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In this paper we demonstrate, that shearing is changing only one parameter of the static loop. By using the shearing factor
N_{s}, linked to the widely used, demagnetization coefficient N
_{D}, we show the one parameter link between the static unsheared and that of the sheared saturation loop, obtained by a non-toroidal, open circuit hysteresis measurement. The paper illustrates the simple relation between open circuit loop data and measured real static saturation data. The proposed theory is illustrated by using the hyperbolic model. For experimental illustration, tests results are used, which were carried out on two closed and open toroidal samples, made of NO Fe-Si electrical steel sheet, mimicking the demagnetization effect of the open circuit VSM measurement. These are both theoretical and experimental demonstrations, that shearing only changes the inclination of the static hysteresis loop. These test results, presented here, agree very well with the calculated results, based on the proposed method.

A practical way of characterising a magnetic material is to measure its static hysteresis loop in a closed magnetic circuit. This usually takes form in a toroid or an Epstein square [

At the beginning we have assumed that, only one of the calculated parameters is affected by the external demagnetization. Test results showed, that this assumption was right. It is possible to work out the un-sheared parameters (equivalent to closed circuit toroidal measurements) from the open circuit results of a VSM results. It will be shown, that this procedure requires only change in the effective slope of the loop with no change in the other major parameters of the modelled sample [

It was initially assumed that, the error coming from the ever present internal demagnetisation and from other sources is smaller than the [

Cullity, who studied the effect of shearing on the internal demagnetization in detail, published the relation specifically to the shape and size of the specimen in graphical form [11-15]. In spite of all the assumptions made, the agreement between the theoretical and experimental results is remarkable. Further experiments are in progress for further verification this simplified method. Work is also extended to include the so far neglected effects to improve on the accuracy.

Preceding the test, the toroidal sample was carefully demagnetized by applying an alternating field of f = 10 Hz with logarithmically decreasing amplitude in 5000 steps from saturation to zero [1,2]. The hysteresis loop was measured with a triangular excitation at a frequency of f = 0.001 Hz and integrated with a Walker integrator (see _{m} therefore at H_{m} = 20 A/m, the rate of change, comes to dH/dt = 4f·H_{m} = 0.08 A/m·s.

The static hystersis loop can only be obtained at low rate of change, particularly when the extremely soft sample has a characteristic square-like hysteresis loop [1,2].

To avoid phase shift between H and B, the current measuring resistance R (see

Although the same test was repeated on ultrasoft, Finemet, from nanocrystalline to Mn-Zn ferrite, NO Fe-Si and low and high carbon steel cores, due to the limited length of the paper, here only the data of one illustrative experiment are included.

The analytical approach is based on the assumption, that in general, particularly in case of soft irons, there are three parallel processes, dominating the overall magnetization process i.e. the reversible and irreversible domain wall movement (DWM), the reversible and irreversible

domain rotation (DR) and the domain wall annihilation and nucleation (DWAN) processes. Although these processes are interlinked, they can be mathematically formulated separately and combined, by using Maxwell’s superposition principle. They individually dominate the low, middle and near saturated region of magnetization and all are supposed to have sigmoid shapes. This model is already used in number of applications [15-18] and it is well documented in the literature therefore, here only a brief summary of the relevant formulation will be given in, canonic form [19,20].

The contribution of the individual processes to the combined hysteresis loop can be described by the following normalized mathematical equations.

for h = h_{m} (3)

Here f_{uk} and f_{dk} are the normalized ascending and descending magnetization functions respectively, h is the field excitation and h_{ck} is the coercivity of the k^{th} process. a_{k} is the amplitude of the component processes present, α_{k} is the slope, n_{S} is the shearing factor (n_{S} is unity for un-sheared loops, see later) and f_{0k} is the integration constant [7,21], while h_{m} represents the maximum field excitation. The index k refers to the individual component processes and n is the total number of processes involved. For most of the magnetic materials used in practice, n equals 3 (see beginning of this section).

