Selection of Measurement Modality for Magnetic Material Characterization of an Electromagnetic Device Using Stochastic Uncertainty Analysis
ABSTRACT Magnetic material properties of an electromagnetic device (EMD) can be estimated by solving an inverse problem where electromagnetic or mechanical measurements are adequately interpreted by a numerical forward model. Due to measurement noise and uncertainties in the forward model, errors are made in the reconstruction of the material properties. This paper describes the formulation and implementation of a timeefficient numerical error estimation procedure for predicting the optimal measurement modality that leads to minimal error resolution in magnetic material characterization. We extended the traditional CramérRao bound technique for error estimation due to measurement noise only, with stochastic uncertain geometrical model parameters. Moreover, we applied the method onto the magnetic material characterization of a Switched Reluctance Motor starting from different measurement modalities: mechanical; local and global magnetic measurements. The numerical results show that the local magnetic measurement modality needs to be selected for this test case. Moreover, the proposed methodology is validated numerically by Monte Carlo simulations, and experimentally by solving multiple inverse problems starting from real measurements. The presented numerical procedure is able to determine a priori error estimation, without performing the very time consuming Monte Carlo simulations.
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ABSTRACT: The magnetic characteristics of the electromagnetic devices' core materials can be recovered by solving an inverse problem, where sets of measurements need to be properly interpreted using a forward numerical model of the device. However, the uncertainties of the geometrical parameter values in the forward model lead to appreciable recovery errors in the recovered values of the material parameters. In this paper, we propose an effective inverse approach technique, in which the influences of the uncertainties in the geometrical model parameters are minimized. In this proposed approach, the cost function that needs to be minimized is adapted with respect to the uncertain geometrical model parameters. The proposed methodology is applied onto the identification of the magnetizing B–H curve of the magnetic material of an EI core inductor. The numerical results show a significant reduction of the recovery errors in the identified magnetic material parameter values. Moreover, the proposed methodology is validated by solving an inverse problem starting from real magnetic measurements.Journal of Magnetism and Magnetic Materials 04/2012; 324(7):1353–1359. · 1.83 Impact Factor
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4564 IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 11, NOVEMBER 2011
Selection of Measurement Modality for Magnetic Material Characterization
of an Electromagnetic Device Using Stochastic Uncertainty Analysis
Ahmed AbouElyazied Abdallh, Guillaume Crevecoeur, and Luc Dupré
Department of Electrical Energy, Systems and Automation, Ghent University, Ghent B9000, 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 and
implementation of a timeefficient numerical error estimation procedure for predicting the optimal measurement modality that leads
to minimal error resolution in magnetic material characterization. We extended the traditional CramérRao bound technique for error
estimation due to measurement noise only, with stochastic uncertain geometrical model parameters. Moreover, we applied the method
onto 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 be
selected for thistest case. Moreover,the proposed methodology is validatednumerically by Monte Carlo simulations,and experimentally
by solving multiple inverse problems starting from real measurements. The presented numerical procedure is able to determine a priori
error estimation, without performing the very time consuming Monte Carlo simulations.
Index Terms—CramérRao bound, inverse problem, magnetic material identification, stochastic uncertainty analysis.
I. INTRODUCTION
A
dustry. In order to precisely predict the machine performance,
themagnetic
 characteristicsofitsmagneticmaterialsneed
to be known. The application of Epstein or single sheet tester
measurements on a separate sheet may result in an inaccurate
approach for performance prediction. Indeed, manufacturing
processes may alter significantly the material characteristics,
see, e.g., [1]. Therefore, the identification of the magnetic
material properties after construction of the EMD is a more
accurate approach. This identification procedure can be im
plemented by solving an inverse problem, based on a coupled
numericalexperimental procedure [2].
Dependingonthenatureofthemeasurementsthatareusedas
inputfor theinverseproblem,a certainresolutionoraccuracyof
the recovered magnetic material parameter values is achieved
[3]. Possible measurements are mechanical measurements
(static torque), local (on a specific part of the geometry) and
global (whole considered geometry) magnetic measurements.
These measurement modalities contain measurement noise
which 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 the
modality that results in the highest accuracy.
