Magnetic noise of induction motors

IEEE Transactions on Power Apparatus and Systems, Vol. PAS-95, no. 1, JanuaryIFebruary 1976

MAGNETIC NOISE OF INDUCTION

John R. Brauer A. 0. Smith Corporation

Milwaukee, Wisconsin

ABSTRACT

A digital computer program is described which predicts the acoustic noise generated by the magnetic field in an induction motor. The motor air gap may have static or dynamic eccentricities, and the squirrel cage rotor may be skewed or unskewed. Besides calculating the sound power in decibels at any motor speed, other important performance parameters such as steady and harmonic (cogging) torques, magnetic side pulls, currents, and the variation of the magnetic field along the stack are computed. The sound power spectrum predicted for a particular motor is compared with laboratory measurements.

INTRODUCTION

Induction motors are used in many home and office appliances which must be quiet. The motor design must not only meet specifications on horsepower, cost, and input power, but also must not exceed the permissable acoustic noise level.

The generation of acoustic noise by electric machines has been under investigation for many years, as shown in a list of 167 papers.l Although these papers are very useful in understanding motor noise, they do not satisfy the need for a way of rapidly predicting the acoustic noise of a particular design. Digital computer programs which predict the output and input performance parameters of induction motors have been developed and are in extensive use. * However, such programs do not usually calculate the acoustic noise generated by motors. A computerized procedure was developed by Erdelyi that accurately predicted sound pressure levels of an induction motor, but it is applicable only to cettain medimsize polyphase motors.3

This paper describes a digital computer program which predicts the magnetic noise of induction motors. Considered first is the computation of air gap magnetic field harmonics. After the detailed magnetic field is known, the stator displacement and the resulting sound power can be computed, as well as the currents and the steady and harmonic torques and side pulls. The paper concludes with calculated results for a typical fractional horsepower motor and their comparison with sound power measurements.

COMPUTING THE AIR GAP MAGNETIC FIELD EARMONICS

Motor Equations

Figure 1 shows the cross section of a typical in-

Machinery Committee of the IEEE Power Engineering Society for presentation at Paper F 75 5694, recommended and approved by the IEEE Rotating

the IEEE PES Summer Meeting, San Francisco, Calif., July 20-25, 1975. Manu- script submitted January 27, 1975; made available for printing May 22, 1975.

Fig.1 Cross section of a squirrel cage induction mot01

duction motor with a skeued or unskewed squirrel cage rotor. The coordinate x is the angle in radians referenced to the stator, and a is the angular coordinate referenced to the rotor. For a constant motor speed 0

x = a + G t (1)

The air gap magnetic field B includes mutual, zig- zag, and skew fluxes, and is found from Ampere's Law applied to the loop A in Figure 1. The rotor and stator currents enclosed by the loop equal the sum of the "F drops along the loop in the air gap and in the rotor and stator cores and teeth x+h I (Ks+Krz>h = H(X,Z,t)g(X,Z,t)+Hts(X,Z,t)ts(X) X

Ks is the stator current density (ampshadian) and &. is the z component of the rotor current density. The effective air gap g between stator and rotor is defined later. Assuming constant permeabilities v (no saturation)

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where B is the air gap magnetic field. Substituting Equations 3-4 in Equation 2 gives

B(x,z,t)=U r [-j(Ks+Krz)dx

The effective air gap g is a function of the stator and rotor tooth shapes, and differs from the geometric air gap because of fringing. Assuming that the rotor slots have negligible effect on B, conformal equations developed by Freeman4 describe the effective gap reciprocal (l/g(x)) as a function of stator tooth and slot widths. Then Equation 6 is approximated as

where Ls is the radial length of a stator tooth.

The air gap reciprocal is affected by any imperfection in the mounting of the rotor. Two types of imperfections are considered here. The first is where the rotor is spinning exactly about its geometric center, but is mounted such that its axis is displaced a constant distance E from the axis of the stator bore center. This static eccentricity of the rotor causes the air gap reciprocal to vary as

rg = ~/(G-Ecos(x-~)) (8)

where 8 is the angle at which the stator and rotor are closest. This function introduces spatial harmonics in the air gap reciprocal and also increases the average value of the reciprocal.

The second possible imperfection is that the rotor spin center differs from its geometric center. The axes of these two centers are assumed parallel and separated by a distance A. Hence

rg = l/(G-Acosx) (9)

gives the coefficients of the resulting gap reciprocal harmonics in a manner as for the static eccentricity. However, this dynamic eccentricity has harmonics with frequency proportional to rotor RPM.

The total gap reciprocal has an average value and harmonics which are the sum of all the above values from the slots and ecaentricities.

