Cross-field demagnetization of stacks of tapes: 3D modeling and measurements

The SuperOx tape contains a Hastelloy substrate, an SC layer, and a silver thin layer; as explained in section 2. However, the model only takes the SC layer into account. Then, the Hastelloy, silver, and cooling medium are treated as a void (or 'air') in the model, since these materials are non-magnetic and (for the metals) their eddy currents are negligible in the studied frequency range (up to 500 Hz). From now on, we refer to 'tape' as the SC layer only. The effective gap is around 200 μm and is slightly different for each studied configuration. The thickness of the SC layer depends on the goal of the study. The general qualitative study of the cross-field demagnetization uses the thickness of 10 μm and the more precise calculation for comparison to experiments uses the real thickness of 1.5 μm. Unless stated otherwise, the frequency of the ripples is taken as 500 Hz. We chose this characteristic frequency because the rotor in electric machines for aviation presents ripple fields of fundamental frequencies of few hundred hertz or higher, since their rotating speed is targeted to a few thousand rpm [6, 22].

We perform most of the calculations here by the MEMEP 3D method [39] based on a variational principle, being the real thickness calculations and comparison to experiments done by this method only. The mathematical formulation uses the ${\bf{T}}$ vector defined as an effective magnetization. The effective magnetization is non-zero only inside the modeling sample, and hence the method does not solve the surrounding air domain. However, the air separation between SC layers in the stack is modeled as a conducting material of high resistivity. The incorporated isotropic power law enables taking n values up to n = 1000 into account. We use n = 200 as an approximation to the CSM and n = 30 as a realistic value for the measurements. The modeling software [40] was developed in C++ and it is enhanced by parallel computation on a computer cluster [35, 41, 42]. The method uses hexahedric elements with a high aspect ratio, up to 5000. Therefore, MEMEP can use the same modeling geometry as the measured samples. The model assumed either isotropic constant J_c, J_c(B) or J_c(B, θ) from measurements, depending on the configuration, with B being the magnitude of the magnetic field (we use the term 'magnetic field' for both ${\bf{B}}$ and ${\bf{H}}$, since for our case ${\bf{B}}={\mu }_{0}{\bf{H}}$) and θ being the angle between ${\bf{B}}$ and the normal of the tape surface (figure 2(b)). The measured I_c of the same kind of tape as the experiments but of 4 mm width for several applied magnetic fields and orientations is in figure 2(a). Some calculations are done in the 2D version of MEMEP [43], to assess the finite-sample effects. All modeling results are 3D, unless stated otherwise.

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Unless stated otherwise, the number of elements in the superconductor are 15 × 15 × 9 per tape, with the last the number of elements in the thickness. We have checked that with this mesh, the calculated cross-field demagnetization is mesh independent. As can be seen in figure 3, the difference in trapped field in one tape for both at the end of relaxation and after 10 cycles is the same for 7 and 9 cells across the thickness. We also checked that the results are insensitive to the number of cells in the tape width and, being this the mesh used for the comparison with experiments. For all calculations, we use a tolerance for J of 0.01% of J_c at zero magnetic field, although we checked that the results do not differ for a tolerance of 0.001%.

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The variational formulation of MEMEP has been extended to take magnetic materials into account, as shown in the 2D examples of [45]. However, magnetic materials are not yet implemented in the 3D version of the software. This is not an issue in this article because the studied tape substrate is non-magnetic.

Here, we use FEM in order to benchmark the MEMEP 3D method. The FEM model is based on the H-formulation of Maxwell's equations implemented in the finite element program COMSOL Multiphysics [46]. Due to the necessity of simulating the air between and around the SC tapes typical of the FEM approach, care had to be taken in building the domains and the mesh. In order to avoid an excessive number of degrees of freedom, an approach based on sweeping a 2D geometry and mesh was followed (see figure 1 of [32] for an example). The external magnetic field was applied on the boundary of the air domains by means of Dirichlet boundary conditions. A magnetic field of B_z = 300 mT was assigned to all simulated domains as an initial condition. For FEM, we use 15 × 15 × 5 elements in each SC tape and a tolerance of 0.5%.

