Research on electromagnetic vibration and noise of transformer under short circuit conditions

B–H curve.

Figure 2 shows the magnetostriction testing platform. Components are as follows: 1, laser ranging probe; 2, positioner; 3, base; 4, clamp; 5, control lever; 6, coil; 7, iron yoke; 8, air pump. The vibration of the core mainly originates from the magnetostriction of the silicon steel sheets. The magnetostriction characteristics of the 20SQG090 silicon steel sheet were measured using a magnetostriction measurement system produced by the German company BRAOCKHAUS.

FIG .2.

Magnetostriction test platform.

This device can generate an alternating magnetic field and can apply adjustable tension and pressure laterally to the sample. Before starting the measurement, both ends of the silicon steel sheet must be clamped, and the sheet should be securely fixed from above. During measurement, one end of the silicon steel sheet is fixed while the other is used as the observation point, with a reflective film attached. As the silicon steel sheet elongates and contracts due to changes in the magnetic field, the laser device of the instrument is used to observe the amount of length change in the silicon steel sheet. Additionally, to prevent bending of the silicon steel sheet during testing—which could affect measurement results—the sheet must be pressed down from above. An isolation platform is also used to prevent interference from other factors, improving measurement accuracy.

The magnetostriction B–λ curve of the silicon steel sheet obtained through measurement is shown in Fig. 3.

FIG .3.

Magnetostrictive B–λ curve.

The saturation phenomenon in the magnetostriction B–λ curve shown in Fig. 3 occurs because, under the influence of the magnetic field, the magnetic moments of the magnetic material gradually align in a particular direction. Once all magnetic moments are fully aligned, any further increase in the magnetic field will not produce additional magnetostrictive effects. This alignment process is gradual, and at a certain magnetic field intensity, most of the magnetic moments are nearly fully aligned. At this point, further increasing the magnetic field intensity results in only a minimal change in the magnetostriction effect, leading to the saturation observed in the magnetostriction B–λ curve.

B. Model validation

This paper constructed a three-dimensional model of the three-dimensional laminated core transformer and performed electromagnetic characteristic calculations to validate the model’s accuracy.

The electromagnetic characteristics calculation results of the three-dimensional coil core transformer under rated operation are shown in Fig. 4. Since the calculated values are phase voltage and phase current, the calculated results are generally the same as the specified results, and the error is <2%, which verifies the accuracy of the finite element model established in this paper.

FIG .4.

Voltage and current under normal operating conditions: (a) primary side voltage; (b) secondary side voltage; (c) primary side current; (d) secondary side current.

III. FINITE ELEMENT CALCULATION MODEL OF THREE-DIMENSIONAL LAMINATED CORE TRANSFORMER

A. Characteristics parameters of silicon steel sheets

The electromagnetic force on the winding is primarily related to the current flowing through the winding and the leakage magnetic field around it.¹² Therefore, studying the distribution of the leakage magnetic field in both normal and fault conditions is essential.

Assume that the cross-sectional view of the winding is shown in Fig. 5, with the inner and outer diameters of the high-voltage and low-voltage windings denoted as a1, b1, a2, and b2, respectively. To calculate the magnetic field at any point P(x, y, z) on the winding, the winding can be divided longitudinally into multiple thin layers, with each layer having a height of ΔH, and the current flowing through each conductor layer is I_c.

I_{C} = \frac{N I}{H} Δ H .

(1)

In the formula, N represents the number of turns in the winding, I is the current flowing through each turn of the winding, and H is the height of the winding.

FIG .5.

Conductor layer and conductor ring.

By dividing the winding radially into multiple thin layers, each thin layer becomes a conductor ring. The current flowing through the cross-section of the conductor ring is I_h, ΔL is the width of the conductor ring, and L is the width of the winding:

I_{h} = \frac{N I}{H L} Δ H Δ L .

(2)

The polar coordinates of a current element (I Δ l) at a point on the conductor ring are

I Δ l = \frac{N I}{H L} Δ H Δ L Δ l,

(3)

Δ l = r [\begin{matrix} e_{y} \cos θ & - e_{x} \sin θ \end{matrix}] Δ θ .

