measuring method for determining the reasons of

Measuring method for determining the reasons ofmagnetically caused structure-borne sound on

electrical drive systemsRoland Lach, Stefan Soter

Lehrstuhl fur Elektrische Antriebe und MechatronikUniversitat Dortmund

GERMANY

Tel:+49-(0)231-7555337; Fax: +49-(0)231-7555336; Email: [email protected]

Abstract— A special signal processing method (order analysis) allows well-defined conclusions from accelerometer measurements on electrical drivesregarding the reasons of structure-borne and airborne sound. In this paperan analysis of acceleration signals of an 5kW induction machine is presen-ted. The special feature is that a projection of sound curves is included inthis analysis. Thus it is possible to assign an oscillation excitation to harmo-nics attributed to the machine design or harmonics of the feeding frequencyconverter.

I. I NTRODUCTION

An electrical drive describes a 3-phase-induction motor fed froma frequency converter. As a result of advancements in powerelectronics and motor design electrical drives have achieved agreat importance in a wide range of industrial applications.The advantages of these systems are for example utilisabilityand flexibility at reasonable costs. A drawback of electrical dri-ves is that under certain conditions unwanted machine behaviorcould occur. This causes an increased temperature of the win-dings, unacceptable vibrations or increased sound (sound powerradiation).

frequency−converter

three−phase netpermanent frequencypermanent voltage

induction−machine

ASM machinealternate frequencyalternate voltage

f Fu ,UFuf 1 ,U1

3~

3~M, ω3~

ASM

Fig. 1. Electrical drive

II. OSCILLATION MEASUREMENT

Different mechanical excitations usually superpose in the ma-chines casing. Thus the vibration measurements can be perfor-med with a small number of acceleration pickups - in many ca-ses a single acceleration pickup is sufficient. Evaluating the re-sults, mechanically caused vibrations can be identified by theirrelation to the rotational speed. Given that noise is carried to themachines casing only for frequencies above a certain excitationfrequency, a minimum rotational speed is required.The conventional measuring method (structure-borne soundmeasurement) performs a frequency analysis of the accelerati-

on pickups: The time domain signal is translated into spectrallines in the frequency domain by a FFT analyzer. In a two di-mensional representation of the spectral amplitudes in respect tothe frequencyfFFT , mechanically excited vibrations can clearlybe identified as they usually exhibit a constant factor in relationto the rotational speed. As an example, tab. I contains a list ofreasons for mechanical excitation and the correlated symptoms(signal identification table).

reasons of mechan. excitation symtoms

twisted machine shaft fn,2 fnmechanical tolerance fnlocal cogging failure fn− i fn, fn + i fn

circular cogging failure i fz, fz/ j, i/ j fz

blade frequency of the fan i fn

TABLE I

SIGNAL IDENTIFICATION TABLE

notations:

fn: Drehfrequenz [Hz]fz: Zahneingriffsfrequenz [Hz]i, j: ganzzahliger Zahler

With individual measurements being recorded at discrete rota-tional speeds, the spectral lines can be displayed in athree di-mensional waterfall diagram of the spectral analysis. In thiscase the individual measurements of a singular rotational speedare located along the y axis in ascending order while the resultsof the FFT analysis (two dimensional spectrum representation)is displayed in the frequency-amplitude plane (xz plane).The three dimensional representation allows for a simple diffe-rentiation betweenrotational speed dependant vibration fre-quencies running diagonally in the xy plane androtationalspeed independent vibration frequenciesrunning parallel tothe rotation speed axis. A distinct raise of an amplitude parallelto the rotation axis is labeled astructural resonanceor a sy-stem eigenfrequencyof the machine. Fig. 2 contains an exam-ple of a three dimensional waterfall diagram. Resonance excita-tions of the machine independent of the rotational speed can beidentified at frequencies offFFT = 400Hzand fFFT = 5900Hzwhile rotational speed independent vibration excitations cannotbe recognized. To scale down the value range, the acceleration

amplitude (magnitude) is displayed as standardised1 decibel va-lues.As the acceleration measurements of the individual rotationalspeeds are recorded with the machine running idle, the y axisscaling corresponds with the base frequencyf1 of the fee-ding frequency converter which is proportional to the rotationalspeed. The method of evaluating accelerometer measurementsis commonly used and considered to be a relatively simple wayto visualize the mechanical eigenfrequencies in the range of themachines revolution speed.

acce

lara

tion

ampl

itude

[Hz]

dB (1.0 |m/s²|)

Quelldatei: zwerte.dat vom 24/10/03 Optionen: Bet Analyse: FFT Analyzer : FFT Spectrum Averaging : Autospectrum

0 1000 2000 3000 4000 5000 6000 7000 8000 900010

2030

4050

60−200−150−100−50

050

100fe

edin

g fre

quen

cy

FFT analysis frequency [Hz]

