Stress zone imaging in steel plates of electrical machines by mostafa.ahmadi.d

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									COMPEL: The International Journal for Computation and Mathematics in
Electrical and Electronic Engineering
Emerald Article: Stress zone imaging in steel plates of electrical machines
Markus Neumayer, Daniel Watzenig, Bernhard Brandstätter



Article information:
To cite this document: Markus Neumayer, Daniel Watzenig, Bernhard Brandstätter, (2011),"Stress zone imaging in steel plates of
electrical machines", COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering,
Vol. 30 Iss: 6 pp. 1938 - 1947
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http://dx.doi.org/10.1108/03321641111168219
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COMPEL
30,6                                    Stress zone imaging in steel plates
                                              of electrical machines
                                                             Markus Neumayer and Daniel Watzenig
1938                                       Institute of Electrical Measurement and Measurement Signal Processing,
                                                       Graz University of Technology, Graz, Austria, and
                                                                                             ¨
                                                                             Bernhard Brandstatter
                                                                Elin Motoren GmbH, Preding/Weiz, Austria

                                        Abstract
                                        Purpose – The purpose of this paper is to demonstrate an inverse problem approach for the
                                        determination of stress zones in steel plates of electrical machines. Steel plates of electrical machines suffer
                                        large mechanical stress by processes like cutting or punching during the fabrication. The mechanical
                                        stress has effects on the electrical properties of the steel, and thus on the losses of the machine.
                                        Design/methodology/approach – In this paper, the authors present a sensor arrangement and an
                                        appropriate algorithm for determining the spatial permeability distribution in steel plates. The forward
                                        problem for stress zone imaging is explained and an appropriate numerical solution technique is
                                        proposed. Then an inverse problem formulation is introduced and the nature of the problem is analyzed.
                                        Findings – Based on sensitivity analysis, different measurement procedures are compared and a
                                        measurement setup is suggested. Further the ill-posed nature of the inverse problem is analyzed by the
                                        Picard condition.
                                        Practical implications – Because of the increased losses due to stress zones, the quantification of
                                        stress effects is of interest to adjust the production process. Stress zone imaging is a first approach for
                                        the application of an imaging system to quantify these material defects.
                                        Originality/value – This paper presents a simulation study about the applicability of an inverse
                                        problem for stress zone imaging and presents first reconstruction results. Further, the paper discusses
                                        several issues about stress zone imaging for the ongoing research.
                                        Keywords Stress zone, Eddy current, Inverse problem, Electric machines, Image sensors, Steel
                                        Paper type Research paper

                                        I. Introduction
                                        During the production process of electrical machines, the magnetic steel plates undergo
                                        several fabrication steps where the material suffers mechanical stress. Especially the
                                        punching causes large mechanical stress in certain areas. The negative influence of
                                        the mechanical stress on the electromagnetic properties of the steel has been reported in
                                        several publications. Nakata et al. (1992) reported on the effects of stress due to cutting.
                                        According to Nakata et al. (1992) the deterioration of the flux density next to the
                                        cutting edge can reach up to 50 percent compared to the raw state of the sheet and this
                                        deterioration can spread up to 10 mm into the material from the cutting edge. In the
                                        works of Saito et al. (2000) and Ossart et al. (2000) the effects due to punching and
                                        bending are reported, with similar impacts on the material properties as cutting.
COMPEL: The International Journal
for Computation and Mathematics in      Hence, a decrease of the magnetic permeability due to production steps is observable. As
Electrical and Electronic Engineering   decreased magnetic properties cause increased iron losses (eddy current and hysteresis)
Vol. 30 No. 6, 2011
pp. 1938-1947                           the significance of this topic has increased in gain and importance due to the efficiency
q Emerald Group Publishing Limited      requirements of electrical machines and due to the fact that power densities steadily
0332-1649
DOI 10.1108/03321641111168219           increase in given active parts volumina. A method to determine the local magnetic
properties of steel sheets was invented by the Austrian E. Werner in the late 1950s of the          Stress zone
last century (Werner, 1957). In this so-called needle probe method, electric currents are       imaging in steel
injected into the steel sheet by two needles. Then, a magnetic field probe (i.e. a Hall probe)
is used to measure the magnetic field strength at the surface of the steel sheet.                         plates
As different magnetic properties cause different field strengths, the signal of the
magnetic probe can be used to quantify the material properties. Senda et al. (2000)
proposed a modified version using several needles. However, as the distance between the                    1939
needles has a lower limit it may not be useful for stress zone analysis in steel sheets
where the stress zones can become tight. As needle probe methods also only provide
local information, the measurement procedure can become time consuming when the
needles have to be attached on several positions. Also the fact that steel plates for
electrical machinery have an insulation layer makes the use of the needle method
unattractive, as the needles may destroy the layer. In this paper we investigate the
possibility of using an inverse problem approach to determine stress zones in steel
sheets by a simulation study. Thus, we place an array of coils above the steel plate.
Then, an AC is injected in one of the coils and the induced voltages in the remaining coils
are measured. This procedure is repeated for every coil. We then formulate an inverse
problem and try to reconstruct the spatial magnetic material parameters given the
simulated data from a known permeability distribution. This paper is structured as
follows. In Section II we define a simulation framework which contains all necessary
parts including the steel sheet and a coil array. In Section III we describe the forward
model and show how we solve it using the finite element method. In Section IV we
describe the inverse problem and present an proper code structure for the computations
as well as an efficient method to solve it. Finally Section V contains first results.