The stator current density Ks in Equation 5 equals the product of winding current times winding turn density ck(x) for all windings

KS is the derivative of the commonly known stator MMP waveform. Since The MMF is a series of step functions, Ck(x) is a series of delta functions, one for each slot occupied by the winding. The Fourier series of a delta function is well known, and thus the harmonics of &(x) are easi,ly calculated for each winding. Typically the largest harmonics are those with order equal to the number of pole pairs, the fifth and sevmth harmonics of these pole pairs, and the pole pairs plus or minus the number of stator slots.

The current Ik in a stator winding equals the applied winding voltage V u plus the induced winding voltage VIk divided by the winding self impedance:

where NB is the number of parallel branches of the winding, RB is the branch resistance, LB is the slot and end turn leakage inductance of the branch, and CB is the capacitance (if any). Calculation of VIk from the air gap magnetic field B is detailed elsewhere.5

The rotor current density Krz in Equation 5 re- ferred to the rotor coordinates is the z component of

where RR is dynamic squirrel cage rotor resistance, including rotor bar resistance as a function of frequency w (skin effect)6 and rotor end ring resistance as a function of the space harmonic (I and frequency, LR is dynamic rotor inductance, including bar inductance as a function of frequency and end ring inductance as a function of 0 , and FR(~) is spatial distri- bution of rotor bars on rotor surface, where the voltage induced in,a rotor bar is from Faraday's L a w

where Y is the skew angle (looking down on rotor stack), u is z/cosY, r) is a-(usin Y)/R, where R is the rotor radius, and L' is L/(2cos Y), where L is the stack length.

Block Diagram Solution

A solution for the air gap field B is sought in terms of the line voltage. Since more B harmonics than the fundamental will be considered it is useful to express the previously developed motor equations in matrix form. To simplify development the 1-4 case is shown here. From Equations 10 and 11 the stator current density is

[KSI = [Til [VMI + [T21 [ V I (14)

The net main and phase voltages, [VM], [ W ] , defined previously5 in terms of line voltage [VL] and field [B], become,

[VMI = [ V L I + [T71 [Bl ; [VI = [ n l + [T81 [Bl (15)

The field [B] is related to stator current density by Equation 5 , which becomes,

(1 - [T31 ([T41 + [T61)) [ B l = IT31 [Ksl (16)

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I p F x T - l 3 / f o r e f f e c t i v e

OUTPUT B(x,z , t ) Ov3

<

I I

I is an iden t i ty ma t r ix . Combining Equations 14, 15 wi th 16 l eads to the des i red so lu t ion for [B] in a single-phase motor,

(1 - [T31 ([TI] [T71 + IT21 [T81 + IT41 + [T61)) [ B l

= IT31 ([TU + [T21) [a1 (17)

The c o e f f i c i e n t matrices [ T l ] through [T8]. w i l l b e considered as t ransfer mat r ices o f b locks 1 through 8 in the b lock d iagram of Figure 2. Equation 16 is represented by blocks 3, 4, and 6 , . which have the mag- n e t i c f i e l d [B] as output and [ K , + %,] as input . Equations 1 2 and 1 3 f o r qz are represented by block 6. Equation 14 for [KS] is represented by blocks 1, 2, ?A, 7B, and 8B in t h e I-@ motor diagram of Fig- ure 2. I f t h e motor is polyphase, then a block is added p a r a l l e l t o b l o c k s 1 and 2 , and a block is added p a r a l l e l t o b l o c k s 7B and 8B.

The input and output to each block is represented as a Four ie r series i n time and space

v = z t i e x p ( j ( k t - - + bz)) I (18)

where A is t h e complex Four ie r ampl i tude coef f ic ien t , w is angular frequency, a i s the space harmonic, o rder , o r number of pole pairs of wave, and 2 is t he skew c o e f f i c i e n t . The argument port ion of V, which contains w, a, and Z, is ca l led the opera t ing vec tor . There are 3 opera t ing vec tors in the block diagram. OV1 is t he ne t vo l t age ac t ing on the s t a to r w ind ings , OV2 is t h e s t a t o r and ro tor cur ren t dens i ty , and OV3 is the magnet ic f ie ld in t he a i r gap.

m

The block diagram solution proceeds by i n i t i -

a l i z i n g a l ~ i c o e f f i c i e n t s t o z e r o . One c o e f f i c i e n t is then set t o 1 . 0 and the ope ra t ing vec to r i npu t t o a block. The output vec tor ob ta ined represents the t r ans fe r func t ion fo r that block . Repea t ing th i s for each coe f f i c i en t i n t u rn l eads t o t he r equ i r ed t r ans - fe r mat r ices . These matrices are e d i t e d t o retain only a cons i s t en t set of the largest harmonic coef- f i c i e n t s .

