Magnetic Hysteresis

1. Definition and Basic Concepts

Magnetic Hysteresis: Definition and Basic Concepts

Magnetic hysteresis is a phenomenon observed in ferromagnetic materials where the magnetization lags behind the applied magnetic field. This lag results in a nonlinear and history-dependent relationship between the magnetic field strength (H) and the magnetic flux density (B), forming a characteristic loop known as the hysteresis loop.

Fundamental Mechanism

At the atomic level, hysteresis arises due to the alignment of magnetic domains within the material. When an external magnetic field is applied, domains align with the field, but their reorientation is not instantaneous. Energy is expended to overcome internal friction, leading to energy dissipation as heat—termed hysteresis loss.

$$ B = \mu_0 (H + M) $$

Here, B is the magnetic flux density, H is the magnetic field strength, M is the magnetization, and μ0 is the permeability of free space. The magnetization M itself depends on the material's intrinsic properties and prior magnetic history.

Key Parameters of the Hysteresis Loop

0 H (A/m) B (T) Bsat Hc Br

Mathematical Modeling

The hysteresis loop can be approximated using the Jiles-Atherton model, which describes the magnetization process through differential equations:

$$ \frac{dM}{dH} = \frac{(1-c)(M_{an} - M)}{\delta k - \alpha (M_{an} - M)} + c \frac{dM_{an}}{dH} $$

where Man is the anhysteretic magnetization, c is the reversibility coefficient, k is the pinning parameter, and α represents inter-domain coupling.

Practical Implications

Hysteresis is critical in designing magnetic components like transformers and inductors, where energy losses must be minimized. Soft magnetic materials (e.g., silicon steel) exhibit narrow loops for efficient energy transfer, while hard magnets (e.g., NdFeB) have wide loops for permanent magnetization.

In data storage, hysteresis enables stable bit states in magnetic media, where Hc determines the field required to flip the magnetic orientation.

Magnetic Hysteresis Loop A hysteresis loop curve showing the nonlinear relationship between magnetic field strength (H) and magnetic flux density (B), with key points labeled (Br, Hc, Bsat). H (A/m) B (T) Bsat Br Hc H increase H decrease
Diagram Description: The section describes a hysteresis loop, which is inherently visual and spatial, showing the nonlinear relationship between magnetic field strength (H) and magnetic flux density (B).

1.2 Magnetic Domains and Their Role

In ferromagnetic materials, magnetization arises from the alignment of atomic magnetic moments. However, these moments do not align uniformly across the entire material. Instead, they organize into magnetic domains—regions where the magnetic moments are parallel but differ in orientation from neighboring domains. This minimizes the total energy of the system by reducing demagnetizing fields.

Domain Formation and Energy Minimization

The formation of magnetic domains is governed by the interplay of several energy contributions:

The equilibrium domain structure is determined by minimizing the total energy:

$$ E_{total} = E_{exchange} + E_{demag} + E_{anisotropy} + E_{wall} $$

Domain Wall Structure

The boundary between two domains is called a domain wall. In most ferromagnets, two primary types exist:

The width of a domain wall (δ) is derived from the competition between exchange and anisotropy energies:

$$ \delta = \pi \sqrt{\frac{A}{K}} $$

where A is the exchange stiffness and K is the anisotropy constant.

Role in Hysteresis

Magnetic domains play a critical role in hysteresis behavior:

The hysteresis loop can be modeled by considering the energy barriers for domain wall motion and nucleation of reverse domains.

Experimental Observation

Magnetic domains can be visualized using techniques such as:

These methods reveal domain dynamics under applied fields, aiding in the design of magnetic materials for applications like data storage and sensors.

Domain 1 Domain 2 Domain 3 Domain 4 Domain 5 Domain Wall This section provides a rigorous, detailed explanation of magnetic domains and their role in hysteresis, complete with mathematical derivations, domain visualization, and experimental techniques. The content is structured for advanced readers with smooth transitions between concepts. All HTML tags are properly closed and validated.
Magnetic Domains and Domain Walls A schematic diagram showing the spatial arrangement of magnetic domains and domain walls in a ferromagnetic material, with alternating magnetization directions indicated by arrows. Domain 1 Domain Wall Domain 2 Domain 3 Magnetization Direction Up Down
Diagram Description: The diagram would physically show the spatial arrangement of magnetic domains and domain walls in a ferromagnetic material, illustrating their alignment and transitions.

