Magnetic hysteresis model using chained plays

Abstract

A magnetic vector hysteresis model is presented using play hysterons as building blocks. The main difference compared to earlier play based magnetic hysteresis models is that instead of forming a superposition of plays with different coercivities, this work uses a set of identical plays which are connected in a chain so that the output of one play acts as input to the next. Like models using a superposition, this yields far more complex minor loop behavior than a single play is capable of. Comparisons are made with measurements on a nonoriented electric steel for BH curves and alternating and rotational losses, showing satisfactory agreement. The approach is found to have both advantages and disadvantages compared to models using superpositions of plays. The problem of determining model parameters becomes simpler and the model can be reversed so either magnetic field or magnetization may be used as known variable. On the other hand, there is no apparent connection to underlying physical processes.

Keywords

Magnetic hysteresis vector hysteresis electrical steel

1. Introduction

Computational models for magnetic hysteresis are relevant for various applications including predicting power losses in soft magnetic cores and calculating demagnetization phenomena in hard magnets. Many hysteresis models exist in the literature, e.g. models using bistable particles as building blocks such as the Preisach model [1], differential equation based models such as the Jiles–Atherton model (JAM) [2] and the Prandtl–Ishlinskii approach of using play and/or stop operators [3]. The last approach which has its roots in Coulomb’s classical hysteresis model for dry friction is widely used in rheology and structural mechanics. It has also to a lesser extent been applied to magnetic hysteresis, often by taking a superposition of plays with different coercivities and invoking an analogy between dry friction and pinning of domain walls [4–6]. We shall make some comparisons to this model which we label the distributed dry friction model (DDFM) with the model proposed here.

1.1. Proposed model

Play hysterons, sometimes called backlashes, are simple hysteresis operators in which the major loop branches are connected by locally reversible lines and the state its fully determined by current values of input and output, such as in Fig. 1. Define P_{k, c} as such a play with major loop width 2k, linear major loop branches with slope 1, and minor loop slope c. We express it mathematically as follows. Suppose that $\begin{eqnarray}\displaystyle \boldsymbol{v}=P_{k,c}[\boldsymbol{u}] & & \displaystyle\end{eqnarray}$ (1) where u and v are generic input and output variables respectively of arbitrary vectorial dimension. Then a small incrementalc change Δ u in input leads to an incremental change Δ v in output according to the algorithm $\begin{eqnarray}\displaystyle {\Delta}\boldsymbol{v}\text{} & =\text{} & \displaystyle c{\Delta}\boldsymbol{u}+{\Delta}s(\boldsymbol{u}-\boldsymbol{v}-c{\Delta}\boldsymbol{u}),\end{eqnarray}$ (2) $\begin{eqnarray}\displaystyle {\Delta}s\text{} & =\text{} & \displaystyle 1-(\max (|k^{-1}(\boldsymbol{u}+\boldsymbol{{\Delta}}\boldsymbol{u}-\boldsymbol{v})|,1))^{-1}.\end{eqnarray}$ (3) Whenever |k⁻¹( u + Δ u − v )| <1, it holds that Δs = 0 giving Δ v = cΔ u .

Fig. 1.

Illustration of play operator P_{k, c} with k = 2, c = 0.3.

Fig. 2.

Comparison of measured and simulated BH curves for isotropic SiFe material, using spiral shaped signal and signal with minor loop excursions respectively.

Fig. 3.

Comparison of measured and simulated alternating and rotational losses for isotropic SiFe material.

A hysteresis model generating more complex minor loop behavior can be constructed by involving a superposition of plays with different widths (coercivities) [3–6]. In this work, we instead take a set of identical such plays and chain them in a sequence so that the output of one is the input to the next: $\begin{eqnarray}\displaystyle \boldsymbol{{\eta}}_{i}=P_{k,c}[\boldsymbol{{\eta}}_{i-1}],i=1,\ldots ,N & & \displaystyle\end{eqnarray}$ (4) with 𝜂₀ ≡ H . It is readily seen that the major loop of 𝜂_i with respect to H has width i × k and incremental slope cⁱ, i.e. slope of small minor loops. Furthermore, as i increases, the minor loop patterns in 𝜂_i with respect to H become increasingly complex. As a final step, 𝜂_N is used as input to an anhysteretic function to get the magnetization: $\begin{eqnarray}\displaystyle \boldsymbol{M}=\boldsymbol{M}_{an}({\eta}_{N}). & & \displaystyle\end{eqnarray}$ (5) Equations (2)–(5) form the proposed model. It uses the same building blocks of simple plays and an anhysteretic function as the DDFM but arranges them in a different manner. As in the DDFM and the J–A model, the anisotropic case is handled by allowing k and c to be matrices and M _an to be anisotropic. Unlike the DDFM, this model can easily be inverted to give H from a known M by reverting the sequence of operations. For a given M , we get 𝜂_N by taking the inverse of the anhysteretic curve. Then we get $\boldsymbol{{\eta}}_{i-1}∼=∼P_{k,c}^{-1}[\boldsymbol{{\eta}}_{i}]$ where we note that the play can be inverted so if v = P_{k, c}[ u ], then u = P_{k, c⁻¹}[ v ], again using Eqs. (2), (3).

