Dynamic properties of micro-magnetic noise in soft ferromagnetic materials

Accepted Manuscript Dynamic properties of micro-magnetic noise in soft ferromagnetic materials A. Stupakov, A. Perevertov PII: DOI: Reference:

S0304-8853(17)33849-0 https://doi.org/10.1016/j.jmmm.2018.02.069 MAGMA 63746

To appear in:

Journal of Magnetism and Magnetic Materials

Please cite this article as: A. Stupakov, A. Perevertov, Dynamic properties of micro-magnetic noise in soft ferromagnetic materials, Journal of Magnetism and Magnetic Materials (2018), doi: https://doi.org/10.1016/ j.jmmm.2018.02.069

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Dynamic properties of micro-magnetic noise in soft ferromagnetic materials A. Stupakova,∗, A. Perevertova a

Institute of Physics of the Czech Academy of Sciences, Na Slovance 2, 18221 Prague, Czechia

Abstract Dynamic response of magnetic hysteresis, magnetic Barkhausen noise and magneto-acoustic emission in a soft ribbon and electrical steels was studied comprehensively. The measurements were performed under controllable magnetization conditions: sinusoidal/triangular waveforms of the magnetic induction and a triangular waveform of the magnetic field. Magnetizing frequency was varied in a wide range: fmag = 0.5 − 500 and 0.1 − 100 Hz for the ribbon and the electrical steels, respectively. Magnetization amplitude was fixed on a near-saturation level Hmax ≃ 100 A/m. Barkhausen noise signal was detected by a sample-wrapping/surface-mounted coil and differently filtered. It was found that intensity of the Barkhausen noise rises approximately as a square root function of the magnetizing frequency. Whereas, level of the magneto-acoustic emission follows the hysteresis loss trend with an additional linear term (classical loss component). Keywords: Magnetization dynamics, Barkhausen noise, Magneto-acoustic emission, Magnetic hysteresis, Electrical steel

∗ Corresponding author. Tel: +420 26605 2114 E-mail: [email protected], URL: www.fzu.cz/˜stupak

Preprint submitted to Journal of Magnetism and Magnetic Materials

February 21, 2018

1

1. Introduction

2

The classical problem of macroscale magnetization dynamics in conven-

3

tional ferromagnetic materials has not been fully understood so far. In partic-

4

ular, magnetic hysteresis in soft magnetic materials is commonly discussed in

5

terms of a phenomenological loss-separation model in spite of growing crit-

6

icism of dubious theory assumptions for a classical loss component and of

7

unclear nature of an excess loss [1, 2]. Promising direction of research that

8

should contribute in elucidation of this classical problem is investigation of

9

the magnetization dynamics on a microscale level by measuring Barkhausen

10

noise (BN) and magneto-acoustic emission (MAE). The BN/MAE signals

11

originate from irreversible motion of the magnetic domain walls (DW) and

12

are detected in the form of magnetic/acoustic impulses. Principal distinc-

13

tion between BN and MAE is that the motion of the 180◦ DWs should not

14

cause any change of the magneto-elastic energy and therefore should not con-

15

tribute to MAE [3, 4]. Local jumps of the magnetic DWs between material

16

pinning sites such as grain boundaries, inclusions and defects are damped by

17

microeddy currents, which give a considerable contribution to the hysteresis

18

losses [5]. Thereby, acquiring extensive knowledge about the magnetization

19

process on the microscopic scale builds a necessary base for deeper under-

20

standing of the dynamic properties and for better prediction of the magnetic

21

losses in electrical transformers and motors.

22

Dynamic properties of BN/MAE have been scantily investigated so far.

23

This is especially true for the measurements under controllable magnetization

24

conditions providing physically interpretable results [4, 6]. In this work,

25

we first present the dynamic data for a nanocrystalline soft ribbon with 2

26

a specific combination of the magneto-mechanical properties making this

27

investigation especially attractive. The principal issue is that a tiny thickness

28

of the ribbon and its high electrical resistivity suppress eddy currents and

29

ensure the homogeneous BN detection. Moreover, the long ribbon can be

30

homogeneously magnetized in a solenoid up to high magnetizing frequencies;

31

and its magnetic field can be easily controlled using the magnetizing current.

32

To ensure consistency of the data and profound analysis of the magnetization

33

dynamics, the measurements are additionally performed for soft electrical

34

steels in another magnetizing configuration.

35

2. Experimental

36

2.1. Ribbon

37

The measurements were performed for an as-cast nanocrystalline Hitperm

38

ribbon with the following composition: Fe54 Co27 Nb7 B12 [7]. It was chosen

39

because of the similar grain structure as industrial ferromagnetic steels. To

40

receive evidence that the data obtained for this ribbon demonstrate a gen-

41

eral trend, other ribbons from the same Hitperm series with different Fe-

42

Co composition and a few industrial samples were additionally tested. The

43

tested ribbon was 1 m long, 7 mm wide and 10 µm thick. It was mag-

44

netized inside a 30-cm long solenoid; the magnetic field H was calculated

45

from the current in the solenoid winding. A sample-wrapping coil of 4 cm

46

length and 2700 turns was positioned in the solenoid center to estimate the

47

magnetization and the BN signal. For this measurement configuration, a

48

49

considerable air induced component ∼ dH/dt was digitally compensated to evaluate the magnetic polarization J. The hardware details and the set-up 3

50

scheme were described in [7]. The magnetization waveform was controlled by

51

a digital feedback with a combined phase-amplitude adjustment, details can

52

be found in [8]. Magnetization conditions, namely the sinusoidal/triangular

53

waveforms of the magnetic polarization J(t) or the triangular waveform of

54

the magnetic field H(t), were controlled in a wide range of the magnetizing

55

frequency fmag = 0.5 − 500 Hz. The measurements without the J(t)/H(t)

56

control, namely at the sinusoidal/triangular waveforms of the magnetizing

57

voltage V (t), were also recorded. Magnetization amplitude was fixed on a

58

near-saturation level of Jmax = 0.9 T or Hmax = 100 A/m.

59

In addition to the measurement with the sample-wrapping coil, BN was

60

alternatively recorded by a small surface-mounted coil also placed in the

61

central part of the solenoid. It was a bobbin coil of 1 cm diameter with

62

800 turns and a pen-type core of ferrite. Its resonance frequency was about

63

60 kHz. The BN signal measured by both coils at different magnetization

64

conditions was filtered within three bandwidths 2–40, 10–40 and 40–80 kHz.

