# Implementation of Energy based Hysteresis Model with Harmonic Balance method

Fabien Sixdenier, Riccardo Scorretti

<figure><img src="image_1.png" /></figure>

## To cite this version:

Fabien Sixdenier, Riccardo Scorretti. Implementation of Energy based Hysteresis Model with Harmonic Balance method. COMPUMAG 2025, Jun 2025, Naples, Italy. hal-05131944

HAL Id: hal-05131944

https://hal.science/hal-05131944v1

Submitted on 26 Jun 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Implementation of Energy based Hysteresis Model with Harmonic Balance method

Fabien Sixdenier¹, Riccardo Scorretti²

¹Universite Claude Bernard Lyon 1, INSA Lyon, ECLyon, CNRS, Ampère, F-69100, Villeurbanne, France

²University of Perugia, Via G. Duranti 96, 06125 Perugia, Italy

Circuit type simulation softwares (like SPICE) are routinely used in power electronics systems to analyze a large variety of topologies and parameter combinations (duty cycle, frequency, input/output voltages, ...). The magnetic components are usually considered linear in these simulations. This article discusses the feasibility of implementing the Energy Based Hysteresis Model (EBHM) into a harmonic balance simulation in order to obtain efficiently, the steady response of a power electronic topology including the saturation and hysteresis effect of the magnetic component.

*Index Terms*—magnetic hysteresis, saturation magnetization, Harmonic analysis

## I. INTRODUCTION

MAGNETIC hysteresis is a fundamental phenomenon in magnetic materials, characterized by a dependence of magnetization on the history of the applied magnetic field. Accurate modeling of this behavior is critical for numerous applications, including power electronics. A number of commercial circuit-type simulation packages offer hysteresis model capabilities for simulation purposes. The Jiles Atherthon (JA) model [1] is implemented in PSPICE® for instance, and the Preisach model (PM) [2] in SABER®. The Energy-Based (EB) hysteresis model [3], on the other hand, has proven to be very accurate if the number of cells (parameters) is chosen large enough. Moreover robust identification protocols [4], [5] exist for that model that have proven their efficiency [6]. The implementation of the EB model into a circuit type simulation software for transient simulations has been made successfully in [7].

However for fast predesign purposes this approach needs several simulations and each simulation needs to reach the steady state conditions. That's where the Harmonic-Balance technique is very interesting because it gives only the steady state answer even for complex problems such as a motor modelled with a 2DFE approach with motion and circuit coupling [8]. However including hysteresis is more complex, because one key difficulty arises from the inherently memory-dependent nature of hysteresis. Thus, while harmonic balance offers a powerful framework for analyzing periodic phenomena, the modeling of magnetic hysteresis within this paradigm continues to present unresolved theoretical and practical challenges. Some potential methodologies to overcome these challenges are for example to use smoothing functions before treating hysteresis [9] or to separate the *H* field into a reversible and an irreversible part [10]. However, in this abstract, only saturation effects, taken into account with the HB technique, of a coil into a typical power electronics applications will be presented.

## II. METHODS

### A. DC/DC "buck" converter output

Our approach will be tested on a classical DC/DC buck converter output composed of an LC filter and a load resistor presented in Fig.1. The whole circuit is supplied by $E(t)$ a voltage source of square waves $(+V_{max}, 0)$ where the frequency $f$, voltage level $V_{max}$ and Duty cycle $DT$, can be changed. This allows to remove the active components of a full DC/DC converter to simplify the study.

<figure><img src="image_3.png" /><figcaption>Fig. 1. Typical output of a DC/DC buck converter</figcaption></figure>

### B. HB Modeling

The temporal ordinary differential system of equation of the previous circuit is given by (1)

$$ \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & C & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} \frac{dv_a}{dt} \\ \frac{dv_b}{dt} \\ \frac{di_e}{dt} \\ \frac{dv_e}{dt} \end{bmatrix} + \begin{bmatrix} 0 & 0 & 1 & -\frac{1}{L} \\ 0 & 1/R & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 1 & -1 & 0 & 0 \end{bmatrix} \cdot \begin{bmatrix} v_a \\ v_b \\ i_e \\ \psi \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ E(t) \\ 0 \end{bmatrix} \quad (1) $$

All the unknowns are put in a vector $\mathbf{X}(t)$ (2). The equation (1) is then rewritten as (3) with $n_t$, the number of time steps in one period $T$

$$ \mathbf{X}(t) = \begin{bmatrix} v_a(t_1), \dots, v_a(t_{n_t}), v_b(t_1), \dots, v_b(t_{n_t}) \\ i_L(t_1), i_L(t_{n_t}), \psi(t_1), \dots, \psi(t_{n_t}) \end{bmatrix} \quad (2) $$

$$ [\mathbf{C}_t] \cdot \left[ \frac{d\mathbf{X}}{dt} \right] + [\mathbf{G}_t] \cdot [\mathbf{X}] = [\mathbf{U}(t)] \quad (3) $$
With $C_t$ representing the action of differential variables and $G_t$ representing links between unknowns (Kirchoff laws, flux vs current link,...). Both $C_t$ and $G_t$ are composed of block-diagonal submatrices of size $n_t$. $U(t)$ is the source term vector. Then (3) is transformed into (4).

$$ (\Omega \cdot C_f + G_f) \cdot X_f = U_f \quad (4) $$

$\Omega$ is a block diagonal matrix formed of several block diagonal submatrices representing the FFT of the differential operator $\Omega = diag\{j\omega_0[\dots, -2I, -I, 0, I, 2I, \dots]\}$ ($\omega_0$ is the fundamental frequency). By introducing $\Gamma$, the discrete Fourier operator and $\Gamma^{-1}$ the inverse discrete Fourier operator, the temporal vectors and matrices are transformed into their equivalent in frequency domain such as (6).