Fig. 2. 3D-Waterfalldiagram

III. M ACHINES AS FREQUENCY RESPONSE SYSTEMS

If the machines operating frequency is in close proximity ofan eigenfrequency an increased vibration amplitude or radia-ted sound power occurs. These effects can negatively affect themachine behavior and even cause permanent damage (bearings,windings overheating).The reasons of this unintended machine behavior can be deter-mined as excitation of the mechanical vibration system (machi-ne construction) in an eigenfrequency. In case of periodic ex-citation in an eigenfrequency the machine generates vibrationswith specific eigenforms:flexural mode and torsional mode.The flexural mode is represented by stationary waves across thecircumference of the machine and it is defined by the number ofnodes i.e. n-nodes flexural vibration or flexural eigenform withmode n respectively.The machine shaft consists of distributed masses, so that the ei-genforms of torsional vibrations are similar defined as n-nodestorsional vibration. The damping in an eigenfrequency of a geo-metric construction achieves a minimum value. Thus an excitati-on with low vibration amplitude involves a high oscillation am-plitude (resonance). In the total speed range of a variable speeddrive it is generally impossible to avoid a resonant excitation inone of the low eigenforms. A operation of the machine at a ,,cri-tical revolution speed” in the range of an eigenfrequency mustbe prevented, as well as a slow ,,run up” or a slow ,,coast down”.

1concerning accelatration:a0 = 1m/s2

IV. M AGNETIC SOUND PHENOMENA

The periodic excitation of the machine can occur for magnetic,mechanical or aerodynamic reasons.

structure−bornesound

mechanical reason aerodynamic reason

air−borne sound

(sound wave propagation: 16 Hz−20 kHz)

sound source: electrical machine

magnetic reason

Fig. 3. Electrical machines as sound sources

The method of machine diagnostics described in fig. 4 can revealmechanical and aerodynamical reasons. If striking spectral linesare situated inside the spectrum of an acceleration signal thenpossible reasons or possible disadvantages can be detected bymeans of signal identification tables (see chapter II).

mechanical reason aerodynamic reason

(impact excitation) (fluid mechanic)

bearing gear outside excitation blower

machine diagnostic from measurements

Fig. 4. Mechanical and aerodynamical reasons

Especially for induction machines with small air gaps vibrationsof varying intensity and sound occur at specific rotation speeds.This relates to the magnetic sound phenomena caused by har-monics and harmonic waves as depicted in fig. 5.

magnetic reason

interaction of stator and rotor fields

converter winding slot saturation eccentricity

calculation form interaction of fields

Fig. 5. Magnetic reasons

The expression ”harmonic waves” describes a large number ofsinusoidal voltages with frequencies that are multiples of the

fundamental frequency. They are directly caused by a non sinu-soidal current feed (frequency converter).Physically, harmonics and harmonic waves can be distinguishedby wave length and propagation speed. The propagation speedof a harmonic wave is lower than the propagation speed of thefundamental wave while a harmonics propagation speed exceedsthat of the fundamental wave. For induction machines, harmonicwaves can primarily be attributed to the winding scheme, slotlayout and occurrences of saturation and eccentricity.

A. Magnetic sound phenomena caused by harmonic waves

Incorporating the explanations of the previous chapter, the oc-currence of magnetic sound phenomena caused by harmonic wa-ves interaction can be summarized as follows. In the air gap, theelectric current feed of the machine is transformed intoradialforcesby harmonic waves interaction.A cyclic force excitation of the machines yoke in the vicinityof a system eigenfrequency increases body vibrations and airsound emission respectively. This radial force can be attributedto special forms of interaction between stator and rotor harmonicwaves or harmonic fields.The measuring method presented in this paper provides additio-nal information about the causes for magnetic structure-bornenoise by representing measurements values (vibration frequen-cies) and precalculated sound frequencies in a diagram. In thefollowing chapter, the derivation for the orders of radial forcecausing magnetic sound excitation is described with the appro-priate sound curves.

V. HARMONIC WAVES AND NUMBERS OF POLE PAIRS

With an three phase winding so evenly distributed that no slot ef-fects occur, an idealized induction machine already displays dis-continuities of the magnetomotive force at zonal borders. Fig. 6depicts the current direction in the winding phases U,V and W.With the three phase current winding, six zones of constant ma-gnetomotive force result along the stators circumference.

τ P

��

��

��

��

��

��

��

��

��

��

Nτ

U+1

U

W

V

W

V

−1

−2

+3+2

−3

γ

StatorRotor

Luftspalt δ

WA−U

WA−W

WA−V

radi

us r

Boh

rung

s−

Fig. 6. 2p-pole machine (not slotted)

The electric loadingα(γ, t) within a winding zone results fromthe amount of magnetomotive force per coil length. The dis-continuities are especially visible in the field excitation curve(fig. 7) where the machines developed view is plotted for diffe-rent points in time t of the three phase current feed.

πp=τ P

πp=τ P

τ P

WA−U WA−WWA−V WA−U

I I 1I 3I 2

== = 0

I 3 I 2

I 1

Maschinenumfang = 2

+1 −3 +2 −1 +3 −2

A

A

γ

Fig. 7. Zonal borders and electric loading curves

This diagram indicates that - despite a sinusoidal current feed ofthe three phase windings with constant frequency (f1 = const) -discontinuities in the field excitation occur, commonly termed astep function.

An evaluation of the discontinuities at the zone borders of thestator strands by means of a fourier transformation implies thatthe electric loading curve can be approximated by a sinusoi-dal fundamental frequency and a theoretically infinite numberof harmonic waves that are multiples of the fundamental fre-quency.