II. Proposed framework
As this work refers to be a first simulation study about the methodology of stress zone
imaging (currently no sensor for stress zone imaging has been built yet) some aspects
about the simulation setup will be carried out first in this section. Figure 1 shows a
sketch of the geometry for the proposed simulation environment. Above a steel plate
coils are arranged in a rectangular grid. The total number of coils is referred as NCoil.
    The steel plate defines the eddy current region V1, whereas the air and the coils are
in the eddy current free region denoted by V2. The surface of the steel plate is referred
as interface surface G12 with the surface normal vector n1, which points from V1
outside into the domain V2. The plate will be divided into a grid of equal-sized
subdomains, where each domain has constant material properties. For the
investigations the following simplifications are made:
    .
       All material quantities like the conductivity s and the permeability m are scalars
       and now deviation into the z-direction is present.
    .
       As a first attempt only the permeability m is reconstructed.
    .
       The conductivity s of the steel plate is not affected by stress and is constant over
       the whole steel sheet.
    .
       No disturbances are present in the surrounding space (except the steel plate
       under investigation).
COMPEL
30,6

                                        0.15

1940                                                     Eddy current free
                                         0.1
                                                            region Ω2
                               z (m)




                                        0.05
                                                                                                  Coil
                                          0


                                       –0.05
                                        –0.1                                                                        0.1
                                                            Eddy current region
Figure 1.                                      0.05         (Steel plate) Ω1                                 0.05
Geometry of the steel plate                             0                                          0
within the finite element                        x (m)                                                    y (m)
                                                                0.05                     0.05
model
                                                                             0.1   0.1

                              The assumption of no deviation in the z-direction of the first point has to be treated by
                              cautiones with respect to the production step in the machine, as, i.e. bending causes a
                              spatial compressing and a spatial stretching of the sheet. Hence, this assumption may
                              not be fulfilled in a real process.
                                  Finally it is assumed that the excitation does not lead to saturation effects.
                              By neglecting saturation effects the forward problem is assumed to be linear. In the
                              case of saturation effects, the forward problem turns into a nonlinear problem, which
                              requires an iterative solution process for the forward problem and thus also requires
                              knowledge about the B-H curve of the steel sheet. For stress zone imaging this is of
                              coarse a tough issue, as the permeability itself is unknown and thus the B-H curve is
                              unknown. Hence, a linear behavior of the system is at least preferable, although this
                              topic opens possibilities for further research. Because of the stage of this work the
                              assumed permeability distributions in the simulation studies cover the whole front side
                              of the plate. This is yet in contrast to real stress zone scenarios, where the stress zones
                              occur on the edges of the steel sheets.

