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Prediction of Transformer Load Noise

M. Kavasoglu*,1, R. Haettel1 and C. Ploetner2 1ABB Corporate Research Sweden, 2ABB Transformers Canada*Corresponding author: 721 78 Västerås Sweden, [email protected]

Abstract: Transformers, as any other industrialproducts, have to fulfill various requirements onnoise levels. Those requirements might besometimes rather stringent and some other timessignificantly easier to fulfill. Therefore, it isessential to use appropriate prediction tools inorder to select a suitable strategy for noisecontrol.The paper will focus on the so called load noisewhich is a typical multiphysics mechanisminvolving electromagnetism, mechanics andacoustics. The finite element model coupling the

different physical fields has been developed byusing COMSOL.Some simplifications have been introduced toreduce problem size and computing time. Theprediction model provides a relatively fast andrather accurate tool to perform parametricstudies. In the paper, the modeling procedure isdescribed and results from parametric studies arepresented.

Keywords: Transformer, windings, acoustics

1. Introduction

Low noise levels are nowadays requested forpower transformers in order to comply withcustomer specifications and environmentallegislations [1,2]. Consequently, manufacturershave to improve the acoustic performances of their products, while, at the same time, limitingthe costs. It is thus of great importance to predictsound levels with a sufficient accuracy at anearly stage of the product design.There are three main sources of noise intransformers: core noise generated bymagnetostriction in the core steel laminations[3], load noise produced by electromagneticforces in the windings [4] and auxiliaryequipment noise due to fans and pumps of thecooling system.The paper describes a method to predict loadnoise which constitutes a typical multiphysicsphenomenon involving electromagnetism,mechanics and acoustics. The interactionbetween the alternating currents in thetransformer windings and the associated stray

field results in varying Lorentz forces generatingwinding vibrations. The mechanical energy isthen transmitted to the tank via the windingclamping system and the insulating oil. The tank acts as a membrane radiating this energy asnoise.

Due to the relative complexity of thestructure and the strong coupling with oil, ananalytical model cannot be used to predict thesound radiation. In general, empirical methodsbased on statistics and power rating are used bymost transformer manufacturers [2]. This

approach presents limitations when applied tonew designs and does not enable accurateparametric studies. Therefore, prediction modelsbased on finite element formulations shall beutilized to describe accurately the complexinteractions of the various design parameters andthe coupling of the physical fields.

In this paper, the governing equations for theelectromagnetic, mechanical and acoustic fieldsare first described in sections 2, 3 and 4. Then,an overview of the modeling procedure is givenin section 5. Some results provided by theprediction model are reported in section 6.

2. Electromagnetic Model

At load condition, the transformer windingscarry their nominal current. During testing, thisis usually achived by short circuiting onewinding and feeding the other winding with a 50Hz or 60 Hz current. The resulting magnetic fieldB induced by the total currents (Amper turns) of both windings is described by using Maxwell'sequations. The magnetic field is considered aspurely sinusoidal and can be written as

B = ∇ × A (1)

where A is the magnetic vector potential.By applying the vector potential formulation, thegoverning equation for the magnetic field isgiven by the relationship

∇ ×ଵ

ఓ∇ × A = J −

డA

డ௧+ ሺv × ∇ × Aሻ (2)

Excerpt from the Proceedings of the COMSOL Conference 2010 Paris

http://www.comsol.com/conf_cd_2011_eu

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where µ is the magnetic permeability, A themagnetic vector potential, J the current density, γ the electrical conductivity and v the conductorvelocity, which is considered here as negligible.Due to the relatively high magnetic permeability

of the transformer core, the boundary conditionsat the core surface are given by

n ൈH ൌ 0 (3)

where n is the vector normal to the core surfaceand H the magnetic field strength.The magnetic flux density can be calculated byusing a predefined current as the excitationsource in the model.

Figure 1. Magnetic flux density inside the windingwindow.