Generation of the opera t ing vec tors and t r a n s f e r matrices is i l l u s t r a t e d by the fol lowing example. F i r s t , qssume a 60 Bz appl ied vol tage, so O V 1 = (377t + Ox + 02). The output of blocks 1 and 2 , OV2, c o n s i s t s of output harmonics of largest ampli tude. A 6 po le 1-4 motor could have

ov2 - 377t f 3x + oz

377t f 9x + oz

OV2 inputs block 3, which outputs OV3. For a 24 s l o t s t a t o r ,

377t + 27x + 0z OV3 = Equation 19 +

377t - 21x + 02 Next OV3 is app l i ed t o b locks 4 and 6. Since

block 4 gives the "F d r o p i n t h e steel cores i t gives an OV2 equal to Equat ion 20. Block 6, however, outputs rotor current harmonics which add new compo- nen t s t o OV2. I f t h e r o t o r is skewed, a l l the output operat ing vectors f rom block 6 must have nonzero Z. Within block 6 the x coordinate is t r a n s f e r r e d t o t h e skewed ro to r coo rd ina te I), and the induced rotor vo l t age is divided by t h e r o t o r impedance to give a r o t o r c u r r e n t i n amps. This cur ren t is then t rans- ferred back t o s ta tor coord ina tes , g iv ing 60 Hz har-

Fig.2 Block diagram of 1-$ o r PSC induction motor. The symbols in each b lock i nd ica t e t he ope ra t ion it performs. Subscript -in winding, prphase winding.

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monics which were induced by and oppose the stator current. New harmonics are also introduced having frequencies

w+i 2n[60 (1-S) (iQ/%) 601 (21)

where Q is the number of rotor slots and Np the number of pole pairs. These frequencies of the magnetic field lead to pressure wave frequencies f 60 Hz higher or lover, which as shown further on, become noise frequencies. If the induced rotor current had no spatial harmonics (as for a cylindrical rotor conductor), no Equation (21) frequencies exist.

For a rotor with Q = 36 bars the block 6 output is r 377t f 3x + Zk3Z

377t f 9x + Zk9Z

377t + 27x + 2272 377t - 21x + z-21z w+lt + 39x + Z3# OV2 = Equation 20 + (22)

w-lt - 33x + z-33z W+2t - 9x + ZmgZ

ue2t + 15x + Z15z

This new OV2 inputs block 3, giving

OV3 = Equation 22 + 377t + 27x + '27' (23) 377t - 21x + z-21z

The new OV3 is input to blocks 4 and 6 and the resulting OV2 is used to generate a newer OV3, etci Loop block 4 - block 6 - block 3 can be circled over and over again, generating several new combinations of u 's , U ' S and 2's. The more times the loop is circled the higher both computing cost and accuracy become. At present the computer program circles the loop 3 times and generates 68 harmonics. After OV3 is up- dated for the last time, the generation of transfer matrices [Tl], [T2], [T3], [T4], and [T6] is com- pleted. OV3 is then input to blocks 7A, 7B and 8B, obtaining transfer matrices [T7] and [Tal. The B coefficients are then found directly from Equation 17 using a standard simultaneous equation solver.

PRESSURE WAVES AND SIDE PULL

Acoustic noise and side pull are produced by the magnetic pressure attracting the stator to the rotor. This pressure is proportional to the square of the air gap magnetic field

where hn is a complex magnetic field amplitude. The P wave arguments are sums and differences of the B wave arguments. Hence the pressure waves include frequencies of 0, 120 Hz, and the slot frequencies

f = 60n(l-s)

where n=O, 1, 2, 3, etc. The orders a of the P waves

are sums and differences of the B wave orders. In single phase motors there are forward and backward main B waves which produce P waves of order zero. In any induction motor P wave orders of -1.1 and/or f3 etc. may be produced. Pressure wave patterns of various orders are shown in Figure 3.

Fig.3 Pressure and displacement patterns for modes 0-6

The skew values of the P waves are the sums and differences of the B wave skews. Often P waves with identical frequency and order, but different skews, will occur. These waves are combined into one wave by adding the two P waves at 5 points evenly spaced along the z axis. The net P wave is assumed to be unskewed with magnitude equal to the RMS value of the 5 sums. The phase angle of the net P wave is assumed equal to the phase angle of the sum at the center of the stack.

Next forward P waves (a > 0) and backward P waves (U < 0) with the same absolute value of u may be combined. It will develop that the acoustic power produced by a P wave is proportional to the square of its amplitude. Thus it can be shown that a pair of forward and backward P waves with same 1 ~ 1 > 1 can be considered as just one P wave of a m p l i t u d e d w where Pf, Pb are forward, backward amplitudes.

