Measurement and analysis of effect of the stress on magnetostriction of electrical steel under harmonic fields

Abstract

A magnetostriction single-sheet tester with an air compressor was developed and the magnetostriction in an electrical steel sheet was measured by a triaxial strain gauge under applied stresses externally and higher harmonic of field. The magnetostriction property’s sensitivity on the applied stress was described. This investigation also tried to understand the relationship between the presence of higher harmonics in the magnetization waveform and those in the magnetostriction waveform under a certain stress. The study results for the magnetostriction under stresses and harmonics show that the appearance of harmonic of magnetic field cannot be ignored, which aggravates the principal strain harmonics.

Keywords

Magnetostriction harmonic fields electrical steel sheet applied stress magnetic flux

1. Introduction

The magnetostriction phenomenon of electrical steel is strongly dependent on stress retained in electrical steels during mechanical process of the motor iron core such as stamping, riveting and assembly. There are some significant researches on the relationships between the magnetostrictive properties and mechanical stress for the optimal designs of electrical machines [1, 2, 3, 4, 5, 6]. However, the cores for the electrical apparatus running in the electric-power grid connected with the nonlinear load may be magnetized with higher harmonics of magnetic field component. This means that the electrical steel in practical applications may stand the higher harmonics of magnetic field, external stress, and alternating and rotational magnetization at the same time.

In this paper, based on a single-sheet tester with an air compressor, the principal strain of magnetostriction in a non-oriented electrical steel sheet under different magnetization directions is measured and analyzed with the stress applied externally. In addition, by applying the third harmonic of magnetic field to the alternating $B$ , the effect of amplitude and phase of the third harmonic on the principal strain is measured and discussed with stress and without stress applied.

Figure 1.

Schematic diagram of measuring system.

Figure 2.

Measurement device with air compressor.

2. Measurement device and approach

A schematic figure of magnetostriction measuring system shown in Fig. 1, which is composed of a magnetizing system with an air compressor and a magnetostriction sensor with a triaxial strain gauge, is developed to measure the magnetostrictive characteristics of electrical steel sheet. In Fig. 1, the NI DAQ is a data acquisition card (PXIe-6368) made in NI company. The BNC adapter is a connection between analog or digital signals and DAQ. In the magnetizing system in Fig. 2, the sample is stretched or pressed by a pneumatic stressing system with an air compressor, which can exert stress with the range of $-$ 10 MPa (compressive stress) $\sim$ 10 MPa (tensile stress) on the sample. The samples are machined by wire cutting into 0.5 mm in depth, 400 mm length, and 30 mm width. To apply the mechanical stress along different angles with regard as the rolling direction (RD), the samples are cut at 30 ${}^{\circ}$ intervals from the RD to transverse direction (TD). A triaxial strain gauge (KFG-10-120-D17-11, KYOWA) is stuck on the surface of the sample shown in Fig. 3 to obtain the magnetostriction strain along three detection directions, which have the angles of 0 ${}^{\circ}$ , 45 ${}^{\circ}$ , 90 ${}^{\circ}$ with respect to the RD. The gauge factor, gauge grid length, gauge resistance at 24 ${}^{\circ}$ C, and adoptable thermal expansion of the triaxial strain gauge are 2.10% $\pm$ 1.0%, 10 mm, 120.4 $\pm$ 0.4 $\Omega$ , and 11.7 ppm/ ${}^{\circ}$ C, respectively. The measured signals from the triaxial strain gauge are delivered to a LabView virtual instrument through a strain bridge box and a strain amplifier.

Figure 3.

Sample with a triaxial strain gauge.