As an example, fig. 8 shows an approximation of a square func-tion (red curve) by a fundamental wave and the third and fifthharmonic. (γ denotes the angle of the machines developed viewon the circumference). Since the ordinal number of an electricalmachine is derived from the type of winding and the number ofpoles, the expressionnumber of pole pairs is commonly used.In the following, the number of pole pairs of the stator is de-notedν whereas ˜µ denotes the rotors number of pole pairs. Thetilde sign ,, ˜ ” indicates that the pole pair numbers relate to themachines fundamental wave.

0 < < 2 πγ

−1.5

−1

−0.5

0

0.5

1

1.5

0 1 2 3 4 5 6

Am

plitu

de

Winkel

sin(x)*4/pi(sin(3*x)/3)*4/pi

(sin(x)+sin(3*x)/3)*4/pi(sin(5*x)/5)*4/pi

(sin(x)+sin(3*x)/3+sin(5*x)/5)*4/pi

Fig. 8. Harmonic analysis for approximation of a square wave function

A. Pole pair number of the stator

On the condition of asymmetrical integral slot winding thenumber of pole pairsν for thefield of the stator windings forany 2p-pole machine (p≥ 2) can be calculated by using the fol-lowing equations:

ν = p(6a+1) with (a= 0,±1,±2, . . .) (1)

ν = p,−5p,7p,−11p, . . . (2)

The fundamental wave withν = 1 and a= 0 can be determinedasν = p. The number of pole pairs of the harmonic waves can becalculated by using eq. (1). The algebraic sign denotes the sen-se of direction of the harmonic wave. A positive algebraic signrepresents that the harmonic wave rotates in the same directi-on as the fundamental wave whereas an negative sign indicatesthat the sense of rotation of the harmonic wave is contrary to thesense of rotation of the fundamental wave.

B. Pole pair number of the rotor

The generation of mechanical torque, structure-borne and air-borne sound can be attributed to the interaction of the electroma-gnetic field of the stator and rotor. The principle of an inductionmachine relies on the induction of voltage in the rotor, whichcauses currents and electromagnetic fields in the short-circuitedturns (squirrel-cage) of the rotor. The currents and fields of therotor depend on the relative differences of the rotational speed,the (fundamental wave)slip s, respectively the operating pointof the machine. Theharmonic wave slipsν can be defined alongthe lines of the fundamental wave slip [4]. This allows the fre-quencies of the rotor currentsf R

ν to be expressed in relation tothe exciting stator current with the frequencyf S

1 :

f Rν = sν · f S

1 (3)

f Rν = (1− ν

p(1−s)) · f S

1 (4)

The frequencies of the fundamental electromagnetic fields of thestator f S

1 and the rotorf R1 are linked by the fundamental wave

slip. According to equation (3) the frequencies of the harmonicelectromagnetic fields of the stator and the rotor are linked ina similar manner by the harmonic wave slip. From now on thenumber of pole pairs of the rotor is referred to asµ respectivelyµ= p·µ. With this nomenclature and the equations it is possibleto deduce conditions which determine which number of polepairs of the rotor ˜µ can be caused by an rotating field of thestator with the number of pole pairsν:1. Number of strands and number of pole pairs of stator and ro-tor winding are equal. Under these conditions the rotor windingonly generates rotating field waves which have the same numberof pole pairs and frequencies as the harmonic waves of the statorwindings.

µ= ν resp. µ= ν (5)

2. Number of strands and number of pole pairs of stator androtor winding are not equal. Under these conditions the numberof pole pairs of the rotating field of the rotor can be determined

by adding the number of pole pairs of the stator and the numberof strands of the rotor m2 in accordance with:

µ = ν+b·m2

pwith (b = 0,±1,±2, . . .) (6)

resp. µ = ν+b·m2 (7)

This represents that a rotational field wave of the stator with thenumber of pole pairsν causes a rotational field wave of the rotorwith the number of pole pairsµ. The number of pole pairsµ ofthe rotor is dependant on the number of strands of the rotor m2.In a squirrel cage induction machine the number of strands ofthe rotor m2 corresponds to the number of barsN2. This meansthat the number of pole pairs of the rotating field of the rotorµaccording to eq. 5 can be determined as:

µ = ν+b·N2

pwith (b = 0,±1,±2, . . .) (8)

resp. µ = ν+b·N2 (9)

VI. CALCULATION OF ELECTROMAGNETIC AIRGAP FIELDS

The occurrence of fundamental and harmonic electromagneticfields in the airgap of a machine can be explained with the num-ber of pole pairs of rotor and stator. The current feed of the sta-tor winding with the frequencyf S

1 causes - according to Equa-tion (2) with a= 0 - a fundamental field of the stator with afrequencyf S

1 and with a6= 0 harmonic fields of the stator linkedto higher numbers of pole pairs. Thefundamental field of thestator windings with the number of polesν = 1 respectivelyν = p causes - according to eq. (9) with b= 0 - a fundamen-tal field with the frequencyf S

1 , the number of pole pairsµ = 1respectively ˜µ = p and with b6= 0 harmonic fields in the rotor.These harmonic fields of the rotor are produced by the voltageinductions in the rotor barsN2. They are referred torotor re-mainder fields. The frequencies of the rotor remainder fieldscan be determined from the harmonic wave slip with eq. (3) andtherefore contain non characteristic frequencies.

