IEEE TRANSACTIONS ON MAGNETICS, VOL. 51, NO. 1, JANUARY 2015 7300210

A Unified Electromagnetic Inverse Problem Algorithm for theIdentification of the Magnetic Material Characteristics of

Electromagnetic Devices Including UncertaintyAnalysis: A Review and Application

Ahmed Abou-Elyazied Abdallh and Luc Dupré

Department of Electrical Energy, Systems, and Automation, Ghent University, Ghent 9000, Belgium

Magnetic properties of the electromagnetic devices (EMDs) core material are reconstructed by solving a coupled experimental–numerical electromagnetic inverse problem. However, the measurement noise, as well as uncertainties of the forward model parametersand structure, may result in dramatic recovery errors in the recovered values of the material parameters. In this paper, we reviewthe use of the electromagnetic inverse problem for the identification of the magnetic material characteristics. The inverse algorithm iscombined with a generic stochastic uncertainty analysis for a priori qualitative error estimation and a quantitative error reduction.The complete inverse methodology is applied to the identification of the magnetizing B–H curve of the magnetic material ofa commercial asynchronous machine. Both numerical and experimental results validate the inverse approach, showing a goodcapability for magnetic material identification in EMDs. The proposed technique is general and can be applied to a wide range ofapplications in the electromagnetic community.

Index Terms— Coupled experimental–numerical inverse problem, error reduction, magnetic material characterization, stochasticanalysis, uncertainty estimation.

I. INTRODUCTION

ELECTRICAL steel sheets are inevitable components ofmost electromagnetic devices (EMDs). Classical mea-

surement techniques, such as Epstein frame measurements, areoften used for identifying their magnetic properties [1]. Duringthe production of EMDs, these electrical steel sheets are sub-jected to different levels of mechanical and thermal stresses,such as punching [2] and shrink-fitting process [3]. In practice,it has been reported in literature that the magnetic mater-ial properties are changed after the manufacturing process[4]–[6], i.e., the characteristics of the magnetic materiallocated inside an EMD may differ from the original ones.Therefore, the properties of the magnetic material need to beidentified in the EMD itself.

Recently, we have reconstructed some important macro-scopic magnetic features, such as a single-valued B–H curve,hysteresis loop, and iron loss parameters [7]–[9], of the corematerial of several rotating electric machines by a hybridnumerical–experimental method [7], [10].

Basically, the magnetic material properties are recoveredusing the inverse methodology by interpreting precisely-defined measurements with a mathematical model of thedevice [11]. The inverse solution is obtained by minimizingiteratively the distance between the measured and simulatedquantities using a specific minimization algorithm [12]. In theideal situation, i.e., measurements are noise-free and physics ofthe EMD is fully described in the model, the actual magnetic

Manuscript received December 12, 2013; revised April 9, 2014; acceptedJune 21, 2014. Date of publication June 25, 2014; date of current ver-sion January 26, 2015. Corresponding author: A. A.-E. Abdallh (e-mail:ahmed.abdallh@ieee.org).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TMAG.2014.2332978

material properties may be correctly obtained [13]. However,in practice on one hand, the measurements are often noisy.On the other hand, the mathematical model is a simplifica-tion of reality, i.e., not all physical phenomena are perfectlymodeled. In addition, the values of some model parameters,especially the geometrical ones, are uncertain. Consequently,retrieving the actual properties of the magnetic material isquestionable [14].

To qualitatively estimate the level of inaccuracy in theinverse problem solution due to noisy measurements and/oruncertainties in the model parameters, we have used thestochastic Cramér-Rao lower bound (sCRB) technique[15], [16]. The obtained results have shown the abilityto design a priori the inverse algorithm. For example,the most efficient measurement modality among differentelectric–magnetic–mechanical measurements, that gives thebest recovery results, is selected by means of the sCRBapproach [15]. In addition, the sCRB was capable of selectinga priori the optimal experimental conditions that lead to thebest inverse problem solution [16].

Furthermore, estimating qualitatively the error in the inverseproblem is not enough, the developed algorithm shoulddecrease that error as well. Therefore, two efficient numericaltechniques, namely the minimum path of the uncertainty(MPU) technique and the stochastic Bayesian approach, havebeen used for minimizing quantitatively the modeling error.In [17], the MPU technique successfully reduced the effectof the modeling error initiated by the uncertain value of theair gap in a switched reluctance motor (SRM). The MPUtechnique successfully enhanced the accuracy of the inverseproblem results by reducing the modeling error caused bythe uncertainty in the air gap value in the SRM. In addition,in [18], the effect of the modeling error caused by the modeling

0018-9464 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

7300210 IEEE TRANSACTIONS ON MAGNETICS, VOL. 51, NO. 1, JANUARY 2015

simplifications of a SRM and an EI electromagnetic coreinductor is diminished using the stochastic Bayesian approach.

In general, inverse problems may be ill-posed problemsthat have non-uniqueness solutions. To well-pose the problem,regularization techniques, such as the Tikhonov regularizationtechnique, need to be used. Otherwise, regularization can beobtained by solving the inverse problem for a multi-objectivefunction (OF) starting from a wide range of experimentalmeasurements. This issue was discussed in [8]. In addition, thetwo proposed techniques, i.e., MPU and Bayesian, are used asregularization techniques, to tackle the non-uniqueness of thesolution originated by the uncertain parameters. It is worthmentioning here that the Bayesian approach is sometimesreferred to as stochastic regularization [19].

The general flow chart of the combined experimental–numerical inverse algorithm for the magnetic material char-acterization of an EMD is shown in Fig. 1.

