# Viscoelastic modeling of magnetic losses in a nanocrystalline core using fractional derivative operators

Benjamin Ducharne, Shengze Gao, Yanhui Gao, Xiaojun Zhao

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Benjamin Ducharne, Shengze Gao, Yanhui Gao, Xiaojun Zhao. Viscoelastic modeling of magnetic losses in a nanocrystalline core using fractional derivative operators. Nonlinear Dynamics, 2025, 10.1007/s11071-025-11226-9. hal-05146607

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# Viscoelastic modeling of magnetic losses in a nanocrystalline core using fractional derivative operators

Benjamin Ducharne · Shengze Gao · Yanhui Gao · Xiaojun Zhao

Received: 29 January 2025 / Accepted: 6 April 2025

© The Author(s) 2025

**Abstract** Nanocrystalline ferromagnetic materials have emerged as advanced solutions for improving magnetic performance, offering reduced core losses and high permeability. However, despite their superior properties, magnetic losses remain critical, particularly under high-frequency conditions. Existing simulation methods, including empirical models, time-dependent hysteresis models, and space-discretized approaches, often fail to accurately capture their complex magnetic properties over wide frequency ranges and amplitudes. This study evaluates the use of fractional derivative operators as innovative tools to model magnetic losses in nanocrystalline ferromagnetic cores. Four simulation approaches are analyzed: an analytical expression of the magnetic losses, two

time-dependent hysteresis models (using first-order and fractional differential equations), and a space-discretized method coupling Maxwell's equations with a material law including a fractional derivative operator. The analytical method provides simplicity and reliable results for total losses but cannot capture temporal or spatial distributions. The lumped hysteresis models, particularly the fractional-order variant, offer improved accuracy by accounting for dynamic effects and frequency dependencies. The space-discretized method is the most robust, achieving the highest precision and providing detailed insights into the local distribution and contributions of magnetic losses. Key findings indicate that fractional derivative operators enable highly accurate simulations of magnetic losses with relatively low derivative orders, reflecting a lower viscous-to-elastic loss ratio in nanocrystalline materials. Additionally, the substantial excess losses observed are attributed to the unique microstructure of the nanocrystalline ribbon, which induces distinctive domain wall motions.

**Keywords** Fractional differential equation · Time-dependent hysteresis models · Space-discretized methods · Domain wall motion · High-frequency applications

<footnote>B. Ducharne (✉)
LGEF, INSA-Lyon, UR682, University Lyon,
69621 Villeurbanne, France
e-mail: benjamin.ducharne@insa-lyon.fr</footnote>

<footnote>B. Ducharne
ELyTMAx IRL3757, CNRS, Univ Lyon, INSA Lyon,
Centrale Lyon, Université Claude Bernard Lyon 1,
Tohoku University, Sendai, Japan</footnote>

<footnote>S. Gao · X. Zhao
Department of Electrical Engineering, North China
Electric Power University, Baoding 071003, China</footnote>

<footnote>Y. Gao
Faculty of Science and Technology, Oita University,
Oita 870-1192, Japan</footnote>
# 1 Introduction

Nanocrystalline ferromagnetic materials emerged from the need to improve the magnetic performance of amorphous alloys, which were first developed in the 1970s. By introducing controlled nanometer-scale crystallization into amorphous precursors, researchers in the late 1980s achieved materials with superior soft magnetic properties, such as reduced core losses and high permeability [1, 2]. The breakthrough came with the development of Fe-based alloys containing small amounts of elements like Nb and Cu, which promote the formation of nanocrystalline grains during annealing. The precise control of the annealing process, particularly temperature and cooling rate, plays a crucial role in determining the final microstructure. Adjustments in these parameters influence the grain size, phase composition, and residual amorphous fraction, all of which directly affect the material's soft magnetic behavior. These materials combine the advantages of both amorphous and crystalline structures. The amorphous matrix suppresses grain boundary effects, reducing eddy current losses and magnetic hysteresis, which are critical for high-frequency applications [3-7]. Meanwhile, the embedded nanocrystalline grains provide high saturation magnetization and improved permeability, enhancing the material's magnetic responsiveness [8].

Beyond their fundamental advantages, nanocrystalline materials are increasingly used in advanced technologies that require efficient, compact, and high-performance magnetic components. In particular, they play a key role in renewable energy systems and electric vehicles, where their low core losses and high permeability contribute to enhanced energy efficiency and miniaturization of power conversion devices. Nanocrystalline transformers, inductors, and chokes are now widely integrated into high-frequency power electronics, improving the performance of inverters, converters, and grid-interface systems.