A. Stator- and rotor induction

In Chapter V the graph of the field excitation of an very sim-plified 2-pole machine has been approximated with an Fourier-progression.The graph of the field excitation of an actual machine can analy-tically be approximated by the sum of fundamental and harmo-nic waves. The influence of slotting, saturation and eccentricityis accounted for with the magnetic conductanceΛ(γ, t). The in-fluence of winding on the other hand is accounted for with theharmonic wave winding coefficientξν from winding zone andwinding tendon [4].By using the wave equation and the equations for determiningthe number of pole pairs of stator and rotor fields it becomes pos-sible to describe thestator induction with the following equa-tions:

bS(γ, t) = ∑ν

bSν(γ, t) = ∑

νBS

ν ·cos(ν γ−ωS1t−ϕS

ν) (10)

The fundamental field of the stator with the pole pair number ˜µand the frequencyf S

1 excitesrotor induction which consists of

the fundamental wave withν = µ andb = 0 and an harmonicwave of the rotor withν = µ andb 6= 0. Theserotor remainderfieldscan be described as follows:

bR(γ, t) = ∑µ=ν,b6=0

BRµ ·cos(µγ−ωR

1t−ϕRµ) (11)

The computation of the resulting airgap field from stator androtor induction requires the values of stator and rotor to be dis-played in the same coordinate system. With this requirement itis possible to determine the transformation from the angle of therotor γ2 to the angle of the statorγ as seen in Figure 9:

γ = β+ γ2 mit β =∫

ωmechdt +β0 (12)

and ωmech=ω1

p(1−s) static conditions

γ =ω1

p(1−s) t +β0 + γ2 (13)

β

δ

γ

γ

2

positive motion

stator

rotor

Fig. 9. Transformation of coorinates

The computation of the amplitude of the magnetic fieldsBRµ of

the rotor remainder fields has to be performed by using therotor valuesβ0 andγ2.In general the rotor position in an induction machine is un-known, as it is dependant on load. Therefore it isimpossibleto conduct the computation by using the aforementioned trans-formation of coordinates.

A.1 Field dampening coefficientdν

It is possible to calculate the amplitudes of the rotor remainderfields by using the approach of field dampening. In this approachSeinsch[2] defined the field dampening coefficientdν. The co-efficient can be calculated from the T equivalent network of aninduction machine for calculating harmonic fields of the win-ding.The method is based on the fact that the rotor drains energy fromthe stator during operation as an induction motor. For this rea-son it is possible to understand the reaction of the rotor field tothe stator field as damping which allows the calculation of theamplitudes of the rotor field from the amplitudes of the statorfield:

BRµ = BS

ν ·(dν−1) · I1,ν

ζν·aµ

aν·ζµ (14)

This equation incorporates the influence of slot width with thereducing factora and the dampening effect of the slot shape withthe factorζ as shown in [2].The reaction from the rotor-field to the stator-field is referred toasprimary armature reaction respectivelyarmature reacti-on of the 1.Order. The rotor remainder fields which are gene-rated by the fundamental field of the stator with the frequencyf S1 cause voltages with non characteristic frequencies in the sta-

tor f Rµ 6= f S

1 . The dampening of the rotor remainder fields by thewindings of the stator is referred to as thesecondary armaturereaction or armature reaction of the 2.Order. The fields of thetertiary armature reaction and under special winding conditi-onsarmature reaction of the 4.Order form accordingly. Anexact analytical computation of these complex interactions canbe performed by using theharmonic fields theorie. The ar-mature reaction of the 4.Order only occurring under specialcombinations of winding and slotting is proven in the originalwriting of theharmonic fields theorie [3]. This shows that thenumber of fields and field-frequencies is finite.

It is possible to calculate the resulting dampened stator windingfields and the resulting dampened rotor remainder fields fromthe amplitudes of the stator fieldBS

ν by using the field dampeningcoefficientdν.

B. Interaction of the fields of stator and rotor

Fundamental and harmonic fields of rotor and stator superimpo-se in the airgap. The fundamental field of stator and rotor pos-sess the same number of pole pairs and the same frequencies andhence produce the fundamental wave torque of the machine. Theharmonic wave interaction as well as the resultingmagneticstructure borne noisecan be calculated analytically for an in-duction machine by using the following simplifications:1. Only the primary armature reaction is taken into account.2. The amplitudes of the rotor field can be computed with thefield dampening coefficientdν [2].3. The tangential forces are neglected since harmonic field in-teractions basically cause radial forces.

VII. R ADIAL FORCE WAVES

With the assumption of ideal iron (µFe = ∞) and a non slotted la-minated sheet package the streamlines of the field proceed radialthrough the airgap and emerge out of respectively enter into thesurface of the iron of stator and rotor perpendicular and oppo-sing. The mechanical interfacial tension (Maxwell’s interfacialforce) for perpendicular out of the iron surface emerging stre-amlines of the field is vectored only in radial direction. The in-terfacial force by Maxwell can be computed from the magneticforce per area:

σr(γ, t) =F(γ, t)

A=

b2(γ, t)2µ0

(15)

b(γ, t) describes the course of the airgap field in the distance andtime domain. This course is dependant on the characteristics ofthe machine in a very complex manner. It is for example depen-dant on the progression of the currents in the windings in thetime domain, the grooving, the saturation and the eccentricity.

