influence of manufacturing tolerances on the electromotive force in permanent-magnet motors

11
5522 IEEE TRANSACTIONS ON MAGNETICS, VOL. 49, NO. 11, NOVEMBER 2013 Inuence of Manufacturing Tolerances on the Electromotive Force in Permanent-Magnet Motors Vicente Simón-Sempere , Manuel Burgos-Payán , and José-Ramón Cerquides-Bueno Electrical Engineering Department, University of Seville, 41092 Seville, Spain Signal Theory and Communications Department, University of Seville, 41092 Seville, Spain In permanent-magnet synchronous motors it is necessary that the induced electromotive force (EMF) is as sinusoidal as possible to achieve a low torque ripple. The techniques used for this purpose require great precision in the positioning and the magnetization of the magnets, which renders them highly susceptible to errors resulting from manufacturing tolerances. This paper analyzes the effect on the EMF of four types of common production errors: magnetization level, position and width of the magnets, and angular displacement between blocks of magnets. To this end, two totally analytical statistical models have been developed: one for the rst harmonics and another for high frequencies, from which design rules for the minimization of the sensitivity to errors can be drawn. The techniques developed here allow the calculation of the deviations in the EMF spectra from the information on tolerances, and reciprocally estimate manufacturing errors from EMF spectra. The proposed techniques have been tested both numerically and experimentally showing good agreement with the developed models. Index Terms—EMF harmonics, magnetization error, manufacturing tolerances, permanent magnets, synchronous motor, torque ripple. I. INTRODUCTION I N the design of any industrial product, it is of prime impor- tance to be aware in advance of the possible differences be- tween the behavior of the designed object and the result after in- dustrialization. The origin of these differences tends to lie in the mechanical manufacturing tolerances and in the variations in the properties of the materials used. This paper studies the inuence from the errors of magnetization, of sizing and positioning of the magnets, and displacement of the blocks of magnets upon the induced electromotive force (EMF) in permanent-magnet syn- chronous motors (PMSM). These errors cause deformation of the spectrum with the consequent appearance of new harmonics, or even the reappearance of those harmonics that should have been explicitly cancelled in the design. Recent studies demon- strate the interest in discovering the origin of magnetization er- rors [1]–[3], and their effects [4], [5], although the latter are only focused on the cogging torque and not on the EMF. Although manufacturing tolerances also generate other sources of error, such as deformations of the stator, and ec- centricities between the stator and the rotor [6], this work is focused on those related to the rotor. In this paper, the errors are modeled by random variables using statistical methods to analyse their effects. To this end, two fully analytical error models have been developed. The rst is linear and describes how the different sources of error inuence the rst harmonics of EMF, wherein lies the practical interest in electric machines. A second nonlinear model enables the interpretation of the effect of imperfections on the high-frequency harmonics. The two models have been successfully tested both numerically and experimentally. For the experimental validation, over seventy motors taken randomly from a production line have been used. Manuscript received March 18, 2013; revised May 06, 2013; accepted June 05, 2013. Date of publication June 19, 2013; date of current version October 21, 2013. Corresponding author: M. Burgos-Payán (e-mail: [email protected]). Color versions of one or more of the gures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identier 10.1109/TMAG.2013.2269906 The results of the analysis can be applied to two elds: to that of design and that of quality control. For the rst eld, the method provides a comprehensive survey of the EMF sensitiv- ities with respect to the various sources of error, which renders it a valuable complement to the methods for the cancellation of EMF harmonics, especially in motors designed with a limited number of slots per pole and phase. In these cases, the stator variables allow little freedom for design, and hence it is neces- sary to resort to modifying certain parameters of the magnets, such as the pole arc, the spacing, and the shape thereof, among others, to achieve a more sinusoidal EMF [7]–[11]. The real effectiveness of these techniques however, depends ultimately on manufacturing tolerances and design sensitivity to these tol- erances. In the eld of quality control processes, the proposed linear model constitutes a valuable tool since it enables the sepa- rate quantication of the different sources of error based on the statistical analysis of the EMF spectrum. The information ob- tained can be used towards improving manufacturing processes and correcting errors in magnetization. II. PROPOSED METHOD A. Modeling the Errors The starting point considered here is a rotor model with sur- face-mounted or interior permanent magnets of the same rated size, consisting of axial blocks of magnets each, (Fig. 1). The nominal angular distribution of the magnets is considered to be identical in all blocks. As usual, it is assumed that both the positions of the magnets and of the blocks have even symmetry with respect to a certain point in the gap. The analysis focuses on the following types of errors: (i) deviations of the remanent ux density of each magnet; (ii) deviations in the positioning of the magnets; (iii) deviations of the nominal width of the magnets; (iv) deviations of the angle of each block. It must also be taken into account that each one of the consid- ered blocks of magnets are independent pieces which are assem- bled separately, which allows expecting that small positioning errors may occur when each block is placed individually. 0018-9464 © 2013 IEEE

