Study of Transformer Core Vibration and Noise Generation Mechanism Induced by Magnetostriction of Grain-Oriented Silicon Steel Sheet

State Key Laboratory of Advanced Power Transmission Technology, Global Energy Interconnection Research Institute Co.Ltd., Beijing 102201, China State Grid Henan Electric Power Research Institute, Zhengzhou 450052, China Beijing Key Laboratory of Environment Noise and Vibration, Beijing Municipal Institute of Labor Protection, Beijing 100054, China State Key Laboratory of Electrical Insulation and Power Equipment, Xi’an Jiaotong University, Xi’an 710049, China


Introduction
Power transformers are critical integral parts of the power grid system, and their safe operation is mandatory for the entire system's reliability. e vibration of transformers affects the power equipment's normal operation, service life, and reliability, while the vibration and noise also have a negative impact on human health and comfort [1][2][3]. e transformer vibration has two main constituents: (i) winding vibration caused by the electromagnetic force generated by the interaction of the current in a winding with leakage flux, and (ii) iron core vibration caused by the magnetostriction forces in the silicon steel sheet (SSS). Under no-load conditions, there is no current in the winding, and the transformer vibration is mainly generated by the core. e vibration signal measured at this time is the core vibration signal, which is induced by the SSS magnetostriction.
So far, most research on the core vibration of transformers is mainly focused on magnetostrictive properties of materials [4][5][6], core vibration, noise testing, vibration transmission characteristics [7][8][9][10][11][12][13][14][15][16][17], and theoretical or numerical modeling [18][19][20][21][22][23][24][25][26]. ere are also some studies focused on signal acquisition for vibration analysis and feature recognition [27][28][29][30][31][32]. Cheng et al. [4] studied magnetic and magnetostrictive properties of different grades of SSS and proposed the selection of grain-oriented SSS in core materials. Wang et al. [6] studied the magnetostrictive dynamic hysteresis characteristics of a grain-oriented electrical steel under different elliptical magnetization loci based on a rotational magnetic properties tester. Moses et al. [7] carried out experimental tests on the surface vibration and noise of transformer cores using a laser vibrometer. Okabe et al. [8] studied the vibration characteristics and mode shapes of the three-phase transformer core under different excitations and analyzed the reasons for the difference between the frequency spectrum and noise spectrum of the magnetostrictive vibration under partial frequency excitation. Ma et al. [15] carried out vibration and noise tests on the inside and outside of a typical 10 kV distribution transformer tank and acquired the acoustic-vibration characteristic and propagation law of the 10 kV distribution transformer. Zhang et al. [16] used the magnetostrictive orthogonal calculation method to simulate the vibration of iron cores of the transformer and shunt reactor models. e calculation and test results showed that Maxwell stress was the main reason of reactor core vibration, and magnetostriction was the main reason of the transformer core. Liu et al. [18] proposed a transformer vibration transmission model based on motion transmission. rough experiments and theoretical calculations, the accuracy of the vibration transmission method was verified, which could be used to analyze the vibration relationship between the transformer core and the casing. Javorski et al. [20] established a calculation method of magnetostrictive force based on experiments and analyzed the magnetostriction of laminated cores in different directions. Ghalamestani et al. [22] calculated the core deformations of a three-phase transformer under a purely sinusoidal magnetization with 50 Hz frequency and a magnetisation with a fifth harmonic component by an twodimensional (2D) finite element technique. Hu et al. [23] calculated the sound pressure level of the radiated noise around the transformer based on the established finite element model combining transient electromagnetic field analysis, mechanical field analysis, and acoustic analysis. Comparison of calculated results and measured data verified that the combined noise-calculating model was applicable for transformer noise prediction. Zhang et al. [24] studied a fast and accurate calculation method for the core vibration of a DFP-270000/500 type super high-voltage generator transformer. Based on the vibration acoustic coupling calculation, the damping coefficient of the core vibration of the transformer was obtained, and the magnetic structural coupling analysis was used for the vibration of the transformer core. Liu et al. [25] established a complete fully coupling modeling chain of such a transformer, from threephase magnetic flux injection to the 2D deformation and acoustic power estimation of the transformer core, taking into account the phase current coupling, magnetic flux excitation (voltage excitation), material anisotropy, magnetostrictive-induced stress, and multilayer homogenization. However, most of the above works are single-stage studies, ignoring the transfer between each stage. erefore, the mechanism of core vibration generation has not been comprehensively described yet.
