# High-Frequency Fractional Predictions and Spatial Distribution of the Magnetic Loss in a Grain-Oriented Magnetic Steel Lamination

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

_{m}can commonly be expressed as the sum of the static hysteresis loss W

_{hyst}, the classical eddy current loss W

_{clas}, and the excess loss W

_{exc}:

_{hyst}(B), also called H

_{stat}(B) later in this paper, is a static contribution obtained from a static hysteresis model (Jiles-Atherton (J-A) model [22], Preisach model [23,24], etc.) in their B-input form. k

_{clas}is a constant depending on the specimen conductivity and geometry, δ is a directional parameter (±1), and g

_{exc}(B) and α

_{exc}(B) are two B-dependent functions that have to be defined for each frequency tested. The comparison simulations/measurements available in [18,21] reveal an excellent behavior of the TSM. Still, the number of parameters to be adjusted for each frequency level (>10 in [21]) is overwhelming, and the predictive capability of TSM is minimal. Eventually, just like STL, TSM does not provide local information.

## 2. Simulation Method Description

_{stat}(B). Still, it will ineluctably lead to inaccurate results as the excess loss contribution will not be considered. In [27], a first-order differential equation equivalent to a viscous behavior was proposed to improve the accuracy of the material law (Equation (5)):

_{i}became frequency dependent, and ρ was the unique parameter accounting for this effect. Still, this dependency was inaccurate, and the domain of validity of the resulting method was restrained to a narrow frequency bandwidth. Later in [19], an improvement in the frequency bandwidth was proposed, but at the cost of B-dependent additional parameters. Equation (6) was first introduced as a generalized equation in which g(B) was one of these parameters:

#### 2.1. Fractional Differential Equation: Physical Interpretation and Resolution

#### 2.2. Combining Equation (8) and Equation (2) for a Simultaneous Resolution

^{n}B/dt

^{n}, and σ by σ’, equivalent to a pseudo conductivity:

_{i}(t), B

_{i}(t), and B(t)). This simulation method was straightforward, and the simulation times were limited; a correct combination of ρ’ and σ’ yielded accurate results on a broad frequency bandwidth. Still, the physical meaning of the anomalous diffusion and σ’ the pseudo conductivity was unclear. The only way to derive Equation (10) was through fractional Maxwell equations. Such equations have already been mentioned in the scientific literature [32,33] but remain complex to justify physically. Therefore, this solution was discarded in this new study.

_{i}(t) and is followed by a local resolution of Equation (8), leading to the local B

_{i}(t). In the last stage, the cross-section flux density B is calculated by averaging the local induction:

_{surf}(B), the resulting simulated hysteresis cycle can be compared to the experimental one. To limit both the discretization and the memory management, such as saving simulation time, we opted for a static contribution H

_{stat}(B) obtained with the derivative static hysteresis model (DSHM), described in [34,35]. This simulation method relies on a 2D interpolation matrix constructed with the columns and rows denoting the discrete values of H and B, and whose terms stand for the dB/dH slope at the corresponding point. As recalled in [34], DSHM can easily switch from H to B-imposed input conditions. To fill the DSHM matrix, experimental first-order reversal curves are promoted, but getting such experimental data is always complex. In this work, we replaced them with simulated first-order reversal curves obtained with the J–A model [36,37]. The J–A model was identified using the limited experimental data available (a saturated and symmetrical quasi-static hysteresis cycle). Additional details about the J–A and the DSHM models can be found elsewhere [34,35,36,37]. The next section describes the experimental setup.

## 3. Experimental Setup Description

^{−1}, were measured and recorded for the range of magnetizing frequency and peak induction. More details about the experimental setup are available in [41]. The following section outlines the model validation process, involving comparisons between simulations and measurements, as well as the developed method for assessing the loss contribution and distribution.

