Experimental set up for magnetomechanical measurements with a closed flux path sample

Mohamad El Youssef; Adrien Van Gorp; Stéphane Clenet; Abdelkader Benabou; Pierre Faverolle; Jean-Claude Mipo

doi:10.1515/phys-2020-0160

Article Open Access

Experimental set up for magnetomechanical measurements with a closed flux path sample

Mohamad El Youssef , Adrien Van Gorp , Stéphane Clenet , Abdelkader Benabou , Pierre Faverolle and Jean-Claude Mipo

Published/Copyright: August 28, 2020

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From the journal Open Physics Volume 18 Issue 1

Abstract

In this article, an experimental procedure is presented to handle magnetic measurements under uniaxial tensile stress reaching the plastic domain. The main advantage of the proposed procedure is that it does not require an additional magnetic core to close the magnetic flux path through the studied sample. The flux flows only in the sample, and no parasitic air gaps are introduced, thus avoiding the use of the H-coil to evaluate the magnetic field, which is often very sensitive and not easy to calibrate. A specimen of nonoriented FeSi (1.3%) sheet (M330-35A) is characterized under uniaxial tensile stress. To validate the proposed procedure, a comparison with the single sheet tester procedure is carried out. The results obtained by the two procedures are in good agreement. Moreover, to illustrate the possibilities offered by the proposed procedure, we confirm some results obtained in the literature. We show that the positive plastic strain leads to a significant degradation of magnetic behavior. An applied tensile stress on a virgin (unstrained) sample leads to a degradation of the magnetic behavior. However, on a pre-strained sample, an applied tensile stress results in reducing the deterioration caused by the plastic strain until a stress value called optimum is attained. Above this threshold, the magnetic behavior re-deteriorates progressively.

Keywords: nonoriented FeSi steel; magnetic characterization; tensile stress; plastic strain

1 Introduction

The behavior of ferromagnetic materials depends not only on their composition but also highly on the mechanical and microstructural properties. It has been clearly shown that the mechanical stress and the plastic strain are the most important sources of degradation of the magnetic behavior [1,2,3]. As manufacturing processes, such as rolling, punching, bending and annealing, modify the mechanical properties [4,5,6,7,8], they also deeply modify the magnetic behavior of ferromagnetic materials. Compared to the raw material, the characteristics of interest for energy conversion, such as the B(H) curves or the iron losses, are almost always degraded. Thus, the performances of the electrical machines, e.g., Slinky Stator, which is obtained by rolling the lamination continuously and determined during the design stage, are always overestimated. In fact, the performances are predicted based on the characteristics of the raw material without considering the degradation due to the manufacturing processes.

To predict the performances of electrical machines more accurately during the design stage, it is necessary to account for the effect of the manufacturing processes. To reach this goal, the relationship between the stress and the plastic strain, on the one hand, and the magnetic flux density and the magnetic field, on the other hand, should be determined. Even if models are available in the literature [9,10], this relationship is often obtained experimentally. In the literature, different experimental methods have been proposed, which are often based on standard characterization devices (single-sheet tester (SST) or the Epstein frame). Hug et al. [11] studied the impact of plastic strain on the magnetic behavior of FeSi laminations. The magnetic measurement device consists of a ferromagnetic yoke with a primary winding, which is in contact with a pre-strained sample. The flux flowing through the sample is measured by a secondary winding located in the middle of the strained specimen. The same principle of measurement was used later with some improvements in ref. [12]. An opposite ferromagnetic yoke was used to reduce the magnetic flux leakage. It leads to better magnetic field distribution in the sample and reduces the impact of eddy currents on the sample parts in contact with the yokes. However, the impact of the parasitic air gap between the yokes and the sample as well as the neglected magnetic field circulation in the yokes leads to a non-negligible error on the magnetic field estimation.

To avoid the parasitic effect of the air gap between the yoke and the sample, an H-coil probe could be used. It allows a direct measurement of the magnetic field in air. However, some practical disadvantages have been reported on its use and have prevented the standardization of this method. In addition, different devices presented in ref. [13] and [14] enable electromagnetic property measurements under elastic tension, which is limited and not adequate for our measurement application. In fact, we are looking for a device that enables one to perform magnetic measurements under elastic and plastic strains.

In this article, we propose an experimental device to characterize the magnetomechanical behavior of the ferromagnetic material under uniaxial tensile stress reaching the plastic domain, without requiring an external magnetic yoke to close the magnetic flux path. The flux flows only in the sample to avoid parasitic air gaps. It also enables avoiding the use of H-coils to evaluate the magnetic field, which are often very sensitive and not easy to calibrate. First, we present the proposed characterization device and the associated measurement procedure. The results obtained using the proposed device are compared with those obtained using an SST characterization method for a validation purpose. Then we present the results of magnetic measurements performed for different tensile stresses and plastic strains.

