Prediction of Noise Generated by Electromagnetic Forces inInduction MotorsM.K. Nguyen1, R. Haettel*,2 and A. Daneryd2

1KTH NADA Stockholm, 2ABB Corporate Research Sweden*Corresponding author: 721 78 Västerås Sweden, romain.haettel@se.abb.com

Abstract: Induction motors, as any otherindustrial products, have to comply with variousrequirements on noise levels. Therefore, it isessential to use an appropriate prediction tool toverify and optimize the design of an inductionmotor with respect to the acoustic performances.The paper will focus on the prediction of themagnetic noise generated and radiated by aspecific motor. The challenge is here in the multi-disciplinary character of the problem. The modelshould consider the mutual coupling betweenelectromagnetic forces, the displacements and theacoustic pressure field.The finite element model coupling the differentphysical fields has been developed by usingCOMSOL. Some simplifications have beenintroduced to reduce the problem size andcomputing time.

Keywords: Induction motor, electromagneticforces, acoustics

1. Introduction

Today, the limitation of noise pollution is amatter of increasing importance in a world wherethe number of equipment generating sound andvibrations is steadily growing. Therefore, lownoise levels are mandatory for many industrialproducts to comply with customer requirementsand environmental regulations. Consequently, itis essential to predict the sound levels of inductionmotors with a sufficient accuracy at an early stageof the product design to select the mostappropriate strategy for noise control.

Three main sources of sound can be identifiedin induction motors: mechanical noise frombearings and unbalance effects, airflow noise dueto the cooling fan and magnetic noise produced byelectromagnetic forces in the motor air gaps.

The paper describes a prediction method forelectromagnetic noise which constitutes a typicalmultiphysics phenomenon involving threephysical fields: electromagnetism, mechanics andacoustics. The interaction between the alternatingelectrical supply and the associated rotating

magnetic fields results in varying forcesgenerating structural vibrations which eventuallyare radiated as sound in the surroundingenvironment.

In some cases, analytical models can be usedadvantageously to provide a rather quickindication of the sound levels generated byinduction motors. However, in general, due to therelative complexity of the motor geometry, thisapproach may present limitations for detailedinvestigations. Therefore, prediction modelsbased on finite element formulation shall beapplied to describe accurately the complexinteractions of the various design parameters andthe coupling of the physical fields.

In this paper, an overview of the modelingprocedure is first given in section 2. Then, thegoverning equations for the electromagnetic,mechanical and acoustic fields are described insections 3, 4 and 5. Some predicted results arereported and discussed in section 6.

2. Modeling Procedure Overview

The prediction of electromagnetic noise froman induction motor is performed in 3 steps.

Figure 1. Overview of the three modeling steps.

In the first step, the motor is modeled in 2D andin time domain until it reaches steady state whereit exhibits a stabilized harmonic pattern. In step 2,the magnetic forces (the radial Maxwell stresstensors in this study) are extracted fromCOMSOL into Matlab for the steady state alongthe stator's inner surface. The magnetic forces arethen transformed by means of Fourier analysis

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from time domain into frequency domain. In thethird step, the stator is modeled with surroundingair in 2D and frequency domain in COMSOL. Inthis step, the Acoustic-Structure interaction isinvestigated to predict the noise level of themotor. Fig. 1 gives an overview of the modelingsteps.

3. Step 1: The Motor Operation

The operating motor is modeled in 2D withthe combination of finite element method andelectrical circuits (lumped element method), asdescribed in [1] and [2].

Electromagnetic field:The lamination effects in the rotor and stator'scores are assumed to be perfect. The stator coilsare assumed to be made of thin wires so that noeddy currents are assumed to be present. Theequations for the motor domains, as presented in[3], are given by

× ( × A) =

⎩⎪⎨

⎪⎧ ezinstatorslots

− A + ezinrotorbars

− A inshaft0incoresandair

(1)

where ez is the unit vector pointing in the z-direction, A = ezthe magnetic vector potential.

