How To Calculate Hysteresis

Calculate magnetic hysteresis loss with precision. Enter your material properties and operating conditions below.

Comprehensive Guide: How to Calculate Hysteresis

Hysteresis represents the lagging of magnetic flux density (B) behind the magnetizing force (H) in ferromagnetic materials. This phenomenon results in energy loss during each magnetization cycle, which is critical to understand for designing efficient electrical machines, transformers, and inductors. This guide provides a detailed explanation of hysteresis calculation methods, practical applications, and optimization techniques.

1. Fundamental Concepts of Hysteresis

A hysteresis loop (B-H curve) illustrates the relationship between magnetic flux density (B) and magnetic field intensity (H). Key parameters include:

• Coercivity (H_c): The magnetic field required to reduce magnetization to zero after saturation
• Remanence (B_r): The remaining flux density when the magnetizing force is zero
• Saturation flux density (B_sat): The maximum flux density achievable
• Hysteresis loss (P_h): Energy dissipated as heat per unit volume per cycle

2. Mathematical Models for Hysteresis Calculation

The most widely used empirical model for calculating hysteresis loss is the Steinmetz equation:

P_h = k × f × B_max^β × V

Where:

• P_h = Hysteresis power loss [W]
• k = Steinmetz coefficient (material-dependent)
• f = Frequency of magnetization [Hz]
• B_max = Maximum flux density [T]
• β = Steinmetz exponent (typically 1.6-2.5)
• V = Volume of magnetic material [m³]

For more accurate calculations, the Modified Steinmetz Equation (MSE) accounts for non-sinusoidal waveforms:

P_h = k_i × f^(α-1) × (ΔB)^β × V

Where k_i is the improved Steinmetz coefficient and α is an additional exponent (typically 1.5-2.0).

3. Material-Specific Steinmetz Parameters

Different ferromagnetic materials exhibit varying hysteresis characteristics. The table below presents typical Steinmetz parameters for common magnetic materials:

Material	k (J/m³)	α	β	Typical Bmax (T)	Typical Applications
Silicon Steel (M3)	0.05	1.6	2.0	1.5-1.8	Power transformers, electric motors
Ferrite (MnZn)	0.008	1.4	2.6	0.3-0.5	High-frequency transformers, inductors
Pure Iron	0.12	1.8	2.0	2.0-2.2	Electromagnets, special-purpose cores
Nickel (78% Ni-Fe)	0.03	1.5	2.4	0.8-1.2	Sensitive relays, magnetic shields
Amorphous Alloy	0.005	1.3	2.5	1.4-1.6	High-efficiency transformers

4. Step-by-Step Calculation Process

1. Determine Material Properties

   Select the appropriate material and identify its Steinmetz coefficients (k, α, β) from manufacturer datasheets or technical literature. For custom materials, these values must be experimentally determined.

2. Measure Operating Conditions

   Identify the maximum flux density (B_max) and operating frequency (f) from your application specifications. For transformers, B_max is typically 60-80% of saturation flux density.

3. Calculate Core Volume

   Measure or calculate the volume of the magnetic core (V) in cubic meters. For laminated cores, use the net iron volume excluding insulation.

4. Apply the Steinmetz Equation

   Plug the values into the appropriate Steinmetz equation. For sinusoidal excitation, use the basic equation. For PWM or other waveforms, use the Modified Steinmetz Equation.

5. Validate Results

   Compare calculated losses with manufacturer-provided loss curves or empirical data. Significant deviations may indicate incorrect material parameters or operating conditions.

6. Consider Temperature Effects

   Hysteresis losses typically increase with temperature. Apply temperature correction factors if operating outside standard conditions (usually 20-25°C).

5. Practical Example Calculation

Let’s calculate the hysteresis loss for a transformer core with the following specifications:

• Material: Silicon steel (M3)
• B_max: 1.5 T
• Frequency: 60 Hz
• Core volume: 0.002 m³
• Steinmetz parameters: k = 0.05, α = 1.6, β = 2.0

Using the basic Steinmetz equation:

P_h = 0.05 × 60 × (1.5)^2.0 × 0.002 = 0.0135 W

This represents the power lost as heat due to hysteresis in the transformer core.

6. Advanced Considerations

For more accurate modeling in real-world applications, consider these additional factors:

• Minor Loops: Partial magnetization cycles create smaller hysteresis loops with different loss characteristics than the major loop.
• DC Bias: The presence of DC magnetization shifts the operating point on the B-H curve, potentially increasing losses.
• Harmonic Content: Non-sinusoidal waveforms require frequency-domain analysis or time-stepped simulations.
• Mechanical Stress: Physical stress on magnetic materials can alter their hysteresis characteristics.
• Aging Effects: Material properties may change over time due to thermal cycling or mechanical vibration.

7. Comparison of Calculation Methods

Method	Accuracy	Complexity	Computational Requirements	Best For
Basic Steinmetz	±20%	Low	Minimal	Quick estimates, sinusoidal excitation
Modified Steinmetz	±10%	Medium	Moderate	Non-sinusoidal waveforms, PWM drives
Generalized Steinmetz	±5%	High	Significant	Complex waveforms, high precision needed
Finite Element Analysis	±2%	Very High	Intensive	Critical designs, 3D field analysis
Empirical Data Fitting	±3%	Medium	Moderate	Material characterization, custom alloys

8. Reducing Hysteresis Losses

Engineers employ several strategies to minimize hysteresis losses in magnetic components:

1. Material Selection

   Choose materials with narrow hysteresis loops (low coercivity). Amorphous alloys and high-grade silicon steels offer superior performance but at higher cost.

