Transmission of Vibrations from Windings to Tank in High Power Transformers

Abstract

This article presents a step-by-step methodology for calculating transformer tank vibrations caused by electromagnetic forces. This approach uses 3D finite element models for both magnetic and structural calculations. Particular attention was paid to the description of momentum transfer between structural and fluid areas of the transformer. The actual geometry of the coils in the phase windings was taken into account. The dominant role of the axial component of the Lorentz force is the main conclusion of the article. The results are given in the form of three-dimensional displacement fields of the transformer tank presented together with the acoustic pressure field in the oil. The theoretical analysis is verified by laser-scanned vibration patterns on the tank wall.

1. Introduction

The windings in power transformers have a complex and heterogeneous structure that is mechanically highly anisotropic and relatively weak, especially in high-voltage units where the amount of insulation materials is substantial. Two main points of interest for the structural analysis of the winding area are the winding’s resistance to buckling under an operational short circuit and its vibrational behavior under a rated load. On first sight, they look similar: in both cases, the displacement field in the coils is enforced by the electromagnetic Lorentz forces. However, because of the completely different values and time profiles of phase currents in these two issues, the methods of analysis and introduced simplifications must also differ. When the question of possible damage to the winding is discussed, the transient electromagnetic state in the first few milliseconds is solved together with coupled structural calculations. The case of possible buckling of inner winding, usually LV, under radial stress is considered in [1]. This paper includes the evaluation of the bending stiffness of a winding disk of multiple-turn laminated construction, the effect of supporting spacers on the deformation of a winding and the plastic deformation of a winding caused by circumferential stress. A similar problem is also solved in [2], where additional moments of force caused by the wiring process are also investigated. The axial stress of electromagnetic origin may also lead to buckling. The position of windings in the core window may be non-symmetric and which leads to additional force components acting on the winding [3]. The important problem here is the solution to stability questions. The key difficulty in this area is the homogenization process of the winding volume, which means the introduction of artificial material with equivalent properties to represent the real object. The nonlinear material constants, in both electromagnetic and mechanical analyses [4], are the main source of the considerable computational effort required for these calculations.

It is easier to analyze the winding vibrations: the electromagnetic and mechanical stress values are so small that the linearization of the material properties can be applied. However, on the other hand, the solid model of the winding is usually coupled with the fluid area because the transformers immersed in oil are the main object of interest. This requires a different mathematical treatment with special attention paid to the smoothness of the boundary between solid and fluid volumes. A regular and possibly simple shape of solid parts of the numerical model is extremely helpful in obtaining the convergence of the solution. Nevertheless, such a calculation is quite long and reported examples of its application in the transformer domain are a good illustration of the progress in numerical tools used in the analysis. The detailed model of the transformer, including the winding structure reduced to its orthotropic equivalent, was analyzed in [5] by means of the coupled magneto-mechanical approach. A similar treatment but extended to the real coil distribution in space may be found in [6]. In both of these works, a two-stage 2D-3D transformer model was used to take into account construction details. The coupling between magnetic, structural and acoustic phenomena inside the transformer tank computed by different software can also be obtained by the external supervising algorithm [7]. The influence of resonance effects of the transformer tank on additional devices attached to it is given in [8]. Nevertheless, the full simultaneous solution of coupled electromagnetic-structural-acoustic fields in a power transformer is not available yet due to geometric and material complexity. The use of several simplified models analyzed in series is necessary, but on the other hand, it helps to find which geometric or material parameters are most responsible for the resultant vibrations.

The significance of winding vibration in power transformers also results from the progress in the manufacture of core lamination [9]. Modern sheets have a low magnetostriction factor, under 1 µm/(mT²) [10], and the load-controlled noise dominates in units with power over 100 MVA [11]. Knowing the path of vibration transmission from the phase windings to the surface of the tank makes it possible to detect possible damage within the winding [12,13,14]. Regardless of the progress in materials technology, the clear tendency of the simultaneous growth of the rated power of transformers, accompanied by the pursuit of the reduction of their costs, means that the development of accurate theoretical models of winding behavior under possible load conditions is still an important issue.