Currently, nanocrystalline ferromagnetic materials dominate the market for high-frequency transformers and inductors (> 5 kHz), with an approximate 70% adoption rate, outperforming Grain-Oriented (GO) and Non-Oriented (NO) electrical steels in these conditions [9, 10]. Their application in power transformers and electric motors is also growing, particularly in electric vehicles, where high-efficiency magnetic cores help reduce energy losses and enhance

performance. Market reports indicate an increasing demand for nanocrystalline materials, driven by the expansion of electrification and the transition to renewable energy sources.

Although reduced compared to classical electrical steel grades, magnetic loss remains a critical issue in nanocrystalline ferromagnetic materials, as it directly impacts their efficiency and performance [12, 13]. While nanocrystalline materials excel at reducing hysteresis and eddy current losses due to their fine grain structure and high electrical resistivity, challenges remain in mitigating anomalous losses, particularly at high frequencies [14]. Additionally, core loss increases with rising operating frequencies and flux densities, which are common in compact, high-power designs. Therefore, developing a robust simulation method to describe magnetic loss in nanocrystalline ferromagnetic materials is essential [12, 15]. Accurately modeling these losses not only advances material science but also has significant implications for optimizing the next generation of energy-efficient electrical devices. Accurate simulations enable researchers and engineers to predict losses under varying operating conditions, aiding in designing more efficient transformers and inductors. Furthermore, simulation tools can guide material development by identifying key parameters that minimize losses while maintaining thermal and mechanical stability. By reducing the reliance on extensive experimental testing, simulation methods accelerate innovation and ensure these advanced materials achieve their full potential [16].

Classical methods used to describe magnetic loss in nanocrystalline materials primarily rely on empirical and semi-empirical models. Bertotti's statistical theory of losses is a widely adopted approach, decomposing total magnetic loss into hysteresis, eddy current, and anomalous components [17, 18]. Hysteresis loss is modeled based on domain wall dynamics, while eddy current loss is calculated using Maxwell's equations and material resistivity. The anomalous loss, unique to materials with fine microstructures like nanocrystalline alloys, is often addressed through phenomenological extensions of classical models [12]. Other methods include Steinmetz equations [19, 20], which provide frequency-dependent loss estimations but usually lack precision at high frequencies. While these approaches offer insights into loss mechanisms, they are typically
limited in accounting for the intricate interplay of
structural and magnetic properties unique to nanocrys-
talline materials, highlighting the need for more
advanced, physics-based modeling techniques. In
addition to classical methods, time-dependent hys-
teresis models have emerged as powerful tools for
describing magnetic losses in nanocrystalline materi-
als, offering a more comprehensive perspective by
simultaneously capturing the evolution of the mag-
netic field and flux density. These models, such as the
Preisach model [21] and Jiles-Atherton model
[22, 23], incorporate the dynamic behavior of domain
walls and magnetic relaxation effects, allowing for a
detailed representation of hysteresis phenomena under
varying operating conditions [24, 25]. By integrating
time-dependent terms, these models account for
frequency-dependent effects and can predict not only
energy losses but also the transient behavior of
magnetic properties. Advanced extensions of these
models, incorporating microstructural parameters and
thermal effects, provide further accuracy [24].

Unfortunately, all these simulation methods face
limitations in accurately capturing losses across a
large frequency bandwidth and extensive amplitude
range [26, 27]. They struggle to simultaneously
account for complex loss due to the dynamic effects
at high frequencies, leading to discrepancies between
predictions and experimental results. Additionally, the
computational complexity of modeling fine-scale
microstructural effects and their interaction with
magnetic fields poses challenges for achieving both
precision and efficiency in simulations.

Experimental observations of nanocrystalline core
loss variations revealed deviations from classical
viscous loss contributions. The measurements indi-
cated that conventional loss models fail to accurately
describe the material's dynamic magnetic behavior.
The dynamic responses exhibit viscoelastic character-
istics that extend beyond purely damping effects.
Therefore, alternative simulation methods suited for
viscoelastic-like behaviors are necessary to accurately
model magnetic losses over a wide frequency range.

Recent advancements in simulation methods uti-
lizing fractional derivative operators have shown
promising potential for simulating viscoelastic phe-
nomena and overcoming the limitations of classical
approaches when applied to typical ferromagnetic
materials used in magnetic energy conversion, such as
various grades of electrical steel [28–31], iron-cobalt

alloys, soft magnetic composites, and low-carbon steel
[31]. By providing an alternative framework for
modeling frequency dependency and introducing an
additional degree of freedom compared to integer-
order derivatives, fractional derivative operators
enable highly accurate simulations of magnetic core
losses across a broad frequency spectrum.