The research presented in this paper aims at taking the mea
surementnoiseandtheuncertaintiesinthenumericalmodelinto
account for error estimation of the recovered magnetic material
properties. Stateoftheart Monte Carlo simulations are able to
switched reluctance motor (SRM) is an electromagnetic
device (EMD), which is widely used, nowadays, in in
Manuscript received May 07, 2010; revised November 14, 2010 and March
17, 2011; accepted April 27, 2011. Date of publication May 10, 2011; date
of current version October 26, 2011. Corresponding author: A. A.E. Abdallh
(email: Ahmed.Abdallh@ugent.be).
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.2011.2151870
achieve this goal but computations may become prohibitive, es
pecially when dealing with time demanding numerical forward
electromagnetic models. We therefore define a mathematical
technique based on the stochastic CramérRao bound (sCRB).
The implemented technique determines the optimal measure
ment modality for magnetic material reconstruction of a SRM.
Anumericalvalidationoftheprocedureiscarriedoutbysolving
multiple 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 a
mathematical model with a set of partial differential equations.
Thismodelisparameterizedbythefollowingmodelparameters:
the unknown parameters
, and the precisely known parameters
an example when dealing with an EMD,
thickness value, the number of excitation windings, etc.
In order to estimate the unknown parameters
problem has to be solved by iteratively minimizing the sum of
the quadratic residuals between the experimental observations
of the magnetic system
, with being the total number of discrete experimental
observations. In other words, the functional
, the uncertain parameters
. As
can be an air gap
, an inverse
and the modelled ones
(1)
needs to be minimized:
(2)
with
being the recovered material parameters. Here,
in (1) are expressed as a single measurement (scalars
,
00189464/$26.00 © 2011 IEEE
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ABDALLH et al.: MEASUREMENT MODALITY FOR MAGNETIC MATERIAL CHARACTERIZATION OF AN ELECTROMAGNETIC DEVICE4565
expressing e.g. the peak value) at the th observation, but can
also be expressed as a set of measurements (vectors expressing,
e.g., the variation in time of the observation). The resolution
of the inverse procedure (2) highly depends on measurements
(measurement noise), modeling accuracy (uncertainties in
forward model) and the definition of the inverse problem, i.e.
.
Here, we use the well known least square nonlinear algo
rithm, LevenbergMarquardt method with line search [4], for
minimizing the cost function. This algorithm is chosen because
the inverse problem is a nonlinear minimization problem which
is also described in a least squares sense where the multiple out
puts are fitted to the experimental measurements.
A. Traditional CramérRao Bound Method (CRB)
In the traditional CRB method, it is assumed that the mod
eling is errorfree and that the identification procedure is only
affected by the errors in the measurements [5]. Several types
of measurement errors can be considered, in particular system
atic and random measurement error. A systematic measurement
error 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
surementerrorisbydefinitionreproducible.Therefore,itcannot
reduced 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 on
themeasuredvaluebymeansofacalibrationprocedure[7].The
results presented in this paper restrict to cases where the mea
surement errors contain only the random component “noise.”
Whencarryingoutmagneticmeasurements,noisecanbecaused
by vibrations of steel sheets, noise originating from excitation
current, noise due to stray field, air flux noise, environmental
noise, etc. [8]. We assume here that when performing multiple
measurements
for a certain modality, these measurements
will follow a Gaussian distribution around a mean value with a
varianceof
.Also,itisacceptableinliteraturetorepresentthe
measurement noise, which is random in nature, by a Gaussian
distribution, see, e.g., [9], [10]. Hence, the measurement noise
is assumed uncorrelated and Gaussian white distributed with
zero mean and a variance of
We propose the use of the CramérRao bound method (CRB)
for quantifying the possible errors on the identified unknown
parameters
. CRB is widelyused in many engineering appli
cations: heat transfer applications [11], biomedical engineering
applications [12],and signalanalysisapplications [5].TheCRB
offers the lower bound of the error within a rather small compu
tational time compared to the wellknow timedemanding tech
niques such as Monte Carlo simulations applied in, e.g., [13],
stochasticfiniteelementmethod[14],polynomialchaosdecom
position [15].
