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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 Permanent link to this document: http://dx.doi.org/10.1108/03321641111168219 Downloaded on: 15-05-2012 References: This document contains references to 9 other documents To copy this document: permissions@emeraldinsight.com This document has been downloaded 165 times. 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The organization is a partner of the Committee on Publication Ethics (COPE) and also works with Portico and the LOCKSS initiative for digital archive preservation. *Related content and download information correct at time of download. The current issue and full text archive of this journal is available at www.emeraldinsight.com/0332-1649.htm 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. Urankar, L. (1980), “Vector potential and magnetic ﬁeld of current-carrying ﬁnite arc segment in analytical form, part I: ﬁlament approximation”, IEEE Transactions on Magnetics, Vol. 16 No. 5, pp. 1283-8. Werner, E. (1957), “Einrichtung zur Messung magnetischer Eigenschaften von Blechen bei Wechselstrom-magnetisierung”, Austrian Patent No. 191015. 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. To purchase reprints of this article please e-mail: reprints@emeraldinsight.com Or visit our web site for further details: www.emeraldinsight.com/reprints