First order (0~1) pressure waves produce one sided magnetic pull between stator and rotor. Con- sider forward and backward first order P waves

The side pull is in the KO direction and equals 2n

F1 = Rz, 1 dx[P.ps(wt-x) + PbCOS(Wti%)]COSX 0

STATOR DISPLACEMENT AND NOISE RADIATION

Each pressure wave causes radial displacement along the stator surface with the same pattern as the pressure wave in Figure 3. The amplitude of the dis-

69

placement is propor t iona l to the ampl i tude T of the pressure wave. For o rde r d of value 2 o r more t h e displacement is7

3 p D4 E 4(1-a2)2 E h 3 $0

where D i s t h e s t a t o r c o r e mean diameter, h t h e sta- t o r c o r e mean thickness , and E t h e s t a t o r c o r e mean modulus of e l a s t i c i t y . $ accounts for the increased displacement due to mechanical resonance, and i s

where n is t h e r a t i o of t h e P wave frequency and 0 t h order s ta tor resonant f requency , and C is t h e s t a t o r core mean damping constant .

The s ta tor resonant f requency is ca lcu la ted us ing equat ions for a r ing having a mass that inc ludes the windings. The inner diameter i s assumed approximately equal to the bot tom of t h e s t a t o r t e e t h . Res- onant f requencies for orders 2 o r more are found using a fdrmula derived elsewhere.8 For m0 the resonant frequency i s

where aG is t he acce le ra t ion of g rav i ty and P is t h e r ing dens i ty .

S ide pu l l c aused by f i r s t o rde r p re s su re waves t ends t o move t h e s t a t o r c o r e and ro to r co re w i th re- spec t t o each o the r as r igid bodies . This motion can occur because of f ini te bear ing or other c learances and because the ro tor shaf t is n o t p e r f e c t l y r i g i d . Knowledge of s i d e p u l l is necessary for optimum design of t h e s h a f t . It might appear tha t the ro tor would move while t h e s t a t o r is he ld f i xed i n p l ace by i t s mountings. However, most s t a t o r mounts are f a r from r ig id . Recogniz ing th i s , the IEEE has s p e c i f i e d t h a t noise measurements on e l ec t r i c mo to r s are t o be made w i t h t h e s t a t o r r e s i l i e n t l y mounted.9 Hence r e s i l i e n t mounting is asslrmed i n t h e n o i s e c a l c u l a t i o n s .

The f i r s t o r d e r s t a t o r d e f l e c t i o n 6 is ca lcu la t ed assuming each bearing i s a spr ing wi th spr ing cons tan t k. Thus the motor cons is t s o f s ta tor and ro to r w i th a sp r ing in between as shown i n Figure 3. The r o t o r s h a f t is cons ide red pe r f ec t ly r i g id he re , so t h a t a l l the re la t ive mot ion takes p lace in the bear ings . Then t h e s t a t o r d e f l e c t i o n is

6 = F1

where m, = s t a t o r mass, mr - r o t o r mass. For k # 0 the denominator will be zero when w is the resonant frequency. The t o t a l s t a t o r and rotor displacement cannot exceed the bearing clearance y, so 6 cannot exceed

(33)

Knowing the v ib ra to ry de f l ec t ions 6 of a l l s t a t o r

modes, t h e sound radiated can be computed. The sound power i n d e c i b e l s , r e f e r e n c e d t o Odb * watt, is calculated using formulas der ived by E l l i son and Moore.1° They assume that a motor with a c y l i n d r i c a l s t a t o r may be approximated as a sphere with a vi- b r a t i n g band. The sound power r ad ia t ed i s a funct ion of the frequency and order of t h e s t a t o r d e f l e c t i o n as shown i n F i g u r e 4 f o r a typical motor. a0

60

40

20

0

-20

-40

rSound power (dB)

f o r motor with diameter =14cm and RMS s t a t o r def lect ion= 1 microinch= 2.54E-~CUI.

/ t 1 I

100 200 400 700 1 K 2K 4K 7K

Frequency (Hz)

Fig.4 Relat ive sound power r a d i a t i o n vs. frequency

TORQUE AND CURRENT CALCULATIONS

Ampere’s fo rce law is used to ob ta in t he t o rque

where [I&] = [T6] [B]. The integrand is

(34)

Only when a, = f an w i l l a T component be produced. The frequencies of the torque components w i l l e q u a l the f requencies o f the zero o rder p ressure waves. The zero frequency torque is the s teady or DC torque. The AC torque components are called harmonic or, a t low speeds, cogging torques. Harmonic torques can some- t imes produce significant structure-borne noise, which i s no t cons ide red i n t h i s pape r .

The s t a t o r c u r r e n t s i n t h e motor are computed from Equation 11 wherein the induced voltage VI i s [T7][B].

TPPICAL RESULTS

The computer program described here has as input the lamination dimensions and winding da ta of a motor and uses the procedures developed above to predict to- t a l performance. Experience has s h m t h a t its pre- d i c t i o n s of s t a t o r c u r r e n t and steady torques compare

70

Elose ly wi th those o f convent iona l equiva len t c i rcu i t Eomputer programs.