2 Characterization procedure

2.1 Sample geometries

The developed device consists of two coils wound on a sample, which forms a closed magnetic path. Its geometry is quite similar to the window frame specimen that was presented in ref. [15]. However, this new version enables accurate magnetic measurements without using H-coils. The magnetic characteristics are determined from the imposed current and the measured voltage on the primary and the secondary windings (Figure 1). With this sample shape, by applying an external force with a tensile test machine, we expect to have a homogeneous stress and strain in the middle parts (MPs). However, the closed parts (CPs) should experience less mechanical stress and strain possible to keep the same characteristics as those of the raw material in this region. Moreover, to limit the effect of cutting, the water jet process has been used to obtain the sample.

Figure 1

The dimensions of the specimen’s geometry (mm) and the corresponding input and output signals required for magnetic and mechanical characterization.

2.2 Calculation principle

With respect to mechanical properties, stress σ is obtained by dividing the measured force by the sample cross-section in Equation (1). While the strain ε, of two legs, is measured directly by strain gauges.

(1) σ = 1 2 F S

where S refers to the section of the sample and F refers to the applied force.

2.2.1 Magnetic field and flux density

For magnetic characterization, magnetic field H is obtained from the imposed current in the primary winding i, using Equation (2) (Ampere’s law), and magnetic flux ∅ is calculated from the measured voltage u, using Equation (3), at the terminals of the secondary winding (Faraday’s law):

(2) H ⋅ l = N 1 ⋅ i

(3) u = − N 2 ⋅ d ∅ d t

where l refers to the magnetic path length of the magnetic circuit, and N ₁ and N ₂ refer to the numbers of the primary and secondary winding turns, respectively. Magnetic flux density B is obtained as follows:

(4) B = ∅ S

For a virgin sample (no stress–no strain), we assume that the sample is isotropic and has homogenous properties. Thus, for a magnetic flux density B at a fixed frequency, the sample has the same virgin reluctance per unit length all along the flux path. We denote the reluctance per unit length as R _v. First, magnetic measurements are performed on the virgin sample (i.e., under no stress). We calculate R _v from current i and magnetic flux ∅ according to the following expression:

(5) R v = N 1 ⋅ i ∅ ⋅ l

Once we have the reluctance per unit length of the virgin material (R _v) in function of the magnetic flux and frequency, it is then possible to estimate the reluctance per unit length in the MP (R _di) corresponding to the sample for a mechanical state di, which corresponds to a given stress and plastic strain levels. In fact, we assume that the reluctances of the CPs remain unchanged and are equal to R _v. Figure 2 presents the magnetic circuit corresponding to the sample, before and during the test, representing a mechanical state di. The magnetic field and the magnetic flux density in the investigated MPs are then determined according to the following calculation:

(6) N ⋅ i = 2 R di ⋅ ∅ ⋅ L di + 2 R v ⋅ ∅ ⋅ L v

(7) R di = ( N ⋅ i − 2 R v ⋅ ∅ ⋅ L v ) / 2 ⋅ ∅ ⋅ L di

(8) H di = R di ⋅ ∅

(9) B di = ∅ S

Figure 2

Equivalent magnetic circuits corresponding to the virgin sample and the sample during test.

2.2.2 Total losses

The total losses in function of the magnetic flux density and the frequency for the virgin material are calculated first from the following equation:

(10) W Tv = 1 T N 1 N 2 ⋅ ∫ i ( t ) ⋅ u ( t ) ⋅ d t

(11) W v = W Tv L

We again apply a procedure similar to that applied to determine the B(H) curve proposed above. The total losses in the CPs are assumed, for given levels of magnetic flux and frequency, to remain unchanged when applying the tensile force. The change in the measured losses is associated with the MPs, which are deformed. Thus, the associated losses in the MPs per unit length can be calculated as follows:

(12) W di = W T − W v ⋅ 2 L v 2 L di

where W _T are the total losses in the sample calculated using Equation (10) when current i and voltage u are measured under stress.

3 Validation

To verify the validity of the proposed approach and, particularly, to check the assumptions made previously for the determination of the B(H) and loss curves associated with a sample subjected to stress, we have made some validation tests. Before carrying out any magnetomechanical measurements, the processes of measuring the mechanical and magnetic characteristics have been checked independently.