The paper presents, in a detailed way, how the winding vibrations are transmitted via oil in the form of the acoustic pressure wave to the transformer tank.

2. Analysis of Magnetic Field and Forces in Phase Windings

Electromagnetic calculations were carried out for a 120 MVA transformer using the numerical model obtained with Simcenter Magnet 2022.1 finite elements (FE) software and presented in Figure 1. It contains half of the transformer structure accompanied by the appropriate symmetry boundary conditions. It means that the small asymmetry of the position of the active part inside the tank was neglected together with the asymmetry of the magnetic field distribution on the opposite walls of the tank. However, the magnetic flux inside the axial channels of transformer windings is not sensitive to that. The case of the short circuit test in rated conditions was analyzed. Such a test is obligatory during the acceptance tests of the transformer, usually, the acoustic emission measurements are also made during it to simulate these phenomena in the conditions of rated operation.

The solution was obtained in terms of time-harmonic 3D analysis using the T-Ω approach, where the nonlinear properties of ferromagnetic parts were included in an approximated way. The Lorentz volumetric forces acting on HV and LV windings were extracted from the solution field and displayed in Figure 1. Due to symmetry, only half of the windings may be considered. This figure shows the amplitudes only, however, the directional properties of these forces in whole windings were included in further analysis.

These forces possess the following properties:

–

The value of both force components does not change along the winding circumference, the small changes from this statement, occurring mostly for the side limbs, may be neglected,

–

The value of the radial component of the volumetric force varies linearly from zero to maximum along the radial direction,

–

The value of the axial component of the volumetric force remains almost constant along the radial direction.

These properties will significantly simplify further analysis. The time dependence of the Lorentz force contains two components—DC and AC, having the same amplitudes. From the point of view of transformer vibrations, the DC part has no meaning, and it may be omitted. Calculation of the volumetric force field f in the winding is performed simply by calculating the vector product of the constant in space current density J (small variations in this parameter close to the top and bottom of the winding were not accounted for here) by the flux density field B

$$f=J\times B$$

(1)

In order to allow the easy transfer of electromagnetic forces into mechanical considerations, some additional tasks are required. The static equivalence between the magnetic stress field σ_ij and force density f is given by

$$div {\sigma }_{ij}=-f$$

(2)

Expanding (2) in cylindrical coordinates and including the axisymmetric properties for the radial component, we have [15]

$$\frac{\partial {\sigma }_{rr}}{\partial r}+\frac{\partial {\sigma }_{rz}}{\partial z}+\frac{{\sigma }_{rr}-{\sigma }_{\theta \theta }}{r}=-f_r$$

(3)

and for the axial one

$$\frac{\partial {\sigma }_{zr}}{\partial r}+\frac{{\sigma }_{zr}}{r}+\frac{\partial {\sigma }_{zz}}{\partial z}=-f_z$$

(4)

The diagonal elements of the Maxwell stress, taking into account B_θ = 0, are given by

$$\left[{\sigma }_{rr},{\sigma }_{\theta \theta },{\sigma }_{zz}\right]=\frac{1}{2{\mu }_0}\left[B_r^2-B_z^2,-B_r^2-B_z^2,-B_r^2+B_z^2\right]$$

(5)

and the remaining off-diagonal element is equal to

$${\sigma }_{rz}=\frac{1}{{\mu }_0}{B}_r{B}_z$$

(6)

Further simplifications may be obtained for the middle zone of the winding where B_r = 0 and B_z = const. The equations of equivalence (3) and (4) are simply

$$\begin{array}{c}{f}_r=-\frac{\partial {\sigma }_{rr}}{\partial r}\\ f_z=0 \end{array}$$

(7)

The surface AC traction f_Sr acting in the radial direction on the middle section of the winding bounded by the outer radii r₁ and r₂ equals to

$$f_{Sr}=\underset{r_1}{\overset{r_2}{{\displaystyle \int }}}{f}_r\left(r,z\right) dr\cong 0.5J{B}_z\left(r_m,z\right)\left(r_2-r_1\right)$$

