What is the B-H curve and how does it determine the characteristics of magnets?

Magnets, as we know, are used in many applications worldwide. To optimize their use in specific instances, we need to understand the behavior of different magnetic materials when magnetized. For example, magnets used in transformer cores need to be able to alter their polarity rapidly when subjected to an alternating current. On the other hand, magnetic materials used in hard disk drives and other data storage applications need to be easily magnetized but retain their magnetism for long periods after the magnetizing field is removed.

The B-H curve, also known as the hysteresis curve, provides a powerful tool for this purpose. This article delves into the intricacies of the B-H curve, elucidating its significance in characterizing the magnetic properties of materials, specifically, their Remanence, Coercivity, Maximum Energy Product, and Permeance.

When a magnetic field is applied to a ferromagnetic material, a magnetic flux is generated within the material. The B-H curve graphically represents the relationship between the applied magnetic field strength (H) and the resulting magnetic flux density (B) within a magnetic material. It serves as a "fingerprint" of the material's magnetic characteristics.

The B-H curve is not linear; it manifests as a double-pointed, wide-bellied graph called the hysteresis loop, which is central to understanding magnetic behavior.

• H (Magnetic Field Strength): This represents the external magnetic field applied to the material as the driving force that attempts to align the magnetic domains within the material. It is measured in Oërsteds (Oe) - in honor of the Danish physicist Hans Christian Oërsted - in CGS units or Amperes per meter (A/m) in SI units. 1 Oërsted ≈ 79.5775 A/m.
• B (Magnetic Flux Density): This quantifies the magnetic field induced within the material due to the applied field and represents the material's response to the applied field. It is measured in Gauss (G) in CGS units or Tesla (T) in SI units, with 1 Gauss = 10^-5 Tesla.

Analyzing the B-H curve reveals several critical parameters that define a material's magnetic properties:

• Origin (0,0): This point signifies the initial state, where both H and B are zero. The material is not magnetized and not subject to any magnetizing force.
• Initial Magnetization Curve (dotted line from the origin to Bs): As the applied field (H) increases, the magnetic flux density (B) rises along a non-linear curve. The slope of the initial magnetization curve is related to the material's Permeability (denoted by the symbol μ), which describes how easily the material can be magnetized.
• Saturation Magnetization (Bs): Beyond a certain point, increasing H does not result in any further increase in B. This plateau represents Saturation Magnetization (Bs), where all magnetic domains within the material are aligned.
• Remanence (Br): When the applied field (H) is reduced to zero, the magnetic flux density (B) does not return to zero. The remaining magnetization is termed Remanence or Residual Magnetism (Br). A high remanence indicates a strong permanent magnet.
• Coercivity (Hc): To nullify the magnetic flux density (B) within the magnet, a reverse magnetic field needs to be applied. The magnitude of the reverse field required to render the material completely non-magnetic (where B=0) is called the Coercivity (Hc) of the material. It quantifies the material's resistance to demagnetization. High coercivity signifies a "hard" magnetic material.
• Loop Shape and Hysteresis Loss: The shape of the hysteresis loop provides further insights:
  • Wide Loop: A wide loop characterizes "hard" magnetic materials, exhibiting high remanence and coercivity. These are suitable for permanent magnet applications.
  • Narrow Loop: A narrow loop is typical of "soft" magnetic materials, possessing low remanence and coercivity. These are easily magnetized and demagnetized, ideal for transformer cores and electromagnets.
  • Enclosed Area: The area enclosed by the hysteresis loop is proportional to the energy lost as heat during each magnetization cycle, known as hysteresis loss.
• Maximum Energy Product (BHmax): This is a magnetic characteristic, independent of the size and shape of a magnet, depending solely on the magnetic material in question. BHmax determines a magnet's grade and is generally expressed in units of MGOe (Mega-Gauss-Oersted). For example, an N52 grade Neodymium magnet has a typical BHmax of 52 MGOe.
  
  In simple terms, a large and a small magnet made from the same magnetic alloy will have the same BHmax, though they may demonstrate vastly different magnetic fields or flux. BHmax is attained when the magnet operates at the highest Induction level (Gauss) and has the smallest volume for a particular pole cross-section. A magnet with a larger volume will produce a larger magnetic field and more flux, even though the magnet may not be operating at BHmax. 
  
  To get the value of (BH)max, the following steps need to be performed:
  • Identify the Second Quadrant
    The B-H curve is a loop, but for determining BHmax, we focus on the portion where the magnetic flux density (B) is positive, and the magnetic field strength (H) is negative. This is the second quadrant of the graph.
  • Find the Point of Maximum Product
    Along the demagnetization curve in the second quadrant, there exists a point where the product of B and H (their absolute values) is the highest. This point represents the BHmax.
  • Calculate (BH)max
    Multiply the absolute value of B at that point by the absolute value of H at that point. The result is the maximum energy product, BHmax.
  • Graphical Representation
    This can be visualized by imagining rectangles inscribed under the demagnetization curve in the second quadrant.
     The rectangle with the largest area corresponds to the (BH)max, and the coordinates of its corners touching the curve give the B and H values for the calculation.

This brings us to the final topic, the Permenace (P) of a magnet. Specifically, the Permeance refers to the amount of magnetic flux available in a magnetic circuit, where a magnet is operating as part of an assembly, e.g., lifting a load or holding a workpiece during machining.

Permeance is calculated using the formula:

A line drawn from the origin with the slope P is called the Load Line of the magnetic circuit, and the point where the load line crosses the B-H curve is called the magnet's Operating Point within that circuit. Different configurations of magnets have different Permeance coefficients and, hence, different Load Lines. An excellent article on calculating the effective Permeance coefficient of a magnetic circuit can be found here.

In short, all the calculations that go into determining the type of magnet required for a specific need, and how it works in a magnetic circuit, starts with identifying the B-H curve for the material, followed by the geometry of the actual magnetic circuit to determine the load line and operating point of the magnet.