Research on Influence of Damping on the Vibration Noise of Transformer

To improve the transformer vibration analysis accuracy, a numerical vibration model of transformer, including the damping effect, is proposed in the paper. According to power transformer structure, the Rayleigh damping model is used to indicate the transformer damping effect. The damping coefficients can be got by analyzing and testing the transformer structure. The modal measurement system of the prototype is constructed and tested to improve and verify the modal analysis method, which can be used to analyze the engineering power transformer that the modal measurement should not be achieved. According to the Rayleigh damping parameters, the vibration and noise of damped and undamped transformers are calculated respectively. Then the effect of the damping on the vibration and noise can be obtained. At last, the vibration and noise of the transformer were tested and compared with the analytic results. Comprehensive analysis shows that the analysis results considering damping can be improved, and are closer to the measured results.

it is of great significance to study the vibration and noise of 23 transformer [1], [2], [3]. 24 The main source of transformer noise is its body vibra-25 tion and cooler. The vibration and noise of the transformer 26 is related to the transformer load, silicon steel sheet mate-27 rial, core structure, magnetic flux density and other factors 28 [4]. In recent years, in order to better design low-noise 29 The associate editor coordinating the review of this manuscript and approving it for publication was Yingxiang Liu . transformers, the academic community has paid more and 30 more attention to improving the accuracy of transformer 31 vibration and noise calculation [5], [6]. By combining the 32 electromagnetic field theory with the elastic theory, a math-33 ematical model of transformer electromagnetic vibration is 34 established [7], [8]. Reference [9] studied the magneto-35 mechanical effects of core transformers with different struc-36 tures, and experiments verified that transformers have high 37 vibration strength. [10], [11] established the electromagnetic 38 mechanical vibration coupling mathematical model consid-39 ering the magnetostrictive characteristics of the transformer, 40 and simulates the vibration and noise of the transformer. 41 Reference [12] proposed to obtain the quantitative relation-42 ship between the magnetostriction characteristics of the core 43 reactor and the noise by systematically evaluating the noise 44 and vibration shape of the simple small transformer core. 45 [13] obtained the relationship between magnetostriction of 46 grain oriented electrical steel (GOES) coils and no-load noise The expressions of transformer mass matrix and stiffness 98 matrix coefficients are obtained by numerical calculation:

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Transformer vibration noise mainly comes from noise caused 104 by core vibration, that is, magnetostrictive effect of core 105 silicon steel sheet and electromagnetic attraction caused 106 by magnetic leakage between the joint of silicon steel 107 sheet and disk. In addition, when transformer is run-108 ning, winding current will generate magnetic leakage in 109 space. Winding under alternating magnetic field will be 110 affected by Lorentz force and cause winding vibration and 111 noise [14].

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Damping, as one of the characteristics of energy dissipa-113 tion in vibration process, is also an important factor affecting 114 the vibration response of transformer.

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Proceed from vibration analysis, this paper comprehen-116 sively interprets the vibration and noise of the transformer 117 through modal analysis, magnetic-mechanical coupling 118 analysis and acoustic field analysis of the prototype 119 transformer.

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Modal analysis is the process of replacing the original finite 122 element node coordinates with vibration coordinates. The fre-123 quency response function of a given input and output position 124 is expressed by modal parameters: In this paper, the motion equation of the prototype trans-127 former's N-degree-of-freedom system is simplified into a 128 finite element elastic system with mass, elasticity and damp-129 ing in the vibration coordinate system, and its motion equa-130 tion is shown in (6): Take the special solution:

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In the magnetic field, the governing equation for the mag-180 netic vector is where A is the magnetic vector potential, J is the cur- In the structural force field, the relationship between stress 191 and strain is During the operation of a power transformer, the trans-194 former produces electromagnetic vibration due to the inter-195 action of electromagnetic fields, among which the main force 196 is the Maxwell force F vmax of the core and the magnetostric-197 tive force F vms , and the Lorentz force F l generated by the 198 windings. The electromagnetic force is calculated according 199 to (15), and the obtained results are analyzed in combina-200 tion with solid mechanics, so as to realize the magnetic-201 mechanical coupling: Among them: where T represents Maxwell stress tensor; σ vms magnetostric-207 tive stress; D represents elastic tensor, which can be obtained 208 from Young's modulus and Poisson's ratio of silicon steel; 209 ε vms is the magnetostrictive strain tensor, which is obtained 210 by interpolating the measure B − λ pp curve; F is the volume 211 force in the structural field.

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The node vibration equation is: where M is the mass matrix, C is the damping matrix, K is the 215 stiffness matrix.

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Combined with the virtual work displacement method, 217 the finite element method is used to discretize the solu-218 tion element, and all the subdivision elements are collected. 219 The magnetic mechanical coupling model can be written as 220 follows: where S is the electromagnetic matrix; u is the displacement 224 to be determined; A is the magnetic vector potential to be 225 found. When the damping effect is not taken into account, 226 the damping term in the vibration equation of the structure 227 is 0.  (19): where ρ is the density of the medium, η is shear modulus,   Table 1:

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Due to the symmetry of the structure of the prototype 288 transformer, the whole transformer model is simplified to a 289 1/2 model, which can simplify the model and improve the cal-290 culation efficiency. Then mesh them, including 7454 domain 291 elements, 3839 boundary elements and 944 edge elements. 292 The results are shown in Figure 1.