65

MAE was detected by a piezoelectric transducer from Dunegan/Endevco,

66

type S9204, placed again in the solenoid center. The amplified MAE signal

67

was filtered in a 100–300 kHz range; the resonance frequency of the MAE

68

transducer was about 150 kHz [4]. The BN/MAE signals were sampled at a

69

2 MS/s rate.

70

2.2. Electrical steels

71

Industrial electrical steels, two grain-oriented (GO) grades and one non-

72

oriented (NO) steel with distinct magnetic parameters, were chosen for the

73

measurements. Steel strips of the Epstein dimensions 300×30×0.25−0.5 mm

74

were magnetized along the rolling direction by a U-shaped transformer yoke 4

75

of the same length and width through a 3 mm air gap; two magnetizing coils

76

were positioned on the yoke poles. The tangential component of the surface

77

magnetic field was measured directly by a Hall sensor at 1.5 mm above the

78

yoke-free sample side. This method of surface field determination ensured

79

a good measurement accuracy for the magnetically open configuration used.

80

By introducing the air gap, a total inductance of the magnetizing circuit

81

decreased, which stabilized the feedback procedure. For this configuration,

82

the air induced component ∼ dH/dt was negligible, so the magnetic induction

83

B = µ0 H + J was estimated directly by a sample-wrapping coil with 100

84

turns. Waveform of B(t) was adjusted to the sinusoidal/triangular shapes

85

in a fmag = 0.5 − 100 Hz range; additional measurements at the sinusoidal

86

V (t) waveform were conducted. Magnetization amplitudes were Bmax =

87

1.5 and 1.2 T for the GO and the NO steels, respectively. Accuracy of

88

the measurement configuration and efficiency of the digital feedback were

89

comprehensively discussed in [8].

90

BN was detected similarly by the sample-wrapping coil and a bigger

91

surface-mounted coil placed in the central part of the strips. It was a bobbin

92

coil of a 16 mm diameter with 1000 turns of fine copper wire inside a Cu-

93

shielded grounded case. A core with a 4 × 4 mm cross-section was made from

94

a laminated GO steel and gently pressed by a spring to the strip surfaces.

95

A resonance frequency of this coil with the GO core was about 120 kHz [9].

96

The BN signal was filtered within three bandwidths 2–70, 20–100 and 70–

97

170 kHz. The MAE signal was detected by the same Dunegan transducer

98

and filtered in a 100–500 kHz range.

5

(a)

(b) 3

W, J/m 1

tri V(t): 50 34+0.688⋅√f - 0.1% tri H(t): 34+0.737⋅√f-0.14%

J, T

ribbon 0.5

ribbon

45

H, A/m -50

50 -0.5

-1

100

sin J(t):

fmag= 5 Hz

50 Hz tri H(t):

500 Hz fmag=50 Hz

40 sin J(t): 34.8+0.384⋅√f +0.4% tri J(t): 34.6+0.32⋅√f + 0.3% sin V(t): 34.7+0.775⋅√f+0.1%

35 0

100

200

300

400

500

fmag , Hz

Figure 1: (a) Typical hysteresis loops J(H) obtained for the tested Hitperm ribbon at the different magnetizing waveforms and frequencies. (b) Dependence of the hysteresis loss W on the magnetizing frequency fmag for the same ribbon at the different magnetizing waveforms. Relations are fitted by the loss-separation function with the negligible linear coefficients.

99

3. Results

100

3.1. Ribbon

101

Typical hysteresis loops measured under the sinusoidal waveform of the

102

magnetic polarization J(t) at three magnetizing frequencies fmag = 5, 50 and

103

500 Hz are shown in Fig. 1(a) together with the loop measured at the tri-

104

angular field H(t) and fmag = 50 Hz. The loops become slightly wider with

105

increasing fmag ; dc coercive field Hc ≃ 12 A/m. Figure 1(b) presents depen-

106

dence of the hysteresis loss W on the magnetizing frequency for the different

107

magnetizing waveforms. The loss dependence for the triangular voltage V (t)

108

and field H(t) are very close to each other because the field generated inside 6

109

110

111

112

113

the solenoid is nearly proportional to the magnetizing voltage (the circuit resistance prevails over its inductive reactance), i.e. H(t) ∼ I(t) ≈ V (t). Other relations are quite different, but all of them are fitted well by the loss-separation formula W = Wdc + a ·

q

fmag + b · fmag , where Wdc is an ex-

trapolated static loss at zero fmag . The classical linear term Wcl = b · fmag is

114

derived from the Maxwell equations under dubious assumptions that a thin

115

and infinite sheet with constant permeability is magnetized unidirectionally

116

and homogeneously. There is still debate about the origin of the square root

117

term, so-called excess loss Wex = a ·

q

fmag , which was added to fit the ex-

118

perimental loss dependence for the electrical steels [1, 2]. The least-square

119

fittings of the experimental loss dependence are shown by the correspond-

120

ing lines. Graph labels give the fitting formulas with the linear coefficient

121

b presented as a percentage ratio to the square root coefficient b/a; the fol-

122

123

124

125

lowing linear term fmag is omitted for the sake of simplicity (e.g. a formula q √ 2 + 10 · f + 1% in the label means the fitting 2 + 10 · fmag + 0.1 · fmag ). It should be noted that the used coefficient ratio b/a does not agree with a corresponding term ratio Wcl /Wex = (b/a) ·

q

fmag . Two terms of the loss-

126

separation formula become comparable Wcl = Wex at a critical magnetizing

127

frequency fcr = (a/b)2 ; so the linear term contribution is negligible in the

128

considered fmag range for the ratio b/a varying on a common error level of

129

a few percents (for b/a = 1, 2 and 5 %, fcr = 10, 2.5 and 0.4 kHz, respec-

130

tively). For the thin ribbons, the classical linear component Wcl is known to

131

be negligible [1, 5]. This is also proved by our data: the ratio b/a reaches a

132

maximal value 0.4% for the sinusoidal J(t). Pearson correlation coefficients

133

of the fits equal R = 0.992 for the controlled J(t) and R = 0.997 for the

7

(a)

U

rms

(b)

, mV

U

rms

, mV

sin J(t)

2

2

sin J(t), filter

tri J(t) sin V(t)

2 - 40 kHz

tri V(t)

10 - 40 kHz

BN, ribbon

1

sample-wrapping

0.28 + 0.117

0

100

sample-wrapping coil

tri H(t), filter

1

2 - 40 kHz

filter 10 - 40 kHz

10 - 40 kHz

coil -0.05 + 0.22

BN, ribbon

tri H(t)

40 - 80 kHz

40 - 80 kHz

-0.08 + 0.16

f + 1.1%

-0.05 + 0.18

f + 2.2%

f + 7.4% f + 0.1%

200

f

mag

, Hz

f

300

0

50

mag

100

, Hz

150

Figure 2: Dependence of the rms value Urms of BN on the magnetizing frequency fmag : (a) for all filtering bandwidths and two magnetizing waveforms, the sinusoidal polarization J(t) and the triangular field H(t); (b) for the medium filtering bandwidth 10–40 kHz and all magnetizing waveforms. The BN signal from the tested Hitperm ribbon is detected by the sample-wrapping coil. Representative relations are fitted by the same loss-separation function with the linear coefficients showed in the labels as the percentage ratio b/a.

