1.2 The Hysteresis Loop and B-H Curve

1.2 The Hysteresis Loop and B-H Curve

Anatomy of the B-H Curve and Hysteresis Loop

The magnetic hysteresis loop (or B-H curve) is the primary graphical fingerprint of a material's magnetic performance under an applied magnetizing force. It characterizes the non-linear relationship between the external magnetic field intensity ($H$) and the resulting internal magnetic induction ($B$).

Coordinate Axes and Units

• Horizontal Axis ($H$ - Magnetizing Force): Also referred to as magnetic field strength or field intensity, $H$ represents the external magnetizing effort applied to the material.
  • In the CGS electromagnetic system, $H$ is measured in Oersteds (Oe), where $1\text{ Oe} = 1\text{ Gilbert per centimeter} = \frac{1000}{4\pi}\text{ A/m} \approx 79.577\text{ A/m}$.
  • In the SI system, $H$ is measured in Amperes per meter (A/m).
• Vertical Axis ($B$ - Magnetic Flux Density): Also known as magnetic induction, $B$ quantifies the total concentration of magnetic lines of force per unit cross-sectional area established inside the material.
  • In the CGS electromagnetic system, $B$ is measured in Gauss (G), where $1\text{ Gauss} = 1\text{ Maxwell per square centimeter}$ ($1\text{ line/cm}^2$).
  • In the SI system, $B$ is measured in Tesla (T), where $1\text{ Tesla} = 1\text{ Weber per square meter}$ ($1\text{ Wb/m}^2$).
  • Conversion: $1\text{ Tesla} = 10,000\text{ Gauss}$ ($1\text{ mT} = 10\text{ Gauss}$).

The Virgin (Initial) Magnetization Curve

When a completely demagnetized ferromagnetic specimen ($B=0, H=0$) is subjected to an incrementally increasing magnetizing force, the resulting flux density traces a unique non-linear path called the virgin or initial magnetization curve:

1. Initial Region (Low $H$): The curve starts at the origin $(0,0)$ with a modest slope. Magnetization proceeds via reversible domain wall displacement. The slope at the origin defines initial permeability ($\mu_i = \lim_{H \to 0} \frac{B}{H}$).
2. Knee Region (Moderate $H$): The curve steepens dramatically as irreversible domain wall movement and Barkhausen jumps occur. Here, the material exhibits its maximum permeability ($\mu_{max}$), where a small increase in $H$ yields a massive surge in flux density $B$.
3. Saturation Plateau (High $H$): The curve bends horizontally as domain rotation completes. The slope flattens out to $\mu_0$. The material reaches magnetic saturation ($B_{sat}$).

The Phenomenon of Hysteresis

The term hysteresis is derived from the ancient Greek word $\dot{\upsilon}\sigma\tau\acute{\epsilon}\rho\eta\sigma\iota\varsigma$, meaning "to lag behind." In magnetic physics, it describes the empirical fact that the magnetic flux density ($B$) lags behind the applied magnetizing force ($H$). Because domain wall movements encounter physical pinning and require finite energy to overcome microstructural obstacles, the reduction of $H$ back to zero does not return $B$ to zero. The magnetization curve follows an entirely different descending trajectory, forming a closed loop upon cyclic reversal.

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Key Parameters: $B_{sat}$, $B_r$, $H_c$, and Loop Energy Loss

1. Magnetic Saturation ($B_{sat}$)

Magnetic saturation represents the maximum achievable magnetic flux density induced within the material under practical magnetizing fields. At $+B_{sat}$, all magnetic domain vectors are oriented parallel to the external field vector. Further increases in $H$ yield no additional domain contribution; the curve increases strictly at the rate of $\Delta B = \mu_0 \Delta H$. In structural carbon steels, saturation flux density typically occurs between $14,000$ and $21,500\text{ Gauss}$ ($1.4$ to $2.15\text{ Tesla}$).

2. Residual Induction ($B_r$) and Retentivity

When the magnetizing force $H$ is progressively reduced from the saturation point back to zero ($H = 0$), the flux density does not collapse to zero. A substantial portion of magnetic flux remains locked inside the material:

• Residual Induction ($B_r$ or Remanence): The actual magnetic flux density remaining in a specific magnetic circuit after the magnetizing force is reduced to zero.
• Retentivity: The maximum value of residual induction that a material can retain when magnetized to full saturation. Retentivity is an inherent material property, whereas remanence depends upon the peak magnetizing force achieved prior to field removal.

3. Coercive Force ($H_c$) and Coercivity

To eliminate the residual induction and drive the internal flux density $B$ back to zero, a reverse magnetizing force must be applied:

• Coercive Force ($H_c$): The reverse magnetizing force (measured in Oersteds or A/m) required to reduce the residual flux density to zero.
• Coercivity: The specific value of coercive force required to reduce the flux density to zero after the material has been magnetized to full saturation. Materials with high coercivity fiercely resist demagnetization and external magnetic shock.

4. Area of the Hysteresis Loop and Energy Dissipation

The enclosed geometric area of the B-H loop possesses direct physical significance: it represents the volumetric energy lost per magnetization cycle: $W_h = \oint H \, dB$ This energy loss is converted irreversibly into thermal energy (heat) as domain walls snap past microstructural pinning obstacles. In alternating current (AC) magnetic particle equipment, materials with broad hysteresis loops will heat up rapidly during continuous testing cycles due to substantial hysteresis losses.

