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4564 IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 11, NOVEMBER 2011

Selection of Measurement Modality for Magnetic Material Characterizationof an Electromagnetic Device Using Stochastic Uncertainty Analysis

Ahmed Abou-Elyazied Abdallh, Guillaume Crevecoeur, and Luc Dupré

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

Magnetic material properties of an electromagnetic device (EMD) can be estimated by solving an inverse problem where electromag-netic or mechanical measurements are adequately interpreted by a numerical forward model. Due to measurement noise and uncertain-ties in the forward model, errors are made in the reconstruction of the material properties. This paper describes the formulation andimplementation of a time-efficient numerical error estimation procedure for predicting the optimal measurement modality that leadsto minimal error resolution in magnetic material characterization. We extended the traditional Cramér-Rao bound technique for errorestimation due to measurement noise only, with stochastic uncertain geometrical model parameters. Moreover, we applied the methodonto the magnetic material characterization of a Switched Reluctance Motor starting from different measurement modalities: mechan-ical; local and global magnetic measurements. The numerical results show that the local magnetic measurement modality needs to beselected for this test case. Moreover, the proposed methodology is validated numerically by Monte Carlo simulations, and experimentallyby solving multiple inverse problems starting from real measurements. The presented numerical procedure is able to determine a priorierror estimation, without performing the very time consuming Monte Carlo simulations.

Index Terms—Cramér-Rao bound, inverse problem, magnetic material identification, stochastic uncertainty analysis.

I. INTRODUCTION

A switched reluctance motor (SRM) is an electromagneticdevice (EMD), which is widely used, nowadays, in in-

dustry. In order to precisely predict the machine performance,the magnetic - characteristics of its magnetic materials needto be known. The application of Epstein or single sheet testermeasurements on a separate sheet may result in an inaccurateapproach for performance prediction. Indeed, manufacturingprocesses may alter significantly the material characteristics,see, e.g., [1]. Therefore, the identification of the magneticmaterial properties after construction of the EMD is a moreaccurate approach. This identification procedure can be im-plemented by solving an inverse problem, based on a couplednumerical-experimental procedure [2].

Depending on the nature of the measurements that are used asinput for the inverse problem, a certain resolution or accuracy ofthe recovered magnetic material parameter values is achieved[3]. Possible measurements are mechanical measurements(static torque), local (on a specific part of the geometry) andglobal (whole considered geometry) magnetic measurements.These measurement modalities contain measurement noisewhich decreases the accuracy of the inverse problem solution.Additionally, the uncertainties of important model parameters,i.e., geometrical parameters, influence the resolution. There-fore, a need exists for a numerical procedure that selects themodality that results in the highest accuracy.

The research presented in this paper aims at taking the mea-surement noise and the uncertainties in the numerical model intoaccount for error estimation of the recovered magnetic materialproperties. State-of-the-art Monte Carlo simulations are able to

Manuscript received May 07, 2010; revised November 14, 2010 and March17, 2011; accepted April 27, 2011. Date of publication May 10, 2011; dateof current version October 26, 2011. Corresponding author: A. A.-E. Abdallh(e-mail: [email protected]).

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

Digital Object Identifier 10.1109/TMAG.2011.2151870

achieve this goal but computations may become prohibitive, es-pecially when dealing with time demanding numerical forwardelectromagnetic models. We therefore define a mathematicaltechnique based on the stochastic Cramér-Rao bound (sCRB).The implemented technique determines the optimal measure-ment modality for magnetic material reconstruction of a SRM.A numerical validation of the procedure is carried out by solvingmultiple inverse problems associated to the different measure-ment modalities using Monte Carlo simulations. Furthermore,an experimental validation is provided by comparing the recov-ered material parameters of the several modalities from the dif-ferent measurements with the actual magnetic material proper-ties of the machine.

II. METHODOLOGY

The behavior of a magnetic system can be represented by amathematical model with a set of partial differential equations.This model is parameterized by the following model parameters:the unknown parameters , the uncertain parameters

, and the precisely known parameters . Asan example when dealing with an EMD, can be an air gapthickness value, the number of excitation windings, etc.

