ISSN 10618309, Russian Journal of Nondestructive Testing, 2012, Vol. 48, No. 1, pp. 23–34. © Pleiades Publishing, Ltd., 2012. Original Russian Text © V.R. Skal’skii, B.P. Klim, E.P. Pochapskii, 2012, published in Defektoskopiya, 2012, Vol. 48, No. 1, pp. 30–43.

MAGNETIC METHODS

Distribution of the Induction of a QuasiStationary Magnetic Field Created in a Ferromagnet by an Attachable Electromagnet V. R. Skal’skii, B. P. Klim, and E. P. Pochapskii Karpenko Physicomechanical Institute, National Academy of Sciences of Ukraine, Ukraine email: [email protected]; [email protected]; [email protected] Received January 17, 2011; in final form, April 15, 2011

Abstract—As a result of numerical calculations, the spatial and time distributions of the induction of a quasistationary magnetic field, which was created in a studied ferromagnet by Ushaped and sole noidal attachable electromagnets (AEMs) at frequencies of 1 and 10 Hz, were obtained. The presence of a phase shift between the amplitude values of the magneticfield induction at different depths in the specimen, which increases with the frequency of the exciting current, was revealed. The dependences of the magneticfield penetration depth into the ferromagnet on the magnetic reversal frequency were studied. It was found that for the frequency range under study the magneticfield penetration depth for both types of AEM is smaller than that for the corresponding stationary case. Keywords: ferromagnet, magnetoelastic acoustic emission, attachable electromagnet (AEM), mag netic field, finiteelement method, magneticfield induction distribution. DOI: 10.1134/S106183091201010X

The magnetoelastic acoustic emission (MAE) method is a promising method for obtaining informa tion on the technical state of a ferromagnetic construction material [1–6]. This emission arises during the magnetization reversal of ferromagnets and is associated with the Barkhausen effect; in structural steels, the MAE is caused primarily by jumps of 90° domain walls [7–12]. The implementation of the MAE method under actual operational conditions of ferromagnetic ele ments of structures provides the magnetization reversal of a certain volume of a diagnosed object. In this case, the locality and depth of the reversely magnetized region are very important. Therefore, calculating the spatiotemporal distribution of the induction of a quasistationary magnetic field, which is created by an attachable electromagnet (AEM) in a ferromagnetic material, is an important problem. STATE OF INVESTIGATIONS Some results of experimental studies of the distribution of the induction of a stationary magnetic field that is created by an AEM in a ferromagnetic specimen are known from references [13–15]. Calculations that were based on the concept of an AEM–ferromagnetic specimen magnetic circuit were also carried out for a stationary case [16–18]. Theoretical approaches with consideration for Maxwell’s equations ensure the calculation of the distribution of a stationary magnetic field, which is created by electromagnets of different configurations in a ferromagnetic system for several simple particular cases [19–22]. The results of numerical calculations of the induction of a stationary magnetic field that is created by a U shaped or solenoidal AEM in a ferromagnetic specimen are also known [23, 24]. However, as was shown by the analysis of literature sources, data that contain the spatial and time distributions of the induction of a quasistationary magnetic field that is created by an AEM in a ferromagnetic specimen are virtually absent [25]. In our investigations, quasistationary magnetic fields are considered because, in most cases, MAE is excited by using magnetizationreversal frequencies ranging from fractions to units of hertz. Fields at such frequencies include the fields of electrotechnical devices that operate at commercial frequencies. In the quasistationary electromagnetic approximation, the frequency of changes in the magnetiza tionreversing field is such that the delay effect (radiation effect) and displacement currents can be disre garded [25]. The predominant influence is exerted on electromagnetic processes by conduction currents, which also include eddy currents induced by an alternating magnetic field, in addition to currents from foreign forces that are induced by applied external stresses (as is observed in a stationary case). The value 23

24

SKAL’SKII et al. Y 5

1

2

3

x O 4

Fig. 1. A planemeridional geometrical model of a solenoidal AEM.

