Simulating the vibration increase of the transformer iron core due to the DC bias

Abstract

Transformers in the power systems are sometimes influenced by the DC bias and produce abnormal vibration or noise. This paper presents a method to simulate the transformer vibration increase due to the DC bias. A three-dimensional finite element model coupling the circuit – magnetic field – mechanical field is established for the simulating. Firstly, the 3D magnetic field in the iron core is calculated through combing the circuit equation and magnetic field equation. Secondly, the magnetostrictive force derived from the magnetic field is used to establish the transient structural force field model, so that the vibration displacement can be calculated. Based on the method, the vibration at different positions of the transformer iron core with different DC bias is investigated. The simulation shows that the vibration of the iron core is quite sensitive to the DC bias. The vibration amplitude significantly increases and the vibration waveform becomes asymmetrical and sharp with the increase of DC current. The odd frequency component, represented by 50 Hz and 150 Hz, appears and grows faster than the 100 Hz and 200 Hz component. The model and the simulation results can be used to guide the structural designing of transformers and promote the fault diagnosis method of transformers based on the vibration signal monitoring.

Keywords

DC bias transformer iron core vibration magnetostrictive finite element method

1. Introduction

Transformers in the power systems are sometimes disturbed by the DC bias, which lead to the half cycle saturation of the iron core, produce the excitation current distortion, and increase the harmonic content, etc. [1]. Additionally, the saturation of magnetic flux causes the enhancement of magnetic expansion in the iron core and the intensified vibration, which not only endangers the safe operation of transformers and affects power system security, but also makes noise pollution to the surrounding environment and influences the physical and mental health of the nearby population [2]. There are two main origins of transformer DC bias. One is geomagnetically induced currents (GIC) in the power networks generated by the magnetic storm. GIC can be up to 100 A in the high voltage power grid in China during strong magnetic storms. GIC can be regarded as the DC current in the AC system because its frequency is 0.0001 $\sim$ 0.01 Hz [3]. The other is the application of the high voltage direct current (HVDC) transmission, leading to the electromagnetic coupling between the electrode and the transformers nearby [4], and making DC current flow through the grounded neutral point to the transformer windings.

In recent years, it has been found many times that the DC bias causes abnormal vibration, transformer faults or even incidents threatening the safety of the power system. For example: in October 2002, the vibration of the transformer in Shanghai 500 kV substation suddenly increased and the noise became larger. The analysis shows that it is caused by GIC in the transmission system [4, 5]. In addition, in December 2002, the noises of two main transformers at Changzhou 500 kV substation reached 91.4 dB with the maximum neutral point DC current of 12.8 A during the commissioning period of a 500 kV HVDC transmission project [5].

At present, there have been several discussions concerning to the vibration of the transformers [6, 7, 8, 9, 10, 11, 12]. Reference [11, 12] offered simple and efficient spectral domain technique for DC bias measurement and compensation, based on detection of the even harmonics of the reactor/transformer magnetization current. However, the research of transformer vibration increase caused by the DC bias is mainly based on the monitoring of the displacement at the oil tank wall. The harm of DC bias to a transformer can be reflected through the monitoring data [13, 14, 15, 16]. Due to the complex structure of the transformers, the measurement of the vibration at the tank wall cannot fully present the vibration characteristics of the iron core. Additionally, the experimental study, in spite of consuming more time and cost, has less help for finding out the relationships between the vibration characteristics and the structure or operation condition of a transformer.

Some researchers tried to establish a model to simulate the vibration characteristics of the iron core under DC bias. For example, reference [17] analyzed the influence of the lamination structure on the resonance of the iron core based on the finite element method (FEM), and reference [18] measured the butterfly curve family of the magnetostrictive in the silicon steel sheet, and carried out the simulation analysis as well as experimental verification on the transformer vibration characteristics under DC bias.

