# Neural Network Modeling of Arbitrary Hysteresis Processes: Application to GO Ferromagnetic Steel

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## Abstract

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## 1. Introduction

^{®}. The resulting neural network (NN)-based model was firstly validated by the reproduction of the remaining measured hysteresis loops, not involved in the identification. The loops simulated by the NN-based model were in this case compared to both those measured and those calculated by the PM. Conclusively, other relevant hysteresis processes were generated by the PM to emulate the typical working excitations to which the ferromagnetic cores are subjected in real applications, such as DC-biased magnetization loops and distorted excitation waveforms. The latter comparative analysis was aimed at testing the capability of the NN-based model in the replication of the results obtained by the PM simulations under an arbitrary excitation field. The possibility to reach a satisfactory degree of accuracy in a wide range of excitations, while saving the computational time and the memory allocation, is an important step toward a future coupling of the hysteresis model with finite-element schemes, for dynamic simulations of magnetic cores.

## 2. Materials and Methods

#### 2.1. Experimental Investigation

^{®}by the authors. The quasi-static hysteresis loop was obtained as the average over n

_{p}periods of the measured signals of H(t) and B(t) at a given fundamental frequency f

_{0}.

_{AR}) between the aspect ratio of the measured signal and that of the pure sinusoidal wave, equal to $\pi /2\sqrt{2}$. In addition, the maximum value of the mean absolute error (MAE

_{max}) between the reference waveform and the measured waveform of B was evaluated to determine the exit condition of the feedback. In addition, the specimen under test was carefully demagnetized before each acquisition, in order to avoid undesired bias values of the magnetic induction, which would lead to an asymmetric hysteresis loop. The parameters set in order to perform the acquisition and the values used in our experimental investigation are illustrated in Table 3.

#### 2.2. Preisach Model Computations

_{up}, H

_{down}.

_{up}, H

_{down}, were equivalently expressed in terms of the interaction field H

_{i}= (H

_{up}+ H

_{down})/2 and the intrinsic coercivity u = (H

_{up}− H

_{down})/2.

^{+}and D

^{−}are the regions of the domain in which the operators have respectively positive (+1) and negative (−1) output, while P is the weight function.

_{i}, representing the nodes on the H field in which the hysterons are located, and u, representing the possible values of the coercive field. The length of the two vectors H

_{i}and u, are indicated respectively with N

_{H}and N

_{U}, while N

_{hyst}= N

_{H}⋅N

_{U}is the total number of hysterons. To optimize the computational performances of the model, it is convenient to locate the operators not uniformly with respect to both the interaction field and the coercivity values. Indeed, it is straightforward to increase the density of operators and then the number of Barkhausen jumps, where the magnetic permeability of the material is higher. The following equations were adopted to define the numerical grid of hysterons:

_{i}, u) has to be found. To allow the identification from a few symmetric hysteresis loops, it is convenient to approximate the weight function with probability distributions, rather than solving the Everett integral in Equation (1) in the analytic form [23].

_{i}and the other depending on only u: $P\left({H}_{i},u\right)={P}_{H}\left({H}_{i}\right)\xb7{L}_{U}\left(u\right)$. The first term is given by the linear combination of two Lorentzian functions, while the second term is a single Lorentzian function: $P\left({H}_{i},u\right)=\left[\alpha {L}_{H1}\left({H}_{i}\right)+\left(1-\alpha \right){L}_{H2}\left({H}_{i}\right)\right]\xb7{L}_{U}\left(u\right)$. The expression of L

_{H}

_{1}, L

_{H}

_{2}, and L

_{U}is described below.

_{1}and L

_{2}with respect to H

_{i}, ${\sigma}_{U}$ describes the standard deviation of L

_{U}with respect to u, and u

_{0}is the most probable value of the intrinsic hysteron coercive field.

_{k}are column vectors with length N

_{hyst}, while M

_{k}is evaluated as a scalar product. Before the computation of M

_{k}, the column vector q

_{k}is determined for the current value of the magnetic field H

_{k}and the previous value q

_{k}

_{−1}. The model was implemented at a low level of abstraction in a computer program, written in Matlab

^{®}.

_{0}= 0.4, 1.0, 1.6 T.