2. Model parameters

Like other magnetic hysteresis models, this one involves material parameters that must be found with the aid of experimental data. It is highly beneficial if parameters can be determined from measurements that are widely available or can be received from a material supplier since performing custom measurements can be a daunting task. Here, we need N, k, c, and the anhysteretic function M _an( H ) (or its inverse). The number of plays N can be chosen as a suitable compromise between a large N giving smoother minor loop shapes and a small N giving faster calculations. Along the ascending and descending branches of the saturation loop along a given axis, 𝜂_N = H ∓ K where we define K ≡ N × k. The loop width 2K is inherited by the M vs H relation, meaning K is equal to the coercivity. This also implies that along the ascending branch of the saturation loop, M = M_an(H − K). Thus both M_an(H) and K can be trivially found if the saturation curve is available. Then define C = c^N which controls the incremental susceptibility and is also proportional to the initial permeability which is sometimes provided by suppliers. One may also for instance adjust C to get the best possible match with loss as function of amplitude. In general, a smaller C makes the loss as a function of magnetization amplitude closer to linear. Numerically, finding the C value that gives the best match with measurements of choice is a simple task if K and M_an(H) have already been determined as above since it leaves only a single variable involved.

In the anisotropic case, K and C may be matrices and their entries can be found from measurements along the principal axes and using the same methods as in the scalar or isotropic case. Finding an anisotropic anhysteretic magnetization function is a challenge not unique to the model presented here. To avoid an excessive number of measurements, approaches such as the multiscale model [7] could be useful to address this.

2.1. Generalization: Variable K

So far, K has been assumed to be a constant scalar or matrix, giving a constant width in the saturation loop. For some materials, like soft ferrites, this can be a good approximation but for others such as grain oriented electrical steel, the width can vary significantly with the magnetization. This effect can easily be incorporated in the model by simply allowing K to be a single valued function of the current value of M. In other regards, the algorithm is completely unchanged. The function K (M) is trivially determined from a measured major loop as the half width at different values of M. The anhysteretic function is then equally easy to find as M = M_an(H − K (M)) along the ascending branch of the saturation loop. The model will then trivially and exactly reproduce the saturation loop.

3. Results and discussion

Comparisons with measurements for an unoriented silicon iron steel material are shown in Figs 2–3. A measured saturation loop was used to determine K (M) and the inverse of M_an(H) and M was used as input variable. Figure 2a shows (B, H) curves for an alternating field with decreasing amplitude, generating high order reversal curves, which are generally well reproduced. Figure 2b shows a field with numerous small minor loop excursions. As in measurements and the DDFM, minor loops are closed. Figure 3 shows comparisons for alternating and rotational losses.

Compared to the DDFM, the presented model has the benefits of simpler determination of parameters, using either H or M as input variable, and more easily representing materials with a large squareness. However, this model has no apparent link to underlying physical phenomena.

Footnotes

Acknowledgements

This work was supported by Sweden’s Innovation Agency.

References

Preisach

, Über die magnetische Nachwirkung, Zeitschrift für Physik94 (1935), 277.

Jiles

D.C.

and Atherton

, Theory of ferromagnetic hysteresis, Journal of Magnetism and Magnetic Materials61 (1986), 48–60.

Visintin

, Differential Models of Hysteresis, Springer, 2013.

Bergqvist

, Magnetic vector hysteresis model with dry friction-like pinning, Physica B: Condensed Matter233(4) (1997), 342–347.

Henrotte

and Hameyer

, A dynamical vector hysteresis model based on an energy approach, IEEE Transactions on Magnetics42(4) (2006), 899–902.

Prigozhin

Sokolovsky

Barrett

J.W.

and Zirka

S.E.

, On the energy-based variational model for vector magnetic hysteresis, IEEE Transactions on Magnetics52(12) (2016), 1–11.

Daniel

Hubert

Buiron

and Billardon

, Reversible magneto-elastic behavior: A multiscale approach, Journal of the Mechanics and Physics of Solids56(3) (2008), 1018–1042.