The p r e d i c t e d s i d e p u l l s and no i se are displayed Ln Table 1 f o r a 5 Hp 2 pole 3-0 motor with 24 s t a t o r s l o t s and 34 unskewed r o t o r b a r s , o p e r a t i n g a t i t s rated speed of 3512 RPM with a s t a t i c e c c e n t r i c i t y E = .003" (.0076 an). The eccen t r i c i ty p roduces s ide p u l l s of frequency 0 (steady pull), 120, 3860, 3980, and 4100 Hz. The p r e d i c t e d p u l l s i n c l u d e t h e e f f e c t s of ro tor cur ren t , and therefore d i f fe r f rom the pu l l values obtained by u s u a l t h e o r i e s . l l

The zero o rder p ressure waves of Table 1 r a d i a t e sound power less than OdB. The 4100 Hz f i r s t o r d e r wave r a d i a t e s 25 dB. Order 2 pressure waves produce s i g n i f i c a n t n o i s e a t 120 Hz and a t 4100 Hz, where a loud 53.79 dB sound occurs. Two order 3 pressure waves produce 15 dB and 16 dB. An order 4 p ressure wave g ives 38 dB at 3980 Hz. Orders 5 and above produce ins igni f icant no ise . The o v e r a l l sound power over the audible f requency range is pred ic t ed t o be 54 dB.

Table 2 shows r e s u l t s f o r a 0.5 Bp 2 po le 1-0 motor with 24 s t a t o r s l o t s , 34 skewed r o t o r b a r s , opera t ing a t 3544 RPM with no eccen t r i c i ty . Note the zero o rder 120 Hz pressure wave which produces 24.4 dB. The rest of the no ise is generated by order 2 waves, which a re i nhe ren t in 2 pole motors, and which radiate sound wel l as shown in Figure 4. The

o v e r a l l sound power is 43.3 dB.

I f t h e sound power must be less than 43.3 dB, then var ious design changes can be examined wi th t he computer model. The motor of Table 2 has t h ree con- c e n t r i c main c o i l s . I f f o u r main c o i l s are used (reducing the s ta tor harmonics) the predicted over- a l l sound power is reduced t o 41.6 dB. However, i f t h e number of ro tor bars is changed t o 29, the over- a l l sound power is increased to 54 .3 dB due t o i n t r o - duc t ion o f f i r s t o rde r p re s su re waves.

The predic ted no ise spec t rum for a 0.35 HP ca- paci tor-run 6 pole motor a t id l e speed is displayed as t h e v e r t i c a l l i n e s in Figure 5. The motor has 24 s t a t o r s l o t s , 36 r o t o r b a r s , a permanently connected 5pf. capacitor, and is assumed t o have a s t a t i c r o t o r e c c e n t r i c i t y E = 0.002" (.05b) and a dynamic e c c e n t r i c i t y A = 0.001'' ( .025m). The noise at f requencies 594 Hz, 714 Hz, and 834 Hz is almost e n t i r e l y due t o f i r s t o r d e r p r e s s u r e waves and es- s e n t i a l l y d i s a p p e a r s i f E and A are zero . The no i se a t 120 Hz, 1428 Hz, and 1548 Hz is due t o z e r o o r d e r pressure waves and v a r i e s l i t t l e wi th eccent r ic i ty . The c a l c u l a t e d f i r s t o r d e r n o i s e h a s t h e e f f e c t s o f t h e f l e x i b l e r o t o r s h a f t i n c l u d e d by using Equation 32 wi th k obtained by a s e p a r a t e s t r u c t u r a l a n a l y - sis of the shaf t . The ca lcu la ted shaf t resonant f re - quency is 1080 Hz.

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OR DE R FREQUFNCV QMS PRFSS NT/W PK OEFL INCH SOUND POWER DE e .. c.0 . .- - .- 0.16595E 0 6 . . 0.79558E-05 . 0 1 2 0 . ~ 0 32679. 0.15671E-Q5

.. _. - . - - - 24.377

2 2 4136.5

. -. - . . . . 4n16.5

2 2 4 4

4 4 4 4 4 4 4 4

..4

5904 8 6024.8. 0.9

120.00 1889.3. __ 2008.3 2128.3

4016.5 1A96.5

4136.5 5904.8 6024.8 6144.8

5.6676 _ _ 2.3274

0.74978E-C2 0 5 R 72 ?E-.@? .

19191. 16445.

. . 146-8? 86.672 140.25 6.7537 26.369 17.121

C m 17326 0.1772P 0.34407E-01

- - . . . . TOTAL -SOUND P O W E R - I S 43.27 DB

MEASURED MOTOR NOISE . .

The acoustic noise of an induction motor was measured for correlation with the theory and calculations of the previous sections. The motor selected was the 0 . 3 5 HF' motor of Figure 5 operating at idle speed.

The teat motor had rotor eccentricities the same as those of the calculations of Figure 5. It was sus- pended from a resilient shock cord near the center of a reverberation r o m at Riverbank Acoustical Labora- tories, Geneva, Illinois. The facilities and measure- mept technique were in conformity with ANSI $1.2-1962.