134

other waveforms.

135

Figure 2 presents similar dynamic dependence of the rms intensity Urms of

136

the one-cycle BN signal detected by the sample-wrapping coil. Low-frequency

137

hysteresis component distinctly distorts the BN output, when fmag value

138

reaches ∼ 1% of a low cutoff frequency of the band-pass filter. Therefore, the

139

measurements at the low filtering bandwidth 2–40 kHz are not representative

140

enough. Constant terms of the loss-separation fits Udc are relatively high:

141

0.05–0.08 mV for the two lower filtering bandwidths and 0.28 mV for the

142

40–80 kHz bandwidth. Level of a background noise can be estimated at the

143

saturation regions, where the BN activity is minimal, i.e. as Urms (Hmax ≃ 8

144

145

±100 A/m), or by reading the empty BN response without applying the voltage into the magnetizing coil, i.e. as Urms (H ≡ 0). Thus determined

146

background noise is several times lower than Udc and also dependent on the

147

filtering: for the two lower bandwidths, the background noise level is about

148

0.01 mV; for the 40–80 kHz bandwidth, it is 0.1 mV [6, 10]. Shape of the

149

magnetizing waveform has a qualitatively similar impact on the rms value

150

and the hysteresis loss: the sinusoidal V (t) and the triangular J(t) give the

151

maximal and the minimal values, respectively. However, quantitatively the

152

waveform shape influence is weaker for Urms . For the two lower filtering

153

bandwidths, the Urms relations divide into two groups for the data obtained

154

with and without the J(t) control (see Fig. 2(b)). However, the difference

155

between these two groups vanishes for the higher filtering bandwidth 40–

156

80 kHz (see Fig. 2(a)). The relative linear coefficients of the loss-separation

157

fits are slightly higher than those for the hysteresis loss: b/a ≃ 1 − 2% for

158

the medium filtering bandwidth 10–40 kHz, which is a still negligible level.

159

For the high filtering bandwidth 40–80 kHz, the linear coefficient is even

160

smaller by an order of magnitude b/a ≃ 0.1%. Correlation coefficients of the

161

fits are R = 0.9998 and 0.9989 for the medium and the high bandwidths,

162

respectively.

163

Figure 3 presents similar dependence of the rms value Urms of BN recorded

164

by the surface-mounting coil with the different filtering bandwidths. Interfer-

165

ence of the low-frequency hysteresis contribution is significantly smaller for

166

the detection by the surface coil, so the measurements with the low filtering

167

168

bandwidth 2–40 kHz can be performed up to fmag ≃ 50 Hz (see Fig. 3(a)). The loss-separation fits of these relations with the correlation coefficients

9

(a)

U

rms

(b)

U

, mV

rms

0.2

surface-mounted coil

0.4

surface-mounted coil

, mV

filter 2 - 40 kHz

filter 10 - 40 kHz

sin J(t)

BN, ribbon

BN, ribbon

tri J(t)

sin J(t)

0.3

sin V(t)

tri J(t)

tri V(t)

sin V(t) tri V(t)

0.1

tri H(t)

0.2

tri H(t) -0.006 + 0.025

-0.013 + 0.05 -0.002 + 0.047

0

10

20

0.1

f - 4.1% f - 5.7%

30

40

f

50

mag

f - 0.9%, R=0.998

-0.018 + 0.041 f

0.4

, R = 0.999

, Hz

f

60

0

100

200

300

400

mag

, Hz

500

Figure 3: Dependence of the rms value Urms of BN on the magnetizing frequency fmag for all magnetizing waveforms and two filtering bandwidths: (a) 2–40 kHz; (b) 10–40 kHz. The BN signal from the tested Hitperm ribbon is detected by the surface-mounted coil. Representative relations are fitted by the same loss-separation function with the linear coefficients showed in the labels as the percentage ratio b/a. In (b) the relation is additionally fitted by a power function.

10

169

R = 0.9998 give the negative linear coefficients b/a ∼ −5%. The rms rela-

170

tions divide into two similar groups with different dependence on the mag-

171

netizing waveform shape. However, the group with the sinusoidal/triangular

172

J(t) demonstrates higher rms values at fmag > 10 Hz in contrast to the re-

173

sults obtained with the sample-wrapping coil (compare Figs. 2(b) and 3(a)).

174

This waveform-induced difference between the rms dependence vanishes at

175

the higher cut-off frequency of the filter (see Fig. 3(b)). The loss-separation

176

fitting results in a negative linear coefficient again, but of a smaller negligible

177

178

179

180

181

level b/a ∼ −1%. An alternative power-law fitting gives a similar curve with the similar correlation coefficient R ≃ 0.999, but the resulting power term 0.4 0.5 fmag is rather far from the expected fmag [11, 12]. Constant terms of the fits

Udc can be again higher than the background noise, whose level is ∼ 3 µV for both filtering bandwidths.