However, these methods have not yet been applied
to nanocrystalline materials. This manuscript evalu-
ates a broad spectrum of simulation techniques,
ranging from empirical models to time-dependent
approaches, to investigate the applicability of frac-
tional derivative operators for simulating power losses
in nanocrystalline magnetic cores. Each simulation
method is described in detail, with results and
conclusions supported by comprehensive comparisons
between simulations and experimental measurements.

## 2 Simulation methods

Due to the specific geometrical properties of the
ferromagnetic nanocrystalline ribbons, a 1D simula-
tion approach (scalar model) is sufficient to represent
the magnetic losses accurately. The ribbon's thin and
elongated structure allows the magnetic flux to align
along its length predominantly, minimizing multidi-
mensional effects and justifying the use of a simplified
scalar method [32]. In this section, which describes
different simulation methods, we will first address the
static, frequency-independent contribution to the
magnetic loss, as it establishes the baseline behavior
of the magnetic core. Following this, we will discuss
the frequency-dependent contributions using both
analytical methods, such as Steinmetz and Bertotti's
Statistical Theory of Losses (STL, [17, 19]), which
express losses as functions of flux density and
frequency, and numerical approaches that solve the
loss behavior directly in the time domain. In each case,
the use of fractional derivative operators will be
considered, and their contributions to improving
accuracy will be discussed.

2.1 Quasi-static-frequency independent-
contribution

The frequency-independent hysteresis contribution
(also called quasi-static contribution) can be accu-
rately simulated using classic quasi-static hysteresis
models like the Preisach or Jiles-Atherton models [21–24]. These approaches effectively capture the reversible and irreversible magnetization processes at low frequencies. However, for materials with very low coercivity, such as the nanocrystalline ribbons studied here (< 1 A·m⁻¹, in the very low-frequency range), the quasi-static hysteresis losses are minimal. This shallow loss profile makes it feasible to approximate the anhysteretic contribution using a sigmoid function (Langevin-Weiss model [33], etc.), which provides a smooth and computationally efficient representation of the magnetization curve. This approach maintains a high accuracy level while simplifying the material’s magnetic behavior description.

## 2.2 Dynamic-frequency dependent-contribution

### 2.2.1 Time fractional derivative: physical interpretation in the context of the ferromagnetic core power loss

Fractional calculus, introduced in the seventeenth century by mathematicians such as Leibniz and de l'Hôpital, extends the concepts of differentiation and integration to non-integer orders, creating a distinctive branch of mathematical analysis [34]. Unlike traditional entire orders derivative approaches, the inherently non-local characteristic of fractional derivative encompasses the whole simulated time range rather than just specific instances [35, 36]. By accounting for extended temporal effects, this property perfectly suits materials with history-dependent behaviors, such as soft ferromagnetic materials.

The choice of fractional derivative definition depends on the application’s specific needs. Trigonometric functions, due to their analyticity, allow for analytical solutions under fractional differentiation, satisfying both the Laplace and Cauchy-Riemann equations [37]:

$$ \frac{d^v}{dt^v} e^{i\omega t} = (i\omega)^v e^{i\omega t} \qquad (1) $$

$$ (i\omega)^v = \omega^v e^{iv\pi/2} \qquad (2) $$

where $v\pi/2$ arises from the argument of $i$ in polar form. The fractional order $v$ captures the balance between viscous and elastic behavior, reflecting the viscoelastic characteristics of the magnetic response. This approach is analogous to viscoelastic modeling in

mechanics, where systems of springs (representing elastic behavior) and dashpots (representing viscous behavior) simulate materials exhibiting combined elastic and viscous properties [38–40]. Using a fractional derivative of order $v$, where $0 < v < 1$, to model a viscoelastic behavior is grounded in the notion that the actual response lies between that of a 0th and 1st-order derivative, somewhere between an elastic solid and a viscous fluid. In this study, the Caputo derivative was chosen due to its suitability for numerical implementation and its adherence to causality principles, which are essential in dynamic hysteresis modeling [41]. Unlike the Riemann–Liouville definition, the Caputo derivative inherently incorporates past states in a physically meaningful way by allowing the use of classical initial conditions, making it computationally convenient for time-dependent simulations. The Caputo derivative is defined as:

$$ D_c^v f(t) = \frac{1}{\Gamma(v-m)} \int_0^t \frac{f^{(m)}(\tau)}{(t-\tau)^{v-m+1}} d\tau, \quad (m-1 < v < m) \qquad (3) $$

where $\Gamma(\cdot)$ is the Gamma function, and $m$ is the smallest integer greater than or equal to $v$ [41]. For numerical implementation, the Caputo derivative is discretized using the Grünwald-Letnikov scheme:

$$ \frac{d^v}{dt^v} f_n \approx \frac{1}{\Delta t^v} \sum_{k=0}^{n} g_k f_{n-k} \qquad (4) $$

where $\Delta t$ is the time step, $g_k$ are the Grünwald weights given by $g_k = (-1)^k \binom{v}{k}$, and $\binom{v}{k}$ is the generalized binomial coefficient. This method is known to have a convergence order of $O(\Delta t)$ for uniform grids, as discussed in [42].