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5522 IEEE TRANSACTIONS ON MAGNETICS, VOL. 49, NO. 11, NOVEMBER 2013

Influence of Manufacturing Tolerances on the Electromotive Force inPermanent-Magnet Motors

Vicente Simón-Sempere , Manuel Burgos-Payán , and José-Ramón Cerquides-Bueno

Electrical Engineering Department, University of Seville, 41092 Seville, SpainSignal Theory and Communications Department, University of Seville, 41092 Seville, Spain

In permanent-magnet synchronous motors it is necessary that the induced electromotive force (EMF) is as sinusoidal as possible toachieve a low torque ripple. The techniques used for this purpose require great precision in the positioning and the magnetization of themagnets, which renders them highly susceptible to errors resulting from manufacturing tolerances. This paper analyzes the effect onthe EMF of four types of common production errors: magnetization level, position and width of the magnets, and angular displacementbetween blocks of magnets. To this end, two totally analytical statistical models have been developed: one for the first harmonics andanother for high frequencies, from which design rules for the minimization of the sensitivity to errors can be drawn. The techniquesdeveloped here allow the calculation of the deviations in the EMF spectra from the information on tolerances, and reciprocally estimatemanufacturing errors from EMF spectra. The proposed techniques have been tested both numerically and experimentally showing goodagreement with the developed models.

Index Terms—EMF harmonics, magnetization error, manufacturing tolerances, permanent magnets, synchronous motor, torqueripple.

I. INTRODUCTION

I N the design of any industrial product, it is of prime impor-tance to be aware in advance of the possible differences be-

tween the behavior of the designed object and the result after in-dustrialization. The origin of these differences tends to lie in themechanical manufacturing tolerances and in the variations in theproperties of the materials used. This paper studies the influencefrom the errors of magnetization, of sizing and positioning of themagnets, and displacement of the blocks of magnets upon theinduced electromotive force (EMF) in permanent-magnet syn-chronous motors (PMSM). These errors cause deformation ofthe spectrumwith the consequent appearance of new harmonics,or even the reappearance of those harmonics that should havebeen explicitly cancelled in the design. Recent studies demon-strate the interest in discovering the origin of magnetization er-rors [1]–[3], and their effects [4], [5], although the latter are onlyfocused on the cogging torque and not on the EMF.Although manufacturing tolerances also generate other

sources of error, such as deformations of the stator, and ec-centricities between the stator and the rotor [6], this work isfocused on those related to the rotor. In this paper, the errorsare modeled by random variables using statistical methods toanalyse their effects. To this end, two fully analytical errormodels have been developed. The first is linear and describeshow the different sources of error influence the first harmonicsof EMF, wherein lies the practical interest in electric machines.A second nonlinear model enables the interpretation of theeffect of imperfections on the high-frequency harmonics. Thetwo models have been successfully tested both numerically andexperimentally. For the experimental validation, over seventymotors taken randomly from a production line have been used.

Manuscript received March 18, 2013; revised May 06, 2013; accepted June05, 2013. Date of publication June 19, 2013; date of current version October 21,2013. Corresponding author: M. Burgos-Payán (e-mail: [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TMAG.2013.2269906

The results of the analysis can be applied to two fields: tothat of design and that of quality control. For the first field, themethod provides a comprehensive survey of the EMF sensitiv-ities with respect to the various sources of error, which rendersit a valuable complement to the methods for the cancellation ofEMF harmonics, especially in motors designed with a limitednumber of slots per pole and phase. In these cases, the statorvariables allow little freedom for design, and hence it is neces-sary to resort to modifying certain parameters of the magnets,such as the pole arc, the spacing, and the shape thereof, amongothers, to achieve a more sinusoidal EMF [7]–[11]. The realeffectiveness of these techniques however, depends ultimatelyon manufacturing tolerances and design sensitivity to these tol-erances. In the field of quality control processes, the proposedlinear model constitutes a valuable tool since it enables the sepa-rate quantification of the different sources of error based on thestatistical analysis of the EMF spectrum. The information ob-tained can be used towards improving manufacturing processesand correcting errors in magnetization.

