# Harmonization and Validation of Jiles–Atherton Static Hysteresis Models

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Basic Model

_{e}) is then expressed in (2) and is analogous to the Weiss mean field experienced by the individual magnetic moment within a ferromagnetic domain [2]. The energy per unit volume of the domain with magnetic moment m is now affected by the effective magnetic field strength (3).

_{e}is defined as:

_{an}as a function of effective magnetic field strength H

_{e}is introduced by Equation (5) and is in the form of a Langevin function, where saturation magnetization is presented by M

_{s}, and a is the variable with dimensions of magnetic field strength that form the shape of anhysteretic magnetization in an acceptable way [2]:

_{pin}can be written as:

_{e}gives the solution (8) describing the magnetization processes. The variable δ was introduced [2] to guarantee that the pinning sites face the changes in magnetization M, meaning that variable δ has the value +1 when H increases (dH/dt > 0) and has the value −1 when H decreases (dH/dt < 0).

_{rev}and irreversible M

_{irr}magnetization (12).

_{rev}:

_{irr}is given by the solution of (10):

## 3. Differences in the Articles

_{new}is introduced (27) and further used in the modified form of the J–A model (28) to keep the same equation topology (17) as in the original article [2]. The new forms of hysteresis loops (Figure 2) were calculated after application of the latest change (28).

_{new}) for [2] are already included in (27) and (28). The results are shown in Figure 2 for Case 1 and in Figure 3 for Case 2.

## 4. Hysteresis Losses

_{hyst}(29):

_{e}, we obtain (30), which describes the magnetization process as already presented in the previous section (8).

_{e}with the use of Equation (4), the differential of hysteresis loss energy density as a function of effective magnetic field strength H

_{e}can be introduced (34).

_{e}is shown in Figure 4 and Figure 5, respectively. The hysteresis losses over one magnetizing period of applied magnetic field strength H

_{e}are presented in Table 6 in the form of value.

## 5. Comments on the Analyzed Hysteresis Models

#### 5.1. Jiles 1986 [2]

_{rev}is zero.

_{rev}is zero. In other cases where variable c is different from zero, the results will be different.

#### 5.2. Jiles 1990 (First Edition) [3]

_{rev}) exists if there is a difference between the anhysteretic curve M

_{an}and the irreversible magnetization M

_{irr}, as presented in (40). On the other hand, in [2], it is deduced that reversible magnetization is the difference between the anhysteretic curve M

_{an}and total magnetization M (39).

_{an}) and total magnetization (M) (as used in [2]). By comparing Equations (39) and (41), it can be concluded that model variable c in [2] and [3] may represent the same phenomena, but they cannot have the same value. According to this fact, (42) must be applied to calculate the value of variable c

_{3}for the model in [3] using the value of variable c

_{2}from the model derived in [2], or vice versa by using (43). The numerical limits of the values of parameter c

_{2}(c defined in [2]) are between 0 and infinity [2]. According to this fact and the form of Equation (43), it can be calculated (and it is also claimed in [7]) that the limits for parameter c

_{3}(c in [3]) are defined between 0 and 1. Applying this new range allows us to better present the amount of reversible magnetization. On the other hand, using this new boundary set (between 0 and 1) for parameter c

_{3}, the physical background of parameter c

_{2}developed in [2] is blurred.

#### 5.3. Jiles 1992 [4]

_{irr}. The author used the results from his previous works [2,3] and applied the irreversible magnetization backward directly to the energy balance equation. In comparison with his previous work [3], the model in [4] has the same results.

#### 5.4. Jiles 1994 [5]

_{rev}is zero, leading to (46). Further on, based on (46), the same differential equation can be derived for irreversible magnetization M

_{irr}, as is written in Table 2 for the model in [5] and used in the models in [2,3,4].

#### 5.5. Ruoyang [7]

## 6. Harmonization of J–A Hysteresis Loop Models

_{new}< 1), thereby bringing it closer to the concept of the variable c (0 < c < 1) as it is used in [3] (40). Following this, the variable c

_{new}becomes more representable in terms of defining the amount of reversible magnetization. This is further reason why the model in [3] is selected as the base for harmonization of all studied hysteresis models.

_{new}and c

_{new}are formed based on the original variables in [2]; they are (52) and (53):

_{e}).

_{rev}does not exist).