In this paper, according to the evolution process from the material to the product, the microstructure and magnetostrictive properties of grain-oriented SSS are tested, and a four-corner model of the core is built to simulate the first-order vibration of the core. Next, the modal, vibration, and noise tests of the 110 kV transformer core product are performed. e in-depth analysis of the mechanism of core vibration generation can provide design ideas and data support for the preparation of low-noise cores.

Mechanism of Iron Core Vibration
e core vibration is mainly caused by the magnetostrictive phenomenon of the SSS and the Maxwell electromagnetic force. Due to the continuous improvement of the lamination method used for the iron core manufacturing and the weftfree adhesive tape application in its binding, the transformer core vibration is mainly controlled by the SSS magnetostriction. When the SSS is subjected to an external magnetic field, its dimensions undergo contraction and/or elongation and recover to the initial values after the external magnetic field removal.
is phenomenon is referred to as the magnetostrictive effect. e magnetostriction ratio is commonly used to reflect magnetostriction, which can be expressed as where ε is the SSS axial magnetostriction ratio, ΔL is the SSS maximum axial expansion or contraction, and L is the SSS original axial dimension. Assuming the supply voltage is U 1 � U s sin ωt, according to the principle of electromagnetic induction, the magnetic induction intensity in the core is where φ is the core magnetic flux, S is the core crosssectional area, and B 0 � (U s /ωNS) is the magnitude of the magnetic induction intensity. When the iron core works in an unsaturated state, the magnetic flux density and the magnetic field strength have a linear relationship. e magnetic field strength in the iron core is where B s is the saturation magnetic induction of the core and H c is the coercive force. Under the effect of an external magnetic field, a smallscale deformation of the SSS caused by magnetostriction satisfies the following relationship: where ε s is the saturation magnetostriction ratio of the SSS. Based on the above relationship, one can derive the maximum axial elongation of the core caused by the SSS magnetostriction as follows: 2 Shock and Vibration erefore, when the transformer is unloaded, the core vibration acceleration caused by the SSS magnetostriction is From equation (6), it can be deduced that the vibration acceleration is proportional to the square of the voltage if the core material and operating temperature remain unchanged: e magnetostriction variation period is twice lower than that of the alternating electromagnetic field, while the fundamental frequency of the core vibration caused by the magnetostriction is twice higher than the electromagnetic field variation frequency ω.
However, in practice, the waveform of the core magnetic flux density is not a standard sine wave. In addition, the magnetic flux distributions and paths of the core outer and inner frames are different, which causes the core vibration signal to contain a large number of higher harmonic components, besides the fundamental frequency component.

Materials and Methods
(1) Materials: the grain-oriented silicon steel sheet (SSS) material used in the test was the B30P105 cold-rolled finished product produced by Baoshan Iron & Steel Co., Ltd., China. (2) Grain-oriented SSS microstructure measurement: microstructure and texture analysis were the basic methods to evaluate grain-oriented SSS. e grainoriented SSS was eroded by hot hydrochloric acid solution and then photographed. e texture was measured with electron backscattered diffractometer (EBSD) attached to Zeiss Ultra 55 field emission scanning electron microscope.
