0.5 mm–1 mm.

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A magnetic field of 10 μT at 1 × 10⁻⁵ Hz was applied along the axial direction of the coil via the Dirichlet boundary condition to test the shielding effect. The external field will induce a screening current in the persistent REBCO coil which generates a reversing magnetic field to shield part of the external field. We define the shielding factor (SF) at the testing point as the specific value, which is the ratio between the resulting field B_r over the external field B₀, SF = (B_r/B₀)× 100%. And therefore, a small SF value indicates a better shielding effect.

Figure 4(a) shows the magnetic intensity at the center of the shielding coils with different topologies. In all of the four models, r₁ is 70 mm and r₂ is 180 mm. The time-dependent applied field of 10 μT at 1 × 10⁻⁵ Hz was plotted with dashed lines. The shielding factors are compared in figure 4(b). It can be observed that the central field of Coil 1–1 is lower than Coil 1/1, which indicates the shielding effectiveness of coils with series connection is better than coils that are placed independently. The shielding effectiveness improves when comparing Coil 1–1, Coil 2–2 and Coil 3–3, with increasing number of turns, the peak resultant field at the central point of Coil 3–3 is 20 nT, which provides the best shielding effect.

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In the model, when r₁ is fixed as 70 mm, the shielding effectiveness then depends on the size of the outer radius (r₂). Figure 5 shows the shielding factor of Coil 1–1 at the center point with different r₂ values. It reveals that the shielding factor decreases when r₂ increases, which means that the larger induction area leads to greater induced reversing field. In addition, when r₂ is larger than 300 mm, the shielding factor turns negative, because the induced reversing field becomes larger than the external field. r₂ = 300 mm can make the shielding factor be nearly zero. Since too large a radius of outer coil would take up much space and increase the volume of the shielding system, r₂ = 300 mm is not an economical choice. In this way, r₂ can be selected as 180 mm, in order to improve the shielding effectiveness by addingturns of coil and may then be effective and space-saving, as demonstrated by Coil 2–2 and Coil 3–3.

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In applications, the size of the shielding space is of great importance. Figure 6 shows the shielding factor along the z-direction (a) and x-direction (b), as well the magnetic field distribution in the x-z plane (c) and x-y plane (d) for Coil 3–3, which shows the best shielding effect at the center of the coil. The tested space is a 50 mm diameter of spherical volume (DSV). This size of space is considered meaningful for practical applications.

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At the center of coil 3–3, the shielding factor is the lowest of 0.2%, and increases along the z-direction, decreasing to negative along the x-direction as shown in figures 6(a) and (b), respectively. In the x-y plane (figure 6(d)) of the shielding area, the size of 1% shielding factor area reaches 20 mm in diameter. However, in the z direction of the x-z plane (figure 6(c)), the size of 1% shielding factor area is less than 10 mm. Moreover, it is observed that the shielding factor increases to 13% 25 mm away from the center of coil 3–3. Therefore the size of the 1% shielding factor zone in the axial direction (z direction) is narrow. Interestingly, there exists a negative shielding factor in some areas, where the reversing induced field of coil is greater than the external field. A negative shielding factor means over shielding. It implies an excess of coil turns may give rise to a negative shielding effect.

To achieve larger available shielding space with 99% shielding factor, especially along the axial direction, a Helmholtz coil is designed, which is believed to be an effective way to improve the uniformity of shielding field. In the present work, a Helmholtz coil consists of double Coil 2–2 is built (namely Coil 2–2AB), also another Helmholtz coil consisting of double Coil 3–3 (namely Coil 3–3 A'B') is built, as shown in figure 7. Generally, for the Helmholtz coil, the distance between two identical coils is the same as the radius of coil. Considering the shielding effectiveness optimization and over shielding effect that occurred, as mentioned above, it is necessary to optimize the distance between coil A and coil B in Coil 2–2AB, also between coil A' and B' in Coil 3–3 A'B'.

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Figure 8 shows the shielding factor in the center of Coil 2–2AB and Coil 3–3 A'B' with different distance. It reveals that the value of shielding factor increases linearly with the distance increasing from 70 mm to 120 mm. In Coil 2–2AB, while the distance is less than 85.78 mm, the value of shielding factor is negative, suggesting that too much of over shielding should be avoided. As the distance is greater than 85.78 mm, the shielding factor increases from 0.01% to more than 7.4%. The applicable distance between Coil A and Coil B appears to be 85.78 mm, while the shielding factor reaches 0.01%. For Coil 3–3 A'B', the shielding factor is negative if the distance is less than 115 mm. This indicates that the reversing induced magnetic field is too strong, and leads to strong over shielding effect which should be avoided. When the distance is 115 mm, the shielding factor of Coil 3–3 A'B' become close to 0, but it will waste more space for the shielding coil in the axial direction and needs wider superconducting tape than Coil 2–2AB. Therefore, Coil 2–2AB is more applicable for magnetic shielding than Coil 3–3 A'B'.

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We further investigate into the spatial distributions of the shielding factor as figure 9 exhibits, in the 70 mm diameter of spherical volume (DSV) at the center of Coil 2–2AB, the optimized distance between Coil A and Coil B is selected as 85.78 mm. Compared with Coil 3–3, the minimum shielding factor of Coil 2–2AB is much lower. Moreover, the size of 1% shielding factor space of Coil 2–2AB is much larger than that of Coil 3–3. Obviously, the shielding factor along the z-direction is much optimized, which is less than 1% along −35 mm to 35 mm (x = 0, y = 0). In the x-y plane, the size of 1% shielding factor area is 40 mm (z = 0) in diameter. In the whole of the 40 mm DSV, the shielding factor is kept below 1%. In the whole of the 70 mm DSV, the shielding factor remains below 4%. Therefore, it is concluded that the design of Coil 2–2AB appears to have better shielding effectiveness than Coil 3–3.

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The shielding factor of the fixed coil is concerned with the external field frequency and the coil resistance. For jointed coils, the shielding effectiveness will reduce in low frequency fields because of jointed resistance. Because the induced electromotive force of the coil is small in low frequency fields, the existence of resistance will significantly reduce the induced current. In other words, it will reduce the reversing induced magnetic field. Since the resistance of joint-free REBCO coil is nearly zero, theoretically speaking, the shielding effectiveness of joint-free coil will not decrease significantly in extremely low frequency magnetic fields. Compared with the report of Bi-2223 shielding coils jointed by YBCO tape with a little welding resistance [16], as figure 10(a) shows, in 10 μT varying external magnetic field, the shielding factor of joint-free coil without resistance (Coil 2–2AB) remains stable from 0.01 Hz to 1000 Hz. However, the shielding factor of jointed coil [16] increases significantly when the frequency is lower than 1 Hz. It indicates that the joint-free architecture is helpful to maintain the shielding effectiveness in low frequency fields.

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The external field intensity determines the induced current intensity of the coil. When the induced current intensity is close to the the critical current intensity of REBCO coil, according to the E-J nonlinear resistivity relationship of HTS, the resistance of coil will increase significantly, it will weaken the induced current, especially under low frequency. Then the shielding effectiveness will decrease (in other words, the shielding factor increases). As figure 10(b) shows, for joint-free coil, the shielding factor of Coil 2–2AB remains about 0.01% in a 1 nT external magnetic field from 1 × 10⁻⁵ Hz to 10 Hz, which means the resulting field is as low as the fT · Hz^−1/2 range. But in a 1 mT external magnetic field, the shielding factor increases with the decrease of frequency. It indicates that if we need to shield higher magnetic fields, the current-carrying capability of superconducting coil should be improved, by using wider tape or more turns, for example.