# Grid-like Vibration Measurements on Power Transformer Tank during Open-Circuit and Short-Circuit Tests

^{1}

^{2}

^{*}

## Abstract

**:**

^{2}, 0.31 mm/s, and 0.42 µm, respectively. The maximum values in short-circuit test are 0.74 m/s

^{2}, 1.14 mm/s, and 1.8 µm, respectively. In the short-circuit test, the frequency component of 100 Hz is dominant. In the open-circuit test, the first few 100 Hz harmonics are significant (100 Hz, 200 Hz, and 300 Hz). In addition to the visualization of RMS values during the open-circuit and short-circuit tests, animations of the vibrations are created. Fourier analysis and phase comparison between frequency components are also used to show vibration animations at dominant frequencies in the spectrum (100 Hz harmonics). The visualization of the vibrations at the tank wall surfaces is transferred into 3D space in such a way that all 15 surfaces are mapped to the spatial coordinates of the surfaces so that a 3D model of the acceleration, vibration velocity, and displacement of the transformer tank is shown.

## 1. Introduction

## 2. Theoretical Considerations

#### 2.1. Propagation of Sound Waves in Solid Plates

_{c}is the frequency where the bending wave speed is equal to the acoustic wave speed in the air (c

_{B}= c = 344 m/s). The same speed and wavelength cause good coupling between these two wave types, which means that the plate efficiently radiates sound at and above this frequency. From Equation (2), it follows that:

_{c}is the material constant. Increasing the frequency of the wave propagation, there is a gradual transition from bending to shear wave. The crossover frequency f

_{s}is defined when bending wave speed c

_{B}becomes equal to the shear wave speed c

_{S}. By equalization of the Equations (2) and (3), the crossover frequency is equal to:

#### 2.2. Normal Modes and Natural Frequencies of a Plate

_{x}x) in the x-direction and sin(k

_{y}y) in the y-direction are used [22]. If there are simple supports at x = 0, x = l

_{x}, y = 0, and y = l

_{y}, the wavenumber k must be equal to:

_{x}is the length of the plate in the x-direction, and l

_{y}is the length of the plate in the y-direction.

_{B}and the mode numbers m and n and replacing bending wavenumber with the expression with the bending wavelength λ

_{B,}the following expression is obtained:

#### 2.3. Analytical Determination of Sound Pressure in the Point Away from the Plane with Measured Vibration Velocity Distribution Considering Incoherent Plane Radiation

^{2}R

_{R}is the real part of the radiation impedance. Estimation of sound power that machines are radiating in operation when their surface is vibrating can be determined by the mean square vibration velocity averaged over the surface [24]. Replacing U

^{2}/2 from Equation (12) with the surface mean square velocity < v

^{2}>

_{S,t}, replacing the piston surface area πa

^{2}with the plate surface S, and R

_{R}with surface radiation efficiency σ, Equation (12) becomes:

_{ref}= 10

^{−12}W, taking logarithms to the base 10, multiplying by 10 of both sides of the equation, and including values for air density ρ and speed of sound in air c into the equation, gives the following expression:

## 3. Measurement Setup

## 4. Results

#### 4.1. Comparison of the Vibration Measurements Using Two Different Voltage Sources

_{ACRMS}; THD: <3% over the whole range) that had significant voltage harmonics at 3 kHz, 6 kHz, and 9 kHz, as can be seen in Figure 5a. These were more prominent for higher voltages when the transformer core was close to saturation because of the nonlinearity of the B-H curve, which caused higher current harmonics. In addition, due to relatively low output power of the source, voltage distortion occurred, and the voltage harmonics appeared depending on the source characteristics (converters inside the source). Due to higher voltage harmonics, the transformer is tested at only 90% of its nominal voltage (the nominal voltage is 10 kV at the HV side and 0.4 kV at the LV side). Consequently, higher excitation voltage harmonics caused high vibration components of 3 kHz and 6 kHz, as shown in Figure 6a.

#### 4.2. Visualization of the Measurement Results Using Matlab Interpolation Functions

#### 4.3. Visualization of the Tank Vibrations in 3D Space

^{2}, 0.31 mm/s, and 0.42 µm, respectively. Maximum values in short-circuit test are 0.74 m/s

^{2}, 1.14 mm/s, and 1.8 µm.