II. PROPOSED METHOD

A. Modeling the Errors

The starting point considered here is a rotor model with sur-face-mounted or interior permanent magnets of the same ratedsize, consisting of axial blocks of magnets each, (Fig. 1).The nominal angular distribution of the magnets is consideredto be identical in all blocks. As usual, it is assumed that both thepositions of the magnets and of the blocks have even symmetrywith respect to a certain point in the gap. The analysis focuseson the following types of errors:(i) deviations of the remanent flux density of each magnet;(ii) deviations in the positioning of the magnets;(iii) deviations of the nominal width of the magnets;(iv) deviations of the angle of each block.It must also be taken into account that each one of the consid-

ered blocks of magnets are independent pieces which are assem-bled separately, which allows expecting that small positioningerrors may occur when each block is placed individually.

0018-9464 © 2013 IEEE

SIMÓN-SEMPERE et al.: INFLUENCE OF MANUFACTURING TOLERANCES ON THE ELECTROMOTIVE FORCE IN PM MOTORS 5523

Fig. 1. (a) Drawing of a rotor with three blocks of six surface-mounted mag-nets, (b) air gap flux density wave, (c) distribution of the blocks and (d) nominalflux density wave of the poles.

Each of these four types of deviations or errors has been mod-eled as zero mean Gaussian random variables, as it is usual inthis type of analysis [2], [4].Fig. 1 shows the air-gap flux density wave for one of the

blocks of magnets and the variables used in the model.In this figure

(1)

where

normally distributed random variablewith 0 mean and variance ;

nominal angular position of the thmagnet inside this block and isits deviation from its nominal value inthe th block;

nominal angular position of the thblock and is its deviation;

nominal width of the poles and isthe relative deviation of the th poleof the th block;

nominal amplitude of the flux densityof the poles and is the relativedeviation of the th pole of the thblock.

As pointed out previously, the objective is to model the ef-fect of the variances of manufacturing errors represented by therandom variables, defined in (1), on the EMF spectrum.For a rotor consisting of blocks of equal axial length, the

induced voltage in a straight conductor is

(2)

where is the angular speed of the rotor, is the radius ofthe rotor, the axial length of the magnetic stake, theair-gap flux density created by the th block (Fig. 1(b)) andis the random variable which describes the angular displacementof said block in relation to the block of reference.The flux density wave of any of the blocks, , can be

expressed as the sum of successive displacements and rever-sals of the nominal flux density of a pole, , as shown inFig. 1(d), whereby at each position has different amplitude andwidth, characterized by the factors described in (1),and , respectively. Thereby

(3)

where is the number of poles or of magnets in a block andis the random variable describing the angular position of the

th magnet.As for the EMF of a section of the winding, , can be

obtained from the EMFs induced in the conductors located inthe slots:

(4)

where is the number of slots that form the section of thewinding and is the angular position of the th stator slot. Thenumber and polarity of the conductors located in each slot are in-dicated with the absolute value and the sign of , respectively.In the following study, it is considered that is aperiodic

and that it takes the value zero outside the interval . Thisapproach does not suppose any loss of generality and permitscontinuous spectra to be used instead of discrete spectra, whichis more appropriated in the analysis of sensitivities and in theinterpretation of results, as discussed below.Expressing (4) in terms of nominal positions and its devia-

tions, and applying the Fourier transform yields

(5)

5524 IEEE TRANSACTIONS ON MAGNETICS, VOL. 49, NO. 11, NOVEMBER 2013

where is the spatial frequency, the Fourier transformof , and the spectrum of the winding distribution

(6)

Although the EMF spectrum, , is a continuous function,it only holds any real meaning for integer values of .

is more easily understood if it is normalized withrespect to the magnitude of the first electric harmonic of fre-quency . The nominal magnitude of the first EMFharmonic can be obtained from (5) by neglecting the effectof errors and by considering that, at this frequency, EMF perpole is at a maximum. This situation happens when the unitaryvectors are all in phase, and the sum ofall these vectors is . Thus, the normalized spectrum is

(7)

The prime in , and indicates that they are normalizedspectra with respect to their first harmonics, respectively.

B. Analysis for Low Frequencies

For the low order harmonics, the following approximationsmay be suitable, due to the reduced value of .(i) The second exponential of (7) can be approximated by thefirst two terms of its Maclaurin series expansion:

(8)

(ii) The term can be approximated by thefirst two terms of its Taylor series expansion around ,yielding

(9)

(iii) Since the deviations of the width of the poles, , aregenerally smaller than magnetization tolerances, , itcan be assumed that

(10)

Substituting (8), (9) and (10) into (7), and neglecting thesecond-order errors, result in

(11)

where

(12)

(13)

(14)

(15)

(16)

is the normalized spectrum of the error-freeEMF and is the function of the positioning of themagnets:

(17)

defines the nominal position of each magnet in theblocks and the nominal position of each block. It isworth bearing in mind that the term also formspart of (16)., , and are the spectra owed to the

errors represented by the variables , , and , respectively.The sum of all these spectra is called