I, A

(a)

U, V

(b)

1.0 –16

0.5

0

0

t1

t2

–0.5

16

–1.0 0

0.08

0.16 t, s

0

0.08

0.16 t, s

Fig. 2. The time dependences of the voltage (a) applied to the AEM winding and the current in it (b) for a sinusoidal signal with an amplitude of 30 V and a frequency of 10 Hz.

of eddy currents in ferromagnetic materials is affected by both the electrical conductivity and the magnetic permeability. Because of the complex character of the initial system of equations of the electromagnetic field, it is impossible to find the analytical solution in a general case, as in the case of a stationary magnetic field; there fore, for calculating the distribution of the induction of a quasistationary magnetic field that is created by a solenoidal and Ushaped AEM in a ferromagnet we use the numerical finiteelement method [26]. The results obtained in these calculations are of applied importance for developing optimal (with respect to their dimensions and consumed power) AEMs and their correct operation for implementing the MAE method. FORMULATION OF THE PROBLEM AND CALCULATION RESULTS AEM with a solenoidal shape. If a studied specimen is specified in the form of a cylinder in the geomet rical model of a solenoidal AEM, the model is characterized by an axial symmetry and its calculation can be reduced to the 2D case. Figure 1 shows a plane meridional geometrical model of a solenoidal AEM for this case. It consists of rodlike magnetic core 1 with a radius r = 6 mm and a height h = 50 mm, winding 2, and specimen 3 of the studied ferromagnetic material with a radius R = 30 mm and a thickness d = 10 mm. RUSSIAN JOURNAL OF NONDESTRUCTIVE TESTING

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25

(b)

8 5 4 3

3 2 7

1 1 Y Z

9

Y

9

X

Z

2

X

7

9 0

8 0.18

7 0.37

6 0.56

5 0.75

0.94 B, T

4

3 1.13

2 1.32

1 1.51

170

Fig. 3. The distribution of the magnitude of the magneticfield induction vector in the XOY plane of the solenoidal AEM model at instants of (a) 0.026 and (b) 0.26 s in the calculation for a sinusoidal exciting current at frequencies of 10 and 1 Hz, respectively.

There is an air gap 4 of height h = 1 mm between the magnetic core and the specimen and air volume 5 surrounds them. The field of the analyzed magnetic system is twodimensional (2D). The calculation was performed by the vector magnetic potential method [27]. The magnetic properties of magnetic core 1 and specimen 3 were mainly described by the magnetiza tion curves of electrical soft 10895 magnetic steel and 30 tool steel, respectively [23]. The relative magnetic permeability of air and the region occupied by the AEM winding was assumed to be unity. The resistivity of the magnetic core and specimen was assumed to be ρ = 10–7 Ω m and the resistivity of the copper winding ρ = 3 × 10–8 Ω m. Specifying the resistivities of the magnetic core and specimen under study means that when the magnetic field changes eddy currents arise in them; the resis tivity of air region 5 is assumed to be infinitely high, thus indicating their absence. When simulating wind ing 2, the initial parameters are the number of turns n = 2000, the winding filling factor, its geometrical dimensions, and the current direction. Calculations were performed at individual points of the time axis with a certain preset constant step during an interval of two periods of the source of the exciting sinusoidal voltage. The calculation yielded the data on the induction of the magnetic field in each unit of the model at each discrete moment of time, as well as the time dependences of the voltage applied to the AEM winding and the current flowing through it. RUSSIAN JOURNAL OF NONDESTRUCTIVE TESTING

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SKAL’SKII et al. B, T

(a)

(b)

B, T

0.4

0.08

0.2 0.04

0 B, T

0.08 (c)

0.16 t, s

0 B, T 0.2

0.08 (d)

0.16 t, s

0.64

0.32

0.1

0

0.8

1.6

t, s

0

0.8

1.6

t, s

Fig. 4. The time dependence of the magnitude of the magneticfield induction vector in the ferromagnetic specimen for a sinusoidal exciting signal with an amplitude of 30 V at frequencies of 10 and 1 Hz, respectively, at depths of (a, c) 1 and (b, d) 5 mm.