(4)

In the formula, Δl is the differential element of the current, r is the inner radius of the conductor layer where the current element is located, e_x, e_y are unit vectors in the direction of the x-axis and y-axis, θ is the angle between Δl and the y-axis, and Δθ is the angle corresponding to the arc length Δl. Thus, the leakage magnetic field intensity at any point in the space around the winding can be obtained:

Δ B = \frac{μ}{4 π} \frac{N I}{H L} Δ H Δ L Δ l \frac{d}{d^{3}} = \frac{k}{d^{3}} Δ l d Δ H Δ L,

(5)

k = \frac{μ N I}{4 π H L},

(6)

Δ l \times d = r Δ θ [\begin{matrix} e_{x} & e_{y} & e_{z} \\ - \sin θ & \cos θ & 0 \\ x_{0} - r \cos θ & y_{0} - r \sin θ & z_{0} - z \end{matrix}] .

(7)

\{\begin{cases} Δ B_{x} = e_{x} \frac{k r}{d^{3}} (z_{0} - z) \cos θ Δ θ Δ H Δ L \\ Δ B_{y} = e_{y} \frac{k r}{d^{3}} (z_{0} - z) (- \sin θ) Δ θ Δ H Δ L \\ Δ B_{z} = e_{z} \frac{k r}{d^{3}} (- y_{0} \sin θ + r - x_{0} \cos θ) Δ θ Δ H Δ L \end{cases}

(8)

By integrating Eq. (8), the leakage magnetic field intensity and its components in various directions at any point within the winding can be determined.

In the formula, ΔB is the magnetic flux density, μ is the magnetic permeability of the medium, L is the distance between the current element at point M and point P in space, d is the vector pointing from M to P, and e_z is the unit vector in the positive direction of the z-axis.

\{\begin{cases} B_{x} = e_{x} \int_{-_{0.5 H}}^{0.5 H} \int_{a}^{b} \int_{0}^{2 π} \frac{k r (z_{0} - z) \cos θ d θ d r d z}{d^{3}} \\ B_{y} = e_{y} \int_{- 0.5 H}^{0.5 H} \int_{a}^{b} \int_{0}^{2 π} \frac{k r (z_{0} - z) (- \sin θ) d θ d r d z}{d^{3}} \\ B_{z} = e_{z} \int_{- 0 \cdot S t}^{0.5 H} \int_{a}^{b} \int_{0}^{2 π} \frac{k r (- y_{0} \sin θ + R - x_{0} \cos θ) d θ d r d z}{d^{3}} \end{cases}

(9)

d = \sqrt{{(x_{0} - r \cos θ)}^{2} + {(y_{0} - r \sin θ)}^{2} + {(z_{0} - z)}^{2}},

(10)

|B| = |B_{x} + B_{y} + B_{z}| = \sqrt{{|B_{x}|}^{2} + {|B_{y}|}^{2} + {|B_{z}|}^{2}} .

(11)

Because the distribution of radial magnetic leakage is characterized by the same size and opposite direction at the top and bottom of the winding,¹³ the axial electrodynamic interaction of the high and low voltage winding causes the compression force inside the winding, as shown in Fig. 6.

FIG .6.

Winding magnetic flux leakage and force.

The calculation formula for electromagnetic force is given by Eq. :

F = B L I N .

(12)

From the above formula, the radial leakage magnetic field intensity B_xy on the winding is given by

B_{x y} = B_{x} + B_{y},

(13)

L = π D .

(14)

In the formula, D is the mean diameter of the winding.

The radial force F_xy and axial force F_z on the winding are

F_{x y} = I_{d} \cdot N \cdot π D \cdot B_{x y},

(15)

F_{z} = I_{d} \cdot N \cdot π D \cdot B_{z} .

(16)

Due to the action of radial electromagnetic force, the high-voltage winding experiences an outward-expanding force.¹⁴ As a result, a tangential force F′ is generated within the winding, with its magnitude given by Eq. and its direction shown in Fig. 7:

F^{'} = \frac{1}{2 π} F_{x y} .

(17)

FIG. 7.

Diagram of tangential forces.

The expression for the tangential tensile stress σ_x is given by Eq. :

σ_{x} = \frac{F_{x y}}{2 π N A_{x}} .

(18)

B. Finite element calculation of inter-turn short circuits in transformer windings

This study uses the finite element analysis method to conduct an in-depth analysis of various operating conditions of the transformer. It examines the normal operating state of the transformer winding as well as the fault conditions involving 1, 3, 7, and 19 turn short-circuits in phase A of the high-voltage winding.