The first study is about in-plane magnetization due to a parallel applied magnetic field. A sketch of the modeling case and the dimensions is shown in the figure 4. The magnetization loop is calculated only for the M_x component, because the applied magnetic field (of amplitude 50 mT and 50 Hz frequency) is along the x axis. We used a thicknesses of the SC layer d in the range from 1–1000 μm for this case. We assume constant J_cd, with J_c = 2.72 × 10¹⁰ A m⁻² for d = 1 μm and an n power law exponent of 30.

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The magnetization increases with tape thickness (figure 5(a)). This is not the case for the CSM, which we approximate as a power law with exponent n = 200 (figure 5(a)). All magnetization loops for different thickness are within 2%, without being identical due to the finite power law exponent (figure 5(b)). However, commercial SC tapes have an n value of around 30 in the self-field. The thickness dependence is due to higher electric fields from the applied magnetic field in thicker tapes. The relatively low n value of 30 allows J > J_c for high local electric fields, and hence the magnetization increases with thickness roughly as 14% for each increase in thickness by a factor 10. This effect does not appear for the CSM, since $| {\bf{J}}|$ is limited to J_c, irrespective to the value of the non-zero electric field. Therefore, we can already see that the sample thickness plays a role in the response to the ripple field. The effects of the thickness are much more important for the cross-field demagnetization (section 4.2), which is also significant for the CSM [27, 47]. In the calculations in the following sections, we assume the real thickness of 1.5 μm for comparison to experiments and 10 μm for the purely numerical analysis.

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Next, we present a more detailed study about thickness influence for the cross-field demagnetization. The model assumes a single tape with thicknesses from 1–100 μm. Here, we make a similar analysis as for the published 2D modeling starting from 10 μm upwards for a single cycle [5] but now for the 3D configuration, starting from 1 μm, and up to 10 cycles, being more relevant for experiments. Now, the critical current density is inversely proportional to the thickness. The J_c of 1 μm tape is J_c = 2.72 × 10¹⁰ A m⁻² and the n value is 30. The tape is magnetized with the perpendicular applied field to the tape surface by the FC method. The field is ramped down with rate 30 mT/s over 100 s with following relaxation of 900 s. Afterwards, a sinusoidal transverse field of 500 Hz is applied along the x axis.

The demagnetization rate greately increases with the thickness (figure 6), even though J_c is inversely proportional to the thickness. The clear thickness dependence shows the importance of the sample thickness, being even more relevant than in section 4.1. Therefore, the thickness in the model is very important for the cross-field demagnetization, where ripples are in the in-plane direction. The model cannot assume thicker films and lower proportionally the critical current density, as already predicted by Campbell et al [25]. Since the SC layer in most ReBCO tapes is of the order of 1 μm, 3D modeling is very challenging due to the high aspect ratio. Most previous works, which are in 2D, assumed unrealistically thicker samples due to numerical issues.

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The next study is about the influence of number of tapes and gap between SC layers in the stack on the initial trapped field. We used the same geometrical parameters as in section 4.1. The SC layer is 10 μm thick with J_c = 2.72 × 10⁹ A m⁻² and n = 30. As shown above, it is not possible to take a larger superconductor thickness and a proportionally lower J_c for cross-field demagnetization and the transverse applied field. However, this simplification can be made for applied fields perpendicular to the tapes [48, 49]. The cause is that the electric field due to the applied magnetic field is roughly uniform in tape thickness and the effect of the self-magnetic field can be averaged over the tape thickness.

The stack is magnetized by the FC method along the z axis. The initial applied magnetic field is 1 T with a ramp down rate of 10 mT/s, because the perpendicular penetration field B_p of one tape is 27.2 mT. Afterwards, we leave a relaxation time of 900 s, which is long enough to reach the stable state of the trapped field. The trapped field decreases logarithmically during relaxation, and hence after a short time the reduction is almost negligible. The trapped field is calculated 1 mm above the top tape, similar to a Hall probe experiment. The probe position is more relevant for commercial application than the magnetic field in the tape or between the tapes. The sketch of the modeling case is shown in figure 7(a).

