electromagnetism - Equation describing magnetic hysteresis

So when you're looking at B-H curves for ferromagnetic substances, you often see these magnetic hysteresis curves, which occur, I gather, largely because of domain formation which has some reversible and some irreversible components:

I've been reading through many papers, web sites and this book (Hysteresis in Magnetism), but I haven't seen any equations for how to generate these curves. I recognize that there may be no easy way to express the entire curve in a single equation, but clearly people are generating these plots, and I think they're doing it from some equations, presumably based on some characteristic parameters like the saturation remanance, the coercive force, etc. Is there anyone out there who can help me with this?

Personally, my first choice would be an equation that actually describes the system given some set of material conditions (subject to some constraints is fine, too) but if that's quite complicated and for the most part, people just stitch together two sigmoid functions until it looks about right, then I'm fine with that answer as well, so long as there's some justification for why this is done in there somewhere.¹

¹_{Full disclosure: I am in thesis deadline mode and I'm currently probably a 5 on the Stanford Sleepiness Scale, so I apologize if some of the words in this post don't make sense - I'll try and edit it if people find that I haven't adequately conveyed my question.}

Answer

The Jiles-Atherton model of ferromagnetism is used in some circuit analysis programs. It may be overkill for this question, but it does give pretty pictures.

I'm going to work in MKS units exclusively.

The model equations are: $$ B= \mu_0 M \quad , \quad M = M_{rev} + M_{irr} $$ where $B$ is the magnetic flux density, and the magnetization $M$ is composed of reversible ($M_{rev}$) and irreversible ($M_{irr}$) components.

Physically, during the magnetization process:

• $M_{rev}$ corresponds to (reversible) magnetic domain wall bending (the S-shaped magnetization curve, but without hysteresis)


• $M_{irr}$ corresponds to (irreversible) magnetic domain wall displacement against the pinning effect (the hysteresis).

These components are calculated according to:

$$ M_{rev} = c (M_{an} - M_{irr}) $$ $$ M_{an} = M_s \left[ \coth \left(\frac{H + \alpha M}{A}\right)-\frac{A}{H+ \alpha M}\right] $$ $$ \frac{dM_{irr}}{dH} = \frac{M_{an}-M_{irr}}{k \delta - \alpha (M_{an}-M_{irr})} $$

Here the anhysteretic magnetization $M_{an}$ represents the magnetization for the case where the pinning effect is disregarded. (This case corresponds to $c=1$, where $M=M_{an}$ and $M_{irr}$ therefore does not contribute to $M$.)

The quantity in the expression for $M_{an}$ in square brackets is the Langevin function $\mathcal{L}$: $$ \mathcal{L}(x) = \coth(x) - \frac{1}{x} \quad , \quad \mathcal{L}(x) \approx \left\{\begin{array}{ccc} \frac{x}{3} & , & |x|<<1 \\ 1 & , & x >> 1 \\ -1 & , & x<<-1 \end{array} \right.$$

and $\delta$ is the sign of the time rate of change of the applied magnetic field $H$: $$ \delta = \left\{ \begin{array}{ccc} +1 & , & \frac{dH}{dt}>0 \\ -1 & , & \frac{dH}{dt}<0 \end{array} \right. $$

$M_{rev}$ can be eliminated from this system of equations to reduce their number by 1: $$ M = c M_{an} + (1-c)M_{irr} $$

The equations for $M, M_{an},$ and $M_{irr}$ are inter-dependent and so are to be solved simultaneously.

There are 5 parameters (listed here together with sample values):

• $M_s$, the saturation magnetization [1.48 MA/m]


• $c$, the weighting of anhysteretic vs. irreversible components [0.0889]


• $\alpha$, the mean field parameter (representing interdomain coupling) [0.000938]


• $A$ sets the scale for the magnetic field strength [470 A/m]


• $k$ sizes the hysteresis [483 A/m]

For the values listed, a crude spreadsheet produced this plot:

The horizontal axis is the applied magnetic field H, in A/m, sweeping from 0 up to 2500, then down to -2500, and then up again to 2500. The vertical axis is the flux density B in T.

This example comes from a 1999 IEEE paper by Lederer et al, "On the Parameter Identification and Application of the Jiles-Atherton Hysteresis Model for Numerical Modelling of Measured Characteristics".

It appears that choosing the parameters to match a given material is a chore, but that's another story...