Stoletov curve approximation for ferromagnets

By itself, the approximation is an approximation to a real function, allowing her to explore analytically, and even to "see" beyond the data provided [1]. In this application we will show a pretty good approximation of the curve Stoletov [2], which allows to analytically study the dependence of magnetic permeability on intensity of magnetic field for ferromagnetic materials is \(\mu(H)\). This need only three points from the graph of measurements. If it is necessary to study the dependence of the relative inductance of the current \(M(I)\), then such points need only two. That's what we're going to approximate, and a more General approach will show below.

The following approximation for the curve Stoletov: \[M(I) = A \exp\left(\frac{-I}{I_0}\right) + \left(B - A \exp\left(\frac{-I}{I_0}\right)\right) \sin\left(\frac{\pi I}{I+I_0}\right) \qquad (1)\] unfortunately, the accuracy of its approximiately poor for calculations which requires minimum deviation from the real curve, it is not applicable. For a more accurate approximation you can use the standard approach, but it will require a power series with 6-7 members, the search of coefficients which will be quite complicated task: \[M(I) = 1 + k_1\,I + k_2\,I^2 + k_3\,I^3 + k_4\,I^4 + k_5\,I^5 + \ldots \qquad (2)\] And even then the accuracy of the approximation will remain low not to mention the prediction ability of the approximating function to predict the behavior of a real function outside of the measurement points.

Based on the studies of real dependences \(\mu(H)\), the author has developed a more simple approximation function, allows you to fairly accurately closer to the real curve and even predict its behavior outside measurements. This will require only two points of the investigated graph — (\(M_m, I_m\)) (\(M_e, I_e\)), and finding the coefficients of the polynomial will not be too difficult.

At first the author had based the following polynomial: \[M(I) = {1 + k_{11} + k_{12} I^2 + k_{13} I^3 \over 1 + k_{21} + k_{22} I^2 + k_{23} I^3} \qquad (3)\] But as a result of experiments proved that it is possible to remove some of the members maintaining the approximation accuracy (the curve Stoletov): \[M(I) = {1 + k_{12} I^2 \over 1 + k_{22} I^2 + k_{23} I^3} \qquad (4)\] in addition, this approach allowed to find the coefficients of this polynomial without using matrices, cubic or even quadratic equations. The coefficients sequentially: first \(k_{23}\), then, on the basis of \(k_{22}\), then \(k_{12}\) \[k_{23} = 2 {M_m - 1 \over M_m\, I_m^3}, \quad M_m \gt 1\] \[k_{22} = {1 \over M_m - M_e} \left( {M_e - 1 \over I_e^2} + k_{23} \left( M_e I_e - \frac32 M_m I_m \right) \right) , \quad M_m \gt M_e \] \[k_{12} = M_m (k_{22} + \frac32 \, k_{23}\, I_m) \qquad (5)\] As you can see, finding the coefficients of the polynomial requires fairly simple arithmetic.

The more General case

In some cases we need to obtain are not relative, and the absolute dependence of magnetic permeability on the current \(\mu(I)\). Then we'll need another point — the initial permeability \(\mu_0\), and instead of points (\(M_m, I_m\)) (\(M_e, I_e\)) to their absolute counterparts: (\(\mu_m, I_m\)) and (\(\mu_e, I_e\)). The General formula (4) then transforms as \[\mu(I) = \mu_0 {1 + k_{12} I^2 \over 1 + k_{22} I^2 + k_{23} I^3} \qquad (6)\] the Formula for finding the coefficients of (5) will remain the same, but they will be some substitutions: \[M_m = {\mu_m \over \mu_0} , \quad M_e = {\mu_e \over \mu_0} \qquad (7)\] Count the current in the magnetic field \(H\) and obtain the dependence of \(\mu(H)\) also is not extremely difficult, however it will require knowledge of the parameters of the real device structure: \[H = {I\, N \over \ell}\qquad (8)\] where: \(N\) is the number of turns in the coil, \(\ell\) — the average length of the magnetic line of the core [3]. Total dependence in this case will be expressed as \[\mu(H) = \mu_0 {1 + h_{12} H^2 \over 1 + h_{22} H^2 + h_{23} H^3} \qquad (9)\] and the new coefficients will be recalculated within thus: \[h_{12} = k_{12} \left({\ell \over N} \right)^2 \quad h_{22} = k_{22} \left({\ell \over N} \right)^2 \quad h_{23} = k_{23} \left({\ell \over N} \right)^3 \qquad (10)\]

Application

If a ferromagnetic material is made for systems where you want to perform calculations on the curve Stoletov (e.g. parametric oscillators), except for initial permeability, their technical characteristics appropriate to add values of three coefficients of \(h_{12}\, h_{22}\, h_{23}\). According to him, thanks to formula (4-10) can recover the curve. After the conducted laboratory researches of a ferromagnet, these ratios can be obtained by the formulas (4), (7), (10).

An example of approximation of the actual measured characteristics of the ferrite ring can be found here: the measured characteristics, and approximation of this curve. In its entirety, the deviation from the actual graph does not exceed 3%. Also, if in the calculator to increase the value of the initial current, you can see the result of the prediction of this approximation.

__The materials used__

© Vyacheslav Gorchilin, 2017

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