- Email: [email protected]
The calculation of iron losses in brushless permanent magnet dc motors
~ ,~ ELSEVIER
Journal of Magnetism and Magnetic Materials 133 (1994) 578-582
journal of magnetism and magnetic materials
The calculation of iron losses in brushless permanent magnet dc motors K. Atallah, D. Howe * Department of Electronic &Electrical Engineering, University of Sheffield, PO Box 600, Mappin Street, Sheffield S I 4DU, UK
Abstract The paper describes a procedure developed for the prediction of the flux density waveforms and iron losses in the stator core of a brushless permanent magnet dc motor. The procedure is applied to a low-cost drive system, and it is shown that the operating condition can have a marked effect on both the magnitude and distribution of the iron loss.
1. Introduction Whilst the mmf and flux density waveforms in most traditional machine formats are essentially sinusoidally time-varying, and appropriate loss data is generally available, the flux density waveforms in brushless permanent magnet dc motors are more or less trapezoidal, a consequence of the magnet mmf and stator winding current waveforms being essentially rectangular. Further, both the iron loss distribution and the total iron loss can vary significantly with the operating condition of a brushless dc drive. In order that the iron loss can be predicted reliably at the design stage, and efficiency measures, such as the use of improved electrical sheet steels, can be accurately quantified, a procedure has been developed [1,2] in which localised flux density waveforms throughout the stator core are determined by undertaking a series of nonlinear magnetostatic finite element analyses as the rotor is rotated incrementally over half an electrical cycle, the instantaneous stator current distribution being deduced from a dynamic simulation of the drive. The hysteresis, eddy current, and excess compo-
* Corresponding author. Fax: +44 (742) 726391.
nents of the iron loss density are then calculated by using appropriate loss constants, which are determined experimentally with a closed-loop computer controlled single-sheet tester. The paper briefly describes the calculation procedure, and compares predicted flux density waveforms and iron loss data with measurements made on a typical low cost brushless drive system, for which the influence of varying the load and the commutation strategy are investigated.
2. Calculation procedure The procedure is applied to a 150 W, four-pole, three-phase, 2000 rpm brushless permanent magnet dc motor equipped with surface-mounted, radially magnetized bonded NdFeB magnets. The stator laminations are fully processed silicon steel, Transil300.
2.1. Field calculation Magnetostatic finite element solutions are obtained for discrete rotor positions equispaced over half an electrical cycle. At each rotor position the current in the stator slots is determined from a dynamic simula-
0304-8853/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0304-8853(94)00212-A
579
K Atallah, D. Howe~Journal of Magnetism and Magnetic Materials 133 (1994) 578-582
B*
o
".=..
I
c
..- , . - " .," .-" ~
¢.)
/ , .e.,P /,,,=.
"cI tO..
~
/
Measured Predicted
.9o
"~ '~. -
• . . . . . " Rotating flux density I ~ ' - - - ~ !aerpendiculat to rolling direction • ........ • Parallel to rolling direction
o
t--
1
2
.
.
01.0
3
Flux density (T) Angle (elec. radians)
Fig. l. Predicted and measured phase current waveforms (normal commutation).
tion of the drive system. The field solutions are then processed to obtain local flux density waveforms. Fig. 1 compares computed and measured phase current waveforms at rated load (normal commutation). 2.2. Iron loss density calculation
The hysteresis loss component is expressed as: Ph = k h fBa,
(1)
where f and B m are the frequency and amplitude of the flux density, and k h and a are determined experimentally. However, when the induction waveform causes minor loops, an empirical correction must be applied [3], giving: P. =
khfB~r(Bm),
(2)
Fig. 3. Comparison of iron loss densities measured under alternating sinusoidal flux densities, both parallel and perpendicular to the rolling direction, and purely rotating flux densities using single-sheet tester. where 0.65 n g(nm) = 1 +nm Y'~ A n i, i=l
and A B i is the change in flux density during the excursion around a minor loop. The eddy current loss component Pd is separated into the classical eddy current component Pc and the excess eddy current component Pc- The classical eddy current loss component is given by: crd21 . [ dB ~2
Pc = I~--~ J T [ "-~ - ]I dt,
(3)
where o-, /5 and d are the electrical conductivity, the mass density and the lamination thickness, respectively. The excess eddy current loss component is given by [4,5]: k e P° =
T f r
dB dt
3/2 dt,
(4)
10
I~ .......... ..... V
13 8
g
"~".~
Measured ~ 100 F~z t Computed at 50 Hz Computed at 100 Hz Measured at S0 Hz
where k~ is determined experimentally from iron loss tests under sinusoidal flux density waveforms spanning a range of frequencies.