In order to estimate the unknown parameters , an inverseproblem has to be solved by iteratively minimizing the sum ofthe quadratic residuals between the experimental observationsof the magnetic system and the modelled ones

, with being the total number of discrete experimentalobservations. In other words, the functional

(1)

needs to be minimized:

(2)

with being the recovered material parameters. Here, ,in (1) are expressed as a single measurement (scalars

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ABDALLH et al.: MEASUREMENT MODALITY FOR MAGNETIC MATERIAL CHARACTERIZATION OF AN ELECTROMAGNETIC DEVICE 4565

expressing e.g. the peak value) at the th observation, but canalso be expressed as a set of measurements (vectors expressing,e.g., the variation in time of the observation). The resolutionof the inverse procedure (2) highly depends on measurements(measurement noise), modeling accuracy (uncertainties inforward model) and the definition of the inverse problem, i.e.

.Here, we use the well known least square nonlinear algo-

rithm, Levenberg-Marquardt method with line search [4], forminimizing the cost function. This algorithm is chosen becausethe inverse problem is a nonlinear minimization problem whichis also described in a least squares sense where the multiple out-puts are fitted to the experimental measurements.

A. Traditional Cramér-Rao Bound Method (CRB)

In the traditional CRB method, it is assumed that the mod-eling is error-free and that the identification procedure is onlyaffected by the errors in the measurements [5]. Several typesof measurement errors can be considered, in particular system-atic and random measurement error. A systematic measurementerror can be defined as a reproducible error that biases the mea-sured value in a given direction [6], i.e., a systematic overesti-mation or underestimation of the true value. A systematic mea-surement error is by definition reproducible. Therefore, it cannotreduced by averaging the values of a large number of measure-ments. However, the reproducible nature of the systematic mea-surement uncertainty makes it possible to estimate the bias onthe measured value by means of a calibration procedure [7]. Theresults presented in this paper restrict to cases where the mea-surement errors contain only the random component “noise.”When carrying out magnetic measurements, noise can be causedby vibrations of steel sheets, noise originating from excitationcurrent, noise due to stray field, air flux noise, environmentalnoise, etc. [8]. We assume here that when performing multiplemeasurements for a certain modality, these measurementswill follow a Gaussian distribution around a mean value with avariance of . Also, it is acceptable in literature to represent themeasurement noise, which is random in nature, by a Gaussiandistribution, see, e.g., [9], [10]. Hence, the measurement noise

is assumed uncorrelated and Gaussian white distributed withzero mean and a variance of .

We propose the use of the Cramér-Rao bound method (CRB)for quantifying the possible errors on the identified unknownparameters . CRB is widely-used in many engineering appli-cations: heat transfer applications [11], biomedical engineeringapplications [12], and signal analysis applications [5]. The CRBoffers the lower bound of the error within a rather small compu-tational time compared to the well-know time-demanding tech-niques such as Monte Carlo simulations applied in, e.g., [13],stochastic finite element method [14], polynomial chaos decom-position [15].

The measurement of a certain magnetic system with actualparameter values , can be represented as

(3)

with being the noise vector. It is thus possible to represent theforward model as with the incorporation of noise, sothat the parameter vector of this model is given by

(4)

We denote the unbiased estimation of these parameters “aftersolving the inverse problem” by . In fact, we can considerthese estimated parameters as unbiased because is includedin this estimator so that the estimation of is no longer biasedby the measurement noise. The Cramér-Rao inequality theoremstates that the covariance matrix of the deviation between thetrue and the estimated parameters is bounded by the inverse ofthe Fisher information matrix [16], [17]:

(5)

is the expected value and can be calculated by

(6)

with being the log-likelihood with respect to data. is a matrix with dimensions. The likelihood of the

data is normally distributed and is given by :

(7)and can be rewritten as

(8)So that the log-likelihood function becomes

(9)When we calculate the derivative to and since we are onlyinterested in the Fisher information matrix associated to , wehave

(10)

Substituting (10) in (6), the Fisher information matrix can bewritten as [18]:

(11)


since , with beingthe measurement variance, .

Usually, the form of the Fisher information matrix is sim-plified by assuming that the measurement noise is uncorrelated,and its variance is independent on the unknown parameters ,i.e. and [19]. Thus, the Fisher informa-tion matrix reduces to the classical form:

(12)Using the inequality of Cramér-Rao bound (5), the lower boundfor the variances of the unknown parameters is givenby [5]:

(13)

expresses the lower bound for the covariance matrix ofthe unknown parameters where the variances of each unknownparameter can be deduced as the diagonal elements of .