(3) Grain-oriented SSS magnetostriction measurement: MST500 magnetostriction measurement system was used to test the magnetostriction coefficient and A-weighted velocity (AWV) noise of grain-oriented SSS according to IEC/TR 62581-2010. e sample dimensions are 100 mm × 600 mm. In the measuring process, one end of the sample is fixed by the clamping part of the device, and the other end of the sample is in the free state. e resolution of the laser is 10 nm/m. A photo of the magnetostriction coefficient measuring instrument is shown in Figure 1. (4) Four-corner core test model: a four-corner iron core model was established to simulate the first-order vibration of the transformer core. e four-corner iron core model adopted a three-step stepwise overlap method; two pieces were stacked on one another and connected by a 45°diagonal joint. e winding was evenly wound on the two arms and was used in series. ere was a total of 220 turns, 800 mm in length, 100 mm in width, and 30 mm in stack thickness. e acoustic vibration analysis system by the B & K Company, Denmark, was used to collect the vibration signals at the five measurement points in the central area and the seam area of the parallel and vertical core planes, as shown in Figure 2. (5) Transformer core measurement: the actual product used a three-phase dual-winding 50MVA 110 kV transformer core to test its vibration and noise characteristics and natural modes. e sound level meter and acoustic vibration analysis system by the B & K Company, Denmark, were used for sound vibration measurement and analysis under the no-load condition, and the modal test of the iron core was performed via a force impact hammer and accelerometer. e noise and vibration measurement points are shown in Figure 3. { } 〈112〉. Figure 6 plots the magnetostriction butterfly curve of the SSS in the rolling direction under a 50 Hz sine period excitation magnetic field. It can be seen that the butterfly curve has left-right symmetry. Its "wing" opening faces downwards, and magnetostriction has both elongation and contraction strains, the latter prevailing over the former ones. is prevalence may be due to the dynamic motion of the domain wall or the re-organization of the magnetic domain structure. Magnetostrictive shrinkage is caused by a 90°unfavorable magnetic domain's magnetic moment rotation. With an increase in the magnetic flux density, the 90°m agnetic domain volume is continuously reduced. When the maximum magnetic flux density reaches 1.7 T, the magnetostriction will no longer increase monotonously and start to drop. e butterfly curve wings narrowed, and the magnetostriction tended to saturation. e magnetostrictive single-value curves under different magnetic densities are shown in Figure 7. It can be seen that, with an increase in the applied magnetic field intensity, the magnetostrictive peak-to-peak value λ p−p , which is defined as the difference between λ p+ and λ p− values under alternating magnetization, gradually increases. When the material magnetic properties tend to be saturated, the magnetostriction also appears to be saturated. From a microscopic perspective, the magnetic domains constituting the material are gradually arranged in parallel to the external field direction under this field action, and the magnetization process also changes with the magnetic domain size. When all magnetic domains form a single domain sample, the magnetization saturation is reached, and the strain of magnetic domains also tends to be saturated. If the magnetic field intensity is further increased, the magnetization will increase slowly, and the variation behavior will no longer be consistent with the sample before saturation. Shock and Vibration flux density is less than 1.7 T, the maximum magnetostrictive lengths are mainly concentrated at 0 and 100 Hz. After 1.7 T, the magnetostrictive lengths at frequencies of 200 and 300 Hz increased significantly. Figure 10 depicts the butterfly curve of the grainoriented SSS in different directions at a magnetic density of 1.3 T under a 50 Hz sinusoidal periodic excitation magnetic field. It can be seen that the butterfly curves in different directions have a left-right symmetry, and magnetostriction directions are not the same, showing high magnetostrictive anisotropy characteristics. Along the rolling direction, the magnetostrictive value is the smallest. In the 45°direction, the magnetostrictive value increases and shows shrinkage. In the vertical direction, the magnetostrictive value is the largest and shows elongation. e maximum value of magnetostriction in the vertical direction exceeds that in the rolling direction by 35 times. e butterfly curves vertical to the rolling direction under different magnetic densities are omitted for brevity. eir variation pattern and