By strain theory, the in-plane magnetostriction $\lambda_{\beta}$ in an arbitrary direction can be derived by

$\lambda_{\beta}=\varepsilon_{x}\cos^{2}\beta+\varepsilon_{y}\sin^{2}\beta+% \gamma_{xy}\sin\beta\cos\beta$ (1)

where $\beta$ is the angle with respect to the RD, $\varepsilon_{x}$ and $\varepsilon_{y}$ are two components of linear strains, and $\gamma_{xy}$ is the shear strain, which can be calculated by

$\left[{{\begin{array}[]{*{20}c}{\varepsilon_{x}}\hfill\\ {\varepsilon_{y}}\hfill\\ {\gamma_{xy}}\hfill\\ \end{array}}}\right]=\left[{{\begin{array}[]{*{20}c}{\cos^{2}\beta_{a}}\hfill&% {\sin^{2}\beta_{a}}\hfill&{\sin\beta_{a}\cos\beta_{a}}\hfill\\ {\cos^{2}\beta_{b}}\hfill&{\sin^{2}\beta_{b}}\hfill&{\sin\beta_{b}\cos\beta_{b% }}\hfill\\ {\cos^{2}\beta_{c}}\hfill&{\sin^{2}\beta_{c}}\hfill&{\sin\beta_{c}\cos\beta_{c% }}\hfill\\ \end{array}}}\right]^{-1}\left[{{\begin{array}[]{*{20}c}{\lambda_{a}}\hfill\\ {\lambda_{b}}\hfill\\ {\lambda_{c}}\hfill\\ \end{array}}}\right]$ (2)

where $\lambda_{a}$ , $\lambda_{b}$ and $\lambda_{c}$ are the measured magnetostriction strain along three detection directions at angles of 0 ${}^{\circ}$ , 45 ${}^{\circ}$ , 90 ${}^{\circ}$ to the RD. The elongation and contraction occurring along the principal strain axis can be derived as

$\lambda_{+},\lambda_{-}=\frac{\varepsilon_{x}+\varepsilon_{y}}{2}\pm\sqrt{% \left({\frac{\varepsilon_{x}-\varepsilon_{y}}{2}}\right)^{2}+\left({\frac{% \gamma_{xy}}{2}}\right)^{2}}$ (3)

The angle $\theta_{\lambda}$ from the principal strain axis to the RD is given as

$\theta_{\lambda}=\frac{1}{2}\tan^{-1}\left({\frac{\gamma{}_{xy}}{\varepsilon_{% x}-\varepsilon_{y}}}\right)$ (4)

Figure 4.

Strain signals measured from three detection angles of the triaxial strain gauge when an alternating $B_{max}=$ 1.2 T applied along the RD with and without the stress. (a) 0 Mpa, (b) 10 Mpa.

3. Measurement and analysis of magnetostriction under different conditions

3.1 Principal strain of magnetostriction under stress

Figure 4 shows the measured magnetostrictive signals along three detection angles of a triaxial strain gauge of 0 ${}^{\circ}$ , 45 ${}^{\circ}$ , 90 ${}^{\circ}$ with respect to the RD when the sample is magnetized along RD with an alternating induction of $B_{max}=$ 1.2 T and the stress are set to be 0 MPa and 10 MPa, respectively. In Fig. 4a, the amplitude of magnetostrictive strain at 90 ${}^{\circ}$ detection angle is 2.161 $\mu$ m/m and the strain is tensile, while that of 0 ${}^{\circ}$ angle has the property of contraction with the peak value of $-$ 2.197 $\mu$ m/m. After applying the tensile stress of 10 MPa to the sample, the magnitude of strain along 90 ${}^{\circ}$ and 0 ${}^{\circ}$ detection angles reduce to 0.538 $\mu$ m/m, 0.385 $\mu$ m/m, respectively, and the elongated strain at 90 ${}^{\circ}$ is bigger than contractive one at 0 ${}^{\circ}$ .

Figure 5.

Principal strain amplitude under different stresses applied on several samples with different magnetization directions. (a) elongated principal strain, (b) contractive principal strain.

In order to investigate the effect of stress on the principal strain, the elongated and contractive principal strain of magnetostriction in one time period are calculated from Eq. (3) by means of measured signals from the triaxial strain gauge. Figure 5 illustrates the variation of amplitude of principal strain with the increase of the stress when the sample is magnetized along 0 ${}^{\circ}$ , 30 ${}^{\circ}$ , 60 ${}^{\circ}$ , and 90 ${}^{\circ}$ with respect to the RD, respectively. Figure 5a and b shows the elongated and contractive principal strain under the purely sinusoidal flux density with a peak value $B_{max}=$ 1.2 T, respectively.

Figure 6.