#### 4.4. Analytical Calculation of Critical, Crossover, and Resonant Frequencies of the Plate

^{2}, steel density ρ = 7.8 × 10

^{3}kg/m

^{3}, Young modulus of steel E = 210 N/m

^{2}, Poisson ratio µ = 0.33, and plate thickness h = 8 × 10

^{−3}m.

_{c}= 1475 Hz, and the crossover frequency is f

_{s}= 126.9 kHz. Resonant frequencies are calculated for every plate and put in Table 2. This calculation takes a few simplifications presenting the transformer plates as perfectly flat and simply supported. Tank plates consist of access-openings, connections for coolers, variations in stiffeners shape, etc. The conditions of a simply supported rectangular plate, which say that the displacement and the moment reaction at the boundaries are zero, are not satisfied for all the plates. Boundaries of plates 1, 2, 3, 14, and 15 are not that simple because there are no stiffeners, and they have angled positions with the neighboring plates.

#### 4.5. Vibrations of Tank Stiffeners and Cover

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Electrical and vibration measurement setup diagram in open-circuit and short-circuit tests.

**Figure 2.**(

**a**) Vibration measurement on the 5 MVA transformer experimental model with measurement points marked with red crosses (on a couple of surfaces crosses highlighted with photo editor) and three accelerometers in the short-circuit test, (

**b**) Top view of the CAD model with numbers presenting transformer surfaces on which measurements are made.

**Figure 3.**Single-sided amplitude spectrum of Signal to Noise Ratio for point 75 at the surface 5 of the transformer.

**Figure 4.**Voltage and acceleration signals of the reference point and some different points on the tank: (

**a**,

**c**) before alignment and (

**b**,

**d**) after alignment.

**Figure 5.**Single-sided amplitude spectrum of voltage and voltage signal in the time domain in the open-circuit test using (

**a**) power converter and (

**b**) regulating transformer.

**Figure 6.**Single-sided amplitude spectrum of vibration acceleration, velocity, and displacement in the open-circuit test using (

**a**) power converter and (

**b**) regulating transformer.

**Figure 7.**(

**a**) CAD model of the surface 5 of the transformer experimental model, and visualization of RMS values of the vibration acceleration, velocity, and displacement on the surface 5 using: (

**b**,

**d**,

**f**) power converter as the source and (

**c**,

**e**,

**g**) regulating transformer as a source.

**Figure 8.**Visualization of RMS values of the vibration velocity on the surface 5 of the transformer using: (

**a**–

**c**) power converter and (

**d**–

**f**) regulating transformer as a source on different frequency components.

**Figure 9.**Visualization of RMS values of the vibration acceleration, velocity, and displacement in 3D space in: (

**a**–

**c**) open-circuit test, and (

**d**–

**f**) short-circuit test.

**Figure 10.**Visualization of RMS values of the vibration velocity in 3D space on the frequency components of 100 Hz, 200 Hz, 300 Hz, and 400 Hz in (

**a**–

**c**,

**g**) open-circuit and (

**d**–

**f**,

**h**) short-circuit test.

**Figure 11.**Visualization of the instantaneous amplitudes of the vibration velocity in 3D space in the time moments: (

**a**) 0 ms, (

**b**) 1.25 ms, (

**c**) 2.5 ms, (

**d**) 3.75 ms, (

**e**) 5 ms, (

**f**) 6.25 ms, (

**g**) 7.5 ms, (

**h**) 8.75 ms, and (

**i**) 10 ms in the short-circuit test.

**Figure 12.**Space and time-averaged mean square vibration velocity of the plate multiplied with surface area for 15 measured transformer surfaces marked in Figure 2b.

**Figure 13.**Stiffener 2 of transformer experimental model on which vibrations are measured with a view to (

**a**) the conservator side (

**b**) the front side and the cooler side.

**Figure 14.**Visualization of the RMS values of the vibration velocity on the stiffener 2 from (

**a**) the conservator side in the open-circuit test (

**b**) the cooler side in the open-circuit test, and (

**c**) the cooler side in the short-circuit test.

**Figure 15.**CAD model of the transformer experimental model in the top view with marked points of vibration measurements (red crosses) on the tank cover.

