ISSN 10618309, Russian Journal of Nondestructive Testing, 2010, Vol. 46, No. 5, pp. 324–332. © Pleiades Publishing, Ltd., 2010. Original Russian Text © V.R. Skal’skii, B.P. Klim, E.P. Pochapskii, 2010, published in Defektoskopiya, 2010, Vol. 46, No. 5, pp. 14–24.

MAGNETIC AND ELECTROMAGNETIC METHODS

Calculation of the Induction of a Constant Magnetic Field Created in a Ferromagnet by an Attachable Electromagnet V. R. Skal’skii*, B. P. Klim**, and E. P. Pochapskii*** Physicomechanical Institute, Ukrainian Academy of Sciences, ul. Nauchnaya, Lviv, 790053 Ukraine *email: [email protected] **email: [email protected] ***email: [email protected] Received July 16, 2009; in final form, December 5, 2009

Abstract—On the grounds of the numerical calculation of the induction of a magnetic field excited in a ferromagnet by scanners in the form of Πshaped and solenoidal electromagnets, it is established that the Πshaped type provides a deeper penetration of a magnetic field into a ferromagnet and its better localization in a specified volume. The induction of a magnetic field appearing under the action of a Πshaped scanner in a ferromagnet decreased with an increase in both the thicknesses of the ferromag net and the gap between the magnetic conductor of a scanner and the surface of a material and also depended slightly on changes in its width; a maximal scanner winding current exists, above which the induction of a magnetic field in a material changes slightly. Key words: magnetic field, electromagnets, magnetic conductor, Barkhausen effect, magnetoacoustic emission, scanner. DOI: 10.1134/S1061830910050025

The method of acoustic emission (AE) has been successfully used for the diagnosis of materials, prod ucts, and construction elements [1, 2]. In some cases, its application is limited, as the additional load required by the standard [3] for testing an object may exceed the allowable level. In this case, gaining infor mation on the state of a ferromagnetic construction material using the method of magnetoelastic acoustic emission (MAE), which appears during the magnetization reversal of a ferromagnet and is connected with the Barkhausen effect, is promising. The first attempts to record elastic vibrations arising under magneti zation reversal and to study the physical nature of this phenomenon were made as early as the 1940s [4], but were continued later [5–8]. MAE is excited by jumpwise changes concerning the positions of domain walls and accompanying magnetostriction effects [9–11]. In construction steels, these jumps primarily involve 90°domain walls. In these materials, the number of individual elastic wave sources is too high to easily record each single displacement of 90°domain walls. Therefore, the MAE signal is averaged over the volume of a reverse magnetized material, even for low magnetization reversal frequencies. The intensity of MAE signals is determined by the type of ferromagnet and is sensitive to structural changes in the material of a tested object and the parameters of MAE signals depend on thermal process ing modes, residual stresses, material hydrogenation, etc. [12–15]. A degraded material changes both its physicomechanical and domain structures, thus influencing MAE parameters [11, 13]. STATE OF THE ART Since ferromagnetic construction elements have different shapes and sizes, for the efficient application of the MAE method it is necessary to perform the magnetization reversal of a certain material volume. To accomplish this, it is necessary to know how to quantitatively evaluate the reversemagnetized volume that is under the action of an external magnetic field and to determine the distribution of the induction of a magnetic field in it. A number of works are devoted to experimental studies on the distribution of magnetic induction in superficial layers of massive bodies that are magnetized by attachable electromagnets (AEMs) [16, 17]. It has been shown that magnetic induction decreases monotonically with the depth of a slab and that it approaches a certain minimal value. In [16], the width be of the tested slab coincides with that of the AEM 324