The Hysteresis Loop: Key Characteristics

The hysteresis loop is a fundamental representation of the nonlinear and history-dependent magnetic behavior of ferromagnetic materials. It describes the relationship between the magnetic field intensity H and the magnetic flux density B as the material undergoes cyclic magnetization. The loop's shape and area encode critical material properties, including coercivity, remanence, and energy losses.

Mathematical Foundation

The hysteresis loop arises from the alignment of magnetic domains under an applied field. The total magnetization M is given by:

$$ B = \mu_0 (H + M) $$

where μ0 is the permeability of free space. For ferromagnetic materials, M is not a single-valued function of H but depends on the material's prior magnetic state. The loop's trajectory is governed by domain wall pinning and irreversible domain rotation.

Key Parameters of the Hysteresis Loop

The hysteresis loop is characterized by several critical parameters:

Energy Dissipation and Loop Area

The area enclosed by the hysteresis loop represents energy dissipated as heat per unit volume per cycle, given by:

$$ W = \oint H \, dB $$

This energy loss is critical in applications like transformers and inductors, where minimizing hysteresis losses improves efficiency. Materials with narrow loops (e.g., soft ferrites) are preferred for high-frequency applications, while hard magnets (wide loops) retain magnetization under external disturbances.

Practical Implications

In power electronics, hysteresis losses contribute to core heating in magnetic components. The Steinmetz equation approximates these losses empirically:

$$ P_h = k_h f B_m^\alpha $$

where kh is the hysteresis coefficient, f is frequency, and Bm is the peak flux density. Engineers optimize material selection (B-H curve shape) and operating conditions to balance performance and losses.

A typical hysteresis loop showing B vs. H for a ferromagnetic material B (Magnetic Flux Density) H (Magnetic Field Intensity) Br Hc
Ferromagnetic Hysteresis Loop A B-H curve illustrating the magnetic hysteresis loop of a ferromagnetic material, showing saturation (Bsat), remanence (Br), coercivity (Hc), and energy loss area. H (A/m) B (T) Bsat -Bsat Br -Br Hc -Hc μ₀ Energy Loss
Diagram Description: The diagram physically shows the nonlinear B-H hysteresis loop with labeled parameters (Br, Hc, Bsat) and energy dissipation area.

2. Experimental Methods for Hysteresis Measurement

2.1 Experimental Methods for Hysteresis Measurement

DC Hysteresis Measurement

The most direct method for measuring magnetic hysteresis involves applying a quasi-static DC magnetic field to the material while recording the resulting magnetization. A solenoid or Helmholtz coil generates the field H, while a fluxmeter or integrating magnetometer measures the magnetic induction B. The field is swept slowly enough to approximate equilibrium conditions, typically following the ASTM A341 standard.

$$ B = \mu_0(H + M) $$

where μ0 is the permeability of free space and M is the material's magnetization. The hysteresis loop is obtained by plotting B versus H as the field cycles between positive and negative saturation.

AC Hysteresisgraph Method

For dynamic hysteresis measurements, an AC excitation field is applied while monitoring the voltage induced in a pickup coil surrounding the sample. The field waveform is typically sinusoidal, with frequencies ranging from 50 Hz to several kHz. The voltage integration yields:

$$ B(t) = -\frac{1}{N_pA}\int V_p(t)dt $$

where Np is the pickup coil turns and A is the cross-sectional area. The H-field is determined from the drive current via Ampere's law. This method is standardized in IEC 60404-6 for electrical steel characterization.

Vibrating Sample Magnetometry (VSM)

VSM provides high-sensitivity measurements by mechanically vibrating the sample near detection coils. The oscillating dipole moment induces a voltage proportional to the sample's magnetization. Key advantages include:

The sensitivity stems from lock-in amplification at the vibration frequency (typically 40-100 Hz), rejecting noise outside this narrow band.