_{irr}and, further, from (12), that M

_{rev}is zero, meaning that variable $c=0$ because M cannot be equal to M

_{an}(if $M={M}_{an},$ it means there are no hysteresis losses). According to these facts, the irreversible magnetization is:

## 7. Measurements

^{−7}to 10

^{−2}) and a (from 10

^{−1}to 10

^{4}) were acquired from [7]. The explanation of the variable c range (0 to 1) has already been described above. Its actual searching area was set between 10

^{−4}and 0.999 to prevent numerical calculation error. According to Jiles’ work [4], the value of the pinning parameter k is near the value of the coercive magnetic field strength. The input bounds were set to 1 and 350 following the applicability for both materials. The value of the parameter M

_{s}was estimated using (9), where B was replaced with the saturation value of magnetic flux density (B

_{s}), and in (9) the magnetic field strength was neglected due to its small value in comparison to the saturation magnetization. According to the measured data for the M270-35A, the saturation point is more than 1.5 T (M

_{s}> 1.19 × 10

^{6}A/m), and 2.2 T (M

_{s}> 1.75 × 10

^{6}A/m) for the FeCo alloy. The actual input limits for variable M

_{s}were then set from 1.1 × 10

^{6}to 2.2 × 10

^{6}A/m. The optimized values of the model’s variables are shown in Table 12. The measured B–H loops and the loops obtained with the harmonized J–A model are presented in Figure 14 for the M270-35A and for the iron–cobalt alloy in Figure 15. Calculated hysteresis loss energy densities in one magnetizing period for all adopted models are shown in Table 13. The modeling results of the harmonized J–A model using variable values for iron–cobalt alloy (Figure 15) show higher accuracy in comparison to the modeled hysteresis loop for M270-35A (Figure 14). The same alternation (mainly at the hysteresis loop knee point) has also been observed before (Figure 5, Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10 in [4]). The error at the knee point can be minimized with various optimization settings (for example, using different fitting constraint functions in PSO), but then higher deviation may occur at the end points and/or in the coercivity area.

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**B–H curves for analyzed J–A models for Case 1 (Table 5).

**Figure 4.**The variation of differential hysteresis energy density during one magnetizing period as a function of applied magnetic field strength for Case 1 (Table 5).

**Figure 5.**The variation of differential hysteresis energy density during one magnetizing period as a function of applied magnetic field strength for Case 2 (Table 5).

**Figure 6.**B–H curves for all modified J–A hysteresis models for Case 1 (Table 5).

**Figure 7.**The variation of hysteresis loss energy density during one magnetizing period as a function of applied magnetic field strength for all studied J–A hysteresis models for Case 1 (Table 5).

**Figure 8.**B–H curves for all modified J–A hysteresis models Case 2 (Table 5).

**Figure 9.**The variation of hysteresis loss energy density during one magnetizing period as a function of applied magnetic field strength for all modified J–A hysteresis models for Case 2 (Table 5).

**Figure 10.**B–H curves for all modified J–A hysteresis models in Case 3 (Table 9).

**Figure 11.**The variation of hysteresis loss energy density during one magnetizing period as a function of applied magnetic field strength for all modified J–A hysteresis models for Case 3 (Table 9).

**Figure 12.**B–H curves for all modified J–A hysteresis models in Case 4 (Table 9).

**Figure 13.**The variation of hysteresis loss energy density during one magnetizing period as a function of applied magnetic field strength for all modified J–A hysteresis models for Case 4 (Table 9).

**Figure 14.**Measured B–H curves (solid line) and B–H curves obtained with the optimization procedure using the harmonized J–A model (Table 11) for M270-35A (dashed line).

**Figure 15.**Measured B–H curves (solid line) and B–H curves obtained with the optimization procedure using the harmonized J–A model (Table 11) for FeCo alloy in the form of a rod (dashed line).

**Table 1.**Five fundamental Equations (8), (2), (5), (12), (13) introduced in the model derivation and the sixth Equation (17) representing the final result of the J–A model in [2] in the form of a differential equation.