As previously mentioned, both techniques were applied intoa rather simple geometry of an EMD, i.e., a SRM and/or anEI core inductor. In addition, the unified uncertainty analysishas not been given yet. The goal of this paper is to reviewthe electromagnetic inverse methodology for the magneticmaterial identification, located in a real EMD with a complexgeometry, using an electromagnetic inverse problem approachcombined with a unified and generic uncertainty analysis.In principle, we aim at recovering the normal magnetiz-ing B–H curve of a magnetic material inside a commercialinduction machine. Notice that, in this paper we focus onelectrical steels. However, the proposed techniques can beeasily extended for other types of magnetic materials [20].

The problem is defined in Section II. Then, studied geom-etry is explained in Section III. Two magnetic measurementmodalities are used and illustrated in Section IV. In Section V,the traditional inverse problems are formulated. The stochastica priori uncertainty estimation approach is briefly explainedand applied into the asynchronous motor application inSection VI. In addition, the MPU and the stochastic Bayesianapproaches are briefly outlined and applied in Sections VIIand VIII, respectively, where the technique is validatedtheoretically and experimentally. Finally, the conclusions aredrawn in Section IX.

II. PROBLEM DEFINITION

The single-valued B–H curve of the induction motors (IMs)magnetic circuit, shown in Fig. 2, is ambiguous and needs tobe explored using the inverse approach.

Two magnetic measurements, which are the input of theinverse problem, are considered in this application, namely,global coupled magnetic flux and local magnetic flux measure-ments in one stator tooth. The accuracy of the inverse problemsolution will depend on the considered measurement type.

In this application, we assume that the values of all geomet-rical model parameters of the IM are precisely known, exceptthe values of the air gap (g) and the number of excitation wind-ings per stator slot (Nss). It is difficult to measure precisely thethickness of the air gap in an asynchronous motor. In addition,it seems quite impossible to quantify precisely the number of

Fig. 1. General flow chart of the combined experimental–numericalinverse algorithm for the identification of magnetic material properties of anEMD [21].

excitation windings in a (commercial) asynchronous motor, ina non-destructive way. The value of the air gap (g) is oftencalculated by subtracting the measured stator inner diameterand the measured rotor outer diameter. However, the numberof excitation windings per stator slot can be calculated by

ABDALLH AND DUPRÉ: UNIFIED ELECTROMAGNETIC INVERSE PROBLEM ALGORITHM 7300210

Fig. 2. Geometry of the studied (36/33 stator/rotor slots) IM. The phasewindings are arranged as follows: (U U −V −V W W −U −U V V −W −W).

measuring the phase resistance and some geometrical para-meters of the wire, i.e., length, diameter, or based on thefilling factor and the wire diameter, or based on the comparisonbetween the measured and simulated phase inductance in thecase of removed rotor. In the latter technique, the number ofturns per stator slot is overestimated if the simulation is carriedout in a 2-D space, i.e., the end-windings are not includedin the simulation. However, all of these techniques are notperfectly accurate. In this application, the true value of Nss isknown by destructively counting the number of turns per statorslot in an exactly similar motor, i.e., Nss = 62. Consequently,the air gap (g) and the number of turns (Nss) are the input ofthe inverse problem, but treated as uncertain parameters.

III. STUDIED GEOMETRY

Fig. 2 shows the cross section of the studied three pair-polesquirrel-cage IM. The motor rated values are 1.5 kW, 50 Hz,3.75 A, 1000 r/min, and 0.77 power factor. The geometricalparameters of the motor are: Sod, Ssod, Ssid, Rod, Rsid, Ssw,Rsw, Shod, with Sod, Ssod, and Ssid being the stator outerdiameter, the stator slot outer diameter, and the stator slotinner diameter, respectively. The Rod and Rsid are the rotorouter diameter and the rotor slot inner diameter. The Ssw andRsw are the width of the stator and rotor slots. The shaftdiameter Shod is 35 mm. The motor axial length with andwithout including the stator end-winding are 140 and 80.7 mm,respectively. The values of the motor geometrical parametersare given in Table I.

IV. MAGNETIC MEASUREMENTS

The following measurements are performed on the geometryof the considered IM at no-load conditions.

TABLE I

VALUES OF THE GEOMETRICAL IM PARAMETERS

A. Global Coupled Magnetic Flux Measurements

The coupled magnetic flux (�) linked with the excitationwinding is obtained by measuring the supply current I andthe voltage V over the main excitation windings

�m(t) =∫ t

0[V (τ ) − RI (τ )]dτ (1)

where R = 5.11 � is the measured resistance of the excitationcoil per phase. The uncertainty in the value of R is notconsidered since the temperature rise is negligible at no-loadconditions. The measurements are done for a set of sinusoidalexcitation voltage waveforms, at the no-load condition, andthe peak values �p,m of the coupled magnetic fluxes �m(t)are recorded.

B. Local Magnetic Flux Measurements

We expect that the measured coupled magnetic flux may beaffected by the stator end-winding, which is not considered inthe numerical model, Section V. To minimize this dependence,we propose to measure the magnetic flux locally around astator tooth, which is less affected by the stator end-winding.To measure the local magnetic flux measurement φlocal,m ,we wound a very thin search coil with four turns aroundone stator tooth. According to Faraday’s law, φlocal,m is thetime integration of the induced voltage over this search coil.The measurements are done for a set of sinusoidal excitationvoltage waveforms, at the no-load condition, and the peakvalues of the local magnetic flux around the stator tooth(φlocal,p,m) are monitored.

