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Power losses in grain-oriented electrical steels
MI
Journal of Magnetism and Magnetic Materials 112 (1992) 33-35 North-Holland
Power losses in grain-oriented electrical steels V l a d i s l a v Wiglasz The Iron and Steel Research Institute, CS-739 51 Dobrd, Czechoslovakia
A theoretical calculation of the specific losses, which occur during ac magnetization in grain-oriented Fe-3% Si electrical steels was made. It was found, that these "theoretical" losses are in good qualitative agreement with experimental results.
I. Introduction Power losses occurring in grain-oriented electrical steels during ac magnetization generally consist of a hysteresis (static) component, and of an eddy-current component. It is known, that the eddy-current loss component in grain-oriented materials is 1.5 to 3 times bigger than the eddycurrent loss component as calculated from the generally known classical equation. Total losses and static hysteresis losses can be measured directly by the usual methods, eddy-current losses are then calculated by subtraction. The ratio of the total eddy-current loss to the eddy-current loss calculated from the classical equation is also often expressed by the ao-called anomaly coefficient. It must be noted - concerning the classical theory of the eddy-current loss - that this theory is valid only under the condition that the dimensions of the domains are considerably smaller than the sheet thickness. When this condition is not met, then the domain structure plays a very important role at the origin of the eddy-current loss [i]. ~xpenmenlally t~tlpulat~u -*" ' "- 1 ~-tlcquc~a~y u"~. i.J.c., n dencies of specific losses per cycle of real grainoriented F e - 3 % Si materials are characterized by two peculiarities:
Correspondence to: Dr. V. Wiglasz, The Iron and Steel Research Institute, CS-739 51 Dobrfi, Czechoslovakia.
a) losses are generally higher than losses calculated by the classical equation, b) the frequency dependence of the losses is nonlinear (according to the Pry-Bean theory it should be linear).
2. Procedure of calculation The basic equation for the solution of this problem was deduced by solving Maxwell's equations. This equation was created for the mathematical description of the magnetization process:
02H
OB
OX 2 = ~'
at '
(1)
where H = magnetic field intensity, B = magnetic induction, y = material conductivity, x = length coordinate and t = time. From eq. (1) another equation for the calculation of the magnetic field may be deduced: O2H OX2
(dB)0H "),' ~ --~=0,
(2)
where B = f ( H ) , dB/dH=f(H) are known (i.e. given or chosen) functions. Eq. (2) is a quasi-linear, partial differential of 2nd order of parabolic type. Since the analytical solution of eq. (2) is not known (with the exception of the case, when dB/dH = constant), the calculation was made numerically by the method of nets.
0304-8853/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
34
V. Wiglasz / Power losses in grain-oriented electrical steels
Fig. 1. Distribution of eddy currents according to the domain model (I~ is vector of magnetic polarization of saturation).
Calculations were realized for the following cases of functional dependence B - f ( H ) , 1. constant permeability B = const x H, 2. nonlinear permeability without consideration of the component of the static hysteresis loss, a dependence B = f ( H ) is expressed by an exponential or power function, 3. nonlinear permeability, including a static hysteresis loss component, dependence of static hysteresis loop B = f ( H ) is expressed by exponential functions. A new model (analogous to the Pry-Bean model) was created for the solution of the magnetization process considering the domain structure of eddy-current losses according to fig. 1; new names of "classical" eddy currents and "domain" eddy microcurrents were introduced. This case was solved by the so called method of the state equations [2].
3. R e s u l t s a n d d i s c u s s i o n
It was found by numerical calculations of the eddy-current loss with consideration of a functional dependence B = f ( H ) and omitting the static hysteresis loss, that the value of the "classical" eddy-current loss component depends on the ~h~,,,~ ,e the U~,.'I./I~.,IIKIK,;III,~K;: A. . . . ~. . . . . B - - j ~-,~ , Hr r X1, 111 T_ s o m e c a s e s . . . . ]bl~.. v ~ t (a very steep dependence B = f ( H ) ) the value of the loss by "classical" eddy currents can be as much as 50% higher than the loss calculated by a classical equation. Also in the case of considering a nonlinear dependence B = f ( H ) , the frequency dependence of the specific eddy-current loss per cycle has an almost linear character.