182

Figure 4 presents the discrete Fourier spectra of the BN signals de-

183

tected by both coils at different magnetizing conditions [9, 13]. For bet-

184

ter visualization, the spectra obtained at the higher magnetizing frequencies

185

fmag = 5 − 50 Hz in Fig. 4(a) are normalized to a level of the original 2 Hz

186

spectrum on the assumption that the rms magnitude increases according to

187

the square root of the magnetizing frequency, i.e. the spectra are divided by

188

Urms / fmag /2. Such fmag -normalized spectra differ in a low-frequency region

189

q

up to f ≤ 20 Hz. This deviation of the spectra from the Urms ∼

q

fmag behav-

190

ior at low frequencies results in the negative linear term of the loss-separation

191

fitting in Fig. 3(a). Moreover, only this low-frequency spectral region is sen-

192

sitive to variations of the magnetizing waveform (see Fig. 4(b)) [6]. The same

193

fmag -normalization of the spectra recorded by the sample-wrapping coil (not

11

(a)

U

rms

1

(f

mag

/2),

(b)

V

U f

mag

BN, ribbon

2 Hz

surface-mounted coil

5 Hz

sin

J(t),

filter 2-40 kHz

rms

1

=

,

V

f

mag

BN, ribbon surface-mounted coil

10 Hz

tri J(t)

tri V(t)

50 Hz 0.5

tri H(t)

0.5

sample-wrapping coil /10

f, kHz 20

sin J(t)

sin V(t)

20 Hz

0

= 2 Hz

filter 2-40 kHz

40

0

10

f, kHz 20

Figure 4: Fourier power spectra of the BN signals from the Hitperm ribbon recorded by the surface-mounted coil at: (a) the sinusoidal polarization J(t) and different magnetizing frequencies fmag = 2 − 50 Hz; (b) different magnetizing waveforms and fixed fmag = 2 Hz. The filtering bandwidth is 2–40 kHz. In (a) the spectra for fmag = 5−50 Hz are normalized to the 2 Hz spectrum on the assumption of the square root rise with the magnetizing p frequency Urms / fmag /2. In (b) the similar spectra recorded by the sample-wrapping coil are shown by the corresponding dashed lines. They are reduced by an order of magnitude to fit the same scale.

12

194

shown in the graph) gives similar spectra deviation in the low-frequency re-

195

gion, but of the opposite sign. This logically leads to the positive linear

196

term of the loss-separation fitting in Fig. 2(a). This low-frequency region up

197

to f ≤ 20 kHz is also sensitive to variations of the magnetizing waveform,

198

however, the spectra obtained with and without the J(t) control are clearly

199

detached for the sample-wrapping data. Next difference between the spectra

200

for both coils is that the spectral magnitudes for the sinusoidal/triangular

201

J(t) are smaller than those obtained without the J(t) control in accordance

202

with the rms dependence in Fig. 2(b).

203

Figure 5(a) demonstrates a nearly linear correlation between the BN rms

204

values and the hysteresis losses at ac magnetizing frequencies fmag > 10 Hz.

205

Number of the individual BN pulses over a magnetization cycle hyperbolically

206

drops with magnetizing frequency (see Fig. 5(b)) [4, 10]. Qualitatively same

207

results are obtained for other magnetizing-sensing conditions.

208

One-cycle rms values of MAE demonstrate qualitatively similar depen-

209

dence on the magnetizing frequency as compared with the hysteresis loss (see

210

Figs. 1(b) and 6(a)). Therefore, there are similar nearly linear correlations

211

between the MAE rms values and the hysteresis losses at fmag > 10 Hz (see

212

Figs. 5(a) and 6(b)). However, a low signal-to-noise ratio leads to a higher

213

scattering of the MAE data. Correlation coefficients of the loss-separation

214

215

fits are of the same level, R = 0.991 − 0.996; but the linear coefficients of the fits are higher b/a ≃ ±1%. It is difficult to perform a detailed spec-

216

tral analysis of the weak MAE signal; the reasonable signal-to-noise ratio

217

is achieved by amplification within the resonance bandwidth of the MAE

218

transducer. Number of the individual MAE pulses also drops hyperbolically

13

(a) U

rms

(b)

, mV

2

f

1

mag

10

3

sin J(t)

ribbon sin

pulse count,

50

BN

40

J(t)

surface-mounted coil

sin V(t)

filter 2 - 40 kHz

tri V(t)

30

= 10 Hz

tri J(t)

BN, ribbon

tri H(t) sample-wrapping coil

20

surface-mounted coil

-0.19 + 52.2

10

f

3

0

W, J/m

36

39

42

f

45

0

10

20

30

40

mag

50

, Hz

60

Figure 5: (a) Correlations between the BN rms values and the hysteresis losses under the sinusoidal J(t) fitted by the linear curves for fmag > 10 Hz. The BN signal from the Hitperm ribbon is filtered within the 10–40 kHz bandwidth. (b) Number of the BN individual pulses over a magnetization cycle versus fmag fitted by the hyperbolical function with R = 0.9998.

14

(a)

2

(b) 2

Urms , µV

sin V(t):-0.04+0.073⋅√f+0.5% tri V(t):-0.03+0.063⋅√f +0.6% tri H(t):-0.15+0.083⋅√f -0.3%

U

rms

,

V

MAE ribbon

MAE ribbon

1

1

sin J(t) f

mag

= 10 Hz

tri J(t) sin V(t) tri V(t)

sin J(t): -0.09+0.065⋅√f -0.9% tri J(t): -0.04+0.048⋅√f -1.7% 0

100

200

300

400

tri H(t)

0

500

fmag , Hz

36

39

42

3

45

48

51

W, J/m

Figure 6: Dependence of the MAE rms value Urms from the Hitperm ribbon on (a) the magnetizing frequency fmag and (b) the hysteresis losses W for the different magnetizing waveforms. In (a) the relations are fitted by the loss-separation function with the linear coefficients showed in the labels as the percentage ratio b/a. In (b) the relations for the sinusoidal J(t) and the triangular H(t) are fitted by the linear curves for fmag > 10 Hz.

219

with magnetizing frequency [4].

220

3.2. Electrical steels

221

Hysteresis loops for the tested electrical steels measured under the sinu-

222

soidal waveform of the magnetic induction B(t) at the magnetizing frequency

223

fmag = 50 Hz are shown in Fig. 7(a) together with the quasi-static loop for

224

the hardest NO steel. The loops are of typical sizes and shapes for this class

225

of ferromagnetic materials [2]. The first GO steel with the dc coercive field

226

Hc ≃ 8 A/m belongs to the softest transformer steels. Figure 7(b) presents

227

dependence of the hysteresis loss on the magnetizing frequency obtained at

228

the sinusoidal B(t). The dependence are well fitted by the above-discussed 15

(a)

(b) 3

450

B, T

W, J/m

400

1

NO, sin V(t): 138+15⋅√f +7.6%

350

NO, sin B(t): 144+8.43⋅√f +17% NO, tri B(t): 150+3.7⋅√f + 41%

300 250

H, A/m

200

-50

50

100

150

150 100

-1

sin B(t), fmag= 50 Hz GO1 NO

sin B(t)

50

GO2 1 Hz

0

20

40

GO1: 50+3.35⋅√f+21% GO2: 47+11⋅√f + 3.8%

60

80

100

120

fmag , Hz

Figure 7: (a) Hysteresis loops B(H) for the tested electrical steels at the sinusoidal induction waveforms B(t) and the magnetizing frequency fmag = 50 Hz. Nearly quasi-static loop obtained at fmag = 1 Hz for the NO steel is shown for comparison. (b) Dependence of the hysteresis loss W on the magnetizing frequency fmag for the tested steels at the sinusoidal B(t). For the NO steel, similar dependence for the triangular B(t) and the sinusoidal V (t) are additionally shown. Relations are fitted by the loss-separation function. Linear coefficients of the fits are shown in the labels as the percentage ratio to the square root coefficients b/a.