V. INVERSE PROBLEM FORMULATION

A. Material and Machine Modeling

A 2-D finite element (2-D FE) model is used for modelingthe studied IM. A 2-D FE method solves the non-linear quasi-static Maxwell’s equation

∇ ×(

1

μ (Az)∇ × Az

)= −Jz . (2)

In the 2-D FE method, the current density is assumed to beperpendicular to the considered plane of the magnetic circuit(J = Jz iz being the current density in the z-direction). Conse-quently, the magnetic vector potential has only one componentin the z-direction A = Az iz . The single-valued non-linearconstitutive relation of the IM magnetic core material, whichdefines the non-linear magnetic permeability μ, is modeledby means of three unknown parameters u = [H0, B0, ν] thatought to be recovered using the inverse approach [22]

H

H0=

(B

B0

)+

(B

B0

)ν

. (3)

7300210 IEEE TRANSACTIONS ON MAGNETICS, VOL. 51, NO. 1, JANUARY 2015

For the validation issue, the true values of the materialproperties are u∗ = [417.57, 1.41, 9.76], which are obtainedusing the concept of the ring core measurements around thestator yoke [23].

B. Traditional Inverse Problem Formulation

In principle, a traditional inverse problem is formulated byiteratively minimizing an OF, which is the Euclidean norm ofthe difference between the simulated and measured responses.Here, the unknown vector u = [H0, B0, ν] can be recoveredby solving any of the following two OFs:

u = arg minu

OFa(u) (4)

u = arg minu

OFb(u). (5)

The inputs of the inverse problems formulated in (4) and (5)are the aforementioned global and local measurements, respec-tively. The output of the inverse problems is the recov-ered parameters of the normal magnetizing B–H curve u.To minimize these two OFs, we use the Levenberg–Marquardtmethod with line search [24]. This algorithm is chosen becausethe inverse problem is a non-linear minimization problem,which is also described in a least-squares sense where themultiple outputs are fitted to the measurements. In this paper,the MATLAB routine of the non-linear least-square algorithm,in an unconstrained way, is used. For constrained problems,evolutionary computation techniques, such as genetic algo-rithm [25], biogeography-based optimization technique [26],and so on, may be used.

The first OF is formulated as follows:

OFa(u) = ‖�p,s(u) − �p,m‖2 (6)

where �p,m and �p,s(u) are the measured and simulated peakvalue of the coupled magnetic flux.

Similarly, the second OF is formulated as follows:

OFb(u) = ‖φlocal,p,s(u) − φlocal,p,m‖2 (7)

where φlocal,p,m and φlocal,p,s(u) are the measured and simu-lated peak value of the local magnetic flux.

It is well-known that solving the inverse problem costs aconsiderable computational time due to its iterative nature.The results presented in this paper are obtained using a2.4 GHz computer with an Intel dual core processor, with 1 GBof RAM. To save computational time, especially for someapplications that have very time demanding forward models,i.e., a 3-D FE model, it is possible to parallelize the numer-ical algorithm by dividing the forward problem into severalsub-problems [21].

VI. A Priori ERROR ESTIMATION

It is well-known that the accuracy of the inverse problemsolution depends on the accuracy of its input, i.e., measure-ments [27]. In the considered application, two measurementmodalities are utilized, which contain by nature measurementnoise. In addition, the accuracy of the inverse problem solutiondepends on the accuracy of the used forward mathematicalmodel. Here, the uncertainties of significant model parameters,

i.e., number of turns per stator slot Nss and air gap g, mayinfluence the accuracy of the recovered solution. Therefore,in this section we use the sCRB method, presented originallyin heat transfer applications [28], to select a priori the bestmeasurement modality that gives the maximum inverse prob-lem solution veracity, considering both the measurement noiseand the geometrical uncertainties. As described in [15], thebest measurement modality is the one that gives minimumestimated uncertainty (EU) by the implementation of the sCRBtechnique.

The sCRB method requires the computation of the sen-sitivity analysis of the forward problem with respect to thesought-after parameters. Due to the fact that these sought-after parameters are not known in advance, fictitious valuesof the unknown parameters ufic need to be assumed toperform the sCRB analysis. Numerical simulations are carriedout based on ufic, and the global/local quantities (�p,m orφlocal,p,m) are recorded. To simulate the measurement noise,the outputs are then corrupted by a Gaussian noise withzero mean and a standard deviation of σn , which is givenby: σn = (δn Wrms) × N [0, 1], where δn is the noise levelin the measurement. Here, we assume δn,local = δn,global.The Wrms is the root mean square of the measured quantity:Wrms = (1/K

∑Kk=1 W 2

k )1/2. The N [0, 1] is a normally dis-tributed random number with zero mean and a unity standarddeviation. It is widely acceptable in literature to representthe measurement noise, which is random in nature, by aGaussian distribution [15], [29], [30]. When using a Gaussiandistribution, the parameters can become positive and negative,while the air gap (g) and the number of turns (Nss) parametersmust remain strictly positive. However, a Gaussian modelis still useful because the stochastic representation of theg and Nss parameters would typically assume a much largermean than a standard deviation. Other distribution functionscan be also used for the uncertainty representation, such asgamma or Weibull distribution functions. However, the CRBis then not applicable anymore and other techniques, e.g.,Monte Carlo simulations [31] or polynomial chaos decom-position [32], are needed.

Similarly, the standard deviation of the uncertain geomet-rical model parameters σb is given by σb = δb μb, whereδb and μb are the uncertainty level and the mean measuredvalues of the two uncertain geometrical parameters, i.e.,μg = 0.275 mm, μNss = 62.

The target of the application of the sCRB is to select a priorithe best input of the inverse problem, that results in the bestoutput accuracy. In addition, we aim at figuring out the mostcritical uncertain parameter.

