# Critical Current Simulation and Measurement of Second Generation, High-Temperature Superconducting Coil under External Magnetic Field

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Experimental Setup

_{r}, which is perpendicular to the surface of the conductor. Similarly, the parallel field is denoted as B

_{z}. These two graphs reveal that the SuperPower is quite different from the SuNAM tape in the magnetic field; that is, critical current of SuperPower tape in the parallel field is smaller than that under the perpendicular field, while normal tapes have a smaller critical current under the perpendicular field, such as the SuNAM tape. According to the 1 μV/cm criterion, the SuNAM tape has a critical current of 217 A in the self-field, while the SuperPower tape has a critical current of 120 A.

_{1}is the lower curve of each tape in Figure 4 and B

_{2}is the higher curve. B

_{10}is the field when the current is half of the critical current on the B

_{1}curve; similarly, B

_{20}is the corresponding field on the B

_{2}curve. k is the ratio between B

_{10}and B

_{20}, which describes the anisotropy of B

_{1}and B

_{2}. All the data we use are based on the measurement of SuNAM tape and SuperPower tape at 77 K. Table 2 summarizes the derived values from the measured data.

## 3. Simulation Model

_{0}, is determined using the tape thickness, and presented in Table 1. Each YBCO layer represents one turn of the pancake coil; the innermost turn is Turn 1, while the outmost turn is Turn 20. Applied current I

_{app}is assigned to each YBCO layer. The superconducting lines are evenly meshed with 50 line elements, while the air region is meshed by triangle elements.

_{i}:

_{i}and J

_{i}are the vector potential and current in the circumferential direction in the ith element, ${f}_{i}^{m}=-{A}_{i}^{m-1}$, C

_{j}is the voltage due to the source in the jth tape, and ${J}_{c}({B}_{\mathrm{r}},{B}_{\mathrm{z}})$ is the critical current density with field dependencies. Thus, Equation (3) means that each tape transports the current of I

_{app}and Equation (4) represents the Bean model [18]. This equation can be optimized by iterations, and it is implemented by using the constrained quadratic minimization procedure in MATLAB. The calculation above only gives the current distribution in the tapes that is assigned to each YBCO layer. The field within and around the coil is solved using the two-dimensional (2D) finite element method (FEM) in cylindrical coordinates. The coil geometry is shown in Figure 5, assuming that the air domain is so large that the magnetic field produced by the applied current decays to zero on the boundary for FEM. A varying current with a sine shape is applied to the coil and it peaks at 80 A. The frequency of the applied current is 1 Hz. Figure 6 shows the current distribution within the coil for four steps: 30 A, peaking current 80 A, −30 A, and 0 A. Each line represents one tape and the z-axis is the magnitude of the current density. From Figure 6b, we can clearly see two regions in the superconducting region: the penetrated region and the unpenetrated region. They are also called the critical region and the subcritical region, consistent with the assumption of Clem [16]. The penetrated region expands from both ends into the central region. In the penetrated region, the current density is bounded by the critical current density. As it was previously pointed out that the critical current decreases in the external field, we can see a clear slope of the current density in the penetrated region. In Figure 6d, the total applied current in each tape is zero; however, we can see that the current density is not homogenously zero. On the contrary, there are both positive and negative currents within the region. This shows the hysteresis nature of the superconducting coil after magnetization.

_{z}exists only in the subcritical region (magnetic fields are parallel to the tape width direction). This is due to the intrinsic property of the superconductor, which expels the magnetic field. This is consistent with W. Yuan’s papers [17,20], which solve the electromagnetic field through the minimization of the (B

_{r})

^{2}within the subcritical region.