Equation (8) | $M={M}_{an}-\delta k\left(\frac{dM}{d{B}_{e}}\right)$ | Magnetization process |

Equation (2) | ${H}_{e}=H+\alpha M$ | Effective magnetic field strength |

Equation (5) | ${M}_{an}={M}_{s}\left(\mathrm{coth}\left(\frac{{H}_{e}}{a}\right)-\frac{a}{{H}_{e}}\right)$ | Anhysteretic magnetization in form of a Langevin function |

Equation (12) | $M={M}_{rev}+{M}_{irr}$ | Total magnetization |

Equation (13) | ${M}_{rev}=c\left({M}_{an}-M\right)$ | Reversible component of magnetization |

Equation (17) | $\frac{dM}{dH}=\frac{1}{1+c}\frac{1}{\delta k/{\mu}_{0}-\alpha \left({M}_{an}-M\right)}\left({M}_{an}-M\right)+\frac{c}{1+c}\frac{dMan}{dH}$ | Final form of hysteresis model |

[2] | $M={M}_{an}-\delta k\left(\frac{dM}{d{B}_{e}}\right)$ | (8) |

[3] | ${M}_{an}\left({H}_{e}\right)=M\left({H}_{e}\right)+k\left(\frac{dM}{d{H}_{e}}\right)$ | (18) |

[4] | ${M}_{irr}={M}_{an}-k\delta \left(\frac{d{M}_{irr}}{d{H}_{e}}\right)$ | (19) |

[5,7] | ${M}_{an}=M+k\delta \left(1-c\right)\frac{d{M}_{irr}}{d{H}_{e}}$ | (20) |

[2] | ${M}_{rev}=c\left({M}_{an}-M\right)$ | (13) |

[3,4,5,7] | ${M}_{rev}=c\left({M}_{an}-{M}_{irr}\right)$ | (21) |

Jiles 1986 [2] | $\frac{dM}{dH}=\frac{1}{1+c}\frac{1}{\delta k/{\mu}_{0}-\alpha \left({M}_{an}-M\right)}\left({M}_{an}-M\right)+\frac{c}{1+c}\frac{d{M}_{an}}{dH}$ | (17) |

Jiles 1990 [3] | $\frac{dM}{dH}=\frac{\left({M}_{an}-{M}_{irr}\right)}{k-\alpha \left({M}_{an}-{M}_{irr}\right)}+c\left(\frac{dMan}{dH}-\frac{d{M}_{irr}}{dH}\right)$ | (22) |

Jiles 1992 [4] | $\frac{dM}{dH}=\left(1-c\right)\frac{\left({M}_{an}-{M}_{irr}\right)}{k-\alpha \left({M}_{an}-{M}_{irr}\right)}+c\frac{d{M}_{an}}{dH}$ | (23) |

Jiles 1994 [5] | $\frac{dM}{dH}=\frac{\left({M}_{an}-{M}_{irr}\right)}{k\delta -\alpha \left({M}_{an}-M\right)}+c\left(\frac{d{M}_{an}}{dH}-\frac{\left({M}_{an}-{M}_{irr}\right)}{k\delta -\alpha \left({M}_{an}-M\right)}\right)$ | (24) |

Ruoyang 2015 [7] | $\frac{dM}{dH}=\frac{{M}_{an}-M}{k\delta -\alpha \left({M}_{an}-M\right)}+\frac{ck\delta}{k\delta -\alpha \left({M}_{an}-M\right)}\frac{d{M}_{an}}{dH}$ | (25) |

Case 1 | Case 2 | |
---|---|---|

a | 1100 A/m | 1100 A/m |

M_{s} | 1.6 × 10^{6} A/m | 1.6 × 10^{6} A/m |

α | 1.6 × 10^{−3} | 1.6 × 10^{−3} |

c | 0.2 | 0.5 |

k; k_{new} | 400 A/m; 502.655 × 10^{−6} Tm | 400 A/m; 502.655 × 10^{−6} Tm |

**Table 6.**The numerical results of hysteresis energy density w

_{hyst}for Case 1 and Case 2 (Table 5).