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poles. The material magnetization depth is nearly equal to the thickness of an AEM pole. It is shown that, for a large area of the slab surface, the magnetic flux propagates not only with depth, but also into side vol umes, and at the ratio of the pole thickness to the pole width of 1 : 3, a slab is reverse magnetized to a depth that is equal to nearly half of the pole thickness. Research on the distribution of the normal and tangential components of the magnetic induction in the slab volume from an AEM should be considered as a major result of [16]. The tangential component of the magnetic induction is symmetrical relative to the transverse AEM symmetry plane and maximal at the internal edges of AEM poles; it decreases slightly towards the symmetry plane and changes its sign under poles. The normal component is symmetrical relative to the intersection line of the axial plane and the surface of a slab and reaches its maximal values under AEM poles, where it also changes its sign. The slab studied in [16] can be only conditionally considered as a massive one, as its width coincides with that of AEM poles. The distribution of the induction of a magnetic field along the depth of a massive slab, which had a large surface area of 250 × 120 mm and a thickness of 60 mm and was made of construc tion highalloy steel with a coercive force Hc = 10.5 A/cm, was studied in [17]. The tangential component Bt of magnetic induction in a slab was measured in the regions of the area coinciding with the transversal AEM symmetry plane at the zenith point of magnetic flux lines in a specimen at the normal component Bn = 0. As a result, the distribution of the tangential component of the magnetic induction along the depth of a slab at different magnetizing currents in the AEM coil was obtained. A number of other works devoted to the magnetization of massive products by attachable electromagnets are also known. In particular, the optimization of the performance characteristics of attachable electromagnets of coercimeters was exper imentally studied in [18]. A magnetic circuit consisting of an attachable electromagnet and a specimen was theoretically investigated in [19–21]. The analytical expressions for estimating the depth of the mag netization of a massive product by an AEM and the dependences allowing one to calculate approximately the parameters of coercimeter fluxgate electromagnets were derived. The theoretical approaches to the calculation of the distribution of a magnetic field excited by differentlyshaped electromagnets in a ferro magnetic system were considered in [22–25]. The analysis of literature sources shows that experimental investigations on the distribution of the mag netic induction in a ferromagnetic medium are rather unique, expensive, and technically complicated. On the other hand, owing to the complexity of the initial set of Maxwell equations, it is impossible to find its analytical solution in a general case. The dependence of the magnetic induction or intensity of a magnetic field at a point on the coordinates of the latter can be obtained only in a closed form or for a small number of simple cases having a partial character or resulting in such complicated expressions that one has to pro cess them with a computer and represent the final result as parametric curves [26]. In most cases, partial differential equations can be solved only with the use of numerical methods [27]. Their principle consists in the discretization of differential equations, i.e., in the presentation of all or some derivatives in the form of approximate expressions (finite differences or elements), thus transform ing differential equations into sets of algebraic equations. For this purpose, the considered region Θ is cov ered by a grid, and all variables are replaced by mesh functions. In other words, the values of variables are investigated not for the entire infinite set of points in the region Θ, but for a certain finite subset G. For the purpose of solving nonstationary problems, the coordinate grid is supplemented with a time grid. The number of algebraic equations (dimension of a discrete problem) is determined by the product of the number of points in the coordinate grid and the number of independent variables in target differential equations. The method of finite elements is widely used to simplify such calculations [27]. Calculation by this method implies the use of lines and curves in a twodimensional problem or planes or curved surfaces in a threedimensional problem to divide the space occupied by a magnetic field into several parts, which have rather small, but finite sizes, and called finite elements. The scalar magnetic potential (magnetostatic case) of each element is represented as a polynomial with coefficients constant within an element. In particular, for the ith triangular element (twodimensional problem) in a Cartesian coordinate system, the potential is represented by a linear polynomial ϕ(i) = a(i) + b(i)x + c(i)y, where a(i), b(i), and c(i) are unknown coefficients [26]. The target of calculation by the men tioned finiteelement method is to find these coefficients for all elements, as they allow us to determine the scalar magnetic potential at any point. The initial data are the known values of potentials or their gra dients at the boundaries. Each jth node belongs simultaneously to at least two finite elements i and k. Therefore, for the common nodes of adjacent finite elements, we can write the equality ϕj(i) = ϕj(k). The main relationships for the formation of a set of calculation equations can be obtained by the method of functional minimization or methods of weighted residuals, such as the least squares method or Galerkin’s RUSSIAN JOURNAL OF NONDESTRUCTIVE TESTING

