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Impact of strain and field ramp functional form on thermomagnetic instabilities in composite Nb3Sn wires with multi-filaments inside the superconducting coil

We theoretically analyze the effects of Cu/SC (superconductor) ratio, strain, and the ramp path on thermomagnetic instabilities of the superconducting coil. By considering the multi-filamentary structures, we find that a lower Cu/SC ratio leads to higher temperature peaks. The strain causes a higher frequency of flux jumps and higher voltage peaks. The temperatures recover to working temperature more difficultly and SC wires quench earlier in the presence of strain. For the jagged ramp cases, few flux jumps occur at the decreasing branch, whereas frequent flux jumps can be observed promptly when the applied current exceeds the pre-existing peak. Our simulated results agree well with experimental observations in Nb3Sn coils. Additionally, unlike the pulsed flux jumps observed in linear ramp cases, giant and prolonged flux jumps are observed at the increasing branch with a sinusoidal ramp path when the applied current is sufficiently large.

However, the composite Nb3Sn wire is subjected to multiple strain states because of the thermal mismatch, manufacture and fabrication process, and high Lorentz force (FL = J  B). Numerous investigations [16][17][18][19][20][21][22][23][24] have revealed that the strain sensitivity of Nb3Sn is more than one order of magnitude higher than in other Nb-based superconductors such as NbTi and Nb, and the strain strongly reduces the critical current density of Nb3Sn. To figure out the relationship between the strain and the critical current density Jc, Ekin designed the first experimental setup to test the dependence of critical current on strains in the 1980s [18]. Since then, plenty of experiments [25][26][27][28][29][30][31] has been done to investigate the effects of strains on the critical current density. Lots of theoretical and numerical research have been done to characterize the strain-dependent physical properties of composite Nb3Sn wires based on constitutive models [4,18,21,[32][33][34][35][36][37], finite element models (FEM) [38][39][40][41], Ginzburg-Landau (GL) theory [42], and density functional theory (DFT) [43][44][45]. The severe degradation in the critical current density Jc of Nb3Sn caused the thermomagnetic instability or great destruction in the superconducting magnet [46].
Except for high critical current density Jc, the stable operation of superconducting magnets also demands other performance targets, such as combating instability, reducing field-ramp-rate-dependent ac loss, and reducting magnetic field induced by screening current [7,11]. All the performance targets require a small filamentary size.
However, the price for a small filamentary size is a relatively low critical current [7]. The filamentary size also has a considerable influence on thermomagnetic instabilities during ramping.
The maximal strain of our Nb3Sn superconducting coil with an applied current of 700 A is about 0.9% which causes a severe decrease in critical current. Although the sensitivities of thermomagnetic instabilities to various electromagnetic and thermal parameters have been studied in experiments and numerical simulations [47][48][49][50][51][52][53][54][55][56][57][58][59][60][61], the effects of strain, filamentary size, and the field ramp functional form on the flux jump are still unclear. These parameters play a crucial role in the flux jump, so it is essential to reveal their influences on the superconducting magnets during the operation.
In this work, we use the representative unit cell (RUC) multi-physics numerical model [47] to numerically explore the sensitivity of flux jump to Cu/SC ratio, strain, and the field ramp functional form. We consider the composite multi-filamentary structures of Nb3Sn wires, the strains resulting from the thermal mismatch and high Lorentz force, and the applied field with a linear field ramp form, jagged field ramp form, or linear field ramp with additional sinusoidal field oscillations. We analyze the remarkable differences between the equivalent and multi-filamentary structures of superconducting wires. Our numerical results indicate that the strain and field ramp functional form significantly impact the thermomagnetic instabilities in Nb3Sn superconducting wires.

Framework of numerical model
At first, we briefly introduce our numerical model. As shown in Fig. 1, the transport currents strictly flow along the composite Nb3Sn wires without a short circuit and produce the surrounding magnetic fields. Based on the field profile calculated by Biot Savart's law, we consider the angular component of magnetic field B// as 0 on the condition that the transversal field Ba is much larger than B//. A representative quadrate unit cell which includes epoxy and one turn of the superconducting wires, is selected to analyze the thermomagnetic instabilities in composite Nb3Sn wires. The three-dimensional model is reduced to the two-dimensional axisymmetric model based on the symmetry of the quadrate unit cell. The diameter of the Nb3Sn superconducting wire is 1.3 mm, and the cross-sectional area of the quadrate unit cell is 1.43×1.43 mm 2 (see Fig. 1). The strain distribution under a transport current of 700 A is obtained by the bidirectional homogenization method [62]. The maximum strain is 0.90 % at the inner radius, and the minimum strain is 0.64 % at the outer one, which is attributed to larger Lorentz force at the inner radius than at the outer one in the superconducting coil.