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Magnetically Soft vs. Magnetically Hard Materials

In NDT engineering and metallurgical classification, ferromagnetic materials are categorized as either magnetically "soft" or magnetically "hard." This distinction depends not on mechanical hardness alone, but on domain mobility and loop morphology.

Magnetically Soft Materials

Magnetically soft materials permit rapid, easy domain wall motion with minimal energetic resistance.

• Loop Characteristics: Narrow, tall, slender hysteresis loop; steep slope (very high permeability $\mu$); low coercive force ($H_c < 10\text{ Oe}$, often $< 1\text{ Oe}$); low retentivity; small enclosed loop area (minimal hysteresis heat loss).
• Representative Alloys: Low-carbon mild steels (AISI 1008, 1018, ASTM A36), commercially pure iron (Armco iron), soft silicon steels (electrical transformer sheet), annealed ductile irons.
• NDT Behavior: Readily magnetized to saturation with modest electrical amperage; loses magnetic flux rapidly upon current cessation; extremely easy to demagnetize with standard AC coil equipment.

Magnetically Hard Materials

Magnetically hard materials contain dense microstructural barriers—such as martensitic needle plates, fine carbide dispersions, high residual lattice stresses, or heavy dislocation forests—that aggressively pin domain walls.

• Loop Characteristics: Wide, broad, squat hysteresis loop; gradual slope (low permeability $\mu$); high coercive force ($H_c > 50\text{ to }100\text{ Oe}$, exceeding $1,000\text{ Oe}$ in permanent magnets); high retentivity; large enclosed loop area (substantial hysteresis heat loss).
• Representative Alloys: High-carbon tool steels (D2, M2, O1), quenched and tempered alloy steels (AISI 4140, 4340, 300M in high-hardness conditions), martensitic stainless steels (AISI 440C), precipitation-hardened steels (17-4 PH H900), permanent magnet alloys (Alnico, Samarium-Cobalt, Neodymium-Iron-Boron).
• NDT Behavior: Requires high magnetizing amperages to reach saturation; retains strong, stable residual magnetic fields; highly resistant to demagnetization, requiring multi-step reversing step-down DC demagnetization cycles.

Summary Comparison Table: Magnetically Soft vs. Hard Materials

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Continuous vs. Residual Magnetization Techniques

The shape and properties of the B-H curve govern the foundational selection between the Continuous Method and the Residual Method in Magnetic Particle Testing.

The Continuous Method

In the continuous method, magnetic particles (dry powder or wet suspension) are applied to the test surface while the magnetizing current is actively flowing, or the suspension application is ceased immediately prior to current termination.

• Physical Mechanism: Inspection takes place at peak magnetic induction ($B$), operating near the upper knee or saturation plateau of the B-H curve.
• Advantages:
  1. Maximum Sensitivity: Because $B$ is maintained at peak operating levels, magnetic flux leakage across tight, shallow, or subsurface flaws is maximized.
  2. Universal Applicability: It is the only reliable method for magnetically soft alloys (low-carbon steels) because their low retentivity causes residual flux to drop precipitously once current stops.
  3. Code Mandate: Major aerospace and structural fabrication codes (e.g., ASTM E1444, ASME Section V Article 7) strictly mandate the continuous method for all primary inspections unless an explicit engineering variance is authorized.

The Residual Method

In the residual method, the component is magnetized by an initial current pulse, the magnetizing current is completely terminated, and magnetic particles are subsequently applied to the surface utilizing only the residual magnetic field ($B_r$).

• Physical Mechanism: Inspection relies entirely on the material's retentivity and coercive force to sustain a leakage field without an active external magnetizing force.
• Strict Limitations:
  1. Material Restriction: Can ONLY be utilized on magnetically hard materials possessing high retentivity ($B_r$) and high coercive force ($H_c$). If attempted on soft low-carbon steel, the residual field drops below the threshold needed to bridge crack air gaps, yielding false-negative results.
  2. Reduced Sensitivity: Even in hard steels, the residual flux density $B_r$ is substantially lower than saturation induction $B_{sat}$. Leakage fields are weaker, making the residual method blind to tight micro-cracks and subsurface discontinuities.
  3. Restricted Use Cases: Primarily utilized for automated, high-throughput screening of small, identical, case-hardened fasteners, bearing balls, or heat-treated tool components where suspension pooling under active current might cause excessive background.

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Level III Engineering Analysis: Demagnetization Physics

Demagnetization is the systematic reversal and progressive reduction of the B-H hysteresis loop down to the origin $(0,0)$.

• A high coercive force ($H_c$) indicates that magnetic domain walls are firmly trapped at microscopic pinning sites.
• An inspector attempting to demagnetize a quenched AISI 4340 landing gear cylinder ($H_c \approx 70\text{ Oe}$) with a portable 60 Hz AC pull-through coil will fail: the skin effect confines the 60 Hz field to the outer millimeter of the part, leaving the internal core heavily magnetized, and the peak field intensity of the coil is often insufficient to overcome $H_c$.
• The Level III must design a demagnetization procedure utilizing low-frequency, reversing step-down direct current (e.g., 0.5 to 1 Hz, 30 to 40 steps, stepping current down from 3,000 Amperes to zero), ensuring full-depth penetration and complete loop collapse.