4 " - ........ ! /
-- !1..-.-....tl ........ .~t........ 1F........
2
0
0
20
40
60
80
10o
(%) Fig. 2. Predicted and measured iron loss density under alternating trapezoidal flux density waveforms.
Fig. 4. Positions of search coils.
IC Atallah, D. Howe/Journal of Magnetism and Magnetic Materials 133 (1994) 578-582
580
Fig. 2 compares computed and measured iron loss densities, for a sample of Transil300, under alternating trapezoidal flux density waveforms characterised by the parameter ~-, which determines the extent of the flattop, where ~-= 0% corresponds to a rectangular waveform and r = 100% corresponds to a triangular waveform. Rotating fluxes also exist in brushless machines. However, to date a model for predicting the associated iron loss has yet to be developed. Nevertheless, experimental investigations [5] have shown that for low to medium induction levels a reasonable approximation is simply to add the losses associated with the corre-
E
sponding orthogonal alternating flux density components. The results shown in Fig. 3 confirm that this is the case for Transil300 up to 1.5 T, despite the inherent anisotropy.
3. Results and discussion The test motor was instrumented with search coils, as shown in Fig. 4, and was tested under both open-circuit and load conditions, with normal, advanced and retarded commutation. Fig. 5 compares computed and measured flux den-
E
Measured Computed
.... > <
<
. . . .
' 1
. . . .
' 2
"
"
• 3
-2
. 0
Angle (el~ r~:fians)
.
.
.
.
. 1
.
.
.
.
.
.
' 2
Angle (elec r~lians)
Search coil A
Search coil B
2~10 *
1X10 ~
Ix10"
....
ComputedMeasur [ ed ,'T
,'7
l .... Measured] Computed -1x10 ~
_~(10 ~
2
I
Angle
(elec radians)
Search coil C
1
2
Angle (eiec radians) Search coil D
Fig. 5. Flux density waveforms in search coils A and B, and flux waveforms in C and D.
K. Atallah, D. Howe/Journal of Magnetism and Magnetic Materials 133 (1994) 578-582
581
30xt0 ~
IXICr~ 1 5X10~
'\ "C
C"
Q)
LL
LL
....
Computed J
Measured ]
....
Measaured
Computed
-1x10 ~
-30xlO ~
1
1
2
2
Angle (elec radians)
Angle (elec radians)
Search coil F
Search coil E 30xlO ~
15x10~
.1.5x10 4
-30xlO" _3.0x10~
-15x10 4
1.5x10-4
0
31~D4
Resulting loci Fig. 6. Radial and circumferential flux waveforms and the resulting locus behind a tooth (search coils E and F).
sity waveforms on full-load (normal commutation) in the tooth body (search coil A), the stator back-iron (search coil B), and both sides of a tooth tip (search coils C and D). It can be seen that armature reaction causes an asymmetry with respect to the tooth axis. Fig. 6 compares radial and circumferential flux components on full-load (normal commutation) at the back of a tooth (search coils E and F), and also shows the resulting flux locus, which exhibits pronounced rotation. Table 1 compares the iron losses on open-circuit with those at rated load under normal commutation. The effect of advancing (i.e. field-weakening) and retarding the commutation by 20 °, whilst maintaining the rated current, is also included. However, it will be seen that on load there is a considerable discrepancy between measured and computed results. The most likely cause is attributable to the measurement technique
employed, viz. the torque-speed/input power method, which inherently limits the achievable accuracy, particularly with distorted voltage and current waveforms. The alternative would be to employ a calibrated calorimetric technique [7]. In addition, the computed results do not account for stator end-effects or rotor losses. Also, because of the high quality of the lamination
Table 1 Iron loss at 2000 rpm Operating condition Open-circuit Normal commutation 20° advanced commutation 20° retarded commutation
Iron loss (pu) Computed
Measured
1.0 1.63 1.12 1.69
0.98
3.64 3.31 3.38
582
K. Atallah, D. Howe/Journal of Magnetism and Magnetic Materials 133 (1994) 578-582
material, the iron loss on full load normal commutation represents only about 15% of the total loss, the copper loss accounting for 75% and friction and windage 10%. Nevertheless, the operating condition can be seen to have a considerable effect on the total iron loss.