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The trapped field increases with the number of tapes as shown in figure 8. The big gap of 200 μm between the SC layers causes saturation of the trapped field. The trapped field decreases with the gap g, and hence SC tapes with thinner Hastelloy, stabilization, and any additional isolating layers are more suitable for applications based on stacks of tapes. The cause of the decrease in the trapped field with increasing the gap is the increase of the overall thickness-to-width ratio, since the contribution of the bottom tapes to the trapped field on the top decreases with the overall thickness. This decay with the thickness is faster in 3D calculations than in cross-sectional 2D. The cause is that in short tapes the magnetic field created by closed loops are dominated by the dipolar term and decays with the distance r as $1/{r}^{3}$, while in 2D the decay is 1/r².

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Next, we cross-check the MEMEP 3D method and the FEM. The comparison case is for a two-tape stack only, for simplicity. The SC layer is 10 μm thin with a 200 μm gap between the layers. The electrical parameters are the same as J_c = 2.72 × 10⁹ A m⁻² and n = 30. The mesh contains 15 × 15 × 5 cells per SC layer. The computing time for this mesh and the cross-field of 240 mT is 3–4 h for the MEMEP 3D method and 14–15 h for the FEM. MEMEP used a computer with i7-4771 CPU at 3.5 GHz and the FEM used a workstation with i7-4960X CPU at 3.60 GHz.

The magnetic moment of the sample at the end of relaxation differs only by 4% between both models. Therefore, the screening current calculations are in very good agreement.

The result of the trapped field 1 mm above the top surface is shown in figure 7(b). The trapped field is normalized by the trapped field at the end of the relaxation time, B₀. The value of B₀ is 32.4 mT for the MEMEP 3D method and 28.6 mT for the FEM. The comparison is in good agreement, even though the trapped field has an 11% difference This difference is due to inaccuracies in the trapped magnetic field by the FEM, due to the relatively coarse mesh in the surrounding air. This could also be the cause of the discrepancy in the magnetic moment. Nevertheless, the demagnetization rate is the same for both methods. This confirms the validity of the MEMEP 3D calculations, being more accurate than FEM for this configuration.

The next study is about the entire cross-field demagnetization process in the stack of five tapes. A sketch of the modeling case is shown in figure 7(a) and it uses the same parameters as in the previous sections. The gap g between the SC layers is 200 μm and the SC layer is 10 μm thick. The time evolution of the magnetizing field, B_az, and the trapped field on the stack, B_t, are shown in figure 9. At 1000 s, the sinusoidal ripple field, B_ax, of 500 Hz is switched on.

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The study of the current density inside the stack is on the x and y cross-section planes, as defined in figure 10. The current density maps (figure 11) are in real scale except the z coordinate. Since the tapes are 10 μm thick and the gap is 200 μm wide, the variations in the thickness of the current density are not visible in the real scale. Therefore, the map contains the real data, but the SC cells are shown with the same height as the the gap cells. Since we use an odd number of cells in the x and y directions, there appears a central cell with zero current density in figures 11(a), (d)–(f), causing the vertical purple line. The study case is with a cross-field amplitude 240 mT.

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The stack at the end of the relaxation is fully saturated with J_x ≈ J_c and J_y ≈ J_c as shown in figures 11(a) and (d), respectively. The J_z component is very small, around 0.001J_c due to the thin film shape. The current density magnitude is below the critical current density J_c, because of flux relaxation.

The applied cross-field at the first positive peak causes small penetration of the screening current from the top and bottom of each individual tape, as can be seen in figure 11(b). The penetration front on the y-plane rewrites the remanent state of the current density J_y to the positive sign from the top and with a negative sign from the bottom of each tape (figure 11(b)). This process can be explained by the Bean model of the infinite thin strip [27] or other numerical 2D modeling [5, 25]. In our stack, the penetration front contains both the J_y and J_z components, even though J_z is very small. The x-plane section of figures 11(d)–(f) shows that the J_x current density from remanent state progressively decreases at each cycle due to the effect of J_y (and also J_z in a lesser amount) caused by the ripple field, which penetrates from the top and bottom of each individual tape. This is the same qualitative behavior found for SC bulks [30]. The oscillations in the current profiles in figures 11(e) and (f) are due to a numerical error.

