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Prediction of power losses in silicon iron sheets under PWM voltage supply
~'~
ELSEVIER
Journal of Magnetism and Magnetic Materials 133 (1994) 140-143
,4~ ,i~ ~
journalof magnetism and magnetic materials
Prediction of power losses in silicon iron sheets under PWM voltage supply M. Amar, R. Kaczmarek *, F. Protat Serr'ice Electrotechnique et Electronique Industrielle, S UPELEC 91192, Gif-Sur-Y~'ette,France
Abstract
The behavior of iron losses in silicon iron steels submitted to a PWM voltage is studied. The influence of modulation parameters (the depth of modulation and the number of eliminated harmonics) is clarified. In particular, the idea of an equivalent alternating pulse voltage that gives the same iron losses as the PWM voltage is established. An estimation formula for iron losses under the PWM voltage is developed based on the loss separation model and the voltage form factor.
1. Introduction
The PWM voltage converters supply electromagnetic devices such as induction motors and inverter transformers. It is well known that the dynamic part of iron losses in magnetic cores is closely linked with the rate of change of flux density. This property is formulated in the loss separation model proposed by Bertotti [1,2]. An application of this model in the case of a PWM voltage supply is proposed. The case of a quarter-wave modulation [4] is tested experimentally with control of the depth of modulation (D) (D = V1/V,~, where Vj and Vm are, respectively, the amplitude of the fundamental component and the maximal level of the PWM voltage) and the number of eliminated harmonics (M). Measured and estimated iron losses in silicon iron sheets are fully discussed.
2. Estimation method
The iron loss separation model, which we briefly summarize below, states that the average total iron losses per cycle is the sum of hysteresis, classical and excess losses components. Hysteresis loss per cycle Ph
are determined from the area of the quasi-static hysteresis loop. They are assumed to be dependent of the peak flux density and independent of frequency. The classical eddy current losses per cycle and per mass unit Pcl are calculated by (o-d2/(12fmv))(B2), where ~, m v and d are respectively the conductivity, the mass per unit volume and the thickness of the lamination. The brackets indicate averaging over the period, f the frequency and /3 the flux density derivative. Excess eddy current losses calculations is developed in [1-3]. These losses per cycle and per mass unit Pc× are obtained by
( ~GV/-~G~oS/(fmv))(IBI 15) where S is the cross section of the lamination, G is a dimensionless coefficient and V0 characterizes the statistical distribution of the local fields of the so-called magnetic objects. Under an alternating pulse voltage of frequency f and pulse width ~-, the iron losses per cycle, at given peak flux density Bm, can be expressed by Eq. (1), as stated by Fiorillo and Novikov in [3].
P~(f) = (o-~2d2/6mv)BZ f and
* Corresponding author.
P~x(f)= ( 8 . 7 ~
0304-8853/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0304-8853(94)00068-3
/mv)Bl~S f f
M Amar et aL/Journal of Magnetism and Magnetic Materials 133 (1994) 140-143
---
Vf
The terms P ~ ( f o ) = ( f o / f ) P ~ ( f ) and P~(f0) = (fV~0/~)Pe~x(f) represent the classical and excess iron losses per cycle under a sinusoidal flux density of peak value B m and of frequency f0 (for example, 50 or 60 Hz). This last formula may be useful in a rapid characterization of the increase of the dynamic iron losses (classical plus excess eddy current components) under a PWM voltage with respect to the reference regime of a sinusoidal voltage (where F c = 1).
v(t) B(t)
0 ,
?/--';i2
x"4~"
141
~ _ i i ~
Time Fig. l. PWM voltage v(t) and resulting flux density B(t).
3. Experimental study are, respectively, the classical and excess iron losses per cycle in a sinusoidal regime of frequency f and of peak flux density Bm.