Figure 2 shows the time dependences of the voltage (a) applied to the AEM winding, which is placed on the cylindrical specimen, and the current in it (b) for a sinusoidal signal. The phase shift between the current and voltage is Δϕ = 360°(t2 – t1)/T ≈ 360°(0.135 – 0.125)/0.1 = 36°, This points to the substantial influence of the reactive component of the electriccircuit impedance, which is caused by the inductance of the AEM–specimen system. Here, T is the oscillation period and t1 and t2 are the instants at which the voltage and current reach the amplitude values. The significant decrease in the phase shift between the current and voltage for a magnetizationreversal frequency of 1 Hz indicates a weaker effect of the inductance of the AEM–specimen system. This results in a larger current amplitude value (~1.5 A against 1.2 A at a frequency of 10 Hz) in this case. Figure 3 shows (in the form of a zone pattern) the distribution of the magnitude of the magneticfield induction vector in the XOY plane of the solenoidal AEM model. The time dependences of the magnitude of the magneticfield induction vector in the ferromagnetic specimen at depths of 1 mm (Figs. 4a, 4c) and 5 mm were constructed (Figs. 4b, 4d) on the basis of the results of the numerical calculation. The comparative analysis of these results shows the decay of the mag neticfield induction with a penetration depth into the specimen and the presence of a phase shift between the amplitude values of the induction at different depths. The latter fact can be explained by the effect of eddy currents that appear in the specimen under the action of an external alternating magnetic field. Figure 5 shows analogous time dependences of the magnitude of the magneticfield induction at the center of the magnetic core. A significant change in the shapes of the curves in comparison to the initial sinusoid can be seen. This follows from the nonlinearity of the characteristic B(H) for the magneticcore material and the influence of its eddy currents. The aforementioned facts of the dependences of the maximum magnitude of the magneticfield induc tion vector and its phase shift on the distance to the specimen surface are represented in the curves that are shown in Fig. 6. It is obvious that the presence of a phase shift as a function of the distance to the spec imen surface must be taken into account when interpreting the experimental results by the MAE methods. The results of the studies of the magnitude of the magneticfield induction vector in the ferromagnetic specimen as a function of the distance to its symmetry axis at different depth and different instants for an exciting sinusoidal signal at different frequencies are presented in Fig. 7. As is seen, the effect of a shift of RUSSIAN JOURNAL OF NONDESTRUCTIVE TESTING

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(a)

1.6

0.8

0.8

0

0.8

(b)

B, T

1.6

1.6 t, s

27

1.6 t, s

0.8

0

Fig. 5. The time dependence of the magnitude of the magneticfield induction vector at the center of the magnetic core for a sinusoidal exciting signal with an amplitude of 30 V at frequencies of 10 and 1 Hz.

B, T

(a)

(b)

P, deg

0.6 80 0.3

40

B, T

(c)

0.8

60

0.4

30

0

4

(d)

P, deg

8 h, mm 0

4

8 h, mm

Fig. 6. The dependence of (a, c) the maximum magnitude of the magneticfield induction vector and (b, d) its phase shift on the distance to the specimen surface for a sinusoidal exciting signal with an amplitude of 30 V at frequencies of 10 and 1 Hz, respectively.

the magneticfield induction peak from the specimen’s center to its edges occurs, which increases with an increase in the field penetration depth. Figure 8 shows analogous dependences for an exciting signal at a frequency of 1 Hz. As is seen, the shift effect is not observed for an instant of 0.26 s (Fig. 8c, 8d). A Ushaped AEM. The 3Dgeometric model specified in Fig. 9 was used in the calculation of the induction of a quasistationary magnetic field created by a Ushaped AEM in a ferromagnet. The model consists of a Ushaped magnetic core 1, two windings 2 and 3, and studied specimen 4 of a ferromagnetic material. There is air gap 6 between the legs of magnetic core 1 and specimen 4; the latter are surrounded by air volume 5 [23]. The vector magnetic potential method was also used to perform the calculation [27]. The AEM wind ings were modeled in the form that is shown in Fig. 10, and the values of the current density were specified by the formula j = (nI/S)sin(ωt), where the number of turns in each winding is n = 2000, S is the winding cross section, I is the current with an amplitude value of 1 A, ω is the cyclic frequency, and t is the time. The number of finite elements obtained during partitioning of the model was ~200000. RUSSIAN JOURNAL OF NONDESTRUCTIVE TESTING

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SKAL’SKII et al. B ×102, T

(a) B, T 0.187

4.249

0.111

2.261

0.035 B, T

0.273 B, T

(c)

0.725

0.084

0.385

0.048

0.045

(b)

(d)

0.012 0

12

24 l, mm

0

12

24 l, mm

Fig. 7. The dependence of the magnitude of the magneticfield induction vector in the ferromagnetic specimen on the distance to the axis of symmetry at depths of (a, c) 1 and (b, d) 5 mm at instants of (a) 0.012 and (b) 0.026 s, respectively, for a sinusoidal exciting signal with an amplitude of 30 V at a frequency of 10 Hz.