Finally, we study in detail the current density maps at the tenth positive peak of the cross-field. At the y-plane, the J_y current density component penetrates slightly more than after the end of the first cycle. The penetration depth is 2 cells from the top and bottom of each layer within the total of 9 cells per SC layer (figure 11(c)). The lowest penetration is in the inner tapes. The cause is that the inductive coupling with the rest of the tapes is the largest. This enhanced inductive coupling slows down the demagnetization decay due to the dynamic magneto-resistance [27, 47], with exponential decay for a large enough time. At the x-plane, the J_x component is almost completely concentrated in three cells in the center of each SC layer (figure 11(f)). The current maps show the slow penetration of the current front and the erasing process of the remnant state with increasing the number of cycles. Although J_z plays a role in the cross-field demagnetization, its value is much below J_c, and hence reducing J_c in the z direction will not cause any significant change.

The trapped magnetic field B_t during the whole process is calculated 1 mm above the top surface (figure 9). After relaxation, we applied cross-fields with two different amplitudes, 40 and 240 mT, in order to see the behavior in the fields below and above the penetration field B_p of the stack. The parallel penetration field B_p,∥ of 1 tape is 17 mT according to the slab CSM [50, 51],

$$\begin{eqnarray}&&\ {B}_{p,\parallel }={\mu }_{0}{J}_{c}d/2.\end{eqnarray} \tag{ 1 }$$

The demagnetization rate for the 40 mT cross-field is very low, with a 3.9% drop of the trapped field after ten cycles (figure 12). The higher cross-field of 240 mT makes a 19.1% reduction of the trapped field. The roughly linear demagnetization is consistent with Brandt's predictions, where there is linear decay for the first few cycles [27]. However, the method of [27] is based on the Bean model and for a single tape. For applied fields above the penetration field, as is the case of both 40 and 240 mT, demagnetization will continue until the entire sample is demagnetized.

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For high demagnetization rates, such as high ripple field amplitudes or thick SC layers, we observe steps in the demagnetization of the time evolution (figures 6 and 12). These steps are also present in bulks (see figure 12 and reference [30]) and are due to the changes in the rate of current density penetration within an AC cycle. Current penetration from the top and bottom layers of each tape slows down at the AC peaks, causing a plateau in the trapped field.

A more detailed trapped field profile is calculated along the (red) "B_t" line above the sample shown in figure 13(a). The profile has the usual symmetric peak at the end of the relaxation time (figure 13(b) black curve). The first positive peak of the cross-field of 240 mT makes a small reduction of the trapped field (figure 13(b) blue curve), since the current density has not changed significantly due to the transverse field. The last positive peak of the cross-field is shown in figure 13(b) as a red line. The trapped field peak is always symmetric without any shift, contrary to cubic bulk samples [30]. The cause of this difference is the thin film shape of the tapes, as follows. For the bulk, the trapped field depends on the current distribution across the thickness, with a higher contribution for J closer to the top surface. Since J does not have mirror symmetry toward the yz-plane, (only inversion symmetry toward the bulk center) the trapped field on the surface is not symmetric. In contrast, the trapped field in the thin films only depends on the average J across the tape thickness, with variations of this dimension being irrelevant. Since the average thickness of J does have mirror symmetry with respect to the yz-plane for each tape of the stack, the trapped field on the surface also presents this mirror symmetry.

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There are two alternatives for high temperature supermagnets: stacks of tapes and bulks. Both candidates broke the world record of a trapped field, at above 17 T with slightly higher values for the stack [1, 2]. However, both behave differently under cross-fields. Therefore, we performed a short simple comparison between them. We used the same geometry for the five-tape stack as was mentioned above. We calculated the engineering current density for the stack J_ce = 160 MA m⁻² and set it as critical current density J_c for the bulk. The samples with their size dimensions are shown in figure 14. We estimated the parallel penetration field of the equivalent bulk from the slab approximation

$$\begin{eqnarray}&&\ {B}_{p,\parallel }\approx {\mu }_{0}{J}_{{ce}}{d}_{{all}}/2,\end{eqnarray} \tag{ 2 }$$

with d_all = 0.85 mm of the overall stack thickness and ${B}_{p,\parallel ,{bulk}}\approx 85.4$ mT.