#### 2.3. Neural Network-Based Model

_{j}is the output of the j-th neurons of the previous layer, w

_{j}is the synaptic weight which connects the neuron to the j-th one of the previous layer, b is the bias value, and f

_{act}is the neuron activation function.

_{k+}

_{1}can be calculated as ${\chi}_{k}\xb7\left({H}_{k+1}-{H}_{k}\right)+{M}_{k}$. The computation of the model output was performed in a closed-loop configuration, in which the λ past values of M, sent as input at each time step k, were those calculated by the model in the previous steps. For this reason, a first input last output (FILO) buffer was required to store the λ floating-point variables M

_{k}, …, M

_{k-λ}. At each step, the variables stored in the buffer were updated (shifted toward the left) for the computation of the successive value of M. Lastly, let us point out that, since the output of the FFNN is the magnetic susceptibility, the model can be easily inverted, allowing coupling with FEM.

^{®}. The program consisted of a simple main script (main_simul_NN.m) in which the magnetic field sequence could be either directly defined by the user as a row vector or loaded from a .txt file.

^{®}function, named “NN_model.m”, to solve the output magnetization. The two files are shared by the authors as Supplementary Materials. Alternatively, exploiting the Neural Network Toolbox of Matlab

^{®}, it is possible to handle the FFNN at a higher level of abstraction, which can be saved as a structure, with defined fields and methods. The same main program can be used to simulate the neural network at the high level of abstraction, calling the method “sim” (=simulate) of the structure instead of calling the function “NN_model.m” in the iterative procedure. However, as shown in Section 3, the computational speed is significantly reduced.

#### 2.4. Training Procedure

## 3. Presentation and Discussion of Results

#### 3.1. Preisach Model (PM) Identification

_{0}= 0.4, 1.0, 1.6 T of the sinusoidal magnetic induction, were involved in the process of identification.

- -
- A classification was made, ordering the individuals on the basis of the cost function value.
- -
- The best individual (on top of the classification) was always copied in the successive generation.
- -
- The best K = 2 was used to generate n_co = 15 individuals by crossing over.
- -
- Each gene of the n_co individuals had a probability P_mut = 15% to mutate.
- -
- Two individuals, obtained from the best K ones as the arithmetic and geometric average of their genes, were directly copied in the next generation.
- -
- The remaining n_co = 3 individuals were again randomly generated.

_{pk}= 1.6 T, in both branches of which a series of 15 asymmetric minor loops was distributed. The magnetic induction at which the minor loops occur is given by ${B}_{IP}\left(i\right)={B}_{pk}-i\mathsf{\Delta}B$, with $\mathsf{\Delta}B=0.2T$ and $i=1,2,\dots ,15$. The sequences of both the magnetic field and the magnetic induction had 3080 samples.

#### 3.2. Neural Network Training

#### 3.3. Simulation of Symmetric Loops

_{max}, was simulated tracing the first magnetization curve from H = 0 to H = H

_{max}, and then applying the measured sequence, opportunely shifted to start from the maximum value. Two periods were simulated, and the second one was extracted to display the hysteresis loop. The NN simulations require a similar definition of the input field, but it was preliminarily verified that the hysteresis loop calculated in the second period was not dependent on the initial magnetization state considered. For this reason, similarly to the case of the Preisach model, the NN-based model simulations could indifferently start from either H

_{init}= H

_{max}and B

_{init}= B

_{max}or from the virgin state.

_{max}= 17 A/m to H

_{max}= 140 A/m. An acceptable accuracy was found in the examined range of H. In particular, if the magnetic field was smaller than about 30 A/m, a slight deviation of the maximum value of B simulated by the models with respect to the measured one was found. The deviations did not appear for higher values of H, as can be seen in Figure 5c,d.

_{max}= 1 T, equal to 14.2 J/m

^{3}for the PM and 8.6 J/m

^{3}for the NN. The specific energy loss W, computed from the experimental loops and those simulated by both the PM and the NN-based model, is plotted versus the maximum value of the magnetic induction in Figure 6.

#### 3.4. Simulation of First-Order Reversal Curves (FORCs)

_{min}(j), B

_{min}(j)), with j = 1, 2, …, 16, to the same maximum value (H

_{max}, B

_{max}) = (140 A/m, 1.6 T). The coordinates of the minimum points, which represent the left corners of the asymmetric loops, were determined on the descending branch of the major loop (B

_{min}= −B

_{max}). The sequences of H and B relative to any of the 16 curves had 1000 points.

