Measurement and modeling of magnetostriction characteristics of highly Grain-oriented electrical steel sheet by using round-type two-dimensional single sheet tester

Abstract

Magnetostriction in highly Grain-oriented electrical steel sheet (ESS) is a primary source of vibration and acoustic noise of power transformers at no-load condition. Especially rotating magnetization, as it occurs in joint area of three-phase power transformer, makes the magnetostriction bigger and more complex than that under alternating magnetization. Meanwhile, magnetic flux density along transverse direction makes a lot bigger magnetostriction than that under rolling direction. In this paper, a new measuring system of the magnetostriction are developed for a highly Grain-oriented electric steel sheet (HGO-ESS) of which magnetic domains are bigger in size and higher in degree of alignment than those of a Grain-oriented one. The magnetostriction characteristics of some HGO-ESS are measured by using the developed system under both alternating and rotating magnetic fields. An improved magnetostriction model is also proposed based on the mechanical elasticity, and its effectiveness is validated through comparisons with experimentally measured results from the measuring system.

Keywords

Electrical steel sheet magnetostriction magnetostriction force transformer

1. Introduction

Vibration and acoustic noise in a power transformer which is usually made of highly Grain-oriented electrical steel sheet (HGO-ESS) is caused especially at no-load condition by magnetostriction of iron core [1, 2]. It is reported that around 40% of the accidents in power transformers are from mechanical problems of which more than half is caused by vibrations [3]. Magnetostriction force in the core laminations, furthermore, is responsible for up to 50% of the total electromagnetic force [4]. In the design of a power transformer, therefore, magnetostriction force should be taken into account not only to reduce vibration and acoustic noise but also to increase reliability. In order for quantitative calculation of the magnetostriction force in a power transformer, magnetostriction characteristics of the HGO-ESS should be measured and modeled, and then adopted into a finite element analysis (FEA).

A square type two-directional (2-D) magnetostriction measurement system which can measure the magnetostriction characteristics under unidirectional, elliptical flux and circular magnetic field conditions was developed by Enokizono et al. [11]. It revealed that the magnetostriction under rotating magnetization of both non-oriented (NO) and Grain-oriented (GO) ESSs is much greater than that under alternating magnetization. A round type 2-D magnetization system was initially developed by Somkun [6] for measurement of magnetostriction and rotational iron loss in ESS. One of the advantages of this type system is that higher magnetization can be achieved but its drawbacks are less magnetic field homogeneity throughout the sample and the difficulty in cutting such a circular shape. In addition, the system is only used for NO and GO SST because the diameter of the disc sample is only 80 mm. So a precise measurement of the anisotropic magnetostriction data up to magnetic saturation level, however, is limited until now to non-oriented (NO) and Grain-oriented (GO) ESSs [5, 6].

Despite strenuous efforts made over the last few decades, the prediction of magnetostriction characteristics still remains a challenging ongoing problem. To include magnetostriction characteristics into magnetostriction force calculation of electrical machine and transformer, a very simple expression was proposed to predict magnetostriction in [7], but this expression cannot describe the hysteresis phenomenon of magnetostriction. In 1999, Lundgren [8], introduced a 2-D magnetostriction model by analogy of mechanical elasticity with history dependence to represent the hysteresis in magnetostriction. Neural networks were used to identify the parameters. However, a large amount of measurement data was required to train the network. Therefore, Somkun et al. [9] proposed a more simple identification method based on the mechanical elasticity theory. Unfortunately, the method cannot apply on the saturation magnetic field condition. However, some electrical devices such as power transformer are usually worked under saturation condition. So, a magnetostriction model which can use for saturation situation is needed.

In this paper, anisotropic magnetostriction data of a HGO-ESS is measured by using a new round-type two-directional single sheet tester (R-2D-SST) up to magnetic saturation level under both alternating and rotating magnetic field conditions. The measured data are, then, modeled via an improved magnetostriction model based on the analogy of mechanical elasticity, and its effectiveness is validated through comparisons with experimentally measured results from the measuring system.

2. Measurement of anisotropic magnetostriction of a highly Grain-oriented ESS

2.1 Single steel sheet measurement system

The measurement system of the magnetostriction is simply two-dimensional SST having three-axial strain gauge on the specimen. A typical configuration of the system is shown in Fig. 1, where both the square- and round-type SSTs can be employed. The conventional square-type SST, as shown in Fig. 2, has its specimen of 80 $\times$ 80 mm, H-coil of 20 $\times$ 20 mm with 150 turns, and B-coil of one turn. In order to prevent the jumping magnetic flux, it has upper and lower magnetic shielding plates.