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The trapped field at 1 mm above the top surface at the end of the relaxation time of 900 s is similar for the stack (71.6 mT) and the bulk (71.4 mT), because of similar parameters. The end of the relaxation time is marked as the 0 cycle in figure 12. The bulk shows a significant drop in the trapped field of around 93% in the first four cycles of the 240 mT cross-field, larger than the bulk penetration field (85 mT). In the case of low cross-field amplitude of 40 mT, the drop is around 16%. For a large number of cycles, the trapped field will decrease until a quasi-stable state is reached, because the ripple field amplitude is below the parallel penetration field of the bulk [27], with the latter dominated by a slow flux creep decay [52]. The stack shows a much slower demagnetization rate for the high cross-field of 240 mT and the trapped field drop in ten cycles is only 19%. The same behavior is observed for the low cross-field of 40 mT with a very low trapped field reduction of 3.9% at ten cycles. Nevertheless, demagnetization should continue until it completely demagnetizes the sample, because the ripple field is above the parallel penetration field of one tape (17mT). Therefore, for the high cross-field cases (ripple field above the parallel penetration field of the bulk) the stack of the tapes is more suitable. However, in the case of low fields the bulk is more suitable because the trapped field after many cycles reaches a non-zero asymptotic value. Nevertheless, if the super-magnet is submitted to a relatively low number of cycles, the stacks of the tapes are preferred in any case. Applications with low frequency ripples and built-in re-magnetization, such as certain low-speed motors and wind turbines, might also favor stacks of tapes, because of less re-magnetization.

The last study is about measurements and comparison with calculations. The stack consists of five tapes and the parameters are listed in the table 1. The details about the sample and the measurements are given in section 2. The measurements are performed for two cross-field frequencies: 1 and 10 Hz.

Table 1.  Input parameters of the measurements and calculation.

Size [mm]	12 × 12 × 0.001 5
${J}_{c,{self}}$ [A/m2]	2.38 × 1010
${B}_{{az},\max }$ [T]	1.0
Ramp rate [mT/s]	10
Relaxation [s]	300
Ec [V/m]	1e-4
${f}_{{ax}}$ [Hz]	0.1,1
Bax [mT]	50, 100, 150
n [-]	30

The demagnetization rate increases with the field (figure 15). The rate per cycle at low frequencies only slightly depends on the frequency, in agreement to [26]. The reason is that the higher frequency of the applied field causes a higher electric field, and hence the current density increases. This reduces both the penetration field and the demagnetization rate [30, 49, 53–55]. The measurements also show increased frequency dependence with field (figure 15). A possible cause might be the decrease of the power law exponent with the applied field amplitude (figure 16(a)). The oscillations observed in the measurements are most likely due to experimental noise.

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For the model, we take three different assumptions for J_c: constant J_c; isotropic J_c(B), with B being the local modulus of the magnetic field; and anisotropic J_c(B, θ). Only the latest approach provides realistic predictions. The model uses a 1.5 μm thin SC layer with a 220 μm gap between them, with the rest of parameters the same as for the measurements listed in table 1.

The coarsest predictions are for constant J_c (figure 17). The measured I_c of the 12 mm wide SuperOx tape is 450 A at the self-field (or J_c = 2.5 × 10¹⁰ A m⁻²). We reduced J_c by 26% to the value J_c = 1.85 × 10¹⁰ A m⁻², in order to get a similar trapped field, 58.9 mT, at the end of the relaxation time as the measurements, 58.1 mT. Naturally, this is an artificial correction of the stack self-field effect, and hence for constant J_c the predicting power is only in regard to the demagnetization rate. However, the predicted demagnetization rate is substantially lower than the measured one. The reason is that the assumed critical current density is too high, because of the missing J_c(B, θ) dependence in the model. The magnetic field reduces the critical current density, and hence it increases the demagnetization rate.

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Another comparison between the model with J_c(B) dependence and measurements is shown in figure 18. The J_c(B) data was measured on the 4 mm wide SuperOx tape (figure 2). The critical current per tape width at the self-field for the measured tape (37.8 A/mm) is roughly the same as the 12 mm wide tape used in the stack (35.8 A/mm), with the latter value the minimum stated one by the producer. The theoretical difference is 5%, which is very small. The average tape I_c of the measured sample could be higher, around 440 or 450 A regarding typical deviations in SuperOx tapes, and hence even more close to that in the calculations. By now, we assume an isotropic J_c(B) dependence, taking the measured J_c(B, θ) values at perpendicular applied field. Then, we assume an isotropic angular dependence in the model. The cross-field is parallel to the tape surface, and hence the actual critical current density is larger than that in the model, also presenting lower reduction under magnetic fields than assumed. This is the reason why the demagnetization rate is overestimated for the high cross-fields of 100 and 150 mT.

