How does magnetic hysteresis work? Mathematical principles

In the first part of this blog post, we explored magnetic hysteresis in ferromagnetic materials as the result of the interaction between an external magnetic field H and the magnetic flux density B. The latter describes the processes inside the ferromagnetic material.

The term hysteresis, from the Greek hysteros meaning delay or lag, refers to the fact that flux density B follows the external field strength H with a time delay. This is because the Weiss domains within the material can only reorient when sufficient energy is available to overcome internal resistance.

Having explained the physical basis of the hysteresis curve in the first part, we will now also use mathematics to analyse its properties in more detail.

Requirements for a mathematical model

A suitable mathematical model should be able to reproduce several properties of real ferromagnetic materials:

• the shape and course of the initial magnetisation curve
• the shape and course of the hysteresis curve
• different curve widths for soft and hard magnetic materials

To produce a hysteresis curve, we need a magnetic field H that periodically changes in magnitude and direction. This can be achieved, for example, using a coil carrying alternating current. The magnetic field then varies sinusoidally over time.

The simple case: no hysteresis

Let us first assume that the magnetic flux density B follows changes in the magnetic field without delay.

Mathematically, this means:

H = H₀ · sin(t)

B = B₀ · sin(t)

In this case, both quantities follow identical sinusoidal patterns over time.

Plotting B against H produces only a linear relationship: when H rises, B also rises; when H falls, B also falls. Graphically, this is a simple inclined line, along which we move up and down as the field varies periodically. Such a relationship exhibits no hysteresis.

Introducing a time delay

To model real magnetic hysteresis, we must account for a time delay between H and B. Mathematically, this is achieved by introducing a phase shift φ:

H = H₀ · sin(t)

B = B₀ · sin(t + φ)

The phase shift φ describes the additional time that flux density B needs to respond to changes in the field H. Consider the relationship between H and B with phase shifts of 1/20 and 1/10 of a period. For an alternating quantity at a frequency of 50 Hz, with a period of 20 ms, these values represent delays of 1 ms and 2 ms respectively in the “response” of flux density B to the “commands” of field strength H.

In other words, plotting B against H now produces a closed curve enclosing an area: a hysteresis curve. Initially, however, this curve is elliptical and does not yet resemble the typical, more “rectangular” hysteresis loops of real materials.

Nevertheless, two important relationships are already apparent:

• A phase shift between H and B produces hysteresis in principle.
• Larger phase shifts lead to broader hysteresis curves.

This already allows us to describe the behaviour of soft and hard magnetic materials mathematically.

Modelling the initial magnetisation curve and saturation

To represent the initial magnetisation curve and saturation behaviour correctly as well, we must abandon the assumption that B also varies sinusoidally. Instead, we assume that the flux density follows a more “box-shaped” waveform. This can be achieved mathematically by combining several sine functions of different orders:

B = B₁ · sin(t) + B₃ · sin(3t) + B₅ · sin(5t) + B₇ · sin(7t) + B₉ · sin(9t)

Superimposing such functions allows us to generate different waveforms, such as triangular, box-shaped or more complex patterns.

In this example, five sine functions were combined because this number was already sufficient to produce a close approximation of a box-shaped waveform. When this flux density is plotted against the magnetic field, the upper-right portion of the curve already closely approximates the real initial magnetisation curve, including saturation.

Combining delay and nonlinearity

To produce realistic hysteresis curves, we must now combine two effects:

• the nonlinear, box-shaped waveform of flux density B
• the time delay between H and B

Mathematically, this gives: B = B₁·sin(t+φ) + B₃·sin(3t+φ) + B₅·sin(5t+φ) + B₇·sin(7t+φ) + B₉·sin(9t+φ)

The phase shift φ again describes the delay between the external magnetic field and the material’s response. This combination produces hysteresis curves that closely resemble those measured in practice.

Why does B have a box-shaped waveform?

The shape can be explained directly by the physical processes within the material. As long as there is insufficient energy to reverse the Weiss domains, the magnetic flux density remains almost constant. This corresponds to the horizontal sections of the curve. Once sufficient energy is available, reorientation occurs very rapidly, corresponding to the steep, almost vertical sections of the curve.

The combination of these two effects produces the typical rectangular shape of many hysteresis curves.

The relationship between energy and curve width

The greater the internal resistance within the material, the more energy is required to reorient the Weiss domains. Accumulating this energy requires a longer time delay, represented mathematically by a larger phase shift φ. The result is a broader hysteresis curve, as is typical of hard magnetic materials.

Materials testing using harmonic analysis

The mathematical representation using several sine functions is far more than a theoretical exercise. It forms the basis of a method used in materials testing: harmonic analysis. The hysteresis curve of a reference material is measured, and the flux density B is then decomposed into its individual sinusoidal components. Each of these sine functions is associated with particular material properties.

For example, a change in the amplitude of a particular component, such as B₉, may indicate a change in the microstructure or material properties.

The fundamental and harmonics

In the expression B = B₁·sin(t) + B₃·sin(3t) + B₅·sin(5t) + B₇·sin(7t) + B₉·sin(9t), sin(t) is called the fundamental, while sin(3t) and sin(5t) are called harmonics.

Analysing these harmonics provides important information about the magnetic behaviour and structure of ferromagnetic materials.

An illustrative comparison

Interestingly, everyone performs a similar analysis every day when listening to music. We immediately recognise whether a note sounds harmonious or dissonant. Dissonance often results from “incorrect” harmonic components in the sound. Our brains therefore unconsciously analyse the harmonic structure of acoustic signals. If these components are right, the sound is pleasant; if they deviate, we find it disturbing.

In a similar way, materials testing uses harmonic analysis to detect deviations in a material’s magnetic behaviour.