B. Stochastic Cramér-Rao Bound Method (sCRB)

Besides the recovery errors due to measurement errors elabo-rated in Section II-A, errors are also introduced by the modelingerror.

Specifically, the accuracy of the modelled response dependsupon the numerical algorithm and the degree of approximationused, e.g., finite difference or finite element, coarse or fine dis-cretization, etc. These errors can be mitigated by using very finediscretizations, inclusion of a better material model, etc. In ad-dition to these errors, the modelled response also exhibits varia-tions which are due to the uncertainties in the uncertain param-eters used in these model calculations. The effect of modelchoice will be presented in Section IV-B2.

The traditional CRB method can be extended when dealingwith stochastic uncertain model parameters , see [11], [20]with an unbiased estimator . The forwardproblem becomes now . The predefined mean valueof the uncertain parameters defines the model .

We assume that the random is Gaussian prior with meanand variance .

This assumption is acceptable because prior informationabout the uncertainty can be provided. For example, a certainmean value of a geometrical parameter of the rotating electricalmachine, as well as a standard deviation can be provided by themanufacturer. When using a Gaussian distribution, the param-eters can become positive and negative, while the geometricalparameters must remain strictly positive. However, a Gaussianmodel is still useful because the stochastic representation of thegeometrical parameters would typically assume a much largermean than a standard deviation. A gamma density distributionneeds to be chosen when the values need to be modeled bya prior with a nonnegative support and where the standarddeviation has a relatively large value compared to the meanvalue, i.e., a Gaussian distribution is not valid anymore.

A uniform distribution is used when no prior information isavailable and is thus not used here because prior informationcan be provided for the geometrical parameters. The CRB isthen not applicable anymore and where other techniques, e.g.,(Monte Carlo simulations [13] or polynomial chaos decompo-sition [15]) are needed.

In order to determine the Fisher information matrix that cor-responds with noise in measurements and uncertainties in modelparameters, the following principle is used. Information is addi-tive: the information from two independent experiments (Fisherinformation matrix from noise and Fisher information matrix

from model parameter uncertainty) is

(14)

Since uncertainties in model parameters and noise in experi-ments are independent, the above can be used.

Using the mathematical expressions in Section II-A and (14),the extended Fisher information matrix is given by [20]:

(15)

According to [20], the effect of the trace term is very small,and can thus be neglected. The extended Fisher information ma-trix, , can then be approximated by

(16)In this situation, a comparison of (12) and (16) suggests that

can be considered as the equivalent noise of the experiment[21]:

(17)

with being the variance of the uncertain parameters, is the sensitivity matrix of the modelled system re-

sponse with respect to the uncertain parameter b:

(18)

So, the lower bound for the variances of the unknown param-eter is [12]:

(19)

In other words, expresses the lower bound for the covari-ance matrix of the unknown parameters where the variances of


Fig. 1. Schematic diagram of the studied 6/4 SRM. � is the air gap at the align-ment condition. Motor depth is 63.5 cm.

each unknown parameter can be deduced as the diagonal ele-ments of .

Notice that for simplicity, we elaborated the theory for scalar, and in (12) and (16). This can be easily rewritten

for vectors, see [20]. Moreover, it is possible to use instead ofGaussian prior for also Gamma prior. needs to be changedthen, see [12].

III. INVERSE PROBLEM FORMULATION

We apply the numerical algorithm of Section II to the fol-lowing test case: magnetic material characterization of a SRM.

A. Studied Geometry and Material Modelling

Fig. 1 shows the schematic diagram of the 6/4 SRM. Thegeometry is characterized by eight geometrical parameters:

, where and are thestator and the rotor pole width, and are the internaland external diameter of the rotor yoke, is the diameter ofair gap, and are the internal and external diameterof the stator yoke, and is the air gap thickness. Only thevalues of 5 parameters are assumed to be importantly un-certain , while the valuesof the other parameters are assumed to be precisely known

. The meanvalues of the uncertain motor geometrical parameters areindicated in Table I. The single-valued nonlinear constitutiverelation of the magnetic material of a SRM, is modelled here bymeans of three unknown parameters[2]:

(20)

B. Objective Functions Formulations

In this paper, we define three different inverse problems inorder to identify the magnetic material parameters of thestudied SRM. In the following section, three objective functions

TABLE IGEOMETRICAL PARAMETERS OF THE STUDIED SRM

s are formulated, which minimize iteratively the quadraticdifference between the measured and simulated quantities (2).The following objective functions are considered:

1) Static Torque Measurements: The first objective functionis based on the static torque profile measurements for

a fixed excitation current . We define the following objectivefunction:

(21)

with being the measured torque for the rotor po-sition, and being the estimated torque, which is givenby [22]:

(22)

where is the co-energy. For the solution of the in-verse problem, we considered rotor angles, i.e.