Sensitivity (±15%) | 10.2 mV/(m/s^{2}) |

Measurement Range | ±490 m/s^{2} |

Frequency Range (±3 dB) | 0.5–10,000 Hz |

Resonant Frequency | 22 kHz |

Broadband Resolution | 3434 µm/s^{2} |

Temperature Range | −54 °C to +121 °C |

Enclosure Rating | IP68 |

Spectral Noise (100 Hz) | 49.1 (µm/s^{2})/√Hz |

**Table 2.**Resonant frequencies and modes of vibrations of 15 transformer plates calculated analytically.

Plate | Resonant Frequency [Hz] (Mode of Vibration) | |||||||
---|---|---|---|---|---|---|---|---|

1, 2, 3, 14, 15 | 193 (1, 1) | 208 (1, 2) | 233 (1, 3) | 269 (1, 4) | 315 (1, 5) | 371 (1, 6) | 437 (1, 7) | 513 (1, 8) |

4, 13 | 312 (1, 1) | 327 (1, 2) | 352 (1, 3) | 388 (1, 4) | 434 (1, 5) | 490 (1, 6) | 556 (1, 7) | 632 (1, 8) |

5, 6, 11, 12 | 63 (1, 1) | 79 (1, 2) | 104 (1, 3) | 140 (1, 4) | 185 (1, 5) | 238 (2, 1) | 241 (1, 6) | 253 (2, 2) |

279 (2, 3) | 307 (1, 7) | 314 (2, 4) | 360 (2, 5) | 384 (1, 8) | 416 (2, 6) | 470 (1, 9) | 482 (2, 7) | |

7, 10 | 118 (1, 1) | 133 (1, 2) | 158 (1, 3) | 194 (1, 4) | 240 (1, 5) | 296 (1, 1) | 362 (1, 7) | 438 (1, 8) |

8, 9 | 62 (1, 1) | 77 (1, 2) | 102 (1, 3) | 138 (1, 4) | 184 (1, 5) | 232 (2, 1) | 240 (1, 6) | 247 (2, 2) |

273 (2, 3) | 306 (1, 7) | 308 (2, 4) | 354 (2, 5) | 382 (1, 8) | 410 (2, 6) | 469 (1, 9) | 476 (2, 7) |

**Table 3.**RMS values of the vibration velocity along the height on the stiffener 2 from the front side in the open-circuit (OC) and short-circuit (SC) test.

y [cm] | RMS Velocity (OC) [mm/s] | RMS Velocity (SC) [mm/s] |
---|---|---|

0 | 0.02 | 0.06 |

−27 | 0.03 | 0.16 |

−54 | 0.04 | 0.21 |

−81 | 0.09 | 0.24 |

−108 | 0.09 | 0.23 |

−135 | 0.10 | 0.20 |

−162 | 0.04 | 0.11 |

−189 | 0.03 | 0.05 |

**Table 4.**RMS values of the vibration velocity on the transformer tank cover in the open-circuit (OC) and short-circuit (SC) tests.

RMS Velocity [mm/s] | c1 | c2 | c3 | c4 | c5 | c6 |
---|---|---|---|---|---|---|

OC | ||||||

r1 | 0.01 | 0.02 | 0.01 | 0.02 | 0.03 | 0.02 |

r2 | 0.06 | 0.06 | 0.04 | 0.03 | 0.09 | 0.02 |

r3 | 0.06 | 0.08 | 0.09 | 0.10 | 0.06 | 0.02 |

r4 | 0.01 | 0.02 | 0.02 | 0.01 | 0.01 | 0.02 |

SC | ||||||

r1 | 0.09 | 0.09 | 0.04 | 0.03 | 0.02 | 0.01 |

r2 | 0.08 | 0.13 | 0.13 | 0.22 | 0.20 | 0.03 |

r3 | 0.05 | 0.39 | 0.26 | 0.29 | 0.08 | 0.03 |

r4 | 0.06 | 0.02 | 0.05 | 0.04 | 0.02 | 0.02 |

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**MDPI and ACS Style**

Petrović, K.; Petošić, A.; Župan, T. Grid-like Vibration Measurements on Power Transformer Tank during Open-Circuit and Short-Circuit Tests. *Energies* **2022**, *15*, 492.
https://doi.org/10.3390/en15020492

**AMA Style**

Petrović K, Petošić A, Župan T. Grid-like Vibration Measurements on Power Transformer Tank during Open-Circuit and Short-Circuit Tests. *Energies*. 2022; 15(2):492.
https://doi.org/10.3390/en15020492

**Chicago/Turabian Style**

Petrović, Karlo, Antonio Petošić, and Tomislav Župan. 2022. "Grid-like Vibration Measurements on Power Transformer Tank during Open-Circuit and Short-Circuit Tests" *Energies* 15, no. 2: 492.
https://doi.org/10.3390/en15020492