The sound power was measured in frequency bands of one third octave and is indicated by the horizontal llnes in Figure 5. Good agreement with the predicted spectrpn is noted in all bands except at 120 Hz and 1000 Hz. The discrepancy at 1000 Hz is believed due to excitation of shaft resonance (at the previously calculated 1080 Hz) by a saturation harmonic of frequency (834 + 2 4 0 ) He. The discrepancy at 120 Hz might be due to saturation. The over41 measured sound power i s 36 dB, the same as the overall calculated sound power.

. .

. ..

40

30

20

10

0

1

006R673F-07 38.393 0 3 73 07E-07 33.682 0 30351E-08 17.102 0.27219F-08, . . . 16.410 0.10e67~-08 8.0173 0.11729E-11 -47.650 6.874R4E-12 -50.025 0 e4726lE-06 0.40504E-06 -98.000

..0.37026E-08~- 0.20401E-08 0.35584E-08 n. 1 A448E-09 0072523E-09 0047425E-09 0.55126E-11 C.57039E-11 O.ll219E-11

6.6746 -7.1831

4.7251 1.2455

-340811 -34.350 -48.330

TABLE 2.

Pressure wavea and sound power of 0.5 HP 1-6 motor.

Iound power (dB)

LEGEND:

f = calculated value

- 0 measured value

--a motor sound power is less than this measured value

-t

2d0

t

- 606 8bO iK

Frequency (Hz)

Fig.5 Sound power spectrum of 0.35 HP motor

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CONCLUSIONS 31 E. Erdelyi, "Predetermination of Sound Pressure Levels of Magnetic Noise of Polyphase Induction

A computer program has been developed which predicts the magnet ic noise of induct ion motors . The f i r s t p a r t o f t h e program calculates the harmonics of t h e air gap magnetic f ield. Knowledge of the f i e l d a l lows p red ic t ions o f s ide pu l l s and torques. The sound power spec t rum pred ic ted for B f rac t iona l horsepower motor wi th eccent r ic ro tor agrees well wi th the measured spectrum.

The program assumes tha t t he mo to r is r e s i l i e n t l y mounted and has a s imple cy l ind r i ca l frame. Where more complex frames o r mounting resonances are encountered t h e r e s u l t s o f t h e program may have to be modif ied by measurements on a s i m i l a r test motor or mounting base.

Work is underway t o i n c i u d e t h e e f f e c t s of satu- r a t i o n .

ACKNOWLEDGEHENTS

The author thanks Dr. R. Y. hodine, J. 8. R a z i e r and A. s. Bickham f o r t h e i r s u g g e s t i o n s and assist- ance.

[ l ] A. J. E l l i s o n and C. J. Moore, "Acoustic Noise and Vibrat ion of Rotat ing Electr ic Machines", - IEE, Vol. 115, NO. 11, pp. 1633-1640, NOV. 1968.

[2] K. J. W a l d s c h i d t , "A Computer Procedure for Single-phase Induction Motor Calculation and Design", IEEE Trans. on Power Apparatus and Sys- L_ tm, Vol. 82, pp. 867-875, Dee. 1963.

Motors" AIEE Trans. (Power Apparatus &nd Systems) , Vol. 74, pp. 1269-1280, Dee. 1955.

(41 E. M. Freeman, "The Calculat ion of Harmonics, Due t o S l o t t i n g , i n the Flux Density Wavefofm of a Dpnamo-Ele&ic Machine", Proc. I&, Vol. 109, No. 6, pp. 581-588, June 1962.

[5] J. R. Brauer, "Saturation Harmonics &nd Current Waveforms of Single-Phase Induction Motors", Trans. Power Apparatus and Systems, Vol. 93, No. 1, pp. 40-44, Jan./Feb. 1974.

[6] M. M. Liwschitz-Garik, "Skin-Effect Bars of Squirrel-Cage Rotors" AIEE Trans. (Power Apparatus aitd Spstem8), Vol. 74, pp. 255-258, August 1958.

[7] P. L. Alger, Induction Machines, 2nd Ed i t ion , New York: Gordon and Breaeh, 1970, p. 369.

[8] B. S. Se ide l and E. A. Erdelyi , "On the Vibra t ion of a Thick Ring In Its Plane'', Trans. "e, Vol. 86, P a r t B , pp. 240-244, August 1964.

[9] "Test Procedure for Airborne Noise Measurements on Rota t ing Elec t r ic Machinery", IEEE No. 85, Feb. 1965, p. 7.

[lO]A. J. E l l i s o n and C. J. Moore, "Calculation of Acoustic Power Radiated by Shor t E lec t r i c Ma- chines", Acubtica, Vol. 21, pp. 10-15, 1969.

I111M. Bradford, "Unbalanced Magnetic P u l l in a 6 Pole - - Induction &tor", Proc. IEE; Vol. 115, No. 11, pp. 1619-1627, NOV. 1968.