.2) Local Magnetic Measurements: The second objective

function is implemented, at a fixed rotor position angle,using the amplitude of the sinusoidal excitation cur-rent and the local magnetic inductionmeasurements, at a specific position on the rotor pole:

(23)

with being the measured peak magnetic inductionvalue of the excitation current and the cor-responding simulated local flux densities given the materialparameter values . The measurements are carried out using asearch coil wound around only one rotor pole.

3) Global Measurements: The third objective functionis implemented, at a fixed rotor position angle

, using global measurements of the excitation currentand the voltage over the exci-

tation winding, according to Faraday’s law [3], where no localmeasurements are used:

(24)

(25)


where is the number of turns of the excitation winding, andis the resistance of the excitation coil. , andare the measured and simulated peak magnetic flux value of the

excitation current, .Based on the estimated error, we are able to determine the

best formulation. Moreover, by implementing the extendedCRB (sCRB) method, we are able to identify the most criticalgeometrical parameter which highly affects the inverse problemaccuracy.

IV. A Priori ERROR ESTIMATION OF A 6/4 SRM

A. Simulation Setup

In order to save the computational time, we built in a firststage, a very fast analytical model of the SRM based on themagnetic reluctance network theory, in which the SRM is ap-proximated by a magnetic network of reluctances and a mag-neto-motive force as a source. The magnetic network model isconstructed in a similar way as described in [23].

Numerical experiments were carried out where we“create” the measurement data ( , , or ), asthe output of the analytical forward model based upona priori chosen fictitious “synthetic” material properties

, corrupted by Gaussian noisewith zero mean and a standard deviation of . The created“noisy” measurement data is used later on as input for theinverse problem and the predefined material properties arethe properties to be reconstructed by the inverse problem.

The standard deviation of the measured quantity , whichcan be the static torque, the local magnetic induction or the mag-netic flux, is given by , whereis the noise-level in the measurement. is the root meansquare of the measured quantity

(26)

is a normally distributed random number with zero meanand a standard deviation of unity.

From our experimental experience, we noticed that the noise-level in the static torque measurements is higher than thenoise-level in the magnetic measurements , due tothe inevitable mechanical misalignment errors. Also, it is worthmentioning that the accuracy of the voltage measurements in

highly depends on the accurate knowledge of the excitationcoil resistance , which is changing during the experiment due tothe changing of the excitation coil temperature. Here we assume“reasonably” . Thisissue is explained in more details in Section V.

Furthermore, the standard deviation of the uncertain geomet-rical model parameters can be expressed similarly to the mea-surement noise as , where and are the uncer-tainty-level and the mean measured values of the five geomet-rical parameters, indicated in Table I, respectively. The uncer-tainty-levels are assumed to be logic, the larger parameter di-mension the lower the uncertainty-level.

B. Results and Discussion

1) Results for an Analytical Model: A wide analysis can beperformed by using a time-efficient analytical model. The re-sults in this subsection are qualitative.

Due to the nonlinearity of the magnetic material characteris-tics, the lower bound of the estimated variance of each parameter

, (19), has to be incorporated into the constitutive relation(20) in order to represent the lower bound of the estimated error:

(27)

where is the percentage lower bound of the estimated error,and is the root mean square of the curve.

It is worth mentioning that can never be calculated inpractice, because the knowledge of the sought-after parameters

is unknown. However, we use the formula, in order toestimate the error in the inverse problem solution in a “qualita-tive” way rather than a “quantitative” way.

Fig. 2 illustrates the values due to measurement noise anduncertainty in the five geometrical parameter values of the SRM,the current is kept constant at the motor rated currentfor , and the rotor is blocked at the alignment condition

for and .It can be observed from Fig. 2 that the accuracy of the in-

verse approach highly depends on the definition or modality ofthe inverse problem (mechanical, local or global magnetic mea-surements) and the considered uncertain model parameter.