16

(a)

U

rms

,

V

(b) 60

BN, filter 20 - 100 kHz

U

rms

sample-wrapping coil

80

,

V

GO2

sin B(t)

tri B(t)

NO

sin B(t)

tri B(t)

40

60 GO1

40

GO2

sin B(t)

tri B(t)

sin B(t)

tri B(t)

f 2 k

sin B(t)

tri B(t)

20

BN, filter 70 - 170 kHz

20

sample-wrapping coil

f +3.5%

3.3+7.36

f +6.1%

2.6+ 5.01

f +4.2%

1.9 + 8.9

f +1%

1.7 + 4.9

f + 3.2%

4.4 + 19.6

0

sin V(t)

NO

2.6+7.37

20

sin B(t)

40

f

f -2%

60

mag

, Hz

f

80

0

20

mag

40

, Hz

60

Figure 8: Dependence of the rms value Urms of BN on the magnetizing frequency fmag for the tested electrical steels and the different magnetizing waveforms. The BN signal is detected by the sample-wrapping coil and filtered within (a) the medium bandwidth 20– 100 kHz and (b) the high bandwidth 70–170 kHz. In (a) the dependence obtained at the low filtering bandwidth 2–70 kHz is additionally shown for the NO steel (the highest green line). Relations are fitted by the same loss-separation function with the linear coefficients showed in the labels as the percentage ratio b/a; correlation coefficients of the fits are R ≃ 0.9998. 229

230

q

loss-separation formula W = Wdc + a · fmag + b · fmag as shown in the graph labels. Pearson correlation coefficients of the fits are R ≃ 0.998. Ratio be-

231

tween the fitting coefficients b/a is on a level of 20 % for the first GO and the

232

NO steels, but is only 4 % for the second GO steel. Variations of the mag-

233

netizing waveform can significantly influence the slope of the loss-frequency

234

dependence as shown in Fig. 7(b) for the NO steel. However, the slope dif-

235

ferences are smaller for the GO steels. For the softest GO steel, there are the

236

same loss-frequency relations for the triangular and the sinusoidal B(t).

17

237

Figure 8 presents the dynamic rms dependence of BN detected by the

238

sample-wrapping coil under different magnetizing waveforms and filtering

239

bandwidths. The linear coefficients of the loss-separation fits for the two

240

highest filtering bandwidths are on a level of b/a ≃ 4%, which is several

241

times higher than those for the tested ribbon (compare with Fig. 2). There

242

is no evident correlation between magnitudes of the linear coefficient b for

243

the hysteresis loss and the BN rms values. Moreover, the coefficient b is

244

negative for the NO steel and the lowest filtering bandwidth. Influence of

245

the magnetizing waveform shape on the BN rms value is practically negligible.

246

247

248

249

250

251

Figure 9 presents similar dynamic dependence of the BN rms value recorded by the surface-mounting coil. The low filtering bandwidths give small negative values of the linear coefficients of the loss-separation fitting, b/a ≃ −1%, which is several times lower than those for the tested ribbon (compare Figs. 3(a) and 9(a)). However, the fittings for the GO steels and the high filtering bandwidth 70–170 kHz give b/a ≃ +4% (see Fig. 9(b)). There is

252

again no evident correlation between the linear coefficients of the BN rms

253

and the loss dependence as well as there is no noticeable influence of the

254

magnetizing waveform shape on the BN rms values.

255

The Fourier power spectra of the BN signal behave correspondingly to

256

the above-shown dynamic dependence of the rms value. There is a simi-

257

lar suppression of the

258

leading to the negative linear coefficient of the loss-separation fits at the

259

low filtering bandwidths (see Figs. 4(a), 9(a) and 10(a)). However, the last

260

power spectrum for fmag = 50 Hz in Fig. 10(a) does not already demon-

261

strate such a suppression in the low-frequency region, but on the contrary, it

q

fmag -normalized spectra in the low-frequency region

18

(a)

U

rms

,

V

150

(b)

GO1

sin B(t)

tri B(t)

sin V(t)

GO2

sin B(t)

tri B(t)

sin V(t)

NO

sin B(t)

tri B(t)

sin V(t)

f20k

sin B(t)

tri B(t)

sin V(t)

U

rms

250

150

BN, filter

V

GO1

200

100

,

sin B(t)

GO2

sin B(t)

NO

sin B(t)

NO

tri B(t)

NO

sin V(t)

BN, filter

2-70 kHz,

70-170 kHz,

100

surface-

50

2 + 12.2

f - 1.1%

1.1+13.7

f -1.3%

2.2+14.8

f -1.8%

3 + 15.1

0

20

40

60

mounted

50

coil f

f -0.5%

80

surface-

mounted

100

mag

, Hz

120

0

20

20+11.

f + 4.2%

18+15.7

f +4.7%

6.3+23.4

f -0.5%

40

60

coil f

80

mag

, Hz

100

Figure 9: Dependence of the rms value Urms of BN on the magnetizing frequency fmag for the tested electrical steels and the different magnetizing waveforms. The BN signal is detected by the surface-mounted coil and filtered within (a) the low filtering bandwidth 2– 70 kHz and (b) the high filtering bandwidth 70–170 kHz. In (a) the dependence obtained at the medium filtering bandwidth 20–100 kHz is additionally shown for the NO steel (the highest green line). Relations are fitted by the loss-separation function with the linear coefficients showed in the labels as the percentage ratio b/a; correlation coefficients of the fits are R ≃ 0.9996.