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Discussion S. P. Venna and R. S. Girgis (University of Saskatchewan, Saskatoon, Sask,Canada): We would like to congratulate the author for attempting to cover all the aspects of magnetic noise of induction machines in a single paper. After having gone through the paper carefully, we observe that the author has confined his efforts to develop a computer program for the calculation of magnetic noise as applied to mainly small and FHP motors. The analytical information adopted in the paper is considerably simplified and therefore it may not deliver accurate results when applied to induction motors in general.

play a key role in the problem of magnetic noise. Therefore, an accurate Both resonance frequencies of a stator and the exciting forces

determination of the values of resonance frequencies as well as the values of exciting frequencies is essential. T h e expressions used by the author for the calculation of resonance frequencies are very ap- proximate and would certainly deliver erroneous results. In order to demonstrate this, a sample of calculated and measured values of resonance frequencies of three different stators are given in the following table for the “zero mode”.

model is not adequate. An investigator should test his method of calculation on as many experimental models, having different constructions and power capacities, as possible [3 ,4] . Since magnetic noise. is always accompanied by bearing and ventilation noise components in actual practice, the discussors feel that a comparison between the calculated magnetic noise and the measured noise does not lead to any verification. Incorporating information about the ventilation and bearing noises in the computer program would, therefore, be very beneficial.

The paper presents a digital computer program, but does not provide any information about the features of the program; e.g. size, cost of each run, computation difficulties, and limitations.

Finally, the discussors would like to draw the attention of the author that presenting the negative values of sound power levels (eg. - 147 db) in Tables (1 & 2) could have been avoided as they do not bear acoustical significance.

REFERENCES

[1 I Frohne, H., “On the Main Parameters Which Determine the Noise Level of Asynchronous Machines” (in German), Doctoral

Experimental Stator No.

2009[21 6170[11 6620[13 Measured Value of Frequency

3 2 1

* Equation (31)

2013 6350 6870 Reference [3]

1890 6190 6630 Reference [l] Calculated Value

1885 8810 8590

L

[*Equation (31) is believed to be wrongly typed, it should take the form [21

131

The above table shows clearly that equation (31) of the paper is erroneous even for model No. 3, in which teeth are absent. Similar [4] results, or even worse, are expected for modes other than the “zero mode”. This is due to the fact that the analysis of reference [8] of the paper is only applicable for smooth single rings without teeth, windings or frame. Actually, stators of electrical machines are much more [5] complicated than a single smooth ring. Moreover, it has already been established by investigators in the field of noise and vibrations that frames of electrical machines play a very important role in the problem [6] of noise [4-71 and that teeth have a major effect on values of resonance frequencies of stators [ 1 ] .

In considering the mechanical response of stators, the paper takes into account only the lowest resonance frequency for each circum- [7] ferential mode of vibration. All other higher frequencies [3, 81 and those created by the presence of the frame 15, 71 are not taken into . account. Accordingly, the program presented in the paper may well [8] deliver false values of noise levels not. only for medium and large sized machines, but also for small machines. The discussors would, therefore,.suggest that more accurate analyses be incorporated in the com- [9] puter program, which would then deliver more realistic values for the resonance frequencies. [ lo1

quency [9] or slip - & double slip-frequency [ 101 magnetic vibrations In his program, the author did not account for the line - fre-

created by eccentricities of rotors. Line-frequency vibrations can be of

machines where the lowest resonance frequency of the stator may well significant magnitudes in the case of less stiff rotors and also in large

lie in the neighbourhood of the 60 Hz line-frequency. Slip- & double slipfrequency vibrations can be responsible for loud audible noise at starting and troublesome structure-borne noise at rated loading con- ditions. Also, the axial vibrations [ 11 ] due to skewing are important and should be considered. In the presence of asymmetrical connections, pulsating torques are produced [ 121 which create torsional vibrations. Both axial and torsional vibrations excite significant radial vibrations

Thesis, Technical University of Hannover, W. Germany, 1959. Bolderl, P., “On the Determination of Vibrations of the Stator of Turbogenerators, Produced by Air-Gap Fields” (in German), Doctoral Thesis, Technical University of Hannover, W. Germany, 1965. Verma, S. P. and Girgis, R. S., “Resonance Frequencies of Electrical Machine Stators Having Encased Construction, Part 11: Numerical Results and Experimental Verification”, IEEE Transactions PAS,Vol. 92,No. 5, Sept./Oct. 1973, pp. 1586-93. Verma, S. P. and Girgis, R. S., “Resonance Frequencies of Electrical Machine Stators Having Encased Construotion, Part I: Derivation of the General Frequency Equation”, IEEE Transactions PAS,Vol. 92,No. 5 , Sept./Oct. 1973, pp. 1577-85. Ellison, A. and Yang, S., “Natural Frequencies of Stators of SmallElectricMachines”.Roc. IEE.Vo1. 118. No. 1. Jan. 1971, pp. 185-190. Jordan, H., Purkermani, M., Raube, W. and Roder, G., “The Effect of a Thin Frame on the Magnetic Noise of Polyphase Machines” (in German), Larmbekampfung, August 1970, pp.