, , are the number of turns, current andcross-section area of a slot. , are the voltageand length across a rotor bar. is the electricalconductivity of the domain materials.In COMSOL, the electromagnetic field ismodeled using Rotating Machinery, Magnetic(rmm) interface. The stator slots domain aremodeled by Multiturn Coils which are excited bycurrent from external circuit. Rotor bar domainsare modeled with SingleTurn Coil which areexcited by voltage from external circuit.The rotor domain and half of the airgap is rotatedusing Prescribed Rotation.

Stator circuit equations:In the stator circuit, the end winding of eachphase is represented by an inductor and a resistor.The coil sides are lumped into one resistor.A sample of the circuit for phase A is shown inFig. 2.By Kirchhoff’s voltage law, the equation for theloop in Fig. 2 is

= + + (2)

where is the voltage of phase A, the current inthe loop, the coil's resistance (including the endwinding), the end winding inductance. isthe total induced voltage across the coil sides (theparts of the coil in the stator core region).

= ∫A+ − ∫A-

(3)

where , , are the number of turns, statorlength and cross-section areas of the stator slots.

is the magnitude of the magnetic vectorpotential in z-direction.A+ ,A- indicate the slots where thecurrents are in ezand −ez directions, respectively,depending on the selected convention.

Figure 2. A sample of stator circuit for one phase.

The stator circuit is modeled by the ElectricalCircuit in COMSOL. To couple with the rmminterface, the induced voltage is modeled by theExternal I vs U component.

Rotor circuit equations:Each segment of the end ring is represented by aninductor and a resistor [1]. The extra part of thebars outside the core is lumped into one resistor ateach end. Fig. 3 shows the circuit for the rotor.

Figure 3. The rotor circuit.

By Kirchhoff’s voltage law, the voltage equationfor loop of , is expressed as

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− + 2 ( − ) + 2 , +2 , = 0 (4)

where is the voltage across the bar in therotor core region and is the bar's current.

is the resistance of the bar outside the rotor coreregion at one end. , are the resistance andinductance of one ring segment.Kirchhoff’s current law gives

= , − , (5)

where , is the current in the ring.The bar's current is:

= − , + = − ∫ + (6)

where , are the voltage and the resistance ofthe bar, is the electrical conductivity of rotorbars and is the cross section of the bar.The rotor circuit is modeled by the ElectricalCircuit in COMSOL. To couple with the rmminterface, the induced current is modeled by theExternal U vs I component.

Rotation equation:The rotational angle of the rotor is expressed by

= rotor − load (7)

where is the moment of inertia of the rotor and the rotational angle. rotor and load are the

rotor and load torques, respectively.Equation (7) is implemented by the Global ODEsand DAEs interface in COMSOL.

Summary:The relationship between the COMSOLinterfaces in step 1 is shown in Fig. 4.

Figure 4. Relationship between COMSOL interfaces inthe model in step 1.

3. Step 2: Fourier Transform

The operating mode of the motor is simulateduntil the steady state is reached, as the motorexhibits harmonic behavior.The radial Maxwell stress tensor is extracted fromCOMSOL to Matlab during the steady state alongthe boundary between the stator and air gap, thentransformed into frequency domain for theacoustic analysis.

4. Step 3: Acoustic-Structure Interaction

The forces applied on the stator generate adisplacement field which in turn implies a soundwave in the air surrounding the structure. In thismodel, only the motor stator and housing aretaken into account. Air is modeled between thestator and housing as well as around the housing.The density of the air is assumed to be constant.The four equations used to describe the acoustic-structure interaction in the frequency domain aregiven by

− u = ⋅ + FVinstator¨ − = 0inair

n ⋅ ( ) = −n ⋅ uatstator-airboundary

⋅ n = natstator-airboundary (8)

The first equation is derived from Newton'ssecond law where is the density of thestructure,u the displacement vector, the stresstensor, FV the body force.In the second equation, is the sound pressureand the speed of sound in air.The last two equations show the continuity of theacceleration and pressure in the normal directionto the Structure-Air boundary.The radial Maxwell stress tensor obtained in thefrequency domain is applied on the stator's innersurface.The Acoustic-Structure interaction is solved infrequency domain. Therefore the above equationsare to be transformed to frequency domain. InCOMSOL, the modeling is done by the Acoustic-Solid Interaction, Frequency interface. Air is builtaround the case for noise investigation. Theoutermost layer of the air is defined as PerfectlyMatched Layer (PML) to absorb the soundradiated from the stator and create free fieldconditions for propagation of sound waves. In

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addition, structural damping are implemented inthe motor stator and housing.