The square of the airgap field has to be replaced by the Fourierseries of the interactions between stator and rotor fields whichresults in equation (15) being transformed into:

σr(γ, t) =[∑bν(γ, t)+∑bµ(γ, t)]2

2µ0

= ∑r

σr ·cos(r γ−ωTont−ϕr) (16)

σr(γ, t) =B2

ν4µ0

· [1+cos(2νγ−2ωνt−2ϕν)]

+B2

µ

4µ0· [1+cos(2µγ−2ωµt−2ϕµ)]

+ · · ·

+Bν Bµ

2µ0· [1+cos((ν± µ)γ− (ων±ωµ)t− (ϕν±ϕµ)]

+ · · · (17)

With regard to the magnetic acoustic noise phenomena not allterms of equation (17) are important. Only the terms responsi-ble for the excitation of the laminated sheet package, i.e. termswhose frequency can lead to an deformation of the laminatedsheet package are of importance. This assumption leads to thedisregard of square termsbecause these force waves only cau-se high ordinal numbers which conform to short wavelengths.It is save to assume that the laminated sheet package possessesenough stiffness against high frequencies or short wavelengthrespectively. This means that waves with high frequencies re-spectively ordinal numbers are unable to deform the laminatedsheet package. force waves with lower ordinal numbers can onlybe generated by the ”mixed product” from equation (17) whichrepresents the addition or subtraction of stator and rotor values.If the existence of the harmonics of stator and rotor is assumedin the following form

Bν(γ, t) = Bν cos(νγ−ωνt−ϕν) (18)

Bµ(γ, t) = Bµcos(µγ−ωµt−ϕµ) (19)

they cause magnetic traction which can be described asradialforce waveof the orderr, the sound frequencyfsound and thephase differenceϕr :

σr(γ, t) = σr cos(r γ−ωTont−ϕr) (20)

The characteristic parameters of the radial force wave can bedetermined as:

amplitude

σr =Bµ · Bν

2µ0(21)

radial force order

r = µ± ν (22)

sound frequency

ωTon = ωµ±ων (23)

phase

ϕr = ϕµ±ϕν (24)

A. Field interactions of low radial force order

The assumption of an armature reaction of the 1.order leadsto four different interactionsA-D between the stator and rotorfields which can cause a radial force of a low ordinal numberand for this reason an excitation of the laminated sheet package.These are the harmonic waves of the stator winding with thenumber of pole pairsν which interact with:

interaction harmonic wave of the rotor (pole pairν)

A rotor remainder fieldsB rotor remainder fields of the saturationC add. rotor remainder fields of the saturation at∆D rotor remainder fields of the eccentricity

TABLE II

FIELD INTERACTIONS OF LOW RADIAL FORCE ORDER

B. Sound equation and Campbell diagram

According to eq. (23) the magnetic excitation for field interacti-ons can be stated as per tab. II. Thefield interaction A is dis-played as an example:

fsound= f1

[|b| ·N2

p(1−s)

]+2 f1

−0(25)

The sound frequencies represent excitation frequencies withwhom the induction machine experiences a mechanical forceexcitation in the airgap. This means sound frequencies repre-sent the frequencies of the radial force waves. According to thenotation thesound frequency fsound can be understood as anlinear equation if the frequency is plotted as the standardizednumber of revolutionsn/nD = (1− s). In this case the slope ofthe straight line can be determined by using the parameters ofthe motorN2, p and f1.The sound equation are straight lines with different slopeswhich result from the sound frequency through the control va-riable of the rotor pole pair numbers|b|. The so-calledreso-nances diagramrespectivelyCampbell-diagram can be con-structed under the condition that the mechanical eigenfrequen-cies of the machine are known2 by using the sound equation.Thestructural resonancesrespectively thecritical speedsareapparent from the Campbell-diagram. These critical speeds arespeeds within a machines speed range at which it can be excitedby resonance as a result of field interaction.The sound frequenciesare dependent on speed and thereforeshow as diagonals in a Campbell-diagram. Theeigenfrequen-ciesof a machine are independent of speed (constant) and the-refore show as lines which are parallel to the standardized speed.

As an example theresonances diagramof the analyzed testmachine is depicted in fig. 10. The stable operating range of theinduction machine is marked with the two lines parallel to theordinate and between idle run (s= 0) and [Bemessungsmoment](s = sN). The following motor parameters were used: Numberof barsN2 = 38, Number of pole pairs p= 4, frequency of the

2either from analytical or computations respectively from vibration measure-ments on the machine

statorf1 = 100Hz, b= 6 (chosen), operation slipsN = 0.973 andthe eigenfrequencies which are calculated from a modal analysisfM0 = 5869Hz (Mode 0), fM2 = 886Hz (Mode 2).

s=0

s_r2

−

s_r0

+s_

r0−

s_r2

+

s=s_

n

freq

uenc

y [H

z]

f_sound−f_sound+ mode 2 mode 0

0

1000

2000

3000

4000

5000

6000

0 0.2 0.4 0.6 0.8 1 1.2 1.4(1−s)=n/n_d

Fig. 10. Campbell-diagram

C. Sound curves and order analysis

The order analysis is an established method for performing fre-quency selective tests on electrical drives with variable speedor speed variations. For an order analysis a signal with chro-nological synchronism is standardized on an signal with speedsynchronism. This means that the calculated FFT-frequency ofa measurement is based on the fundamental frequency of thatmeasurement. The result of this operation is aspeed indepen-dent spectrumas an function of the order.The order has no unit and is in general defined as quotient of(excitation-) frequency and fundamental frequency:

ord =ff1

(26)