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method. The output data supplemented with boundary conditions and energy dependences result in a set of algebraic equation, which allow calculation of the polynomial coefficients for all finite elements. Many software products exist that perform the finite element method, for example, FEMLAB, ELCUT, COSMOSM, FEMM, ANSYS, etc. Among the listed software, the ANSYS software package is most widely used. Its ANSYS/Emag module serves for simulating electromagnetic processes. FORMULATION OF THE PROBLEM AND THE METHOD OF ITS SOLUTION Let us calculate the distribution of the induction of a stationary magnetic field excited by the AEM of a scanning probe (scanner) in a ferromagnetic rectangular parallelepiped with a height d and a square base of side w using a numerical method. Consider a stationary case, as MAE is generally excited with the use of frequencies within the interval from several fractions of hertz to several hertz. This approach makes it possible to apply the magnetostatic approximation and study the influence of the AEM shape and the sizes of a studied object on the value and direction of the magnetic induction vector B at each point of the latter. The geometrical models of attachable electromagnets used in our calculations, namely, Πshaped and solenoidal ones, are shown in Figs. 1 and 2, respectively. The former consists of a Πshaped magnetic con ductor 1 and two windings 2 and 3 and is placed on a studied ferromagnetic specimen 4. The air gap 6 exists between the poles of magnetic conductor 1 and studied specimen 4 and the air medium 5 around them. RUSSIAN JOURNAL OF NONDESTRUCTIVE TESTING

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B, T 1 2

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Fig. 3. Dependence of the induction of a magnetic field B on its intensity H for the materials of (1), magnetic conductor (steel 10895) and (2), ferromagnetic specimen (steel 30).

The latter consists of the rodlike magnetic conductor 1 and winding 2 and is also placed on the studied ferromagnetic specimen 3. The air gap 5 exists between the magnetic conductor 1 and the studied speci men 3 and the air medium 4 around them. The field of the analyzed magnetic systems is threedimensional. The calculation was performed by the method of scalar magnetic potential with the use of SOLID96 eightnode finite elements appropriate for this method [28]. As the grid of finite elements was generated in the free mode, these elements were used in their partial form, i.e., in the form of fournode tetrahedrals. The relationship between the induction and intensity of a magnetic field at any point of a magnetic conduc tor and a tested specimen was specified by the normal magnetization curves of their materials (Fig. 3), i.e., elec trical softmagnetic steel 10895 (curve 1) and construction steel 30 (curve 2), respectively. The relative magnetic permeabilities of the air and the volume occupied by the windings were taken as unity. The windings were modeled with a special Racetrack current element, whose input parameters are its current and geometrical sizes and the number of winding turns [28]. For the models of a Πshaped and solenoidal electromagnets, we specified numbers of winding turns n of 2000 and 4000 in each winding, respectively, and a current I of 1.0 A, thus providing equal magnetomotive forces in them. We also specified the opposite direction of current in the windings of a Πshaped electromagnet. The thickness d of a ferro magnetic specimen was 10 mm. Theoretically, the magnetic fields of the considered magnetic systems extend to infinity. However, as studies show, at a sufficiently short distance from a system, the intensity of a magnetic field is nearly zero [26]. Therefore, in the case of our models, we can consider a convex figure of finite sizes, in particular, a rectangular parallelepiped, instead of an infinite space. Due to these considerations, we did not use INFIN final elements intended for modeling boundaries stretching to infinity. By default, in the calculation by the method of the scalar magnetic potential with the use of the ANSYS software package, the magnetic flux was parallel to the external surfaces bounding the model. A zero mag netic potential was also taken for the node located on the lower boundary surface near the point at which its diagonals intersect. From this point, the values of the magnetic potentials at other points are counted. When partitioning a model, for the purpose of optimizing the number of obtained finite elements, we specified their sizes from 0.5 and 0.8 mm in the air gaps and in a magnetic conductor and specimen, respectively, and to 2.0 mm in air, taking the sizes of the appropriate regions and the probable rate of change in the sought characteristics of a magnetic field into account. To reduce the number of elements, we also took the symmetry of the models into consideration, in particular, the symmetry of a Πshaped electromagnet relative to the XOZ plane. The number of finite elements obtained in this way approached 50000. The calculations were preformed by the method of differential scalar potential (DSP) [26, 28], which is a variant of the method of scalar potential. This method is applied for the case of open magnetic circuits, i.e., for magnetic circuits with air gaps like those used in our work. As a result of the calculation, we deter mined the induction of a magnetic field at each node of a model. RUSSIAN JOURNAL OF NONDESTRUCTIVE TESTING