A. sCRB Method

In this section, we explain briefly the sCRB technique.Simply, the sCRB approach uses the principle of the sensi-tivity analysis of the forward model response with respectto the unknown parameters to calculate the so-called Fisherinformation matrix (FIM) [28]. The inverse of the FIM givesinformation about the uncertainty level in the inverse problemsolution. To take the effect of the uncertain geometrical

ABDALLH AND DUPRÉ: UNIFIED ELECTROMAGNETIC INVERSE PROBLEM ALGORITHM 7300210

Fig. 3. EU values using the sCRB due to the uncertainty in g and themeasurement noise, for the two considered OFs.

parameters into account, the extended FIM, M, is introducedas follows:

(M)lm ∼=K∑

k=1

[{∂k

∂um

}T

V −1k

{∂k

∂ul

}], l, m = 1, . . . , p (8)

with and p being the forward model response and thenumber of unknown parameters u = [H0, B0, ν], i.e., p = 3in this specific application. The Vk is the total equivalent noise

Vk = �k G�Tk + Sk (9)

where G and Sk are the covariance matrix of the measurementnoise and the geometrical model uncertainty, respectively. The�k is the sensitivity matrix of the forward problem to theuncertain parameters, calculated numerically using the finitedifference technique

�k,g = ∂k/∂g, �k,Nss = ∂k/∂ Nss. (10)

The lower bound of the variance of the unknown parametersσ 2

u,M can then be obtained by the inversion of M

σ 2u,Mii

≥ (M−1)

ii i = 1, . . . , p. (11)

Hence, the EU is calculated for the sake of the comparativestudy, which can be defined as follows [16]:

EU =∣∣∣∣RMSBH(ufic + σu,M )

RMSBH(ufic)− 1

∣∣∣∣ × 100% (12)

where RMSBH is the root mean square of the B–H curve,i.e., RMSBH = (

∑Kk=1 B2(Hk)/K )1/2. Due to the fact that EU

depends on the fictitious values of the unknown parameters,EU gives only a qualitative indication of the results. In thefollowing analysis, ufic = [250, 1.2, 8]. For more detailsabout sCRB [15], [28].

B. Numerical Experiments

In this section, we show some numerical results of thesCRB analysis due to the measurement noise coupled with theuncertainty in the value of g or the uncertainty in the valueof Nss. A fictitious material characteristic is considered forthe numerical experiment. Figs. 3 and 4 show the EU valuesfor the two considered OFs due to the measurement noisecombined to the uncertainty in g or the uncertainty in thevalue of Nss, respectively.

Fig. 4. EU values using the sCRB due to the uncertainty in Nss and themeasurement noise, for the two considered OFs.

Fig. 5. Recovered B–H characteristics based on the estimated sCRBvalues for the two OFs compared with the original characteristic due tothe uncertainty in g (δg = 10%) and the measurement noise (δn,local =δn,global = 1%).

Fig. 6. Recovered B–H characteristics based on the estimated sCRBvalues for the two OFs compared with the original characteristic due to theuncertainty in Nss (δNss = 10%) and the measurement noise (δn,local =δn,global = 1%).

Consequently, Figs. 5 and 6 show the recovered B–Hcharacteristics based on the estimated sCRB values for thetwo OFs compared with the original fictitious characteristicdue to the measurement noise (δn,local = δn,global = 1%) andthe uncertainty in g (δg = 10%) or the uncertainty in Nss(δNss = 10%), respectively.

C. Comparative Study

It is clear from Figs. 3 and 5 that the inverse problem basedon the second OF (local magnetic measurements) gives slightlybetter results than the inverse problem based on the first OF(global magnetic measurements), i.e., there is no appreciable

7300210 IEEE TRANSACTIONS ON MAGNETICS, VOL. 51, NO. 1, JANUARY 2015

difference between the results of the two OFs in this case.On the other hand, it is clear from Figs. 4 and 6 that theinverse problem based on the first OF is dramatically affectedby the uncertainty in Nss, but the one based on the secondOF is less affected by the uncertainty in Nss. In addition, it isclear from Figs. 3 and 4 that the uncertainty in Nss givesworse inverse problem results compared with the uncertaintyin g. Therefore, based on the sCRB analysis, we show theimportance of using the accurate value of Nss for the inverseproblem accuracy.

VII. ERROR REDUCTION: MPU TECHNIQUE

In this section, we show the reduction of the error arisingfrom the uncertain values of some geometrical model para-meters, i.e., g and Nss, due to the implementation of theMPU technique, which is presented originally in biomedicalapplications [33], [34]. In the following, we briefly presentthe new formulation of the inverse problems using the MPUtechnique.

A. First OF: MPU Formulation

The traditional first OF (OFa) is reformulated as follows.

1) Uncertainty in g

OFa,MPU,g(u)

=∥∥∥∥�p,s(u) + αa

(∂�p,s(u, g)

∂g

)− �p,m

∥∥∥∥2

(g=g•).

(13)

2) Uncertainty in Nss

OFa,MPU,Nss (u)

=∥∥∥∥�p,s(u) + βa

(∂�p,s(u, Nss)

∂ Nss

)− �p,m

∥∥∥∥2

(Nss=N•ss)

.

(14)

3) Combined uncertainties in g and Nss

OFa,MPU,g,Nss(u)

=∥∥∥∥�p,s(u) + αa

(∂�p,s(u, g)

∂g

)

+ βa

(∂�p,s(u, Nss)

∂ Nss

)− �p,m

∥∥∥∥2

(g=g•, Nss=N•ss)

.

(15)

B. Second OF: MPU Formulation

Similarly, the traditional second OF (OFb) is reformulatedas follows.

1) Uncertainty in g

OFb,MPU,g(u)

=∥∥∥φlocal,p,s(u) + αb

(∂φlocal,p,s(u, g)

∂g

)

− φlocal,p,m

∥∥∥2

(g=g•). (16)

2) Uncertainty in Nss

OFb,MPU,Nss (u)

=∥∥∥φlocal,p,s(u) + βb

(∂φlocal,p,s(u, Nss)

∂ Nss

)

− φlocal,p,m

∥∥∥2

(Nss=N•ss)

. (17)

3) Combined uncertainty in g and Nss

OFb,MPU,g,Nss(u)

=∥∥∥φlocal,p,s(u) + αb

(∂φlocal,p,s(u, g)

∂g

)

+ βb

(∂φlocal,p,s(u, Nss)

∂ Nss

)− φlocal,p,m

∥∥∥2

(g=g•,Nss=N•ss)

.