We proposed at the beginning, that the changes in the shape of the hystetesis loop measured on a closed toroid can be described by an appropriate change in the α_{k}n_{S} product (see Equation (2)). Here α_{k} represents the slope of the major loop, measured on a toroid sample (for n_{S} = 1, n_{D} = 0) in the model. For demonstration, we used the measurements on a toroidal sample made of NO Fe-Si soft magnetic material. The geometrical details of the toroids used, in the experiment, were: D_{ex} = 25 mm, D_{in} = 15 mm and thickness d = 0.5 mm [

In the first part of the experiment, the toroid was magnetised to B_{m} = 1.68 T by applying a linearly variable field of H_{m} = 2750 A/m maximum amplitude for measuring the hysteresis curve. Then, two 0.5 mm air gaps were cut in the magnetic circuit at a slow rate and the measurement was repeated. Great care was taken not to introduce additional side effects at cutting. This was for the simulation of an open-loop VSM measurement with the same N_{D} and N_{S}. The two hystetesis loops (before and after cutting) are shown in

The iteration yielded the following normalised (lower case letters) and corresponding physical parameters (capital letters) for the best fit:

a_{1} = 1.17, a_{2} = 0.14, a_{3} = 0.35α_{1} = 4.5, α_{2} = 1.2, α_{3} = 0.143h_{c}_{1} = 0.115, h_{c}_{2} = 0.45, h_{c}_{3} = 0.575h_{m} = 7.5.

Corresponding to: _{}

A_{1} =1.17 T, A_{2} = 0.14 T, A_{3} = 0.35 TH_{c}_{1} = 44 A/m, H_{c}_{2} = 164.9 A/m, H_{c}_{3} = 210.75 A/m.

H_{m} = 2750 A/m.

The normalization: 1b = 1T, 1h = 382.6 A/m.

By altering the shearing coefficient n_{s}, in the iteration and using the new values of α_{1}·n_{S} = 0.28, α_{2}·n_{S} = 0.073, and α_{3}·n_{S} = 0.0087, we obtained the closest fit to the sheared hystetesis loop. The best fit appeared to be, when n_{S} was around 0.061.

From the geometry of the magnetic circuit, by using the magnetic Ohm’s law, N_{D} (see Equation 7 for the relation between N_{D} and N_{S}) the demagnetization factor, can be calculated by using Equations (4) and (5). We assumed here, that the amplitude of the magnetization vector is the same in the iron as in the air gap. So:

when μ_{ir} relative permeability is used and it is assumed that l_{a} l_{i} then

where

is the effective magnetic field excitation.

or in normalised form

Here l_{i} and l_{a} are the mean paths of iron and air respectively, μ_{i} and μ_{a} represent the permeability of iron and air (free space) and A is the cross section area of the iron core. It shows that, when N_{S} the shearing coefficient is unity (no shearing), then the N_{D} demagnetization factor is zero. The expression in (9) represents the effective permeability.

We must remind the reader of the following: N_{D} is given traditionally in a value with unity dimension, (i.e. when both the H and B measured in A/m). When different unitary system is used then N_{D} has a different physical dimension and must be normalised (see n_{D} as normalised N_{D}).

By using the geometrical details of the toroid, given above with 1 mm effective gap, the physical shearing coefficient numerical value will come to

here the initial permeability, calculated from the measured static hysteresis loop data of the soft iron sample, was around μ_{i} = 1000.

In the second experiment the closed toroidal sample was magnetized into deeper saturation by applying a high H_{m} = 4125 A/m maximum excitation. Following that, two 1.5 mm cut (effective 3 mm gap) were made in the toroid and the measurement was repeated.

This time however the maximum excitation was increased to approach B_{m} = 1.68 T, achieved without the gap.

The loops were modelled by using the same numerical parameters at a larger h_{m} maximum field excitation. The resulted loops (measured and calculated) are depicted in

Other additional factors like the internal, shape demagnetization factors and/or stresses were taken as negligibly small.