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CALCULATION RESULTS AND THEIR INTERPRETATION The calculated distribution of the magnitude of the induction vector of a magnetic field over the XOZ plane is shown in Fig. 4 in the form of a zone field pattern [26], for which the entire interval of a plotted value is divided by default into nine parts (zones). From the arrangement of appropriate zones in Fig. 4, it can be seen that the magnetic field created by a Πshaped electromagnet with a certain value of its induction penetrates deeper into a ferromagnetic specimen in comparison with that excited by a solenoi dal electromagnet. The calculated distributions of the B magnitude and the components Bx and Bz of the induction vector of a magnetic field are illustrated in Figs. 5–7. The comparison of the results given in Fig. 5 shows that the magnitude of the magneticfield induction vector in the neighborhood of the point near the axis of an electromagnet at a depth of 6.0 mm for a Πshaped AEM is equal to approximately 0.45 T (Fig. 5b) and higher than that of 0.26 T for a solenoidal AEM (Fig. 5d). The comparison of the results given in Fig. 6 shows that the component Bx of the induction vector of a magnetic field excited by a Πshaped AEM (Figs. 6a and 6b) has the highest value between the poles of a magnetic conductor and reaches approximately 0.45 T at a depth of 6.0 mm in a specimen. In the case of a solenoidal AEM (Figs. 6c and 6d) in the neighborhood of the point near the electromagnet axis at a depth of 6.0 mm in a specimen, this parameter is close to zero and grows up to a value of ±0.2 T, as the distance from the axis reaches 9.0 mm. From the results shown in Fig. 7, it follows that, for a Πshaped AEM (Figs. 7a and 7b), the component Bz of the induction vector of a magnetic field is zero in the neighborhood of the point near the electro magnet axis and grows to a value of ±0.23 T at a depth of 6.0 mm in a specimen under the poles of a mag netic conductor, as the distance from the electromagnet axis increases. For the solenoidal AEM (Figs. 7c and 7d) in the neighborhood of the point near its axis, the corresponding value of magnetic induction is nearly 0.25 T at a depth of 6.0 mm in a specimen. The aboveconsidered calculation results on the induction of a magnetic field excited by AEMs of two types in a studied ferromagnetic specimen allow us to conclude that a [Pi]shaped AEM provides a deeper penetration of a magnetic field into a ferromagnet and its better localization in a specified volume of a specimen. On these grounds, we also studied the distribution of a magnetic field created by a Πshaped AEM. In particular, Fig. 8 illustrates the dependence of the component Bx of the induction vector of a RUSSIAN JOURNAL OF NONDESTRUCTIVE TESTING

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x, mm Fig. 7. Distribution of the component Bz of the induction vector of a magnetic field excited by (a, b) [Pi]shaped and (c, d) solenoidal AEMs in a ferromagnetic specimen at depths of (a, c), 1.0 mm and (b, d), 6.0 mm along the line parallel to the OX axis.

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Fig. 8. Dependence of the component Bx of the induction vector of a magnetic field excited by a Πshaped AEM at points located along the OZ axis on (a), the specimen thickness d; (b), specimen width w; (c) air gap thickness g; and (d), winding current I at depths of (1) 1.0 and (2) 4.0 mm in a specimen and (3), in a magnetic conductor. RUSSIAN JOURNAL OF NONDESTRUCTIVE TESTING