4. Modeling Parameters

A specific motor is modeled in COMSOL in thisstudy and the parameters used for the simulationsare presented in Table 1.

Table 1. Motor parameters

Power supply 3 phase, 388V (nominal)

Number of poles 4 poles

Stator 48 slots

Rotor Squirrel cage, 38 straightbars

Load No load

Parameters used for the solver in order to modelthe operating motor are shown in Table 2.

Table 2. Solver parameters for step 1.

Method Generalized alpha

Time step Fixed (0.0001s)

Nonlinear solver Newton method

6. Prediction Results

The 2D model of the motor described in section 4was created and run according to the procedureexplained in section 2. The calculations wereperformed at the electrical frequency of 50 Hzcorresponding to a rotation speed of 1500 rpm forthe motor. The total force amplitudes obtained byintegration of the Maxwell stress tensor on thestator-airgap line at each frequency are given inFig. 5.

Figure 5. The total electromagnetic forces with respectto frequency.

To explain the acoustical behaviour of the motor,the mode shapes of the motor are determined in

2D by using the solid mechanics interface inCOMSOL. Neither air is present, nor externalforces are applied on the stator in this particularmodel. A few mode shapes calculated for thestator with housing and the stator without housingare shown in Fig. 5. The different resonancefrequencies obtained for both configurations arepresented for the same mode shapes.

a1) 1155 Hzb1) 916 Hz

a2) 3361 Hz b2) 2455 Hz

a3) 5281 Hz b3) 4402.62 Hz

a4) 5264 Hz b4) 5924 HzFigure 6. Mode shapes of stator with housing (a1-a4)and stator without housing (b1-b4).

The electromagnetic forces applied on the motorstructure consisting of stator and housing providethe sound power level presented in Fig. 7. Theclear peak observed at 1150 Hz in the soundpower level plot corresponds to an amplificationimplied by the resonance identified at 1155 Hz, asshown in Fig. 6. As a matter of fact, the total forceat 1150 Hz does not present a particularly highlevel in Fig. 5. However, it can still provide themain contribution to the total sound power levelwhen it is combined with a resonance.Sound pressure and displacement fields aredisplayed for the stator with housing at 1150 Hzin Fig. 8. In this picture, the mode shape of themotor determined at 1155 Hz can be clearlyidentified.

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Figure 7. Sound Power Level for the motor operatingat 1500 rpm.

Figure 8. Total Sound Pressure field (root meansquare) and displacement field. The color bar on the leftis for the pressure (in Pa) and on the right is for thedisplacement (in m).

7. Conclusions

A method for predicting the noise levelsgenerating by an induction motor was presented.The electromagnetic FE-model of the motor isfirst solved in the time domain, then the resultingforces are transformed to the frequency domain inorder to perform the acoustic analysis.It was demonstrated that the motor operating at1500 rpm is particularly noisy at the particularfrequency of 1150 Hz close to a natural frequencyidentified at 1155 Hz.The prediction tool can already be used toinvestigate the impact of various designparameters on the sound power levels.The present work will be continued by extrudingthe 2D mechanical model into a 3D model in orderto perform more accurate investigations andvalidate the predicted results with experimentaldata.

8. References

1. J. Güdelhöfer, R. Gottkehaskamp, and A.Hartmann. Numerical calculation of the dynamicbehavior of asynchronous motors with COMSOLmultiphysics. In Proceedings of the 2012COMSOL Conference in Milan, 2012.2. Javier Martinez, Anouar Belahcen, and AnteroArkkio. A 2d fem model for transient and faultanalysis of induction machines. PrzegladElektrotechniczny, 88(7B):157–160, 20123. Sami Kanerva. Simulation of Electricalmachines, Circuit and Control systems usingFinite element method and System simulator.PhD thesis, Helsinki University of Technology,2005

9. Acknowledgements

M. Hanke (KTH), J. Oppelstrup (KTH) and B.Larsson (ABB) are greatly acknowledged for theircontribution to this work.

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