(18)

For the characterization of errors, the mean square value(MS), or second-order moment with respect to the originof , and the probability density function (PDF) of itsmodule should be obtained. The MS is a statistic of easy inter-pretation and algebraic manipulation, while the PDF allows thedetermination of the probability of occurrence of a particularvalue.Initially, the MS value of the errors is determined by means

of the following notation:

(19)

where is the expected value.The MS value of is obtained from (18), by taking into

account that the components of are independent fromeach other and of mean zero

(20)

The MS values of each of the terms that form part ofare now calculated.1) Deviations in the Remanent Flux Density of Magnets: The

MS value of the spectrum (13) owing to the errors in the fluxdensity is

(21)

SIMÓN-SEMPERE et al.: INFLUENCE OF MANUFACTURING TOLERANCES ON THE ELECTROMOTIVE FORCE IN PM MOTORS 5525

where and are the real and imagi-nary components of the double summation of (13), respectively

(22)

These equations can be expressed as a linear transformationfrom the random variables :

(23)

The matrix of the covariances of the variables and ,, can be obtained from the transformation matrix, , and

from the covariance matrix of the random variables :

(24)

If the random variables are independent from each otherand of equal variance, then

(25)

and therefore

(26)

The double summation of the covariances and is zerodue to the consideration of even symmetry in the positioning ofthe magnets previously mentioned. Furthermore, given that therandom variables have a null mean, ,therefore

(27)

By substitution of (27) into (21) and by taking (26) into ac-count, the MS value of the spectrum is obtained as

(28)

2) Deviations in the Positioning of the Magnets: Working inthe same way on of (14), the following expressions forthe covariances and for the MS are attained:

(29)

(30)

3) Deviations in the Width of the Magnets: Working on thespectrum of (15) in the same way as in 1) above, thefollowing expression is obtained for the MS value:

(31)

4) Deviations in the Positioning of the Blocks: The expan-sion is similar to that carried out in 2) above.

(32)

and for the MS value:

(33)

5) Overall Effect of the Errors: The total normalized MSvalue, , is obtained as the sum of the partial MSs (28),(30), (31) and (33), and yields

(34)

On the other hand, MS value of the total spectrum and MSof the error-free spectrum are obtained from (11) and (12),respectively

(35)

(36)

Fig. 2 shows the components that form the error-free EMFspectrum (36) in the upper section, and the trajectories of thedifferent sources of error (34) in the lower section. The effectof spectrum in the upper section and on the components ,

and as well as that of their derivative on can clearlybe observed.

5526 IEEE TRANSACTIONS ON MAGNETICS, VOL. 49, NO. 11, NOVEMBER 2013

Fig. 2. Contribution of the different sources of error to EMF spectrum in thelow-frequency range.

Fig. 3. Model for the normalized input error for harmonics of low frequency.

An alternative model for the errors is that shown in Fig. 3,where all the errors are added together at the same point.To this end, a new function has been introduced into

the trajectory of the component

(37)

In Fig. 3, acts on the same point of the system as doesin Fig. 2. Hence, can be considered as the error as-

sociated to the spectrum of the positioning of the magnets andalso as the normalized input error, which is indicated with thesubindex “ ” in its identifying variable and in its components.The normalized input error is therefore

(38)

6) Distribution of the Deviations: Once the MS value of theerrors has been obtained, it is interesting to determine the prob-ability that the module of the EMF of a determined harmonicexceeds a given value. The question can be approached by ob-taining the PDF of the module of a vector, , whose com-ponents are independent Gaussian random variables, of meanzero, but with different variances.Starting from a bivariate normal PDF in Cartesian coor-

dinates, , by converting to polar coordinates,, and by integrating with respect to in order to obtain

, the following expression is attained:

(39)

where is the modified Bessel function of the first kind andzeroth order.

Fig. 4. Probability density function and distribution function of the module, ,in terms of and .

TABLE IPROBABILITY OF OCCURRENCE FOR THE WORST CASE

The function takes the form of a Rayleigh PDF whenthe variances are equal, and the form of a half-normal PDFwhenone of the variances is zero. In (39), the probability of the oc-currence of a determined value of the module, , is obtainedfrom the variances and . It is straightforward to showthat the sum of these variances coincides with the MS value ofthe module:

(40)

where the mean values of and are considered to be zero.Fig. 4 shows the forms that the functions of density and of

distribution acquired for different values of and , for. As can be seen the spread of errors between the real

and imaginary components has little effect on the distributionfunction, thereby enabling a reference value to be taken in termsof the total error but not of its components.To this end, the worst-case scenario is taken from the enve-

lope of functions , contained in Table I.Table I shows that the probability that the value of

obtained from a single sample is greater than the MS value ob-tained from (34), is 0.368. It also shows that the probability thatthe value of is greater than the triple of the MS valueobtained from (34), is .