magnetostrictive characteristics are the same as those along the rolling direction: they both increase in magnetostriction as the magnetic flux density increases until saturation. However, at the same magnetic flux density, the magnetostriction   Shock and Vibration of the grain-oriented SSS in the vertical direction is significantly larger than that in the parallel (rolling) direction. Figure 11 shows the AWV noise curves of grainoriented SSS in different directions with different magnetic flux densities under 50 Hz sinusoidal periodic magnetic field excitation. Limited by the equipment power, the maximum field strength in the vertical and 45°directions is 1.6 T. It can be seen that the AWV noise level of grain-oriented SSS varies in different directions. Under the same magnetic density, the AWV noise is the smallest along the rolling direction, slightly increases in the 45°direction, and is the largest in the vertical direction. e latter AWV noise is nearly twice higher than that in the rolling direction. e rate of increase of the AWV noise value of grain-oriented SSS is not affected by sample saturation. Before and after the sample is saturated, the rise of the AWV noise value is consistent with an increase in the magnetic flux density. Although the increase rate of the magnetostrictive peak-to-peak value λ p−p becomes smaller after saturation, it does not increase or even decrease slightly after the sample becomes saturated. However, with the emergence of the saturation phenomenon, the harmonic content contained in the magnetostrictive waveform increases, and the contribution to the AWV noise value is relatively large, so the AWV noise value increases.

Vibration Characteristics of the Four-Corner Iron Core
Model. Figure 12 shows the vibration time domain and vibration spectrum of the four-corner iron core model SSS at a magnetic density of 1.3 T. It can be seen that the time-domain vibration signal is not a standard sinusoidal waveform, but a distorted waveform superimposed by various frequency signals. However, it is still possible to determine the core vibration signal as a periodic signal with a frequency of 100 Hz. e main occurrence locations of vibration frequency peaks at each measurement point and the total effective value of the vibration acceleration are shown in Table 1. e main vibration peak frequencies in different directions are obviously different. In addition to the fundamental frequency, there are also well-pronounced high-frequency components, which are consistent with the theoretical analysis results. In addition, 450 and 650 Hz odd-order harmonics also appeared in the seam area (points 3 and 4), which may be caused by the combined effect of the cyclic excitation in the common area and the electromagnetic force between gaps.
e effective values of vibration acceleration vertical to the core plane (a 1 and a 3 ) are higher than those parallel to the core plane (a 2 , a 4 , and a 5 ). Figure 13 shows the vibration characteristics of measurement points 2, 3, and 5 under different magnetic densities. It can be seen that, with an increase in the magnetic density, the peak intensity of the main vibration frequency continues to increase, new frequency vibration peaks appear, and the vibration waveform becomes more complicated.
e fundamental frequency vibration acceleration amplitude exhibits a linear relationship with the squared magnetic density, as shown in Figure 14. at is, the fundamental frequency vibration acceleration has a linear relationship with the squared voltage, which is consistent with the foregoing observation that the core vibration is proportional to the squared operating voltage. Figure 15 shows the acceleration value of each measurement point under different magnetic densities. As the magnetic density increases, the acceleration value of each measuring point increases accordingly. It shows that the vibration vertical to the core plane is always higher than that parallel to it. In the same direction, the vibration in the central area of the SSS is significantly smaller than that in the seam area of the edge (a 1 < a 3 and a 2 < a 4 ). In the plane parallel to the iron core, the vibration phase in the rolling direction (a 5 ) is similar to that in the central area (a 2 ) in the     Shock and Vibration vertical (rolling) direction at a small-scale magnetic density. But as the magnetic density increases, the vibration vertical to the rolling direction is gradually greater than that grainoriented along this direction. As mentioned earlier, the microstructure and texture of various regions of SSS are not significantly different, in contrast to the SSS magnetostriction in different directions. erefore, the main reason for the obvious difference between various regions is the SSS magnetostriction.
