3D modeling and simulation of 2G HTS stacks and coils

When considering a stack of tapes carrying the same current, as it is the case for the cross section of a coil, integral constraints can be used. In that way, the net current within each conductor is set to a given value. Considering the stack of n_c tapes presented in figure 2(a), a set of integral constraints of the form

$$\begin{equation} \int _{C_{k}}\boldsymbol {J}(x,\bar {y},z,t)\boldsymbol {\cdot } \hat {{\boldsymbol {s}}_{{t}}}\, {\rm d}x\, {\rm d}z=f_{k}(\bar {y},t), \quad\qquad k\in \{ 1,2,\ldots ,n_{{\rm c}}\} \label {eq:IntC3D-disc} \end{equation} \tag{ 1 }$$

guarantees the requested current share. Here J is the current density, $f(\bar {y},t)$ is the current imposed to each tape and $\hat {{\boldsymbol {s}}_{{t}}}$ is a unitary vector defined locally as being tangential to the tapes in the stack, pointing in the intended direction of the current flow. One can note here that the position $\bar {y}$ where the constraint is placed is arbitrary, as the high resistivity of the air or insulation domain will prevent the current from flowing outside the superconducting domains. This makes f_k uniform and equation (1) can be rewritten as:

$$\begin{equation} \int _{C_{k}}\boldsymbol {J}(x,\bar {y},z,t)\boldsymbol {\cdot } \hat {{\boldsymbol {s}}_{{t}}} \,{\rm d}x\, {\rm d}z=f_{k}(t),\quad\qquad k\in \{ 1,2,\ldots ,n_{{\rm c}}\}. \label {eq:IntC3D-disc-1} \end{equation} \tag{ 2 }$$

Use of (1) for the anisotropic homogenous-medium approximation (see figure 2(b)) is not possible as there are no longer any separated conductors that could be individually considered. Therefore, a different approach should be used to ensure the desired current distribution. For this purpose, one can consider the homogenized stack as being composed of infinitely thin tapes. Then, the following condition:

$$\begin{equation} \int _{C}\boldsymbol {J}(\bar {x},\bar {y},z,t) \boldsymbol {\cdot } \hat {{\boldsymbol {s}}_{{t}}}\, {\rm d}z= g(\bar {x},\bar {y},t)\label {eq:IntC3D} \end{equation} \tag{ 3 }$$

can be used to impose a current $g(\bar {x},\bar {y},t)$ to every one of these thin tapes. One must note again that in the case of stacks and coils where individual tapes share the same current, g becomes uniform. Therefore, the aforementioned constraint becomes:

$$\begin{equation} \int _{C}\boldsymbol {J}(\bar {x},\bar {y},z,t) \boldsymbol {\cdot } \hat {{\boldsymbol {s}}_{{t}}}\, {\rm d}z=g(t).\label {eq:IntC3D-1} \end{equation} \tag{ 4 }$$

In the following subsections three different ways to impose this condition are presented.

2.1.1. 2D integral constraint

A first approach is the direct enforcement of (4). Numerically, this requires defining an integral constraint in 2D. In this way, the constraint should specify the total current flowing tangentially to the tapes in the stack or winding in the intended direction of the current flow. For an infinite stack as in the one presented in figure 2, this tangential direction is parallel to the $\hat {\boldsymbol {y}}$ vector, so $\hat {{\boldsymbol {s}}_{{t}}}= {\hat {\boldsymbol {y}}}$. In the case of a racetrack coil whose straight section of length 2 y₀ is aligned with the ${\hat {\boldsymbol {y}}}$ direction as the one in figure 3, $\hat {{\boldsymbol {s}}_{{t}}}$ is given by the following expression:

$$\begin{equation} \hat {{\boldsymbol {s}}_{{t}}}=\left \{ \begin{array}{@{}cl} \displaystyle \frac {1}{\sqrt {x^{2}+(y-y_{0})^{2}}} \{ -(y-y_{0}),x\} & y_{0}\leq y\\ \displaystyle \{ 0,\, \mbox {sign}(x)\} & -y_{0}\leq y<y_{0}\\ \displaystyle \frac {1}{\sqrt {x^{2}+(y+y_{0})^{2}}} \{ -(y+y_{0}),x\} & y<-y_{0}. \end{array}\right .\label {eq:tang} \end{equation} \tag{ 5 }$$

One can note that circular planar coils can also make use of (5) by setting y₀ = 0. In general, expressions for other, more complicated, windings or stack-like structures can easily be found provided that the packing of the tapes is uniform. Constraint (4) can then be imposed by means of Lagrange multipliers.