### 2.2.2 Analytical expression of the ferromagnetic core power loss using fractional derivative

Under alternating magnetization conditions, the total magnetic energy losses, $W_{alt}$, in a nanocrystalline ribbon can be decomposed into two contributions:

$$ W_{alt} = W_{statalt} + W_{dynalt} \qquad (5) $$

$W_{statalt}$ represents the frequency-independent quasi-static energy losses (discussed in Sect. 2.2). $W_{statalt}$ is
classically estimated by calculating the area of the hysteresis loop $B_a(H_{surf})$, where $B_a$ is the flux density averaged across the specimen cross-section, and $H_{surfstat}$ is the low-frequency excitation field measured at $f_{stat} = 1/T_{stat}$. This contribution, often referred to as hysteresis energy losses, is calculated as:

$$W_{statalt} = \int_{0}^{T_{stat}} \frac{dB_a}{dt} \cdot H_{surfstat} dt \quad (6)$$

This contribution approaches zero for nanocrystalline ribbons and can reasonably be assumed negligible.

$W_{dynalt}$ is the dynamic energy losses (frequency-dependent). $W_{dynalt}$ reflects the material's limitations in responding to rapid magnetic changes. This includes both excess energy losses related to magnetic domain reorganization and classical eddy current energy losses resulting from circulating currents induced by changing magnetic fields. In Bertotti's Statistical Theory of Losses (STL, [16, 17]), these frequency-dependent contributions are calculated separately; however, using fractional derivatives, a unified expression provides reliable predictions. For a given $B_a$, the dynamic excitation field $H_{surfdyn}$ is expressed as the sum of $H_{surfstat}$ and a dynamic term $H_{dyn}$, where $H_{dyn}$ is the product of a constant $\rho$ (dependent on material geometry and properties and of unit A·V⁻¹·m²·s⁻¹) and the v-th order time-fractional derivative of $B_a$:

$$H_{surfdyn} = H_{surfstat} + H_{dyn} \quad (7)$$

$$H_{dyn} = \rho \cdot \frac{d^v B_a}{dt^v} \quad (8)$$

The dynamic energy losses contribution $P_{dyn alt}$ is then given by:

$$W_{dynalt} = \int_{0}^{T} \frac{dB_a}{dt} \cdot H_{dyn} dt = \int_{0}^{T} \frac{dB_a}{dt} \cdot \rho \cdot \frac{d^v B_a}{dt^v} dt \approx W_{alt} \quad (9)$$

Notably, $B_a$ is the only time-dependent variable in this equation. If $B_a$ follows a trigonometric form, as standardized in soft ferromagnetic materials characterization, we can derive an analytical expression for $W_{alt}$:

$$B_a = B_M \sin(\omega t) \quad (10)$$

$$W_{alt} \approx \rho \pi B_M \omega^v \sin\left(v \frac{\pi}{2}\right) \quad (11)$$

Additional details about this simulation method can be found elsewhere [31].

### 2.2.3 Time-dependent methods

In this subsection, dedicated to describing the time-dependent hysteresis model for simulating magnetic power loss in a nanocrystalline magnetic core, we distinguish two approaches:

* The lumped approach assumes that magnetization is homogeneous across the cross-section. The resulting simulations represent an average behavior. While this method is more straightforward and requires fewer computational resources, it fails to accurately reproduce the local behavior within the cross-section of the simulated ribbon.

* The distributed approach combines the Maxwell diffusion equation with a material law including a fractional derivative operator. Although it demands extensive computational resources, it provides detailed insights into the local time variations of magnetic quantities, including the spatial distribution of all magnetic loss contributions.

A—Lump approach using a fractional differential equation

The lump approach extends the quasi-static hysteresis contribution (e.g., Preisach model, Jiles-Atherton model) to account for frequency dependence by incorporating a fractional viscoelastic-type dynamic. A simple implementation of this approach is given by Eq. 12. It involves combining Eqs. 7 and 8. This combination results in a fractional differential equation, which can be solved in the time domain.

$$\rho \cdot \frac{d^v B_a}{dt^v} = H_{surfdyn} - H_{surfstat} \quad (12)$$

As discussed in sub-Sect. 2.1, for nanocrystalline ribbons, the quasi-static term $H_{surfstat}$ can be reasonably expressed using an anhysteretic contribution modeled by a sigmoid function, as suggested in [33]. Additional details about the lump approach can be found elsewhere [25, 43].
B—Space discretized method using a material law including a fractional derivative operator.