2. Operating Point Optimization

   Operate at the lowest practical flux density that meets performance requirements. Reducing B_max from 1.7T to 1.5T can reduce losses by 30-40%.

3. Core Geometry

   Design cores to minimize volume while maintaining required inductance. Toroidal cores often provide better magnetic paths than E-I cores.

4. Lamination Techniques

   Use thin laminations (0.1-0.35mm) with proper insulation to reduce eddy current losses that compound with hysteresis losses.

5. Heat Treatment

   Proper annealing can improve grain orientation and reduce dislocation density, lowering hysteresis losses by 10-20%.

6. Flux Shaping

   Implement control strategies to maintain sinusoidal flux waveforms, avoiding harmonic-rich operating conditions.

9. Measurement Techniques

Accurate hysteresis loss measurement requires specialized equipment:

• Epstein Frame: Standardized test fixture for laminated materials (IEC 60404-2). Provides highly repeatable measurements under controlled conditions.
• Single Sheet Tester: Evaluates individual lamination sheets, useful for quality control in manufacturing.
• Ring Core Tester: Measures toroidal samples with uniform magnetic paths, offering high accuracy for research applications.
• Wattsmeter Method: Direct electrical measurement of input power minus copper losses to determine core losses.
• Thermal Methods: Calorimetric measurement of temperature rise to infer losses, useful for complete assemblies.

10. Industry Standards and Regulations

Several international standards govern hysteresis loss measurement and reporting:

• IEC 60404-6: Magnetic materials – Methods of measurement of the magnetic properties of magnetically soft materials at frequencies in the range 20 Hz to 100 kHz using ring specimens.
• ASTM A977/A977M: Standard test method for magnetic properties of high-coercivity permanent magnet materials using hysteresisgraph techniques.
• JIS C 2550: Japanese standard for testing methods of magnetic properties of electrical steel sheets and strips.
• IEEE Std C57.12.80: Standard terminology for power and distribution transformers, including loss definitions.

11. Common Calculation Errors and Pitfalls

Avoid these frequent mistakes when calculating hysteresis losses:

1. Incorrect Material Parameters

   Using generic Steinmetz coefficients instead of manufacturer-specific data can lead to errors exceeding 30%. Always verify parameters with material certifications.

2. Ignoring Waveform Effects

   Applying the basic Steinmetz equation to PWM or other non-sinusoidal waveforms may underestimate losses by 50% or more.

3. Volume Calculation Errors

   Forgetting to account for lamination stacking factor (typically 0.95-0.98) can lead to 2-5% underestimation of losses.

4. Temperature Dependence

   Failing to adjust for operating temperature can cause errors, as hysteresis losses typically increase by 0.2-0.5% per °C.

5. DC Bias Neglect

   Even small DC components (1-2% of AC excitation) can increase hysteresis losses by 10-15% due to asymmetric loop traversal.

6. Mechanical Stress Effects

   Installation stresses or vibration can degrade magnetic properties by 5-10%, increasing losses beyond calculated values.

12. Software Tools for Hysteresis Calculation

Several professional software packages assist with hysteresis loss calculation and magnetic design:

• FEMM (Finite Element Method Magnetics): Open-source 2D finite element analysis tool for magnetic problems.
• JMAG: Commercial FEA software with advanced material modeling capabilities.
• Flux: Altair’s electromagnetic simulation software with specialized modules for core loss calculation.
• COMSOL Multiphysics: General-purpose FEA with AC/DC module for magnetic simulations.
• PSIM: Circuit simulator with built-in magnetic core loss models.
• MathWorks MATLAB: With specialized toolboxes for magnetic analysis and custom script development.

13. Emerging Research in Hysteresis Modeling

Recent advancements in hysteresis modeling include:

• Machine Learning Approaches: Neural networks trained on extensive material data can predict hysteresis behavior with higher accuracy than empirical models.
• Multi-scale Modeling: Combining atomic-scale simulations with macroscopic models to better understand material behavior.
• Dynamic Hysteresis Models: Time-domain models that account for rate-dependent effects in high-frequency applications.
• Vector Hysteresis Models: 2D and 3D models that capture rotational magnetization effects in electrical machines.
• Thermal-Electromagnetic Coupling: Integrated models that simultaneously solve magnetic and thermal equations for more accurate loss prediction.

Authoritative Resources on Hysteresis Calculation

For further study, consult these authoritative sources:

• National Institute of Standards and Technology (NIST) – Magnetic Materials Group provides extensive research data and measurement standards for magnetic materials.
• U.S. Department of Energy – Magnetics Manufacturing Consortium – Research on advanced magnetic materials and loss reduction techniques for energy-efficient applications.
• NASA Electronic Parts and Packaging (NEPP) Program – Technical reports on magnetic materials for space applications, including radiation effects on hysteresis properties.
• IEEE Magnetics Society – Publishes the IEEE Transactions on Magnetics, featuring cutting-edge research on hysteresis modeling and measurement techniques.