The Lorentz forces f v applied on the windingscan be determined from the magnetic fieldcalculations. Those forces are given by theexpression

f vൌ J ൈ B (4)

where J is the current density and B the magneticfield.The forces can also be written as a function of

the magnetic vector potential A

f v ൌ െ൬డA

డ௧ െV

v

ൈ ሺ ൈA

ሻ൰ ൈ ሺ ൈA

ሻ

(5)

The Lorentz forces provide the coupling termbetween the electromagnetic field and thewinding motion described by the mechanicalmodel.

Figure 2. Lorentz forces appearing on windings.

3. Mechanical Model

In the mechanical model, the linear elasticityequations are used to describe the motion of thewindings. The equation of motion is given by theexpression

∙ f v ൌ u ሷ (6)

where σ is the stress tensor, f v the volume forces, ρ the density and ü the acceleration vector.The volume forces f v in the equation of motionrepresent the Lorentz forces applied on thewindings.The stress tensor σ is a function of the straintensor ε as given in Hooke's law

ൌ (7)

where cijkl

is the stiffness tensor and εkl

the straintensor.The strain tensor ε is a function of thedisplacementൌ ଵ

ଶሾ u ሺ uሻ்ሿ (8)

where u is the displacement vector.The rather complex structure of the winding issimplified to an equivalent cylinder presentingorthotropic properties. In addition, the rotationalsymmetry of the winding is used to provide athree dimensional axisymmetric model. Thegoverning equations have thus to be written for

the two dimensional axisymmetric model incylindrical coordinates. The relationship betweenstrain and displacement in the two dimensionalaxisymmetric case is given by

ൌ డ௨

డ , ఏ ൌ ௨

, ௭ ൌ డ௨

డ௭ , ௭ ൌ

ଵ

ଶቀడ௨

డ௭ డ௨

డቁ , ఏ ൌ 0, ఏ ൌ ௭ఏ ൌ 0 (9)

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There is no displacement and no shear strain inthe circumferential direction.By combining the previous equations, arelationship in cylindrical coordinates is obtained

for the equation of motion

െSଵ்ሾ ሿSଶ ൌ (10)

where S1 and S2 are two operators defined as

Sଵ ൌడ

డ ଵ

0

0 డ

డ௭െ ଵ

0

డ

డ௭

డ

డ ଵ

Sଶ ൌడ

డ0

0 డ

డ௭ଵ

0

డ

డ௭

డ

డ

(11)

and [C ] is the stiffness tensor of an orthotropicmaterial [5].By introducing the volume forces in the equationof motion, the displacement of the orthotropicwindings can be determined at first for theaxisymmetric conditions without the oil loading.

Figure 3. Motion of windings subjected to Lorentzforces.

4. Acoustic Model

The displacement of the windings generatessound waves in the surrounding oil which in turnimplies a surface pressure on the structure.Sound waves propagating in fluid, such as oiland air, are described by the equation

ଶ െ ଵ

మ

డమ

డ௧మൌ 0 (12)

where p is the acoustic pressure and c the speed

of sound. The fluid is onsidered here as non-viscous.The fluid loading of the windings gives a firstboundary condition and is defined as

f p ൌ െ n (13)

where p is the fluid pressure and n a vectornormal to the structure surface.The second boundary condition is given by theparticle velocity continuity between structure andfluid, thus yielding

v f

ൌ∂u

డ௧ ∙n (14)

where v f is the particle velocity of the fluid and u the displacement vector.The relationship between pressure and velocity isobtained by using the linear equation of motionyielding

డu

డ௧ൌ െ (15)

where ρ is the fluid density and u thedisplacement vector.

Figure 4.Drawing of the transformer model.