3.1 Mechanical aspect

The verification of the mechanical hypotheses requires to consider several points. First, let us consider the hypothesis of having non-deformed zones in the CPs to consider that there is no modification of the reluctance in that area during the magnetomechanical characterization. To validate this assumption, the mechanical problem is simulated using a two-dimensional finite element model with Abaqus. The problem consists of a full constrained side of the sample while the other side is under displacement in the x direction. The displacement value is equal to 2 mm. In addition, the impact of strain rate is neglected; thus, the velocity of the displacement is not considered (Figure 3a).

Figure 3

(a) A schematic of the applied boundary conditions and (b) the stress and equivalent plastic strain distribution for an applied force using the finite element method.

Figure 3b presents the distribution of the stress in the principle direction (x) and the plastic equivalent strain overall the sample. The results suggest that the stress and the plastic strain are almost homogeneous throughout the MPs, which is needed in this region of interest. Note that the stress is negligible (below 3 MPa) in the middle of the CP (Figure 3b, blue area). Thus, the magnetic properties of this area are not modified during the tensile test. At the extremities of the CP, we can see a stress field with a maximum level (230 MPa) below the one in the MPs (300 MPa), while the plastic strain distribution in these square regions is well localized at their diagonals. This will lead to magnetic property degradation and will generate an error on the magnetic field estimation. Thus, the error should be quantified. Note that the stress distribution in the other direction is negligible.

To estimate the generated error on the magnetic field, we have separated the corner areas from the CPs (Figure 4a). The magnetic circuit of the sample considering the corners’ reluctance is presented in Figure 4b. Two extremum cases could be considered. The first case is presented in Figure 4c, representing our assumption. The reluctance at the corner is equal to that of the CP. The second case is presented in Figure 4d, which shows that the reluctance at the corner is equal to that of the MP.

Figure 4

a) The geometry of the sample, b) the equivalent magnetic circuit with R _v < R _c < R _di and the equivalent reluctance of the magnetic circuit where, (c) R _c = R _v; and (d) R _c = R _di.

Assume that the stress magnitude in the corners is equal to the stress in the MPs with an intensity level equal to the yield stress (240 MPa). Based on the results presented in ref. [16], the maximal permeability of the raw material is two times higher than the permeability at yield stress. As permeability is proportional to reluctance, the reluctance at yield stress is two times higher than that of the raw material. Thus, the equivalent reluctances of the magnetic circuits are then 340R _v in the first case and 366R _v in the second case. The real equivalent reluctance is in between. Consequently, if we consider that the magnetic properties in the CP regions are not modified during the tensile test, we can guarantee an error over the magnetic field of less than 8%.

3.2 Magnetic aspect

To evaluate the accuracy of magnetic measurements, the device is compared with a standardized characterization method. As the sample width is quite small (5 mm), the impact of the cutting method could be more pronounced than for a regular SST sample, where the size of the sample, in our case, is 200 mm × 50 mm. Thus, some degradation should be expected. To evaluate this degradation due to the cutting technique, a set of 10 strips of 5 mm (the width of the MPs), cut by water jet and joint together, is characterized with the SST device on a Brockhaus MPG200D equipment. If the cutting process has no effect, then the two B(H) curves obtained with the 10 strips and the reference sample should be the same.

In Figure 5, three normal curves have been superposed:

the one corresponding to the sample characterized by the developed device
the one obtained on the 10 strips with the SST
the one obtained on the reference sample with the SST

Figure 5

Comparison of the normal curve obtained by using the proposed device with the normal curves obtained by using SST.

When we compare the two samples characterized with the SST, a degradation introduced by the cutting can be noted. However, the normal curve resulting from the developed device is in good agreement with the set of strips characterized by the SST. It means that our device enables one to characterize the B(H) curve in an equivalent way as an SST. Furthermore, to evaluate the procedure of reconstructing the B(H) curve under deformation as presented in Section 2, we deformed three samples according to the following plastic strains 0.45%, 0.75% and 1.75%; then, we measured the B(H) curve for each of them using the SST. In addition, we, for almost the same plastic strain, measured the B(H) curves on our device by applying the proposed procedure. The normal curves obtained for both characterization methods and the three plastic strain levels are compared in Figure 6. Note that the normal curves are almost superposed for the three plastic strains. Therefore, we can consider that the proposed procedure enables one to measure B(H) under axial stress.

Figure 6

Comparison between normal curves of samples deformed by tensile test characterized by the proposed device (dashed lines) and an SST (continuous lines).

4 Impact of mechanical characteristics on magnetic properties

Here we present the magnetic measurements performed under different mechanical states, di (a given stress and plastic strain values), using the developed experimental device.