For example, the uncertainty in the air gap thickness value“always” gives the worse results irrespective of the modalitytype, as shown in Fig. 2(a). The first modality, which is basedon torque measurements, is less influenced by the uncertaintyin compared to the second and third modality. This is because

is carried out at a wide range of rotor angles ,however, and are carried out at a fixed rotor angle

. Also, it is clear that the estimated error due to uncertaintiesin and is higher than the uncertainties in and .

Fig. 2(b) depicts the due to measurement noise only foreach modality. As mentioned above, the , which is basedon local magnetic induction, results in a very small due toits very small measurement noise.

In case of estimating the due to both the measurementnoise and the uncertainties in the five geometrical parametervalues, Fig. 2(c), it is clear that the is generally the mostaccurate modality. Again, is the worst inverse problemmodality for all geometrical parameters except for , which ishighly influenced by more than .

Since the air gap is the most critical model parameter, weshow further only the effect of the uncertainty in the value onthe . Fig. 3 shows the effect of the local magnetic inductionmeasurement noise-level and uncertainty-level on the . Itcan be observed that the slightly increases with increasingthe measurement noise-level, and dramatically increases withincreasing uncertainty-level. These results reveal that the ac-curate estimation of the air gap thickness value is much more im-portant than the measurement noise. Similar results with highervalues are obtained for the other modalities, i.e., and .


Fig. 2. The estimated error due to the measurement noise and five different geometrical uncertainties of the studied SRM. (� � � � for �� , � � � for ��and �� ). (a) G: uncertainty in � only, (b) S: measurement noise only, (c) V: measurement noise and uncertainty in �.

Fig. 3. The effect of local magnetic induction measurement noise-level and airgap uncertainty-level on the estimated error.

In order to decide which setup configuration is the best forthe identification process, the results are obtained for different

in , and different in and .In practice, the accurate determination of the static torque

curves of a SRM “experimentally” is not an easy task [24], asany mechanical misalignment leads to large errors in the calcu-lations. Moreover, the resolution of the measured static torquedepends on the excitation current; the higher excitation current,the higher the measurement accuracy. The best current value forstatic torque measurements is approximately situated around therated current value of the SRM.

Fig. 4 depicts the effect of the rotor angle on the forand , due to the uncertainty in value only. It is clear

that the error is appreciably high at the alignment rotor position. In particular, the measurements at

results in small , compared to . This canbe explained due to the dominance of the air gap parameterat compared to . At thelatter region, the magnetic path length in the air increases, and

Fig. 4. The effect of the rotor angle on the estimated error, for �� and �� ,due to the uncertainty in the air gap thickness value.

hence the value is less dominant, i.e., ,is the approximate stray flux length. This result means

that the proposed numerical procedure not only selects the mostaccurate modality, but also determines at which rotor angle themeasurements, for and , have to be carried out.

Fig. 5 shows the recovered - curve for the three suggestedinverse problem modalities compared to the original one basedon the . Again, it is clear that the is the best inverseproblem modality.

2) Results for a Numerical Model: It is well-known thatthe values depend on the accuracy of the SRM model.Therefore, we recalculate the for a more accurate numer-ical model based on the finite element method, so as to havea better quantitative estimation. Fig. 6 shows a comparison be-tween the results obtained for analytical and numerical models


Fig. 5. The recovered �-� curve based on the different modalities (�� ,�� , or�� ), compared to the original one based on the fictitious data � �� .

Fig. 6. The comparison between the analytical and numerical model, due tothe measurement noise and the air gap uncertainty.

due to the measurement noise and air gap uncertainty; a similartrend is obtained with different values.

Also, in order to study the effect of the synthetic input dataon the selection procedure of the best inverse problem modality,we tested two different synthetic material characteristics withhigher and lower - curve compared to the one that has beenused in the previous analysis. The analytical model of the SRMis used in this investigation. Again, we observed that the ampli-tude value of the depends on the synthetic data and haveagain the same trend, see Fig. 7, i.e., the best inverse problemmodality is the same. In order to study the effect of altering onlyone element of on the estimated , we calculated thefor six different parameter values. At each time only one ele-ment is changed while the other two elements are kept fixed,see Table II. Again, the best inverse problem modality isin spite of the input fictitious parameter values . These re-sults mean that the estimation of the most appropriate inverseproblem modality and the most critical geometrical parametersdoes not depend here on neither the model accuracy nor the syn-thetic input parameter values. This feature proves the applica-bility of the proposed methodology as a priori inverse problemmodality selection.