To observe key parameters such as the spatial magnetic field, radial leakage magnetic field, and axial leakage magnetic field inside the A-phase winding, the red line position in Fig. 8 was selected as the observation point during the analysis. Detailed simulation calculations and data analysis revealed that, under normal operating conditions, the magnetic field distribution inside the winding is uniform, with a relatively low leakage magnetic field. However, under the fault conditions of 1, 3, 7, and 19 turn short-circuits in the high-voltage winding phase A, the magnetic field distribution inside the winding shows abnormalities, and the intensity and distribution range of the leakage magnetic field change significantly.

FIG. 8.

Winding magnetic flux leakage observation position.

In Fig. 9, the vertical axis represents the height of the winding, while the horizontal axis indicates the magnetic flux density on the inner side of the A-phase high-voltage winding. During normal operation of the transformer, the spatial leakage magnetic field reaches its maximum at the center of the winding, up to 27 mT, and gradually decreases toward both ends. The radial leakage magnetic field, on the other hand, is at its maximum at both ends of the winding, reaching ∼6 mT, with the radial flux density at zero in the middle; the curve is symmetrical about the origin. The distribution of the axial flux density is roughly the same as that of the spatial flux density but in the opposite direction.

FIG .9.

The magnetic field distribution diagram on the inner path of the winding under normal operation and different short circuit turns: (a) spatial leakage magnetic field distribution under normal operation; (b) spatial leakage magnetic field distribution under different short-circuit scenarios; (c) radial leakage magnetic field distribution under normal operation; (d) radial leakage magnetic field distribution under different short-circuit scenarios; (e) axial leakage field distribution under normal operation; (f) axial leakage field distribution under different short circuit turns.

However, under inter-turn short-circuit conditions, the spatial flux density curve of the winding becomes distorted due to the short-circuit current, exhibiting two peaks that increase with the number of short-circuited turns. These two peaks are located at the ends of the short-circuited turns, while the leakage magnetic field intensity at the center of the winding suddenly decreases. This phenomenon is related to the radial flux density in the inter-turn short-circuit state.

In Fig. 9(d), it can be observed that the radial flux density in the winding first increases, then suddenly reverses direction, and decreases, with the two maximum values corresponding to the flux density at both ends of the short-circuited turns, maintaining symmetry about the origin. The maximum axial flux density occurs at the center of the winding, while the axial flux density at the ends of the short-circuited turns approaches zero. The axial flux density of the non-short-circuited turns is opposite to that of the short-circuited turns, with the waveform being symmetrical about the midpoint of the winding.

IV. THREE-DIMENSIONAL WOUND CORE TRANSFORMER VIBRATION AND NOISE MULTIPHYSICS FIELD ANALYSIS

A. Transformer vibration simulation and analysis

By adding fixed constraint conditions at the bottom of the transformer core, the displacement variation of the core’s vibration is observed under different numbers of short-circuited turns in the A-phase high-voltage winding.

As shown in Fig. 10, the deformation of the transformer core can be observed, where the fixed constraint applied at the bottom causes deformation primarily in the mid-upper region of the core. Further analysis reveals that during transformer operation, the deformation of the core is mainly induced by the force of the internal magnetic field. In normal operating conditions, the internal magnetic field is evenly distributed, resulting in uniform deformation of the core. However, when an inter-turn short circuit occurs, the resulting short-circuit current disrupts the magnetic field balance, causing uneven deformation. This imbalance leads to greater vibration and deformation on one end of the core than the other, creating an uneven deformation pattern that exacerbates core vibration.

FIG .10.

Core deformation profile under normal operation and different short circuit turns: (a) normal operation; (b) shorted turn (1 turn); (c) shorted turn (3 turns); (d) shorted turn (7 turns); (e) shorted turn (19 turns).

As shown in Fig. 11, as the number of short-circuited turns increases, the short-circuit current also rises, leading to a stronger electromagnetic force on the core. This increased force results in larger displacement amplitudes and higher vibration frequencies of the core, which intensifies the noise and vibration of the transformer, potentially causing equipment damage in severe cases.

FIG. 11.

Core displacement waveform under normal operation and different short circuit turns: (a) time-domain waveform; (b) frequency-domain waveform.

Then, the displacement variation of the winding vibration is observed under different numbers of short-circuited turns. Figure 12 shows the deformation of the short-circuited turns, where it can be seen that after an inter-turn short circuit occurs, the short-circuited turns experience an inward force and displacement.

FIG. 12.

Deformation trend of short circuit turns.