However, in reference [18], the simulation model and the establishment procedure of the simulation model has not been fully discussed, and using the silicon steel magnetostrictive strain as the displacement of the iron core may bring significant error in the model. Reference [19] presented a model for simulating the vibration of the iron core and windings under the DC bias. In this model, it is necessary to know the permeability change of the medium with the volume density. But reference [20] has been proved that it is very difficult to obtain. Therefore, the simulation of the transformer vibration under the DC bias needs further research.

In this paper, we have established a three – dimensional circuit – magnetic field – mechanical field coupling model of the transformer by using the finite element method, and used it to analyze the vibration characteristics of the transformer core under different DC biases. Firstly, we combined the circuit equation and magnetic field equation, then discretized the weighted residual equations to the algebraic equations, so that we can compute the 3-D magnetic field in the transformer core. Secondly, we derived the equivalent magnetostrictive force of the iron core from the elastic theory and established the transient structural force field numerical model, which calculate the vibration displacement of the transformer core. Finally, we numerically simulated the iron core vibration with different DC bias. The model in this paper can be used to guide the structural designing of the transformer so that we can reduce the vibration and noise. Meanwhile, it can promote the development of the fault diagnosis method of transformers based on the vibration signal monitoring.

Figure 1.

The circuit and magnetic field coupling model with DC bias.

2. Coupling model of the circuit with the magnetic field

The origin of DC bias is the potential difference between the neutral points of the transformers. When establishing the equivalent model, we use the normal AC voltage source and the offset DC voltage source on the neutral point as the excitation. The coupling model of circuit and magnetic field is shown in Fig. 1, where

$U_{d}$ is the equivalent DC voltage;

$u_{s1}$ , $u_{s2}$ and $u_{s3}$ are three-phase AC voltage respectively;

$Z_{0}$ , $Z_{1}$ and $Z_{2}$ are the equivalent impedance of the power source, the three phase load at the medium voltage side and the three-phase load at the low-voltage side, respectively;

$i_{1}$ , $i_{2}$ and $i_{3}$ are the three-phase currents in the primary windings respectively.

2.1 Circuit equation

When both medium voltage and low voltage windings are open, in the circuit composed of the three-phase and the neutral point in the primary windings, the equation can be written as:

$\displaystyle u_{sj}\left(t\right)+U_{d}-\left({R_{sj}i_{j}\left(t\right)+R_{j% }i_{j}\left(t\right)+L_{j}\frac{di_{j}\left(t\right)}{dt}}\right)=N\left({% \frac{d\Phi_{j}\left(t\right)}{dt}}\right),j=1,2,3$ (1)

Where

$\Phi$ ${}_{j}$ ( $j=$ 1, 2, 3) is the flux in the three-phase core column;

$R_{j},L_{j}$ , $R_{sj}$ are the three-phase winding resistance, linear inductance, and the power source internal resistance, respectively;

$N$ is the number of turns of the primary windings.

2.2 Magnetic field model

According to the fundamental electro-magnetic theory, the nonlinear magnetic field within a transformer can be described as:

$\displaystyle\nabla\times\frac{1}{\mu}\times{\bm{A}}={\bm{J}}$ (2)

Where

A is the magnetic vector potential, which can be set to zero at the outside surface of the transformer tank;

J is the current density, which can be obtained from the coupling calculation of the circuit Eq. (1);

$\mu$ is the permeability of the material. The permeability of the silicon steel sheet is determined from its B-H curve. The permeability of the tank wall, the transformer oil, and other parts are set to fixed value according to the material properties.

The 3D magnetic field within a transformer can be calculated by the edge finite element method. The edge element interpolation function is chosen as

$\displaystyle{\bm{A}}=\sum\limits_{n=1}^{n_{n}}{M_{n}(x,y,z)}{\bm{A}}_{n}$ (3)

Where

$M_{n}$ ( $n=$ 1, 2, $\ldots$ , $n_{n}$ ) is the basis function sequence, formed by relevant element shape function;

$n$ is the number of base functions;

$n_{n}$ is the total number (total edge number).