When considering a functional dependence B = f ( H ) , including the static hysteresis loss component (in this case a static hysteresis loop was expressed with the use of exponential functions), it was found that the hysteresis component of the total specific loss per cycle is frequency dependent and that the size of the eddy-current loss component depends on the steepness of the nonlinearity of the static hysteresis loop. This result - surprising as it may look at the first glance - is completely different from the concepts described in the literature, where it is usually supposed, that the hysteresis specific loss component is directly proportional to the frequency of the magnetization. In case of consideration of the influence of the domain structure on the calculation of the eddycurrent loss component, it was found that the total specific loss by eddy currents are closer to the real values which are usually obtained by measurement of these materials in practice. Total specific losses by eddy currents are determined by the sum of specific losses due to "'classical" and "domain" eddy currents, and they can be influenced by the change of the main domain structure (i.e. by reduction of the distance of 180° domain walls by various physical methods), which results in reduction of the size of "domain" eddy microcurrents and consequently also in the reduction of specific losses by eddy currents. It was found at the same time, that the eddy-current loss component - as calculated by the method introduced in this paper - differs substantially from the results obtained on the basis of the Pry-Bean model. Another conclusion drawn from theoretical calculation of the eddy-current loss component is the fact, that results of specific loss as obtained by the measurement at the harmonical and then triangl~ (or generally nonharrnonic_~!) course of the total magnetic induction flux cannot be used for the calculation of the distribution of the total loss on the hysteresis component and eddy-current loss component. During ac magnetization, which is effected by the motion of domain walls, in all the cases described above occurs a bending of domain wails as a result of the performance of induced eddy
," Wiglasz / Power losses in grain-oriemed electrical steels DE-MAGNETIZED STATE:
0
®
B?(t)=O
0
®
I~i
L..L direction B?(t}: 0 ..
t -~.0-
.0
oat:0 l~(t)=0 T
al e- i "=i _{._ -, -O
9! F=
-O
( I .
.
,
.
, ,
-O"
oat =
I x , / ( s,(t=l.06T t=T
.
,
', 1-1/Bp{t}:l.5
T
=ot ,,,
- ( ~ IJ ~ ',/
--'~-.
'
tI - ~ ~.I
't-'-
I__2
l
..- ~ t = 4
Bp(t)=l.06T
,¢=, ~k~ Bp{t)= 0T
Fig. 2. Shape of the domain walls in the sample Fe-3% Si in some points of the magnetization cycle at f = 50 Hz. Bp = 1.5 T, 2 d = 2 L =0.0003 m and at the harmonical course Bp(t) (dependence B = f ( H ) : ~m,,, = 0.182 H rn -I, ~min = 0.0025 H m-l).
currents. It can be undoubtedly said, that in conductive ferromagnetic materials a bending of domain walls occurs always during ac magnetization. The size of this bending is proportional to the value of induced eddy currents, and increases with the increase of the thickness of the material, its conductivity, amplitude of the magnetic induction and frequency of the ac magnetization. An example of the shape of the domain walls of grain-oriented F e - 3 % Si material in some points of the magnetization cycle is given in fig. 2 for the case of the magnetization frequency of 50 Hz, amplitude of average magnetic induction 1.5 T and distance of domain walls 2L = 0.0003 m. It is possible to make a conclusion on the basis of realized numerical calculations of losses in grain-oriented F e - 3 % Si materials with an ideal (model) domain structure, that both peculiarities of the frequency dependence of the specific loss per cycle, obtained by experimental measurement
35
(as was described in the introduction) are caused by dynamic behaviour of the domain structure. Anomaly ad a) is caused by the presence of the domain structure, which results during ac magnetization in the creation of "domain" eddy microcurrents around each moving domain wall. The value of the loss caused by these micro currents is approximately the same as a component of the loss causes by "classical" eddy currents. That is why in the area of industrial frequencies ( = 50 Hz) the total specific losses caused by eddy currents can be 2-3 times higher as compared with the theoretical value caicul"ted by the classical equation, which correspL, nds qualitatively to the values obtained experiraentally in real F e - 3 % Si grain-oriented materials. Anomaly ad b) is connected also with the presence of the domain structure, especially with the distance of domain walls of the main (strip) domain structure. Experimental data published in technical literature (e.g. refs. [3,4]) prove the fact that the distance of the domain walls of the main domain structure is frequency-dependent. Increase of the magnetization frequency leads to the reduction of the distance between the domain walls. This process is discontinuous in individual grains and the frequency-dependence of the specific loss per cycle for individual isolated grains could be described by a broken line. In a macroscopic scale this process seems therefore to be continuous and represents in fact an "average" value of all the grains, participating in the ac magnetization. As a result of what was said above, the frequency-dependence of the specific loss per cycle has a nonlinear course (which is usually found in real grain-oriented F e - 3 % Si materials).
References [1] R.H. Pry, and C.P. Bean, J. Appl. Phys. 291 (1958) 532. [2] F. Koufil and K. Vrba, Teorie Neline~irnich Obvodu (SNTL, Prague, 1982) p. 141. [3] K.J. Overshott and S. Hill, in: Proc. EPS Conf. SMM2 (Cardiff, 1975)p. 115. [4] J.W. Shilling, in: Proc. EPS Conf. SMM2 (Cardiff, 1975) p. 92.