Figure 1.

2-D square and round type SST system.

Figure 2.

Square type SST.

The magnetic domains of a HGO-ESS are in general bigger in size and higher in degree of alignment than those of non-oriented and grain-oriented ones [10]. In order to measure the magnetic and magnetostriction properties in average sense, therefore, the HGO-ESS requires wider region of uniform magnetic field than other ESSs. According to our experience, with the conventional square-type SST, the $B$ -waveform control to get sinusoidal $B$ -waveform is very difficult.

A new round-type 2-D SST which is designed in this paper to have broader region of uniform field and allows a larger specimen than the square-type SSTs [9, 11] is shown in the Fig. 3 which has its specimen of diameter 162.5 mm, H-coil region of 40 $\times$ 40 mm with 130turns, and B-coil of one turn. The system, to remove the jumping-flux and apply uniform magnetic field on the specimen, has upper and lower magnetic shielding plates made of NO ESS at the positions of 2 mm apart from the specimen, respectively. In the system, the $B$ -coils threaded through small holes at the center of the specimen, and the $H$ -coil is located at the surface of the specimen. The three-axial strain gauge is attached to the central part of the specimen to measure the mechanical strain. The specifications of the square- and round-type SSTs are compared in Table 1.

Table 1

Specifications of the developed R-2D-SST system

Item	Proposed	Conventional
Type	Round	Square
Specimen	Circular $d=$ 162.5 mm	Square 80 $\times$ 80 mm ${}^{2}$
B-coil width	40 mm	20 mm
H-coil region	40 $\times$ 40 mm ${}^{2}$	20 $\times$ 20 mm ${}^{2}$
Shield	Upper and lower	None

${}^{*}$ The symbol $d$ stands for diameter.

Figure 3.

Round type SST.

The strain in a plane surface is measured by using a three-axial strain gauge, SKF-5964 from KYOWA, Japan, of which gauge length, gauge factor, gauge resistance at 24 ${}^{\circ}$ C and adoptable thermal expansion are 15 mm, 2.02 $\pm$ 2.0%, 350.0 $\pm$ 2.5 $\Omega$ and 10.8 ppm/ ${}^{\circ}$ C, respectively. The relative positions of $B$ - and $H$ -coils, and the strain gauge on the specimen are shown in Fig. 3.

In this system a feedback method is selected to control the applied current of the exciting coils, as in a typical SST, in order to ensure that the exact magnetic field applied on the specimen. And the $B$ -, $H$ - and magnetostriction waveforms are measured simultaneously.

In all measurements, magnetic flux densities are controlled using digital feedback to be elliptic defined with three parameters as shown in Fig. 4: maximum value $B_{\max}$ , inclination angle $\varphi$ , and axis ratio $\alpha$ between the major and minor axes. The axis ratios of $\alpha=$ 0 and $\alpha=$ 1 give an alternating and pure rotating magnetic fields, respectively. In the measurement of the strain, data for 50 periods are measured and filtered with ideal low-pass filter which does not have any phase delay, and then averaged to increase the signal to noise ratio.

Figure 4.

Definition of elliptic B-waveform.

The normal strain ( $\varepsilon_{x}$ , $\varepsilon_{y}$ ) and shear strain ( $\gamma_{xy}$ ) can be calculated from the measured strains $\varepsilon_{a}$ , $\varepsilon_{b}$ , $\varepsilon_{c}$ as follows:

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

where $\theta_{a}$ , $\theta_{b}$ and $\theta_{c}$ are the three gauges angles to the RD, respectively. $\varepsilon_{a}$ , $\varepsilon_{b}$ , $\varepsilon_{c}$ are strains measured from each strain gauge, respectively. Then, arbitrary in-plane magnetostriction $\varepsilon$ ( $\theta$ , $t$ ) can be obtained by

$\displaystyle\varepsilon(\theta,t)=\varepsilon_{x}(t)\cos^{2}\theta+% \varepsilon_{y}(t)\sin^{2}\theta+\gamma_{xy}(t)\sin\theta\cos\theta$ (2)

where $\theta$ is an arbitrary angle with respect to the $x$ axis.