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Since the magnetic properties and magnetostrictive prop-294 erties are inherent properties of the material, the data of 295 the magnetic properties and magnetostrictive properties of 296 the silicon steel sheet when the transformer is running are 297 brought into the model for calculation. First, the input exci-298 tation of the transformer is set by the current calculated 299 from the no-load rated voltage provided by the prototype 300 transformer manufacturer. In this paper, the 26A AC power 301 supply is used as the input excitation of the transformer, and 302 the windings are set to be uniform and multi-turn. Because the 303 prototype works at power frequency, the frequency domain 304 is set to 50Hz.After the designed circuit meets the operating 305 conditions of the transformer, carry out numerical simulation 306 of electromagnetic vibration and noise.

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Through the modal experiment platform and electromag-308 netic vibration measurement platform, the modal parame-309 ters, electromagnetic vibration and noise of the three-phase 310 VOLUME 10, 2022     Table 2 and the modal test system for 320 transformer specimens is shown in Figure 2.   few orders of modal state have a more significant effect on 333 the system vibration, the first six orders of modal frequency 334 were taken as the main object of study in this paper. The first 335 six orders of the transformer are shown in Figure 4.

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Combined with the simulated experimental calculation 337 data, after the modal test completed the data acquisition at 338 each measurement point, the collected excitation force signal 339 and the response signal collected by the three-way acceler-340 ation s-ensor were imported into the X-MODAL/DSP signal 341 proces-sing system respectively, and the first six orders of the 342 transf-ormer calculated and measured eigenfrequencies were 343 collat-ed asshown in Table 3.  (3) and (4)    with the change of excitation frequency. By analyzing the 386 electromagnetic vibration of a 10kVA/380V three-phase 387 transformer, the electromagnetic vibration of transformer 388 under different damping conditions is studied in this paper. 389 To reflect the influence of modal damping on vibration it 390 is first necessary to carry out a vibration analysis of the trans-391 former. On the basis of a correct calculation of the magnetic 392 field, the analysis of electromagnetic vibration is started.

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The specific parameters of the experimental prototype first 394 need to be entered into the simulation software to calcu-395 late the core flux density and coil current density in the 396 magnetic field in the transformer prototype as shown in 397 Figure 7.

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Based on the calculations obtained in the magnetic field, 399 the magnetostrictive strain is converted into magnetostrictive 400 stress using the stressstrain relationship of elastodynamics, 401 which is brought into the vibration calculations as a load, 402 resulting in the vibration of the transformer as shown in 403 Figure 8.

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This paper is based on the magneto-mechanical cou-405 pling model, which is further extended to establish an analyt-406 ical model of the sound field of the prototype. The purpose 407 of the model is to accurately calculate the magnitude of 408 the vibration noise generated by the prototype transformer. 409 VOLUME 10, 2022      line is closer to the experimental values.Therefore, it is easy to 428 draw a conclusion that the electromagnetic vibration analysis 429 of transformers is affected by damping effect to some extent, 430 and considering damping effect can make the calculation of 431 electromagnetic vibration more accurate.   The comparison data are shown in Table 4.   2) The accuracy of calculation will be improved with the 477 addition of damping effect.

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In this paper, the distribution of electromagnetic vibra-479 tion and noise of transformer considering damping effects 480 is studied. These opinions play an important role in improv-481 ing the calculation accuracy of transformer electromagnetic 482 vibration and noise, accurately predicting the noise level of 483 transformer products, and researching more effective method 484 of vibration reduction and noise reduction. He received the bachelor's degree in electri-573 cal engineering and automation from the Hebei 574 Normal University of Science and Technology, 575 in 2019. He is currently pursuing the master's 576 degree in electrical engineering with Tiangong 577 University.

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His research interests include numerical anal-579 ysis of engineering electromagnetic fields, vibra-580 tion reduction, and reduction of electromagnetic 581 energy equipment noise technology. During his master's degree, he won a 582 Freshman Scholarship and a Third-Class Scholarship, in 2020. In April 2021, 583 he has published an invention patent ''Method for Active Noise Reduction 584 of Electrical Equipment,'' which has been disclosed. His research interests 585 include numerical analysis of engineering electromagnetic fields and multi-586 physics coupling.

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LAN LU was born in November 1995. She received 588 the bachelor's degree in electrical engineering and 589 automation from the School of Science and Tech-590 nology, North China Electric Power University, 591 in 2018. She is currently pursuing the master's 592 degree in electrical engineering with Tiangong 593 University. Her research interests include numer-594 ical analysis of engineering electromagnetic field 595 and vibration and noise reduction technology of 596 electromagnetic energy equipment. During her 597 master's degree, she won a Freshman Scholarship and a Third-Class Scholar-598 ship. In September 2021, she published a utility model patent ''A Phononic 599 Crystal Sound Isolator for Noise Reduction of Electrical Equipment.'' At 600 present, the electromagnetic vibration and noise of engineering electrical 601 equipment are mainly studied and studied.