(8)

where r_m is the mean value of radii r₂ and r₁. The volumetric force at one end equal to zero. This traction is compensated by the hoop stress appearing in windings along their circumference—the outer winding, usually, HV is elongated, and the inner one is compressed. A different situation exists for the axial stress; in both windings, the electromagnetic axial traction is self-compensated, creating the compression of the winding structure. Assuming once again that means axial stress appears in points having the mean radius, we may calculate the axial traction applied to the winding part between coordinates (0,z) as

$$f_{Sz}=\underset{0}{\overset{z}{{\displaystyle \int }}}{f}_z\left(r_m,z\right) dz$$

(9)

The self-compensation of axial forces concerns the case with the symmetric placement of phase windings on the core. When there is a slight asymmetry in the position of the windings on the core limb, an additional force component appears, causing the windings to move in the form of a rigid body. That displacement is limited by the properties of the pressboard support, and this issue was not considered here. Integral (9) has no analytic expression, and it must be calculated numerically. The volumetric force fields shown in Figure 1 extracted and integrated along lines with the mean radii for both windings give the distributions of the mean traction in radial and axial directions along the winding’s height, displayed in Figure 2. Some perturbations visible in the picture of radial traction were caused by a much smaller number of finite elements in the radial direction.

3. Numerical Model of Radial Duct in Windings

The main windings in high-power transformers are usually wired with a Continuously Transposed Cable (CTC) and have a dozen cooling radial ducts. The outlook of the section of such a winding is presented in Figure 3.

The majority of radial ducts are understood as the volumes limited by symmetry planes (r,z) of subsequent spacers and symmetry planes (r,θ) of subsequent coils that have the same dimensions in the z-direction. Each of them has two additional planes of symmetry cutting out a quarter of the duct volume. Inside this sub-volume, we have different materials: copper wires (half of CTC), axial and radial spacers and transformer oil filling the empty space between them. This sub-volume was modeled by the ANSYS system as the coupled structural-fluid continuum and solved in terms of a linear harmonic problem.

The structural part was represented by the 3D solid elements having three unknown displacements {u} in each node of the mesh. The solved set of the equation has the general form

$$\left(-{\omega }^2\left[M\right]+j\omega \left[C\right]+\left[K\right]\right)\left\{u\right\}=\left\{F\right\}$$

(10)

where [M], [C] and [K] denote matrices of mass, damping and stiffness, {F} is the nodal force vector and ω means the angular frequency.

The fluid mesh was created with 3D acoustic elements solved in terms of scalar acoustic (velocity) potential Φ in element nodes. The acoustic state variables pressure p_a and particle velocity v_a are defined as

$$\begin{array}{c}{p}_a=\rho \frac{d\mathsf{\Phi }}{dt}\\ {\mathrm{v}}_a=\nabla \mathsf{\Phi }\end{array}$$

(11)

where ρ is the oil density. The state variables and potential Φ fulfill the wave equation

$${\nabla }^2{p}_a=\frac{1}{c^2} \frac{d^2{\mathrm{p}}_{\mathrm{a}}}{d{t}^2}$$

(12)

where c is the sound velocity. Introducing the FE approximation, it is possible [16] to convert (12) into the form in (10). The ANSYS 2022 software recognizes the fluid-structure boundary where the fluid has additional unknowns describing the fluid displacements. Therefore, definition (11) allows the preservation of continuity of normal stress and normal velocity in both continua. The geometry of an analyzed model of the radial duct is displayed in Figure 4. The model starts from the outer surface of the tertiary winding (r = 0.41 m) and ends on the outer surface of the Tap winding (r = 0.76 m). These windings are shell-type without radial ducts. Tertiary winding is placed almost directly on the core, so it may be treated by the fluid as the stiff boundary. The Tap winding is fixed by the appropriate symmetry boundary conditions on its (r,z) ending planes.