Magnetostriction characteristics are measured using the developed R-2D-SST system under both alternating and rotating magnetic field conditions for HGO specimen. Figure 5 shows the in-plane magnetostriction calculated by Eq. (2) when magnetic fields are along different angles, under alternating and rotating magnet field, respectively. From the figure, it is found that the magnetostriction is only 1 $\sim$ 3 $\mu$ m/m when B is 1.9 T along rolling direction while, along the transverse direction, it is 30 $\sim$ 40 $\mu$ m/m when B is 1.2 T. It is also shown that the magnetostrictions under rotating magnetic field are slightly bigger than those under alternating one. Therefore, the amplitude of magnetostriction is strongly depend on the direction of B.

Figure 5.

Arbitrary angle in-plane magnetostriction at the max B position, solid line is under rotating field when $\alpha=$ 0.3, dash line is under alternating field when $\alpha=$ 0.

Figure 6.

Transformer model measurement system.

Figure 7.

Measurement results of the transformer measurement system.

Figure 8.

Parameters $p_{x}$ , $p_{y}$ , $d_{x}$ and $d_{y}$ as functions of magnitude ( $B$ ) and inclination angle ( $\varphi$ ) of applied magnetic field.

2.2 Magnetostriction measurement system of transformer model

Figure 6 shows a model transformer magnetostriction measurement system. The transformer is made of HGO-ESS (30PH105) and has six laminations. In the experiment, to have the same magnetic flux distribution with a real transformer, three-phase sinusoidal voltages of 100 (V ${}_{L-L}$ ) are applied to the windings of 356 turns per phase. Some three-axial strain gauges are attached on the surface of the model to measure the magnetostriction at some points as shown in Fig. 6. The peak value of tensile and compressive value at every measurement point are shown in Fig. 7. From the figure, the magnetostriction at the T-joint zone which have rotating magnetic fields is much bigger than that at the leg zone which have alternating fields.

3. Magneto-elasticity model of magnetostriction

3.1 Improved magneto-elasticity model of magnetostriction

Magnetostriction arises from the movement and rotating of magnetic domain walls because of the magnetic force. The analogy with the in plane stress and strain relationship in mechanics, therefore, gives a magneto-elasticity model for magnetostriction.

The mechanical elasticity model describes the relationship between the mechanical stress and strain through Hook’s law as follows:

$\displaystyle\varepsilon=\frac{\sigma}{E}$ (3)

where $\sigma$ and $E$ are mechanical stress and Young’s modulus, respectively. Analogy to magnetostriction gives similar relationship between magnetic stress and strains as follows:

$\displaystyle\varepsilon=\frac{1}{P}\frac{B^{2}}{2\mu_{0}}$ (4)

where $P$ is the equivalent magnetic modulus, and $\mu_{0}$ is the magnetic permeability of free space. Introducing the hysteretic property of the magnetostriction, the two-direction magneto-elasticity model can be written as follows for the isotropic materials under 2-D magnetization [8]:

where $\tau$ is time constant for controlling phase of magnetostriction, $P_{x}$ , $P_{y}$ and $\xi_{x}$ , $\xi_{y}$ are the magnetic moduli and Poisson’s ratios along rolling direction (RD) and transverse direction (TD), respectively, and $G_{xy}$ is the shear modulus. It should be noted that all the parameters, $P_{x}$ , $P_{y}$ and $G_{xy}$ , in this model are constant without regard to the magnitude and inclination angle of magnetic flux density as explained in [9]. In this model, a sinusoidal magnetic flux density always gives sinusoidal magnetostriction contrary to the experimental phenomena that even sinusoidal magnetic flux density induces non-sinusoidal magnetostriction when the magnetic induction is near to saturation. This model, therefore, is reported not applicable when the magnetization is close to saturation [8].