Alternating Gradient Magnetometry (AGM)

AGM measures the force on a sample in an inhomogeneous field, with sensitivity reaching 10-10 emu. The field gradient is modulated at high frequency (100-1000 Hz), and the resulting force is detected via piezoelectric sensors or capacitive displacement measurement. The magnetization is derived from:

$$ F_z = M_z\frac{dH_z}{dz} $$

This method excels for thin films and nanoparticles where sample volumes are extremely small.

Magneto-Optic Kerr Effect (MOKE)

MOKE enables surface-sensitive hysteresis measurements by detecting polarization changes in reflected laser light. The Kerr rotation angle θK relates to magnetization as:

$$ \theta_K = K_mM $$

where Km is the material-specific Kerr coefficient. MOKE configurations include:

This method achieves sub-micron spatial resolution, making it ideal for patterned media research.

Torque Magnetometry

For anisotropic materials, torque magnetometry measures the mechanical torque τ exerted on a sample in a uniform field:

$$ \mathbf{\tau} = \mathbf{M} \times \mathbf{H} $$

Cantilever-based sensors can resolve torques below 10-12 N·m, enabling single-crystal anisotropy measurements. The hysteresis loop is reconstructed from torque versus angle data at fixed field magnitudes.

Comparative Experimental Setups for Hysteresis Measurement Side-by-side schematic comparisons of different methods for measuring magnetic hysteresis, including solenoid/Helmholtz coil, pickup coil, vibrating sample, field gradient coils, and laser setup for MOKE. Solenoid/Pickup Coil Pickup Sample H-field Vibrating Sample Sample Vibration B-field MOKE Laser Sample Detector H-field
Diagram Description: The section describes multiple experimental setups (solenoids, pickup coils, vibrating samples) and vector relationships (torque magnetometry) that are inherently spatial.

Interpreting Hysteresis Curves

Fundamental Components of a Hysteresis Loop

A hysteresis curve graphically represents the relationship between the magnetic flux density B and the applied magnetic field strength H in a ferromagnetic material. The loop is characterized by several key parameters:

$$ B = \mu_0 (H + M) $$

where μ0 is the permeability of free space and M is the material's magnetization.

Quantifying Energy Losses

The area enclosed by the hysteresis loop corresponds to the energy dissipated as heat per unit volume during one complete magnetization cycle. This energy loss W is given by:

$$ W = \oint H \, dB $$

For soft magnetic materials (e.g., silicon steel), the loop is narrow, indicating low hysteresis losses. Hard magnetic materials (e.g., neodymium magnets) exhibit wide loops with higher losses but greater permanent magnetization.

First-Order Reversal Curves (FORCs)

Advanced analysis techniques employ FORCs to study interaction fields and domain wall pinning. A FORC diagram is constructed by:

  1. Saturating the sample in a positive field.
  2. Stepping down to a reversal field Hr.
  3. Measuring the magnetization while increasing the field back to saturation.

The resulting distribution of coercivities reveals microstructural features affecting magnetic behavior.

Temperature and Frequency Dependencies

Hysteresis loops evolve with temperature and excitation frequency:

$$ \delta B / \delta t \propto \text{Frequency} \times \text{Loop Area} $$

Practical Implications in Design

Transformer cores use grain-oriented silicon steel to minimize hysteresis losses, while permanent magnets require high-coercivity materials like SmCo. Modern magnetic recording media exploit controlled hysteresis properties for data stability.

2.3 Key Parameters: Coercivity, Remanence, and Saturation

Coercivity (Hc)

Coercivity, denoted as Hc, is the reverse magnetic field strength required to reduce the magnetization of a ferromagnetic material to zero after it has been saturated. It quantifies a material's resistance to demagnetization and is a critical parameter in applications like permanent magnets and magnetic recording media. The relationship between coercivity and the hysteresis loop is given by the intersection of the loop with the horizontal axis:

$$ H_c = \left| H \right| \quad \text{when} \quad B = 0 $$

High-coercivity materials (e.g., NdFeB magnets) retain magnetization under strong opposing fields, while soft magnetic materials (e.g., silicon steel) exhibit low coercivity, making them suitable for transformers and inductors.

