图 1  (a)单根$ {\text{Nb}}_3{\text{Sn}} $超导线横截面几何形状及相关尺寸(单位: mm)示意图, 选取了超导线横截面中间的线段$ A-A $来展示超导线的电流密度和磁通密度分布. 超导线置于沿着y方向的垂直交变磁场中, 交变磁场形式如图(b)所示

Figure 1.  (a) Schematic diagram of the section cross of single $ {\text{Nb}}_3{\text{Sn}} $ superconducting wire (unit: mm). We select a line $ A-A $ in the middle of the cross section to show the current density and flux density distribution of the superconducting wire. The superconducting wire is exposed to a perpendicular magnetic field along y axis. The form of the alternating magnetic field is shown in panel (b)

图 2  完全耦合和完全不耦合情况的发热功率随时间变化曲线, 插图展示在曲线图$P \text-t$上某时刻的电流密度图

Figure 2.  Developments of thermal power versus time between fully coupled and uncoupled cases, the insets show the contour of current density at time indicated in the $P\text-t$ curves

图 3  为了验证本文采用的H法的准确性, 分别使用H法和$H\text-\varphi$法模拟三根超导芯丝在$ 2.5 $和$ 5\; {\text{s}} $时的(a)电流密度$ J_z $、(b)磁通密度$ B_y $, 以及(c)三根芯丝的发热功率随时间的变化

Figure 3.  In order to verify the H formula adopted by this paper, (a) the current density $ J_z $, (b) flux density $ B_y $ at $ 2.5 $ and $5\; {\rm{s}}$, and (c) the variations of total thermal power of the three filaments versus time were conducted both by H and $H \text-\varphi$ formulas

图 4  由本文使用的H法计算得到的归一化磁化曲线, 与参考文献[33]中的实验结果符合较好

Figure 4.  The normalized magnetization curve calculated by the H method used in this paper is matched with the experimental results in Ref. [33]

图 5  超导线发生跳跃时的(a)平均电场$ \bar{E_z} $和(b)平均温度T随时间的变化曲线; (c)和(d)分别展示了上升阶段$\bar{E_z}\text-t$或者$T\text-t$曲线对应时刻超导线中心$ A-A $线的电流密度$ J_z $和磁通密度$ B_y $分布; (e)和(d)分别展示了下降段电流密度$ J_z $和磁通密度$ B_y $分布, 插图①—④分别展示了在$T\text-t$曲线上对应时刻单根芯丝的电流密度和磁通密度图

Figure 5.  The variations of (a) average electric field $ \bar{E_z} $ and (b) average temperature T as a function of time when flux jump occurs in the superconducting wire. (c) and (d) show the distributions of current density $ J_z $ and flux density $ B_y $ at the center of superconducting wire in ascending branch at oscillating times indicated in $\bar{E_z}\text-t$ or $T\text-t$ curve. (e) and (d) show the current density $ J_z $ and flux density $ B_y $ distributions in the descending branch. The insets ①–④ show the contours of current density and flux density of a single filament at times indicated in the $T\text-t$ curve

图 6  超导线发生跳跃时的(a)平均电场$ \bar{E_z} $和(b)平均温度T随时间的变化曲线, (c)和(d)分别展示了上升段阶段$\bar{E_z}\text-t$或者$T\text-t$曲线上未震荡和发生震荡对应时刻电流密度$ J_z $和磁通密度$ B_y $分布; (e)和(d)分别展示了下降段未震荡和发生震荡时的电流密度$ J_z $和磁通密度$ B_y $分布, 插图①—④分别展示了在$T\text-t$曲线上对应时刻整根超导线的电流密度和磁通密度图

Figure 6.  The variations of (a) average electric field $ \bar{E_z} $ and (b) average temperature T as a function of time when flux jump occurs in the superconducting wire. (c) and (d) show the distributions of current density $ J_z $ and flux density $ B_y $ at the center of superconducting wire in ascending branch at no oscillating and oscillating times in $\bar{E_z}\text-t$ or $T\text-t$ curve. (e) and (d) show the current density $ J_z $ and flux density $ B_y $ distributions in the descending branch. The insets ①–④ show the contours of current density and flux density of the whole superconducting wire at times indicated in the $T\text-t$ curve.