The measurement of a certain magnetic system with actual
parameter values
, can be represented as
.
(3)
with
forward model as
that the parameter vector of this model is given by
being the noise vector. It is thus possible to represent the
with the incorporation of noise, so
(4)
We denote the unbiased estimation of these parameters “after
solving the inverse problem” by
these estimated parameters as unbiased because
in this estimator so that the estimation of
by the measurement noise. The CramérRao inequality theorem
states that the covariance matrix of the deviation between the
true and the estimated parameters is bounded by the inverse of
the Fisher information matrix
. In fact, we can consider
is included
is no longer biased
[16], [17]:
(5)
is the expected value and can be calculated by
(6)
withbeing the loglikelihood 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 loglikelihood function becomes
(9)
When we calculate the derivative to
interested in the Fisher information matrix associated to
have
and since we are only
, we
(10)
Substituting (10) in (6), the Fisher information matrix can be
written as [18]:
(11)
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4566 IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 11, NOVEMBER 2011
since
the measurement variance,
Usually,theformoftheFisherinformationmatrix
plified by assuming that the measurement noise is uncorrelated,
and its variance is independent on the unknown parameters
i.e.
and
tion matrix
reduces to the classical form:
, with
.
being
issim
,
[19]. Thus, the Fisher informa
(12)
Using the inequality of CramérRao bound (5), the lower bound
for the
variances of the unknown parameters
by [5]:
is given
(13)
expresses the lower bound for the covariance matrix of
the unknown parameters where the variances of each unknown
parameter can be deduced as the diagonal elements of
.
B. Stochastic CramérRao Bound Method (sCRB)
Besides the recoveryerrors due to measurement errors elabo
rated inSection IIA,errors are also introducedby themodeling
error.
Specifically, the accuracy of the modelled response depends
upon the numerical algorithm and the degree of approximation
used, e.g., finite difference or finite element, coarse or fine dis
cretization, etc. These errors can be mitigated by using veryfine
discretizations, 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 model
choice will be presented in Section IVB2.
The traditional CRB method can be extended when dealing
with stochastic uncertain model parameters
with an unbiased estimator
problembecomesnow
of the uncertain parameters
defines the model
We assume that the random
and variance.
This assumption is acceptable because prior information
about the uncertainty can be provided. For example, a certain
mean value of a geometrical parameter of the rotating electrical
machine, as well as a standard deviation can be provided by the
manufacturer. When using a Gaussian distribution, the param
eters can become positive and negative, while the geometrical
parameters must remain strictly positive. However, a Gaussian
model is still useful because the stochastic representation of the
geometrical parameters would typically assume a much larger
mean than a standard deviation. A gamma density distribution
needs to be chosen when the values need to be modeled by
a prior with a nonnegative support and where the standard
deviation has a relatively large value compared to the mean
value, i.e., a Gaussian distribution is not valid anymore.
, see [11], [20]
. The forward
.Thepredefinedmeanvalue
.
is Gaussian prior with mean
A uniform distribution is used when no prior information is
available and is thus not used here because prior information
can be provided for the geometrical parameters. The CRB is
then 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
respondswithnoiseinmeasurementsanduncertaintiesinmodel
parameters, the following principle is used. Information is addi
tive: the information from two independent experiments (Fisher
information 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 IIA and (14),
the extended Fisher information matrix
is given by [20]:
(15)
According to [20], the effect of the trace term is very small,
andcanthusbe neglected.TheextendedFisherinformationma
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
with respect to the uncertain parameter b:
,
sponse
(18)
So, the lower bound for the
eter
is [12]:
variances of the unknown param
(19)
In other words,
ance matrix of the unknown parameters where the variances of
expresses the lower bound for the covari
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ABDALLH et al.: MEASUREMENT MODALITY FOR MAGNETIC MATERIAL CHARACTERIZATION OF AN ELECTROMAGNETIC DEVICE4567
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 of
Gaussian prior for
also Gamma prior.
then, see [12].
