Determination of Local Magnetic Material Properties Using an Inverse Scheme

The precise knowledge of material properties is of utmost importance for motor manufacturers to design and develop highly efficient machines. Due to different manufacturing processes, these material properties can vary greatly locally and the assumption of homogeneous material parameters for the electrical steel sheets is no longer feasible. The goal of our research project is to precisely determine these local magnetic material properties using a combined approach of measurements, numerical simulations, and the applications of inverse methods. In this article, we focus on the identification of the local linear permeability of electrical sheets considering cutting edge effects. In doing so, the electrical sheets are divided into subdomains, each assigned with a linear magnetic material model. The measurement data are generated artificially by solving the magneto-static case using the finite element (FE) method and overlay these data with a Gaussian white noise. Based on the measured and simulated data, we apply our inverse scheme to determine the parameters of the linear material model. To ensure solvability of the ill-posed inverse problem, a Tikhonov regularization is used and the regularization parameter is computed via Morozov’s discrepancy principle.


I. INTRODUCTION
V ARIOUS cutting processes, e.g., punching, water cutting or laser cutting, are an essential part in the production chain of electrical steel sheets.It is generally known that these processes lead to a deterioration of the magnetic properties, especially in the area of the cutting edges (see [1] and [2]), and can significantly reduce the efficiency of electrical machines.Therefore, the accurate knowledge of the local magnetic properties is of utmost importance for motor manufacturers and must be taken into account during the design and development process.
A widely used approach to determine the influence of cutting edges is to increase the cutting length to bulk material ratio (see [3] and [4]).This is achieved by cutting an electrical steel sheet into several smaller strips, such that the total width of all strips aligned next to each other matches the original sheet size.Varying the strip width leads to different cutting length to bulk material ratio combination, which are then measured with a single sheet tester (SST) or Epstein frame.
In our approach, we combine methods based on measurements, numerical simulations, and inverse schemes to determine the magnetic material behavior locally.To generate measured data, a sensor-actuator (SA) system is developed, capable of locally exciting the electrical steel sheets and measure the magnetic flux density.Using an appropriate SA model, numerical data are produced using the finite element (FE) method.Based on the measured and numerical data, an inverse problem is solved to determine the search-for unknown parameters of an assumed material model.In this article, the general methodology is presented and the accuracy as well as convergence of our approach is investigated.Therefore, we restrict ourselves to a linear (no dependence on the field itself) magnetic material behavior for the electrical steel sheets.The measured data are artificially generated by forward simulations solving the magnetic field for the magneto-static case applying the FE method and overlay this data with a Gaussian white noise.Starting with an initial guess for the unknown material parameter and considering the artificially generated measurement data, the inverse scheme is applied to determine the search-for parameter for the considered material model.Due to the ill-posedness of the inverse problem, a Tikhonov regularization [5] is applied to ensure solvability.

II. SA MODEL
The SA system in Fig. 1 is considered to be a stacked iron core and is excited with two excitation coils.The system is capable of magnetizing the electrical sheets and measuring the magnetic flux density locally, using a sensor array.The sensor array includes S Hall-and/or GMR-sensors and measures the x-, y-, and z-components of the magnetic field.Two electrical steel sheets (Sample 1 and Sample 2) are placed concisely together along the cutting edge (see Section II-B).Since both samples originate from the same batch and unchanged cutting process parameters are considered, identical and symmetric material behavior for sample 1 and sample 2 can be assumed.
For the generation of measurement data, sample 1 and sample 2 are measured at N different positions along the x-direction.Due to the high change in material behavior in the vicinity of cutting edges, the density of measurement positions in this area are much higher than in the bulk material.For the sake of completeness, between each measurement position, the sheets are demagnetized to ensure, no residual magnetism is present (assumption for the numerical simulation).The resulting measuring data contain the three magnetic field components for each sensor and measurement position B meas x,i, j , B meas y,i, j , and B meas z,i, j with i = 1, 2, . . ., S the sensor positions and j = 1, 2, . . ., N the measurement positions.Based on these data, the amplitude of the magnetic field density is computed using the euclidean norm and results in the measurement data