C. Behavior at High Frequency

For high values of the frequency, , the low-frequency modelbased on the linear approximation , is nolonger suitable. However, in this case, it is possible to considerthat the argument of the complex number behavesas a random variable uniformly distributed between and .Returning to the analysis in (7), the additional phase terms rep-resented by the power , and the factor

are, in this case, irrelevant. The effects of and of on

SIMÓN-SEMPERE et al.: INFLUENCE OF MANUFACTURING TOLERANCES ON THE ELECTROMOTIVE FORCE IN PM MOTORS 5527

the amplitude can also be considered as negligible in compar-ison with those that introduce as an argument of . In thisway, for high values of , the voltage spectrum can be expressedas

(41)

where is a random variable, with uniform distribution in theinterval .The MS value of is

(42)

The linearity property of the operator of the expected valueand the independency of the terms of the respectiverandom variables and , yields

(43)

Therefore, only those terms that verify contribute to-wards the summation. Hence

(44)

Furthermore,

(45)The introduction of the change of variables into

(45) and the consideration that , yields

(46)

The integral (46) can be interpreted as the MS value of thefunction multiplied by a Gaussian window function cen-tered at , whose width is dependent on the frequency, as shownin Fig. 5.If follows a sinusoidal law, its MS value then runs parallel

to its envelope, , with a value , as shown in Fig. 5.In this case, (44) takes the form

(47)

In order to compare this value with the MS value of the error-free spectrum at high frequency, a conventional configurationwith equidistant magnets and a small angular displacement be-tween the blocks has been considered. By making use of (17), it

Fig. 5. Interpretation of (46): Windowing of with a Gaussian function.

can be shown that, for odd electrical harmonics, , for, and hence, from (36)

(48)From the comparison of (47) and (48), it can be concluded

that the MS value of passes from being fluctuant in theerror-free spectrum (48) to become nearer to its mean value,divided by factor , in the presence of errors, as appears in(47).Finally, the relationship between the MS value of from

(47) and the largest attainable value of is obtained, whichis useful for the experimental validation of the high frequencymodel, in Section V.From (41), the largest attainable value of at high fre-

quency can be obtained as

(49)

and from (47) and (49):

(50)

This expression relates the largest attainable squared value ofand its mean squared value for any .

III. ANALYSIS OF RESULTS AND NUMERICAL VERIFICATION

In this section, the main conclusions of the analysis per-formed are summarized and the models are tested by means ofnumerical simulation.

A. Analysis of Results

From the diagram of blocks of Fig. 2, the following can bededuced.(a) The spectrum acts as a filter for all the errors except

for the errors in the width of the magnets, characterized by. Therefore, the zeros of the spectrum enable all

the errors to be cancelled out except those related to thewidth of the magnets.

(b) Function appears in none of the trajectories of theerrors, and hence cannot be used for their cancellation.The zeros of the spectrum , defined in (17), cancelout only those errors of displacement between blocks,characterized by .

5528 IEEE TRANSACTIONS ON MAGNETICS, VOL. 49, NO. 11, NOVEMBER 2013

Fig. 6. Normalized input error of a PMSM with and for thefirst 50 spatial harmonics.

(c) The effects of the error source, , depend on the deriva-tive of and not on as in the other cases. Fromthis it can be deduced that, for a complete cancellation ofthe errors at a precise frequency, it is necessary that both

and its derivative contain at least one zero at thatprecise frequency.

(d) In the linear zone, all the components of error increasewith the frequency, with the exception of , which doesnot depend on the frequency.From the analysis performed in Section II-C):

(e) In the zone of high frequency, the errors stop behavingas linearly additive and completely transform the EMFspectrum. The result is a smoothed spectrum in that themaxima and the zeros have disappeared.

B. Numerical Verification1) Verification for Low Frequency: Fig. 6 shows the normal-

ized input error of a PMSM obtained in analytic form and bymeans of numeric simulation.The considered motor has four poles and eight magnets, dis-

tributed in two blocks shifted rad in order to eliminate theseventh EMF electric harmonic (14 spatial).The considered flux density wave of the poles, , is

taken as a pulse of 169.2 electrical degrees of width. The fol-lowing errors are considered: , ,

and . The graphs in Fig. 6 show eachof the components of the normalized input error, and the totalresult for the first 50 spatial harmonics.For the simulation, the Monte Carlo method has been

used with an input vector of pseudorandom components, by averaging the MS values of the spectra.