4. Conclusions It has been shown that the iron loss in brushless dc motors can be markedly affected by the load condition. It has also been shown that a calculation procedure based on coupling a drive system simulation to magnetostatic finite element analyses, and incorporating single-sheet test results, enables the prediction of the flux density waveforms and the total iron loss under any
operating condition of the drive. However, further experimental validation is required.
References [1] K. Atallah, Z.Q. Zhu and D. Howe, IEEE Trans. Magn. 28 (1992) 2997. [2] K. Atallah, Z.Q. Zhu and D. Howe, Proc. ICEM, Manchester, 1992, p. 1025. [3] J.D. Lavers, P.P. Biringer and H. Hollitscher, IEEE Trans. Magn. 14 (1978) 386. [4] G. Bertotti, IEEE Trans. Magn. 24 (1988) 621. [5] F. Fiorillo and A. Novikov, IEEE Trans. Magn. 26 (1990) 2094. [6] T. Yamaguchi and K. Narita, Elec. Eng. Jpn 96 (1976) 15. [7] B. Baholo, P.H. Mellor, D. Howe and T.S. Birch, J. Magn. Magn. Mater. 133 (1994) 433 (this volume).
Journal of Magnetism and Magnetic Materials 133 (1994) 578-582
journal of magnetism and magnetic materials
The calculation of iron losses in brushless permanent magnet dc motors K. Atallah, D. Howe * Department of Electronic &Electrical Engineering, University of Sheffield, PO Box 600, Mappin Street, Sheffield S I 4DU, UK
Abstract The paper describes a procedure developed for the prediction of the flux density waveforms and iron losses in the stator core of a brushless permanent magnet dc motor. The procedure is applied to a low-cost drive system, and it is shown that the operating condition can have a marked effect on both the magnitude and distribution of the iron loss.
1. Introduction Whilst the mmf and flux density waveforms in most traditional machine formats are essentially sinusoidally time-varying, and appropriate loss data is generally available, the flux density waveforms in brushless permanent magnet dc motors are more or less trapezoidal, a consequence of the magnet mmf and stator winding current waveforms being essentially rectangular. Further, both the iron loss distribution and the total iron loss can vary significantly with the operating condition of a brushless dc drive. In order that the iron loss can be predicted reliably at the design stage, and efficiency measures, such as the use of improved electrical sheet steels, can be accurately quantified, a procedure has been developed [1,2] in which localised flux density waveforms throughout the stator core are determined by undertaking a series of nonlinear magnetostatic finite element analyses as the rotor is rotated incrementally over half an electrical cycle, the instantaneous stator current distribution being deduced from a dynamic simulation of the drive. The hysteresis, eddy current, and excess compo-
* Corresponding author. Fax: +44 (742) 726391.
nents of the iron loss density are then calculated by using appropriate loss constants, which are determined experimentally with a closed-loop computer controlled single-sheet tester. The paper briefly describes the calculation procedure, and compares predicted flux density waveforms and iron loss data with measurements made on a typical low cost brushless drive system, for which the influence of varying the load and the commutation strategy are investigated.
2. Calculation procedure The procedure is applied to a 150 W, four-pole, three-phase, 2000 rpm brushless permanent magnet dc motor equipped with surface-mounted, radially magnetized bonded NdFeB magnets. The stator laminations are fully processed silicon steel, Transil300.
2.1. Field calculation Magnetostatic finite element solutions are obtained for discrete rotor positions equispaced over half an electrical cycle. At each rotor position the current in the stator slots is determined from a dynamic simula-
0304-8853/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0304-8853(94)00212-A
579
K Atallah, D. Howe~Journal of Magnetism and Magnetic Materials 133 (1994) 578-582
B*
o
".=..
I
c
..- , . - " .," .-" ~
¢.)
/ , .e.,P /,,,=.
"cI tO..
~
/
Measured Predicted
.9o
"~ '~. -
• . . . . . " Rotating flux density I ~ ' - - - ~ !aerpendiculat to rolling direction • ........ • Parallel to rolling direction
o
t--
1
2
.
.