[2], step 1 | $\frac{dM}{dH}=\frac{1}{1+c}\frac{1}{\delta k/{\mu}_{0}-\alpha \left({M}_{an}-M\right)}\left({M}_{an}-M\right)+\frac{c}{1+c}\frac{d{M}_{an}}{dH}$ | $\mathit{k}\to {\mathit{k}}_{\mathit{n}\mathit{e}\mathit{w}}=\mathit{k}{\mathit{\mu}}_{\mathbf{0}}$ | $\frac{dM}{dH}=\frac{1}{1+c}\frac{1}{\delta {k}_{new}/{\mu}_{0}-\alpha \left({M}_{an}-M\right)}\left({M}_{an}-M\right)+\frac{c}{1+c}\frac{d{M}_{an}}{dH}$ |

[2], step 2 | $\frac{dM}{dH}=\frac{1}{1+c}\frac{1}{\delta {\mathit{k}}_{\mathit{n}\mathit{e}\mathit{w}}/{\mu}_{0}-\alpha \left({M}_{an}-M\right)}\left({M}_{an}-M\right)+\frac{c}{1+c}\frac{d{M}_{an}}{dH}$ | Changed according to our model derivation check (in original paper, possible typos) | $\frac{dM}{dH}=\frac{1}{1+c}\frac{1}{\delta {\mathit{k}}_{\mathit{n}\mathit{e}\mathit{w}}/{\mu}_{0}-\alpha \left({M}_{an}-{\mathit{M}}_{\mathit{i}\mathit{r}\mathit{r}}\right)}\left({M}_{an}-{\mathit{M}}_{\mathit{i}\mathit{r}\mathit{r}}\right)+\frac{c}{1+c}\frac{d{M}_{an}}{dH}$ |

[2], step 3 | $\frac{dM}{dH}=\frac{1}{1+c}\frac{1}{\delta {\mathit{k}}_{\mathit{n}\mathit{e}\mathit{w}}/{\mu}_{0}-\alpha \left({M}_{an}-{\mathit{M}}_{\mathit{i}\mathit{r}\mathit{r}}\right)}\left({M}_{an}-{\mathit{M}}_{\mathit{i}\mathit{r}\mathit{r}}\right)+\frac{c}{1+c}\frac{d{M}_{an}}{dH}$ | $\mathit{c}\to {\mathit{c}}_{\mathit{n}\mathit{e}\mathit{w}}=\frac{\mathit{c}}{\mathit{1}-\mathit{c}}$ 0 < c <1 | $\frac{dM}{dH}=\frac{1}{1+{\mathit{c}}_{\mathit{n}\mathit{e}\mathit{w}}}\frac{1}{\delta {\mathit{k}}_{\mathit{n}\mathit{e}\mathit{w}}/{\mu}_{0}-\alpha \left({M}_{an}-{\mathit{M}}_{\mathit{i}\mathit{r}\mathit{r}}\right)}\left({M}_{an}-{\mathit{M}}_{\mathit{i}\mathit{r}\mathit{r}}\right)+\frac{{\mathit{c}}_{\mathit{n}\mathit{e}\mathit{w}}}{1+{\mathit{c}}_{\mathit{n}\mathit{e}\mathit{w}}}\frac{d{M}_{an}}{dH}$$\phantom{\rule{0ex}{0ex}}{k}_{new}=k\cdot {\mu}_{0}\phantom{\rule{0ex}{0ex}}{c}_{new}=\frac{c}{1-c}$ |

[3] | $\frac{dM}{dH}=\frac{\left({M}_{an}-{M}_{irr}\right)}{k-\alpha \left({M}_{an}-{M}_{irr}\right)}+c\left(\frac{d{M}_{an}}{dH}-\frac{d{M}_{irr}}{dH}\right)$ | Stays the same | $\frac{dM}{dH}=\frac{\left({M}_{an}-{M}_{irr}\right)}{k-\alpha \left({M}_{an}-{M}_{irr}\right)}+c\left(\frac{d{M}_{an}}{dH}-\frac{d{M}_{irr}}{dH}\right)$ |

[4] | $\frac{dM}{dH}=\left(1-c\right)\frac{\left({M}_{an}-{M}_{irr}\right)}{k-\alpha \left({M}_{an}-{M}_{irr}\right)}+c\frac{d{M}_{an}}{dH}$ | Stays the same | $\frac{dM}{dH}=\left(1-c\right)\frac{\left({M}_{an}-{M}_{irr}\right)}{k-\alpha \left({M}_{an}-{M}_{irr}\right)}+c\frac{d{M}_{an}}{dH}$ |