From the graphs, it can be deduced that the component ofthe magnetization error, , is independent of the frequency,whereas those of the position of the magnet, , and the dis-placement of the blocks, vary linearly with the frequency, witha slope of 20 dB per decade.The component of the error due to the width of the poles, ,

exerts a negligible effect on the first harmonics and presents amaximum at the 17th electrical harmonic (34th spatial).The resulting takes the form of the envelope of its com-

ponents. Therefore, for the first harmonics, tend towards

Fig. 7. MSs of error and error-free spectra for a PMSM with ,for high frequency.

the value of , which is the dominant effect. In the middlefrequency range, the value of evolves towards , and itis almost equal to at .The zero imposed in the seventh electric harmonic by means

of the positioning of the blocks fails to introduce any changeinto the graph of , as stated in Section III-A b).2) Verification for High Frequency: In order to determine

the validity of the model for high frequency, the spectra witherrors (47) and error-free (48) are obtained through simulationfor one phase of a three-phase motor with , ,

, and .A full-pitch winding with one slot per pole and phase is

considered in such a way that, for the odd electric harmonics,. In Fig. 7, the spectra with errors and error-free in

the odd electric harmonics are shown.Up to the 30th spatial harmonic, both error and error-free

spectra almost completely coincide, but from this frequency on-wards the spectrum with error diminishes, and remains 9 dBbelow the envelope of . This value coincides with theanalytical value from (47), since .Figs. 6 and 7 show the good agreement between the analytical

and the numeric results, which enables the experimental verifi-cation to be tackled as covered in Section V.

IV. PROCEDURE FOR ERROR MEASUREMENT

This section describes the experimental method to determinethe variances of each of the sources of error separately fromthe statistical analysis of the EMF spectra of a set of permanentmagnet motors. The estimation of the variances is performed intwo stages. In the first step, the normalized input error, , isobtained, and the 4 variances are then identified by making useof the analytical model (38). In the case when the rotor has asingle block of magnets, the number of variances to be deter-mined is reduced to 3.

A. Determining the Normalized Input Error

From the diagram in Fig. 3:

(51)

SIMÓN-SEMPERE et al.: INFLUENCE OF MANUFACTURING TOLERANCES ON THE ELECTROMOTIVE FORCE IN PM MOTORS 5529

TABLE IICOMPARISON BETWEEN DEVIATIONS OBTAINED WITH THE

DIRECT METHOD AND THOSE VIA EMF RECORDS

By making use of (35) and of (36), (51) becomes

(52)

is known and analytical since it represents the posi-tion of the design of the magnets, but in general remainsunknown and has to be estimated. A first estimator can be per-formed by averaging of the EMF spectra from different samples,considering that . This estimator requires the errorto have zero mean, which only holds true in the low-frequencyzone, as discussed in Section II-C. Fortunately, the harmonics ofinterest in electric machines are those of low frequency, wherethe analytical linear model can be used.

B. Separation of the Components of Normalized Input Error

In order to estimate the value of each of the variances sepa-rately, a linear system of equations based on (38) is formed,by taking a different value of for each equation and the cor-responding value of obtained from the experiments. The-oretically the system is determined if is equal to the numberof unknown variances. However, given that the variables con-tain errors, it is convenient to form an overdetermined systemof equations in order to apply adjustment techniques that mini-mize the error in the estimated values of the variances.Equation (38) involves function , which in turn depends

on through (37) and can be obtained from (12)

(53)

where the approximation has also been applied.

V. EXPERIMENTAL VALIDATION

In order to experimentally validate the models, various se-ries of experiments on four sets of motors have been performed.Motors under test are driven by an auxiliary motor at constantspeed, and the EMF of one phase of each motor have been reg-istered through a 14-bit data acquisition system.The four sets of tested motors are:Set #1: consists of 35 units of a pre-series of three-phasebrushless motors of 100 W, of four poles with straight sur-face-mounted magnets (Fig. 8(a)).Set #2: consists of 35 motors of characteristics equalto those above, but of the current production series(Figs. 8(b) and 8(c)).

Fig. 8. Rotors of the set #1 (a) and set #2 (b); 35 rotors of the set #2 (c); rotorsof set #3 (d) and #4 (e).

Set #3: corresponds to eight prototypes of 250 W brush-less motors of interior Nd-Fe-B magnets, with eight poles(Fig. 8(d)).Set #4: consists of eight prototypes of 180 W, BLDCmotors with Nd-Fe-B bonded magnets, with six poles(Fig. 8(e)).