                              III. Forward problem
                              By the term forward problem the computation of the underlying eddy current problem
                              in the steel plate is meant. For the computation of the forward problem an A,V 2 F
                              formulation with Coulomb Gauge (Biro and Preis, 1989) is used to solve the Maxwell’s
                              equations. Thereby, A and V denote the magnetic vector potential and the electric
                              scalar potential, which are used to describe the field quantities in V1. F denotes the
                              magnetic scalar potential, which is used to describe the field in the region V2 .
                              The magnetic field H in V2 is expressed by:
                                                                    H ¼ T 0 2 7F;                                     ð1Þ
where T0 acts as excitation term and is given by:                                                                              Stress zone
                                                 7 £ T 0 ¼ J T;i ;                                                  ð2Þ    imaging in steel
                                                                                                                                    plates
where JT,i denotes the impressed current density in the transmitter coil (ith coil).
Hence, T0 is the Biot-Savart field of the coil. Given the solution in the eddy current
free region, the induced voltage in the other receiver coils can be computed using
Faraday’s law given by:                                                                                                              1941
                                        Z
                         U Coil;k ¼ jvm      ðT 0 2 7FÞdG;                        ð3Þ
                                                         GCoil;k

for k ¼ 1. . .NCoil and k – i. GCoil,k in equation (3) denotes the cross-section of the kth
receiver coil. Beside the fact that the A,V 2 F formulation requires less memory than
for example the A,V 2 A formulation, the decision for the A,V 2 F formulation was
due to the fact, that the geometry of the coils does not have to be modeled explicitly
within the finite element structure as it has for example to be done when the A,V 2 A
formulation is used. However, also an Ar,V 2 Ar formulation provides the feature
of neglecting the coil geometry, but we decided for the A,V 2 F formulation.
A remaining problem of the A,V 2 F formulation is the fact of numerical errors in the
edges of the plate. As the A,V 2 F formulation was used for both types of simulations
(generation of test data and reconstruction), all models have the same property and
hence this errors should not dominate. However, for further investigations the effects of
formulation-specific faults and errors have to be of concern.

A. Implementation
For the defined geometry the A,V 2 F formulation is implemented by means of finite
elements. The Ritz-Galerkin equations are given by Biro and Preis (1989):
  Z                                               Z                                       Z
        7 £ N i · m 21 7 £ A n dV þ                    7·N i m 21 7·A n dV þ                  N i ·jvsðA n þ 7nn ÞdV
    V1                                             V1                                      V1
  |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
         Z              I1                      Z                   I2                                               ð4Þ
    2            7 £ N i ·n 1 Fn dG ¼                N i · ðT 0 £ n 2 ÞdG ði ¼ 1...n1 Þ;
         G12                               G12
                          Z
                                  7N i · jvs ðA n þ 7nn ÞdV ¼ 0             ði ¼ 1. . .n2 Þ;                        ð5Þ
                            V1
                              Z                       Z
                          2       7N i · m7Fn dV 2          N i · ðn 1 · 7 £ A n ÞdG
                               V2                       G12
                                   Z                                                                                ð6Þ
                              ¼2        7N i · mT 0 dV ði ¼ 1. . .n3 Þ;
                                      V2

where n1,2,3 are the numbers of variables to solve for each quantity. Because of the fact,
that only the permeability is reconstructed, only the integrals I1 and I2 in equation (4)
are effected by a material update. Evaluating the Ritz-Galerkin equations results in
NCoil linear equation systems of the form:
                                           Ku i ¼ f i       i ¼ 1. . .N Coil                                        ð7Þ
COMPEL   where K is the common stiffness matrix and fi and ui are the individual right-hand side
         vectors and the solution vectors for each excitation coil, which correspond to the
30,6     measurement procedure. The stiffness matrix K for the A,V 2 F formulation has the
         attribute of being not diagonally dominant due to the entries of the surface integrals.
         The Biot-Savart field T0 for each coil was computed by an analytical expression, which
         allows a fast evaluation of the right-hand side integrals in equations (4) and (6),
1942     compared to a numerical evaluation of T0, where numerical integration techniques are
         required. This becomes an issue for changing geometries, where a reevaluation of T0
         becomes necessary. For simple coil geometries the evaluation of the Biot-Savart field
         is trivial. For more complex geometries Urankar (1980) provided a number of
         publications with analytical solutions of T0 for more general coil forms. As the stiffness
         matrix K of the eddy current problem has typically a high-condition number, an
         iterative solver in combination with some preconditioning has to be used to solve
         equation (7). After solving equation (7) for each coil the induced voltages in the receiver
         coils are computed by evaluating equation (3), which results in (NCoil 2 1) numbers.
         These results are then stored into the vector U.