_{min}> 10 A/m, were almost negligible.

_{max}− H

_{min}(j), for j = 1, 2, …, 16, in Figure 8. Since the material is almost saturated when H = H

_{max}, in the range ΔH < 120 A/m, the asymmetric loops were quite narrow and the energy losses were lower than 10 mJ per m

^{3}. However, for 120 A/m < ΔH < 175 A/m, the energy losses sharply increased to about 50 mJ per m

^{3}. For higher values of ΔH, the loops tended to be symmetric and equal to the major cycle. In the range of ΔH examined, the energy losses predicted by the NN-based model were very close to those obtained by the Preisach simulations.

#### 3.5. Two-Tone Excitation Waveforms

_{0}= 100 A/m, and SPP = 1000.

_{0}of the fundamental component and the number SPP of samples per period are fixed, two final degrees of freedom characterize the magnetic field sequence: the modulation index m and the phase displacement φ of the third harmonic with respect to the fundamental.

^{3}for m = 0, to 60.5 J/m

^{3}for m = 0.3. The NN-based model tended to overestimate the losses, but the absolute error was always lower than 5 J/m

^{3}, reflecting a maximum relative displacement of 9.3%, found for m = 0.24. The displacement between the two models turned out to be an increasing function of m, as well as of the minor loop area. If the modulation index was above 0.15, the area of the minor loops became higher than the area of those present in the training set, and the increase in error was expected. However, it must be specified that such high values of the modulation index are not expected in practical applications.

^{3}, for φ = 0 to 54.9 J/m

^{3}, for φ = 180°. The energy losses calculated by the neural network were slightly higher than those predicted by the Preisach model, with a maximum displacement, found for φ = 0, equal to 9.1%. In this case, it is interesting to note that, for small values of the phase lag angle, the error propagation during the closed=loop calculation reflected a slight overestimation of the coercive field. This is the main reason behind the upward deviation of the energy losses computed by the NN model in the range of $\phi \in \left[0,\pi /2\right]$. For higher values of the angle, the percentage deviation was lower than 5%.

^{®}Core™ i7-2670 QM @ 2.20 GHz, 8 GB of RAM memory, and 64-bit operating system. The calculation speed, expressed in terms of the number of samples processed per second, was evaluated for the PM, the NN model at a high level of abstraction, and the NN at a low level of abstraction. The latter case, with 4638 samples processed per second, turned out to be the fastest approach. However, it was found out that the high-level implementation of the NN, exploiting the Neural Network Toolbox of Matlab

^{®}, was slower than the low-level implementation of the Preisach model. The sample rate found for the neural network model at the high level of abstraction was 106 samples/s, against the 120 samples/s of the PM. It can be concluded that, in order to exploit the numerical effectiveness of the neural network, a suitable low-level implementation is necessary. The RAM memory occupied by the PM is 5.76 × 10

^{5}floating variables, to store the location (H

_{i}) and the intrinsic coercivity (u) of all the hysterons plus 2.88 × 10

^{5}integer variables, to store their status at each sample step. It has to be also mentioned that some techniques have been developed recently [26] to reduce the memory occupation of the PM. On the other hand, the RAM memory required for the NN, implemented at the low level of abstraction, only consists of 529 floating variables: 480 for the synaptic weights and 49 for the neuron bias values.

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Epstein measurements for the M4T27 steel: sinusoidal waveforms of the magnetic induction (

**a**) and waveforms of the applied magnetic field (

**b**).

**Figure 2.**Block diagram of the NN-based model, having 2λ + 1 inputs and n hidden layers (

**left panel**). Illustration of the model output as a function of the evolution of the past λ points swept on the H–M plane (

**right panel**).

**Figure 3.**Graphical user interface developed by the authors for the implementation of the training procedure.

**Figure 4.**Closed loop calculation of the training set performed by the six trained networks with the number of neurons per each hidden layer equal to 6 (

**a**), 8 (

**b**), 10 (

**c**), 12 (

**d**), 14 (

**e**), and 16 (

**f**).

**Figure 5.**Symmetric hysteresis loops for different H

_{max}= 17 A/m (

**a**), 26 A/m (

**b**), 45 A/m (

**c**), 140 A/m (

**d**), obtained by the measurements (black dashed line), compared with those calculated by both the Preisach model (blue continuous line) and the neural network model (red dots).