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The last comparison here uses the measured J_c(B, θ) dependence. Now, the model agrees very well with the measurements (figure 19). The lowest deviation of the trapped field at the last cycle is 0.2% for 50 mT and 1.2% for the 100 mT cross-field. The highest cross-field amplitude causes the highest demagnetization in the calculation, with the difference of 4.0%. For low demagnetization, the demagnetization rate, β, is more relevant than the trapped field, which we can define as

$$\begin{eqnarray}&&\beta \equiv [{B}_{t}({t}_{{\rm{i}}})-{B}_{t}({t}_{{\rm{f}}})]/({t}_{{\rm{f}}}-{t}_{{\rm{i}}})\end{eqnarray} \tag{ 3 }$$

with t_i and t_f the time at the beginning and end of the demagnetization process, respectively. For our configuration, which is similar to that in other works [5, 24, 25, 28], the error regarding this quantity is more strict than for the trapped field (table 2), resulting in errors below 20% for ripple fields of 100 mT or below.

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We also study the effect of the measured n(B, θ) dependence (figure 16(a)) compared to the constant n = 30 assumption. The demagnetization rate is only slightly changed with n(B, θ) dependence (figure 16(b)). The local J increases with decreasing the n value, and hence it changes the demagnetization rate. There is a slight reduction of the demagnetization rate at the first few cross-field cycles. However, later on the demagnetization rate overlaps with the constant n curve and slightly increases. The demagnetization rate is more influenced by n(B, θ) for cross-fields above 50 mT.

There are several reasons for the reduction in the accuracy of the model. The J_c(B, θ) data covers correctly only the J_c in the y-plane position (figure 10), where the current is perpendicular to the magnetic field. However, at the x-plane, the current density presents a large component parallel to the applied magnetic field in the x direction. This is the so-called force-free configuration [35], where J_c should be different than the typical J_c(B, θ) measurements with ${\bf{B}}$ always perpendicular to the transport direction. Since measurements show that J_c in force-free configuration is often higher [56, 57], this could explain the overestimated cross-field demagnetization in the model. Measurements of solid angle dependence, J_c(B, θ, ϕ) with ϕ the angle of ${\bf{B}}$ with the current density, are scarce for any type of sample [56] and missing for this particular tape. The cause is the complexity of the measurements, requiring a double goniometer [58–61]. The model uses J_c(B, θ) data for both components J_x and J_y, and hence there is a discrepancy between the model and the real measurements in the highest cross-fields. The discrepancy between the measurements and calculations could also be due to variations in the magnetic field dependence of J_c between the measured one and the SuperOx tapes in the stack.

Finally, we check the relevance of 3D modeling, in contrast to cross-sectional 2D computations. Since this is entirely a geometrical effect, the constant J_c assumption is sufficient. For the calculations, we use the same conditions as those in the previous section, about comparison to experiments.

As can be seen in figure 20, the trapped field for 2D modeling is around 20% lower at the end of the relaxation (400 s). In addition, there are also substantial differences in the relative demagnetization rate, as defined in equation (3). 2D modeling predicts a fasterdemagnetization rate by 33%, 35%, and 39% for 50, 100, and 150 mT ripple field amplitudes, respectively, as calculated for the 20th cycle. This will worsen the agreement with experiments, where the 3D model already overestimates the demagnetization rate for 100 and 150 mT, possibly caused by neglecting force-free effects. Since 2D modeling also inherently neglects force-free effects, geometrical and force-free errors accumulate, which might cause a deviation as large as 60% for the 20th cycle and high ripple fields.

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In addition, 2D modeling cannot study the parallel component of ${\bf{J}}$ to the ripple field. This component, the x component in figures 10 and 11, has roughly the same contribution to the trapped field as the y component, the latter being the only one computed in 2D modeling. 2D modeling also disables the possibility to take force-free effects into account, potentially incurring to further errors.