Theoretical analysis of the transport critical-state ac loss in arrays of superconducting rectangular strips

We present a systematic theoretical study of current profiles, magnetic field lines, and hysteresis ac loss in linearly arranged arrays of rectangular superconducting strips subjected to a transport current. Results are obtained by means of numerical calculations assuming the critical-state model with a constant critical current density. Because finite filament thickness and magnetic coupling effects are considered, we can provide some useful hints as to how to arrange the filaments in order to reduce the ac loss in actual superconducting tapes.

Figure 1

Current fronts for horizontal arrays made of three strips with aspect ratio $a∕b=20$ and separations $d_x∕a=0.02$ (a), 0.2 (b), and 4 (c). The current fronts correspond to $I∕I_c=0.1$, 0.2, 0.4, 0.6, and 0.8 from outer to inner. The vertical axis has been expanded and strip separations have been drawn as the same for better visualization.

Figure 2

Current fronts (thick lines) and magnetic flux lines [actually they are $A_c-A\left(\mathbf{r}\right)$ level curves; see text] for horizontal arrays made of three strips with aspect ratio $a∕b=20$ and separation $d_x∕a=0.2$ (a) and $d_x∕a=1$ (b) carrying a transport current $I=0.6I_c$. Only the right half part of the tape is plotted. The $A_c-A\left(\mathbf{r}\right)$ variation between level curves is the same for both plots.

Figure 3

Normalized ac loss $2\pi Q∕{\mu }_0{I}_c^2$ as a function of the normalized ac field amplitude $I_m∕I_c$. Solid lines correspond to horizontal arrays made of three strips with $a∕b=20$ and $d_x∕a=0,0.2,1,10,\infty$ in the arrow direction, the dashed curve to a rectangular strip with $a∕b=20$, the dotted curve to the analytical formulas for ellipse (Ref. 15), and the dash-dotted curve to a thin strip (Ref. 15).

Figure 4

Current fronts for horizontal arrays with $n_{s,x}=2$ (a) and 9 (b). Current profiles correspond to $i=0.1,0.2,0.4,0.6$ (solid lines) and $i=0.8,0.9$ (dash lines) from outer to inner. The cross shows the kernel position. Only the right half is plotted for space reasons.

Figure 5

Normalized ac loss $2\pi Q∕{\mu }_0{I}_c^2$ as a function of the normalized ac field amplitude $I_m∕I_c$. Solid lines correspond to horizontal arrays with $a∕b=20$, $d_x∕a=0.2$, and $n_{s,x}=1$, 2, 3, 5, and 9 in the arrow direction and the dotted line is for an ellipse (Ref. 15). Inset shows the ac loss for a wider range of normalized current amplitude.

Figure 6

Current fronts in vertical arrays made of three strips with aspect ratio $a∕b=20$ and separations $d_y∕a=0.02$, 0.2, and 4 (a,b,c). The current fronts correspond to $i=I_m∕I_c=0.1$, 0.2, 0.4, 0.6, and 0.8 from outer to inner. The vertical axis has been expanded and strip separations have been drawn as the same for better visualization.

Figure 7

Current fronts (thick lines) and $A_c-A\left(\mathbf{r}\right)$ level curves as magnetic flux lines for vertical arrays made of three strips with aspect ratio $a∕b=20$ and separation $d_y∕a=1$ (a) and $d_x∕a=0.2$ (b) carrying a transport current $I=0.6I_c$. The $A_c-A\left(\mathbf{r}\right)$ variation between level curves for these figures is the same as for Fig. 2.

Figure 8

Normalized ac loss $2\pi Q∕{\mu }_0{I}_c^2$ as a function of $I_m∕I_c$. Solid lines correspond to a vertical array made of three strips with $a∕b=20$ and $d_y∕a=0,0.2,1,10,\infty$ in the arrow direction, the dash curve to a rectangular strip with $a∕b=20$, the dot one to the analytical formula for ellipse (Ref. 15), and the dash-dotted curve to a thin strip (Ref. 15).

Figure 9

Normalized ac loss as a function of normalized current amplitude. Solid lines correspond to a vertical array made of three strips with $a∕b=50$ and $d_y∕a=0,0.2,2,\infty$ in the arrow direction, the dashed curve to a rectangular strip with $a∕b=50$, and the dotted curve to an ellipse. The inset shows the current fronts for a vertical array with $a∕b=50$ and $d_y∕a=0.2$ (solid lines) and a rectangular strip with $a∕b=50∕3$ (dotted lines). The strip separation for the array has been removed and the vertical axis has been expanded for better visualization.

Figure 10

Normalized ac loss as a function of normalized current amplitude. Solid lines correspond to vertical arrays of strips with $a∕b=20$, $d_y∕a=0.2$, and $n_{s,y}=1,2,3,5,9$ in the arrow direction and the dotted curve is for an ellipse. The inset shows the ac loss for a wider range of normalized current amplitude.

Figure 11

Normalized ac loss as a function of normalized current amplitude. Solid lines correspond to vertical arrays of strips with $a∕b=20$, $d_y∕a=1$, and $n_{s,y}=1,2,3,9$ in the arrow direction and the dotted curve is for an ellipse. The inset shows the ac loss for a wider range of normalized current amplitude.

Figure 12

Current fronts (thick lines) and $A_c-A\left(\mathbf{r}\right)$ level curves as magnetic flux lines for a $3\times 3$ matrix array of strips with aspect ratio $a∕b=20$, $d_y∕a=0.2$ and $d_x∕a=0.2$ (a) and $d_x∕a=1$ (b). In both cases the array carries a transport current $I=0.6I_c$. The $A_c-A\left(\mathbf{r}\right)$ variation between level curves for both cases is the same.

Figure 13

Normalized ac loss as a function of normalized current amplitude. Solid lines correspond to a $3\times 3$ matrix array of strips with $a∕b=20$ and several strip separations. In (a) matrices have the same vertical separation $d_y∕a=0.2$ and several horizontal ones $d_x∕a=0,0.2,2,\infty$ in the arrow direction, whereas in (b) separations are $d_x∕a=0.2$ and $d_y∕a=0.2,0,2,\infty$ in the arrow direction.

Figure 14

Sheet current profiles for a horizontal array of two strips with $a∕b=100$, $d_x∕a=2$ (a) and $d_x∕a=0.2$ (b), and $i=0.1$, 0.2, 0.4, 0.6, and 0.8 in the arrow direction. Solid lines correspond to our numerical results, while dashed lines correspond to the analytical formulas in Ref. 18 for a double thin strip.