         IV. Inverse problem
         Summarizing the parameter of interest (i.e. the permeability) into the parameter
         vector j, an approach to solve the inverse problem is given by:
                                          n                              o
                            j * ¼ arg min kU ðjÞ 2 U meas jj2 þ akL jjj2 ;
                                                            2          2              ð8Þ
                                         j

         where Umeas denotes the measured voltages and U(j) denotes the computed output of
                                       2
         the model. The term akL jjj2 is a regularization term, which is necessary, since the
         inverse problem is of ill-posed nature. Defining the complex residual vector rc by:
                                             r c ¼ U ðjÞ 2 U meas ;                              ð9Þ
         Equation (8) is solved iteratively using a Gauß-Newton scheme given by:
                                                            À              Á
                            j kþ1 ¼ j k 2 sðJJ T þ aL T LÞ21 Jr þ aL T Lj k ;                  ð10Þ
         where s denotes the parameter of a line search algorithm. r and J are the real-valued
         residual vector and the real-valued Jacobian, which are given by:
                                             h À Á      À Á iT
                                                 T
                                       r ¼ R rc        I rT
                                                          c      ;                        ð11Þ
                                           h                      i
                                      J ¼ Rð7jk r c Þ Ið7jk r c Þ :                       ð12Þ


         A. Computation of the Jacobian
         The Gauß-Newton scheme requires the computation of the derivatives of the elements
         of rc with respect to the elements of j. As a numerical evaluation of the derivatives (e.g.
         by computing the difference quotient) is too time consuming an analytic scheme given
                                                       ¨
         by the adjoint variable approach (Brandstatter, 1999) is applied. Given a function C:
                                               C ¼ Cðj; uðjÞÞ;                                 ð13Þ
of the variables j, where u(j) is the solution of a linear equation system:                     Stress zone
                                   K ðjÞuðjÞ ¼ f ðjÞ;                               ð14Þ    imaging in steel
where again all quantities can depend on j. The total derivative of C with respect to
                                                                                                     plates
the elements of j is given by:
                                                 !        ! !
                      dC ¼
                            ›C
                                 dj i þ g T ›f
                                                   2
                                                     ›K
                                                           u dj i ;             ð15Þ                  1943
                            ›j i            ›j i     ›j i
where g is the solution of an adjoint problem given by:
                                              ›C
                                     K Tg ¼      :                                  ð16Þ
                                              ›u
Hence, the evaluation of the derivatives by the adjoint variable approach only requires
the solution of the adjoint problem (17). For the given inverse problem (8) the functions
C are the entries of the residual vector rc. Hence the derivatives of each function C
with respect to j form the column vectors of the complex-valued Jacobian matrix.
B. Code structure for fast computations
As the solution process of the inverse problem requires several evaluations of the
forward problem an effective code structure is mandatory for fast calculations. This is
because of the comparatively large computation times for the forward problem.
However, with respect to further developments also an efficient update scheme of the
stiffness matrix is preferable.
    In a typical finite element scheme the Ritz-Galerkin equations (or rather each
individual integral) (4)-(6) are first evaluated for each finite element forming the
element matrices Kelem,i. Finally the common stiffness matrix K is assembled for the
element matrices. However, for the given inverse problem only the integrals I1 and I2
are effected by an update of j, as it is only the aim to reconstruct the permeability
distribution. Thus, a stiffness matrix Kini is assembled given by:
                                                   
                                                   
                                        X          
                                K ini ¼   K elem;i 
                                                       :                         ð17Þ
                                        i          
                                                  K I 1;2 ¼0

To obtain the full stiffness matrix K only the matrices obtained by the integrals I1 and
I2 have to be added:
                                  K ¼ K ini þ K I 1;2 :                             ð18Þ
From the Ritz-Galerkin equations (4)-(6) it can be seen that the Biot-Savart field T0 is
only present in the right-hand side terms. Thus, the NCoil right-hand side vectors can be
precomputed. The same holds for the right-hand side vectors of the adjoint problem
which are needed to compute the Jacobian.