Journal of Magnetism and Magnetic Materials 112 (1992) 33-35 North-Holland
Power losses in grain-oriented electrical steels V l a d i s l a v Wiglasz The Iron and Steel Research Institute, CS-739 51 Dobrd, Czechoslovakia
A theoretical calculation of the specific losses, which occur during ac magnetization in grain-oriented Fe-3% Si electrical steels was made. It was found, that these "theoretical" losses are in good qualitative agreement with experimental results.
I. Introduction Power losses occurring in grain-oriented electrical steels during ac magnetization generally consist of a hysteresis (static) component, and of an eddy-current component. It is known, that the eddy-current loss component in grain-oriented materials is 1.5 to 3 times bigger than the eddycurrent loss component as calculated from the generally known classical equation. Total losses and static hysteresis losses can be measured directly by the usual methods, eddy-current losses are then calculated by subtraction. The ratio of the total eddy-current loss to the eddy-current loss calculated from the classical equation is also often expressed by the ao-called anomaly coefficient. It must be noted - concerning the classical theory of the eddy-current loss - that this theory is valid only under the condition that the dimensions of the domains are considerably smaller than the sheet thickness. When this condition is not met, then the domain structure plays a very important role at the origin of the eddy-current loss [i]. ~xpenmenlally t~tlpulat~u -*" ' "- 1 ~-tlcquc~a~y u"~. i.J.c., n dencies of specific losses per cycle of real grainoriented F e - 3 % Si materials are characterized by two peculiarities:
Correspondence to: Dr. V. Wiglasz, The Iron and Steel Research Institute, CS-739 51 Dobrfi, Czechoslovakia.
a) losses are generally higher than losses calculated by the classical equation, b) the frequency dependence of the losses is nonlinear (according to the Pry-Bean theory it should be linear).
2. Procedure of calculation The basic equation for the solution of this problem was deduced by solving Maxwell's equations. This equation was created for the mathematical description of the magnetization process:
02H
OB
OX 2 = ~'
at '
(1)
where H = magnetic field intensity, B = magnetic induction, y = material conductivity, x = length coordinate and t = time. From eq. (1) another equation for the calculation of the magnetic field may be deduced: O2H OX2
(dB)0H "),' ~ --~=0,
(2)
where B = f ( H ) , dB/dH=f(H) are known (i.e. given or chosen) functions. Eq. (2) is a quasi-linear, partial differential of 2nd order of parabolic type. Since the analytical solution of eq. (2) is not known (with the exception of the case, when dB/dH = constant), the calculation was made numerically by the method of nets.
0304-8853/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
34
V. Wiglasz / Power losses in grain-oriented electrical steels
Fig. 1. Distribution of eddy currents according to the domain model (I~ is vector of magnetic polarization of saturation).
Calculations were realized for the following cases of functional dependence B - f ( H ) , 1. constant permeability B = const x H, 2. nonlinear permeability without consideration of the component of the static hysteresis loss, a dependence B = f ( H ) is expressed by an exponential or power function, 3. nonlinear permeability, including a static hysteresis loss component, dependence of static hysteresis loop B = f ( H ) is expressed by exponential functions. A new model (analogous to the Pry-Bean model) was created for the solution of the magnetization process considering the domain structure of eddy-current losses according to fig. 1; new names of "classical" eddy currents and "domain" eddy microcurrents were introduced. This case was solved by the so called method of the state equations [2].
3. R e s u l t s a n d d i s c u s s i o n
It was found by numerical calculations of the eddy-current loss with consideration of a functional dependence B = f ( H ) and omitting the static hysteresis loss, that the value of the "classical" eddy-current loss component depends on the ~h~,,,~ ,e the U~,.'I./I~.,IIKIK,;III,~K;: A. . . . ~. . . . . B - - j ~-,~ , Hr r X1, 111 T_ s o m e c a s e s . . . . ]bl~.. v ~ t (a very steep dependence B = f ( H ) ) the value of the loss by "classical" eddy currents can be as much as 50% higher than the loss calculated by a classical equation. Also in the case of considering a nonlinear dependence B = f ( H ) , the frequency dependence of the specific eddy-current loss per cycle has an almost linear character.