In the SSS central area, the magnetic field distribution is continuous and uniform, and the magnetic field direction is definite and consistent. In contrast, the magnetic field distribution in the edge seam area is intermittent, and the presence of rotating magnetic flux causes the scattering of the magnetic field direction in this area in multiple directions. At the same time, during the deformation process in the microstructure of the seam area, grains are deformed and rotate more freely because they are less hindered and restrained by the surrounding grains. In addition, due to the mutual compression between the SSS in the common area, deformation behavior other than magnetostriction may occur. ese factors cause the seam area to have completely different magnetostrictive characteristics from the central area, which will inevitably lead to low and high vibration parameters of the core central and seam areas, respectively, as well as differences in their vibration spectrum characteristics. At the same time, microdeformation vertical to the core plane is generated.
Under the magnetic field effect, the magnetostrictive direction of the core SSS mainly coincides with the SSS rolling direction and the magnetic field direction. However, both ends of the SSS are constrained by six degrees of freedom. erefore, the magnetic field causes the SSS magnetostriction in its direction, but the material deformation cannot be released from both ends. It causes microbending and torsional deformation in the middle part   Shock and Vibration of the SSS, and the component of this deformation in the direction vertical to the SSS plane will cause periodic deformation between laminations.
is periodic microdeformation behavior induces vibrations in a direction vertical to the SSS plane. It can be seen that the core vibration in the direction vertical to the core plane is the result of the joint deformation of the edge seam area of the core SSS along this direction and the microdeformation between laminations in the central region. e vibration parallel to the core plane is the result of the SSS deformation in the magnetic field direction.

Modal, Noise, and Vibration Characteristics of the Actual
Transformer Core. Figure 16 is a photo of a modal test of a 110 kV transformer core. Modal test results are shown in Table 2. It can be seen that, in the vicinity of the natural frequency of the transformer core, it is easy to cause the equipment resonance and increase its overall vibration and noise levels. Figure 17 shows the first six natural frequency mode characteristics of the transformer core. It can be seen that the first-order mode of the transformer core is mainly characterized by the vibration deformation of the overall structure and the local torsion as the auxiliary. e secondorder mode is mainly characterized by the bending and torsion deformation and twist deformation of the structure. e third-order mode is mainly characterized by the torsional deformation of the structure. e fourth-order mode is mainly characterized by the bending and torsion deformation of the structure. e vibration modes above the fifthorder one show the combined effect characteristics of torsional deformation and bending deformation. As the core structural order rises, its modal characteristics gradually change from bending deformation to multiorder bending and torsional deformation, showing more complex deformation characteristics. e experimental field diagrams of the vibration and noise test of the 110kv transformer core components under no-load conditions and the frequency spectrum characteristics of some measurement points are shown in Figures 18  and 19 . e frequency of the vibration acceleration peak vertical to the core plane mainly appears at 100, 200, 300, and 500 Hz. ere are also certain peaks at 50, 150, 250, 350, 400, 600, 700, 800, and 900 Hz. e frequency of the vibration acceleration peak parallel to the core plane mainly appears near 300 and 500 Hz. ere are also certain peaks at 100, 200, 400, 600, 700, 800, and 900 Hz. e vibration spectrum characteristics of different positions and directions vary. e total effective value of the vibration acceleration at each point is shown in Figure 20. e amplitude of vibration in the direction parallel to the core plane is generally lower than that in the vertical direction. In the same direction, there is a gradually decreasing trend from both ends to the center. Due to the differences in the manufacturing process, capacity, and testing of the four-corner core model and the actual core products, the main vibration spectrum in each area is slightly different, but they are mainly based on the 100 Hz fundamental frequency and higher harmonic frequencies.