PtPotlSe(f, 7") = Ph +
4 1 PcSl(f) 2 1 P~x(f) - + "rr2 7- f w v~ ~/f
(1) Let us consider the PWM voltage v(t) (Fig. 1), the peak value of which is Vm the frequency of which is f and the pulse number per half period of which is n, with ¢i the width of the ith pulse. The flux density B ( t ) resulting in the load core has a rise time equal to the sum of pulse widths over a half period (~';7"i) and a time derivative /~ equal to zero or to +_2Bm/YTlr i. Knowing the form of/~, we can explicitly calculate the classical and excess iron losses components as formulated above. Then the average total PWM iron losses per cycle, at given Bm, are as follows:
etPotm ( f , ~.~'ri) 4 1 P~(f) = Ph + ,rr2 y~],ri ~
2 1 + 'rr ~
eeSx( f ) ff
(2)
A comparison between Eqs. (1) and (2) shows that, at the same Bm, iron losses under PWM voltage are the same as those under an alternating pulse voltage with a pulse duration equal to I2~'7"i (called equivalent pulse voltage). This is a particular case of a general rule that we state as follows: iron losses induced by two PWM schemes of 2n and 2m pulses per period respectively, are the same if Y'.~ri = Er~r:, where r i (r~) is the width of ith (jth) pulse in the first (second) scheme. Eq. (2) can be rewritten, at given Bin, as a function of the frequency and the form factor coefficient of the PWM voltage Fc = 2 / ( - n - ~ ) (defined by the ratio between the form factor of the PWM voltage and that of the sinusoidal voltage):
s
2f
PtPotm(f, Fc) = P h + Pcl(f0)Fc Too +PeSx(f°)Fc
V~0 " (3)
The general formulae presented above were tested experimentally in the case of quarter-wave modulation. The PWM signal is numerically computed and generated with control of modulations parameters. Experiments are carried out on a normalized Epstein frame supplied by a large bandwidth power amplifier. Iron losses are measured by means of a numerical precision wattmeter adapted to loads of low power factor. Two types of 3% silicon iron Epstein samples (high permeability grain-oriented (GO) type M2-H and non-oriented (NO) type M-19 of thickness 0.30 and 0.35 mm, respectively) are explored. Measurements of the area of the quasi-static hysteresis loop and of the total iron losses due to a sinusoidal flux density of frequency f0 = 50 Hz as well as a computation of the classical component allow us to separate iron losses at operating peak flux density. Fig. 2 shows the measured iron losses for different peak flux densities under a PWM voltage with eight harmonics eliminated and three fixed values of the depth of modulation. Compared with the reference case of a sinusoidal voltage, the PWM voltage increases losses in the lamination. This increase is greater for a low value of D than for the high values and it is more important in the grain-oriented material than in an non-oriented one. Iron losses are also measured under equivalent pulse voltage, they confirm the link between iron losses and the sum of pulse widths of the PWM voltage over a half period. A relative increase of the iron losses due to the PWM as defined by Eq. (4) is presented in Fig. 3. Iron loss under PWM voltage
}
e ( % ) = 100 Iron loss under sinusoidal voltage - 1 .
(4) The evolution of • in function of the modulation parameters can be explained as follows: low (high) values of the depth of modulation imply low (high) pulse time widths and hence, low (high) rise times of the flux density. The form factor is deteriorated (improved) and losses increase (decrease). However, a
M. Amar et al. /Journal of Magnetism and Magnetic Materials 133 (1994) 140-143
142
so
t
a
70
G.O. 3% FeSi
f=5OHz
i 50 40 3o oz
8O
a
G.O. 3% FeSi
5O
2
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~
x
x
5 .
Z
20
o z0
10
10
i
0 016
0.8
1
1.2
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1.6
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1.8 120
120 N.O. 3% FeSi
1.
f=5OHz
100
100
8O
8O
60
60 O Z O
40
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20 0.2
0.4
0.6
0.8
1
1.2
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D=0.5:--D=O.8:--
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60
v tO
B = 1.5T
.3 T
100
150
200
FREQUENCY (Hz)
80
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N.O. 3% FeSi
50
large eliminated harmonics band slightly increases the time widths of pulses and as a consequence, iron losses decrease slightly. The grain-oriented material is more sensitive than the non-oriented one to the variation of the modulation parameters.
a
200
20
1.6
Fig. 2. M e a s u r e d i r o n losses vs. p e a k flux d e n s i t y in G O (a) a n d N O (b) s a m p l e s u n d e r s i n u s o i d a l v o l t a g e (S: ), P W M v o l t a g e w i t h d i f f e r e n t d e p t h s o f m o d u l a t i o n ( D = 0.3 ( + ) , 0.5 ( * ) , 0.8 ( 0 ) , w i t h M = 8) a n d a n e q u i v a l e n t p u l s e v o l t a g e (1, 2, 3: ) f o r e a c h d e p t h value.
M=8
150
~ b
PEAK FLUX DENSITY (T)
300
100 FREQUENCY (Hz)
PEAK FLUX DENSITY (r)
Z O
x
X
~40 o~
,
00.4
C~
1 ~ 1.7T
7O
Fig. 4. Estimation of iron losses ( ) in GO (a) and NO (b) samples under PWM voltage (with D = 0.8 and M = 8) compared with measurements (o). Iron losses measured under equivalent pulse voltage are plotted with (×).