The vibration analysis was performed separately for radial and axial excitations having a frequency of 120 Hz. The pressure representing the radial forces was applied to surfaces between oil and windings in the main insulation axial channel. The axial pressure was forced on the surfaces of both windings belonging to the boundary of the whole model. The pressure values were taken from the traction distributions shown in Figure 2 at half of the winding height. The oil pressure on the inner surface of the Tap winding is the main parameter determining the transmission of vibrations to the walls of the tank, and its space profile is shown in Figure 5. The oil pressure in the axial channel between the HV and Tap windings created by the radial excitation is about two thousand times smaller than at the axial case. This effect is due to the significantly different stiffness of the materials limiting the deformation in both cases. When we have the radial excitations, they are practically fully balanced by the hoop stress in the windings and the oil area is almost not deformed. In the case of axial forces, the displacement values follow the properties of radial spacers and oil, for which the elastic moduli are many times smaller than for copper.

Additional effects appear when the frequency of traction varies. Having the same amplitude as before, the axial case was solved for three frequency values. The results are given in Figure 6.

When increasing the frequency of excitation, a new spatial component appears, which has the form of a standing wave across the radial length of the channel and the magnitude grows with the frequency. For sufficiently high frequency, here 900 Hz, its magnitude rapidly falls down and simultaneously, the phase is reversed into an opposite value. This phenomenon was caused by a resonance form occurring inside the oil channel where the elasticities and masses of the low voltage winding (axial vibrations) and Tap winding (radial vibrations) were coupled through the oil. The maximum vibrations occur at 365 Hz. The shape of this eigenmode is presented in Figure 7, and it is in good agreement with the forced mode shown in Figure 5. This means that the contribution of the higher frequencies that are always present in the force spectrum will be enhanced.

4. Numerical Model of Transformer Tank

The FE model of the transformer tank consists of a rectangular steel tank having walls 10 mm thick equipped with vertical stiffeners of 150 × 10 mm and nine magnetic shields of 1650 × 450 × 30 mm connected to the tank in three points along each vertical edge. The shields are “transparent” for the oil interaction, increasing the stiffness and mass of the tank wall only. This simplification was dictated by the need to maintain the regularity of the FE mesh in the oil area, which is of decisive importance from the point of view of the stability of the solution to the entire task. Inside the tank exists the set of Tap windings having the null displacement conditions on surfaces inside the core windows. The outlook of the structural part of the tank is shown in Figure 8. The interior of the tank shown in Figure 9 is filled with acoustic elements modeling the oil. Its volume is limited to the area outside the HV windings and the core yokes, which means that the outer surfaces of these parts represent infinitely stiff media. The model is excited on the outer surfaces of the HV windings, where the known values of the normal oil particle velocity v_ak were introduced in form

$${\mathrm{v}}_{ak}\left(t\right)=V_m{e}^{j\left(\omega t-{\phi }_k\right)}$$

(13)

where ω is angular frequency and ϕ_k = 0, ±2π/3, respectively. This mode of the exciting results from the movements of the oil particles inside the radial channels, as was calculated above.

The magnitude of the velocity V_m has been chosen in such a way that the mean value of the resultant pressure extracted along the winding circumference in the duct between HV and Tap is equal to the value of 6100 Pa calculated previously in the model of the radial duct. The velocity distribution is applied to oil along the cylindrical surface a bit shorter than the winding height, which gives the same value of excitation force. The comparison of these two approaches is shown in Figure 10.

The explanation of this procedure is as follows. First, by inserting any value of the vibration velocity on the outer surface of the HV winding, say V_m0 = 1, and solving the model, we have a certain sound pressure distribution. Taking the spatially averaged pressure value P_a0 along the line shown in Figure 10b, we obtained a specific acoustic impedance Z_a0 for the requested system of excitation

$$Z_{a0}=\frac{P_{a0}}{V_{m0}}$$

(14)

The desired velocity amplitude V_m giving approximately the pressure P_am calculated with a single radial channel is then simply

$$V_m=\frac{P_{am}}{Z_{a0}}$$

(15)

The next two figures show the instantaneous fields of the tank wall velocity and the sound pressure on the tank wall.