(a)

B, T 0.539

0.086

0.291

0.054

0.043

0.023 (c)

B, T

(b)

B, T

(d)

B, T

0.684

0.182

0.364

0.114

0.046

0.044 0

12

24 l, mm

0

12

24 l, mm

Fig. 8. The dependence of the magnitude of the magneticfield induction vector in the ferromagnetic specimen on the distance to the axis of symmetry at depths of (a, c) 1 and (b, d) 5 mm at instants of (a) 0.12 and (b) 0.26 s, respectively, for a sinusoidal exciting signal with an amplitude of 30 V at a frequency of 1 Hz.

The calculation was performed during one period of operation of the excitation source at individual points positioned with a certain preset constant step on the time axis. As a result of this calculation, the following parameters were determined: the magneticfield induction at each unit of the model and at each discrete moment of time and the time dependence of the current density flowing through the model wind ings. Figure 11 shows the specified direction of the current density in the windings of the Ushaped AEM. The currents in the windings of the Ushaped AEM have opposite directions. RUSSIAN JOURNAL OF NONDESTRUCTIVE TESTING

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(b)

30 1 5

5 3

10 10

60

2

Y Z

1 4

X

4 2 6

Z Y

3

6 X

Fig. 9. The geometric model of the Ushaped AEM: (a) the cross section in the XOY plane and (b) the top view.

Z Y X

Fig. 10. General view of the Ushaped AEM winding.

The time dependences of the currentdensity magnitude in the windings of the Ushaped AEM were obtained for the signals at frequencies of 10 and 1 Hz at a current density in the windings of j = 8.68 MA/m2. Figure 12 shows the results of calculating the distribution of the magnitude of the magneticfield induction vector in the XOZ plane of the model. The results obtained from the numerical calculation were used to plot the time dependences of the magnitude of the magneticfield induction vector in the ferromagnetic specimen at different depths and frequencies (Fig. 13). The analysis of the above results shows that the magneticfield induction decays with the depth and a phase shift occurs between the amplitude values of the induction at different depths for a frequency of 1 Hz (a, b). A much larger phase shift between the amplitude induction values is observed at different depths for a frequency of 10 Hz (c, d). Figure 14 shows analogous time dependences of the magneticfield induction magnitude at the center of the magnetic core. Changes in the shapes of the curves in comparison to the initial sinusoid is observed as a result of a nonlinearity of the characteristic B(H) for the magneticcore material and the effect of the eddy currents in the magnetic core. Figure 15 shows the dependences of the maximum magnitude of the magneticfield induction vector on the distance to the specimen surface. The comparison of the behavior of these curves shows that the magnitude of the induction vector decays with the depth into the specimen more slowly for the 1Hz exciting current (Fig. 15a, from 1.15 to 0.38 T) than for the 10Hz exciting current (Fig. 15b, from 1.46 to 0.18 T). RUSSIAN JOURNAL OF NONDESTRUCTIVE TESTING

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Fig. 11. Directions of the current density in the windings of the Ushaped AEM.

(a)

(b)

9 1 1

2 3 5

4

9 0

8

7

6

2 6 7 8 9 5

3 4

5

4

3

2

6 7 8

1

0.244 0.489 0.733 0.978 1.222 1.467 1.711 1.956 2.200 B, T

Fig. 12. The distribution of the magnitude of the magneticfield induction vector in the XOY plane of the model at instants of (a) 0.025 and (b) 0.25 s in the calculation for sinusoidal exciting signals at frequencies of 10 and 1 Hz, respectively.