The separation and calculation of α_{k} and n_{S} from the experimentally obtained α_{k}·n_{S} product, requires the presence of two invariants in the transformation.

It is imperative to look at the two loops (un-sheared and sheared) at the same maximum magnetization (or as near as possible). Under this condition we can claim to have the necessary two invariants i.e. the coercivity and the maximum magnetisation.

It can be shown, that the two smaller processes (DR, DWAM) have often a negligible effect on the overall coercivity and experience shows that in most applications,

particularly for soft irons, it is accurate enough in most cases to use the equations describing the dominant process. The proposal is valid, however, when the DR and DWAM processes are not negligible. All resultant parameters, required for the transformation can be calculated (see Appendix) from the parameters of the processes involved. It is enough therefore to demonstrate the process by using one major component [

By using (2a), we can write for the sheared and the un-sheared dominant loop:

(n_{S} being unity in the first case)

From this, the shearing coefficient n_{S} may be calculated as:

where h_{m}_{1} and h_{m}_{1s} are the amplitudes of field excitations necessary to achieve the same maximum magnetization in the un-sheared and the sheared sample respectively.

In soft steel, when the coercivity h_{c} is small relative to the peak excitation, it is often good enough to use the ratio of the maximum field, from the saturation loop, measured by VSM. In the knowledge of n_{S} the value of α_{κ} could be calculated from the measured product.

The external shearing coefficients, calculated from the measured data was n_{s} = 0.065 for the first experiment (1 mm gap) and n_{s} = 0.02 for the second one (3 mm gap). Some allowances should be made to the inherent internal demagnetization present in all type of measurement [

From the second invariant, the ratio between the arctanh values of the remanences (sheared and un-sheared) (see Equation 2(b)) is also equal to n_{S}.

The measured remanence of the 1 mm gap sample was 0.032 T at B_{m} = 1.41 T, while the corresponding modelled figure was 0.035 T. In the case of the 3 mm gap sample, the measured and the modelled remanence figures were 0.0085 T and 0.0066 T respectively, taken at B_{m }= 1.5 T. Based on these figures, B_{r} the projected remanence of the loop, transformed back onto the unsheared (n_{S} = 1 and n_{D} = 0 ) plane came to 0.636 T.

It can be shown also, that the transformation does not change the area enclosed by the hysteresis loop, therefore sheared or un-seared, it has the same hyteretic losses.

It was proven, that shearing and/or un-shearing will not affect most of the static characteristic, material parameters of the soft sample. It is also shown, that the external demagnetization affects one parameter only, namely the effective slope (i.e. n_{S}α) of the hysteresis loop. From the sample’s geometry the shearing coefficient can often be calculated. With the proposed method and the knowledge of N_{D} [_{D}) and calculate the equivalent closed circuit static parameters.

The transformation is based on the two invariants, the coercivity and the maximum excitation.

This model independent transformation can be an useful tool in practice, in experimental work as well as in theoretical applications. The very close agreement between the theoretical and experimental results, the relative simplicity of the procedure and accuracy demonstrates the applicability of the proposed method in practical cases.

The single equivalent tanh function can be calculated from (14), (15), (16), and (17) as shown in (18), (19), (20) and (21).

where the new integration constant is:

The remanence of the resultant hysteresis of the two process magnetization is described from (1b) and (2b) (for h = 0) as:

The maximum magnetization amplitude a_{0} is:

From (15) α_{0}h_{c}_{0} can be calculated. From the firs derivative of (2b) by h comes:

In the knowledge a_{0} and α_{0} the h_{c}_{0} can also be calculated.

and

From (15), (16), (17), (18) and (19) all parameters can be calculated for

and

as functions of the field excitation h.

This calculation can be repeated n – 1 times, for the equivalent set of parameters, where n is the number of processes in magnetization.