needs to be changed
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. The
geometry is characterized by eight geometrical parameters:
, where
and
and
are the internal
is the diameter of
are the
stator and the rotor pole width,
and external diameter of the rotor yoke,
air gap,
and
of the stator yoke, and
values of 5 parameters are assumed to be importantly un
certain
of the other parameters are assumed to be precisely known
are the internal and external diameter
is the air gap thickness. Only the
, while the values
. The mean
values of the uncertain motor geometrical parameters are
indicated in Table I. The singlevalued nonlinear constitutive
relation of the magnetic material of a SRM, is modelled here by
means of three unknown parameters
[2]:
(20)
B. Objective Functions Formulations
In this paper, we define three different inverse problems in
order to identify the magnetic material parameters
studied SRM. Inthefollowing section,threeobjective functions
of the
TABLE I
GEOMETRICAL PARAMETERS OF THE STUDIED SRM
s are formulated, which minimize iteratively the quadratic
difference between the measured and simulated quantities (2).
The following objective functions are considered:
1) Static Torque Measurements: The first objective function
is based on the static torque profile measurements for
a fixed excitation current . We define the following objective
function:
(21)
with
sition,and
by [22]:
being the measured torque for the
beingtheestimatedtorque,whichisgiven
rotor po
(22)
where
verse problem, we considered
is the coenergy. For the solution of the in
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
rent
and the local magnetic induction
measurements, at a specific position on the rotor pole:
sinusoidal excitation cur
(23)
with
value of the
responding simulated local flux densities given the material
parameter values . The measurements are carried out using a
search coil wound around only one rotor pole.
3) Global Measurements: The third objective function
is implemented, at a fixed rotor position angle
, using global measurements of the excitation current
and the voltage
tation winding, according to Faraday’s law [3], where no local
measurements are used:
being the measured peak magnetic induction
excitation current and the cor
over the exci
(24)
(25)
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4568 IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 11, NOVEMBER 2011
where
is the resistance of the excitation coil.
are 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 extended
CRB (sCRB) method, we are able to identify the most critical
geometrical parameter which highly affects the inverse problem
accuracy.
is the number of turns of the excitation winding, and
, and
.
IV. A Priori ERROR ESTIMATION OF A 6/4 SRM
A. Simulation Setup
In order to save the computational time, we built in a first
stage, a very fast analytical model of the SRM based on the
magnetic reluctance network theory, in which the SRM is ap
proximated by a magnetic network of reluctances and a mag
netomotive force as a source. The magnetic network model is
constructed in a similar way as described in [23].
Numerical experiments were
“create” the measurement data (
the output of the analytical forward model based upon
a priori chosen fictitious “synthetic” material properties
carriedout
, or
where we
, ), as
, corrupted by Gaussian noise
with zero mean and a standard deviation of
“noisy” measurement data is used later on as input for the
inverse problem and the predefined material properties
the properties to be reconstructed by the inverse problem.
The standard deviation of the measured quantity
canbethestatictorque,thelocalmagneticinductionorthemag
neticflux,
isgivenby
is the noiselevel in the measurement.
square of the measured quantity
. The created
are
, which
,where
is the root mean
(26)
isanormallydistributedrandomnumberwithzeromean
and a standard deviation of unity.
From our experimental experience, we noticed that the noise
level in the static torque measurements
noiselevel in the magnetic measurements
the inevitable mechanical misalignment errors. Also, it is worth
mentioning that the accuracy of the voltage measurements in
highlydependsontheaccurateknowledgeoftheexcitation
coilresistance ,whichischangingduringtheexperimentdueto
thechangingoftheexcitationcoiltemperature.Hereweassume
“reasonably”
issue is explained in more details in Section V.
Furthermore, the standard deviation of the uncertain geomet
ricalmodelparameters
canbeexpressedsimilarlytothemea
surement noise as
, where
taintylevel and the mean measured values of the five geomet
rical parameters, indicated in Table I, respectively. The uncer
taintylevels are assumed to be logic, the larger parameter di
mension the lower the uncertaintylevel.
is higher than the
, due to
. This
andare the uncer
B. Results and Discussion
1) Results for an Analytical Model: A wide analysis can be
performed by using a timeefficient analytical model. The re
sults in this subsection are qualitative.