(18)

In (13)–(18), αa , βa , αb , and βb are the fitting constantsthat can be obtained from the linear or plane fitting for oneor two uncertain parameters, respectively. The fitting is doneby setting the OF equal to zero [35]. The g• and N•

ss are themean values of g and Nss that are used in the direct model.In the following sections, the MPU technique is tested bothnumerically and experimentally.

C. Numerical Results

Numerical results are obtained by simulating the mea-surements as the output of the direct model based onthe same fictitious data as in the sCRB analysis usingg = 0.275 mm, Nss = 62. For each uncertain model parame-ter, we solve several inverse problems starting from differentassumed values of the uncertain parameter. Accordingly, therecovery error (RE) is calculated as follows [17]:

RE =

⎛⎜⎜⎝

∫ Hmax

0Brecovered.dH

∫ Hmax

0Bfictitious.dH

− 1

⎞⎟⎟⎠

2

× 100% (19)

with Hmax being the maximum considered magnetic field forthe comparison between the two B–H curves, here Hmax =2000 A/m. In the following sections, the MPU techniqueis implemented for reducing the effect of the uncertaintyin the two considered geometrical parameters in each OFformulation. It is clear from Fig. 7 that the error is appre-ciably decreased due to the implementation of the MPUtechnique.

1) Uncertainty in the Value of the Air Gap Thickness g:Fig. 7(a) and (b) shows a considerable reduction in the relativeerror values when the MPU technique is implemented, due tothe uncertainty in g value for both OFs. In addition, it is clearthat the values of the two errors in both OFs are comparable,which validate the results obtained using the sCRB method(Fig. 3).

ABDALLH AND DUPRÉ: UNIFIED ELECTROMAGNETIC INVERSE PROBLEM ALGORITHM 7300210

Fig. 7. Relative error values for the traditional and the MPU OF formulations,due to the uncertainty in the value of the air gap thickness g. (a) Based on OFa .(b) Based on OFb .

Fig. 8. Relative error values for the traditional and the MPU OF formulations,due to the uncertainty in the number of turns per stator slot (Nss). (a) Basedon OFa . (b) Based on OFb.

2) Uncertainty in the Number of Turns Per Stator Slot Nss :Fig. 8(a) and (b) shows a considerable reduction in the relativeerror values when the MPU is used, due to the uncertaintyin Nss value for both OFs. Indeed, the error in the inverseproblem based on the first OF is much greater than the errorin the inverse problem based on the second OF. In addition,the comparison of Figs. 7 and 8 gives an evidence that theeffect of the uncertainty in Nss is higher than the effect ofthe uncertainty in g. Again, these results validate the resultsobtained using the sCRB method, see the corresponding resultsshown in Fig. 4.

3) Combined Uncertain Model Parameters g and Nss :In this case, we assume that the two model parameters areuncertain. Therefore, the combination of 25 inverse problemsare solved for each OF formulation starting from differentassumed values of g and Nss. Fig. 9(a) and (b) shows therelative error values due to the combined effect of the twouncertain model parameters. Again, the inverse problem basedon OFb gives improved results in comparison with the OFa

based inverse problem.

D. Experimental Validation

In this section, real measurements are used to solve theinverse problem, where the IM is modeled using the 2-D FEwith a fine mesh density. Due to the fact that we use the accu-rate value of Nss, we do not implement the MPU technique

Fig. 9. Relative error values for the traditional and the MPU OF formulations,due to the combined effect of the two uncertain model parameters g and Nss.(a) Based on OFa . (b) Based on OFb .

Fig. 10. Identified B–H curves starting from local magnetic measurements(OFb) using the traditional and the MPU techniques compared with theoriginal characteristics. These traditional and MPU results are obtained usingthe 2.4 GHz computer and take ∼8 and 16 h, respectively.

for reducing the effect of the uncertainty in Nss. We restrictourselves to the uncertainty in g. Two inverse problems, basedon, i.e., the traditional and the MPU formulation, are solvedfor each measurement modality.

Fig. 10 shows the recovered B–H curves for both inverseproblem formulations, starting from local magnetic measure-ments (OFb), compared with the original characteristic. It isclear that results are improved, in this specific case, whenthe MPU technique is implemented. Furthermore, as it wasmentioned in [35], the MPU technique is capable of estimat-ing the value of the uncertain parameter with a reasonableaccuracy, i.e., grecovered = gassumed + α. Here, the value ofthe recovered air gap using the MPU technique is 0.309 mm,which is assumed later as being the accurate value of theair gap.

On the other hand, Fig. 11 shows the recovered B–Hcurves for both inverse problem formulations, starting fromglobal magnetic measurements (OFa), compared with theoriginal characteristic. In the traditional inverse problem, weuse the recovered value of the air gap, i.e., g = 0.309 mm.In this way, the model parameters are assumed to be correct.Here, the MPU reduced the RE, which is not caused by theuncertainties in the model parameters, as shown in Fig. 10.Indeed, the error in this specific case is originated fromneglecting the 3-D effect and the corresponding end-winding

7300210 IEEE TRANSACTIONS ON MAGNETICS, VOL. 51, NO. 1, JANUARY 2015

Fig. 11. Reconstructed B–H curves starting from global magnetic measure-ments (OFa) using the traditional and the MPU techniques compared with theoriginal characteristics. These traditional and MPU results are obtained usingthe 2.4 GHz computer and take ∼8 and 16 h, respectively.

effects in the 2-D FE model. The reduction in the RE could beexplained as follows. The neglected end-winding effects in the2-D FE model can be compensated to some extent by changingsome model parameters, i.e., air gap and/or number of turns.Therefore, the MPU reduces the RE by virtually adapting thetrue values of the model parameters to fit the experimentalmeasurements. It is worth mentioning here that the recoveredvalue of g, in this specific case, is not the accurate value.