The space discretization method involves solving the Maxwell diffusion equation (Eq. 13) and a hysteretic dynamic material law like the one shown in Eq. 12 simultaneously.

$$ \sigma \frac{\partial B}{\partial t} = \nabla^2 H \qquad (13) $$

The specific geometry of the nanocrystalline ribbons (where the thickness d is significantly smaller than the width and length) allows the diffusion equation to be solved in one dimension (1D, Eq. 14) using finite differences while maintaining accuracy.

$$ \sigma \frac{\partial B}{\partial t} = \frac{\partial^2 H}{\partial x^2} \qquad (14) $$

For the material law to be solved simultaneously with the diffusion equation, Eq. 12 is reformulated to isolate $dB/dt$:

$$ \frac{\partial B}{\partial t} = \frac{d^{1-\nu} \left( \frac{H-H_{stat}(B)}{\rho} \right)}{dt^{1-\nu}} \qquad (15) $$

The next step involves concatenating Eqs. 14 and 15 in a single equation (Eq. 16) [44, 45]. Then, the space-dependent term (left term in Eq. 16) is solved using a finite differences discretization, and the time-dependent one (right term) is expressed with the discretized Caputo derivative's definition. The resulting equation Eq. 17 is exclusively constituted of terms equivalent to magnetic fields $H$:

$$ \frac{\partial^2 H}{\partial x^2} = \sigma \cdot \frac{d^{1-\nu} \left( \frac{H-H_{stat}(B)}{\rho} \right)}{dt^{1-\nu}} \qquad (16) $$

$$ \frac{H(x-r,t) - 2H(x,t) + H(x+r,t)}{r^2} = \sigma \cdot \frac{1}{\Gamma(\nu)} \sum_{k=0}^{n} \frac{(n+1)^\nu - n^\nu}{\Delta t^{1-\nu}} \cdot \left( \frac{H(x, t_{n-k}) - H_{stat}(B(x, t_{n-k}))}{\rho} \right) \qquad (17) $$

Since $H(x,t)$ and $B(x,t)$ are symmetric about the plane $x = d/2$, a resolution of Eq. 17 on the segment $[0, d/2]$ is enough, and a Neumann condition can be applied for the $x = d/2$ node. In this condition, $r$ the space discretization in Eq. 17 is worth $d/[2(N-1)]$, where $N$ is the number of nodes. Additional details about this simulation method can be found elsewhere

[28, 46]. Similarly, information about the convergence, convergence order and stability analysis of the Caputo definition can be found in [35, 47, 48].

## 3 Experimental conditions

### 3.1 Experimental setup description

A dedicated experimental setup was designed to measure the hysteresis cycles under various magnetic field amplitudes and frequencies. This setup operates under sinusoidal magnetic flux densities as international standards require [49]. Both the excitation and sensor coils had five turns. The system integrated a signal generator, a power amplifier, and a negative feedback mediation system. The excitation frequency range of the instrument spanned from 3 Hz to 20 kHz. Due to the power supply limitations, ideal DC excitation was not achievable; therefore, hysteresis loops measured at 3 Hz were considered as quasi-static hysteresis loops in this work. Additional information about the experimental results can be found elsewhere [50].

### 3.2 Tested specimens description

The tested specimen was a nanocrystalline ribbon provided by Junda Company (Hefei, China), with the grade designation 1K107B. This type of ribbon is designed for use in advanced common mode chokes, high-frequency power transformers, precision current transformers, magnetic amplifiers, and magnetic shielding sheets. The specimen was ring-shaped, with a density of 7180 kg·m⁻³ and an electrical conductivity of 0.769 10⁶ S·m⁻¹ [51]. The ring inner radius was 50.1 mm, the outer radius was 79.9 mm, and the ribbon thickness was 25 µm. The final magnetic properties of nanocrystalline ribbons depend on fabrication conditions such as the initial amorphous state, annealing profile, and cooling rate. A controlled annealing process ensures the formation of ultra-fine grains (~ 10–20 nm) embedded in an amorphous matrix, optimizing the balance between low core losses and high permeability. Variations in annealing temperature or duration can lead to excessive grain growth or the emergence of secondary crystalline phases, which degrade magnetic performance by increasing coercivity and hysteresis losses.
# 4 Experimental and simulation results

