Critical state in finite type-II superconducting rings

We study the magnetic behavior of a finite superconducting ring in the presence of a uniform applied field directed along its axis by means of the critical-state model and the minimization of magnetic energy. We systematically study the dependence of the magnetization and the ac susceptibility upon the geometry of the ring and develop an approximate analytical expression for the case of narrow rings of any aspect ratio. Besides, we show how the critical-current density of the superconductor can be obtained from magnetization measurements and conclude that ring geometry is a very convenient one for such a purpose. In particular, we present an expression for the full penetration field in the case of finite rings, which allows us to find the value of the critical current from the value of the magnetic field at just one point on the axis of the ring.

Figure 1

Calculated penetration profiles for twelve rings at different applied fields. Upper line (a)-(b)-(c)-(d): $\gamma =5$, center line (e)-(f)-(g)-(h): $\gamma =1.0$, and bottom line (i)-(j)-(k)-(l): $\gamma =0.2$. Left column (a)-(e)-(i): $\delta =0.8$, center left column (b)-(f)-(j): $\delta =0.5$, center right column (c)-(g)-(k): $\delta =0.2$, and right column (d)-(h)-(l): $\delta =0$ (bulk cylinder). In all cases, the applied field $H_{\mathrm{a}}$ increases from zero in steps of $\Delta H_{\mathrm{a}}=0.1J_c{R}_{\text{out}}$. Only a semiplane of constant angle is plotted. In each figure, the left axis corresponds to the rotation axis. The length dimensions in the $\gamma =5$ sample have been reduced by a factor of 2.5.

Figure 2

Approximate field lines for a ring with $\gamma =0.2$ and $\delta =0.8$. Applied fields are (a) $H_{\mathrm{a}}∕J_c{R}_{\text{out}}=0.1$ and (b) $0.02$. Gray regions correspond to current penetrated regions.

Figure 3

Calculated penetration field for different $\delta$ values as a function of $\gamma$. The different symbols correspond to the numerically calculated penetration field (see text) for different $\delta$’s. Solid lines correspond to the approximate expression of Eq. (2). Dotted lines correspond to the approximation for thin ring [Eq. (6)] whereas dashed lines correspond to the infinite ring limit [Eq. (3)].

Figure 4

(a) Calculated $z$ component of the magnetic field at the center of the top surface of the ring $(z=L∕2,\rho =0)$ as a function of the applied field. Different lines correspond to different rings: from thinner to thicker line $\gamma =0.2$, 1.0, and 2.0. The lines marked with open and solid symbols correspond to $\delta =0.8$ and 0.5, respectively. The dotted lines correspond to the field created by fully penetrated rings (analytically calculated). They provide an easy visualization of the kink in each curve (see text).

Figure 5

Experimental and calculated (a) magnetic field and (b) its second derivative as a function of the applied field at the point $z=L∕2,\rho =0$. Points correspond to experimental data and solid curves to calculated data.

Figure 6

Calculated magnetization loops for different rings. (a) Results for fixed $\gamma$ and different $\delta$ [$\delta =0.0$ (inmost), 0.2, 0.5, and 0.8 (outmost)]. (b) Results for fixed $\delta$ and different $\gamma$ [$\gamma =5$ (inner), 1, 0.2 (outer)].

Figure 7

Calculated real and imaginary susceptibility and modeled susceptibility Eqs. (23, 25) as a function of the ac field amplitude.