In Fig. 4, estimated iron loss values under a P W M voltage (from Eq. (3)) are compared with the measured values over the frequency band of 25 to 200 Hz. The form factor coefficient is directly obtained from the computed voltage. The proposed formula gives a very satisfactory prediction. The input parameters needed for its application are the P W M voltage form factor coefficient and the Ph, P~l(fo) and P~x(fO) iron losses per cycle components at operating peak flux density and frequency f0 = 50 Hz. Iron losses are also measured under an equivalent alternating pulse voltage with pulse duration equal to ]~]'z i. This test verifies the idea of equivalent pulse voltage over this explored frequency interval.
G.O.
40
4. Conclusion
10o 20
0
0.2
0.4
0.6
0.8
0
N.O.
i
A
M D Fig. 3. Relative increase of iron losses due to a PWM voltage with respect to the sinusoidal voltage in function of D (a) and M (b) at 1.5 T and 50 Hz.
The iron losses induced by a P W M voltage supply decrease when the depth of modulation increases. These losses are influenced to a lesser extent by the harmonics elimination criteria. With the number of the eliminated harmonics growing, the iron losses decrease for small numbers and stabilize thereafter. These re-
M. Amar et al. /Journal of Magnetism and Magnetic Materials 133 (1994) 140-143
suits are due to the direct impact of the modulation parameters on pulse widths. Iron losses under a P W M voltage can be referred back to those due to a simple alternating pulse voltage of a pulse width equal to E~3-i. An application of this approach gives a very satisfactory prediction of power losses in silicon iron sheets under quarter-wave P W M voltage. Other magnetic materials and types of modulation will be tested in the future.
143
References [1] G. Bertotti, J. Appl. Phys. 57 (1985) 2110. [2] G. Bertotti, IEEE Trans. Magn. 24 (1988) 621. [3] F. Fiorillo and A. Novikov, 1EEE Trans. Magn. 26 (1990) 2904. [4] H.S. Patel and R.G. Hoft, IEEE Trans. Indust. Appl. 9 (May/June 1973) 310.
ELSEVIER
Journal of Magnetism and Magnetic Materials 133 (1994) 140-143
,4~ ,i~ ~
journalof magnetism and magnetic materials
Prediction of power losses in silicon iron sheets under PWM voltage supply M. Amar, R. Kaczmarek *, F. Protat Serr'ice Electrotechnique et Electronique Industrielle, S UPELEC 91192, Gif-Sur-Y~'ette,France
Abstract
The behavior of iron losses in silicon iron steels submitted to a PWM voltage is studied. The influence of modulation parameters (the depth of modulation and the number of eliminated harmonics) is clarified. In particular, the idea of an equivalent alternating pulse voltage that gives the same iron losses as the PWM voltage is established. An estimation formula for iron losses under the PWM voltage is developed based on the loss separation model and the voltage form factor.
1. Introduction
The PWM voltage converters supply electromagnetic devices such as induction motors and inverter transformers. It is well known that the dynamic part of iron losses in magnetic cores is closely linked with the rate of change of flux density. This property is formulated in the loss separation model proposed by Bertotti [1,2]. An application of this model in the case of a PWM voltage supply is proposed. The case of a quarter-wave modulation [4] is tested experimentally with control of the depth of modulation (D) (D = V1/V,~, where Vj and Vm are, respectively, the amplitude of the fundamental component and the maximal level of the PWM voltage) and the number of eliminated harmonics (M). Measured and estimated iron losses in silicon iron sheets are fully discussed.
2. Estimation method
The iron loss separation model, which we briefly summarize below, states that the average total iron losses per cycle is the sum of hysteresis, classical and excess losses components. Hysteresis loss per cycle Ph
are determined from the area of the quasi-static hysteresis loop. They are assumed to be dependent of the peak flux density and independent of frequency. The classical eddy current losses per cycle and per mass unit Pcl are calculated by (o-d2/(12fmv))(B2), where ~, m v and d are respectively the conductivity, the mass per unit volume and the thickness of the lamination. The brackets indicate averaging over the period, f the frequency and /3 the flux density derivative. Excess eddy current losses calculations is developed in [1-3]. These losses per cycle and per mass unit Pc× are obtained by
( ~GV/-~G~oS/(fmv))(IBI 15) where S is the cross section of the lamination, G is a dimensionless coefficient and V0 characterizes the statistical distribution of the local fields of the so-called magnetic objects. Under an alternating pulse voltage of frequency f and pulse width ~-, the iron losses per cycle, at given peak flux density Bm, can be expressed by Eq. (1), as stated by Fiorillo and Novikov in [3].