This paper proposes an improved magnetostriction model which is applicable up to saturation level by introducing non-linear parameters based on the analogy with plane mechanical elasticity as follows:

$\displaystyle\tau\frac{d}{dt}\left[{{\begin{array}[]{*{20}c}{\varepsilon_{x}}% \hfill\\ {\varepsilon_{y}}\hfill\\ {\gamma_{xy}}\hfill\\ \end{array}}}\right]+\left[{{\begin{array}[]{*{20}c}{\varepsilon_{x}}\hfill\\ {\varepsilon_{y}}\hfill\\ {\gamma_{xy}}\hfill\\ \end{array}}}\right]=\frac{1}{2\mu_{0}}\left[{{\begin{array}[]{*{20}c}{1% \mathord{\left/{\vphantom{1{p_{x}}}}\right.\kern-1.2pt}{p_{x}}}\hfill&{{-\xi_{% x}}\mathord{\left/{\vphantom{{-\xi_{x}}{p_{y}}}}\right.\kern-1.2pt}{p_{y}}}% \hfill&0\hfill\\ {{-\xi_{y}}\mathord{\left/{\vphantom{{-\xi_{y}}{p_{x}}}}\right.\kern-1.2pt}{p_% {x}}}\hfill&{1\mathord{\left/{\vphantom{1{p_{y}}}}\right.\kern-1.2pt}{p_{y}}}% \hfill&0\hfill\\ 0\hfill&0\hfill&1\hfill\\ \end{array}}}\right]\left[{{\begin{array}[]{*{20}c}{B_{x}^{2}}\hfill\\ {B_{y}^{2}}\hfill\\ {{B_{x}^{2}}\mathord{\left/{\vphantom{{B_{x}^{2}}{d_{x}}}}\right.\kern-1.2pt}{% d_{x}}+{B_{y}^{2}}\mathord{\left/{\vphantom{{B_{y}^{2}}{d_{y}}}}\right.\kern-1% .2pt}{d_{y}}}\hfill\\ \end{array}}}\right]$ (6a) $\displaystyle p_{x}=p_{x}(B,\varphi),\quad p_{y}=p_{y}(B,\varphi),\quad d_{x}=% d_{x}(B,\varphi),\quad d_{y}=d_{y}(B,\varphi)$ (6b)

where the parameters $\tau$ , $\xi_{x}$ and $\xi_{y}$ are set to 0.0004, 2 and 0.5, respectively, using the identification method in [9], and $d_{x}$ and $d_{y}$ are RD and TD components of shear modulus, respectively. The identification of the non-linear parameters is as follows.

Figure 9.

Magnetostrictions under alternating and rotating magnetic fields where solid and dashed lines are calculated from improved and original models, respectively, and symbols are measured results.

Piecewise linear shape functions $p_{x}$ , $p_{y}$ , $d_{x}$ and $d_{y}$ are assumed to be non-linear functions of the magnitude ( $B$ ) and inclination angle ( $\varphi$ ) of the applied magnetic field, respectively. And for each angle, $p_{x}$ , $p_{y}$ , $d_{x}$ and $d_{y}$ can be expressed as:

$\displaystyle p_{x}(B_{j})=p_{x}(B_{j-1})+\mu_{px,j}(B_{j}-B_{j-1})/\Delta B$ $\displaystyle p_{y}(B_{j})=P_{y}(B_{j-1})+\mu_{py,j}(B_{j}-B_{j-1})/\Delta B$ (7) $\displaystyle d_{x}(B_{j})=d_{x}(B_{j-1})+\mu_{dx,j}(B_{j}-B_{j-1})/\Delta B$ $\displaystyle d_{y}(B_{j})=d_{x}(B_{j-1})+\mu_{dy,j}(B_{j}-B_{j-1})/\Delta B,j% =1,...,M$

Therein, $B_{j}=j\Delta B$ , $\Delta B=B_{s}/M$ , $B_{s}$ is the saturation magnetic flux density value, $M$ is the discrete points number of the piecewise linear shape. $\mu_{px}$ , $\mu_{py}$ , $\mu_{dx}$ , $\mu_{dy}$ will be get from trial and error method through compare the modeled and measured results from $j=$ 1 to $M$ . Figure 8 shows non-linear characteristics of the parameters which are obtained through trial and error.

3.2 Modeling results

With the non-linear parameters, the normal strains are simulated via Eq. (4) and compared in Fig. 9 with measured results. It is shown that the proposed model with non-linear parameters is applicable up to magnetic saturation level.

4. Conclusion

A new round-type two-directional single sheet tester and an improved modeling algorithm for magnetostriction of highly Grain-oriented electrical steel sheet are proposed. The developed measuring system, owing to its wide region of uniform magnetic field, can be applied to the measurement of highly Grain-oriented electrical steel sheet as well as non-oriented and Grain-oriented ones. The suggested magnetostriction model having non-linear parameters is also validated through comparisons with experimentally measured results.

Footnotes

Acknowledgments

This research was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education (2017R1D1A3B03034633).

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