图 7  在$ T_0 = 4.2\ {\text{K}} $和幅值为$ 0.7\ {\text{T}} $时不同频率下超导线的平均温度随时间的变化

Figure 7.  the variations of average temperature of a superconducting wire with time for different frequencies at the amplitude of 0.7 T and $ T_0 = 4.2\ {\text{K}} $.

图 8  在$ T_0 = 4.2\ {\text{K}} $下, 不同幅值下超导线发生初次磁通跳跃的磁场阈值$B_{{\rm{th}}}$随频率f的变化(图中双斜线表示不发生磁通跳跃的临界点)

Figure 8.  The variations of magnetic field threshold of the initial flux jump $B_{{\rm{th}}}$ versus frequency f at different amplitudes at $ T_0 = 4.2\ {\text{K}} $ (The double slash lines in the figure indicate critical points where flux jump does not occur)

图 9  在$ T_0 = 4.2\ {\text{K}} $和频率为$ 0.3\ {\text{Hz}} $时, 不同幅值下超导线的平均温度随时间的变化, 插图①—④分别展示了在$T\text-t/t_{0}$曲线上对应时刻整根超导线的电流密度图

Figure 9.  The developments of average temperature of a superconducting wire as a function of time for different amplitudes at the frequency of 0.3 Hz and $ T_0 = 4.2\ {\text{K}} $.The insets ①–④ show the contours of current density of the whole superconducting wire at times indicated in the $T\text-t/t_{0}$ curve

图 10  在$ T_0 = 4.2\ {\text{K}} $下, 不同频率时初次发生磁通跳跃的磁场阈值$B_{{\rm{th}}}$随幅值$B_{{\rm{a}}1}$的变化(图中双斜线表示不发生磁通跳跃的临界点)

Figure 10.  At $ T_0 = 4.2\ {\text{K}} $, the variations of magnetic field threshold of the initial flux jump $B_{{\rm{th}}}$ with the amplitude $B_{{\rm{a}}1}$ at different frequencies (The double slash lines in the figure indicate critical points where flux jumping does not occur)

图 11  (a)在$ T_0 = 4.2\ {\text{K}} $时超导线发生磁通跳跃的频率和幅值的临界阈值曲线, 其中黑色实线所包围的蓝色区域为超导线发生磁通跳跃的区域, 其余灰色区域为磁通平稳穿透的区域. (b)—(g)分别展示了图(a)中6个点对应情况下超导线的平均电场和平均温度随时间的变化, 插图$ a_1 $—$ c_2 $表示$ \bar{E_z}-t $曲线的最高点(黑色实心圆点)对应时刻超导线的电流密度

Figure 11.  At $ T_0 = 4.2\ {\text{K}} $, (a)the critical threshold of frequency and amplitude(black curve)when flux jump will happen in a superconducting wire. The blue region is the region where flux jump occurs, and the rest gray region is the region where smooth penetration occurs; (b)–(g) show the variations of the average electric field and temperature of the superconducting wire with time under the corresponding conditions of the 6 points in (a); Panels $ a_1 $–$ c_2 $ show the current density of the superconducting wire at times (black solid dots) corresponding to the highest points of the $ \bar{E_z}-t $ curves

图 12  幅值为$ 0.7\; {\rm{T}} $, 频率为$ 0.3\; {\rm{Hz}} $情况下, 不同背景温度时超导线的平均温度随时间的变化

Figure 12.  At the amplitude of 0.7 T and the frequency of 0.3 Hz, the variations of average temperature of the superconducting wire with time for different background temperatures

图 13  不同频率和幅值下初次发生磁通跳跃的磁场阈值$ B_{{\rm{th}}} $随背景温度$ T_0 $的变化(图中双斜线表示不发生磁通跳跃的临界点)

Figure 13.  The variations of magnetic field threshold of the initial flux jump $ B_{{\rm{th}}} $ with the background temperature $ T_0 $ at different frequencies and amplitudes (The double slash lines in the figure indicate critical points where flux jump does not occur)