Due to the nonlinearity of the magnetic material characteris
tics,thelowerboundoftheestimatedvarianceofeachparameter
, (19), has to be incorporated into the constitutive relation
(20)inordertorepresentthelowerboundoftheestimatederror:
(27)
where
and
It is worth mentioning that
practice, because the knowledge of the soughtafter parameters
is unknown. However, we use the
estimate the error in the inverse problem solution in a “qualita
tive” way rather than a “quantitative” way.
Fig.2illustratesthe
valuesduetomeasurementnoiseand
uncertaintyinthefivegeometricalparametervaluesoftheSRM,
the current is kept constant at the motor rated current
for , 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 of
the 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 modality
type, as shown in Fig. 2(a). The first modality, which is based
on torque measurements, is less influenced by the uncertainty
in
compared to the second and third modality. This is because
iscarriedoutatawiderangeofrotorangles
however,
and arecarriedoutatafixedrotorangle
. Also, it is clear that the estimated error due to uncertainties
in
andis higher than the uncertainties in
Fig. 2(b) depicts the due to measurement noise only for
each modality. As mentioned above, the
on local magnetic induction, results in a very small
its very small measurement noise.
In case of estimating the
noise and the uncertainties in the five geometrical parameter
values, Fig. 2(c), it is clear that the
accurate modality. Again,
modality for all geometrical parameters except for , which is
highly influenced by
more than
Since the air gap is the most critical model parameter, we
show further only the effect of the uncertainty in the
the
. Fig. 3 shows the effect of the local magnetic induction
measurement noiselevel and
can be observed that the
slightly increases with increasing
the measurement noiselevel, and dramatically increases with
increasing
uncertaintylevel. These results reveal that the ac
curateestimationoftheairgapthicknessvalueismuchmoreim
portant than the measurement noise. Similar results with higher
values are obtained for the other modalities, i.e.,
is the percentage lower bound of the estimated error,
is the root mean square of the
can never be calculated in
curve.
formula, in order to
,
and.
, which is based
due to
due to both the measurement
is generally the most
is the worst inverse problem
.
value on
uncertaintylevel on the. It
and.
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ABDALLH et al.: MEASUREMENT MODALITY FOR MAGNETIC MATERIAL CHARACTERIZATION OF AN ELECTROMAGNETIC DEVICE4569
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 noiselevel and air
gap uncertaintylevel on the estimated error.
In order to decide which setup configuration is the best for
the identification process, the results are obtained for different
in , and differentin
In practice, the accurate determination of the static torque
curves of a SRM “experimentally” is not an easy task [24], as
any mechanical misalignment leads to large errors in the calcu
lations. Moreover, the resolution of the measured static torque
depends on the excitation current; the higher excitation current,
thehigherthemeasurementaccuracy.Thebest currentvaluefor
statictorquemeasurementsisapproximatelysituatedaroundthe
rated current value of the SRM.
Fig. 4 depicts the effect of the rotor angle on the
and , due to the uncertainty in
that the error is appreciably high at the alignment rotor position
. In particular, the measurements at
results in small, compared to
be explained due to the dominance of the air gap parameter
at compared to
latter region, the magnetic path length in the air increases, and
and.
for
value only. It is clear
. This can
. At the
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 most
accurate modality, but also determines at which rotor angle the
measurements, for
and
Fig.5showstherecovered

inverse problem modalities compared to the original one based
on the
. Again, it is clear that the
problem modality.
2) Results for a Numerical Model: It is wellknown that
the
values depend on the accuracy of the SRM model.