Thus the Galerkin weighted residual equation can be written as:

$\displaystyle\int\!\!\!\int\!\!\!\int_{V}{\frac{1}{\mu}}\left({\nabla\times M_% {m}}\right)\cdot\left({\nabla\times M_{n}}\right){\bm{A}}_{n}dV=\int\!\!\!\int% \!\!\!\int_{V}{M_{m}}{\bm{J}}dV$ (4)

Where $M_{n}$ is a sequence of basis functions, and the weight function is the same as the basis function.

We take the weight function Eq. (3) into the weighted residual Eq. (4). Then the weighted residual equation can be discretized into algebraic equations for all the weight function. Therefore, with any given value of J in Eq. (2), we can figure out the vector magnetic field A at all the nodes or edges, and then obtain the magnetic field B.

Assuming the original magnetic flux in the iron core is zero, we can calculate the exciting current $i$ from Eq. (1) with the predetermined AC and DC voltage sources, then substitute it into Eq. (2). With the help of Eqs (3) and (4), the magnetic flux in the iron core $\Phi$ can be obtained. Then substitute it into Eq. (1), we can get a new value of the exciting current. Repeat that procedure, and a stable state will be reached after iterating 4–5 industrial frequency cycles. Thus the distribution of the magnetic field in the iron core can be obtained.

In the coupling model of circuit and magnetic field, the nonlinear of magnetic flux saturation is mainly derived from the B-H curve of the silicon steel sheet, which can be obtained from the data of the silicon steel sheet provided by the manufacturer.

3. Coupling model of magnetic field and mechanical field

3.1 Iron core vibration mechanism

The main sources of the iron core vibration are the magnetostrictive of the silicon steel sheet, and the electromagnetic force produced by the magnetic leakage at the joint of the silicon steel sheet and among the laminations. Moreover, the vibration caused by the magnetostrictive is more important [21].

It is assumed that the mass of a particle in the transformer core is $m$ , and the motion equation of the particle can be expressed as follows:

$\displaystyle m\ddot{x}\left(t\right)+f\left[{\dot{x}\left(t\right),x\left(t% \right),t}\right]=f_{em}\left(t\right)+f_{ms}\left(t\right)+mg$ (5)

Where

$x(t)$ is the displacement of the iron core particle;

$f$ [(t),x(t),t] is the internal force generated by the iron core;

$f_{em}(t)$ is the electromagnetic force for the iron core particle;

$f_{ms}(t)$ is the magnetostrictive force of iron core;

mg is the gravity.

According to the investigations which have been made earlier [13, 14], the magnetostrictive force is much higher than the electromagnetic force. Therefore, we ignored the impact of the electromagnetic force when discussing the iron core vibration.

3.2 Calculation of the magnetostrictive force

When the iron core is magnetized, the silicon steel sheet will have the magnetostriction. This phenomenon could be described through the following relation: magnetostrictive force caused by the internal magnetic field of the silicon steel sheet makes the deformation of the silicon steel sheet. The Magnetostrictive force can be obtained based on the elastic mechanics. In the numerical calculation of the three-dimensional mechanical vibration of the iron core, the discrete FEM model is used to replace the original continuous model. Therefore, each FEM unit in the iron core model is satisfied with the following assumptions in the elastic field:

The silicon steel sheet is homogeneous and isotropic elastoplastic, which means that the elastic constants of the objects do not change with the coordinate and direction of the grain.

The iron core is completely elastic and obeys the Hooke’s law.

According to the basic principle of the elastic mechanics, we can obtain the force of the unit from the stiffness matrix and the displacement of the elements. Neglecting the shear strain of the silicon steel sheet, we can obtain the equivalent magnetostrictive force of the element by the generalized Hooke law.

