Field and current distributions and ac losses in superconducting strips

In this paper I discuss analytic and numerical calculations of the magnetic-field and sheet-current distributions in superconducting strips of width $2a$ and arbitrary thickness $2b$ at the center when the cross-section is an ellipse, a rectangle, and a shape intermediate between these limits. Using critical-state theory, I use several methods to determine the functional dependence of the ac transport-current losses upon $F=I/I_c$, where $I$ is the peak alternating current and $I_c$ is the critical current, and I discuss how this dependence can be affected by the cross-sectional shape, aspect ratio, and a flux-density-dependent critical current density $J_c\left(B\right)$.

Figure 1

Cross-sections considered in this paper, all representing superconducting strips of width $2a$ and thickness $d=2b$ at the center: an ellipse of semimajor axis $a$ and semiminor axis $b$, a rectangle of length $2a$ and height $2b$, and an intermediate shape, described in the text.

Figure 2

Plots of (a) $B_y\left(x,0\right)/{\mu }_0{J}_{c}a$ and (b) $B_x\left(0,y\right)/{\mu }_0{J}_{c}a$ vs $x/a$ for $c/a=0$, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0 in a strip of rectangular cross-section of dimensions $2a\times 2b$, where $b=a/2$. The corresponding values of $y_0$, $\beta$, $S_I/S_c$, and $F=I/I_c$ are given in Table .

Figure 3

The upper five solid curves show values of ${\Delta }_x=\left(a-c\right)/a$ vs $F=I/I_c$ for a strip of rectangular cross-section $\left(2a\times 2b\right)$, numerically calculated as described in the text for $b/a=1$, 0.1, 0.01, 0.001, and 0.0001. The corresponding dashed curves show results using the conformal-mapping method described in Sec. 3C2. The lowest solid curve shows ${\Delta }_x=1-\sqrt{1-F^2}$, obtained from Eq. (20) in Sec. 2B, Norris’s result for a very thin strip of rectangular cross-section (Ref. 1).

Figure 4

Contour plot of $A_z\left(x,y\right)$ vs $x/a$ and $y/a$ for $c/a=0.8$, $y_0/a=0.372$, $\beta =2.629$, $S_I/S_c=0.534$, and $F=I/I_c=0.466$ in a strip of rectangular cross-section of dimensions $2a\times 2b$, where $b=a/2$. The flux front, described by $y=\pm y_I\left(x\right)$ [Eq. (6)] and shown as the bold curve, surrounds the area $S_I$. The contours correspond to magnetic-field lines, which circulate around $S_I$ in the counterclockwise direction.

Figure 5

Plots of $H_y\left(x,0\right)$ and $H_x\left(x,-ϵ\right)$ (on the bottom of the strip) calculated from Eqs. (15, 16, 17, 18, 19, 20) for $a=1$, $\alpha =0$, 2.474 (dashed), and $\infty$, and $F=I/I_c=0.75$, for which $c/a=0.500$, 0.579, and 0.661.

Figure 6

Plots of $L\left(1\right)$ vs $b/a$ at $I=I_c$ calculated from Eqs. (4, 5, 25, 26, 27, 28) for $\alpha =0$ (elliptical cross-section, $S_c=3.14ab$), values of $\alpha$ corresponding to intermediate shapes and cross-sectional areas (see Fig. 1), $\alpha =1.186$ $\left(S_c=3.36ab\right)$, $\alpha =2.474$ $\left(S_c=3.57ab\right)$, $\alpha =5.961$ $\left(S_c=3.79ab\right)$, and $\alpha =\infty$ (rectangular cross-section, $S_c=4ab$). The cross-sectional area for $\alpha =2.474$ is the average of the areas of the elliptical and rectangular cross-sections. For small $b/a$ the curves asymptotically approach the corresponding thin-film limits calculated in Sec. 3D, shown as dashed lines.

Figure 7

Plots of the loss fraction $f_y\left(F\right)=L_y\left(F\right)/L\left(F\right)$ vs $F=I/I_c$ calculated from Eqs. (36, 37) for $b/a=0.0001$, 0.001, 0.01, 0.03, 0.1, 0.3, and 1 for conductors with an elliptical cross-section $\left(\alpha =0\right)$.

Figure 8

The upper solid curves display plots of the loss function $L$ vs $F=I/I_c$ calculated from Eq. (53) for a strip of rectangular cross-section with relative dimensions $b/a=1$, 0.1, 0.01, 0.001, and 0.0001. The bottom solid curve shows the Norris result [Eq. (44)] for a very thin strip of rectangular cross-section. For comparison, the upper dashed line shows $F^3/6\pi$ and the lower dashed curve shows $F^4/6\pi$.

Figure 9

The upper solid curves display plots of the loss function $L$ vs $F=I/I_c$ numerically calculated from Eqs. (29, 30, 31, 32, 33, 34, 35) for a strip of rectangular cross-section with relative dimensions $b/a=1$, 0.1, 0.01, 0.001, 0.0001, and 0.00001. The bottom solid curve shows the Norris result [Eq. (44)] for a very thin strip of rectangular cross-section. For comparison, the upper dashed line shows $F^3/6\pi$ and the lower dashed curve shows $F^4/6\pi$.

Figure 10

Plots of the loss function $L$ vs $F=I/I_c$ in the thin-film limit calculated from Eq. (62) for strips of different cross-sectional shapes characterized by the parameter $\alpha =1$, 3, 10, 30, 100, and 1000 (top to bottom) in Eq. (1). The top dashed curve shows the Norris result for an elliptical cross-section [Eq. (36)], and the bottom dashed curve shows the Norris result for a thin strip of rectangular cross-section [Eq. (44)] (Ref. 1).

Figure 11

The solid curves show plots of $L_{eff}\left(F_p\right)/L\left(F_p\right)$ for $\gamma =10$ and values of $\alpha =0$ (elliptical cross-section), 10, 90.3, and $\infty$ (rectangular cross-section) in the thin-film limit, where $L_{eff}\left(F_p\right)$ is given by Eq. (71) and $L\left(F_p\right)$ by Eq. (62). The dashed line shows the corresponding ratios when $\gamma =0$ and $J_c$ is independent of $B$.