A. Numerical Model to Consider Edge Effects
The deterioration of the material parameters due to the cutting edges is pronounced within a small range of millimeters up to maximum centimeters and depends strongly on the used cutting technique as well as cutting process parameters.To appropriately represent the large material change in this area during the simulation, each electrical steel sheet is divided into M non-equidistant distributed subdomains m , where the subdomain size in the immediate vicinity of the cutting edges is much smaller than in the bulk material (see Fig. 2).
In the numerical simulation, the electrical steel sheet is generally subject to a certain material behavior that is characterized by a corresponding material model μ (linear, nonlinear, or hysteretic model).Depending on the chosen model, a different number of selectable parameters defines the material model μ = f (a 1 , . . ., a K ) and must be determined, respectively, such that, the model behavior fits the real material behavior.In order to take the cutting edge effects into account, the selected material model is assigned to each subdomain, whereby the model parameters for each subdomain a 1,m , . . ., a K ,m can be selected independently.Thus, the searched-for parameter vector reads as p = (a 1,1 , . . ., a K ,1 , a 1,2 , . . ., a K ,2 , . . ., a 1,M , . . ., a K ,M ) T .The advantage of this approach is that no adaptation of the material model is necessary to take into account factors influencing the magnetic material behavior, e.g., residual stresses, microstructure, etc., since these are inherently included in the model parameter for each subdomain.

B. Sample Arrangement and Magnetization of the Electrical Steel Sheets
For the proposed method, it is crucial that a flux density change is clearly present due to the degradation of magnetic properties caused by the cutting edges.For this purpose, we consider the following arrangements and evaluate the magnetic flux densities at the measurement point s 1 for the case without (homogeneous linear relative permeability µ exact r,5 in Table II) and with cutting edge effects, using the sample model as shown in Fig. 5 and the permeabilities µ exact r,m listed in Table II.The arrangement with only one sheet shows hardly any magnetization in the vicinity of the cutting edges (see Fig. 3).Thus, the change in flux density is caused predominantly geometrically.Therefore, changing the material properties in the nonmagnetized area does not lead to any significant field change.Consequently, the sensitivity (∂ B/∂ μ) can be classified as very small.
For the second case, two sheets are placed parallel along the cutting edges.As shown in Fig. 4, with this arrangement, the magnetization is clearly pronounced in the cutting edge area.Therefore, a high sensitivity can be assumed.
The assumptions made regarding the sensitivity are confirmed by computing the relative change of the magnetic flux density due to neglected and considered cutting edge effects Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.I.

III. INVERSE SCHEME
The inverse scheme calculates the searched-for parameter vector p based on the measured magnetic flux densities B meas and the simulated magnetic flux densities B sim .Therefore, a nonlinear least squares problem has to be solved to find the optimal parameter p opt , such that the error norm between B meas and B sim is minimized.Due to the inevitable measurement noise in the data, difficulties in solving the nonlinear root squares problem occur.More precisely, small perturbations in the measurement data have a pronounced extreme negative effect on the computed parameters and cause the solution strategy to diverge.From a mathematical point of view, this can be stated as an ill-posed problem.To overcome this problem, a Tikhonov regularization is applied to ensure convergence.In doing so, the minimization problem reads with F i (x j , p) = B sim i (x j , p) − B meas i (x j ), N p the number of measurement positions, N s the number of sensors position, B sim i (x j , p) the simulated magnetic flux density, B meas i (x j ) the measured magnetic flux density, α the regularization parameter, A the magnetic vector potential and J the electric current density.Finding the optimal parameter p opt of the minimization problem, ( 2) is performed iteratively, using a quasi-Newton scheme [6] with I the identity matrix, q the search direction, p ref a priori information, λ the line search parameter (determined by Armijo rule) and B the approximated Jacobian using Broyden's update formula Since an iterative solution strategy is used, a stopping criterion must be defined.Therefore, the following error norm is used: A. Regularization Parameter The choice of an appropriate regularization parameter is crucial for finding an optimal solution during the iterative procedure.Since, an a priori upper bound δ for the error norm with B exact the exact data without noise, is available, the discrepancy principle of Morozov is used.Therefore, an initial regularization parameter α init is chosen, such that the regularization term is pronounced compared to the error term ∥F i (x j , p)∥ 2 2 .This is necessary due to the initial poorly approximated Jacobian B, which shows a very high condition number.Otherwise, the optimization procedure diverge.During the iteratively solved minimization problem, α init is decreased in each iteration step by until the following condition is fulfilled For all computations, a = 0.9 and α init = 1 has been chosen.