VIII. ERROR REDUCTION: BAYESIAN

APPROXIMATION ERROR APPROACH

In the experimental validation results presented inSection VII-D, we use a 2-D FE model of the IM withfine mesh discretizations. However, this model is highly timedemanding, especially when it is used in the iterative inverseproblem scheme. On the other hand, a relatively lighter coarsemodel can be used to speed up the inverse problem. Sincethese coarse models are not accurate as fine models, one mayexpect an error in the inverse problem solution. Recently, theBayesian approximation error approach has been presented,originally in biomedical applications [36], [37], for reducingthe error in the inverse problem solution arising when incor-porating a coarse model as an alternative of a fine model.

A. Fine and Coarse Models

For the analysis, three models are built; a 2-D FE fine modeland two relatively coarse models, defined by model-x andmodel-y. For simplicity, we change only the geometry of thestator and rotor teeth. In model-x , the round parts in the tips ofthe stator and rotor teeth are neglected. However, in model-y,we only simplify the stator geometry (Fig. 12). Simplifyingthe geometry reduces the computational time, but decreasesthe accuracy of the model. It is worth mentioning that veryfast surrogate models, such as Kriging-based models, can becoupled with the Bayesian approach as presented in [38].

B. Modeling Error Quantification

The modeling error between the fine and each coarsemodel is quantified stochastically using the procedure

Fig. 12. Magnified schematic diagram of the geometry of the IM. (a) Finemodel. (b) Coarse model-x . (c) Coarse model-y.

TABLE II

MEAN AND THE STANDARD DEVIATION VALUES (×10−3), AND THE

CORRESPONDING TRUE (T ) AND FALSE (F ) CASES

OF THE MODELING ERROR FOR THE COARSE

MODEL-x AND MODEL-y

described in [18] and [20] forward calculations are carriedout, for each of the three models, based on random mater-ial parameters u generated by the latin hypercube samplingtechnique [39]. For simplicity, we study only the inverseproblem with the local magnetic measurements input, i.e.,based on OFb. The responses of these forward calculationsare then compared. Table II depicts the modeling errors, interms of their mean and standard deviation values, for the twoconsidered coarse models. The modeling errors are then usedto estimate the coarseness level using the model coarsenesscriterion [18], which is based on the average fidelity of thediscrete status of each observation point. Based on the true(T ) and false (F) cases, shown in Table II, we calculate(ηmodel-x = 3/8 = 37%) and (ηmodel-y = 6/8 = 75%).

Since ηmodel-x and ηmodel-y are, respectively, lower andhigher than the coarseness limit, i.e., 50%, we know a priorithat the model-y is much finer than model-x , as well as muchcloser to the fine model. On the basis of this investigation,we expect that the implementation of the Bayesian approachwill not give a reasonable error reduction in model-x as inmodel-y, which will be validated experimentally in the nextsections.

C. OF Formulation

In this section, the traditional second OF, OFb is reformu-lated according to the Bayesian approach. Since we restrictourselves to the modeling error due to the used model,we use in the simulation the correct value of Nss, i.e.,Nss = 62. In addition, the recovered air gap value from the

ABDALLH AND DUPRÉ: UNIFIED ELECTROMAGNETIC INVERSE PROBLEM ALGORITHM 7300210

Fig. 13. Retrieved B–H curve, based on OFb inverse problem, for the threemodels with and without the implementation of the Bayesian approximationerror approach compared with the original characteristics, g = 0.309.

MPU technique, i.e., g = 0.309 mm is utilized, so that thegeometrical model parameters are correct.

Equations (5) and (7), which are repeated here for clarity

uTraditional = arg minu

OFb, Traditional(u) (20)

OFb, Traditional(u) = ‖φlocal,p,s(u) − φlocal,p,m‖2 (21)

are reformulated as follows:

uCompensated = arg minu

OFb, Compensated(u) (22)

OFb, Compensated(u) = ‖Lm × (φloal,p,s(u)

− φlocal,p,m(u) − μm)‖2 (23)

where Lm is the Cholesky factor of the modeling errorcovariance, i.e.,

(���

2m

)−1 = LTmLm . μm and ���

2m are the

mean vector and the covariance matrix of the modeling error,respectively [36].

D. Experimental Validation

Several inverse problems are solved, with noise-free mea-surements assumption en = 0, for each model, and thenthe identified single-valued B–H curves are compared.Equation (20) is solved for the three computer models, how-ever, (22) is solved only for the two relative coarse models,i.e., model-x and model-y, because no modeling error isconsidered in the fine model. Only one traditional inverseproblem is solved for the fine model based on (20). However,for each coarse model, in addition to the traditional inverseproblem based on (20) in which the modeling error is notdiminished, two inverse problems based on (22) in which themodeling error is reduced, are solved.

Fig. 13 shows the recovered B–H curves compared with theoriginal characteristics. It is obvious from this figure that theimplementation of the Bayesian approximation error approachdecreases the effect of the modeling error, which validatesexperimentally the approach. The error reduction in model-yis larger than the one in model-x as expected previ-ously. The computational time of the Bayesian approxi-mation error compared with the traditional one is shownin Table III.

TABLE III

APPROXIMATE COMPUTATIONAL TIME, IN MINUTES, REQUIRED WHEN

USING (21) AND (23) FOR THE COARSE MODEL-x AND MODEL-y

COMPARED WITH THE TIME REQUIRED FOR THE FINE MODEL,

ASSUMING SAME NUMBER OF ITERATIONS, I.E., n = 50

IX. CONCLUSION

In this paper, we applied the proposed unified inversemethodology coupled with a generic uncertainty analysis forexploring the unknown magnetic material properties inside acommercial asynchronous machine.