V. Analysis and results
This section provides first results and analysis about the results obtained with the
simulation environment. Within the simulation environment a steel sheet of 10 £ 10 cm
was modeled. The steel sheet was subdivided into 9 £ 9 subdomains resulting in
COMPEL        an inverse problem with a number of unknown of 81. Above the plate an array of
30,6          5 £ 5 coils was placed. The conductivity of the sheet was set to s ¼ 106 Sm2 1. The
              plate has the dimensions of 100 £ 100 £ 1 mm. For the simulations linear eight-nodal
              hexahedral (brick) elements were used, to discretize the problem domain.

              A. Picard condition
1944          For a first analysis about the behavior of the inverse problem, the so-called Picard
              condition (Hansen, 1998) can be evaluated. The Picard condition states, that for discrete
              inverse problems of form Ax ¼ b, the existence of a best approximate and bounded
              solution is given, if the Fourier coefficients ju T bj with respect to the singular functions
                                                               i
              decay fast enough relative to the singular values si. For the computation of the
                                        T
              Fourier coefficients ju i bj, a singular value decomposition of the matrix A has to
              be performed, given by A ¼ USV T. Then the vectors ui are the column vectors of the
              matrix U and the singular values si are the diagonal entries of the matrix S. For the
              nonlinear inverse problem (8), the Picard condition can be verified for the linearization
              (10), where A ¼ (JJ T þ aL TL) represents the Gauss-Newton system matrix and b is
              the gradient of equation (8) in equation (10).
                 Figure 2 shows the Picard plot for a typical permeability distribution. For this case,
              the regularization parameter a has been set to zero. One can see that the Fourier
              coefficients stay almost constant while the trend of the singular values has a fast
              decay. Hence, the Picard condition is not fulfilled, which means, that the problem has
              no bounded solution. Hence, the problem requires a regularization, which indicates that
              the inverse problem is of ill-posed nature.
                   10
                                                                                                  σi



                    5




                    0
                        0         10    20        30        40       50        60        70       80


                        × 10–13
                    2
                                                                                                T
                                                                                              |u b|
                                                                                                i
                  1.5

                    1

                  0.5


Figure 2.           0
                        0         10     20       30        40        50       60        70       80
Picard plot
                                                             i
B. Sensitivity analysis                                                                            Stress zone
Figure 3(a) shows a column vector of the Jacobian J plotted for the individual pixels.         imaging in steel
Such a plot is referred to as a sensitivity map. The transmitter coil is positioned in the
lower left corner of the plate, while the receiver coil is positioned in the upper right                plates
corner of the plate. One can see that by this measurement setup, the sensitivity is
distributed over a large area of the plate. In contrast, Figure 3(b) shows the sensitivity
map when only the actual transmitter coil is used for sensing. In this case the sensor                     1945
only provides spatial information in a close distance around the active coil. This is
essentially given by the flow of the eddy currents. Hence the principle to measure the
induced voltages in the other coils is better suited for the inverse problem of stress zone
imaging, as it provides more information about the permeability distribution.

C. Reconstruction results
Finally, Figure 4 shows first reconstruction results. Figure 4(a) and (c) shows the true
permeability distributions. For the domains concerning to the black pixels the relative
permeability has been set to 10 £ 103, whereas in the perturbated region the
permeability has been set to 5 £ 103. Figure 4(b) and (d) shows first reconstruction
results. For regularization we used a discrete version of a second-order differential
operator. Hence, a smoothness assumption on the solution is incorporated. For the
reconstruction and the generating the test data the same finite element model has been
used. One can see that the reconstruction results indicate the true material distribution.
However, as the reconstruction results do not reach the material values of the original
distribution yet, another scale has to be used for the color. Although the original material
distribution for the first is a symmetric figure with respect to the center of the plate,
Figure 4(b) shows an unsymmetrical result. This is a typical behavior of for ill-posed
inverse problems, in the case, that the regularization parameter a is too low. On the other
hand, for the reconstruction result shown in Figure 4(d), the weight of the regularization
is slightly too high and hence, the result may appear as too smooth.