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magnetic field at points located along the OZ axis of a Πshaped AEM on the ferromagnet thickness d (Fig. 8a) and width w (Fig. 8b), the width of the air gap g (Fig. 8c), and the winding current I (Fig. 8d). Consequently, as the thickness of a ferromagnet increases from 5.0 to 25.0 mm, the induction of a mag netic field decrease along the OZ axis, and the rate of decrease becomes rather lower at great thicknesses (Fig. 8a). From the obtained results (Fig. 8b) it follows that the component Bx of the induction vector of a magnetic field at the points along the OZ axis depends slightly on the specimen width within its interval from 60.0 to 120.0 mm. When the width of an air gap between the magnetic conductor of an AEM and a studied ferromagnetic specimen changes within an interval of 1.0–8.0 mm, the component Bx of the induction vector of a magnetic field decreases from 0.7 to 0.2 T and from 0.53 to 0.16 T at depths of 1.0 and 4.0 mm measured along the OZ axis, respectively (Fig. 8c). From the dependence of the component Bx of the induction vector of a magnetic field on the AEM winding current, it can be seen that a maximal value of the winding current (1.0 A in our case) exists, above which the induction of a magnetic field changes slightly, and it is not reasonable to select the current value from this interval (Fig. 8d). The results we obtained with the use of numerical methods agree basically with the known literature data [16–25]. CONCLUSIONS The numerical calculation of the induction of a magnetic field excited in a ferromagnetic specimen by [Pi]shaped and solenoidal AEMs allows us to draw the following conclusions. (1) A [Pi]shaped AEM provides a deeper penetration of a magnetic field into a ferromagnet and its better localization in a specified volume. Thus, for a solenoidal AEM, the induction of a magnetic field at a depth of 6.0 mm in a ferromagnet is 0.26 T, and for a Πshaped AEM, the analogous parameter is 0.45 T, thus exceeding the former value by 1.73 times. (2) The induction of a magnetic field created by a Πshaped AEM in a ferromagnet decreases with the increasing width of the ferromagnet and the thickness of the air gap between the magnetic conductor of an AEM and the surface of the material and also depends slightly on the change in the ferromagnet width. For example, a increase in the thickness of a ferromagnet from 5.0 to 25.0 mm produces a 2.5fold decrease in the induction of a magnetic field at a depth of 4.0 mm, and the rate of decrease becomes lower at greater thicknesses. The change in the width of the air gap between the magnetic conductor of an AEM and the studied specimen from 1.0 to 8.0 mm gives a 3.3fold decrease in the induction of a magnetic field at a depth of 4.0 mm in a ferromagnet. (3) For the considered types of AEMs, a maximal value of the winding current of 1.0 A exists, above which the induction of a magnetic field in a ferromagnet changes slightly. REFERENCES 1. Skal’s’kii, V.R. and Koval’, P.M., Akustichna emisiia pid chas ruinuvannia materialiv, virobiv i konstruktsii. Metodolohichni aspekti vidboru ta obrobki informatsii (Acoustic Emission in the Destruction of Materials, Prod ucts, and Constructions. Methodological Aspects of Information Gathering and Processing) Lviv: Spolom, 2005. 2. Klim, B.P., Mikitin, G.V., Pochapskii, E.P., and Bukhalo, O.P., Aspects of the Pickup of an AcousticEmission Signal, Tekhn. Diagn. Nerazrush. Kontr., 2000, no. 3, pp. 17–23. 3. DSTU (National Standard of Ukraine) 4227–2003: Guidelines on the AcousticEmission Diagnosis of Extra Haz ardous Objects, 2003. 4. Vonsovskii, S.V. and Shur, Ya.S., Ferromagnetizm (Ferromagnetism), Moscow–Leningrad: OGIZ, 1948. 5. Lord, A.E.Jr., Acoustic Emission, in Physical Acoustics: Principles and Methods, Mason, W.P. and Thurston, R.N., Eds., New York: Academic Press, 1975, vol. 11, pp. 253–289. 6. Kolmogorov, V.N., Some Researches on the Acoustic Emission in Ferromagnetic Materials, Tez. dokl. Vses. nauch.tekhn. seminara (Proc. of AllRuss. Sci.Tech. Seminar), Khabarovsk, 1972. 7. Ono, K. and Shibata, M., Magnetomechanical Acoustic Emission of Iron and Steels, Mater. Evaluation, 1980, vol. 38, pp. 55–61. 8. Shibata, M. and Ono, K., Magnetomechanical Acoustic Emission: a New Method of Nondestructive Stress Measurement, NDT International, 1981, vol. 14, no. 5, pp. 227–234. 9. Ranjan, R., Jiles, D.S., and Rastogi, P.K., Magnetoacoustic Emission, Magnetization, and Barkhausen Effect in Decarburized Steels, IEEE Trans. Magn., 1986, vol. MAG22, no. 5, pp. 511–513. 10. Volkov, V.V., Kumeishin, V.F., Chernokhovskii, M.Yu., et al., On the Acoustic Emission of Reversely Magne tized Ferromagnets, Defektoskopiya, 1986, no. 4, pp. 21–28. RUSSIAN JOURNAL OF NONDESTRUCTIVE TESTING