P~(f) = (o-~2d2/6mv)BZ f and
* Corresponding author.
P~x(f)= ( 8 . 7 ~
0304-8853/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0304-8853(94)00068-3
/mv)Bl~S f f
M Amar et aL/Journal of Magnetism and Magnetic Materials 133 (1994) 140-143
---
Vf
The terms P ~ ( f o ) = ( f o / f ) P ~ ( f ) and P~(f0) = (fV~0/~)Pe~x(f) represent the classical and excess iron losses per cycle under a sinusoidal flux density of peak value B m and of frequency f0 (for example, 50 or 60 Hz). This last formula may be useful in a rapid characterization of the increase of the dynamic iron losses (classical plus excess eddy current components) under a PWM voltage with respect to the reference regime of a sinusoidal voltage (where F c = 1).
v(t) B(t)
0 ,
?/--';i2
x"4~"
141
~ _ i i ~
Time Fig. l. PWM voltage v(t) and resulting flux density B(t).
3. Experimental study are, respectively, the classical and excess iron losses per cycle in a sinusoidal regime of frequency f and of peak flux density Bm.
PtPotlSe(f, 7") = Ph +
4 1 PcSl(f) 2 1 P~x(f) - + "rr2 7- f w v~ ~/f
(1) Let us consider the PWM voltage v(t) (Fig. 1), the peak value of which is Vm the frequency of which is f and the pulse number per half period of which is n, with ¢i the width of the ith pulse. The flux density B ( t ) resulting in the load core has a rise time equal to the sum of pulse widths over a half period (~';7"i) and a time derivative /~ equal to zero or to +_2Bm/YTlr i. Knowing the form of/~, we can explicitly calculate the classical and excess iron losses components as formulated above. Then the average total PWM iron losses per cycle, at given Bm, are as follows:
etPotm ( f , ~.~'ri) 4 1 P~(f) = Ph + ,rr2 y~],ri ~
2 1 + 'rr ~
eeSx( f ) ff
(2)
A comparison between Eqs. (1) and (2) shows that, at the same Bm, iron losses under PWM voltage are the same as those under an alternating pulse voltage with a pulse duration equal to I2~'7"i (called equivalent pulse voltage). This is a particular case of a general rule that we state as follows: iron losses induced by two PWM schemes of 2n and 2m pulses per period respectively, are the same if Y'.~ri = Er~r:, where r i (r~) is the width of ith (jth) pulse in the first (second) scheme. Eq. (2) can be rewritten, at given Bin, as a function of the frequency and the form factor coefficient of the PWM voltage Fc = 2 / ( - n - ~ ) (defined by the ratio between the form factor of the PWM voltage and that of the sinusoidal voltage):
s
2f
PtPotm(f, Fc) = P h + Pcl(f0)Fc Too +PeSx(f°)Fc
V~0 " (3)
The general formulae presented above were tested experimentally in the case of quarter-wave modulation. The PWM signal is numerically computed and generated with control of modulations parameters. Experiments are carried out on a normalized Epstein frame supplied by a large bandwidth power amplifier. Iron losses are measured by means of a numerical precision wattmeter adapted to loads of low power factor. Two types of 3% silicon iron Epstein samples (high permeability grain-oriented (GO) type M2-H and non-oriented (NO) type M-19 of thickness 0.30 and 0.35 mm, respectively) are explored. Measurements of the area of the quasi-static hysteresis loop and of the total iron losses due to a sinusoidal flux density of frequency f0 = 50 Hz as well as a computation of the classical component allow us to separate iron losses at operating peak flux density. Fig. 2 shows the measured iron losses for different peak flux densities under a PWM voltage with eight harmonics eliminated and three fixed values of the depth of modulation. Compared with the reference case of a sinusoidal voltage, the PWM voltage increases losses in the lamination. This increase is greater for a low value of D than for the high values and it is more important in the grain-oriented material than in an non-oriented one. Iron losses are also measured under equivalent pulse voltage, they confirm the link between iron losses and the sum of pulse widths of the PWM voltage over a half period. A relative increase of the iron losses due to the PWM as defined by Eq. (4) is presented in Fig. 3. Iron loss under PWM voltage
}
e ( % ) = 100 Iron loss under sinusoidal voltage - 1 .