Electromagnetic field and heat conduction
The A-V formulation [63,64] of magnetoquasistatic Maxwell equations is employed to describe the electromagnetic properties of the SC wire. Here, we give a brief introduction to the A-V formulation. The electric field of an axially symmetric system at one point can be expressed as: is the critical electric field. is characterized by flux creep exponent 0 , temperature , and critical temperature . c is the upper critical magnetic field when = 0 K. is the critical current density characterized by , , and , where is the strain along the circumferential direction. In one superconducting wire, the total screen current induced by the external field is zero, which means that the filaments in the unit cell are fully coupled [66]. We assemble the state equations for each element.
We can obtain the field and current distributions in the unit cell by equations (11) and (12).
To obtain the temperature distribution of unit cell at any time, the governing equations of heat diffusion are given by: The specific heat and the thermal conductivity are temperature-dependent. ℎ is nonzero only at the edges of the unit cell, which approximatively denotes the capabilities to take heat away. We assume that ℎ, , and are all proportional to 3 [67]. The experimental data of critical current density versus strain for Oxford wires are obtained in Ref. [68], as shown in Fig. 2. The fourth-order Polynomial and second-order Fourier functions are used in the curve-fitting. We choose the fourth-order Polynomial function, and (ε) under loading and recovery can be expressed as:

Mechanical equivalent treatment and Physical parameters
(ε) = −2.81x10 7 4 + 5.95x10 5 3 − 6783 2 + 3.88 + 1.00 (16) (ε) = 1.09x10 7 4 − 2.70x10 5 3 − 9.28 2 − 0.0054 + 0.72 (17) Considering the representative unit cell contain all the components (Nb3Sn, Cu, Epoxy), we take the superconducting coil as a geometric configuration which array the unit cell alone the r and z directions. So, the representative volume element (RVE) method is more effective than fractional volume method and self-consistent method in calculating the anisotropic equivalent mechanical parameters of the superconducting coil. At first, we use finite element method to establish a representative unit cell model, that each component material is isotropic. Then, the stress and strain distribution of RVE under different displacements can be obtained by applying unit axial and unit shear strain, respectively. The anisotropic equivalent mechanical parameters of the superconducting coil can be calculated by taking the mean stresses and strains under different strain states into the constitutive relation of RVE.
The consistency of mechanical response under equivalent parameters and actual material parameters validates the reliability of equivalent parameters. We calculate the displacement of the equivalent superconducting coil under the Lorentz force. We select the displacement data of several unit cells and re-apply these data to the actual composite structure to obtain the detailed strain distribution at sub-regions [62]. We assume that the equivalent material has always been in the elastic stage, and there exists no fracture of filaments during ramping.  At first, we focus on the difference between the equivalent structure and the actual structure. Note that Nb3Sn and epoxy are the only two materials for the equivalent structure. Both the two structures have the same 0 . The thermal conductivity and the specific heat for the equivalent wire are given by SNb3Sn and SCu represent the percentages of Nb3Sn and Cu in the SC wire, respectively. There are apparent differences in the power profile, the voltage, and the temperature during operation, as shown in Fig. 4. Noticeable heating can be observed all over the equivalent SC wire during ramping, while it can only be observed in the filaments for the multi-filamentary structure (see Fig. 4(a)). The immense power of the equivalent structure leads to a rapid temperature rise (see Fig. 4(c)) compared with the actual structure. The rapid temperature rise narrows the range of flux jumps, which is attributed to the that a small flux creep exponent at the high temperature promotes flux creep. As shown in Fig. 4(b), the regularity of voltage jumps of the actual structure is different from the equivalent structure. The frequent flux jump in the equivalent structure (9 peaks before 180 A, while 7 peaks before 180 A for the actual structure) results in minor magnetic pressure build-up and small amplitudes of the voltages during ramping. There is no doubt that it is vital to investigate the flux jump in the actual structure based on the above results.  Fig. 5(a). The capability to carry high-loss current decreases with increasing Cu/SC ratio (see Fig. 5(b)) due to the decrease of the Nb3Sn fraction as the Cu fraction increases. The amplitudes of voltage peaks are higher in the Cu/SC ratio of 4.79. The frequency of flux jumps decreases with SC content, and the amplitude of temperature peaks increases with SC content (see Fig. 5(c)). These are attributed to the that high ac loss at large SC fraction causes rapid temperature rise, then restrains the flux jump. In the actual operation of superconducting magnets, considerable strains due to the fabrication process cause significant degradation in the superconductivity. The critical current density Jc reduces nonlinearly and reversibly within certain limits. Therefore, it is vital to understand the effect of the reduction of Jc on flux jump. We chose the SC wire located in the center of the superconducting coil with ℎ 0 of 60 W/m 3 ·K. The ramp rate is 1 A/s, and the field ramp rate is 0.0064 T/s. We assume a fixed strain result from the fabrication process and thermal mismatch on the SC wire. Fig. 6 shows the considerable difference between cases where the effect of strain on Jc is considered or not. Both larger amplitudes of voltages peaks and higher frequency of flux jumps are observed in the SC wire subjected to strain (see Fig. 6(a)), which is attributed to the reduction of Jc requires less steep magnetic field attenuation so that more flux vortices penetrate into the SC wire. As the frequent flux jumps, the temperature is more challenging to return to the working temperature. As illustrated in Fig. 6(b), the quench current decreases from 1400 A to 800 A because of the strain. The strain restrains the temperature peaks because of the reduction of Jc, which causes lower heating power. It is not appropriate for any analysis of flux jumps in magnets exposed to high currents to ignore the influence of strain. Except for the effects of the manufacture and fabrication process, the thermal contraction differences between materials on superconducting properties, the Lorentz forces during operation also are considered. We investigate the thermomagnetic instabilities in SC wires with a jagged field ramp form. Fig. 7(a) shows that the transport current along with the SC wire increases by 80 A and then decreases by 40 A with a ramp rate of 1 A/s. When the reduction of transport current in decreasing branches is as small as a specific value, Fig. 7(b) indicates that concentrated flux jumps occur as the current exceeds the previous current peaks, and almost few flux jumps and corresponding temperature jumps occur as the applied current decrease. Our numerical results are consistent with the experiment results (see Fig. 7(c)) observed in Nb3Sn coils. Compared to the case with the linear field ramp form in Fig. 6(b), the temperature can recover to the ambient temperature after flux jumps, promoting the superconducting magnet's stable operation and high capacity to carry current. It also inspires us to optimize the operation of superconducting magnets from the field ramp functional form. To figure out the influence of field ramp functional form on thermomagnetic instabilities in the presence of current-dependent strain, we consider linear field ramp with additional sinusoidal field oscillations and change both the amplitude and the frequency of the sine wave. The SC wire is close to the edge, and the ramp functional forms are shown in Fig. 8(a), where the transport current is increased from 0 A to 700 A. Remarkable differences are observed between Fig. 8(b) and Fig. 6(b). Giant and prolonged flux jumps are observed at the increasing branch when the applied current is sufficiently large. In contrast with the linear field ramp cases, the temperature in the SC wire can recover to working temperature more easily. Unlike the case in Fig. 6(b), dense flux jumps exist at the branches with a high field ramp rate corresponding to the rapid accumulation of magnetic pressure (see Fig. 8(c)).