[5] | $\frac{dM}{dH}=\frac{\left({M}_{an}-{M}_{irr}\right)}{k\delta -\alpha \left({M}_{an}-M\right)}+c\left(\frac{d{M}_{an}}{dH}-\frac{\left({M}_{an}-{M}_{irr}\right)}{k\delta -\alpha \left({M}_{an}-M\right)}\right)$ | Changed according to our model derivation check (in original paper, possible typos) | $\frac{dM}{dH}=\frac{\left({M}_{an}-{M}_{irr}\right)}{k\delta -\alpha \left({M}_{an}-{\mathit{M}}_{\mathit{i}\mathit{r}\mathit{r}}\right)}+c\left(\frac{d{M}_{an}}{dH}-\frac{\left({M}_{an}-{M}_{irr}\right)}{k\delta -\alpha \left({M}_{an}-{\mathit{M}}_{\mathit{i}\mathit{r}\mathit{r}}\right)}\right)$ |

[7] | $\frac{dM}{dH}=\frac{{M}_{an}-M}{k\delta -\alpha \left({M}_{an}-M\right)}+\frac{ck\delta}{k\delta -\alpha \left({M}_{an}-M\right)}\frac{d{M}_{an}}{dH}$ | Stays the same | $\frac{dM}{dH}=\frac{{M}_{an}-M}{k\delta -\alpha \left({M}_{an}-M\right)}+\frac{ck\delta}{k\delta -\alpha \left({M}_{an}-M\right)}\frac{d{M}_{an}}{dH}$ |

Case 3 | Case 4 | |
---|---|---|

a | 1100 A/m | 1100 A/m |

M_{s} | 1.6 × 10^{6} A/m | 1.6 × 10^{6} A/m |

α | 1.6 × 10^{−3} | 1.6 × 10^{−3} |

c; c_{new} | 0; 0 | 0.9; 9 |

k; k_{new} | 400 A/m; 502.655 × 10^{−6} Tm | 400 A/m; 502.655 × 10^{−6} Tm |

**Table 11.**Five fundamental Equations (18), (2), (5), (12), (21) and the sixth Equation (22) representing the final result of the proposed harmonized J–A model for further research works.

Equation (18) | ${M}_{an}\left({H}_{e}\right)=M\left({H}_{e}\right)+\delta k\left(\frac{dM}{d{H}_{e}}\right)$ | Magnetization process |

Equation (2) | ${H}_{e}=H+\alpha M$ | Effective magnetic field strength |

Equation (5) | ${M}_{an}={M}_{s}\left(\mathrm{coth}\left(\frac{{H}_{e}}{a}\right)-\frac{a}{{H}_{e}}\right)$ | Anhysteretic magnetization (for example, a Langevin function) |

Equation (12) | $M={M}_{rev}+{M}_{irr}$ | Total magnetization |

Equation (21) | ${M}_{rev}=c\left({M}_{an}-{M}_{irr}\right)$ | Reversible component of magnetization |

Equation (22) | $\frac{dM}{dH}=\frac{\left({M}_{an}-{M}_{irr}\right)}{\delta k-\alpha \left({M}_{an}-{M}_{irr}\right)}+c\left(\frac{dMan}{dH}-\frac{d{M}_{irr}}{dH}\right)$ | Final form of hysteresis model |

**Table 12.**Sets of model variable values for the harmonized model B–H loop for two measured samples (M270-35A, FeCo alloy).

M270-35A | FeCo Alloy | |
---|---|---|

a | 21.16 A/m | 502.6 A/m |

M_{s} | 1.23 × 10^{6} A/m | 1.960 × 10^{6} A/m |

α | 5.71 × 10^{−5} | 7.241 × 10^{−4} |

c | 1 × 10^{−5} | 0.528 |

k | 29.60 A/m | 251 A/m |

**Table 13.**Hysteresis losses (w

_{hyst}) in one magnetizing period for two measured samples (M270-35A and FeCo alloy).

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**MDPI and ACS Style**

Rupnik, U.; Alić, A.; Miljavec, D.
Harmonization and Validation of Jiles–Atherton Static Hysteresis Models. *Energies* **2022**, *15*, 6760.
https://doi.org/10.3390/en15186760

**AMA Style**

Rupnik U, Alić A, Miljavec D.
Harmonization and Validation of Jiles–Atherton Static Hysteresis Models. *Energies*. 2022; 15(18):6760.
https://doi.org/10.3390/en15186760

**Chicago/Turabian Style**

Rupnik, Urban, Alen Alić, and Damijan Miljavec.
2022. "Harmonization and Validation of Jiles–Atherton Static Hysteresis Models" *Energies* 15, no. 18: 6760.
https://doi.org/10.3390/en15186760