Fig. 7. The comparison among different fictitious input data using the analyt-ical model, due to the measurement noise and the air gap uncertainty.

TABLE IITHE VALUES OF �� DUE TO ALTERING ONLY ONE ELEMENT IN �

3) Monte Carlo Validation: All the results presented in thispaper represent the “lower bound” for the variance of the esti-mated magnetic parameter values for any unbiased estimator.Here, we present the Monte Carlo simulation results to validatethe results obtained using the sCRB. For each modality, we solve3000 inverse problems (1000 for each case) with the analyticalforward model for measurement noise, or uncertainty for thevalue or both. The air gap value and/or the measurements aredistorted by zero mean white gaussian noise, using a randomnumber generator with the same uncertainty and noise levels asthe one used in the analysis. For each modality and a specificcase, we estimated the magnetic material parameters using thenonlinear least-squares approach. The comparison of the sCRBwith Monte Carlo simulations is done in a similar way as de-scribed in [25], in which the lower bound for the variance of theunknown parameters obtained by sCRB is compared to the rootmean square error of the Monte Carlo simulation results. Theoverall results shown in Fig. 8 reveal that the computationallyfast sCRB results are always lower than the “time-demanding”Monte Carlo simulation results, which validates the proposedmethodology.

V. EXPERIMENTAL VALIDATION OF THE STOCHASTIC

PROPOSED METHODOLOGY

In order to validate “experimentally” the obtained resultsusing the sCRB, the three different inverse problems are solvedstarting from real measurements.


Fig. 8. Comparison between the sCRB and Monte Carlo (MC) simulation re-sults.

A. Experimental Setup

1) Static Torque Measurements Setup: The static torque mea-surements (used for ) are carried out in a similar way as de-scribed in [26]. In this experiment, the static torque of the SRMis measured at a specific excitation current and rotor angular po-sition by balancing the coupled beam, which is attached to therotor shaft, see Fig. 9. The screw mechanism is installed on therotor shaft so that the deflection angle of the beam can be set.First, at the fully alignment condition, the beam is adjusted atthe horizontal level. When the beam is set to a certain deflectionangle by the screw mechanism, and the stator phase is energizedby a DC power supply, the beam tends to move towards the hor-izontal level. However, by hanging the appropriate mass, whichwas precisely weighted by a digital scale, the beam stabilizes atthe preset deflection angle. Fig. 9 illustrates schematically thestatic torque measurements setup.

Static torque characteristics can be simply calculated as

(28)

where is the measured static torque (N.m), is thehanging mass (kg), is the gravitational acceleration

, is the distance along the beambetween the center of the rotor shaft and the hanging mass point

, and is the deflection angle from the horizontallevel (mechanical degree).

The static torque is measured for three excitation currents (4,8, and 12 A), and for 10 rotor angles . Allmeasurements are repeated five times to ensure better results.From these measurements, the mean torque profile values andstandard deviations can be easily obtained. The mean torqueprofile is the average of the five measurements

, and the standard deviation is calculated

as . The av-erage standard deviation of the torque for all rotor angles at 12A is e.g. 0.1290 N.m. Fig. 10 shows the mean static torque mea-surements and the corresponding error bars. The mean torquevalues are used for solving the inverse problem.

2) Local Magnetic Induction Measurements Setup: Thequasi-static magnetic measurements are performed with a

Fig. 9. The schematic diagram for the static torque measurements. A: Sim-plified diagram of the stator, B: Simplified diagram of the rotor, C: Circulardisk with 144 holes (each hole represents 2.5 mechanical degrees), D: Screwmechanism, and E: Balancing beam. Step(1): at the fully alignment condition,the stator phase is excited by DC current. Step(2): at absence of the excitationcurrent, set the rotor to a certain deflection angle � by the screw mechanism.Step(3): stator phase is excited and rotor tends to rotate to the alignment posi-tion, but the appropriate mass weight fix the rotor at the preset deflection angle.