$\displaystyle f_{ms}\left(t\right)=\left[{\begin{array}[]{l}\sigma_{x}\\ \sigma_{y}\\ \sigma_{z}\\ \end{array}}\right]=K\cdot\varepsilon=\frac{E\left({1-\alpha}\right)}{\left({1% +\alpha}\right)\left({1-2\alpha}\right)}\left[{{\begin{array}[]{*{20}c}1\hfill% &{\frac{\alpha}{1-\alpha}}\hfill&{\frac{\alpha}{1-\alpha}}\hfill\\ {\frac{\alpha}{1-\alpha}}\hfill&1\hfill&{\frac{\alpha}{1-\alpha}}\hfill\\ {\frac{\alpha}{1-\alpha}}\hfill&{\frac{\alpha}{1-\alpha}}\hfill&1\hfill\\ \end{array}}}\right]\left[{{\begin{array}[]{*{20}c}{\varepsilon_{x}}\hfill\\ {\varepsilon_{y}}\hfill\\ {\varepsilon_{z}}\hfill\\ \end{array}}}\right]$ (6)

Where

$\sigma_{x}$ , $\sigma_{y}$ , $\sigma_{z}$ are the unit normal stress;

$K$ is the element stiffness matrix;

$E$ is the elastic modulus of silicon steel sheet;

$\alpha$ is Poisson’s ratio of the silicon steel sheet;

$\varepsilon_{x}$ , $\varepsilon_{y}$ , $\varepsilon_{z}$ are the strain obtained by the interpolation of the magnetostrictive curve.

3.3 Coupling of magnetic field and mechanical field

The magnetostrictive curve of the silicon steel sheet along the rolling direction (RD) and the perpendicular direction (TD) can be expressed as the following:

$\displaystyle\left\{{{\begin{array}[]{*{20}c}{\lambda_{rd}=f_{rd}(B)}\hfill\\ {\lambda_{td}=f_{td}(B)}\hfill\\ \end{array}}}\right.$ (7)

Where

$\lambda_{td}$ and $\lambda_{rd}$ are respectively the magnetostrictive rate at TD and RD direction;

$B$ is the magnetic flux density.

For the example of the upper iron yoke, the lower left corner is supposed to be the origin, and the direction of the iron yoke is extended to the $x$ axis. For any individual unit, the deformation is:

$\displaystyle\left[{{\begin{array}[]{*{20}c}{\varepsilon_{x}}\hfill\\ {\varepsilon_{y}}\hfill\\ {\varepsilon_{z}}\hfill\\ \end{array}}}\right]=\left[{{\begin{array}[]{*{20}c}{f_{rd}\left({\left|{B_{x}% }\right|}\right)\bullet\left|{\Delta_{x}}\right|}\hfill\\ {f_{td}\left({\left|{B_{y}}\right|}\right)\bullet\left|{\Delta_{y}}\right|}% \hfill\\ {f_{td}\left({\left|{B_{z}}\right|}\right)\bullet\left|{\Delta_{z}}\right|}% \hfill\\ \end{array}}}\right]$ (8)

Where

$\Delta_{x}$ , $\Delta_{y}$ , $\Delta_{z}$ are the unit length in the $x, y, z$ direction, respectively;

$B_{x}$ , $B_{y}$ , $B_{z}$ are the components of the magnetic field strength in the $x, y, z$ direction, respectively.

Taking Eqs (7) and (8) into Eq. (6), the equivalent unit magnetostrictive force is:

