# Numerical Investigation of High-Temperature Superconducting-Coated-Conductors Subjected to Rotating Magnetic Fields

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## Abstract

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## 1. Introduction

## 2. Model Description

#### 2.1. H-Formulation

_{0}is the permeability of free space, and µ

_{r}is the relative permeability. The resistivity in Equation (4) is constant for normal materials, but it varies with the current density in superconductors. It usually takes the form of the power law, as follows:

_{0}is the characteristic electric field strength, J

_{c}is the critical current density, and n represents the steepness of the shift from superconducting to the normal state, usually called the power factor. As n reaches infinity, the power law approaches the critical state model [23], also known as Bean’s critical state model when n approaches infinity. Combining Equations (1)–(5), one obtains the partial differential equation (PDE) in terms of the variable H as follows:

_{r}= 1. Furthermore, when J has a parallel component to B, the direction of the current density relative to the magnetic field affects the critical current density, resulting in force-free effects. As a result, high-accuracy models based on Jc(B) and anisotropy dependence are required. To account for the Jc(B) dependency, Kim’s model [25] is also used to consider such an isotropic behavior, as shown below in Equation (7);

#### 2.2. Rotating Magnetic Fields

_{0}is the magnetic flux density amplitude, ω is the angular frequency, φ denotes the phase shift angle, and τ is a time constant of 0.05 s. A purely circular rotating field is produced, which is often achieved by adjusting the phase angle of the second source by 90 degrees (i.e., π/2). When a phase angle is shifted along the axis, the magnetic field’s starting value is not equal to zero, which causes an issue with the initial values in FEA and prevents the model from converging. As a result, an exponential step function is used in the second field source in Equation (10) to establish a transitory start in Equation (9). Figure 3 shows the representation of a HTS CC subjected to alternating and rotating magnetic fields while carrying a transport current.

## 3. Results and Discussion

#### 3.1. AC Loss Analysis

#### 3.2. Flux Density Maps

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**A Vertical Stack of HTS CC and individual layers of sliver layer, superconducting layer, substrate, and two layers of copper stabilizers.

**Figure 2.**Homogeneous model of the vertical stack of CC, where the real topological characteristics of the tapes are “washed out”.

**Figure 3.**HTS CC stack subjected to alternating and rotating magnetic fields while transport current is applied.

**Figure 4.**AC loss in HTS stack subjected to an alternating and rotating magnetic field with transport currents of (

**a**) 0 A, (

**b**) 10 A, (

**c**) 20 A and (

**d**) 50 A.

**Figure 5.**Magnetic flux density maps for stacks of HTS CC when subjected to rotating magnetic field of 20 mT without any transport current.

**Figure 6.**Magnetic flux density maps for stacks of HTS CC when subjected to a rotating magnetic field of 20 mT while carrying a transport current of 20 A.

**Figure 7.**Magnetic flux density B maps for stacks of HTS CC when subjected to a rotating magnetic field of 20 mT while carrying a transport current of 50 A.

Parameter | Symbol | Value |
---|---|---|

Insulation/air gap | h_{l} | 200 µm |

Copper layer | h_{cu} | 40 µm |

Substrate layer | h_{c} | 50 µm |

Silver layer | h_{ag} | 2 µm |

HTS (YBCO) layer thickness | h_{HTS} | 1 µm |

Unit thickness | D | 293 µm |

Tape width | a | 4 mm |

Air/insulation resistivity | ρ_{Ins} | 1 Ω·m |

Silver resistivity | ρ_{Ag} | 2.70 nΩ·m |

Copper resistivity | ρ_{Cu} | 1.97 nΩ·m |

Substrate resistivity | ρ_{Subs} | 1.25 µΩ·m |

Permeability of free space | μ_{0} | 4π × 10^{−7} H.m^{−1} |

Number of tapes in stack | C | 64 |

Power factor | n | 38 |

Critical current density | J_{c0} | 108 A·mm^{−2} |

Characteristic electric field | E_{0} | 10^{−4} V m^{−1} |

Kim’s model arbitrary parameter | B_{0} | 0.0041 T |

Kim’s model arbitrary parameter | m | 0.5 |

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**MDPI and ACS Style**

Soomro, W.A.; Guo, Y.; Lu, H.; Jin, J.; Shen, B.; Zhu, J.
Numerical Investigation of High-Temperature Superconducting-Coated-Conductors Subjected to Rotating Magnetic Fields. *Solids* **2022**, *3*, 569-577.
https://doi.org/10.3390/solids3040036

**AMA Style**

Soomro WA, Guo Y, Lu H, Jin J, Shen B, Zhu J.
Numerical Investigation of High-Temperature Superconducting-Coated-Conductors Subjected to Rotating Magnetic Fields. *Solids*. 2022; 3(4):569-577.
https://doi.org/10.3390/solids3040036

**Chicago/Turabian Style**

Soomro, Wafa Ali, Youguang Guo, Haiyan Lu, Jianxun Jin, Boyang Shen, and Jianguo Zhu.
2022. "Numerical Investigation of High-Temperature Superconducting-Coated-Conductors Subjected to Rotating Magnetic Fields" *Solids* 3, no. 4: 569-577.
https://doi.org/10.3390/solids3040036