Here, we see similar shapes for the real pressure and imaginary velocity components, as well as an inverted pair. From a fluid point of view, this means that the acoustic field inside the tank does not create any propagating wave, and only near-field disturbances are observed. The critical frequency f_cr below which the acoustic wave cannot propagate may be estimated for the idealized case of a cylindrical source of height L vibrating in a stiff baffle toward the infinite fluid [15]

$$f_{cr}=\frac{c}{2L}$$

(16)

where the sound velocity in oil is c = 1480 m/s, and the winding height is L = 1.64 m. It gives f_cr = 451 Hz, which is well above the excitation frequency of 120 Hz. The presence of obstacles, such as the tank wall or adjacent windings in the real case, will increase the f_cr value.

Velocity patterns move horizontally due to the phase shift between the axial forces acting in the phase windings on adjacent core columns. They contain both traveling and standing wave elements. Having in mind the harmonic relation between the displacement u and velocity v

$$v=j\omega u$$

(17)

Field Im(v_n), shown in Figure 11b, also represents the field −Re(u_n), which, in turn, is almost identical after the sign change to the pressure field, Re(p_a), displayed in Figure 12a. The lack of a significant phase shift between pressure and displacement means that resonance effects observed at the tank wall at a given frequency of excitation are of second-order in the analyzed case.

Some comments should be added to the value of pressure magnitude obtained in the calculations. The tank wall is also excited by the leakage field, which creates normal stress directly on the magnetic shields’ surfaces. This stress, not accounted for in the presented analysis, is almost in-phase with the pressure transmitted via oil because the phase currents are their common origin. The places on the tank wall where magnetic stress occurs are sketched in Figure 13. The slight phase shift occurring between magnetic and acoustic components results from wave phenomena present in the oil because the pressure at a given point on the tank wall depends not only on the component produced by the nearest pair of phase windings but also on the other windings carrying currents displaced in phase and located at a different distance from this point.

The amplitude (AC component) of magnetic stress σ_m is about 150 Pa, which results [17] from

$${\sigma }_m=\frac{B_m^2}{4{\mu }_0}$$

(18)

where B_m is the amplitude of the magnetic flux density close to the shield surface. Magnetic stresses occur slightly further from the top and bottom of the wall than acoustic stresses, thus creating larger moments of force bending the tank wall. Comparing it with amplitudes equal to 485 Pa presented in Figure 12 and Figure 13 means that both effects are of similar significance.

5. Experimental Validation

The vibration velocity was measured with the PSV400 laser scanning vibrometer during the short-circuit test of analyzed transformer. The results are shown in Figure 14. It should be mentioned that the calculated and measured Realis time points are not exactly the same, as no special triggering was used during the experiments.

The measured patterns of vibrations are not as clear as the calculated ones. It is probably caused by the connection between the shielding system and the tank wall, which, in reality, is not as precise as in the theoretical model. The welding technology applied in the real transformer for the assembly of shields may create additional local prestress, and also, the resultant number of connection points between the shield box and the wall remains unknown. The lack of calculations of the magnetic forces applied directly to the tank, which is shifted in the phase in relation to the acoustic excitation, also affects the shape of the vibration isolines, which was discussed above. The values and shape of measured velocities of the tank surface are in acceptable agreement with calculations because the dominated vibrations on the tank wall appear close to the bottom and top of the wall, as is also indicated by the calculations. Local differences in the magnitude of the measured and calculated velocity fields are about 30%, which is almost nothing on the dB scale commonly used in vibration analysis. However, the high value of the local maximum shown in Figure 14b, which is almost twice as large as the calculated amplitudes, has not been confirmed by theoretical considerations.

6. Conclusions

This article describes, in detail, the method of transferring winding vibrations caused by electromagnetic forces to the transformer tank. It was found that axial forces are responsible for most of the vibration energy appearing on the tank surface. The pressure of acoustic origin on the tank wall results mainly from the compression of the oil inside the radial channels in the phase windings. Therefore, the forced vibration analysis requires the use of 3D numerical models of the transformer in both the magnetic, structural and acoustic domains. The basic problem is the matching of finite element grids modeling acoustic phenomena in the oil area close to the structural elements of the transformer. Due to the numerical complexity of the analyzed problem, a direct approach is still not possible, and it is necessary to take several intermediate steps. This approach, however, allows for a deeper insight into the physics of the considered phenomena, which, in turn, can explain the origin of excessive vibrations and emitted acoustic noise in real transformers.