                                           × 10–16                                  × 10–14
                                            10                                        0

                                                                                      –2


                                            5                                         –4

                                                                                      –6

                                                                                      –8
                                            5
                                                                                      –10

                                                                                      –12
                                            –5
                                                                                                          Figure 3.
                                                                                                    Sensitivity maps
Notes: (a) Sensitivity for different coils; (b) sensitivity for one coil
COMPEL
30,6


1946




                                  (a) True Permeability distribution                (b) Reconstructed distribution




Figure 4.
Exemplary reconstruction
result
                                  (c) True Permeability distribution                (d) Reconstructed distribution


                           VI. Conclusion and outlook
                           In this paper stress zone imaging in steel plates of electrical machines using an inverse
                           problem approach was presented and first simulation-based results have been
                           presented. As this work is referred as a simulation study about the principal
                           applicability of the proposed approach, further work has to cover several important
                           topics in order to develop an applicative system for the production process. A first major
                           step for future research has to deal with real measurements on steel plates and
                           comparative simulation studies of stress zone effects, in order to gain trust in the
                           expected effects and computational methods. In this context also the use of another
                           eddy current formulation for correct computations is of concern. As real stress zones
                           occur in the region near the boundary of the steel sheet, next research focus will be put on
                           inverse problems with specific geometries. Further the formulation and incorporation of
appropriate prior knowledge will be of topic, i.e. since sharp material transitions are of                   Stress zone
interest a total variation approach seems more suitable. In this context also a statistical
formulation of prior knowledge can be of interest, although a statistical inversion
                                                                                                         imaging in steel
approach may yet be too computationally expensive.                                                                plates
References
Biro, O. and Preis, K. (1989), “On the use of the magnetic vector potential in the finite-element                   1947
       analysis of three-dimensional eddy currents”, IEEE Transactions on Magnetics, Vol. 25
       No. 4, pp. 3145-59.
         ¨
Brandstatter, B. (1999), “Optimal design with adjoint state approaches”, PhD thesis, Graz
       University of Technology, Graz.
Hansen, P.C. (1998), Rank-deficient and Discrete Ill-posed Problems: Numerical Aspects of Linear
       Inversion, SIAM, Philadelphia, PA.
Nakata, T., Nakano, M. and Kawahar, K. (1992), “Effects of stress due to cutting on magnetic
       characteristics of silicon steel”, IEEE Translation Journal on Magnetics in Japan, Vol. 7
       No. 6, pp. 453-7.
Ossart, F., Hug, E., Hubert, O., Buvat, C. and Billardon, R. (2000), “Effect of punching on electrical
       steels: experimental and numerical coupled analysis”, IEEE Transactions on Magnetics,
       Vol. 36 No. 5, pp. 3137-40.
Saito, A., Yamamoto, T. and Iwasaki, H. (2000), “Magnetization properties and domain structures
       of grain-oriented silicon steel sheets due to bending stress”, IEEE Transactions on
       Magnetics, Vol. 36 No. 5, pp. 3078-80.
Senda, K., Kurosawa, M., Ishida, M., Komatsubara, M. and Yamaguchi, T. (2000), “Local
       magnetic properties in grain-oriented electrical steel measured by the modified needle
       probe method”, Journal of Magnetism and Magnetic Materials, Vol. 215-216, pp. 136-9.
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About the authors
Markus Neumayer received an MSc in Electrical Engineering from Graz University of Technology in
2008 and is currently a Research and Teaching Assistant at the Institute of Electrical Measurement
and Measurement Signal Processing. His research interests include physical modelling, inverse
problems, model based measurement and signal processing. Markus Neumayer can be contacted at:
neumayer@TUGraz.at
   Daniel Watzenig received an MSc in Electrical Engineering in 2002 and a PhD in 2006 both
from Graz University of Technology, Austria, in 2002 and 2006, respectively. In 2009, he received
the venia docendi. He is currently heading the Automotive Electronics and Control Group of the
Virtual Vehicle Research Center in Graz.
                       ¨
   Bernhard Brandstatter received an MSc in Electrical Engineering in 1996 and a PhD in 1999
both from the Graz University of Technology. He received the venia docendi in 2003. He is
currently the Head of the R&D Department of ELIN EBG Motoren GmbH in Weiz, Austria and
Distinguished Lecturer at the Graz University of Technology.

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