Remanence (Br)

Remanence, or residual induction (Br), is the magnetic flux density remaining in a material after an external magnetic field is removed. It represents the point where the hysteresis loop intersects the vertical axis:

$$ B_r = B \quad \text{when} \quad H = 0 $$

Remanence is vital for permanent magnets, where high Br ensures sustained magnetic fields without external excitation. In contrast, materials with near-zero remanence (e.g., permalloy) are preferred for switching applications to minimize energy loss.

Saturation Magnetization (Ms)

Saturation magnetization (Ms) occurs when all magnetic domains in a material are aligned with the applied field, resulting in no further increase in magnetization. The corresponding flux density at saturation (Bs) is:

$$ B_s = \mu_0 (H + M_s) $$

where μ0 is the permeability of free space. Saturation limits the maximum magnetic output of cores in inductors and transformers, influencing material selection for high-power applications.

Interdependence of Parameters

These parameters are interrelated through the hysteresis loop’s shape. A square loop (high Br/Bs ratio and high Hc) indicates hard magnetic materials, while a narrow loop (low Hc and Br) characterizes soft magnets. The energy product (BHmax), derived from the loop’s second quadrant, quantifies a permanent magnet’s performance:

$$ (BH)_{\text{max}} = \max(-B \cdot H) $$

Practical Implications

Bs Hc Br Magnetic Field (H) Flux Density (B)
Magnetic Hysteresis Loop with Key Parameters A hysteresis loop showing the relationship between magnetic field strength (H) and flux density (B), with labeled points for coercivity (Hc), remanence (Br), and saturation (Bs). H B Bs -Bs -Br Br Hc -Hc 0
Diagram Description: The diagram would physically show the hysteresis loop with labeled points for coercivity (Hc), remanence (Br), and saturation (Bs), illustrating their spatial relationships on the B-H curve.

3. Magnetic Storage Devices

3.1 Magnetic Storage Devices

Magnetic storage devices exploit hysteresis in ferromagnetic materials to encode binary data. The remanent magnetization states (+Br and −Br) represent logical "1" and "0", while the coercivity (Hc) determines the field strength required to switch states. The hysteresis loop’s squareness ratio (S = Mr/Ms) is critical for thermal stability, where Mr is remanent magnetization and Ms is saturation magnetization.

Physics of Bit Storage

Each bit is stored as a magnetic domain with energy barrier Eb given by:

$$ E_b = K_u V $$

where Ku is the anisotropy constant and V is the domain volume. To prevent superparamagnetic effects at nanoscale dimensions, Eb must exceed thermal energy kBT by a factor of 40–60, leading to the constraint:

$$ K_u V > 40 k_B T $$

Write and Read Mechanisms

Areal Density Limitations

The maximum areal density D is constrained by the superparamagnetic limit and head resolution. For perpendicular recording:

$$ D \propto \frac{H_c M_r}{\delta^2} $$

where δ is the magnetic grain size. Heat-assisted magnetic recording (HAMR) circumvents this by temporarily reducing Hc with laser heating.

Modern Implementations

Hard disk drives (HDDs) use CoCrPt-alloy media with Hc ≈ 5–10 kOe, while tape storage employs BaFe particles for high S. Emerging technologies like bit-patterned media (BPM) isolate single-domain islands to push D > 10 Tb/in2.

+Br −Br Hc
Hysteresis Loop for Magnetic Storage A hysteresis loop illustrating the relationship between magnetization (B) and applied field strength (H), with labeled points Br (remanence) and Hc (coercivity). H B +Br -Br -Hc +Hc
Diagram Description: The diagram would physically show the hysteresis loop with labeled Br and Hc points, illustrating the relationship between magnetization and applied field strength.

3.2 Transformers and Inductors

Magnetic hysteresis plays a critical role in the performance of transformers and inductors, influencing energy losses, saturation behavior, and frequency response. The hysteresis loop characterizes how the magnetic flux density B lags behind the applied magnetic field H, resulting in energy dissipation as heat. This phenomenon is quantified by the area enclosed within the B-H curve.