## 4. Simulation Method Settings and Validation up to 1 kHz and 1.7 T

_{stat}(B): DSHM + J–A models), while the other two are for the frequency-dependent contribution. The quasi-static contribution parameters M

_{s}= 1.36 × 10

^{6}A·m

^{−1}, a = 1.5 A·m

^{−1}, k = 11.5 A·m

^{−1}, c = 0.06, and α = 7 × 10

^{−6}were determined based on the optimization of the relative Euclidean difference criteria (red(%), Equation (18)) comparing measured and simulated quasi-static saturated hysteresis loops (f = 1 Hz, B

_{m}= 1.7 T).

#### 4.1. Comparisons Simulation/Measurement, Model Validation

#### 4.2. Model Exploitation, Spatial Losses Distribution

_{i}(H

_{i stat}) and dynamic B

_{i}(H

_{i}) hysteresis cycles. Figure 4 shows those cycles for nodes 1–10 in the case of Figure 2b, where f = 1 kHz and B

_{m}= 1.7 T.

_{hysti}(Equation (23)) and excess losses W

_{exci}(Equation (24)). Then, starting from node 1, the local classic losses W

_{clasi}are calculated from the difference between W

_{tot}and a virtual W

_{tot N=}

_{2–10}that would consider only nodes 2 to 10 for the average induction and H

_{1}for the excitation field. The difference between W

_{tot}and W

_{tot N=}

_{2–10}corresponds to W

_{tot1}, the total loss at node 1; W

_{clas}

_{1}is obtained by subtracting W

_{hyst1}and W

_{exc}

_{1}:

^{2}− dependent losses); this method does not provide any local information, nor can it be used to discriminate the classic and the excess losses. It was therefore discarded from the validation process.

## 5. Model Predictions

#### 5.1. Sinus B-Imposed Simulation

^{−1}) centered around the value of H at t = t − dt. The H value that leads to the targeted B is conserved and becomes H(t). Then, the process is rerun for the next H(t + x.dt). This method only converges for minimal discretization steps, increasing the simulation time inevitably.

_{M}(t) is the measured output (at time t), $\u03f5$(t) = y

_{G}(t) − y

_{M}(t) is the error, and x(t) is the system input. K

_{p}, K

_{I}, and K

_{D}are the proportional, integral, and derivative gains, respectively.

_{p}. Its implementation is very straightforward, and, like classic PID, it can be very robust with the right choice of K

_{p}.

#### 5.2. From 2 to 10 kHz Predictions

_{a}(H

_{surf}) predictions and the associated losses in space distributions.

_{m}= 1.7 T, f = 10 kHz on a 0.29 mm thick GO electrical steel (M2H) using a single sheet tester in a single-shot mode measurement to limit heat increase. No information is provided in the paper about using a cooling system, nor do we know how the imposed waveform for the single shot is predetermined. The only information provided is that the magnetomotive force (MMF) drop to the flux closing yoke was compensated. Then, they propose a method to predict the loss distribution. Figure 11 compares their predictions to ours.

_{hyst}and W

_{class}contributions is visible. The slight increase in W

_{hyst}, when f increases in [44], can be attributed to a higher level of B on the edge layers, but this behavior does not happen in our simulation method, where the static contribution illustrated in Figure 4b is always very saturated, even in a lower level of frequency. The most noticeable difference comes from the evolution of W

_{exc,}which increases at a much lower rate in [44] than in our simulation method. The low level of W

_{exc}in [44] is attributed to the significant skin effect in the high-frequency range (f > 200 Hz) and the absence of magnetization in the center of the tested lamination. At 1 kHz, where our simulation method has been validated through comparison with experimental results, our W

_{exc}contribution is already three times larger than that of [44]. This difference could be attributed to the different grades of oriented grain electrical steel tested. Unfortunately, in [44], no experimental result at 1 kHz could validate this hypothesis. Finally, a slight deviation in excess loss is observable in our simulation results for the high level of f. This observation should be confirmed experimentally in future studies. Although differences can be observed in the prediction of the excess loss frequency dependency (Figure 11), the close behavior of the classic and hysteresis losses, combined with the excellent simulation results shown in Figure 2 and Figure 3, can be considered as a solid validation of our simulation method.