## 4.1 Semi-empirical method: analytical expression of the ferromagnetic core power loss using fractional derivative

The first set of simulation-to-measurement comparisons, shown in Fig. 1, highlights the accuracy of the semi-empirical method described in sub-Sect. 2.2.1 and defined by Eq. 11. Applying the Relative Euclidean Distance percentage (RED (%)) metric, as given in Eq. 18, to all Fig. 1 experimental data results in 6.8%. This result is good, considering the wide range of frequencies and flux densities tested.

$$ \text{RED}(\%) = \frac{100}{q} \cdot \sum_{i=1}^{q} \frac{\sqrt{(P_{\text{meas}_i}(B_{M_i}) - P_{\text{sim}_i}(B_{M_i}))^2}}{\sqrt{(P_{\text{meas}_i}(B_{M_i}))^2}} \quad (18) $$

The optimal values of v and ρ were determined through an optimization process that involved testing a wide range of values for each parameter and retaining the combination that yielded the lowest RED (%) value. Table 1, adapted from [31], summarizes the experimental and simulation data obtained using the simulation method described in Section II for a wide range of tested materials. The RED (%) values were calculated using both alternating and rotational data for the materials in [31], whereas only alternating data were used for the nanocrystalline specimens tested in this study. [31] demonstrated that ρ was significantly

<figure><img src="image_2.png" /><figcaption>**Fig. 1** Simulation/measurement comparisons for the alternative energy loss $W_{alt}$ on a vast range of frequency (f) and maximum flux density ($\max(B_a)$) ($v = 0.67, \rho = 0.0098$)</figcaption></figure>

linearly dependent to $\sigma d^2$, and an expression was established to quantify this dependency:

$$ \rho \propto p_1 \sigma d^2 + p_2 \quad (19) $$

with $p_1 = 0.135$ and $p_2 = 0.05$. It is noteworthy that applying Eq. 19 to the nanocrystalline specimens tested in this study yields $\rho = 0.05 \text{ A} \cdot \text{V}^{-v} \cdot \text{m}^{2v-1}$. Although this value does not exactly match the result obtained from the optimization process, it falls within the same range, a contrast to the other materials tested in [31], as shown in Table 1. This observation reinforces the universality of the Eq. 19 expression and its implication, namely, the potential to reduce the number of dynamic parameters to v. Figure 2 illustrates this by plotting the value of ρ for all the materials listed in Table 1. The bottom-right-hand-side plot provides a zoomed-in view of ρ values approaching zero. This plot clearly shows that the blue line, corresponding to Eq. 19, overestimates $p_2$. For greater accuracy, $p_2$ should be closer to 0.01.

## 4.2 Time-dependent methods

A—lump approach using a fractional differential equation

The second set of simulation-to-measurement comparisons, presented in Fig. 3, highlights the limited accuracy of the simulation method described in sub-Sect. 2.2.3—A and defined by Eq. 12 when v is set to 1. In this case, Eq. 12 simplifies to a first-order differential equation, with ρ serving as the sole degree of freedom for the simulation. In Eq. 12, the dynamic term behaves a dumping element inducing viscous losses. The optimal value of ρ was determined by minimizing the RED (%) (Eq. 18), applied to all the comparisons in Fig. 3 and throughout the complete hysteresis cycles. The optimization process resulted in a minimum RED of 23.8% for ρ = 0.0006.

Figure 4 presents the same comparisons, but with v now considered as a parameter. As a result, Eq. 12 transformed into a fractional differential equation, where the dynamic term exhibited a combined behavior of dumping and elasticity, leading to viscoelastic losses. The ratio between elastic and viscous losses depends on the value of v: for large v, the dynamic term primarily behaves as a viscous element, while for small v, it behaves like an elastic element.
<table><caption>**Table 1** Experimental and simulation data for a vast range of ferromagnetic materials [31]. In red, the results for the specimens tested in this study</caption><thead><tr><th>Material</th><th>Electrical conductivity (S·m<sup>-1</sup>)</th><th>Density (kg·m<sup>-3</sup>)</th><th>Thickness (m)</th><th>ρ (-)</th><th>ν (-)</th></tr></thead><tbody><tr><td>FeCo</td><td>2.27 × 10<sup>6</sup></td><td>8120</td><td>0.201 × 10<sup>-3</sup></td><td>0.0165</td><td>0.934</td></tr><tr><td>NOFeSi</td><td>1.92 × 10<sup>6</sup></td><td>7650</td><td>0.194 × 10<sup>-3</sup></td><td>0.0536</td><td>0.8355</td></tr><tr><td>GOFeSi</td><td>2.12 × 10<sup>6</sup></td><td>7650</td><td>0.27 × 10<sup>-3</sup></td><td>0.024</td><td>0.938</td></tr><tr><td>SMC</td><td>1.43 × 10<sup>4</sup></td><td>7370</td><td>22 × 10<sup>-3</sup></td><td>0.973</td><td>0.95</td></tr><tr><td>LC steel</td><td>1.99 × 10<sup>6</sup></td><td>7850</td><td>0.64 × 10<sup>-3</sup></td><td>0.27</td><td>1</td></tr><tr><td>Nanocrystalline</td><td>0.769 × 10<sup>6</sup></td><td>7180</td><td>0.025 × 10<sup>-3</sup></td><td>0.0098</td><td>0.67</td></tr></tbody></table>