$\displaystyle f_{ms}(t)=\frac{E\left({1-\alpha}\right)}{\left({1+\alpha}\right% )\left({1-2\alpha}\right)}\left[{{\begin{array}[]{*{20}c}1\hfill&{\frac{\alpha% }{1-\alpha}}\hfill&{\frac{\alpha}{1-\alpha}}\hfill\\ {\frac{\alpha}{1-\alpha}}\hfill&1\hfill&{\frac{\alpha}{1-\alpha}}\hfill\\ {\frac{\alpha}{1-\alpha}}\hfill&{\frac{\alpha}{1-\alpha}}\hfill&1\hfill\\ \end{array}}}\right]\times\left[{{\begin{array}[]{*{20}c}{f_{rd}\left({\left|{% B_{x}}\right|}\right)\bullet\left|{\Delta_{x}}\right|}\hfill\\ {f_{td}\left({\left|{B_{y}}\right|}\right)\bullet\left|{\Delta_{y}}\right|}% \hfill\\ {f_{td}\left({\left|{B_{z}}\right|}\right)\bullet\left|{\Delta_{z}}\right|}% \hfill\\ \end{array}}}\right]$ (9)

In the same way, we can obtain the distribution of the equivalent magnetostrictive force in any other structural parts. Taking them into Eq. (5), we can obtain the displacement, acceleration and other mechanical quantities at each position.

4. Simulation example analysis

In this paper, we take the case of a three-phase five column transformer with the model of SFSZ10-180000/220. The nominal voltage of the transformer is 230 kV /121 kV /11 kV. The connection group is YNyn0d11. The type of silicon steel sheet of its iron core is 30ZH120 (Japan), and the core lamination coefficient is 0.97. The 3-D transformer model for the simulation is shown in Fig. 2, and the detailed assembly drawing of the iron core and windings is shown in Fig. 3.

Figure 2.

Three-dimensional model of the transformer (A, B and C are three testing points selected on the iron core).

Figure 3.

Assembly drawing of the iron core and the windings. (1 – left side column, 2 – A phase column, 3 – medium voltage winding, 4 – upper iron yoke, 5 – high voltage winding, 6 – the right side column, 7 – lower iron yoke).

Figure 4.

Distribution of the vibration displacement of the transformer.

The simulation steps are shown below: firstly, the magnetic flux density of each unit is obtained by the field-circuit coupling simulation with the help of ANSYS software, and then the equivalent magnetostrictive force of each unit is calculated by the field calculator. Secondly, we plugged the magnetostrictive force into the transient analysis field, imposing the constraint conditions on the iron core, and then computed the vibration of the iron core. In the external circuit of the model, the middle and low voltage windings are both set to open circuit to simulate the no-load operation. The DC voltage is adjustable for changing the DC bias current.

4.1 Influence of DC bias on the vibration amplitude of the iron core

The distribution of the vibration displacement in the iron core is shown in Fig. 4 when the DC bias current in the high voltage winding is 0 A and 2 A. Figure 4a shows that the maximum value of the vibration displacement is 2.295 $\mu$ m when there is no DC bias. When the DC current is 2 A, the maximum vibration displacement is 3.460 $\mu$ m which is 51% higher than that without the DC bias. It can be seen from the comparison that the vibration of the iron core is quite sensitive to the DC bias. Even small DC bias is enough to cause obvious increase on the vibration and noise on a transformer.

Figure 5.

Relationship between the maximum amplitude of the vibration and the magnitude of the bias current.

In Fig. 5, the vibration amplitude increases of the iron core are shown with the increase of the DC current. The vibration amplitude is represented by the maximum displacement of testing point A, B and C. As shown in Fig. 2, A, B and C point are on the top of the left side column, A phase column and B phase column.

It can be seen that the vibration amplitude is approximately a linear relationship with the growth of DC bias before it reaches a turning point, and the growth rate of the vibration amplitude decreases thereafter because the iron core is deeply saturated.

Figure 5 also shows the vibration amplitude increase due to the DC bias at different position of the iron core. The DC bias has more influence on the vibration amplitude at the point A, and has less influence on the vibration amplitude at the point C. When the DC bias is exerted, the magnetic flux in flanked columns has maximum increments, so the enhanced vibration in the side columns causes the increasing of yoke lateral displacement. Therefore, the nearer the point closes to the side column, the faster the vibration amplitude grows.

Figure 6.

Comparison of the vibration displacement waveform when the DC bias is 0 A and 2 A.