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2.1.2. Use of anisotropic resistivity

One alternative procedure is to use an anisotropic resistivity tensor that provides a very high resistivity in the direction normal to the tapes' surface. Therefore, the E–J relationship can be expressed as:

$$\begin{equation} {{\boldsymbol {E}}_{\Vert }}=\rho _{\Vert }\boldsymbol {J}_{\Vert } \quad\qquad \mbox {and} \quad\qquad {\boldsymbol {E}_{\bot }}=\rho _{\bot }\boldsymbol {J}_{\bot }. \label {eq:ParPerp} \end{equation} \tag{ 6 }$$

Here, ρ_∥ corresponds to the homogenized resistivity of the superconducting stack and ρ_⊥ is an artificially imposed high resistivity used to prevent current flow in the direction normal to the tapes' surface.

2.1.3. Manual discretization of the homogenized bulk

As mentioned before, in this work, all calculations are made using the H-formulation of Maxwell's equations [16]. To account for the magnetic flux density dependence of the critical current density J_c, the following expression is used:

$$\begin{equation}J_{\rm c}(B_{\parallel },B_{\perp })=\frac {J_{{\rm c}_{0}}\, f_{\rm HTS}}{\Big [1+\sqrt {(B_{\parallel }k)^{2}+B_{\perp }^{2}} \Big /B_{\rm c}\Big ]^{b}}\label {eq:JcB} \end{equation} \tag{ 7 }$$

here B_∥ and B_⊥ are respectively the parallel and perpendicular components of the magnetic flux density. The parameters J_c0, k, B_c and b have the respective values of 49 GAm⁻², 0.275, 32.5 mT and 0.6. These parameters correspond to a characterization (not reported here) for the J_c(B_∥, B_⊥) for the HTS tape used in [17] using an elliptic fit. This corresponds to a self-field critical current I_c of 160 A at 77 K. The remaining parameter, f_HTS, is the volume fraction of the superconducting material in the homogenized bulk[10]. In what follows, we will refer to J_c(B_∥, B_⊥) simply as J_c(B). To describe the E–J relationship, a power law is used:

$$\begin{equation} \boldsymbol {E}=E_{0}\left |\frac {J}{J_{\rm c}(B_{\parallel },B_{\perp })} \right |^{n}\frac {\boldsymbol {J}}{|J|}\label {eq:PL} \end{equation} \tag{ 8 }$$

here, the critical electrical field E₀ at which J = J_c(B_∥, B_⊥) is set equal to 1 μV cm⁻¹. The exponent n = 21 in the power law is used to describe how abrupt the transition from the superconducting to the normal state is.

To validate the proposed strategy, the case of a stack consisting of 50 4 mm-wide tapes having a total height of 2 cm was considered. Assuming a thickness for the superconducting layer of each tape of 1 μm, this corresponds to a volume fraction, f_HTS of 2.5 × 10⁻³. AC currents at 50 Hz were imposed in each of the tapes in the stack. The straight stack was modeled following two different approaches: a 2D model for large stacks as described in [7] which takes into account the 50 different conductors in the original layout of the stack, and the 3D homogenized model as described in the previous section. Since the conductivity of the superconducting layers is much higher than that of the other materials involved, and no magnetic substrate is used, only the superconducting layers were simulated. As a mean to compare both models, AC losses were computed for different current amplitudes. The results are shown in figure 4. It is important to note the good agreement over the large range of applied currents and over more than two orders of magnitude for the calculated AC losses.