The RE, due to measurement noise and modeling errors,is first qualitatively estimated and then quantitatively reduced.To this end, the following techniques are used: 1) the sCRB;2) the MPU; and 3) the Bayesian approximation error.

From the results presented in this paper, it seems that thesCRB method gives a good qualitative view for the inverseproblem results. It shows, in this specific application, that theinverse problem based on the global measurements is worsethan the one based on the local measurements. In addition,the sCRB method shows that the uncertainty in the numberof turns per stator slot is much more important than theuncertainty in the air gap thickness value, in this specificapplication. These results are validated using the MPU tech-nique, which is applied profitably to reduce the RE causedby geometrical model uncertainties. In addition, the effect ofmodeling error, due to the simplification of the used model, onthe inverse problem solution is decreased using the Bayesianapproximation error approach. The proposed inverse procedureis general and can be applied into other electromagneticinverse problems.

ACKNOWLEDGMENT

This work was supported by the Special Research FundBijzonder Onderzoeksfonds through Ghent University, Gent,Belgium.

REFERENCES

[1] F. Fiorillo, Measurement and Characterization of Magnetic Materials.San Diego, CA, USA: Academic, 2004.

[2] E. Araujo, J. Schneider, K. Verbeken, G. Pasquarella, and Y. Houbaert,“Dimensional effects on magnetic properties of Fe-Si steels due tolaser and mechanical cutting,” IEEE Trans. Magn., vol. 46, no. 2,pp. 213–216, Feb. 2010.

[3] D. Miyagi, N. Maeda, Y. Ozeki, K. Miki, and N. Takahashi, “Estimationof iron loss in motor core with shrink fitting using FEM analysis,” IEEETrans. Magn., vol. 45, no. 3, pp. 1704–1707, Mar. 2009.

[4] A. Abdallh and L. Dupré, “The influence of magnetic material degra-dation on the optimal design parameters of electromagnetic devices,”IEEE Trans. Magn., vol. 50, no. 4, pp. 1–4, Apr. 2014.

7300210 IEEE TRANSACTIONS ON MAGNETICS, VOL. 51, NO. 1, JANUARY 2015

[5] B. Vaseghi, S. A. Rahman, and A. M. Knight, “Influence of steelmanufacturing on J-A model parameters and magnetic properties,” IEEETrans. Magn., vol. 49, no. 5, pp. 1961–1964, May 2013.

[6] O. Nakazaki, Y. Kai, T. Todaka, and M. Enokizono, “Iron loss propertiesof a practical rotating machine stator core at each manufacturingstage,” Int. J. Appl. Electromagn. Mech., vol. 33, nos. 1–2, pp. 79–86,2010.

[7] A. Abdallh, P. Sergeant, and L. Dupré, “A non-destructive methodologyfor estimating the magnetic material properties of an asynchronousmotor,” IEEE Trans. Magn., vol. 48, no. 4, pp. 1621–1624, Apr. 2012.

[8] A. Abdallh, P. Sergeant, G. Crevecoeur, and L. Dupré, “An inverseapproach for magnetic material characterization of an EI coreelectromagnetic inductor,” IEEE Trans. Magn., vol. 46, no. 2,pp. 622–625, Feb. 2010.

[9] A. Abdallh and L. Dupré, “Magnetic material characterization usingan inverse problem approach,” in Advanced Magnetic Materials,L. Malkinski, Ed. Rijeka, Croatia: InTech, May 2012.

[10] A. Abdallh, P. Sergeant, G. Crevecoeur, and L. Dupré, “Magneticmaterial identification of a switched reluctance motor,” Int. J. Appl.Electromagn. Mech., vol. 37, no. 1, pp. 37–49, 2011.

[11] O. Dorn and D. Lesselier, “Introduction to the special issue on electro-magnetic inverse problems: Emerging methods and novel applications,”Inverse Problems, vol. 26, no. 7, pp. 070201-1–070201-4, 2010.

[12] A. Tarantola, Inverse Problem Theory and Methods for Model ParameterEstimation. Philadelphia, PA, USA: SIAM, 2004.

[13] D. Ioan and M. Rebican, “Extraction of B-H relation based on the inversemagnetostatic problem,” Int. J. Appl. Electromagn. Mech., vol. 13,nos. 1–4, pp. 329–334, 2002.

[14] J. Kaipio and E. Somersalo, Statistical and Computational InverseProblems (Applied Mathematical Sciences vol. 160). New York, NY,USA: Springer, 2005.

[15] A. Abdallh, G. Crevecoeur, and L. Dupré, “Selection of measurementmodality for magnetic material characterization of an electromagneticdevice using stochastic uncertainty analysis,” IEEE Trans. Magn.,vol. 47, no. 11, pp. 4564–4573, Nov. 2011.

[16] A. Abdallh, G. Crevecoeur, and L. Dupré, “Optimal needle placementfor the accurate magnetic material quantification based on uncertaintyanalysis in the inverse approach,” Meas. Sci. Technol., vol. 21, no. 11,pp. 115703-1–115703-16, 2010.

[17] A. Abdallh, G. Crevecoeur, and L. Dupré, “A robust inverse approachfor magnetic material characterization in electromagnetic devices withminimum influence of the air gap uncertainty,” IEEE Trans. Magn.,vol. 47, no. 10, pp. 4364–4367, Oct. 2011.

[18] A. Abdallh, G. Crevecoeur, and L. Dupré, “A Bayesian approach forstochastic modeling error reduction of magnetic material identificationof an electromagnetic device,” Meas. Sci. Technol., vol. 23, no. 3,pp. 035601-1–035601-12, 2012.