Speeds which result from constant frequencies show on the or-der diagram as curves with the function1Ord . This is due to thefact that they are linked to the fundamental frequency of the re-spective speeds. In the order diagram (figure 11) an electricfrequencyni with the excitation frequencyf = i · f1, i.e. withthe harmonici of the fundamental frequency is defined as:

ni =i · f1 ·60sec

Ord= nOS (27)

In the order diagram the number of pole pairs of the test machinewith a f = 100Hz-feed (i = 2, respectivelyf2 = 2 ·50Hz) andnD = 1500min−1 can be calculated to p= Ord = 4 accordingly.The sound frequencies consist of the fundamental frequencyf1and the machine specific term (N2/p) multiplied with the controlvariable|b| of the rotor pole pair number according to eq. (25).It is necessary to describe the slip in relation of the order to beable to transform sound equation into sound curves. The speednb is used to relate slip and order. With eq.( 27) it results in:

nb =|b| · f1 ·60sec

Ord(28)

spee

d [1

/min

]

0

500

1000

1500

2000

2500

3000

0 2 4 6 8 10 12 14Ord

i=1 i=2 i=3 i=4

Fig. 11. order diagram of electrical frequencies

With this the dependency on slip(1− s) can be expressed asdependency on the order:

s =nD−nb

nD= 1− nb

nD

s = 1−|b|· f1·60sec

Ordf1·60sec

p

= 1− |b| ·pOrd

(29)

(1−s) =|b| ·pOrd

(30)

The sound curves which are derived from the sound equationaccording to eq. (25) are subsequently referred to as speeds ofharmonicsnOW. They are a function of the speed and can be cal-culated by applying eq. (27) to an induction machine and usingeq. (30):

nOW =f1 ·60secp·Ord

[|b| ·N2

p· |b| ·p

Ord

]+ 2 f1·60secp·Ord

−0

nOW =[

f1 ·60sec·b2 ·N2

p·Ord2

]+ 2 f1·60secp·Ord

−0(31)

n+OW =

f1 ·60sec·b2 ·N2

p·Ord2 +2 f1 ·60sec

p·Ord(32)

n−OW =f1 ·60sec·b2 ·N2

p·Ord2 (33)

With these equations it becomes possible to display the measu-rement results of anorder analysis - which are contrary to thespectral analysisindependent of speed and are explicit alloca-ted to frequencies - along with the precalculated sound frequen-cies in one diagram, the so-calledresonance-diagram of theorder analysis.In this diagram the harmonics of the frequency converter can bedistinguished from the harmonics of the machine. This can bedone by analyzing the curve progression according to the trans-formation of eq. (27) and eq. (31):

harmonics harmonic waves

nOS∼1

OrdnOW ∼ 1

Ord2

The harmonic frequencies can generally be computed witheq. (27):

fOS= f1 · iOS=Ord ·n60sec

(34)

The sound frequencies of field interactionsA, C andD (static) -according to tab. II - can be displayed as:

fsound= f1

[|b| ·N2

p(1−s)

]+2 f1

−0≡ fsound∗ (35)

The sound frequencies of field interactionB - the interactionof the rotor remainder fields of saturation with the fields of thestator windings can be calculated from:

fsound= fsound∗ +2 f1 (36)

Therefore the frequencies of the harmonicsfOW are:

fOW = f1 · iOW = f1 ·fsound∗

f1= fsound∗ =

Ord ·n60sec

(37)

Since the frequencies of the harmonics are composed of the sumor the difference of the interacting fields it is necessary to havethese evaluated separately. The frequencies of the field interac-tionsA, C andD (static) can be computed as follows:

f +OW =

Ord ·n60sec

−2 f1 (38)

f−OW =Ord ·n60sec

(39)

Field interactionB can be computed according to:

f +OW =

Ord ·n60sec

−4 f1 (40)

f−OW =Ord ·n60sec

−2 f1 (41)

Res2

Res1

Res3

spee

d [1

/min

]

0

200

400

600

800

1000

1200

1400

1600

0 50 100 150 200 250 300 350 400Ord

n_OW+, b=6n_OW−, b=6

n_OS, f=5800Hzn_OS, f=5900Hz

Fig. 12. resonant diagram of order analysis

Fig. 12 shows the resonance diagram of the order analysis forthe induction motor used in Chapter. VIII to validate the measu-rement procedure.As this representation corresponds with the translation of theCampbell diagram (fig. 10 ) into the order analysis, the same

motor parameters are used. Additionally, a frequency conver-ter feed with a fundamental frequencyf1 = 100Hz and a syn-chronous pulse frequency offT = 3kHz is assumed. This pro-duces mechanical revolution speeds of doubled revolution fre-quencies thus, the exemplary harmonic revolution speeds ofnOS

are drawn atf = 5800Hzand f = 5900Hz.In the case of order analysis measurements displaying resonancepoints that correspond with the precalculated sound curves, a de-termination of the excitation frequency is possible as well as adifferentiation between converter excitation and structural reso-nance of the machine.