19

(a)

U

rms

(f

mag

/2),

(b)

V

f

mag

U

=

rms

(f

mag

/2),

V

f

mag

=

2 Hz 0.2

BN, GO2, sin

surface-mounted coil filter 2 - 70 kHz

2 Hz 0.2

5 Hz

B(t)

5 Hz

10 Hz

10 Hz

20 Hz

20 Hz

50 Hz

50 Hz

0.1

0.1

BN, GO2, sin

B(t)

surface-mounted coil filter 70 - 170 kHz f, kHz

0

20

40

f, kHz 60

60

80

100

120

140

160

Figure 10: Fourier power spectra of the BN signals recorded by the surface-mounted coil at different filtering bandwidths: (a) 2–70 kHz; (b) 70–170 kHz. The measurements are performed for the second GO steel at the sinusoidal B(t) waveform. Similar to Fig. 4(a), p the spectra for fmag = 5 − 50 Hz are normalized to Urms / fmag /2. 262

263

shows a small rise as compared with the 20 Hz spectrum at higher frequencies f ≃ 20 − 60 kHz. This rise becomes apparent at frequencies ∼ 100 kHz lead-

264

ing to the positive coefficient of the loss-separation fits at the high filtering

265

bandwidth (see Figs. 9(b) and 10(b)). Peaks of the spectra near 120 kHz is

266

determined by the resonance of the surface-mounted coil [9]. For such high

267

frequencies, the signal-to-noise ratio is worse; the spectra are noisier with

268

clear spikes of disturbing harmonics [13]. Dynamic dependence of the BN

269

pulse count for the electrical steels are similar to that for the ribbon (see

270

Fig. 5(b)).

271

For the softest GO steel, good linear correlations between the BN rms

272

values and the hysteresis loss are observed for different measurement condi-

273

tions in an ac magnetization range fmag > 5 Hz (see Fig. 11(a)). For the

20

(a) 200

U

rms

,

(b) U

V

rms

250

GO1, sample coil, 20-100 kHz 3

W, J/m

150

GO2, surface coil, 2-70 kHz

100

BN sin

surface coil, 20-100 kHz

200

B(t)

surface coil, 70-170 kHz

150 f

mag

f

= 5 Hz

mag

= 2 Hz

100

50

BN, NO sin

50 -138+22.6

0

V

surface coil, 2-70 kHz

GO1, surface coil, 2-70 kHz GO1, surface coil, 70-170 kHz

,

sample coil, 20-100 kHz

W -1.5%

-31.6 + 0.84 W

60

80

100

120

3

0 150

W, J/m

140

200

-270+28.7

W -1.3%

-581+64.9

W -1.9%

250

300

350

B(t) 3

W, J/m

400

Figure 11: Correlations between the BN rms values and the hysteresis loss for the tested electrical steels. The measurements are made by both coils at the sinusoidal B(t) and different filtering bandwidths. The data for the GO steels are fitted by the linear curves for fmag > 5 Hz. The data for the second GO and the NO steels are fitted by the loss-separation function for fmag > 2 Hz; correlation coefficients of the fits are R ≃ 0.998.

21

(a)

U

rms

15

,

(b)

V

U

rms

15

sin B(t)

sin V(t)

10

10 4.7+0.63 5 + 0.44 f

40

60

MAE GO1 sin B(t) tri B(t)

0.72

sin V(t)

, R = 0.9999

f

20

-0.8 + 0.12 W, R =0.9965

f + 9.4%, R =0.9995

5 0

V

-1.2 + 0.11 W, R =0.9968

MAE GO1

tri B(t)

,

80

mag

5

, Hz

100

3

W, J/m

60

80

100

120

140

Figure 12: (a) Dependence of the rms value Urms of MAE on the magnetizing frequency fmag for the softest GO steel and the different magnetizing waveforms. The dependence is fitted by the loss-separation and the power functions with the comparable correlation coefficients R. (b) Linear correlations between the MAE rms values and the hysteresis loss for the same GO steel.

274

second GO steel, this correlation bends down to the loss scale and is bet-

275

ter fitted by the same loss-separation function with a small linear coefficient

276

277

b/a = −1.5%; the correlation coefficient of this fit is R = 0.9977 versus the linear fit R = 0.993. For the NO steel, the correlations should be also fitted

279

by the loss-separation function with the similar ratio b/a (see Fig. 11(b)). √ The observed W -like relations are consistent with our previous results for

280

other electrical steels with similar magnetic properties [14].

278

281

Figure 12 presents dependence of the MAE rms value on the magnetizing

282

frequency and the hysteresis loss for the softest GO steel. There is again

283

no evident influence of the magnetizing waveform shape on the rms value.

284

The linear coefficient of the loss-separation fit Urms (fmag ) is relatively high,

285

b/a = 9.4%, but it is still twice lower than that for the corresponding depen22

286

287

dence of the hysteresis loss (compare Figs. 7(b) and 12(a)). The constant term Udc ≃ 5 µV is much higher than that for the ribbon measurements

288

because of the wider filtering bandwidth (compare with Fig. 6(a)). The cor-

289

relations between the MAE rms values and the hysteresis loss demonstrate

290

the same linear trend, but in the whole fmag range in contrast to the BN data

291

(compare Figs. 11(a) and 12(b)). For the second GO steel, very similar results

292

293

294

q

are obtained: Urms = 4.5 µV+0.98· fmag +11% and Urms = −1.44+0.121·W for the sinusoidal B(t) (see Fig. 4 in our recent work [4]). First, there are comparable ratios b/a ≃ 10% for the MAE dependence Urms (fmag ) for both

295

GO steels despite of quite different ratios b/a = 21 and 3.8% for the corre-

296

sponding loss data. Second, a good linear correlation of the MAE rms values

297

with the hysteresis loss for the second GO steel contradicts the corresponding

298

BN data (see Fig. 11(a)).

299

4. Discussion

300

Hysteresis losses of magnetic ribbons are known to increase nearly proq

301

portional to

302

it is a consequence of their tiny thickness, high resistivity or specific DW

303

structure [1, 5]. The studied nanocrystalline ribbon follows the same trend

304

and the thickness is expected to be an important factor determining such

305

a loss behavior (see Fig. 1(b)). Anyway, this factor is definitely important

306

for investigation of the BN response: the tiny thickness ensures homogeneous

307

magnetization and detection. But even under these measurement conditions,

308

intensity of the BN signal behaves differently from the hysteresis loss. For the

309

case of the sample-wrapping coil, a considerable positive level of the linear

fmag , although there is still no a clear explanation whether

23

310

coefficient b can be explained by influence of the high-frequency harmonics of

311

the hysteresis induction signal since the high-frequency filtering decreases the

312

linear term of the loss-separation fits to a negligible level (see Fig. 2). How-

313

ever, a negligible negative level of b for the case of the sample-mounted coil

314

is evidently caused by a suppression of large BN pulses in the low-frequency

315

part of the power spectra (see Figs. 3 and 4). These largest pulses are only

316

sensitive to variations of the magnetizing waveform and are logically assigned

317

to the magnetization reversal at the coercive field region, where considerable

318

jumps of the 180◦ DW occur (see Fig. 1(a)). With increasing magnetizing

319

frequency, number of the BN pulses is significantly reduced due to a pulse

320

overlapping (see Fig. 5(b)). Such an overlapping is assumed to occur with a

321

small loss in the total BN intensity, just as a peak of the differential magnetic

322

permeability loses in its height with fmag increase [6, 15].