[19] A. Emery, E. Valenti, and D. Bardot, “Using Bayesian inference forparameter estimation when the system response and experimental con-ditions are measured with error and some variables are considered asnuisance variables,” Meas. Sci. Technol., vol. 18, no. 1, pp. 19–29, 2007.

[20] J. De Bisschop, A. Abdallh, P. Sergeant, and L. Dupré, “Identificationof demagnetization faults in axial flux permanent magnet synchronousmachines using an inverse problem coupled with an analytical model,”IEEE Trans. Magn., to be published.

[21] A. Abdallh, “An inverse problem based methodology with uncertaintyanalysis for the identification of magnetic material characteristics ofelectromagnetic devices,” Ph.D. dissertation, Dept. Elect. Energy Syst.Autom., Ghent Univ., Leuven, Belgium, 2012.

[22] P. Sergeant, G. Crevecoeur, L. Dupré, and A. Van den Bossche,“Characterization and optimization of a permanent magnet synchronousmachine,” Int. J. Comput. Math. Elect. Electron. Eng., vol. 28, no. 2,pp. 272–285, 2009.

[23] M. Oka, M. Kawano, K. Shimada, T. Kai, and M. Enokizono, “Evalua-tion of the magnetic properties of the rotating machines for the buildingfactor clarification,” Przeglad Elektrotech. (Elect. Rev.), vol. 87, no. 9B,pp. 43–46, 2011.

[24] D. Marquardt, “An algorithm for least-squares estimation ofnonlinear parameters,” SIAM J. Appl. Math., vol. 11, no. 2, pp. 431–441,1963.

[25] G. Crevecoeur, “Numerical methods for low frequency electromag-netic optimization and inverse problems using multi-level techniques,”Ph.D. dissertation, Dept. Elect. Eng., Ghent Univ., Leuven, Belgium,2009.

[26] M. Sayed, A. Abdallh, and L. Dupré, “A modified migration modelbiogeography evolutionary approach for electromagnetic device multiob-jective optimization,” in Proc. 13th Workshop Optim. Inverse ProblemsElectromagn. (OIPE), Sep. 2014, pp. 20–21.

[27] G. Crevecoeur, D. Baumgarten, U. Steinhoff, J. Haueisen, L. Trahms,and L. Dupré, “Advancements in magnetic nanoparticle reconstruc-tion using sequential activation of excitation coil arrays using mag-netorelaxometry,” IEEE Trans. Magn., vol. 48, no. 4, pp. 1313–1316,Apr. 2012.

[28] A. Emery, A. Nenarokomov, and T. Fadale, “Uncertainties in parameterestimation: The optimal experiment,” Int. J. Heat Mass Transf., vol. 43,no. 18, pp. 3331–3339, 2000.

[29] N. von Ellenrieder, C. Muravchik, and A. Nehorai, “Effects of geometrichead model perturbations on the EEG forward and inverse problems,”IEEE Trans. Biomed. Eng., vol. 53, no. 3, pp. 421–429, Mar. 2006.

[30] L. Beltrachini, N. von Ellenrieder, and C. Muravchik, “General boundsfor electrode mislocation on the EEG inverse problem,” Comput. Meth.Programs Biomed., vol. 103, no. 1, pp. 1–9, Jul. 2011.

[31] U. Ausserlechner, “Inaccuracies of giant magneto-resistive angle sensorsdue to assembly tolerances,” IEEE Trans. Magn., vol. 45, no. 5,pp. 2165–2174, May 2009.

[32] R. Gaignaire, G. Crevecoeur, L. Dupré, R. Sabariego, P. Dular, andC. Geuzaine, “Stochastic uncertainty quantification of the conductivity inEEG source analysis by using polynomial chaos decomposition,” IEEETrans. Magn., vol. 46, no. 8, pp. 3457–3460, Aug. 2010.

[33] B. Yitembe, G. Crevecoeur, L. Dupré, and R. Van Keer, “EEG inverseproblem solution using a selection procedure on a high number ofelectrodes with minimal influence of conductivity,” IEEE Trans. Magn.,vol. 47, no. 5, pp. 874–877, May 2011.

[34] B. Yitembe, G. Crevecoeur, R. Van Keer, and L. Dupré, “Reducedconductivity dependence method for increase of dipole localizationaccuracy in the EEG inverse problem,” IEEE Trans. Biomed. Eng.,vol. 58, no. 5, pp. 1430–1440, May 2011.

[35] A. Abdallh, G. Crevecoeur, and L. Dupré, “Impact reduction of theuncertain geometrical parameters on magnetic material identification ofan EI electromagnetic inductor using an adaptive inverse algorithm,”J. Magn. Magn. Mater., vol. 324, no. 7, pp. 1353–1359, Apr. 2012.

[36] A. Nissinen, L. Heikkinen, and J. Kaipio, “The Bayesian approxima-tion error approach for electrical impedance tomography—Experimentalresults,” Meas. Sci. Technol., vol. 19, no. 1, pp. 015501-1–015501-9,2008.

[37] T. Tarvainen et al., “An approximation error approach for compensatingfor modelling errors between the radiative transfer equation and the dif-fusion approximation in diffuse optical tomography,” Inverse Problems,vol. 26, no. 1, pp. 015005-1–015005-18, 2010.

[38] A. Abdallh and L. Dupré, “Stochastic modeling error reduction usingBayesian approach coupled with an adaptive Kriging based model,” Int.J. Comput. Math. Elect. Electron. Eng., vol. 33, no. 3, pp. 856–867,2014.

[39] F. Viana, G. Venter, and V. Balabanov, “An algorithm for fast optimalLatin hypercube design of experiments,” Int. J. Numerical Methods Eng.,vol. 82, no. 2, pp. 135–156, 2010.