4.1 Methodology

To reach different mechanical states, we applied almost the same principle of the method used in ref. [12]. A cyclic loading is applied; each cycle is represented by four states. The loaded one, illustrated by “State 1” label in Figure 7, is represented by the sample that is submitted to an increasing tensile stress in the elastic domain. During this step, for each magnetic measurement, the deformation is kept constant, which implies a constant loading stress. As measurements are carried out for different magnetomotive forces at different frequencies, the period of magnetic characterization, for a given stress and plastic strain, is around 25 min. Thus, the relaxation phenomenon is considered negligible in the elastic domain. The next state, “State 2,” is performed in the elastoplastic domain. It consists of applying an increasing force to reach a well-defined total strain. Then, in “State 3,” we maintain the deformation. In the plastic domain, the relaxation stress cannot be ignored. Thus, we wait 20 min, an estimated duration to get a stabilized stress, before magnetic characterization. Finally, “State 4” consists of a progressive unloading until no stress is applied on the sample. This entire experimental protocol is repeated for other plastic strain states. Note that the magnetic characterization is performed at 50 Hz with an applied magnetic field that reaches around 4,500 A/m. The mechanical measurements are performed with stresses in the range of 0–185 MPa and plastic strain values up to 3%.

Figure 7

The experimental procedure for mechanical stress application.

4.2 Magnetic behavior of pre-strained specimens under elastic stresses

In Section 4.1, we presented the considered procedure to span different mechanical states corresponding to a given tensile stress σ and plastic strain ε _p. Magnetic measurements are carried out at different mechanical states. As an example, Figure 8a presents the required magnetic field to reach a magnetic flux density of 1 T at 50 Hz in function of the mechanical state. Focusing on the plastic strain, we note that the required magnetic field increases with plastic strain ε _p. We can also note that the degradation is more pronounced for low plastic strain levels. Figure 8b presents the evolution of the magnetic field’s gradient over the plastic strain; we note that its value at 0.5% ε _p (∼94%) is much higher than the gradient value (∼3%) at 3.3% ε _p.

Figure 8

(a) The evolution of the required magnetic field, to reach a flux density level of 1 T, with stress for a frequency of 50 Hz and (b) the evolution of the magnetic field’s gradient.

In addition, we can denote that the magnetic behavior under plastic strain cases in the unload state is more degraded than those under the tensile load. To reach the same flux density level, a higher magnetic field level is required. The evolution of the required magnetic field with the applied stress is in accordance with the observations made by Iordache et al. [12]. We notice that the magnetic behavior improves, when a tensile stress is applied, up to a limit stress level corresponding to a minimal required magnetic field. Beyond this level, the magnetic behavior deteriorates gradually again (Figure 9a). In addition, we note that the optimal stress level increases with the plastic strain level (Figure 9b). For a 0.37% plastic strain, the optimal stress is around 71 MPa; whereas for a 3.3% plastic strain, it is around 157 MPa. We can conclude that this optimal stress value is a function of the plastic strain.

Figure 9

(a) The evolution of the required magnetic field, to reach a flux density level of 1 T, with stress for a frequency of 50 Hz and (b) the evolution of optimal stress with plastic strain.

5 Conclusion

This work presents, first, a new experimental procedure allowing magnetomechanical measurements on electrical steel strip samples under an applied uniaxial (tensile) stress approaching and exceeding the macroscopic elastic limit. The principle consists of characterizing a sample, with a closed magnetic path and without a parasitic air gap, based on the flux-metric method using two winding coils (primary and secondary). The device has been developed with the capacity to measure the magnetic hysteresis loops under any magnetic field waveform. The comparison of the measurements with those obtained on a standard SST shows that the proposed device ensures good reliability and accuracy of the measurements.

Second, an in situ experimental investigation of a nonoriented FeSi sheet (M330-35A) is reported. Several conclusions can be drawn: a slightly applied tensile stress, on a pre-strained material; in the unloaded case (stress = zero), the magnetic properties are more deteriorated compared to that in the loaded case. In that case, a tensile stress reduces the deterioration of the magnetic behavior performed by the plastic strain until an optimum stress value. Beyond this, the behavior tends to re-deteriorate gradually. We have extracted from the measurement the optimal stress in function of the plastic strain to be applied, which enables one to optimize the magnetic characteristics.

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Received: 2020-01-30

Revised: 2020-05-07

Accepted: 2020-05-18

Published Online: 2020-08-28

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/phys-2020-0160

Keywords for this article

nonoriented FeSi steel; magnetic characterization; tensile stress; plastic strain

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