The obtained quantitative characteristics emphasize a strong influence of eddy current at high frequencies of the exciting current. As a result of these calculations, the dependences of the maximum magnitude of the magneticfield induction vector in the ferromagnetic specimen on the distance to the axis of symmetry of the Ushaped AEM model were determined at different depths at different instants (Figs. 16 and 17). It follows from these figures that for an exciting signal at a frequency of 1 Hz, the maximum magnetic field induction is localized virtually at the specimen center, while the effect of a shift of the maximum magneticfield induction from the specimen center to its edges with an increase in the penetration depth is observed at 10 Hz. RUSSIAN JOURNAL OF NONDESTRUCTIVE TESTING

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(a)

B, T 0.72

0.4

0.40

0.2

31

(b)

0.08 0 B, T

0.4 (c)

0.8

t, s

0 B, T

1.6

0.32

0.8

0.16

0

0.4

0.8

t, s

0

0.4 (d)

0.8 t, s

0.4

0.8 t, s

Fig. 13. The time dependence of the magnitude of the magneticfield induction vector in the ferromagnetic specimen at depths of (a) 4 and (b) 7 mm for a sinusoidal exciting signal at a frequency of 1 Hz and at depths of (c) 4 and (d) 5 mm for a signal at a frequency of 10 Hz.

B, T

(a)

(b)

B, T

1.8

2

1.0

1

0.2 0

0.4

0.8 t, s

0.08 t, s

0.04

0

Fig. 14. The time dependence of the magnitude of the magneticfield induction vector in the magnetic core for sinusoidal exciting signals at frequencies of (a) 1 and (b) 10 Hz.

On the basis of the results of our calculations (additional calculations for an excitation frequency of 5 Hz were performed) and the data from [23] for a stationary magnetic field, we plotted the dependences of the ferromagneticspecimen magnetization depth on the frequency of the exciting signal for a (1) sole noidal and (2) Ushaped AEM (Fig. 18). The results were compared using the known [28] formula δ =

2ρ  , μ 0 μω

(1)

(where μ0 is the magnetic constant, μ is the relative magnetic constant, and ω is the cyclic frequency), on whose basis theoretical curve 3 was constructed. The character of curves 1 and 2 allows one to conclude that, for magnetizationreversal frequencies of 1 Hz or higher, the ferromagneticspecimen magnetization depth for both AEMs is smaller than that obtained upon magnetization with a constant field (7.5 and 3.2 mm for a 1Hz magnetization frequency against 9.5 and 4.8 mm for the constant field for the Ushaped and solenoidal AEMs, respectively). This RUSSIAN JOURNAL OF NONDESTRUCTIVE TESTING

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B, T

(b)

B, T

1.1

1.2

0.9 0.8 0.7 0.4

0.5 0.3 2

0

8 h, mm

6

4

0

2

8 h, mm

6

4

Fig. 15. The dependence of the magnitude of the magneticfield induction vector on the distance to the specimen surface for sinusoidal exciting signals at frequencies of (a) 1 and (b) 10 Hz.

B, T 0.788

(a)

B, T 0.093

0.424

0.053

0.060

0.013

B, T 0.776

(b)

(c)

B, T 0.446

(d)

0.230

0.396

0.016 –24 –12

0

12

0.014 24 l, mm –24 –12

0

12

24 l, mm

Fig. 16. The dependence of the maximum magnitude of the magneticfield induction vector in the ferromagnetic speci men on the distance along the OX axis at depths of (a, c) 1 and (b, d) 5 mm at instants of 0.05 and (b, d) 0.25 s for a sinu soidal exciting signal at a frequency of 1 Hz.

fact confirms that the range of the magnetizationreversal frequencies of interest does not reduce to a sta tionary case and it is necessary to perform such calculations in each particular case (the magnetization reversal frequency and the shape and dimensions of the specimen). From the comparison of curves 1 and 2 in Fig. 18, it is seen that it is desirable to use a Ushaped AEM for deeper magnetization reversal of a ferromagnetic specimen when conducting studies using the MAE method. The departure of theoretical curve 3 from curves 1 and 2 in Fig. 18 can be hypothetically explained by the following conditions that are imposed during derivation of formula (1): constancy of the relative mag netic permeability of the specimen μ = const, planeness of the electromagnetic wave that is incident on the specimen, and the shape of the specimen in the form of an infinite halfspace [28]. The theoretical curve that is often used in calculations of the magneticfield penetration depth confirms that, in each par ticular case, the numerical calculation methods yield the best results in evaluating this parameter. RUSSIAN JOURNAL OF NONDESTRUCTIVE TESTING

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(a)