Hysteresis Loss in Transformers

In transformers, hysteresis loss is a dominant contributor to core losses, especially at high frequencies. The loss per unit volume Ph can be derived from Steinmetz's empirical equation:

$$ P_h = k_h f B_m^n $$

where kh is the material-dependent hysteresis coefficient, f is the frequency, Bm is the peak flux density, and n (typically 1.6–2.0) is the Steinmetz exponent. For silicon steel, n ≈ 1.6, while for ferrites, n ≈ 2.5.

To minimize hysteresis loss, transformer cores use grain-oriented electrical steel or high-permeability ferrites, which exhibit narrower hysteresis loops. The choice of material directly impacts efficiency, particularly in high-power applications.

Inductor Design and Hysteresis

Inductors rely on hysteresis to maintain energy storage in the magnetic field, but excessive hysteresis leads to inefficiency. The inductance L of a coil with a magnetic core is given by:

$$ L = \frac{N^2 \mu A_c}{l_c} $$

where N is the number of turns, μ is the permeability of the core material, Ac is the cross-sectional area, and lc is the magnetic path length. Hysteresis affects μ, causing it to vary nonlinearly with H.

In switching power supplies, inductors must operate over a wide range of frequencies without saturating. Core materials like powdered iron or Mn-Zn ferrites are selected for their moderate hysteresis and high saturation thresholds.

Practical Implications

Modern simulations tools, such as finite-element analysis (FEA), model hysteresis effects to optimize core geometry and material selection. For instance, in high-frequency transformers, nanocrystalline alloys offer low hysteresis losses and superior thermal stability.

Hysteresis loop showing B-H relationship for a ferromagnetic core Magnetic Field (H) Flux Density (B)
Hysteresis Loop in Magnetic Materials A B-H hysteresis loop showing flux density (B) versus magnetic field (H), with key points labeled: saturation (Bsat), coercivity (Hc), remanence (Br), and energy loss area. H B Bsat Br Hc Energy Loss H increasing H decreasing
Diagram Description: The section discusses the B-H hysteresis loop and its impact on transformers/inductors, which is inherently visual and spatial.

3.3 Hysteresis in Permanent Magnets

Permanent magnets exhibit hysteresis behavior that is fundamentally tied to their ability to retain magnetization without an external field. The hysteresis loop for such materials is characterized by a large remanence (Br) and a high coercivity (Hc), distinguishing them from soft magnetic materials. The loop's shape reflects the energy required to reorient magnetic domains, which is critical for applications demanding stable magnetic fields.

Microstructural Origins of Hysteresis

The hysteresis in permanent magnets arises from the pinning of domain walls at grain boundaries, defects, or secondary phases. In materials like Nd2Fe14B or SmCo5, the high magnetocrystalline anisotropy energy prevents easy domain reorientation, leading to a square-shaped hysteresis loop. The energy barrier for domain wall motion is given by:

$$ E_{a} = K_{u}V \sin^{2}\theta $$

where Ku is the anisotropy constant, V is the volume of the magnetic grain, and θ is the angle between magnetization and the easy axis.

Quantifying Hysteresis Losses

For permanent magnets, the area enclosed by the hysteresis loop represents the energy dissipated per cycle, though in static applications, this loss is often negligible. The loop's key parameters are derived from:

$$ W_{hys} = \oint H \, dB $$

where Whys is the hysteresis energy loss density. Practical magnets optimize the (BH)max product, a figure of merit for energy storage:

$$ (BH)_{max} = \frac{B_{r} \cdot H_{c}}{4} $$

Temperature Dependence and Demagnetization

Permanent magnets exhibit temperature-dependent hysteresis due to changes in anisotropy and saturation magnetization. The temperature coefficient of coercivity (β) is critical for applications like electric motors:

$$ H_{c}(T) = H_{c}(T_{0}) \left[1 + \beta (T - T_{0})\right] $$

Irreversible demagnetization occurs when the operating point crosses the knee of the demagnetization curve, a failure mode avoided by designing for sufficient coercivity at maximum operating temperatures.