323

For the electrical steels, the BN dynamic behavior is qualitatively similar

324

to the above-discussed ribbon case despite the quite different dependence of

325

the hysteresis loss [10]. There is no any correlation between the parameters

326

of the loss-separation fits for the hysteresis losses and the BN rms values.

327

There is much weaker influence of the magnetizing waveform shape on the

328

BN rms value as compared with the hysteresis loss. Moreover, the dynamic

329

rms dependence for three tested steels are very close to each other, which

330

indicates a comparable level of the BN activity. For the low filtering band-

331

width, the BN spectra demonstrate similar suppression of the low-frequency

332

333

334

BN signal at f ≤ 20 kHz giving the negative linear term of the loss-separation fits (see Figs. 8(a), 9(a) and 10(a)). Qualitatively similar results with higher b/a ≃ −7% are obtained in our previous work [6] for a harder microalloyed 24

335

336

steel with dc Hc = 210 A/m. There is only one significant distinction from the ribbon data. This is the high-frequency BN activity at f ∼ 100 kHz lead-

337

ing to the positive b coefficient at the high filtering bandwidth, which can not

338

be already explained by the high-frequency harmonics of the hysteresis signal

339

(see Figs. 8(b) and 9(b)). This high-f BN activity is determined by short BN

340

pulses and can be logically assigned to the hysteresis loop knees, where the

341

90◦ DW reorganization mostly occurs. These are prior and completion phases

342

of the magnetization reversal, where nucleation, growth and subsequent an-

343

nihilation of the 90◦ domains on the grain boundaries and the sheet surface

344

take place. These domains are needed to close the internal magnetic flux

345

of the rapidly moving 180◦ domains, thereby minimizing the magneto-static

346

energy [16]. With increase of the magnetizing frequency, these knee-regions

347

of the hysteresis loops do not significantly change, only the magnetization

348

reversal occurs at higher magnetic fields producing wider B(H) loops (see

349

Figs. 1(a) and 7(a)). However, a number of the magnetic domains is known

350

351

352

353

to rise at high fmag ≥ 40 Hz for this class of materials [17]. Increasing number of the 90◦ DWs can cause this additional BN activity at f ∼ 100 kHz giving the positive linear coefficient b for the high filtering bandwidth. The principal issues under discussion are validity and physical grounds q

354

of the observed dynamic dependence with the

355

use the loss-separation formula to fit the micro-magnetic signals simply by

356

analogy with the hysteresis data. There is still no suitable theory for de-

357

scribing the obtained dynamical properties of BN/MAE, which we can rely

358

on [18, 19, 20]. We also try to use the power function proposed in [11, 12] as

359

a fitting alternative. It provides a similar accurate fitting of the experimental

25

fmag dominant term. We

360

data, but the power coefficient can significantly deviate from the expected

361

value 0.5, even if the corresponding loss-separation fitting gives a small ra-

362

tio b/a (see Fig. 3(b) and 12(a)). Analysing the BN spectra in Figs. 4 and

363

10, we are inclined to stick with the opinion about the dominating role of

364

365

the a ·

q

fmag component with the above-discussed deviations, which can be

introduced by the linear term ±b · fmag . Physical background should be of

366

similar origin as proposed to explain the excess loss Wex component. Rapid

367

irreversible changes of local magnet moments caused by the DW motion are

368

detected by the induction coils with an unknown transfer coefficient [19]. Af-

369

ter the filtering (it should be mentioned that a coil with many turns of wind-

370

ing has a self-capacitance and is a LC filtering circuit in fact), the detected

371

packet of the electrical impulses is symmetrical in polarity, so evaluation of

372

the rms value is a routine method to estimate the intensity of the micro-

373

magnetic signals [9]. These sensing-evaluation procedures could also have

374

an influence on the resulting dynamic dependence, but we again stick with

375

the opinion about the micro-magnetic origin of the observed Urms ∼

q

fmag

376

behavior. The local DW motion does not only give rise to the BN activity,

377

but also to micro eddy-currents, which are considered to define the excess

378

379

loss Wex ∼

q

fmag term [1, 2].

Small negative ratios b/a ≃ −1% for the MAE J(t)-controlled dependence

380

of the ribbon in Fig. 6(a) can be similarly explained by the overlapping of

381

the largest MAE pulses because the magnetization rates near Hc are min-

382

imal under the J(t) adjusted waveforms [6]. Much higher positive ratios

383

384

b/a ≃ 10% for the MAE dependence of the GO steels as compared with the corresponding BN relations (see Figs. 8, 9(b) and 12(a)) can be explained

26

385

by a more intensive MAE response to the 90◦ DW activity. As mentioned in

386

Introduction, theoretically MAE should not be sensitive to the motion of the

387

180◦ DWs at all [3, 4]. The next factor, which can be responsible for higher

388

MAE ratios b/a, is the fact that the MAE waves freely propagate through

389

the whole material, whereas the surface-mounted coil picks up the BN signal

390

from the steel surface only [4, 9]. However, the thickness of the GO steel

391

sheets is still small enough to ensure relatively homogeneous magnetization

392

and BN detection.

393

Finally, let us discuss the observed correlations between the micro-magnetic

394

rms values and the hysteresis losses. A main difference between these pa-

395

rameters is caused by the reversible component of the magnetization reversal

396

describing by the loss-separation terms Wdc and Wcl ∼ fmag . The constant

397

terms Wdc /Udc and different near-dc magnetization dynamics on the macro-

398

and the micro-scales are probably responsible for a lack of the correlation at

399

400

401

low fmag ≤ 2 − 10 Hz. At higher dynamic range, the correlations actually proves the physical coupling of the micro-magnetic signals (local DW motions causing the detected magnetic pulses) with the exchange losses Wex ∼

q

fmag

402

assigned to the micro eddy currents [1]. This assumption is experimentally

403

confirmed by the linear correlations in case of all MAE data and the BN

404

data for the two softest and thinnest materials, the ribbon and the first GO

405

steel (see Figs. 5(a), 6(b), 11(a) and 12(b)). The square-root correlations for

406

the two harder and thicker materials, the second GO and the NO steels, are

407

408

defined by a near-linear dependence W ∼ fmag at high fmag > 20 Hz (see Figs. 7(b) and 11).

