# Critical State Theory for the Magnetic Coupling between Soft Ferromagnetic Materials and Type-II Superconductors

^{*}

## Abstract

**:**

## 1. Introduction

**H**or the magnetic vector potential

**A**(amongst other PDE models) [66]—are doomed to find the same local solution at the SC domain regardless of whether the SC is sheathed by an SFM or not. This is because the attained numerical solution still represents the simplest and most mathematically valid response for the SC, which simply neglects any possible magnetostatic coupling between this and the SFM, unless it had been explicitly included in the numerical formulation.

## 2. Multipole Expansion of the CST in Rounded SC-SFM Heterostructures

## 3. SC vs. SC-SFM Metastructures: Differences on the Current Density and Magnetic Field Profiles

## 4. Experimental Evidences and General Map of AC-Losses for SC-SFM Heterostructures

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

2D | Two dimensional |

AC | Alternating Current |

CST | Critical State Theory |

DC | Direct Current |

EXT | Exterior domain, outside of the SC-SFM metastructure |

FEM | Finite Element Methods |

MOL | Magneto Optical Layer |

SC | Superconductor or Superconducting |

SFM | Soft-Ferromagnet or Soft-Ferromagnetic Material |

PDE | Partial Differential Equation |

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**Figure 1.**Pictorial representation of the analysed Superconducting (SC) Soft-Ferromagnetic (SFM) metastructure. The main plot shows the distribution of current density, $\mathbf{J}$, in the SC under self-field conditions for an applied transport current ${I}_{tr}={I}_{c}sin\left(\omega t\right)$, with $\omega t=\pi /4$ (red shadowed area), and the relative coordinates for a finite-element ${J}_{i}({r}_{i},{\varphi}_{i})$ as a reference for Equations (5)–(18). The left-plots show, for illustration, the time dynamics of the superconducting current density $\mathbf{J}$ within the SC-SFM metastructure. Red and blue areas correspond to distributions with ${\mathbf{J}}_{i}=\pm {J}_{c}{\hat{u}}_{z}$, respectively. White areas correspond to regions with no $\mathbf{J}$, and the yellow area corresponds to the SFM sheath where no ${I}_{tr}$ is to flow.

**Figure 2.**Dynamics of the norm of magnetic flux density $\left|\mathbf{B}\right|$ in units of $({\mu}_{0}/4\pi ){J}_{c}{R}_{SC}$ in (

**a**) an SC wire (without SFM sheath) of radius ${R}_{SC}=1$ (in arbitrary units) whose cross section lies on the plane $xy$ and which is subjected to an applied transport current ${I}_{tr}={I}_{c}sin\left(\omega t\right)$. (

**b**) shows the same distribution of field but in a 2D representation that focus on the local flux dynamics inside the SC, $|{\mathbf{B}}_{SC}|$. Analogously, (

**c**) shows the flux distribution $|{\mathbf{B}}_{SC-SFM}|$ for the SC-SFM metastructure with ${R}_{SFM}=1.5{R}_{SC}$ and ${\mu}_{r}=46$, where the impact of the SFM on the SC can be seen clearer in the 2D representation shown in (

**d**), that is, the bottom pane of subplots. The time interval between columns is $\Delta t=(\pi /4){\omega}^{-1}$, such that the dynamics shown between the second column $(\omega t=3\times \pi /2)$ and last column $(\omega t=\pi /2)$ represents the minimum hysteresis period for the calculation of AC Losses, in accordance with the distribution of profiles of current density shown in Figure 1.