When considering a functional dependence B = f ( H ) , including the static hysteresis loss component (in this case a static hysteresis loop was expressed with the use of exponential functions), it was found that the hysteresis component of the total specific loss per cycle is frequency dependent and that the size of the eddy-current loss component depends on the steepness of the nonlinearity of the static hysteresis loop. This result - surprising as it may look at the first glance - is completely different from the concepts described in the literature, where it is usually supposed, that the hysteresis specific loss component is directly proportional to the frequency of the magnetization. In case of consideration of the influence of the domain structure on the calculation of the eddycurrent loss component, it was found that the total specific loss by eddy currents are closer to the real values which are usually obtained by measurement of these materials in practice. Total specific losses by eddy currents are determined by the sum of specific losses due to "'classical" and "domain" eddy currents, and they can be influenced by the change of the main domain structure (i.e. by reduction of the distance of 180° domain walls by various physical methods), which results in reduction of the size of "domain" eddy microcurrents and consequently also in the reduction of specific losses by eddy currents. It was found at the same time, that the eddy-current loss component - as calculated by the method introduced in this paper - differs substantially from the results obtained on the basis of the Pry-Bean model. Another conclusion drawn from theoretical calculation of the eddy-current loss component is the fact, that results of specific loss as obtained by the measurement at the harmonical and then triangl~ (or generally nonharrnonic_~!) course of the total magnetic induction flux cannot be used for the calculation of the distribution of the total loss on the hysteresis component and eddy-current loss component. During ac magnetization, which is effected by the motion of domain walls, in all the cases described above occurs a bending of domain wails as a result of the performance of induced eddy
," Wiglasz / Power losses in grain-oriemed electrical steels DE-MAGNETIZED STATE:
0
®
B?(t)=O
0
®
I~i
L..L direction B?(t}: 0 ..
t -~.0-
.0
oat:0 l~(t)=0 T
al e- i "=i _{._ -, -O
9! F=
-O
( I .
.
,
.
, ,
-O"
oat =
I x , / ( s,(t=l.06T t=T
.
,
', 1-1/Bp{t}:l.5
T
=ot ,,,
- ( ~ IJ ~ ',/
--'~-.
'
tI - ~ ~.I
't-'-
I__2
l
..- ~ t = 4
Bp(t)=l.06T
,¢=, ~k~ Bp{t)= 0T
Fig. 2. Shape of the domain walls in the sample Fe-3% Si in some points of the magnetization cycle at f = 50 Hz. Bp = 1.5 T, 2 d = 2 L =0.0003 m and at the harmonical course Bp(t) (dependence B = f ( H ) : ~m,,, = 0.182 H rn -I, ~min = 0.0025 H m-l).
currents. It can be undoubtedly said, that in conductive ferromagnetic materials a bending of domain walls occurs always during ac magnetization. The size of this bending is proportional to the value of induced eddy currents, and increases with the increase of the thickness of the material, its conductivity, amplitude of the magnetic induction and frequency of the ac magnetization. An example of the shape of the domain walls of grain-oriented F e - 3 % Si material in some points of the magnetization cycle is given in fig. 2 for the case of the magnetization frequency of 50 Hz, amplitude of average magnetic induction 1.5 T and distance of domain walls 2L = 0.0003 m. It is possible to make a conclusion on the basis of realized numerical calculations of losses in grain-oriented F e - 3 % Si materials with an ideal (model) domain structure, that both peculiarities of the frequency dependence of the specific loss per cycle, obtained by experimental measurement
35
(as was described in the introduction) are caused by dynamic behaviour of the domain structure. Anomaly ad a) is caused by the presence of the domain structure, which results during ac magnetization in the creation of "domain" eddy microcurrents around each moving domain wall. The value of the loss caused by these micro currents is approximately the same as a component of the loss causes by "classical" eddy currents. That is why in the area of industrial frequencies ( = 50 Hz) the total specific losses caused by eddy currents can be 2-3 times higher as compared with the theoretical value caicul"ted by the classical equation, which correspL, nds qualitatively to the values obtained experiraentally in real F e - 3 % Si grain-oriented materials. Anomaly ad b) is connected also with the presence of the domain structure, especially with the distance of domain walls of the main (strip) domain structure. Experimental data published in technical literature (e.g. refs. [3,4]) prove the fact that the distance of the domain walls of the main domain structure is frequency-dependent. Increase of the magnetization frequency leads to the reduction of the distance between the domain walls. This process is discontinuous in individual grains and the frequency-dependence of the specific loss per cycle for individual isolated grains could be described by a broken line. In a macroscopic scale this process seems therefore to be continuous and represents in fact an "average" value of all the grains, participating in the ac magnetization. As a result of what was said above, the frequency-dependence of the specific loss per cycle has a nonlinear course (which is usually found in real grain-oriented F e - 3 % Si materials).
References [1] R.H. Pry, and C.P. Bean, J. Appl. Phys. 291 (1958) 532. [2] F. Koufil and K. Vrba, Teorie Neline~irnich Obvodu (SNTL, Prague, 1982) p. 141. [3] K.J. Overshott and S. Hill, in: Proc. EPS Conf. SMM2 (Cardiff, 1975)p. 115. [4] J.W. Shilling, in: Proc. EPS Conf. SMM2 (Cardiff, 1975) p. 92.