Effects of field ramp functional form on flux jump in the presence of strain
As illustrated in Fig. 8(c), the statistics of voltage peaks under linear field ramp with additional sinusoidal field oscillations are significantly less than those under a linear field ramp form. Additionally, we investigate the characteristics of thermomagnetic instabilities in the SC wire during decreasing branches. We consider the irreversible degradation of critical current density due to the strain. Fig. 9(a) illustrate the decreasing branches with different ramp rate (-1.0 A/s, -0.5 A/s and -0.25A/s). The / and ℎ 0 are 0.0064 T/A and 60 W/m 3 ·K, respectively. As shown in Fig. 9(b), the statistics of flux jumps during decreasing branch increase with decreasing field ramp rate (16 and 20). The temperature is easier to recover to the ambient temperature after flux jumps, so a slow field ramp rate promotes the protection of the superconducting magnet. The voltage peaks during increasing branches are generally larger than those during decreasing branches (see Fig. 9(c)). Both flux jumps and reverse flux jumps are observed when the applied current is reduced from 1000 A to 0 A. At the decreasing branch, the frozen flux in the inner regions [74] and the reverse temperature distribution in contrast with the cases in the increasing branch cause a lower Jc region in the interior of the Nb3Sn wire (see Fig. 9(d)). The reverse current profile and high resistivity region due to the lower Jc lead to the reverse voltage signals, as shown in Fig. 9(c).

IV. Conclusion
In this paper, an appropriate theoretical method mentioned in Ref. [47] is adopted to analyze the effects of Cu/SC ratio, strain, and the field ramp functional form on thermomagnetic instabilities of the superconducting coil. The approximate strain profile across the cross-section of the superconducting coil is obtained based on the equivalent mechanical model by simplifying the materials for the coil to an equivalent anisotropic material. We illustrate that the Cu/SC ratio, strain, and field ramp functional form play significant roles in thermomagnetic instabilities. A smaller Cu/SC ratio leads to higher temperature peaks, and a larger proportion of Cu causes high voltage peaks.
Additionally, the strain causes a higher frequency of flux jumps and higher voltage peaks. The temperature in the SC wire is more challenging to return to the working temperature, and the SC wire quench earlier due to the existence of strain. For the ramping magnetic field with a jagged ramp form, few flux jumps and temperature jumps occur at decreasing branches, whereas frequent flux jumps can be observed promptly again when the applied current exceeds the pre-existing peak value. Our numerical results agree well with the experiments. Moreover, remarkable differences between the linear field ramp forms and the linear field ramp with additional sinusoidal