## 4. Discussion

#### 4.1. Model Validation

#### 4.2. Comparison of Two Different Tapes

_{3}nanocolumnar particles in the film. These BaZrO

_{3}nanocolumns are aligned in parallel to the c-axis of REBCO (Rare Earth Barium Copper Oxide). The result is that the flux pinning for H//c is greatly enhanced. When the pinning from the nanocolumns is strong enough, the I

_{c}(H//c) becomes higher than the I

_{c}(H//ab). So, it is of practical interest to analyze the difference when this new SuperPower tape is applied in pancake application under a DC magnetic field compared with a normal magnetic field dependency tape, such as the SuNAM tape. This field dependency is approximated by a modified Kim model, which is shown in Figure 4 and Table 2.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**Schematic view of the experimental setup; the cylinders represent the magnet poles. The coil is fixed by a non-magnetic nut in the center of the two magnet poles.

**Figure 4.**Measured field dependency for SuNAM tape and SuperPower tape. (

**a**) B

_{1}represents the lower curve and B

_{10}is the field when the current is half of the critical current on the B

_{1}curve; (

**b**) B

_{2}represents the upper curve and B

_{20}is the field when the current is half of the critical current on the B

_{2}curve.

**Figure 5.**A coil with N tapes: the width is w and the thickness is t

_{0}. The radius of the coil is r

_{0}. YBCO superconductor layers are presented by thicker lines.

**Figure 6.**Current distribution for 20 turns with an external field of −0.2 T: (

**a**) ramping up to 30 A; (

**b**) ramping up to 80 A; (

**c**) ramping down to −30 A; (

**d**) ramping up to 0 A. Tape No. 1 is in the innermost turn.

**Figure 7.**Magnetic field distribution and flux line plot of a 20-turn coil with an external field of −0.2 T. The white contour lines represent the directions. (

**a**) Ramping up to 30 A; (

**b**) ramping up to 110 A; (

**c**) ramping down to −30 A; (

**d**) ramping up to 0. The rectangle outlined in black represents the region occupied by superconductors.

**Figure 8.**Comparison between measured and simulated critical current for SuNAM (

**a**) and SuperPower (

**b**) coil with 20 turns in different external fields. The measured critical currents are defined by 0.5 μV/cm in E-I curves.

**Figure 9.**The current distribution when the applied current is 107.5 A and the external field is anti-phase 0.2 T.

**Figure 10.**The current distribution when the applied current is 101 A and the external field is in-phase 0.2 T; the inner turn of the SuNAM 20-turn coil is fully penetrated.

**Figure 11.**E-I curves of different sections with the SuNAM coil with in-phase and anti-phase 0.2 T magnetic fields.

**Figure 13.**Perpendicular field (

**a**) and parallel field (

**b**) distribution for 20-turn coil. This coil has a uniform current distribution and each tape transports a current of 100 A.

Tape Yype | HCN4045 | SCS4050 |
---|---|---|

Manufacturer | SuNAM | SuperPower |

Tape I_{c} | 217 A | 120 A |

Tape thickness | 0.20 mm | 0.20 mm |

Inner diameter | 56 mm | |

Total turns | 20 | |

Tape width | 4 mm | |

Insulation | Kapton tape |

Type of Tapes | SuNAM | SuperPower |
---|---|---|

B_{1} | Perpendicular | Parallel |

B_{10} (T) | 0.140 | 0.115 |

B_{2} | Parallel | Perpendicular |

B_{20} (mT) | 0.360 | 0.180 |

k | 0.140/0.360 | 0.115/0.180 |

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**MDPI and ACS Style**

Yu, D.; Liu, H.; Zhang, X.; Gong, T.
Critical Current Simulation and Measurement of Second Generation, High-Temperature Superconducting Coil under External Magnetic Field. *Materials* **2018**, *11*, 339.
https://doi.org/10.3390/ma11030339

**AMA Style**

Yu D, Liu H, Zhang X, Gong T.
Critical Current Simulation and Measurement of Second Generation, High-Temperature Superconducting Coil under External Magnetic Field. *Materials*. 2018; 11(3):339.
https://doi.org/10.3390/ma11030339

**Chicago/Turabian Style**

Yu, Dongmin, Huanan Liu, Xinhe Zhang, and Taorong Gong.
2018. "Critical Current Simulation and Measurement of Second Generation, High-Temperature Superconducting Coil under External Magnetic Field" *Materials* 11, no. 3: 339.
https://doi.org/10.3390/ma11030339