Field and current distributions and ac losses in a bifilar stack of superconducting strips

In this paper, I first analytically calculate the magnetic-field and sheet-current distributions generated in an infinite stack of thin superconducting strips of thickness $d$, width $2a⪢d$, and arbitrary separation $D$ when adjacent strips carry net current of magnitude $I$ in opposite directions. Each strip is assumed to have uniform critical current density $J_c$, critical sheet-current density $K_c=J_{c}d$, and critical current $I_c=2a{K}_c$, and the distribution of the current density within each strip is assumed to obey critical-state theory. I then derive expressions for the ac losses due to magnetic-flux penetration both from the strip edges and from the top and bottom of each strip and I express the results in terms of integrals involving the perpendicular and parallel components of the magnetic field. After numerically evaluating the ac losses for typical dimensions, I present analytic expressions from which the losses can be estimated.

Figure 1

Infinite bifilar stack: a stack of thin $(d⪡2a)$ superconducting strips of infinite length in the $z$ direction, with those at $y=0,\pm 2D,\pm 4D,\dots$ carrying current $I$ in the $+z$ direction and those at $y=\pm D,\pm 3D,\dots$ carrying the same current in the $-z$ direction.

Figure 2

Contours of constant $\mathfrak{R}\mathcal{G}(x+iy)$ calculated from Eqs. (6, 7) for a stack of thin superconducting strips of width $2a$ (thick lines) and spacing $D$. The contours correspond to magnetic field lines, and in this figure when the current in is the $+z$ direction, the field lines circulate in a counterclockwise direction around the strips centered at $(x,y)=(0,-2D)$, (0,0), and $(0,2D)$, and in a clockwise direction around the strips centered at $(0,-D)$ and $(0,D)$. Here, $D=a=2c$.

Figure 3

Plots of the real (solid) and imaginary (dashed) parts of $\mathcal{H}(x-iϵ)=H_y(x,0)+i{H}_x(x,-ϵ)$ calculated from Eq. (6) or Eqs. (8, 9, 10, 11, 12) for a stack of thin superconducting strips of width $2a$ and spacing $D$. Here, $a=1$, $c=0.5$, and $D=1$.

Figure 4

Plots of $c∕a$ vs ${(I∕I_c)}^2$ (solid) determined from Eq. (13) for $D∕a=0.1$, 0.3, 1, 3, and 10 (top to bottom). The dashed lines show the corresponding linear slopes for ${(I∕I_c)}^2⪡1$ [Eq. (15)].

Figure 5

Plots of $K_z\left(x\right)∕K_c$ vs $x∕a$ for $I∕I_c=0.5$ determined from Eqs. (11, 12, 13) for $D∕a=0.1$ $(c∕a=0.989)$, 1 $(c∕a=0.926)$, and 10 $(c∕a=0.868)$.

Figure 6

Plots of the dimensionless function describing edge losses $L_e$ vs $F=I∕I_c$ for $D∕a=0.1$, 0.3, 1, and 10 (bottom to top). The solid curves were obtained from Eq. (21) and the dashed lines from Eqs. (15, 26). On this scale the solid curve for $D∕a=10$ is indistinguishable from the Norris result $L_2$ [Eq. (22)] for an isolated strip (Ref. 20).

Figure 7

Plots of the dimensionless function describing top and bottom losses $L_{tb}$ vs $F=I∕I_c$ for $d∕a=0.001$ and $D∕a=0.1$, 0.3, 1, and 10 (bottom to top). The solid curves were obtained from Eq. (35). The corresponding dashed curves were generated from Eqs. (15, 37, 39), but on this scale, they are indistinguishable from the solid curves except for $D∕a=10$ very close to $I∕I_c=1$.

Figure 8

Plots of the dimensionless function $L=L_e+L_{tb}$ describing the sum of the edge losses $L_e$ [Eq. (21)] and top and bottom losses $L_{tb}$ [Eq. (35)] vs $F=I∕I_c$ for $D∕a=0.1$, 0.3, 1, and 10 (bottom to top) and $d∕a=0.001$. The solid curves display those portions of $L$ for which $F>F_X$ and $L_e>L_{tb}$, and the dashed curves display those portions for which $F<F_X$ and $L_e<L_{tb}$. The dotted curves show $L$ calculated from the sum of the approximate expressions for $L_e$ [Eq. (26)] and $L_{tb}$ [Eq. (37)].

Figure 9

Plots of $F_X$, the value of $F=I∕I_c$ for which the edge losses [Eq. (21)] are equal to the top and bottom losses [Eq. (35)], vs $D∕a$ for several values of $d∕a$. For given values of $D∕a$ and $d∕a$, when $F>F_X$, edge losses exceed top and bottom losses, but when $F<F_X$, top-and-bottom losses exceed edge losses.