<figure><img src="image_3.png" /><figcaption>**Fig. 2** ρ vs. σd² for all the materials listed in Table 1 [31]</figcaption></figure>

The combination yielding the best simulation results (v = 0.66 and ρ = 0.016) was determined again through the minimization of the RED (%). Under this configuration, the RED (%) decreased to 9.89%. Notably, the relatively low value of v is significant (a similar observation can be made for Table 1 in the case of the semi-empirical method). This indicates a relatively less dominant contribution of viscous losses in nanocrystalline materials. Such a phenomenon is likely associated with the limited influence of pinning sites and the unique microstructure of these materials, which modifies the behavior of domain wall motions and the associated losses.

B—Space discretized method using a material law including a fractional derivative operator.

The third set of simulation-to-measurement comparisons, presented in Fig. 5, highlights the improved accuracy obtained with the simulation method

described in sub-Sect. 2.2.3 – B and defined by Eq. 17. In this configuration, the RED (%) calculated in a similar way, reached 8.2% with v = 0.6 and ρ = 0.02.

Notably, this simulation method provides additional insights, such as the distribution of magnetic losses across the cross-section of the tested specimen. A detailed description of the approach used to calculate the contributions and distribution of magnetic losses is provided in Sect. 4.2 of [28]. The distribution, for f = 5 kHz and max(Bₐ) = 1.2 T, is presented in Fig. 6b. The corresponding positions within the specimen's cross-section are illustrated in Fig. 6a. By summing these local predictions, a total energy loss of 44.6 J·m⁻³ is obtained. This value matches the experimental results (46.3 J·m⁻³) shown in Fig. 1 for the same values of f and max(Bₐ).

As discussed in the introduction, under these working conditions, the amount of classical losses is only 8.4 J·m⁻³, whereas the excess losses reach 36.2 J·m⁻³, confirming their dominant contribution. Regarding their distribution, the classical losses decrease almost continuously from the surface to the center of the ribbon. In contrast, the excess losses are relatively homogeneously distributed across the cross-section. This homogeneity is further evidenced by the consistent maximum flux density of the corresponding local hysteresis cycles Bᵢ(Hᵢ) shown in Fig. 6c. Conversely, coercivity decreases continuously, reflecting the reduction in classical losses.

C – Predicted results in the high frequency range (f = 20 kHz)

In the last set simulation-to-measurement comparisons presented in Figs. 7a, 7b and 7c all the time-dependent methods discussed in this sub-section were
<figure><img src="image_4.png" /><figcaption>**Fig. 3** Simulation/measurement comparisons for the average hysteresis cycle $B_a(H_{surf})$ on a vast range of frequency and maximum flux density using Eq. 12 with $v = 1$ and $\rho = 0.0006$</figcaption></figure>

tested under high-frequency conditions. However, due to equipment limitations, the maximum average flux density was limited to 0.3 T.

The enhanced accuracy of the space-discretized method described in sub-Sect. 2.2.3 – B becomes particularly evident in this final set of comparisons. The frequency effect is substantial, especially on the excess losses, which account for 78.6% of the total losses. The distribution of these losses, shown in Fig. 7d, reveals a similar trend for the classic contribution, while the excess contribution appears less homogeneous, with a lower magnitude at the center of

the nanocrystalline ribbon, where the magnetization is more constrained.

To evaluate the computational cost of the proposed methods, we analyzed the performance of the space-discretized approach. Despite requiring the simultaneous resolution of the hysteresis contribution and the fractional derivative formulation, the method remains computationally efficient. The 1D resolution is sufficient to capture the relevant magnetic loss mechanisms, and the limited thickness of the nanocrystalline ribbon allows accurate results with only $N = 20$ spatial discretization points. Furthermore, even with
<figure><img src="image_5.png" /><figcaption>**Fig. 4** Simulation/measurement comparisons for the average hysteresis cycle $B_a(H_{surf})$ on a vast range of frequency and maximum flux density using Eq. 12 with $v = 0.66$ and $\rho = 0.016$</figcaption></figure>

a high temporal resolution of 100,000 points per period, a full hysteresis cycle simulation completes in less than 1 s. These factors ensure that the method remains practical for large-scale simulations and industrial applications requiring high-frequency magnetic loss modeling.