Erdelyi, E. and Horvay, G., “Vibration Modes of Stators of Induction Motors”, A.S.M.E. Trans., Vol. 24, [E], 1957, pp.39 45. Hohmann, F., “Resonance Frequencies of StatorCores of Polyphase Machines by Considering the Vibration Behaviour of Teeth”, ETZ-A,Bd. 93, 1972,pp. 82-85. Robinson, R. C., “Line-Frequency Magnetic Vibrations of A.C. Machines”, A.I.E.E., Trans., Vol. 82, Part 111, 1963, pp. 675-79. Summers, E. W., “Vibrations in 2-pole Induction Motors Related to SliD-Freauencv”. A.I.E.E. Trans., Vol. 74. Part 111,

81-87.

~~ . - I

1955, pp.69-75. Subramanyan, V., “Axial Forces in Induction Motors with Skewed Slots”, Proc. I.E.E., Vol. 118, No. 12, Dec. 1971, pp. 175940. Verma, S. P., “TorqueCalculations of Polyphase Asynchronous Machines with Asymmetrical Connections”, Acta Technica CSAV, NO. 1, 1971, pp. 1-21.

of certain frequencies. In the light of the above discussions, the dis- [3] which may give rise to a new set of pure tones or add to the noise

crepancies between the computed and the experimental values as John R. Brauer: - I thank the discussers for their interest in this paper reported in Figure (5) may be easily explained. and for the extensive discussion they have written.

Small machines are generally not suitable for checking the validity The discussers contend that the computer program is not applicable of magnetic noise theories. Also, testing such theories on only one to medium or large induction motors because of the assumptions made

Manuscript received August 12,1975. Manuscript received September 25,1975.

74

in the paper on the mechanical response of the stator. They are correct only to the extent that large induction motors often have frames which are more complicated than a simple cylinder. This is noted in the con-

induction motors do have the simple cylindrical frame assumed by the clusions of the paper. Most medium, small, fractional and sub-fractional

program. Also, medium sized and smaller motors have dimensions such that tooth resonances and nonfundamental resonant frequencies can be neglected [Reference 3 of the paper J . for the resonant frequency of the zero order mode. Unfortunately the

I thank the discussers for pointing out the typing error in Eq. 31

paper’s length prevented complete definition of the ring used to represent the vibrating stator. The density p of the ring is calculated by concentrating the total stator mass into the ring of thickness h equal to the yoke thickness plus one-tenth of the tooth length t,. The effective diameter D is calculated as (outer diameter - h) + 0.9. Using these defiiitions in Eq. 31, the program calculates 6347 Hz for stator No. 1 and 6094 Hz for stator No. 2. These values agree well with the measured

the frequency of stator No. 3 because it has no teeth. frequencies reported by the discussers. The program could not calculate

The program makes the usual assumption [reference 3 of the paper] that vibrational displacements are negligiile compared to the air gap, and thus line frequency vibration is not calculated. Slip and double slip frequency pulsations produced by dynamic rotor eccentricities are calculated by the program, contrary to the discusser’s statement. Also calculated by the program, but not mentioned in the paper (again due to length limitation), are the fundamental and harmonic axial forces of skewed motors.

are generally not suitable for checking the validity of magnetic noise I must disagree with the discussors’ statement that small machines

theories. In many ways small motors are easier to test than large ones. They are conveniently transported to and energized in acoustic labora- tories. They can easily be resiliently mounted in accordance with the

difficult with large machines. IEEE test procedure by suspending from a rubber cord, which is very

I agree that sound measurements on additional motors would be desirable, but accurate tests are expensive. In fact, one benefit of the computer program is avoiding the expense of building and testing a multiplicity of prototype motors.

Regarding the incorporation of bearing and ventilation noise in the program, the paper and the program titles state that they calculate only magnetic noise. If mechanical noise is significant, it can easily be added (logarithmically) to the predicted magnetic noise. In many motors, including the one tested in the paper, ventilation and bearing noise is negligible compared with the magnetic noise.

I am pleased to answer the discussers’ questions on the operation of the program. It is available on A. 0. Smith’s IBM 370/165 computer for commercial applications. Because several disk devices are used for working storage by the program, the region of core occupied is only 186,000 bytes (186K). The cost of a complete analysis at one motor speed is approximately $65. Program input consists of eight cards needed for calculations of basic performance [reference 2 of the paper] plus three cards giving eccentricities etc. needed for noise prediction. Input data is checked for errors by the program to avoid computational difficulties.

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Magnetic noise of induction motors