IV. NUMERICAL RESULTS
As an example, electrical steel sheets with dimensions 32 × 0.5 × 20 mm are characterized.The affected area due to cutting is w CE = 5 mm.As described in Section II-A, each steel sheet is decomposed into ten subdomains (see Fig. 5), with x m ∈ [0.5, 1, 1.5, 2, 11] in mm.Furthermore, a linear material behavior is considered and the material model assigned for each subdomain is defined by with µ r,i the relative permeability for the subdomain i and µ 0 the permeability in vacuum.
Since symmetrical material properties are assumed, the searched-for parameter vector of the minimization problem reads p = (µ r,1 , µ r,2 , µ r,3 , µ r,4 , µ r,5 ) T . (13) In total, the electrical steel sheets are measured at six different positions with the SA system, whereby the measurement  To test the proposed inverse scheme, it is assumed, that the exact relative permeabilities µ exact r as well as a priori information regarding the material change are known.Due to the inverse procedure, an initial guess of the relative permeabilities are necessary.All the considered values for the initial µ init r , reference µ ref r (a priori information) and exact µ exact r relative permeability for each subdomain are listed in Table II.
The permeabilities (µ init r , µ ref r , and µ exact r ) for the subdomain 5 are equal.This is based on the assumption, that the material properties of the bulk material are known from SST or Epstein measurements.Thus, this parameter is assumed to be constant during the iterative procedure.
The measurement data B meas are generated artificially by 3-D forward simulations solving the magnetic field for the magneto-static case for the exact permeabilities.Furthermore, the generated data are overlaid by a Gaussian white noise with 10% standard deviation.The results of the FE simulation for the exact and noisy measurement data are shown in Fig. 6.
Based on the given data, the searched-for parameter p is computed, using the proposed method and the results are shown in Fig. 7. Overall a smooth convergence of the parameters can be observed.The remaining error of the parameters is due to the following points.First, due to the measurement noise, only a solution in the vicinity of the real solution (without noise) can be found.Second, depending on the regularization parameter, the regularization term and subsequently the a priori information are included in the calculated parameters.

V. CONCLUSION
In this article, a methodology for determining the local magnetic material properties taking into account cutting edge effects is described.Therefore, a concept of a SA system is proposed, capable to magnetize the electrical steel sheets and measure the magnetic flux density locally, using 3-D Hall sensors.Based on the measured and simulated data, an inverse problem is solved to determine unknown the parameters of a defined material model.This is performed by a quasi-Newton method, where the Jacobian is approximated via a Broyden update formula.Due to the ill-posedness, a Tikhonov regularization is used to ensure solvability of the iterative solution strategy.The methodology is applied to an example considering a spatially varying linear magnetic peremability.The results show a fast and smooth convergence of the model parameter.For future work, this procedure is tested for a nonlinear material model.

Fig. 1 .
Fig. 1.Concept of the SA system with two electrical steel sheets (sample 1 and sample 2).s i the number of sensors in the sensor array.

Fig. 2 .
Fig. 2. Example of sample discretization into M subdomains m (color coded), each subdomain assigned with a material model μm .The sample width is w S and w CE is the affected area due to cutting.x m is the length of the subdomains.
FOR CONSIDERED ARRANGEMENTS with B CE the norm of the magnetic flux density taking into account cutting edge effects and B const the norm of the magnetic flux density considering homogeneous material behavior.The results are shown in Table

Fig. 5 .
Fig. 5. Measurement position 1 of the SA system with six sensors for the characterization of electrical steel sheet with ten subdomains, each subdomain assigned a linear material model μlin m .

Fig. 6 .
Fig. 6.Measurement data B meas exact (without noise) and with noise for number of sensors N S = 6 and number of measurement positions N P = 6.

Fig. 7 .
Fig. 7. Convergence behavior and error of the searched-for parameter p.

TABLE II CONSIDERED
INITIAL µ init r , REFERENCE µ ref r AND EXACT µ exact