Analytical Expression for Hysteresis Loop Area EDIT:  I've updated the title and restatement of my question.
Is there a known analytical solution for the area of a hysteresis loop?  I've found that a hysteresis loop can be described analytically using parametric equations, however performing contour integration has not bee successful.
I’ve been working with a system that exhibits hysteresis and I’ve found that the more common models do not work for me.  I am wondering if anyone is aware of other models that might be out there for hysteresis.  Thus far, I’ve tried the Preisach model and Jiles-Atherton model, neither of which allow me to simulate the types of loops that I work with.
I’ve gone the the Physics, Mathematics, Electrical Engineering, Mathematica, and some other StackExchange communities, but haven’t found anything beyond Preisach or Jiles-Atherton being mentioned as potential approaches to modeling/simulating hysteresis.  I found some information on the VINCH model, the Bouc–Wen model, and the Bouc–Wen–Baber–Noori model, but none of them fit what I am trying to model.  This model really needs to be analytical rather than computational.  Some of the models use functions like sign[x], and so forth, that aren't going to work for my system.
I’ve been messing with the idea of using a parametric approach, that is to say, having both quantities that are shown in the hysteresis plot (for instance, M and H for ferromagnetism, or D and E for ferroelectricity, etc.), both being dependent on some other variable.
I’ve found one paper that analytically handles this type of model by Lapshin, R.V., Analytical Model for the Approximation of Hysteresis Loop and Its Application to the Scanning Tunneling Microscope. Review of Scientific Instruments, 1995. 66(9): p. 4718-4730 (DOI: 10.1063/1.1145314). The model shows some interesting results, however the hysteresis loops that I am trying to model are more complex than those in the publication.
Actually, I am considering trying to use a superposition of solutions to try and construct a solution appropriate for my system.  However, my attempts have yet to work.
What I am trying to model is the following: a system in hysteresis that allows for both linear and non-linear "slopes", as well as linear and non-linear phase lags.
Here are the pieces that you see in the graph above:
(1)  A linear (reversible), in-phase component, same slope over the entire range (BLUE)
(2)  A linear (irreversible), out-of-phase component (RED)
(3)  A non-linear (reversible), in-phase component, one slope near origin, saturates far from origin (MAGENTA)
(4)  A non-linear (irreversible), out-of-phase component (GREEN)
If this was a circuit model, and were plotting charge vs. voltage (q vs V), while driving the circuit sinusoidally, (1) would be simple linear resistor, (2) would be a linear capacitor, (3) would be a non-linear resistor (perhaps one that saturates above a certain voltage, but not a memristor), and (4) would be a non-linear capacitive element that is bistable with very constant remanence and coercivity, not sure what it would be called though, perhaps an "ideal hysteron".
If you add these four components together, you can get something resembling a proper hysteresis loop:

Notice that the hysteresis loop can also be rotated in phase space.
Has anyone seen a model that would allow this?  Also, is there some way to correlate these components to real physical properties for a hysteresis loop?
Thanks!
 A: "Efficient Estimation of Elliptical Hysteresis with Application to the Characterization of Heat Stress"
F. Yang & A.M. Parkhurst.
I have been reading this paper and working with R package. I am trying to model a sensor. I am trying to make a predictive model. Their package is available on CRAN now. Could be a good reference. 
A: Hysteresis loops of Leaf, Crescent (Boomerang), and Classical types
Dear Colleague!
Recently, I have published the article “An improved parametric model for hysteresis loop approximation” that deals with the problem.
Abstract
A number of improvements have been added to the existing analytical model of hysteresis loops defined in parametric form. In particular, three phase shifts are included in the model, which permits us to tilt the hysteresis loop smoothly by the required angle at the split point as well as to smoothly change the curvature of the loop. As a result, the error of approximation of a hysteresis loop by the improved model does not exceed 1%, which is several times less than the error of the existing model. The improved model is capable of approximating most of the known types of rate-independent symmetrical hysteresis loops encountered in the practice of physical measurements. The model allows building smooth, piecewise-linear, hybrid, minor, mirror-reflected, inverse, reverse, double, and triple loops. One of the possible applications of the model developed is linearization of a probe microscope piezoscanner. The improved model can be found useful for the tasks of simulation of scientific instruments that contain hysteresis elements.
R. V. Lapshin, An improved parametric model for hysteresis loop approximation, Review of Scientific Instruments, vol. 91, iss. 6, no. 065106, 31 pp., 2020 (DOI: 10.1063/5.0012931)
The article can be downloaded at https://doi.org/10.1063/5.0012931
Take attention to the “Supplementary material” that accompanies this article:

*

*r.v.lapshin_hysteresis_loop_mathcad_2001i_worksheets_ver.03.01.2020.zip (34 MB)
https://aip.scitation.org/doi/suppl/10.1063/5.0012931/suppl_file/r.v.lapshin_hysteresis_loop_mathcad_2001i_worksheets_ver.03.01.2020.zip


*r.v.lapshin_hysteresis_loop_readable_mathcad_2001i_worksheets_ver.03.01.2020.pdf (76 MB)
https://aip.scitation.org/doi/suppl/10.1063/5.0012931/suppl_file/r.v.lapshin_hysteresis_loop_readable_mathcad_2001i_worksheets_ver.03.01.2020.pdf
The supplementary material includes zip archive with Mathcad 2001i worksheets, where all aspects of the original and improved parametric models of hysteresis loop are considered in detail (definitions, proofs, illustrating graphs, comments, notes). Those who do not have Mathcad software may take advantage of the enclosed readable Mathcad worksheets as a PDF-document. Due to restriction on the length of the article, it presents only the most common hysteresis loops. If the required loop is absent in the article, it makes sense to search in the supplementary material.
Best regards,
Rostislav Lapshin
Rostislav V. Lapshin, Ph. D.
Institute of Physical Problems
