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The Measure of Magnetic Material Characterizations in Ring SamplesI.Branovitsky, G.Razmyslovitch
Institute of Applied Physics of National Academy of Sciences of Belarus
16, Academicheskaya Str., Minsk, 220072, Belarus
By transformation the Gelder inequality, taking into account the Marenin-Shteimets formula it can be shown, that for real materials, including electric steels, (having a@2), g>1, that is a nonuniform magnetic induction distribution in ferromagnet always results in increasing a magnetic loss in it. For example, when testing ring samples, as it is known, the magnetizing field strength H is varied along a whole ring width during its magnetization by a toroidal coil inversely with the running value of the radius r. This causes a nonuniform distribution of a magnetic induction along the ring width, though the rules require measuring a magnetic loss in uniform magnetizing fields and induction conditions. The difference between a value of a magnetic loss in one and the other of these cases defines a test error, due to a nonuniform induction distribution in a sample:
A form of the magnetic induction distribution function along a radial ring direction depends on both a distribution of a magnetic field strength H in this direction and magnetic material properties. The distribution of the magnetic field strength H along a ring width, has the form
Having in mind that Bm=m om Hm, where m o is the magnetic constant, m is the magnetic material permeability, putting a =2 and considering an induction change along a radial direction of the ring, it can be obtained from (1), when a magnetic permeability does not depend on a field (m(H)=const):
Here . In ferromagnetic materials, as known, the value of a magnetic permeability depends on a magnetic field strength, that is m(H)¹const. It the case of µ(H)= a + eH from (1) it is obtained:
where Hav is the magnetic field strength along the average ring radius rav.
The expression (5) was used for calculating the coefficient value g in real materials while testing them on ring samples. For this purpose by the least squares method a line approximation of experimental dependences m(H) was done for isotropic electric steel in the field intervals, defined by the relation Hmax/Hmin=r with the average amplitude induction values in the ring, equal 0,4; 1,0; 1,5 T. In the every case the values of the approximation coefficients a and b (considering their sings) were found and were substituted in (5) and then the appropriate values of g and also d, were found. In fig.1, a and b, as an example, there are the dependences of a control error of a magnetic loss in a ring of isotropic steel with different magnetic properties on the ratio of the external and internal ring radiuses.
|Fig 1: The r dependence of the magnetic loss control error.|
From the above it follows that for a maintenance of a permissible error while measuring magnetic loss in a ring samples the requirements, imposed on the limiting ratio value of the maximum and minimum radiuses of the ring, depend on magnetic material properties and magnetic fields regions in which tests is carried out.
Let us consider now the influence a nonuniform magnetic induction distribution due to a skin effect on the magnetic loss value.
As is known, the distribution of an a.c. magnetic field strength along a thickness of a plate magnetized by the frequency w takes the form :
where Ho the magnetic field strength on a plate surface; m is the half-thickness of a plate; is the skin effect depth; n is the specific electric resistance of a plate material. Again consider two cases: 1) a magnetic permeability of a plate material does not depend on a magnetic field strength m=const; 2) a magnetic permeability has the linear dependence on a magnetic field strength.
Accepting b(1) a=2 , we write this expression for one - dimensional case as:
For m =const the distribution of the amplitude induction value along the plate thickness it can be written as
Where Bmo is the amplitude induction value for y, tending to m. For a calculation of the integral I2 located in the (7) expression denominator we transform the expression (8), using Taylor series and limiting by two first members of the expansion, i.e. We obtain
Integrating the expression (7) and taking into account (9), we obtain:
Here A = 1/[2(ch2km + cos2km)] ; x = 2km ; j (x )=sh(x )/x ; y (x )=sin(x )/x .
Having in view (10) and denoting j (x )+ y (x )= T(x ), we get the expression for g :
Making similar calculations for the case, when a magnetic permeability is linear dependent on a magnetic field strength, we get
In fig.2 a, b the frequency dependency of the coefficient g for electric steel samples of a different thickness is presented. In so doing the dependencies of fig.2 are constructed for the case: (a) when m=const, (the expression (11)) and (b) - when m(H)¹const (the expression (12)). Comparing these plots one can see that an influence magnetic properties on the coefficient value g is more profoundly expressed for m (H)¹const. This is due to the greater nonhomogenity of magnetic induction distribution along a sample thickness for m(H)¹const.
|Fig 2: The frequency dependence of anomalous magnetic loss on the skin effect in the samples of thickness, mm: 1 - 0,27; 2 - 0,30; 3 - 0,35; 4 - 0,50.|
In such a manner the implemented analysis shows that the skin effect can grately affect electric steel performances in particular its magnetic loss even for small frequencies. This fact must be took into account both when testing the materials and using them in cores of electric machines.
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