## 6. Conclusions

_{m}= 1.7 T, eddy currents resulting from the penetration equation fail to prevent magnetization of the lamination center (node 10), highlighting a significant aspect for further exploration.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**(

**a**) Comparisons simulation/measurement for f = 1 Hz, B

_{m}= 1.7 T, (

**b**) comparisons simulation/measurement for f = 1 kHz, B

_{m}= 1.7 T.

**Figure 3.**(

**a**) Comparisons simulation/measurement for f Є [1–1000] Hz, B

_{m}= 1.7 T, (

**b**) comparisons simulation/measurement for f = 1000 Hz, B

_{m}Є [1–1.7] T, (

**c**) comparisons simulation/measurement for f Є [1–1000] Hz, B

_{m}= 1.3 T.

**Figure 4.**(

**a**) Local dynamic hysteresis loops (f = 1000 Hz, B

_{m}= 1.7 T), (

**b**) local static hysteresis loops (f = 1 Hz, B

_{m}= 1.7 T).

**Figure 8.**(

**a**) Comparison between the targeted and the simulated flux density at the end of the P−ILC process (f = 2 kHz), (

**b**) H(t) related surface field.

**Figure 9.**(

**a**) f = 2, 5 and 10 kHz, B

_{m}= 1.7 T, B(H) simulation predictions, (

**b**) Spatial loss distribution for f Є [2–10] kHz.

**Figure 11.**Comparison of frequency-dependent loss contribution predictions between those gathered from [44] and those obtained using the method in this paper.

**Table 1.**Quantitative comparison (uncertainty) based on the red(%) and the FF(%) criterion for all Figure 3 simulation results (in black, the red is calculated for all contributions; in red, just the dynamic contribution) (Top table: B

_{m}= 1.7 T; bottom table: f = 1 kHz).

Quasi-Static | f = 50 Hz | f = 100 Hz | f = 200 Hz | f = 400 Hz | f = 800 Hz | f = 1000 Hz | Av. | |

red(%) | 13.1 | 7.3 | 10.1 | 9.6 | 8 | 2.7 | 1 | 7.4 |

red(%)—dyn. cont. | - | 2.4 | 8.6 | 8.6 | 7.2 | 1.8 | 0.2 | 4.8 |

FF(%) | 0.22 | 0.118 | 0.111 | 0.068 | 0.028 | 0.003 | 0.005 | 0.0875 |

Bm = 1 T | Bm = 1.3 T | Bm = 1.5 T | Bm = 1.7 T | Av. | ||||

red (%) | 2.4 | 6 | 5 | 1 | 3.6 | |||

FF(%) | 0.002 | 0.002 | 0.004 | 0.005 | 0.0033 |

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## Share and Cite

**MDPI and ACS Style**

Ducharne, B.; Hamzehbahmani, H.; Gao, Y.; Fagan, P.; Sebald, G.
High-Frequency Fractional Predictions and Spatial Distribution of the Magnetic Loss in a Grain-Oriented Magnetic Steel Lamination. *Fractal Fract.* **2024**, *8*, 176.
https://doi.org/10.3390/fractalfract8030176

**AMA Style**

Ducharne B, Hamzehbahmani H, Gao Y, Fagan P, Sebald G.
High-Frequency Fractional Predictions and Spatial Distribution of the Magnetic Loss in a Grain-Oriented Magnetic Steel Lamination. *Fractal and Fractional*. 2024; 8(3):176.
https://doi.org/10.3390/fractalfract8030176

**Chicago/Turabian Style**

Ducharne, Benjamin, Hamed Hamzehbahmani, Yanhui Gao, Patrick Fagan, and Gael Sebald.
2024. "High-Frequency Fractional Predictions and Spatial Distribution of the Magnetic Loss in a Grain-Oriented Magnetic Steel Lamination" *Fractal and Fractional* 8, no. 3: 176.
https://doi.org/10.3390/fractalfract8030176