B, T

0.622

0.113

0.350

0.057

0.078 B, T

0.001 B, T

(c)

1.004

0.167

0.544

0.091

0.084 –24 –12

0

12

33

(b)

(d)

0.015 24 l, mm –24 –12

0

12

24 l, mm

Fig. 17. The dependence of the magnitude of the magneticfield induction vector in the ferromagnetic specimen on the distance along the OX axis at depths of (a, c) 1 and (b, d) 5 mm at instants of 0.005 and (b, d) 0.025 s for a sinusoidal excit ing signal at a frequency of 10 Hz.

h, mm 8 2 3 4

1

0

4

8

f, Hz

Fig. 18. The dependence of the magnetization depth for the ferromagnetic specimen on the frequency of the exciting sig nal for the (curve 1) solenoidal and (curve 2) Ushaped AEM; (3) theoretical curve.

CONCLUSIONS As a result of the numerical calculation of the induction of a quasistationary magnetic field that is cre ated in a studied ferromagnet by Ushaped and solenoidal AEMs at frequencies of 1 and 10 Hz the follow ing results were obtained: (i) For frequencies of 1– Hz, the ferromagneticspecimen magnetization depth for both types of AEMs is smaller than that obtained in a stationary case (7.5 and 3.2 mm for a 1Hz magnetization frequency against 9.5 and 4.8 mm for the constant field for the Ushaped and solenoidal AEMs, respectively); (ii) A phase shift occurs between the amplitude values of the magneticfield induction at different depths, which increases with an increase in the frequency of the excitation current and must be considered when interpreting the MAE results; (iii) As in the case of a stationary magnetic field, the structure of the Ushaped AEM permits the deeper penetration of the magnetic field into the ferromagnet and its better localization in a specified vol ume, as compared to the solenoidal AEM. RUSSIAN JOURNAL OF NONDESTRUCTIVE TESTING