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Having successfully tested the proposed 3D homogenization with a 2D stack, the aim is set in addressing more complicated problems. Just like infinitely long straight stacks can be modeled in 2D, circular coils can be modeled using an asymmetrical approach, yielding again a 2D model. Therefore, we have chosen to use the proposed homogenization technique for a design that can not be modeled with 2D tools: racetrack coils. Racetrack coils, such as the one shown in figure 3 have two straight and two round sections. Taking advantage of the symmetry planes, only one eighth of the coil was considered for modeling and simulation. Material properties used for simulating the racetrack coil were the same as the ones described for the stack in the previous section. The model considered a 50 turn single racetrack coil wound using 4 mm wide tape with geometrical parameters y₀ = 7.5 cm,  r₁ = 1.5 cm and r₂ = 3.5 cm as shown in figure 3. Just like before, a volume fraction, f_HTS of 2.5 × 10⁻³ was considered. AC currents at 50 Hz were imposed to the homogenized coil.

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Figure 5 shows the magnitude of the magnetic flux density at the three symmetry planes of the racetrack coil at a peak current value I of 100 A. The highest magnetic flux density is localized in the inner part of the round section, with a value of 234 mT. In the internal region of the coil, close to the coil's plane, the magnitude of the field is almost uniform with a value of about 10 mT. However, clear differences are observed in both the straight and the round sections of the coil, showing the true three dimensional structure of the model.

Normalized current density distributions J/J_c(B) for both zero crossing (a) and peak current (b) values are shown in figure 6 for the case of a transport current of 100 A. One can note how the homogenization method allows for calculating the current distribution in any place within the stacked tapes. At a first glance, it might seem that the current distributions for each particular case (a) and (b) are rather uniform in the direction of the coil's winding. However as it will be shown that is not necessarily the case. Figure 7 shows the current distributions close to peak value for a transport current of 120 A for two particular locations: the middle of the straight section (top right) and the middle of the round section (bottom right). First, one can note that the current distributions are not symmetric, they have larger critical regions close to the coil's inner side. Secondly, this effect is considerably bigger in the cross section corresponding to the round part. This means that the current is limited by the innermost turn of the coil in the round section. Looking at figure 5 it is easy to see how this relates to the high magnetic flux density in this region of the coil and the fact that a J_c(B) relationship was used.

So far, the model presented here has proved to be useful to calculate magnetic field, current distribution and critical current for a coil under current transport conditions. All of this within a 3D framework.

Finally, AC losses were computed for different current amplitudes at 50 Hz for a 50 turn coil made with geometrical parameters y₀ = 7.5 cm, r₁ = 3.5 cm and r₂ = 5.5 cm. These correspond to the coil frame designed as part of the Superwind project at DTU [18]. These estimates are presented in figure 8. The dashed line shows a power law fit of the form Q = 2.92 (I/I_c)^3.58. Here, Q is the AC loss in J/cycle, I is the amplitude of the current and I_c the self field critical current of the tape (160 A). A coil of similar size was studied in [7] by means of a 2D planar model. Although the tape characteristics in [7] were different, in said work a similar exponent (3.6351) in the power law fit is reported. For the purpose of comparison, losses were also calculated using both 2D planar and 2D axisymmetric models. The 2D planar model assumed two opposing stacks of 50 tapes each separated by 7 cm. Losses per unit volume were computed and scaled to the volume of the racetrack coil. The 2D axisymmetric model assumed a circular coil of 50 tapes with inner radius of 3.5 cm and outer radius of 5.5 cm. Again, losses per unit volume were computed and scaled to the volume of the racetrack coil.

As shown in figure 8, losses computed using a 2D axisymmetric model provided good agreement only for a limited range of currents but diverged for large I/I_c values. This behavior can be easily understood since for a given current amplitude, the field in the innermost turn of a circular coil is larger than that of its corresponding racetrack coil. Therefore, as the current increases, the innermost turn of the circular coil will saturate faster than that of the racetrack coil, hence yielding much larger losses. On the other hand, the 2D planar model showed good agreement with the 3D homogenization method. For the reasons presented above, this agreement is expected to decrease for larger currents when the effect of the round section of the racetrack coil becomes more important or for coils with smaller straight sections.