4.2 Influence of the DC bias on the vibration of the iron core

The comparison of the vibration waveform with or without DC bias in the time domain is shown in the Fig. 6. The waveform presents the vibration displacement of the point A in Fig. 2, which is on the top of the side column. It can be seen from the Fig. 6 that the vibration amplitude of the iron core is relatively smaller and the vibration waveform of the positive and negative half cycle in one excitation period is symmetrical when the DC current is zero. As a comparison, when the DC current is 2 A, the vibration amplitude is significantly increased, besides that the vibration waveform became asymmetrical and sharp, which implicate a change at the vibration spectrum.

Figure 7.

Comparison of the vibration displacement frequency spectrum when the DC bias is 0 A and 2 A.

The comparison of the vibration displacement spectrum with or without the DC bias in the frequency domain is shown in the Fig. 7. The displacement comes from the point A in Fig. 2. It can be seen from Fig. 7a that the main vibration spectrum component without the DC bias is 100 Hz, and a little of even harmonics, and the odd harmonics are almost negligible. When the DC current is 2 A, the spectral components, which are shown in Fig. 7b, appear obvious 50 Hz and 150 Hz components, and the energy of higher frequency component is obviously increased so that the frequency spectrum is getting more complicated.

Figure 8.

Relationship between the spectrum distribution of the vibration displacement and the DC bias.

4.3 Vibration spectrum characteristics with different DC bias

The relationship between the spectrum distribution of the vibration displacement and DC current is shown in Fig. 8. With the intrusion of DC current, the vibration frequency component is no longer concentrated on the 100 Hz. The odd times frequency component, represented by the 50 Hz and 150 Hz, appear and increase faster with the enlargement of DC current than the component of 100 Hz and 200 Hz. This simulation result is accord with the theoretical analysis, and it provide a criterion to judge if there is a core flux saturation due to the DC bias from the abnormal noise of a transformer.

5. Conclusions

This paper establishes a three-dimensional FEM coupling model of the circuit – magnetic field – mechanical field to simulate the transformer vibration increase due to the DC bias. Based on the method, we investigated the transformer core vibration under different DC current, which shows the significant influence of the DC bias on the vibration of a transformer.

The vibration of the iron core is quite sensitive to the DC current. Even small DC current is enough to cause obvious increase of the vibration and noise. The vibration amplitude appears approximately a linear increase with the growth of DC bias before the iron core is deeply saturated.

Because the cross section of the side column is only half of the phase column, the position near the side column has more vibration increment than that of the core column under the influence of DC bias. Therefore, we need to design the side column more carefully.

The 100 Hz component is the main component of the vibration spectrum without the DC bias. With the increase of DC current, the vibration amplitude significantly increases and the vibration waveform becomes asymmetrical and sharp. The odd frequency component, represented by 50 Hz and 150 Hz, appears and grow faster than the 100 Hz and 200 Hz component. This provide a criterion to judge if there is a core flux saturation due to the DC bias from the abnormal noise of a transformer and may be helpful for fault diagnosis and condition maintenance of power transformers.

Footnotes

Acknowledgments

This work was supported by National Key R&D Program of China (2016YFC0800103) and National Natural Science Foundation of China (51677068).

References

Boteler

D.H.

Shier

R.M.

Watanabe

and Horita

R.E.

, Effects of geomagnetically induced currents in the BC Hydro 500 kV system, IEEE T Power Deliver 4(1) (1989), 818–823.

Liu

C.M.

Liu

L.G.

and Yang

, Monitoring and Modeling Geomagnetically Induced Currents in Power Grids of China. in: Asia-pacific Power and Energy Engineering Conference, 2009, pp. 1–4.

Liu

C.M.

Liu

L.G.

and Pirjola

, Geomagnetically Induced Currents in the High-Voltage Power Grid in China, IEEE Transactions on Power Delivery 24(4) (2009), 2368–2374.

Zheng

K.D.