$$ X_f = \Gamma \cdot X_t \quad (5) $$

$$ G_f = \Gamma \cdot G_t \cdot \Gamma^{-1} \quad (6) $$

The matrix of the linear system of (4) is the Jacobian of the Harmonic Balance $J_f$ system and has to be reevaluated in the case of non-linear simulations. To do so, inverse Fourier transforms are performed on $X_f$ to obtain $X_t$ and evaluate the temporal non linearities at each time step. $C_t$ and $G_t$ are then reevaluated and retransformed into $C_f$ and $G_f$ to update $J_f$. This means that $L(t)$ is not constant during a period and has to be updated. To take into account the saturation we simulate the magnetization of the core with a simple Langevin function (7) with a saturation magnetization of $Ms = 310983$ A/m and $h_0 = 10.8903$ A/m, the shape parameter. These last parameters approximate the anhysteretic magnetization of a N30 ferrite material. Regarding the shape of the coil, the assumptions are that it is made of a toroidal core with a constant cross section $A_e$ and magnetic path length $\ell_m$ and that there is no leakage flux.

$$ M(H) = M_s \left( \coth \left( \frac{H}{h_0} \right) - \frac{h_0}{H} \right) \quad (7) $$

### III. RESULTS

All results obtained by our approach were compared to LT-spice simulations once the steady state has been reached (400 periods). The linear case was simulated with the conditions resumed in the table III. The non-linear case also, except that $L$ was no more constant and the same model (Langevin function) has been used in LTspice and the HB technique.

<table><thead><tr><th>V<sub>max</sub> (V)</th><th>f (kHz)</th><th>DT</th><th>L (μH)</th><th>C(μ F)</th><th>R(Ω)</th></tr></thead><tbody><tr><td>10</td><td>10</td><td>0.5</td><td>557</td><td>100</td><td>10</td></tr></tbody></table>

Fig. 2, shows the voltages at points a and b and the current $i_L$ versus time. First, The voltage at point b and the current $i_L$ are very well retrieved for a limited number of harmonics. One can see the influence of this limited number of harmonics on the voltage waveform at point a for our approach. This last voltage can better match with SPICE measurements by increasing the number of harmonics $n_h$. In the non-linear case, the LTspice crashed systematically for this case because of the

too severe peak of current during the transient state. However the HB technique shows the right behaviour. Indeed because of the saturation of the core the current in the coil has a peak much higher than in the linear case. This current also induces a voltage ripple at point b higher than in the linear case. These very well known phenomena occur when the coil saturates and is well captured by the simulation.

<figure><img src="image_4.png" /><figcaption>Fig. 2. voltage waveforms at the point a and b and current flowing into the inductor L for a number of harmonics $n_h = 16$</figcaption></figure>

### IV. CONCLUSION

In the extended version, more details about the algorithm will be given as well as how the hysteresis will be treated. Additionally, some metrics about the calculation cost and accuracy will be given especially regarding the number of harmonics.

### REFERENCES

[1] D. Jiles and D. Atherton, "Ferromagnetic hysteresis," *IEEE Transactions on Magnetics*, vol. 19, no. 5, pp. 2183-2185, 1983.

[2] F. Preisach, "Über die magnetische nachwirkung," *Zeitschrift für Physik*, vol. 94, no. 1, pp. 277-302, 1935.

[3] F. Henrotte, A. Nicolet, and K. Hameyer, "An energy-based vector hysteresis model for ferromagnetic materials," *COMPEL*, vol. 25, no. 1, pp. 71-80, 2006.

[4] K. Jacques, S. Steentjes, F. Henrotte, C. Geuzaine, and K. Hameyer, "Representation of microstructural features and magnetic anisotropy of electrical steels in an energy-based vector hysteresis model," *AIP Advances*, vol. 8, no. 4, p. 047602, 2018.

[5] R. Scorretti and F. Sixdenier, "An analytical formula to identify the parameters of the energy-based hysteresis model," *Journal of Magnetism and Magnetic Materials*, p. 168748, 2021.

[6] M. R. Longhitano, F. Sixdenier, R. Scorretti, C. Geuzaine, F. Henrotte, and L. Krähenbühl, "Comparison of identification protocols of a static hysteresis model," in *Compumag 2019*, Paris, France, Jul. 2019. [Online]. Available: https://hal.science/hal-02063610

[7] F. Sixdenier, R. Scorretti, N. Davister, C. Geuzaine, and F. Henrotte, "Energetic based hysteresis model implementation in LTspice," in *Compumag 2022*, Cancun, Mexico, Jan. 2022. [Online]. Available: https://hal.science/hal-03540757

[8] E. Scolaro, L. Alberti, R. V. Sabariego, and J. Gyselinck, "Harmonic balance applied to a 2d nonlinear finite-element magnetic model with motion and circuit coupling," *International Journal of Numerical Modelling: Electronic Networks, Devices and Fields*, vol. 37, no. 4, p. e3275, 2024. [Online]. Available: https://onlinelibrary.wiley.com/doi/abs/10.1002/jnm.3275

[9] L. P. Miguel, R. de Oliveira Teloli, and S. da Silva, "Some practical regards on the application of the harmonic balance method for hysteresis models," *Mechanical Systems and Signal Processing*, vol. 143, p. 106842, 2020. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S0888327020302284

[10] W. Cheng and Y. Saito, "Modeling and analysis of hysteresis by harmonic balance method," *Journal of Applied Physics*, vol. 117, no. 17, p. 17D143, 04 2015. [Online]. Available: https://doi.org/10.1063/1.4918337