**Figure 6.**Specific energy losses computed from the experimental loops (black dashed line with squared marker) are compared with those calculated from both the Preisach model (blue continuous line with cross markers) and the neural network model (red continuous line with dots) simulations.

**Figure 7.**Simulation of first-order reversal curves: comparison between the Preisach model (blue continuous line) and the neural network (red dots) for B

_{min}= 1.20 T (

**a**), 0.5 T (

**b**), 0 T (

**c**), −0.5 T (

**d**), −1.0 T (

**e**), −1.20 T (

**f**).

**Figure 9.**Comparison between the hysteresis loops computed by the Preisach model and the neural network under two—tone magnetic field waveforms, for different values of the phase angle φ = 0° (

**a**), 60° (

**b**), 120° (

**c**), 180° (

**d**) and constant modulation index m = 0.24.

**Figure 10.**Specific energy loss under two-tone magnetic field waveforms: curves calculated from the Preisach model and the neural network under either different values of m and constant φ = 0° (

**a**) or different values of φ and constant m = 0.24 (

**b**).

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

Thickness | d | 27 mm |

Assumed density | δ | 7650 kg/m^{3} |

Silicon content | p_{Si} | 3.0% |

Lamination factor (min) | LF | 0.95 |

Polarization @ 800 A/m | B_{800} | 1.80 T |

Equipment | Model |
---|---|

Power amplifier | Kepco BOP 36-5 |

Current probe | Rhode & Schwarz RT-ZC03 |

Voltage probe | Tektronics TPP0101 |

Data acquisition module | NI USB 6363 BNC Type |

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

Fundamental frequency | f_{0} | 1 Hz |

Sample rate for the A/D conversion | SR | 500 samples/s |

Number of periods for the average | n_{p} | 40 |

Maximum number of iterations | max_iter | 20 |

Maximum MAE allowed for convergence | MAE_{max} | 1.8% |

Maximum error for the aspect ratio | MAE_{AR} | 1.5% |

**Table 4.**Preisach parameters and values of the cost function obtained after the optimization via the genetic algorithm (GA) and the pattern search algorithm (PSA).

Parameter | ${\mathit{\sigma}}_{\mathit{H}1}$ | ${\mathit{\sigma}}_{\mathit{H}2}$ | ${\mathit{\sigma}}_{\mathit{U}}$ | ${\mathit{u}}_{0}$ | $\mathit{\alpha}$ | ${\mathit{f}}_{\mathit{e}\mathit{r}\mathit{r}}$ |
---|---|---|---|---|---|---|

GA | 8.60 A/m | 70.31 A/m | 6.50 A/m | 7.35 A/m | 0.655 | 0.0276 |

PSA | 8.60 A/m | 78.31 A/m | 6.51 A/m | 7.35 A/m | 0.645 | 0.0255 |

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## Share and Cite

**MDPI and ACS Style**

Quondam Antonio, S.; Bonaiuto, V.; Sargeni, F.; Salvini, A.
Neural Network Modeling of Arbitrary Hysteresis Processes: Application to GO Ferromagnetic Steel. *Magnetochemistry* **2022**, *8*, 18.
https://doi.org/10.3390/magnetochemistry8020018

**AMA Style**

Quondam Antonio S, Bonaiuto V, Sargeni F, Salvini A.
Neural Network Modeling of Arbitrary Hysteresis Processes: Application to GO Ferromagnetic Steel. *Magnetochemistry*. 2022; 8(2):18.
https://doi.org/10.3390/magnetochemistry8020018

**Chicago/Turabian Style**

Quondam Antonio, Simone, Vincenzo Bonaiuto, Fausto Sargeni, and Alessandro Salvini.
2022. "Neural Network Modeling of Arbitrary Hysteresis Processes: Application to GO Ferromagnetic Steel" *Magnetochemistry* 8, no. 2: 18.
https://doi.org/10.3390/magnetochemistry8020018