Therefore, we recalculate the
ical model based on the finite element method, so as to have
a better quantitative estimation. Fig. 6 shows a comparison be
tween the results obtained for analytical and numerical models
,
, have to be carried out.
curveforthethreesuggested
is the best inverse
for a more accurate numer
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4570 IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 11, NOVEMBER 2011
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 to
the measurement noise and the air gap uncertainty.
due to the measurement noise and air gap uncertainty; a similar
trend is obtained with different
Also, in order to study the effect of the synthetic input data
on the selection procedure of the best inverse problem modality,
we tested two different synthetic material characteristics with
higher and lower
 curve compared to the one that has been
used in the previous analysis. The analytical model of the SRM
is used in this investigation. Again, we observed that the ampli
tude value of the
depends on the synthetic data and have
again the same trend, see Fig. 7, i.e., the best inverse problem
modality is the same. In order to studytheeffect ofaltering only
one element of
on the estimated
for 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 is
in spite of the input fictitious parameter values
sults mean that the estimation of the most appropriate inverse
problem modality and the most critical geometrical parameters
doesnotdependhere onneitherthemodelaccuracynorthesyn
thetic input parameter values. This feature proves the applica
bility of the proposed methodology as a priori inverse problem
modality selection.
values.
, we calculated the
. These re
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 II
THE VALUES OF ?? DUE TO ALTERING ONLY ONE ELEMENT IN ?
3) Monte Carlo Validation: All the results presented in this
paper represent the “lower bound” for the variance of the esti
mated magnetic parameter values
Here, we present the Monte Carlo simulation results to validate
theresultsobtainedusingthesCRB.Foreachmodality,wesolve
3000 inverse problems (1000 for each case) with the analytical
forward model for measurement noise, or uncertainty for the
value or both. The air gap value and/or the measurements are
distorted by zero mean white gaussian noise, using a random
number generator with the same uncertainty and noise levels as
the one used in the analysis. For each modality and a specific
case, we estimated the magnetic material parameters using the
nonlinear leastsquares approach. The comparison of the sCRB
with Monte Carlo simulations is done in a similar way as de
scribed in [25], in which the lower bound for the variance of the
unknown parameters obtained by sCRB is compared to the root
mean square error of the Monte Carlo simulation results. The
overall results shown in Fig. 8 reveal that the computationally
fast sCRB results are always lower than the “timedemanding”
Monte Carlo simulation results, which validates the proposed
methodology.
for any unbiased estimator.
V. EXPERIMENTAL VALIDATION OF THE STOCHASTIC
PROPOSED METHODOLOGY
In order to validate “experimentally” the obtained results
using the sCRB, the three different inverse problems are solved
starting from real measurements.
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ABDALLH et al.: MEASUREMENT MODALITY FOR MAGNETIC MATERIAL CHARACTERIZATION OF AN ELECTROMAGNETIC DEVICE4571
Fig. 8. Comparison between the sCRB and Monte Carlo (MC) simulation re
sults.
A. Experimental Setup
1) StaticTorqueMeasurementsSetup: Thestatictorquemea
surements (used for
) are carried out in a similar way as de
scribed in [26]. In this experiment, the static torque of the SRM
is measured ata specificexcitationcurrentand rotor angularpo
sition by balancing the coupled beam, which is attached to the
rotor shaft, see Fig. 9. The screw mechanism is installed on the
rotor shaft so that the deflection angle of the beam can be set.
First, at the fully alignment condition, the beam is adjusted at
the horizontal level. When the beam is set to a certain deflection
angle bythescrew mechanism,and thestatorphaseis energized
by a DC power supply, the beam tends to move towards the hor
izontal level. However, by hanging the appropriate mass, which
was precisely weighted by a digital scale, the beam stabilizes at
the preset deflection angle. Fig. 9 illustrates schematically the
static torque measurements setup.
Static torque characteristics can be simply calculated as
(28)
where
hanging mass (kg),
is the measured static torque (N.m),
is the gravitational acceleration
, is the distance along the beam
between the center of the rotor shaft and the hanging mass point
, and is the deflection angle from the horizontal
level (mechanical degree).
The static torque is measured for three excitation currents (4,
8, and 12 A), and for 10 rotor angles
measurements are repeated five times to ensure better results.
From these measurements, the mean torque profile values and
standard deviations can be easily obtained. The mean torque
profile is the average of the five measurements
, and the standard deviation is calculated
is the
. All
as
erage standard deviation of the torque for all rotor angles at 12
A is e.g. 0.1290 N.m. Fig. 10 shows the mean static torque mea
surements and the corresponding error bars. The mean torque
values are used for solving the inverse problem.