However, the vibration behavior is completely in accordance with the vibration behavior in different regions of the above four-corner core model. e noise spectrum characteristics (vertical and parallel to the core plane direction) are depicted in Figures 21(a) and 21(b), respectively. e peak frequencies with the highest noise vertical to the core plane mainly appear near 100, 200, and 500 Hz.
ere are also obvious noise peaks at such frequencies, such as 50, 80, 300, 400, and 600 Hz. For the noise spectra parallel to the iron core plane, the highest noise peak frequencies mainly appear near 300 and 500 Hz. ere are also obvious noise peaks near such frequencies as 50, 80, 100, 200, 400, and 600 Hz.
It can be seen that the core vibration and noise spectra in the same direction have significant acoustic vibration correlation characteristics. In the direction vertical to the core plane, the frequencies of the extreme amplitudes of vibration and noise mainly appear near 100, 200, and 500 Hz. ere are also certain peaks near such frequencies as 50, 300, and 400 Hz. In the direction parallel to the core plane, the frequencies of extreme amplitudes of vibration and noise mainly appear near 300 and 500 Hz. Peaks also appeared near such frequencies as 100, 200, and 400 Hz. e noise at the relevant frequency is mainly attributed to the core vibration.   Figure 22 shows the relationship between core modes, vibration, and noise. e first-order vibration of the core inherently includes frequencies near 50, 100, and 200 Hz, and its mode characteristics are mainly bending and torsional deformation in the direction vertical to the core plane. e analysis of the vibration spectrum in two directions shows that only the former vibration direction has a maximum peak near frequencies of 100 and 200 Hz. e noise spectrum also has peaks of maximum noise near 100 and 200 Hz, indicating that the core modal characteristics strongly influence its vibration and noise characteristics, which may cause structural resonance at these frequencies. It can be seen that the core modal characteristics, vibration, and noise also have a close correlation.
Based on the above analysis, three low-noise core design points are given. One is to select a low magnetostrictive silicon steel sheet that matches the core compression force to  reduce the vibration amplitude in all directions of the core. e second is to change the lapping mode or spacing of the core seam area to reduce the vibration amplitude at the seam areas. e third is to use noise-reducing damping materials or improve the structure to change the modal characteristics of the core to keep it away from the main operating frequency band in the energized state. Avoid the occurrence of equipment resonance under the power-on operation state, and reduce the damage ability of the resonance to the equipment, thereby ensuring the safe operation of the equipment and improving the environmental protection level of the equipment.

Conclusions
In this paper, the magnetostrictive characteristics of grainoriented silicon steel sheets (SSS) were tested. A four-corner core model platform was constructed to simulate the firstlevel vibration of the core. Finally, noise, vibration, and modal tests were performed on the actual 110 kV core product. is work provided a detailed analysis of the core vibration mechanism from the material to the product. e main conclusions are as follows: (1) e texture structure characteristics and crystal orientation distribution of the oriented SSS's central and edge seam areas are consistent. As the magnetic flux density of grain-oriented SSS increases, the magnetostriction in each direction grows until saturation, and the AWV noise value continues to increase without being affected by saturation. At the same magnetic flux density, the magnetostriction vertical to the rolling direction is significantly larger than that in the rolling direction, showing significant anisotropy. (2) ere are significant differences in the vibration of various areas of the four-corner core model. Vibrations vertical to the core plane exceed those parallel to it. Vibration in the central area in the same plane is significantly lower than that in the edge seam area. is difference is caused by the different microscopic deformation contributed by various regions with completely different magnetostrictive characteristics.
(3) e vibration behavior of the actual 110 kV iron core product is shown to be the same as that of the fourcorner iron core model. In the same direction, there is a gradually decreasing trend from both seams to the center, and the vibration and noise spectra have significant acoustic vibration correlation characteristics. Its modal values also have a close correlation with its vibration and noise characteristics.
In the future, the vibration amplitude of the core can be reduced by choosing the low magnetostrictive silicon steel sheet which matches the core compression force, changing the lapping mode or spacing of the core seam area, increasing the noise-reducing damping materials, or improving the structure.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.