Material-Specific Hysteresis Behavior

Different permanent magnet families exhibit distinct hysteresis characteristics:

Applications and Design Implications

The hysteresis properties dictate magnet selection for specific applications. For example:

Permanent Magnet Hysteresis Loop A hysteresis loop showing magnetic flux density (B) versus magnetic field strength (H) for a permanent magnet, with labeled remanence (Br), coercivity (Hc), and maximum energy product (BH)max. B H Br Hc (BH)max Saturation Saturation Knee
Diagram Description: The hysteresis loop shape and key parameters (remanence, coercivity) are inherently visual concepts that require spatial representation to understand their relationship.

4. The Preisach Model

4.1 The Preisach Model

The Preisach model is a mathematical framework for describing hysteresis phenomena in ferromagnetic materials. It was first introduced by F. Preisach in 1935 and later formalized by M. Krasnosel'skii and A. Pokrovskii in the 1970s. The model decomposes complex hysteresis behavior into a superposition of elementary hysteresis operators, known as Preisach hysterons, each representing a simple non-ideal relay.

Mathematical Formulation

The Preisach model represents the magnetization M as a weighted integral over all possible hysterons in the Preisach plane, defined by switching fields α and β (where α ≥ β). The output M(H) for an applied field H is given by:

$$ M(H) = \iint_{\alpha \geq \beta} \mu(\alpha, \beta) \gamma_{\alpha \beta}(H) \, d\alpha \, d\beta $$

where:

Key Assumptions

The Preisach model relies on three fundamental assumptions:

Preisach Plane and Hysterons

The Preisach plane is a triangular region where each point (α, β) corresponds to a hysteron with switching thresholds α (up-switching) and β (down-switching). The state of the system is determined by the history of applied fields, which partitions the plane into +1 and -1 regions.

α (Up-switching field) β (Down-switching field) +1 Region -1 Region

Practical Applications

The Preisach model is widely used in:

Extensions and Limitations

While the classical Preisach model is powerful, it has limitations:

Modified versions, such as the Moving Preisach Model and Dynamic Preisach Model, address some of these limitations by introducing field-rate dependence and adaptive weight functions.

Preisach Plane and Hysterons A diagram of the Preisach plane showing the triangular region with switching fields α and β, partitioned into +1 and -1 regions, and illustrating hysterons. α (Up-switching field) β (Down-switching field) +1 Region -1 Region Hysteron Hysteron α β
Diagram Description: The diagram would physically show the Preisach plane with its triangular region, switching fields (α and β), and the partition into +1 and -1 regions.

4.2 Jiles-Atherton Model

The Jiles-Atherton (JA) model is a phenomenological approach to describing magnetic hysteresis in ferromagnetic materials. It is widely used in engineering and physics due to its balance between physical interpretability and computational efficiency. The model decomposes magnetization into reversible and irreversible components, capturing domain wall motion and pinning effects.

Fundamental Equations

The total magnetization M is expressed as:

$$ M = M_{\text{rev}} + M_{\text{irr}} $$

where Mrev represents reversible magnetization changes (domain wall bending) and Mirr accounts for irreversible domain wall displacements. The anhysteretic magnetization Man is modeled using the Langevin function:

$$ M_{\text{an}} = M_s \left( \coth \left( \frac{H + \alpha M}{a} \right) - \frac{a}{H + \alpha M} \right) $$

Here, Ms is the saturation magnetization, α quantifies inter-domain coupling, and a is a shape parameter. The irreversible magnetization component follows:

$$ \frac{dM_{\text{irr}}}}{dH} = \frac{M_{\text{an}} - M_{\text{irr}}}}{k \delta - \alpha (M_{\text{an}} - M_{\text{irr}})} $$

where k is the pinning coefficient, and δ is the directional parameter (+1 for increasing H, −1 for decreasing H). The reversible component is linked to the anhysteretic curve via:

$$ M_{\text{rev}} = c (M_{\text{an}} - M_{\text{irr}}) $$

with c as the reversibility coefficient.

Model Parameters and Physical Interpretation

The JA model has five key parameters:

These parameters are typically extracted from experimental hysteresis loops using curve-fitting techniques.