Finally, we conducted a sensitivity analysis to evaluate how variations in key parameters ($v$ and $\rho$ affect the simulation outcomes). For this analysis, we computed the RED% between experimental and simulated data as a function of $v$ and $\rho$, considering

both the analytical expression and the space-discretized method. The results are shown in Fig. 8.

For the analytical expression, the RED(%) curve exhibits a V-shape with a high slope, indicating that the method is highly sensitive to variations in both $v$ and $\rho$. This is particularly true for $\rho$, which plays a dominant role in adjusting the loss predictions. In the case of the space-discretized method, the V-shape trend is still present, but the slope is less steep. This suggests that the method is less sensitive to variations in $v$ and $\rho$, making it more robust in practical applications.
<figure><img src="image_6.png" /><figcaption>**Fig. 5** Simulation/measurement comparisons for the average hysteresis cycle $B_a(H_{surf})$ on a vast range of frequency and maximum flux density solving Eq. 17 with $v = 0.67$ and $\rho = 0.0098$</figcaption></figure>

**5 Conclusion**

Fractional derivative operators are highly effective mathematical tools for simulating viscoelasticity. In this study, these operators were applied to analyze magnetic losses in a nanocrystalline ferromagnetic core. Four approaches were investigated:

* A semi-empirical method based on an analytical expression of the magnetic losses.

* A time-dependent hysteresis model using a lumped approach, where the average flux density is calculated by solving a first-order differential equation.

* A time-dependent hysteresis model using a lumped approach, where the average flux density is calculated by solving a fractional differential equation.

* A space-discretized method, in which the Maxwell equation for magnetic field diffusion is solved alongside a fractional differential equation representing the material behavior.

The semi-empirical method offers simplicity and accurate results but cannot capture the time evolution of magnetic quantities or the spatial distribution of magnetic losses.
<figure><img src="image_7.png" /><figcaption>Fig. 6 a Space discretization of the simulated nanocrystalline ribbon cross-section. b Distribution of the magnetic losses throughout the specimen cross-section at $f = 5$ kHz and $\max(B_a) = 1.2$ T. c Local hysteresis cycles $B_i(H_i)$</figcaption></figure>

The lumped approaches also produce accurate results, with the model incorporating fractional derivatives as the dynamic term demonstrating superior performance.

The space discretized method is the most complex to implement but delivers the highest accuracy and provides detailed information, including the local distribution and contributions of magnetic losses.

For all methods tested, the best results were achieved with relatively low fractional derivative orders, suggesting a lower viscous-to-elastic ratio compared to other typical materials. Notably, the excess losses are significant. While the limited thickness of the ribbon may contribute to this, the most plausible explanation lies in the unique micro-structure of the nanocrystalline ribbon, which induces
<figure><img src="image_8.png" /><figcaption>**Fig. 7** **a** Simulation/measurement comparisons for the average hysteresis cycle $B_a(H_{surf})$ at $f = 20$ kHz and $\max(B_a) = 0.1, 0.2$ and 0.3 T, using Eq. 12 with $v = 1$ and $\rho = 0.0006$. **b** Same comparisons, using Eq. 12 with $v = 0.66$ and $\rho = 0.016$. **c** Same</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>**Fig. 8** Distribution of the magnetic losses throughout the specimen cross-section associated to Fig. 7c' simulations at $f = 20$ kHz and $\max(B_a) = 0.3$ T</figcaption></figure>

distinctive domain wall motions and substantial excess losses.

Finally, it is noteworthy that the optimal combination of dynamic parameters $\rho \approx 0.01$ and $v \approx 0.66$, remains almost consistent across the different simulation methods. This consistency confirms that the viscoelastic effect associated with this parameter combination is independent of the simulation method.

Consequently, it can be determined using the simplest simulation method and preserved for more complex ones.

**Author contribution** BD wrote the main manuscript, conducted the simulations, and developed the models. SG, YG, and XZ reviewed the manuscript and provided the experimental data.
<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /><figcaption>**Fig. 8** a RED%(ρ) for the analytical expression method. b RED%(v) for the analytical expression method. c RED%(ρ) for the space discretized method. d RED%(v) for the space discretized method</figcaption></figure>

**Funding** The authors have not disclosed any funding.

**Data availability** No datasets were generated or analysed during the current study.

**Declarations**

**Conflict of interest** The authors declare no competing interests.

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