20230604
Finding Coefficients for the Stoletov Curve
For the first time, the three coefficients of the Stoletov curve were obtained in this paper.
They allow you to completely restore this curve or plot the dependence of the induction on the magnetic field strength.
These dependencies are necessary for designers to determine the magnetic permeability, or induction in the coil, for any value of current or magnetic field strength.
There is a way to build the Stoletov curve and find its coefficients using special stand.
The disadvantage of this approach is a very limited range of measured cores, because larger sizes require a more powerful booster for the stand.
Also, this method involves several dozen measurements and their inclusion in a special table.
In this paper, the author proposes a new, simpler way to find these coefficients.
To do this, you need to assemble a breadboard with a regular key on a mosfet transistor, and connect it to a standard signal generator.
In series with the coil L it is necessary to connect a resistor R with a relatively low resistance,
on which to measure the voltage (current through the coil) using an oscilloscope.
The stand itself, and the measurement process, is described in detail in this paper.
With its help, it is necessary to make only three current measurements according to known time values, using cursor measurements provided in any modern device (Fig. 1).
Then, using the measured or reference data of the core, and the parameters of the coil itself, these coefficients can be calculated.
Actually, this material is devoted to the mathematics of such a calculation.
Here we will consider a coil with a ferromagnetic closed core without a gap (Gapped).
For open cores, it will be necessary to additionally introduce the corresponding demagnetizing factor [1].
And the calculation of coefficients for cores with a gap will, apparently, be devoted to a separate study.
Fig.1. Definition of three measuring points on the oscillogram (coil current versus time)

This work appeared as a result of a rather long theoretical and practical preparation.
Therefore, here the author will often refer to previously derived formulas, so as not to clutter up this material with these calculations.
Readers can always doublecheck their conclusion by clicking on the links provided.
Find coefficients
If the inductance of the coil (with a core) did not change depending on the current passing through it, then everything would be calculated very simply, and this work would not make sense.
In fact, such a dependence exists: it has a very unusual form [2], greatly complicates the calculation formulas, but it, under certain conditions, allows you to get some energy increase,
which we will talk about in future works.
The Stoletov curve for ferromagnets, and hence the inductance of the coil \(L\) with such a core,
can be described by a simple polynomial (4) developed by here:
\[L = L_0 {1 + k_{12} I^2 \over 1 + k_{22} I^2 + k_{23} I^3} \tag{1}\]
Here: \(L_0\) is the initial inductance of the coil (without current in it),
\(I\)  current through the coil,
\(k_{12}\, k_{22}\, k_{23}\) are the coefficients of the Stoletov curve, which we have to find.
There you can also find the derivation of one of the coefficients  \(k_{23}\), through the maximum value of the curve \(M_m\):
\[ k_{23} = 2 {M_m  1 \over M_m\, I_a^3} \tag{2}\]
For further calculations, it is necessary to introduce the maximum inductance \(L_a\), which can be achieved in the coil at a current \(I_a\).
But we can determine these parameters by the first measuring point 1, which is approximately in the middle of the first current section (Fig. 1):
\[ M_m = {L_a \over L_0}, \quad I_a = I_1 \tag{3}\]
The maximum inductance itself is found from expression (17), taken from here:
\[L_a = {t_1\, R \over \ln \left( {U \over U  I_1 R} \right) } \tag{4}\]
Then we can find the first coefficient for the Stoletov curve as follows:
\[ k_{23} = 2 {L_a  L_0 \over L_a\, I_1^3} \tag{5}\]
From formula (1.8) derived by here, one can immediately find the following factor:
\[ k_{12} = k_{23} {3\, U \over R} \left( 1  \mathrm{e}^{\tau R / (3 L_0)} \right) \tag{6}\]
Here we have added the correction factor \(3\) to the numerator, and to the exponent; he refines this formula, which he managed to find out as a result of practical measurements.
In formula (6), one more parameter appears  the core saturation time τ, which is approximately between t_{2} and t_{3} (Fig. 1).
More precisely, it is determined by formulas (1416) taken from here,
which in turn have been detailed in this appendix.
If for some reason the core saturation time cannot be found using the proposed method,
but the core saturation induction \(B_s\) is known, then this time can be calculated approximately
according to formulas (2.8) or (2.9) from of this work.
At the same time, the accuracy of finding the coefficients \(k_{12}, k_{22}\) will also decrease slightly,
on the other hand, the search for all three coefficients for the Stoletov curve can be performed using only one measurement, the first measurement point.
It remains to find the last coefficient.
To do this, we take the original formula and transform it in relation to the first point of our measurements
\[{1 + k_{12} I_1^2 \over 1 + k_{22} I_1^2 + k_{23} I_1^3} = {L_a \over L_0} \tag{7}\]
from which we derive the last missing coefficient:
\[ k_{22} = {1 \over I_1^2} \left( {L_0 \over L_a} \left(1 + k_{12} I_1^2 \right)  1  k_{ 23} I_1^3 \right) \tag{8}\]
or
\[ k_{22} = {L_0 \over L_a} \left({3 \over I_1^2} + k_{12} \right)  {3 \over I_1^2} \tag {9}\]
All three coefficients are thus found.
Recheck
Using the three already known coefficients for the Stoletov curve, you can again get the current \(I_1\) for rechecking.
Recall that at this current the inductance of the coil is maximum: \(L=L_a\).
It is derived from (1), and already known coefficients, by solving the cubic equation [4]:
\[ I_1 = \sqrt[3]{ C + \sqrt{C^2 + D} } + \sqrt[3]{ C  \sqrt{C^2 + D} }
\\
C = {k_{12}  k_{22} \over k_{12}\, k_{23}}, \quad D = {1 \over k_{12}^3}
\tag{10}\]
Taking this current, you can once again remeasure the time \(t_1\) on the oscillogram, and then recalculate \(L_a\) using formula (4), which increases the accuracy of obtaining the desired coefficients by an order of magnitude.
Conclusions
In this work, the author proposed a method for determining three coefficients for the Stoletov curve from three points on the oscillogram (Fig. 1).
These coefficients are found sequentially: first \(k_{23}\) by formula (5), then \(k_{12}\)  by formula (6), and then \(k_{22} \)  by formula (9).
These coefficients are applicable, as invariants, to describe the operation of a coil with a core.
The coefficients for describing the Stoletov curve directly for a ferromagnetic core (without taking into account the coil) \(h_{12}\, h_{22}\, h_{23}\) can be obtained by formula (10)
from this work.
The technique was tested and corrected on a special bench
and showed fairly accurate results.
As a result of its development, certain patterns of behavior of ferromagnets, and hence the coefficients for the Stoletov curve, were revealed,
depending on the frequency (speed) of the transient process.
At the moment, the author recommends that these coefficients be indicated in the passport of the ferromagnetic core for three ranges of its operation:
low frequency, mid frequency and high frequency.
For these three ranges, the coefficients for the same core will be slightly different.
Such an indication is necessary for more accurate calculations of generators in which the operating frequency of the coil core or the transient time is known.
Calculator
All together was collected by the author in a specialized calculator,
where you can quickly calculate the desired coefficients, view the resulting graphs and even find some of the missing parameters of the coil and its core.
In the same place, in the "Details" section, you can find the methodology presented here, painted in steps.