(4) The evolution of • in function of the modulation parameters can be explained as follows: low (high) values of the depth of modulation imply low (high) pulse time widths and hence, low (high) rise times of the flux density. The form factor is deteriorated (improved) and losses increase (decrease). However, a
M. Amar et al. /Journal of Magnetism and Magnetic Materials 133 (1994) 140-143
142
so
t
a
70
G.O. 3% FeSi
f=5OHz
i 50 40 3o oz
8O
a
G.O. 3% FeSi
5O
2
X
~
x
x
5 .
Z
20
o z0
10
10
i
0 016
0.8
1
1.2
1.4
1.6
50
1.8 120
120 N.O. 3% FeSi
1.
f=5OHz
100
100
8O
8O
60
60 O Z O
40
~
20 0.2
0.4
0.6
0.8
1
1.2
1.4
40
100
D=0.5:--D=O.8:--
G.O.
b
60
v tO
B = 1.5T
.3 T
100
150
200
FREQUENCY (Hz)
80
o-~ 2oo
N.O. 3% FeSi
50
large eliminated harmonics band slightly increases the time widths of pulses and as a consequence, iron losses decrease slightly. The grain-oriented material is more sensitive than the non-oriented one to the variation of the modulation parameters.
a
200
20
1.6
Fig. 2. M e a s u r e d i r o n losses vs. p e a k flux d e n s i t y in G O (a) a n d N O (b) s a m p l e s u n d e r s i n u s o i d a l v o l t a g e (S: ), P W M v o l t a g e w i t h d i f f e r e n t d e p t h s o f m o d u l a t i o n ( D = 0.3 ( + ) , 0.5 ( * ) , 0.8 ( 0 ) , w i t h M = 8) a n d a n e q u i v a l e n t p u l s e v o l t a g e (1, 2, 3: ) f o r e a c h d e p t h value.
M=8
150
~ b
PEAK FLUX DENSITY (T)
300
100 FREQUENCY (Hz)
PEAK FLUX DENSITY (r)
Z O
x
X
~40 o~
,
00.4
C~
1 ~ 1.7T
7O
Fig. 4. Estimation of iron losses ( ) in GO (a) and NO (b) samples under PWM voltage (with D = 0.8 and M = 8) compared with measurements (o). Iron losses measured under equivalent pulse voltage are plotted with (×).
In Fig. 4, estimated iron loss values under a P W M voltage (from Eq. (3)) are compared with the measured values over the frequency band of 25 to 200 Hz. The form factor coefficient is directly obtained from the computed voltage. The proposed formula gives a very satisfactory prediction. The input parameters needed for its application are the P W M voltage form factor coefficient and the Ph, P~l(fo) and P~x(fO) iron losses per cycle components at operating peak flux density and frequency f0 = 50 Hz. Iron losses are also measured under an equivalent alternating pulse voltage with pulse duration equal to ]~]'z i. This test verifies the idea of equivalent pulse voltage over this explored frequency interval.
G.O.
40
4. Conclusion
10o 20
0
0.2
0.4
0.6
0.8
0
N.O.
i
A
M D Fig. 3. Relative increase of iron losses due to a PWM voltage with respect to the sinusoidal voltage in function of D (a) and M (b) at 1.5 T and 50 Hz.
The iron losses induced by a P W M voltage supply decrease when the depth of modulation increases. These losses are influenced to a lesser extent by the harmonics elimination criteria. With the number of the eliminated harmonics growing, the iron losses decrease for small numbers and stabilize thereafter. These re-
M. Amar et al. /Journal of Magnetism and Magnetic Materials 133 (1994) 140-143
suits are due to the direct impact of the modulation parameters on pulse widths. Iron losses under a P W M voltage can be referred back to those due to a simple alternating pulse voltage of a pulse width equal to E~3-i. An application of this approach gives a very satisfactory prediction of power losses in silicon iron sheets under quarter-wave P W M voltage. Other magnetic materials and types of modulation will be tested in the future.
143
References [1] G. Bertotti, J. Appl. Phys. 57 (1985) 2110. [2] G. Bertotti, IEEE Trans. Magn. 24 (1988) 621. [3] F. Fiorillo and A. Novikov, 1EEE Trans. Magn. 26 (1990) 2904. [4] H.S. Patel and R.G. Hoft, IEEE Trans. Indust. Appl. 9 (May/June 1973) 310.