Fig. 12 shows three possible resonance pointsfRes1, fRes2

and fRes3. The curve progressions indicate thatRes1 (n+OW,

800min−1, 51Ord) and Res2 (n+OW, 400min−1, 75Ord) are

structural resonances of the machine caused by radial forces ofa harmonic field interaction, whereasRes3 (nOS, 1400min−1,250Ord) turns out to be a structural resonance of the converterfeed.The assumptions• machine frequency f1=100Hz• harmonic waves (HW) are not caused by rotor remainderfields of the saturation (field interactionB)• idle run with decreased voltage• harmonic waves are attributed to the tone frequenciesf +

OW(sum of pole pair numbers of rotor and stator)allow the eigenfrequencies to be calculated out of Fig. 12:

fRes1 =800min−1 ·51Ord

60sec−2·100Hz= 480Hz


60sec−2·100Hz= 300Hz


60sec= 5833.3Hz

VIII. A PPLICATION OF THE MEASUREMENT METHOD

The vibration measurements are performed with a accelerati-on pickup and aBr uel & Kjær vibration measurement sy-stem. The measuring setup consists of the testing machine (8pole 5kW induction machine) and a load machine according tofig. 13.

Fig. 13. test bench

To allow an order analysis, a measurement of the revolutionspeed is required. This is done with an inductive sensor that is

located below the nut of the coupling, generating one triggerimpulse per shaft revolution.

Fig. 14. inductive sensor

A conventional vibration measurement evaluated by a spectralanalysis (fig.. 15) shows the structural resonances of the machi-ne. However, it does not allow the exciting frequencies to beidentified as the spectral lines are blurred because of their rota-tional speed dependency.

Quelldatei: up3DAutoZ.txt vom 19/02/04 Optionen: Real 3D Analyse: FFT Analyzer : FFT Spectrum Averaging : Autospectrum

0 2000

4000 6000

8000 10000

12000 14000

Hz 200 400

600 800

1000 1200

1400

Drehzahl

0

100

200

300

400

500

600

Re(m/s2)

Fig. 15. waterfall diagram of spectral analysis

Also, it is not possible to differentiate between structural reso-nance caused by the converter feed (harmonics) and structuralresonance originating from the machine construction (harmonicwaves).

A. Application of the order analysis

Incorporating the revolution speed measurement (induction sen-sor), the time base signals that are picked up by the accelerome-ter during idle run across the revolution speed range are used foran order analysis. With an appropriate data evaluation the mea-surement values that produced the waterfall diagram of fig.. 15by a spectral analysis can also be used to create the waterfalldiagram of the order analysis according to fig. 16.As the spectra are determined in dependency of the rotationalspeed, the blurring of the spectral analysis is prevented, allowi-ng the structural resonances of individual rotational speeds to

Quelldatei: up3DOrderAutoZ.txt vom - Optionen: Real 3D Analyse: Order Analyzer : FFT Spectrum Averaging : Autospectrum

0 50 100 150 200 250 300 350 400Order 200

400 600

800 1000

1200 1400

Drehzahl

0

100

200

300

400

500

600

Re(m/s2)

Fig. 16. waterfall diagram of order analysis

be identified as well as the exact frequency of the exciting vi-brations at the resonance points. Also, the theoretically deduceddifference of the curve forms of structural resonances caused byharmonic waves and harmonics is visible.

B. Evaluation with sound curves

A 2D illustration with the acceleration spectra being plottedagainst the rotational speed - each curve representing one order- allows the frequencies to be determined.

0

100

200

300

400

500

600

0 200 400 600 800 1000 1200 1400 1600

Re(

m/s

2 )

Drehzahl

Quelldatei: up3DOrderAutoZ.txt vom - Optionen: Real 2D Analyse: Order Analyzer : FFT Spectrum Averaging : Autospectrum

46.047.048.053.054.055.0

277.0278.0279.0280.0

Fig. 17. zoom of an order analysis (2D)

According to fig. 17, the structural resonance of the machinecaused by the frequency converter (harmonics) can be attributedto the ordersOrd = 278 andOrd = 279 at a rotational speed ofapprox.n= 1250min−1. This yields the exciting frequencies of:

fOS,1 =1250min−1 ·278Ord

60sec= 5791.7Hz

fOS,2 =1250min−1 ·279Ord

60sec= 5812.5Hz

For lower orders, structural resonances can be recognized for ro-tational speeds ofn= 750min−1 (Ord = 54) andn= 1350min−1

(Ord = 47). These resonances are harmonic waves with a cur-ve form of 1/Ord2. Incorporating the theory for sound equationcalculation and transformation into sound curves, the frequen-cies of these structural resonances can be determined as well asthe field interaction causing the resonance.Plotting the sound curves caused by field interactions into therotational speed-order plane of the waterfall representation ofthe order analysis results in a large number of possible excitati-ons of different pole pair numbers of rotor and stator fields. Witha appropriate selection of possible radial force orders (see [4]),this number can be considerably reduced.According to fig. 18, the excitation with the frequencyfOW,1 atthe rotational speed ofn = 750min−1 is caused by an excitati-on of the coil fields with rotor remainder fields (field interacti-on A). Excitations with the frequenciesfOW,2 and fOW,3 at thehigher rotational speeds ofn = 1300min−1 andn = 1350min−1

(according to fig. 19) can be attributed to saturation phenomena.