Fig. 10. The measured static torque profiles.

sinusoidal current excitation at 1 Hz so to have a negligiblepresence of eddy current effects in the magnetic core. Theobject under test is demagnetized between two successivemeasurements. The same setup condition is applied for fluxlinkage measurements, see Section V-A3.

The local magnetic induction measurements (used for )are carried out using a search coil wound around a specificrotor pole with number of turns . The voltageinduced over the search coil is integrated analogously to ob-tain the magnetic induction according to Faraday’s law. In this


Fig. 11. The measured local magnetic induction characteristics.

Fig. 12. The measured flux linkage characteristics.

measurement setup, the rotor shaft is blocked mechanically ata specific rotor angle. The measurements are done for 12 ex-citation currents , and for 10 rotor angles

. Fig. 11 shows the local magnetic induc-tion measurements at different rotor angles and excitation cur-rents, which is used for solving the inverse problem.

3) Flux Linkage Measurements Setup: The global measure-ments (used for ) are carried out by recording the excitationcurrent and the voltage over the excitation winding inthe time domain. The resistance of the excitation winding ismeasured and coupled with and in order to obtain thelinkage flux using (24). The measurements are executed for 12excitation currents , and for 10 rotor angles

. Fig. 12 shows the flux linkage measure-ments at different rotor angles and excitation currents, which isused for solving the inverse problem.

It is clear from Figs. 10–12 that the ripple, which indicatesthe noise in the measurements, in local magnetic induction mea-surements is lower than the ripple in the flux linkage measure-ments. The reason for that might be the error in the value of theexcitation coil resistance, and the error in the numerical integra-tion. On the other hand, the noise level in the mechanical statictorque profile is the highest one. That is due to the difficulty ofmeasuring the deflection angle.

Fig. 13. The recovered material characteristics based on the different inverseproblem modalities compared to the original characteristics.

B. Inverse Problem Implementation

Three inverse problem are solved, one for each modality.Then, the identified magnetic characteristics (single valued

- curve) are compared with the original normal magne-tizing - curve of the material, which is measured using theIEEE standard 393-1991 [27], in which a magnetic ring coreis fully and uniformly wound with two windings, an excitationwinding and a measurement winding. The magnetic fieldstrength and magnetic induction are obtained using Ampere’slaw and Faraday’s law, respectively [2]. The original -curve is fitted by (20), which results in the actual material pa-rameter values .

Fig. 13 depicts the recovered magnetic properties for eachinverse problem modality compared to the original characteris-tics. In this figure, the inverse problem is solved for at 12 A,and for and at . Fig. 14 shows the valuesfor each inverse problem. It is clear from Fig. 13 and its cor-responding Fig. 14 that the inverse problem based on the statictorque measurement results in the worst identificationresults. However, both and result in quite acceptableidentification results. The results based on is a little bitworse than the results based on due to the error in the exci-tation coil resistance value. The recovery error shown in Fig. 14could be explained due to the improper value of the air gap usedin the inverse problems. For the more accurate magnetic ma-terial identification results, the value of the air gap thicknessshould be included in the inverse problem as described in [2].The results presented in Figs. 13 and 14 validate the theoreticalresults.

VI. CONCLUSION

In this paper, we proposed a numerical methodology to esti-mate the error in the recovered material properties of a SRM.The results presented in this paper discuss qualitatively andquantitatively the most accurate measurement modality thatleads to minimal error resolution of material properties, takinginto account the effects of measurement noise and geometricalmodel uncertainties. It is shown that the inverse problem basedon local magnetic induction measurements at misalignmentcondition is the most appropriate modality. Furthermore, we


Fig. 14. The errors in the recovered material characteristics.

noticed that the accurate estimation of the air gap thicknessvalue is an important factor for increasing the resolution of theinverse problem’s solution accuracy. Finally, the theoreticalresults are validated numerically by the time consuming MonteCarlo simulation, and experimentally by solving three inverseproblems starting from the three different measurements.

ACKNOWLEDGMENT

The authors gratefully acknowledge the financial support ofprojects GOA07/GOA/006, and IAP-P6/21 funded by the Bel-gian Government. G. Crevecoeur is a postdoctoral researcher forthe “Fund of Scientific Research Flanders” (FWO).

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