27

409

410

5. Conclusion The dynamic behavior of magnetic Barkhausen noise and magneto-acoustic

411

emission for the typical soft ferromagnetic materials is primarily determined

412

by a square root of the magnetizing frequency. However, a supplementary

413

linear term could be added to fit the experimental dependence of the rms

414

intensities of these micro-magnetic signals (by analogy with the hysteresis

415

loss separation). For the Barkhausen noise data, this linear term is mostly

416

negligible. Its sign, however, is opposite for different regions of the Fourier

417

spectrum. It is negative for the largest impulses at the frequencies up to

418

f ≤ 20 kHz, but seems to be positive for the short impulses at f ∼ 100 kHz.

419

For the magneto-acoustic emission, the linear terms of the fits become con-

420

siderable and comparable with the classical hysteresis loss components. This

421

gives a good linear correlation between the hysteresis loss and the rms inten-

422

sity of the magneto-acoustic emission. No direct correlation is found between

423

the fitting parameters for the micro-magnetic and the macro-hysteresis re-

424

sponses, however, related physical factors (local domain wall motion and

425

attendant micro eddy currents) are assumed to be responsible for the square

426

root rise of the dynamic dependence. Influence of the magnetizing waveform

427

shape on the micro-magnetic rms intensities is nearly negligible as compared

428

with that on the hysteresis loss.

429

Acknowledgment

430

The authors appreciate the financial support of the Czech Science Founda-

431

ˇ tion (GA CR) under projects 13-18993S and 14-36566G. The authors thank

28

432

ˇ Dr. P. Svec from Institute of Physics SAS in Bratislava and Dr. P. Anderson

433

from Cardiff University for sample provision.

434

435

[1] A.J. Moses, Energy efficient electrical steels: Magnetic performance prediction and optimization, Scripta Mater. 67 (2012) 560–565.

436

[2] S.E. Zirka, Y.I. Moroz, R.G. Harrison, K. Chwastek, On physical as-

437

pects of the Jiles-Atherton hysteresis models, J. Appl. Phys. 112 (2012)

438

043916.

439

440

[3] M. Guyot, V. Cagan, The magneto-acoustic emission (invited), J. Appl. Phys. 73 (1993) 5348–5353.

441

[4] A. Stupakov, O. Perevertov, M. Landa, Dynamic behaviour of magneto-

442

acoustic emission in a grain-oriented steel, J. Magn. Magn. Mater. 426

443

(2017) 685-690.

444

445

[5] C.D. Graham Jr, Physical origin of losses in conducting ferromagnetic materials (invited), J. Appl. Phys. 53 (1982) 8276–8280.

446

[6] A. Stupakov, O. Perevertov, V. Zablotskii, Dynamical properties of mag-

447

netic Barkhausen noise in a soft microalloyed steel, IEEE Trans. Magn.

448

51 (2015) 6100204.

449

450

ˇ [7] O. Stupakov, P. Svec, Three-parameter feedback control of amorphous ribbon magnetization, J. Electr. Eng. 64 (2013) 166–172.

451

[8] A. Stupakov, O. Perevertov, V. Zablotskii, A system for controllable

452

magnetic measurements of hysteresis and Barkhausen noise, IEEE

453

Trans. Instrum. Measur. 65 (2016) 1087–1097. 29

454

[9] O. Stupakov, J. Pal’a, T. Takagi, T. Uchimoto, Governing conditions

455

of repeatable Barkhausen noise response, J. Magn. Magn. Mater. 321

456

(2009) 2956-2962.

457

[10] O. Stupakov, Y. Melikhov, Influence of magnetizing and filtering fre-

458

quencies on Barkhausen noise response, IEEE Trans. Magn. 50 (2014)

459

6100104.

460

[11] A. Zhukov, M. V´azquez, J. Vel´azquez, C. Garc´ıa, R. Valenzuela,

461

B. Ponomarev, Frequency dependence of coercivity in rapidly quenched

462

amorphous materials, Mater. Sci. Eng. A A226–228 (1997) 753–756.

463

[12] R. Gr¨ossinger, N. Mehboob, D. Suess, R. Sato Turtelli, M. Kriegisch, An

464

eddy-current model describing the frequency dependence of the coerciv-

465

ity of polycrystalline galfenol, IEEE Trans. Magn. 48 (2012) 3076–3079.

466

[13] M. Vashista, V. Moorthy, Influence of applied magnetic field strength

467

and frequency response of pick-up coil on the magnetic Barkhausen noise

468

profile, J. Magn. Magn. Mater. 345 (2013) 208-214.

469

[14] O. Stupakov, Local non-contact evaluation of the ac magnetic hysteresis

470

parameters of electrical steels by the Barkhausen noise technique, J.

471

Nondestruct. Eval. 32 (2013) 405–412.

472

473

[15] O.V. Perevertov, Application of the Preisach model to the magnetization process in steels, J. Phys. D: Appl. Phys. 35 (2002) 2467–2471.

474

[16] O. Perevertov, J. Thielsch, R. Sch¨afer, Effect of applied tensile stress

475

on the hysteresis curve and magnetic domain structure of grain-oriented

476

transverse Fe-3%Si steel, J. Magn. Magn. Mater. 385 (2015) 358-367. 30

477

478

479

480

[17] T.R. Haller, J.J. Kramer, Observation of dynamic domain size variation in a silicon-iron alloy, J. Appl. Phys. 41 (1970) 1034–1035. [18] G. Biorci, D. Pescetti, Frequency spectrum of the Barkhausen noise, J. Appl. Phys. 28 (1957) 777–780.

481

[19] G. Bertotti, F. Fiorillo, M.P. Sassi, Barkhausen noise and domain struc-

482

ture dynamics in Si-Fe at different points of the magnetization curve, J.

483

Magn. Magn. Mater. 23 (1981) 136-148.

484

[20] R.G. Harrison, Accurate mean-field modeling of the Barkhausen noise

485

power in ferromagnetic materials, using a positive-feedback theory of

486

ferromagnetism, J. Appl. Phys. 118 (2015) 023904.

31

Highlights Intensity of Barkhausen noise rises as a square root function of the magnetizing frequency Level of magneto-acoustic emission is proportional to the hysteresis loss Magnetizing waveform has small influence on intensities of these micro-magnetic signals