2) Local Magnetic Induction Measurements Setup: The
quasistatic magnetic measurements are performed with a
. The av
Fig. 9. The schematic diagram for the static torque measurements. A: Sim
plified diagram of the stator, B: Simplified diagram of the rotor, C: Circular
disk with 144 holes (each hole represents 2.5 mechanical degrees), D: Screw
mechanism, 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 excitation
current, 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 negligible
presence of eddy current effects in the magnetic core. The
object under test is demagnetized between two successive
measurements. The same setup condition is applied for flux
linkage measurements, see Section VA3.
The local magnetic induction measurements (used for
are carried out using a search coil wound around a specific
rotor pole with
number of turns
induced over the search coil is integrated analogously to ob
tain the magnetic induction according to Faraday’s law. In this
)
. The voltage
Page 9
4572 IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 11, NOVEMBER 2011
Fig. 11. The measured local magnetic induction characteristics.
Fig. 12. The measured flux linkage characteristics.
measurement setup, the rotor shaft is blocked mechanically at
a specific rotor angle. The measurements are done for 12 ex
citation currents
. 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 carriedoutby recordingtheexcitation
current
and the voltage over the excitation winding
the time domain. The resistance of the excitation winding
measured and coupled with
linkageflux
using(24).Themeasurementsareexecutedfor12
excitation currents
. Fig. 12 shows the flux linkage measure
ments at different rotor angles and excitation currents, which is
used for solving the inverse problem.
It is clear from Figs. 10–12 that the ripple, which indicates
thenoiseinthemeasurements,inlocalmagneticinductionmea
surements is lower than the ripple in the flux linkage measure
ments. The reason for that might be the error in the value of the
excitation coil resistance, and the error in the numerical integra
tion. On the other hand, the noise level in the mechanical static
torque profile is the highest one. That is due to the difficulty of
measuring the deflection angle.
, and for 10 rotor angles
in
is
andin order to obtain the
, and for 10 rotor angles
Fig. 13. The recovered material characteristics based on the different inverse
problem 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 the
IEEE standard 3931991 [27], in which a magnetic ring core
is fully and uniformly wound with two windings, an excitation
winding and a measurement winding. The magnetic field
strength and magnetic induction are obtained using Ampere’s
law 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 each
inverse problem modality compared to the original characteris
tics.Inthisfigure,theinverseproblemissolvedfor
and for
andat
for each inverse problem. It is clear from Fig. 13 and its cor
responding Fig. 14 that the inverse problem based on the static
torque measurement
results in the worst identification
results. However, both
and
identification results. The results based on
worse than the results based on
tation coil resistance value. The recovery error shown in Fig. 14
could be explained due to the improper value of the air gap used
in the inverse problems. For the more accurate magnetic ma
terial identification results, the value of the air gap thickness
should be included in the inverse problem as described in [2].
The results presented in Figs. 13 and 14 validate the theoretical
results.

.
at12A,
values. Fig. 14 shows the
result in quite acceptable
is a little bit
due to the error in the exci
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 and
quantitatively the most accurate measurement modality that
leads to minimal error resolution of material properties, taking
into account the effects of measurement noise and geometrical
model uncertainties. It is shown that the inverse problem based
on local magnetic induction measurements at misalignment
condition is the most appropriate modality. Furthermore, we
Page 10
ABDALLH et al.: MEASUREMENT MODALITY FOR MAGNETIC MATERIAL CHARACTERIZATION OF AN ELECTROMAGNETIC DEVICE4573
Fig. 14. The errors in the recovered material characteristics.
noticed that the accurate estimation of the air gap thickness
value is an important factor for increasing the resolution of the
inverse problem’s solution accuracy. Finally, the theoretical
results are validated numerically by the time consuming Monte
Carlo simulation, and experimentally by solving three inverse
problems starting from the three different measurements.
ACKNOWLEDGMENT
The authors gratefully acknowledge the financial support of
projects GOA07/GOA/006, and IAPP6/21 funded by the Bel
gianGovernment.G.Crevecoeurisapostdoctoralresearcherfor
the “Fund of Scientific Research Flanders” (FWO).
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