5. Modeling Procedure Overview

In this section, the procedure to obtain thecomplete model is described step by step. Thetwo dimensional magnetic field is first calculatedfor the windings around the core limb. The

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associated Lorentz forces applied on thewindings can then be determined from thecurrents and the resulting stray fields. The twodimensional finite element model isimplemented and run in the COMSOL AC/DC

module. The harmonic forces can be utilized asthe driving term in the equation of motion whichis solved for oil loaded windings in twodimensions by means of the COMSOL StructuralAnalysis and Acoustics modules. The resultingvibration field is extended to a three dimensionalrepresentation by using the COMSOL featuredesignated Extrusion Coupling Variables. Thefield is then mapped onto a three dimensionalmodel of the transformer core, tank andenvironment. If a three-phase transformer isinvestigated, a 120 degree phase shift betweenthe three windings can be implemented in the

model. By using the COMSOL Acousticsmodule, the sound transmission paths such asinsulating oil, tank walls and air surrounding thetransformer are modeled by finite elements andthe sound pressure levels can be determined atany positions in oil or air.

6. Prediction Results

The load noise of a single phase transformeris investigated. Some of the transformercharacteristics are indicated in Table 1.

Table 1. Transformer characteristics.

Power Rating 145 MVA

Tank Length 3,8 m

Tank Width 2,7 m

Tank Height 3,5 m

A model of this single phase transformer wascreated and run in COMSOL according to theprocedure described in section 5 to obtain thesound levels emitted in the air surrounding thetransformer, as shown in Fig. 5.The sound power level of the transformer wasmeasured according to the applicable IEC

standard [2] and was calculated by using theprediction model. The deviation betweenmeasured and calculated sound power levels wasfound to be less than 2 dB. This can beconsidered as an acceptable match betweenexperimental and predicted results.

Figure 5. Tank wall displacement and sound pressurefield outside the tank.

Sensitivity studies have been performed toinvestigate the effects of some materialproperties and design parameters on the soundlevels.For example, it was found that tank thicknessvariations of 1 mm can imply sound power leveldifferences of nearly 5 dB. This rather largediscrepancy in sound power can be attributed tothe impact of the tank eigenfrequencies on soundtransmission.Furthermore, changes of 20% in the Young'smodulus of the radial spacers in the windingsresult in sound power level differences of approximately 1,5 dB.The simulations also reveal that the sound power

level varies with more than 1 dB if the Young'smodulus of insulation paper in the windings isaltered by 20%.The parametric studies clearly show the complexinteractions of the design parameters and theeffects of the material properties on the soundpower levels. The sensitivity tests also underlinethe need of reliable input data to ensure accuratepredictions. In particular, the dynamic propertiesof non-linear materials shall be properlydetermined by means of experimental methods.

7. Conclusions

A finite element model to predict transformerload-noise was developed by using COMSOL.This prediction tool can be used to demonstratethe influence of various material properties andgeometrical parameters on the sound powerlevels.

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8. References

1. G. Ramsis, J. Anger, D. Chu, The Sound of Silence Designing and Producing SilentTransformers, ABB Review, 2/2008, Pages 47-51

(2008)2. IEC 60076-10-1, Power Transformers – Part10 – 1: Determination of Sound Levels –Application Guide (2005)3. B. Weiser, H. Pfützner, J. Anger, Relevance of Magnetostriction and Forces for the Generationof Audible Noise of Transformer Cores, IEEE

Transactions on Magnetics, Volume 36, Pages3759-3777 (2000)4. M. Rausch, M. Kaltenbacher, H. Landes, R.Lerch, J. Anger, J. Gerth and P. Boss,Combination of Finite and Boundary ElementMethods in Investigation and Prediction of Load-

Controlled Noise of Power Transformers, Journal of Sound and Vibration, Volume 250,Pages 323-338 (2002)5. O. Rand, V. Rovenski, Analytical Method inAnisotropic Elasticity with SymbolicComputational Tools, Birkhäuser, Boston, 2005

9. Acknowledgements

Y. Zhang, S. Sander-Tavallaey, J. Anger, J.Oppelstrup and M. Hanke are greatlyacknowledged for their contribution to this work.

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