Liu

C.M.

and Wan

, Experiment and Research of the Influence of Direct-current Magnetic Bias on Transformer, Jiangsu Electrical Engineering 23 (2004), 1–5,

Liu

L.Y.

and Xie

X.W.

, Analysis of Increase of Noise of 500 kV Transformer, High Voltage Engineering 68(Supplement S11) (2005), 2514–2518.

Moses

A.J.

, Measurement of magnetostriction and vibration with regard to transformer noise, IEEE T Magn 10(2) (1974), 154–156.

Kitagawa

Ishihara

Todaka

and Nakasaka

, Analysis of structural deformation and vibration of a transformer core by using magnetic property of magnetostriction, Electr Eng Jpn 172(1) (2010), 9–26.

Ertl

and Landes

, Investigation of load noise generation of large power transformer by means of coupled 3D FEM analysis, Compel International Journal of Computations and Mathematics in Electrical 26(3) (2007), 788–799.

Wakabayashi

and Enokizono

, Analysis of vector magnetic property and two-dimensional magnetostriction in electrical steel sheets, International Journal of Applied Electromagnetics and Mechanics 44(3) (2014), 317–330.

10.

Ghalamestani

S.G.

Vandevelde

Dirckx

J.J.J.

Guillaume

and Melkebeek

J.A.A.

, Magnetostrictive deformation of a transformer: A comparison between calculation and measurement, International Journal of Applied Electromagnetics and Mechanics 44(44) (2014), 295–299.

11.

Vukosavić

S.N.

and Perić

L.S.

, High-Precision Sensing of DC Bias in AC Grids, IEEE Transactions on Power Delivery 30(3) (2015), 1179–1186.

12.

Pjevalica

Petrovic

Pjevalica

and Teslic

, Experimental Detection of Transformer Excitation Asymmetry through the Analysis of the Magnetizing Current Harmonic Content, Elektronika Ir Elektrotechnika 22(2) (2016), 43–48.

13.

Bai

B.D.

Liu

and Wang

J.Y.

, Simulation and Experimental Research of Transformer Vibration under DC Magnetic Bias, Transactions of China Electrotechnical Society 28(Sup. 2) (2013), 427–433.

14.

Chen

Q.H.

H.B.

and He

J.L.

, Field Monitoring and Analysis on Vibration and Noise of 500 kV Electrical Transformer under DC Current Biasing, High Voltage Apparatus 43(8) (2009), 3591–3597.

15.

Guo

Huang

Tang

and He

W.L.

, Analysis on 500 kV Power Transformer Vibration Under DC Magnetic Biasing, Power System Technology 36(3) (2012), 70–75.

16.

Yang

Liu

Yang

Jadoon

and Zhang

, Analysis of DC bias vibration of transformer core based on electromagnetic force field coupling, International Journal of Applied Electromagnetics and Mechanics 50(2) (2016), 297–309.

17.

Kubiak

and Witczak

, Vibration analysis of small power transformer, Compel International Journal of Computations and Mathematics in Electrical 29(4) (2010), 1116–1124.

18.

Wang

Bai

Liu

and Ma

, Research on Vibration and Noise of Transformers under DC Bias, Transactions of China Electrotechnical Society 30(8) (2015), 56–61.

19.

Wang

J.G.

, K.Mao Duan

Yang

Y.M.

and Ke

L.I.

, Simulation and test of transformer vibration under DC bias, Electric Machines and Control 19(1) (2015), 58–67.

20.

Reyne

Sabonnadiere

J.C.

Coulomb

J.L.

and Brissonneau

, A survey of the main aspects of magnetic forces and mechanical behaviour of ferromagnetic materials under magnetization, IEEE T Magn 23(5) (1987), 3765–3767.

21.

Weiser

Hasenzagl

Booth

and Pfützner

, Mechanisms of noise generation of model transformer cores, J Magn Magn Mater 160(7) (1996), 207–209.