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11. Buttle, D.J., Scruby, C.B., Yakubovics, J.P., and Briggs, J.A., Magnetoacoustic and Barkhausen Emission: Their Dependence on Dislocation in Iron, Phil. Mag., 1987, vol. 55, no. 6, pp. 717–734. 12. Gorkunov, E.S., Khamitov, V.A., Bartenev, O.A., et al., Magnetoelastic Acoustic Emission in Thermally Pro cessed Construction Steels, Defektoskopiya, 1987, no. 3, pp. 3–9. 13. Gorkunov, E.S, Bartenev, O.A., and Khamitov, V.A., Magnetoelastic Acoustic Emission in Ferrosilicon Single Crystals, Izv. Vuzov, Ser.: Fizika, 1986, no. 1, pp. 62–66 [Rus. Phys. J. (Engl. Transl.), 1986, vol. 29. no. 1, pp. 56–59]. 14. Boltachev, V.D., Golovshchikova, I.V., Ermakov, A.E., and Dragoshanskii, Yu.N., Barkhausen Effect and Mag netoacoustic Emission in the FeAl, FeCo, and FeSi Alloys, Fiz. Met. Metalloved., 1992, no. 12, pp. 59–67. 15. Nazarchuk, Z.T., Skal’s’kyi, V.R., Klym, B.P., et al., Influence of Hydrogen on the Changes in the Power of Barkhausen Jumps in Ferromagnets, Fiz.Khim. Mekh. Mater., 2009, vol. 45, no. 5, pp. 49–54 [Mater. Sci. (Engl. Transl.), 2009, vol. 45, no. 5, p. 663]. 16. Mikheev, M.N., Topography of the Magnetic Induction in Products Locally Magnetized with Attachable Elec tromagnets, Izv. Acad. Nauk SSSR, Otd. Tekhn. Nauk, 1943, nos. 3–4, pp. 68–77. 17. Bida, G.V., On the Depth of Magnetization of Massive Products with an Attachable Electromagnet and the Depth of Testing Their Operational Properties, Defektoskopiya, 1999, no. 9, pp. 70–81. 18. Bida, G.V. and Sazhina, E.Yu., Optimization of the Performance Characteristics of Attachable Electromagnets, Defektoskopiya, 1996, no. 5, pp. 92–99. 19. Bida, G.V., Mikheev, M.N., and Kostin, V.N., Determination of the Sizes of an Attachable Electromagnet Des tined for the Nondestructive Testing of the Depth and Hardness of CaseHardened Layers, Defektoskopiya, 1984, no. 8, pp. 10–16. 20. Fridman, L.A., Tabachnik, V.P., and Chernova, G.S., Magnetization of Massive Ferromagnetic Products with Attachable Electromagnets, Defektoskopiya, 1977, no. 4, pp. 104–112. 21. Zakharov, V.A., On the Theory of Attachable Magnetic Devices with a Magnetic Conductor, Defektoskopiya, 1978, no. 3, pp. 75–81. 22. Gazizova, G.G., Guseinova, T.I., Kaganov, Z.G., and Fradkin, B.M., Calculation of the Magnetic Field of an Attachable Electromagnet with a ΠShaped Core, Defektoskopiya, 1981, no. 3, pp. 71–76. 23. Shchur, M.L., Shleenkov, A.S., and Shcherbinin, V.E., On the Formation of the Field and Induction in Ferro magnetic Media, Defektoskopiya, 1983, no. 10, pp. 11–18. 24. Pechenkov, A.N., Calculation of the ThreeDimensional Magnetic Field Produced by a Circular DCCarrying Coil of Rectangular Cross Section, Defektoskopiya, 2006, no. 9, pp. 65–71 [Rus. J. Nondestr. Test., (Engl. Transl.), 2006, vol. 42, no. 9, pp. 610–615]. 25. Matyuk, V.F., Osipov, A.A., and Strelyukhin, A.V., Descriprion of the Magnetization of SoftMagnetic Hollow Cylindrical Rods in a Uniform QuasiStatic Magnetic Field. Normal Magnetization Curve, Tekhn. Diagn. Ner azrush. Kontr., 2008, no. 1, pp. 31–34. 26. Bul’, O.B., Metody rascheta magnitnykh sistem elektricheskikh apparatov. Programma ANSYS (Methods for Cal culating the Magnetic Systems in Electric Devices. ANSYS Software), Moscow: ACADEMIA, 2006. 27. Bakhvalov, N.S., Zhidkov, N.P., and Kobel’kov, G.G., Chislennye metody (Numerical Methods), Moscow: Lab. baz. znanii, 2000. 28. Documentation for ANSYS. Release 11.0.

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