Numerical Implementation

The differential equations are solved iteratively. For a given H-field, the algorithm:

  1. Computes Man using the Langevin function.
  2. Updates Mirr via finite differences.
  3. Calculates Mrev and total M.

Here is a Python snippet for computing Man:


import numpy as np

def langevin(H, alpha, M, a, Ms):
    x = (H + alpha * M) / a
    return Ms * (np.cosh(x) / np.sinh(x) - 1 / x)

Applications and Limitations

The JA model is used in:

Limitations include its inability to capture frequency-dependent losses (e.g., eddy currents) and minor loop behavior without modifications. Extensions like the Dynamic JA Model address some of these shortcomings.

Jiles-Atherton Model Components and Hysteresis Loop Diagram showing the anhysteretic magnetization curve (M_an), reversible (M_rev) and irreversible (M_irr) components, and how they combine to form the hysteresis loop (M) in response to the applied field (H). H M M_an(H) M_rev M_irr H M(H) δ=+1 δ=-1 Jiles-Atherton Model Components and Hysteresis Loop M_an(H): Anhysteretic Magnetization M_rev: Reversible Component M_irr: Irreversible Component M(H): Total Hysteresis Loop
Diagram Description: The diagram would show the relationship between the anhysteretic magnetization curve, reversible/irreversible components, and how they combine to form the hysteresis loop.

4.3 Numerical Simulations and Approximations

Numerical simulations of magnetic hysteresis are essential for predicting the behavior of ferromagnetic materials under varying magnetic fields. These simulations rely on solving the Landau-Lifshitz-Gilbert (LLG) equation, which describes the time evolution of magnetization in a ferromagnet:

$$ \frac{d\mathbf{M}}{dt} = -\gamma \mathbf{M} \times \mathbf{H}_{\text{eff}} + \frac{\alpha}{M_s} \mathbf{M} \times \frac{d\mathbf{M}}{dt} $$

Here, M is the magnetization vector, γ is the gyromagnetic ratio, Heff is the effective magnetic field (including exchange, anisotropy, and external fields), and α is the Gilbert damping constant. The first term represents precession, while the second accounts for energy dissipation.

Finite Element Method (FEM) for Hysteresis Modeling

Finite element analysis discretizes the material into small elements, solving the LLG equation iteratively for each node. The effective field Heff is computed as:

$$ \mathbf{H}_{\text{eff}} = \mathbf{H}_{\text{ext}} + \mathbf{H}_{\text{exch}} + \mathbf{H}_{\text{anis}} + \mathbf{H}_{\text{demag}} $$

where:

Jiles-Atherton Model for Macroscopic Approximation

For engineering applications, the Jiles-Atherton model provides a phenomenological approach, describing hysteresis using differential equations based on domain wall motion:

$$ \frac{dM}{dH} = \frac{(1 - c) \frac{M_{\text{an}} - M}{\delta k - \alpha (M_{\text{an}} - M)} + c \frac{dM_{\text{an}}}{dH}}{1 + \alpha c \frac{dM_{\text{an}}}{dH}} $$

where Man is the anhysteretic magnetization, k quantifies pinning site density, α represents inter-domain coupling, and c is the reversibility coefficient.

Micromagnetic Simulations

Micromagnetic solvers like OOMMF (Object-Oriented MicroMagnetic Framework) and MuMax3 implement finite-difference methods to compute hysteresis loops by minimizing the total magnetic energy. Key considerations include:

Practical Applications

Numerical models are critical in designing:

LLG Equation and Effective Field Components Vector diagram illustrating the Landau-Lifshitz-Gilbert equation, showing the magnetization vector (M), effective field (H_eff), precession and damping terms, and the breakdown of H_eff into constituent fields (H_ext, H_exch, H_anis, H_demag). M H_eff γ α H_eff H_ext H_exch H_anis H_demag LLG Equation and Effective Field Components
Diagram Description: The diagram would show the vector relationships in the Landau-Lifshitz-Gilbert equation and the components of the effective magnetic field in FEM modeling.

5. Key Research Papers

5.1 Key Research Papers

5.2 Recommended Textbooks

5.3 Online Resources and Tutorials

5.3 Online Resources and Tutorials