Fourier analysis of principal strain of magnetostriction when the 3rd harmonic of magnetic field is overlapped with the fundamental one without stress applied. (a) effect of magnitude of the 3rd harmonic, (b) effect of phase of the 3rd harmonic.

Figure 7.

Fourier analysis of principal strain of magnetostriction when the 3rd harmonic of magnetic field is overlapped with the fundamental one under stress applied. (a) 8 MPa, (b) $-$ 8 MPa.

In Fig. 5a, with the increase of the tensile stress from 0 Mpa to 10 Mpa, for a particular magnetization sample such as 30 ${}^{\circ}$ magnetized angle, the elongated principal strain reduces gradually from about 2.6 $\mu$ m/m to 1.5 $\mu$ m/m. However, compressive stress intensified the magnetostriction so that the elongated principal strain increases from 2.6 $\mu$ m/m to nearly 3.0 $\mu$ m/m when the compressive stress increass from 0 Mpa to 10 Mpa. Moreover, for a particular stress, e.g. $-$ 4 Mpa, the increasing of the magnetized angle of the sample results in the rise of the amplitude of magnetostriction from 2.296 $\mu$ m/m to 6.721 $\mu$ m/m. The contractive principal strain in Fig. 5b are symmetrical with those as Fig. 5a.

3.2 Principal strain under both harmonic of magnetic field and stress

To investigating the effect of harmonic field under different stresses on the magnetostrictive property, the magnetic flux density applied to the specimen is controlled into a distortedly sinusoidal waveform including a 50 Hz fundamental component and a third harmonic during the measurement. The phase delay of the third harmonic of magnetic field with respect to the fundamental component ranges from 0 ${}^{\circ}$ to180 ${}^{\circ}$ while the percentage of third harmonic accounting for fundamental component is chosen as be from 2% to 10%. The measured principal strain waveform of magnetostriction can be transformed into a fundamental component of 100 Hz and several even higher harmonics by Fourier series analysis.

Figure 6 shows the effect of amplitude ratio and phase angle of the third harmonic $B_{3}$ to fundamental one $B_{1}$ on the 100 Hz, 200 Hz and 300 Hz harmonics of principal strain when the sample is magnetized along the RD with the fundamental component 1.0 T of magnetic field without any applied stress. It can be seen that the presence of the third harmonic of magnetic field causes the obvious increase of 200 Hz and 300 Hz harmonics of magnetostrictive strain. However, in Fig. 6b the 100 Hz, 200 Hz and 300 Hz harmonic component of magnetostrictive strain decrease with the increase of the phase of the third harmonic of magnetic field, and the 100 Hz harmonic reduces faster than the 200 Hz one.

After exerting the $-$ 8 MPa and 8 MPa stress on the sample, the effect of higher harmonic of magnetic field on the principal strain is investigated once more, as shown in Fig. 7. When the mechanical stress and harmonic of magnetic field appear at the same time, the principal strain of magnetostriction happen to change. When applying 8 MPa to the sample, by comparing Figs 6 and 7a, we can see that the magnetostriction decrease and 100 Hz component decreases faster obviously than other two harmonics. However, when the compressive stress $-$ 8 MPa is applied to the sample in Fig. 7b, all the higher harmonics of magnetostriction become bigger with the increase of the percentage of the 3 ${}^{\rm rd}$ harmonic of magnetic field and the 100 Hz component increases fastest.

4. Conclusions

Magnetostriction of non-oriented electrical steel along different magnetized directions is measured and its dependence on the stress is discussed. Compressive stress intensified the magnetostriction and the amplitude of magnetostriction is increasing gradually with the increase of magnetization angle. With an alternating magnetic flux density magnetization combined with a third harmonic under stress applied, the 100 Hz and 200 Hz harmonics of magnetostrictive strain increase with the increase of the percentage ratio of the third harmonic obviously and decrease with the increase of the phase delay $\theta_{3}$ . The applied stress also has effect on the harmonic portion. As a result, in order to evaluate the vibration and noise in machines and transformer cores effectively, the appearance of harmonic of magnetic field cannot be ignored, which aggravates the principal strain harmonics.

Footnotes

Acknowledgments

This work was supported by National Natural Science Foundation of China under Grant 51777128.

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