A. Verification of the Normalized Input Error

In the first place, the normalized input errors (52) of thesets #1 and #2 of the motors are obtained from the EMFmeasurements. Once is obtained, the variances ,and and are calculated with the method described inSection IV-B.In order to verify the precision in the calculation of the above

variances, the actual errors from the air-gap flux density of eachmotor have been directly measured. To this end, a single-toothsearch coil is used as a probe following the procedure detailedin the Appendix as Direct Method.Table II shows the results obtained from the Direct Method,

as well as those obtained from the proposed method using EMFmeasurements.Finally, Fig. 9 compares the normalized input error obtained

from applying expression (52) to the EMF measurements,labeled as “measured”, with that obtained from the analyticalmodel (38), labeled as “analytical”. In the last case, the vari-ances have been measured with the technique described in theAppendix.

B. Verification for High Frequency

At this point, the relationship (50) between the maximum at-tainable value of and its mean value is verified. Al-though the maximum attainable value cannot be directly mea-sured from a limited number of samples, it is possible to esti-mate it based upon the maximum sample value observed oversamples. From (41), it is possible to obtain a statistical rela-

tionship, , between the maximum attainable value of

5530 IEEE TRANSACTIONS ON MAGNETICS, VOL. 49, NO. 11, NOVEMBER 2013

Fig. 9. Normalized input error obtained experimentally (“measured”) and bymeans of the linear model (“analytical”) for sets #1 and #2 of the motors.

Fig. 10. Relationship between the largest attainable value of , andthe expected maximum sample value for samples at high frequency, with thenumber of magnets, , as a parameter, obtained numerically.

Fig. 11. Relationship between the estimated maximum possible value ofand its mean value in the high frequency zone for odd electric

harmonics obtained experimentally.

on measuring samples and the theoretical maximum value. Inthis work, has been obtained numerically, and it is shown inFig. 10 in terms of the total number of magnets, , and interms of .Fig. 11 shows the relationship between the estimated max-

imum possible value of and its mean value in the highfrequency zone, for sets #1 and #2 of the motors.The estimated maximum possible value is derived from di-

viding the maximum values from 35 samples by . As can beseen, the graphs of the two motor sets tend towards 9 dB, which

Fig. 12. Histogram for the fundamental, 3rd, 7th and 11th electric harmonicsof experimentally obtained for the Set #1 of the motors.

corresponds to the value of (50) for . Set #1 reachesthis value first due to its greater level of error.

C. Verification of the Probability Distribution FunctionsThe PDF of the module of a determined harmonic depends on

the relationship between the variances of its real and imaginarycomponents. The resulting distribution is Rayleigh when bothvariances are equal or a half-normal distribution when one ofthem is zero, as shown in Fig. 4.The real and imaginary components of harmonic voltages are

given by (26), (29) or (32), according to the type of error. Fromthe application of these expressions to the configurations of themotors of sets #1 and #2, it can be verified that the real part ofthe spectrum is due to the errors of positioning of themagnets, and the imaginary part is due to the deviations in theamplitude of the flux density and in the width of the poles.As indicated in Section III-B, in the low-order harmonics, er-

rors of magnetization amplitude dominate over the rest; therebycausing the real component of error to become almost zero,which leads to a half-normal distribution. At higher frequen-cies, new components of error appear, with real and imaginaryparts, which gives rise to a Rayleigh distribution.In Fig. 12(a), the histogram of the fundamental and the third

electric harmonic of the Set #1 of the motors is shown, while inFig. 12(b), the 7th and 11th harmonics of the same set of motorsare given. The similarity of the first histogram with the half-normal PDF, and that of the second histogram with the RayleighPDF confirm the results.

D. Other ConfigurationsFinally, Fig. 13 summarizes the results of the experiments

performed on the motors of sets #3 and #4.Set #3 is composed of eight pre-series motors with interior

magnets of Nd-Fe-B and eight poles, and Set #4 also has eightpre-series motors, with a ring of isotropic-bonded Nd-Fe-Bmagnets with six poles. The motors of Set #3 exhibit fewererrors than those of Set #4 due to the fact that their interiormagnets perform the guiding of the flux more efficientlythanks to the rotor core. Furthermore, the magnetization of theisotropic magnets of Set #4 tends to contain more errors that ofthe anisotropic magnets [1].

VI. EXAMPLE APPLICATION TO THE DESIGN

In certain designs it is necessary to introduce zeros in deter-mined frequencies of the EMF spectrum in order to obtain a

SIMÓN-SEMPERE et al.: INFLUENCE OF MANUFACTURING TOLERANCES ON THE ELECTROMOTIVE FORCE IN PM MOTORS 5531

Fig. 13. Normalized input error of Set #3 of the motors (eight interior-magnetmotors with eight poles), and of Set #4 (8 Nd-Fe-B ring-bonded-magnet motorswith six poles).