**Figure 3.**Dynamics of the norm of magnetic flux density over the radial directions $(r,\pi /4)$ and $(r,0)$ for different magnitudes of the applied AC transport current, ${I}_{tr}={I}_{c}sin\left(\omega t\right)$, it measured inside the SC during (

**a**,

**b**) the first ramp of the applied current (top legend-box) and (

**c**,

**d**) the peak-to-peak hysteretic period (bottom legend-box) as described in Figure 1 and Figure 2. The panel of subplots at the right shows the corresponding profiles for the first ramp of current at (

**e**) inside the SFM and (

**f**) outside the SC-SFM wire, respectively. Dashed-dot lines at each subplot refer to the left axes $|{\mathbf{B}}_{SC}|$ showing classical Bean’s behaviour, whilst the solid lines must be read accordingly with the right hand axes $|{\mathbf{B}}_{SC-SFM}|$. The arrows show the ’time’ evolution of the field profiles, and units for $\mathbf{B}$ are $({\mu}_{0}/4\pi ){J}_{c}{R}_{SC}$.

**Figure 4.**Magnetic flux difference between an SC wire of radius ${R}_{SC}$ and the equivalent SC-SFM metastructure with ${R}_{SFM}=1.5{R}_{SC}$ in units of $({\mu}_{0}/4\pi ){J}_{c}{R}_{SC}$. Solid and dotted lines show the numerical results obtained by the extended CST along two different radial directions, being ${0}^{\circ}$ the line over the x-axis at $y=0$ in Figure 1, and ${45}^{\circ}$ the $xy-$ plane diagonal. Two set of curves are shown corresponding to self-field conditions with ${I}_{tr}={I}_{c}sin\left(\omega t\right)$, when $\omega t=3\pi /16$ and $\pi /4$ (i.e., ${I}_{tr}/{I}_{c}\simeq 0.5556$ and $0.7071$, respectively). For qualitative comparison, solid symbols and dashed lines show the raw and segmented-regression fitted data extracted from MOI measurements [39,41,43,80] reported for the Fe-sheathed MgB${}_{2}$ monocore displayed at the subplot (a), where the ovals highlight the regions were an anomalous “elevation” and “dip” of the magnetic flux have been observed. All other insets show the calculated 2D local distribution of magnetic flux density at different instants of the AC current for the SC (top) and SC-SFM (bottom) wires. Equally sized ovals as in (a) are displayed, highlighting thence how the extended CST allows a straightforward explanation of the non-conventional patterns for the local magnetic flux density in the SC-SFM metastructures.

**Figure 5.**(

**a**) Hysteretic losses ratio between SC-SFM metastructures $\left({L}_{SC-SFM}\right)$ and the AC losses produced by an isolated SC wire $\left({L}_{SC}\right)$ of cylindrical cross section as a function of the relative magnetic permeability ${\mu}_{r}$ of the SFM with ${R}_{SFM}=1.5{R}_{SC}$. (

**b**) The top inset shows the well known analytical solution for ${L}_{SC}$ as a function of ${I}_{tr}$ [47,51,60]. (

**c**) The bottom outer inset shows the numerical tendency of the non-dimensional factor ${\overline{R}}_{\mu 1}$ (Equation (16)) valid for any radius of the SFM sheath up to ${R}_{SFM}=10{R}_{SC}$ and with magnetic permeabilities from ${\mu}_{r}\sim 1$ up to $1\times {10}^{5}$. (

**d**) Finally, the inner inset shows the dependence of the metastructure losses $\left({L}_{SC-SFM}\right)$ for different amplitudes of the transport current ${i}_{tr}$ in units of ${I}_{c}$, and the relative magnetic permeability of the SFM sheath ${\mu}_{r}$.

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Fareed, M.U.; Ruiz, H.S.
Critical State Theory for the Magnetic Coupling between Soft Ferromagnetic Materials and Type-II Superconductors. *Materials* **2021**, *14*, 6204.
https://doi.org/10.3390/ma14206204

**AMA Style**

Fareed MU, Ruiz HS.
Critical State Theory for the Magnetic Coupling between Soft Ferromagnetic Materials and Type-II Superconductors. *Materials*. 2021; 14(20):6204.
https://doi.org/10.3390/ma14206204

**Chicago/Turabian Style**

Fareed, Muhammad U., and Harold S. Ruiz.
2021. "Critical State Theory for the Magnetic Coupling between Soft Ferromagnetic Materials and Type-II Superconductors" *Materials* 14, no. 20: 6204.
https://doi.org/10.3390/ma14206204