Quelldatei: up3DOrderAutoZ.txt vom - Optionen: Real 3D ton A-kurve zoomx Analyse: Order Analyzer : FFT Spectrum Averaging : Autospectrum

’A,b=-1,r=-6,nu+mu’’A,b=1,r=-2,nu+mu’’A,b=-2,r=4,nu+mu’’A,b=-2,r=-4,nu-mu’

’A,b=2,r=4,nu-mu’’A,b=-3,r=6,nu-mu’’A,b=3,r=2,nu+mu’’A,b=3,r=-6,nu-mu’’A,b=-4,r=0,nu+mu’’A,b=-5,r=2,nu-mu’’A,b=5,r=6,nu+mu’’A,b=5,r=-2,nu-mu’

’A,b=-6,r=-4,nu+mu’’A,b=6,r=-4,nu+mu’’A,b=-7,r=6,nu+mu’’A,b=-7,r=-2,nu-mu’

’A,b=7,r=2,nu-mu’

0 20

40 60

80 100

Order 200 400

600 800

1000 1200

1400

Drehzahl

0

100

200

300

400

500

600

Re(m/s2)

Fig. 18. waterfall diagram of order analysis with field interactionA sound cur-ves

Quelldatei: up3DOrderAutoZ.txt vom - Optionen: Real 3D ton B-kurve zoomx Analyse: Order Analyzer : FFT Spectrum Averaging : Autospectrum

’B,b=-1,r=2,nu+mu’’B,b=-1,r=-6,nu-mu’’B,b=1,r=6,nu+mu’’B,b=1,r=-2,nu-mu’’B,b=-2,r=4,nu-mu’’B,b=2,r=-4,nu+mu’

’B,b=-3,r=-2,nu+mu’’B,b=3,r=2,nu-mu’

’B,b=-4,r=0,nu-mu’’B,b=4,r=0,nu+mu’

’B,b=-5,r=-6,nu+mu’’B,b=5,r=6,nu-mu’

’B,b=-6,r=4,nu+mu’’B,b=-6,r=-4,nu-mu’’B,b=6,r=4,nu+mu’’B,b=6,r=-4,nu-mu’’B,b=-7,r=6,nu-mu’

0 20

40 60

80 100

Order 200 400

600 800

1000 1200

1400

Drehzahl

0

100

200

300

400

500

600

Re(m/s2)

Fig. 19. waterfall diagram of order analysis with field interactionB sound curves

The frequencies are calculated forfOW,1 with b= 6, r =−4 andν+ µ and for fOW,2 and fOW,3 with b =−7, r = 6 andν− µ:

fOW,1 =750min−1 ·54Ord

60sec−2·100Hz= 475Hz

fOW,2 =1300min−1 ·47Ord

60sec−2·100Hz= 813,3Hz

fOW,3 =1350min−1 ·47Ord

60sec−2·100Hz= 857,5Hz

C. Review of the measuring method

To illustrate the differences between conventional structure bor-ne noise analysis and the measuring method presented in thispaper - order analysis utilizing precalculated sound curves - thecalculated exciting frequencies causing structural resonances inthe machine are compiled in the following table:

critical speed notation frequency reason750min−1 fOW,1 475Hz fild harmonics (A)813min−1 fOW,2 813.3Hz field harmonics (B)858min−1 fOW,3 857.5Hz field harmonics (B)1250min−1 fOS,1 5791.7Hz harmonics1250min−1 fOS,2 5812.5Hz harmonics

TABLE III

RESULT OF MEASURING METHOD

These remarkably exact results are validated by the evaluationof the conventional structure borne noise analysis shown in thewaterfall diagram of the spectral analysis in fig. 15.As demonstrated in this paper, minor modifications of an exi-sting vibration measurement system allow an order analysis tobe performed.With the determined structural resonances run along the orderaxis the cause for the structural resonance can be determinedinstantanously as well as the respective frequency.Utilising precalculated sound curves, the cause for a resonancecan be attributed to a specific field interaction in the machinesair gap.

REFERENCES

[1] Jordan, H.Gerauscharme Elektromotoren, Verlag W. Girardet, Essen 1950[2] Seinsch, H. O.:Oberfelderscheinungen in Drehfeldmaschinen, B.G. Teub-

ner, Stuttgart, 1992[3] Oberretl, K.:Die Oberfeldtheorie des Kafigmotors unter Berucksichtigung

der durch die Ankerruckwirkung verursachten Statoroberstrome und derparallelen Wicklungszweige, Archiv fur Elektrotechnik, Band 49 (1965),Seiten 343 ff., Springer-Verlag Berlin, Heidelberg, New York

[4] Lach, R. Magnetische Gerauschemission umrichtergespeister Kafiglaufer-Asynchronmaschinen, Diss. Universitat Dortmund, 2004

[5] Lach, R. H.; Soter, S.Individual Approach to an Efficient Minimization ofVibration and Noise Stimulation for Inverter-Fed Drives, PCIM Conference,17.-19.03.2004, Shanghai, China

[6] Lach, R. H.; Soter, S.Sound source location measuring system for a syste-matic design modification of electrical machines, PCIM Conference, 25.-27.05.2004, Nuernberg, Germany

[7] Lach, R. H.; Soter, S.Experimental determination of systemeigenfrequen-cies on electrical dives, EPE, 02.-04.09.2004, Riga, Latvia

[8] Heimbrock, A.Analyse der Oberschwingungsverluste zweipoliger Indukti-onsmaschinen am Pulsumrichter, Dissertation, Universitat Hannover, 2003

measuring method for determining the reasons of

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