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REFERENCES 1. Vonsovskii, S.V. and Shur, Ya.S., Ferromagnetizm (Ferromagnetizm), Moscow: OGIZ, 1948. 2. Lord, A.E., Acoustic emission, Phys. Acoust., 1975, vol. 11, pp. 253–289. 3. Kolmogorov, V.N., Some Investigations of Acoustic Emission in Ferromagnetic Materials, Tez. dokl. Vsesoyuzn. nauch.tekhn. seminara (Abstracts of Papers of AllUnion Scientific and Technical Seminar), Khabarovsk, 1972, pp. 18–19. 4. Ono, K. and Shibata, M., Magnetomechanical Acoustic Emission of Iron and Steels, Mater. Evaluation, 1980, vol. 38, pp. 55–61. 5. Shibata, M. and Ono, K., Magnetomechanical Acoustic Emission – a New Method of Nondestructive Stress Measurement, NDT International, 1981. 6. Ranjan, R., Jiles, D.C., and Rastogi, P.K., Magnetoacoustic Emission, Magnetization and Barkhausen Effect in Decarburized Steels, IEEE Trans. Magn., 1986, MAG22, no. 5, pp. 511–513. 7. Volkov, V.V., Kumeishin, V.F., Chernikhovskii, M.Yu., et al., On Acoustic Emission of Reversely Magnetized Ferromagnets, Defektoskopiya, 1986, no. 4, pp. 21–28. 8. Buttle, D.J., Scruby, C.B., Yakubovics, J.P., and Briggs, J.A., Magnetoacoustic and Barkhauzen Emission: Their Dependence on Dislocation in Iron, Phil. Mag, 1987, vol. 55, no. 6, pp. 717–734. 9. Gorkunov, E.S., Bartenev, O.A., and Khamitov, V.A., Magnetoelastic Acoustic Emission in Single Crystal of Sil icon Iron, Izv. Vuzov. Ser. Fizika, 1986, pp. 62–66. 10. Gorkunov, E.S., Khamitov, V.A., Bartenev, O.A., et al., Magnetoelastic Acoustic Emission in Thermally Treated Construction Steels, Defektoskopiya, 1987, no. 3, pp. 3–9. 11. Boltachev, V.D., Golovshchikova, I.V., Ermakov, A.E., and Dragoshanskii, Yu.N., Barkhausen Effect and Mag netoacoustic Emission in FeAl, FeCo, and FeSi alloys, Fiz. Met. Metalloved., 1992, no. 12, pp. 59–67. 12. Nazarchuk, Z.T., Skal’s’kyi, V.R., and Klym, B.P., et al., Influence of Hydrogen on the Changes in the Power of Barkhausen Jumps in Ferromagnets, Materials Science, 2009, vol. 45, no. 5, pp. 663–669. 13. Mikheev, M.N., Topography of Magnetic Induction in Items Locally Magnetized by Attachable Electromag nets, Izv. AN SSSR. OTN, 1943, no. 3, pp. 68–77. 14. Bida, G.V., Magnetization Depth of Massive Articles with an Attachable Electromagnet and the Testing Depth of Operational Properties, Defektoskopiya, 1999, no. 9, pp. 70–81. 15. Bida, G.V. and Sazhina, E.Yu., Optimization of Operational Characteristics using Attachable Electromagnets, Defektoskopiya, 1996, no. 5, pp. 92–99. 16. Bida, G.V., Mikheev, M.N., and Kostin, V.N., Determining the Dimensions of an Attachable Electromagnet Designed for Nondestructive Testing of the Depth and Hardness of SurfaceHardened Layers, Defektoskopiya, 1984, no. 8, pp. 10–16. 17. Fridman, L.A., Tabachnik, V.P., and Chernova, G.S., Magnetization of Massive Ferromagnetic Articles using Attachable Electromagnets, Defektoskopiya, 1977, no. 4, pp. 104–112. 18. Zakharov, V.A., To the Theory of Attachable Magnetic Devices with Magnetic Cores, Defektoskopiya, 1978, no. 3, pp. 75–81. 19. Gazizova, G.G., Guseinova, T.I., Kaganov, Z.G., and Fradkin, B.M., Calculation of the Magnetic Field of an Attachable Electromagnet with a Ushaped Core, Defektoskopiya, 1981, no. 3, pp. 71–76. 20. 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. 21. Pechenkov, A.N., Calculation of a 3D Magnetic Field of a Round Coil with a Rectangular Cross Section and a Direct Current, Defektoskopiya, 2006, no. 9, pp. 65–71 [Rus. J. Nondestr. Test. (Engl. Transl.), 2006, vol. 42, no. 9, pp. 610–616]. 22. Matyuk, V.F., Osipov, A.A., and Strelyukhin, A.V., Description of the Process of Magnetizing Hollow Cylindri cal Rods from Soft Magnetic Materials in a Homogeneous QuasiStatic Magnetic Field, Main Magnetization Curve, Tekh. Diagnost. Nerazr. Kontr., 2008, no. 1, pp. 31–34. 23. Skal’skii, V.R., Klim, B.P., and Pochapskii, E.P., Calculation of the Induction of a Constant Magnetic Field Created in a Ferromagnet by an Attachable Electromagnet, Defektoskopiya, 2010, no. 5, pp. 14–24 [Rus. J. Non destr. Test. (Engl. Transl.), 2007, vol. 43, no. 4, pp. 324–332]. 24. Kostin, V.N., Lukinykh, O.N., Smorodinskii, Ya.G., et al., Simulation of Field and Inductance Spatial Distri bution in Locally Magnetized Massive Objects and Optimization of UShaped Transducer Design, Defektosko piya, 2010, no. 6, pp. 13–21 [Rus. J. Nondestr. Test. (Engl. Transl.), 2010, vol. 46, no. 4, pp. 403–410]. 25. Bul’, O.B., Metody rascheta magnitnykh sistem elektricheskikh apparatov. Programma ANSYS (Methods for Cal culating Magnetic Systems of Electrical Apparatuses), Moscow: ACADEMIA, 2006. 26. Bakhvalov, N.S., Zhidkov, N.P., and Kobel’kov, G.G., Chislennye metody (Numerical Methods), Moscow: Lab oratoriya Bazovykh Znanii, 2000. 27. Documentation for ANSYS. Release 11.0. 28. Kifer, I.I., Ispytanie ferromagnitnykh materialov (Tests of Ferromagnetic Materials), Moscow: Gosenergoizdat, 1962. RUSSIAN JOURNAL OF NONDESTRUCTIVE TESTING

Vol. 48

No. 1

2012