# A Numerical Comparison between Preisach, J-A and D-D-D Hysteresis Models in Computational Electromagnetics

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Materials

#### 2.2. Hysteresis Models

**H**and the magnetization

**M**of the ferromagnetic material. The magnetic flux density

**B**is then obtained from the constitutive equation given by:

_{0}is the vacuum permeability.

#### 2.2.1. CPM Model

_{1}, h

_{2}) is the Preisach distribution function and m(h

_{1}, h

_{2}, H(t)) represents the elementary relay operators which can take the values +1 or −1. Depending on the history of the applied magnetic field, two domains can be distinguished on the Preisach triangle: one where the operators are switched down (h

_{1}) and one where the operators are switched up (h

_{2}). The boundary between domains is called staircase line.

_{r}= dH, where d is a constant and added to the magnetization produced by the Preisach model. It should be noted that an improved closed form with additional polynomials was proposed in [32], but for the sake of comparison, the original version was hereby adopted to reduce the parameter numbers.

#### 2.2.2. J-A Model

_{an}is the anhysteretic magnetization that is computed using Langevin’s polynomial [7], while M

_{s}, α

_{J}, a

_{J}, k and c

_{J}are the five J-A model parameters, which represent the saturation magnetization, the inter-domain coupling coefficient, the effective domain density, the energy-dissipative features in the microstructure and the reversibility coefficient, respectively [35].

#### 2.2.3. D-D-D Model

_{D}and β are the three D-D-D parameters. In particular, the former affects the amplitude of the magnetization, whereas α and β influence the slope and width of the obtained hysteresis loop, respectively. It should be noted that the proposed D-D-D model is conceptually similar to the Preisach model but mathematically closer to the J-A model, with an even simpler ODE to be solved and two parameters less to be identified. Since the latter are straightforward to determine, a tentative approach has been undertaken.

## 3. Results

#### 3.1. Accuracy

_{m}and A

_{c}are the measured and calculated areas, respectively. The second one is presented to elaborate the goodness of fit of the proposed models (i.e., the Pearson correlation coefficient), which is expressed as [24]:

_{m}is the total number of sampled measurement points, B

_{m}

_{,k}is the k-th measured point, B

_{c}

_{,k}is the k-th calculated point and B

_{m}

_{,avg}is the average value of all measured points. The value range of r

^{2}is [0, 1], and the closer it is to 1, the more accurate the fitting is.

#### 3.2. Computational Time

## 4. Discussion

_{asc}= 100 samples for each branch has been deemed as a good trade-off between accuracy and CT.

_{asc}is shown to affect the accuracy, even for the D-D-D model, the number of points N

_{c}= 2 N

_{asc}used to discretize the phase shift vector ωt would affect the accuracy. Indeed, it has been experienced that a number N

_{c}< 100 yielded unclosed and asymmetric loops. This could happen with coarse measurement points (N

_{m}< 100) in evaluating the Pearson correlation coefficient (i.e., N

_{c}= N

_{m}). In such a case, an oversampling of both measured and simulated points will avoid this issue.

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Comparison between measured (solid) and calculated (dashed colored) hysteresis loops. (

**a**) MnZn ferrite. (

**b**) Fe-Si loop 11. (

**c**) NdFeB at 80 °C. (

**d**) NdFeB at 27 °C.

Material | Model | Parameter Number ^{1} | ||||
---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | ||

MnZn ferrite | CPM | 0.049 | −8.73 | 17.495 | 0.95∙10^{−3} | - |

J-A | 3.178∙10^{5} | 1.099∙10^{−7} | 12.649 | 12.448 | 0.844 | |

D-D-D | 2.74∙10^{5} | 5.5 | 0.116 | - | - | |

Fe-Si loop 11 | CPM | 0.044 | 69.12 | 23.212 | 4∙10^{−4} | - |

J-A | 1.23∙10^{6} | 1∙10^{−4} | 47 | 66 | 0.99 | |

D-D-D | 1.194∙10^{6} | 300 | 0.07 | - | - | |

NdFeB at 80 °C | CPM | 6.6∙10^{−3} | 4.56∙10^{5} | 1.49∙10^{5} | 0.5∙10^{−3} | - |

J-A | 1.18∙10^{6} | 0.46 | 1.25∙10^{5} | 5.24∙10^{5} | 0.05 | |

D-D-D | 1.13∙10^{6} | 12.5 | 0.3 | - | - | |

NdFeB at 27 °C | CPM | 0.016 | 1.24∙10^{6} | 5.44∙10^{4} | 0.2∙10^{−4} | - |

J-A | 0.954∙10^{6} | 1.1 | 1.45∙10^{5} | 1∙10^{6} | 1∙10^{−6} | |

D-D-D | 0.909∙10^{6} | 20 | 1.02 | - | - |

^{1}Parameter number reads as follows: CPM: a, b, c and d; J-A: M

_{s}, α

_{J}, a

_{J}, k and c

_{J}; D-D-D: γ, α

_{D}and β.

**Table 2.**Numerical comparison of the considered models. Bolded values are the most performant metrics.

Material | Model | e (%) | r^{2} | CT (s) |
---|---|---|---|---|

MnZn ferrite | CPM | 4.05629 | 0.99785 | 0.09182 |

J-A | 1.51375 | 0.99948 | 0.15134 | |

D-D-D | 4.53686 | 0.99718 | 0.00685 | |

Fe-Si loop 11 | CPM | 15.24575 | 0.91456 | 0.08678 |

J-A | 5.0458 3 | 0.99212 | 0.22523 | |

D-D-D | 4.91653 | 0.97532 | 0.00976 | |

NdFeB at 80 °C | CPM | 0.19651 | 0.99927 | 0.07080 |

J-A | 9.62853 | 0.93845 | 0.26254 | |

D-D-D | 2.03716 | 0.99921 | 0.00766 | |

NdFeB at 27 °C | CPM | 2.18167 | 0.99584 | 0.07430 |

J-A | 31.31527 | 0.79565 | 0.35150 | |

D-D-D | 2.16800 | 0.99872 | 0.00660 |

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**MDPI and ACS Style**

De Santis, V.; Di Francesco, A.; D’Aloia, A.G.
A Numerical Comparison between Preisach, J-A and D-D-D Hysteresis Models in Computational Electromagnetics. *Appl. Sci.* **2023**, *13*, 5181.
https://doi.org/10.3390/app13085181

**AMA Style**

De Santis V, Di Francesco A, D’Aloia AG.
A Numerical Comparison between Preisach, J-A and D-D-D Hysteresis Models in Computational Electromagnetics. *Applied Sciences*. 2023; 13(8):5181.
https://doi.org/10.3390/app13085181

**Chicago/Turabian Style**

De Santis, Valerio, Antonio Di Francesco, and Alessandro G. D’Aloia.
2023. "A Numerical Comparison between Preisach, J-A and D-D-D Hysteresis Models in Computational Electromagnetics" *Applied Sciences* 13, no. 8: 5181.
https://doi.org/10.3390/app13085181