Fig. 14. Permanent magnets utilized to get a zero (a) and two zeros (b) at; and device for the spectrum capture (c).

more sinusoidal EMF wave. The sensitivity of these zeros to themanufacturing tolerances must therefore be kept under a min-imum. In these cases, in accordance with Section III-A c), it isnecessary to act upon the two trajectories of the error in Fig. 2,by imposing zeros both in and in its derivative. This con-dition means creating double zeros in at the objective fre-quencies. Obviously these zeros must be created by using onlythe geometry of the poles.In order to experimentally compare the robustness of a double

zero against manufacturing errors, two types of magnets havebeen built for the same surface-mountedmagnet motor with fourpoles: the first with a zero in the 7th electrical harmonic andthe second with a double zero at the same frequency. The zeroof the first type of magnet is obtained by shortening the polearc, thereby rendering it equal to mechanical rad. The firstzero of the second type of magnet is also obtained by makingthe pole arc equal to mechanical rad, and the second zeroby introducing an axial skew of the magnets in the form of astep of mechanical rad, and by using the proceduredescribed in [10]. In Fig. 14, the shape of the first type of magnetis shown in (a), and the second type in (b).For the experiments to be carried out, two sets of skewed

magnets were constructed, and other two sets with straight mag-nets. In the first sets of each type, a deviation is intentionallyintroduced over the nominal value of the pole arc of 3%, andof 3% in the second sets, as indicated in Table III.Fig. 15 shows the air-gap flux-density spectra for the

two types of magnets. The flux density wave, , was

TABLE IIIDIMENSIONS OF THE MAGNETS UTILIZED IN THE TEST

Fig. 15. Air gap flux density spectra of a set of magnets containing a singlezero and another set with a double zero at .

experimentally obtained from the voltage induced in asingle-tooth-winding search coil, following the proceduredescribed in Section V-A. The spectral interpolation techniqueis then employed for the continuous representation of theair-gap flux-density spectra. As can be seen, the amplitude ofthe spectrum around are lower and more flattened forthe geometry with double zero than for the case with a singlezero. This result confirms that the geometry with double zerois less sensitive to manufacturing tolerances than that with asingle zero.

VII. CONCLUSION

A statistical study of the effects of manufacturing tolerancesupon the EMF spectrum in PMSM has been performed. Theresults of such study constitute two models for the character-ization of the error in the EMF spectrum; one linear for thelow-order harmonics, and a second nonlinear, for the high-fre-quency range.The models have been validated by means of simulation, and

in experimental form through a set of tests involving 70 motors.The linear model allows predicting the changes in the EMF

spectrum as a consequence of the errors of magnetization, ofpositioning of the magnets and of the blocks, and of the widthof the poles.Furthermore, a procedure has been developed which permits

to determine the variance of each of the errors to be determinedbased on a statistical analysis of the EMF spectrum.The analysis herein enables to obtain design rules for the re-

duction of sensitivity to mechanical and magnetic tolerances.Based on these rules, a procedure for the reduction of sensi-

tivities in critical design has been presented and experimentallyverified.

5532 IEEE TRANSACTIONS ON MAGNETICS, VOL. 49, NO. 11, NOVEMBER 2013

Fig. 16. Experimental measure of the deviations of the wave of induction.

APPENDIXDIRECT METHOD FOR THE MEASUREMENT OF ERRORS

In order to measure the deviations of the magnets to obtainits associated variances in a direct form, the induction in theair gap of each one of the motors has been registered. To thisend, a single-tooth search coil is used as a probe, whose inducedvoltage, once integrated and scaled, enables the waveform offlux density of each motor, , to be known [12]. In compar-ison with the use of the Hall-effect sensor [13], the method hasthe advantage of axially averaging the flux density in the air gap.This is a valid procedure provided that the magnets are straight,as in this case.Fig. 16 shows one of the experimentally obtained waves,, which illustrates the procedure used for the measurement

of errors.The pole width is hereby taken as the difference between

the angles corresponding to the points where the flux density is50% of its maximum (1 and 2 in Fig. 16). The positioning of thepoles, , is assumed as the mean value between these points.The mean value of between two passes through zero is

assumed to be the flux density in each of the poles, , and it isrepresented by the peak-to-peak value of its integral.This method has been used because it performs well on an

induction wave, , rather rectangular-shaped (Fig. 16). Itsutility should be checked for motors with other waveforms.

ACKNOWLEDGMENT

This work was supported in part by the Spanish MECand the European Commission (ERDF—European Regional

Development Fund) under Grant ENE2011-27984 and bythe Government of Andalusia under Research Project Ref.P-09-TEP-5170. The authors wish to express their gratitudeto Internacional Hispacold, S.A. for their collaboration andsupport.

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