As shown in Fig. 13, the deformation of the winding under different numbers of short-circuited turns is analyzed. The significant short-circuit electromagnetic force generated during inter-turn short circuits causes large displacement deformations in the winding. It is observed that during normal operation of the transformer, the high-voltage winding experiences an outward force and displacement. In contrast, during an inter-turn short circuit, the short-circuited turns are subjected to an inward short-circuit force and displacement deformation. Under inter-turn short-circuit fault conditions, the greater the number of short-circuited turns, the larger the electromagnetic force generated, resulting in stronger displacement deformation in the winding and subsequently increasing the noise generated.

FIG .13.

Wing deformation profile under normal operation and different short circuit turns: (a) normal operation; (b) shorted turn (1 turn); (c) shorted turn (3 turns); (d) shorted turn (7 turns); (e) shorted turn (19 turns).

Figure 14 shows the time-domain and frequency-domain waveforms of the vibration displacement at a point on the short-circuited winding. Due to the alternating current flowing through the winding, the winding undergoes periodic vibrations at a frequency twice that of the current. In addition, the amplitude of the winding’s vibration after an inter-turn short circuit is several times that of normal operation. Under normal conditions, the vibration amplitude of the winding is only 0.3 μm, but when one turn is short-circuited, the vibration amplitude increases to 0.6 μm. As the number of short-circuited turns increases, the vibration amplitude of the winding further increases.

FIG. 14.

Winding displacement waveform under normal operation and different short circuit turns: (a) time-domain waveform; (b) frequency-domain waveform.

B. Transformer noise simulation and analysis

The vibration of transformers generates noise. Finite element analysis (FEA) is conducted on transformer noise under normal operating conditions and varying numbers of short-circuited turns.

Figure 15 shows the noise distribution when different numbers of inter-turn short circuits occur in the middle of the A-phase high-voltage winding. As the number of short-circuited turns increases, the vibration amplitude of the core and winding also increases, resulting in higher noise levels.

FIG .15.

Noise distribution under normal operation and different short circuit turns: (a) normal operation; (b) shorted turn 1 turn; (c) shorted turn 3 turns; (d) shorted turn 7 turns; (e) shorted turn 19 turns.

Figure 16 depicts the noise level measured at a point 1 m outside the transformer. It shows that in cases of low turn short circuits, such as 1 or 3 shorted turns, the impact on transformer noise is minimal, producing only slight noise without significantly affecting normal operation. However, with moderate turn short circuits, like 7 turns, the noise becomes more pronounced and may impact the surrounding environment. When high turn short circuits occur, the transformer generates very noticeable noise that can affect the surrounding environment and may pose health risks to nearby personnel. In addition, high turn short circuits may lead to transformer damage, disrupting its normal operation.

FIG. 16.

Noise at 1 m outside the transformer under short turns with different numbers of short circuits.

V. CONCLUSION

This study first measured the magnetic and magnetostrictive properties of the silicon steel sheets used in transformers. Then, based on a 50 kV A open-type stereo wound-core transformer, an equivalent circuit model and a finite element model were established. Research was conducted on voltage, current, electromagnetic force, leakage magnetic field, vibration, and noise under inter-turn fault conditions to analyze the effects of different short-circuit turn numbers on transformer performance. Finally, experimental verification of the analysis results was performed.

Using fundamental theories of finite element analysis and numerical principles of electromagnetic fields, the B–H curve and magnetostriction curve for silicon steel sheets were obtained. A steady-state model of the stereo wound-core transformer was developed, with simulation results compared to rated values to verify model accuracy. Equivalent circuit and finite element models of the transformer winding inter-turn short circuit were established, comparing the influence of different short-circuit turn numbers on the core’s magnetic flux density magnitude and distribution. It was found that at peak fault current, the core magnetic flux density at the fault position decreased, while the flux density at non-fault positions increased. Simulation analysis of core vibration, winding vibration, and noise changes under varying short-circuit turn conditions revealed that the transformer’s vibration and noise increased with the number of shorted turns.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Zhongxiang Li: Writing – original draft (equal). Wenxing Sun: Conceptualization (equal). Duanjiao Li: Data curation (equal). Dezhi Chen: Writing – review & editing (equal). Linglong Cai: Funding acquisition (equal). Xian Yang: Methodology (equal). Shuo Jiang: Project administration (equal). Ziyuan Xin: Validation (equal). Xianghui Chang: Software (equal). Sunjing Gong: Software (equal).

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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