01.0
3
Flux density (T) Angle (elec. radians)
Fig. l. Predicted and measured phase current waveforms (normal commutation).
tion of the drive system. The field solutions are then processed to obtain local flux density waveforms. Fig. 1 compares computed and measured phase current waveforms at rated load (normal commutation). 2.2. Iron loss density calculation
The hysteresis loss component is expressed as: Ph = k h fBa,
(1)
where f and B m are the frequency and amplitude of the flux density, and k h and a are determined experimentally. However, when the induction waveform causes minor loops, an empirical correction must be applied [3], giving: P. =
khfB~r(Bm),
(2)
Fig. 3. Comparison of iron loss densities measured under alternating sinusoidal flux densities, both parallel and perpendicular to the rolling direction, and purely rotating flux densities using single-sheet tester. where 0.65 n g(nm) = 1 +nm Y'~ A n i, i=l
and A B i is the change in flux density during the excursion around a minor loop. The eddy current loss component Pd is separated into the classical eddy current component Pc and the excess eddy current component Pc- The classical eddy current loss component is given by: crd21 . [ dB ~2
Pc = I~--~ J T [ "-~ - ]I dt,
(3)
where o-, /5 and d are the electrical conductivity, the mass density and the lamination thickness, respectively. The excess eddy current loss component is given by [4,5]: k e P° =
T f r
dB dt
3/2 dt,
(4)
10
I~ .......... ..... V
13 8
g
"~".~
Measured ~ 100 F~z t Computed at 50 Hz Computed at 100 Hz Measured at S0 Hz
where k~ is determined experimentally from iron loss tests under sinusoidal flux density waveforms spanning a range of frequencies.
4 " - ........ ! /
-- !1..-.-....tl ........ .~t........ 1F........
2
0
0
20
40
60
80
10o
(%) Fig. 2. Predicted and measured iron loss density under alternating trapezoidal flux density waveforms.
Fig. 4. Positions of search coils.
IC Atallah, D. Howe/Journal of Magnetism and Magnetic Materials 133 (1994) 578-582
580
Fig. 2 compares computed and measured iron loss densities, for a sample of Transil300, under alternating trapezoidal flux density waveforms characterised by the parameter ~-, which determines the extent of the flattop, where ~-= 0% corresponds to a rectangular waveform and r = 100% corresponds to a triangular waveform. Rotating fluxes also exist in brushless machines. However, to date a model for predicting the associated iron loss has yet to be developed. Nevertheless, experimental investigations [5] have shown that for low to medium induction levels a reasonable approximation is simply to add the losses associated with the corre-
E
sponding orthogonal alternating flux density components. The results shown in Fig. 3 confirm that this is the case for Transil300 up to 1.5 T, despite the inherent anisotropy.
3. Results and discussion The test motor was instrumented with search coils, as shown in Fig. 4, and was tested under both open-circuit and load conditions, with normal, advanced and retarded commutation. Fig. 5 compares computed and measured flux den-
E
Measured Computed
.... > <
<
. . . .
' 1
. . . .
' 2
"
"
• 3
-2
. 0
Angle (el~ r~:fians)
.
.
.
.
. 1
.
.
.
.
.
.
' 2
Angle (elec r~lians)
Search coil A
Search coil B
2~10 *
1X10 ~
Ix10"
....
ComputedMeasur [ ed ,'T
,'7
l .... Measured] Computed -1x10 ~
_~(10 ~
2
I
Angle
(elec radians)
Search coil C
1
2
Angle (eiec radians) Search coil D
Fig. 5. Flux density waveforms in search coils A and B, and flux waveforms in C and D.
K. Atallah, D. Howe/Journal of Magnetism and Magnetic Materials 133 (1994) 578-582
581
30xt0 ~
IXICr~ 1 5X10~
'\ "C
C"
Q)
LL
LL
....
Computed J
Measured ]
....
Measaured
Computed
-1x10 ~
-30xlO ~
1
1
2
2
Angle (elec radians)
Angle (elec radians)
Search coil F
Search coil E 30xlO ~
15x10~
.1.5x10 4
-30xlO" _3.0x10~
-15x10 4
1.5x10-4
0
31~D4
Resulting loci Fig. 6. Radial and circumferential flux waveforms and the resulting locus behind a tooth (search coils E and F).
sity waveforms on full-load (normal commutation) in the tooth body (search coil A), the stator back-iron (search coil B), and both sides of a tooth tip (search coils C and D). It can be seen that armature reaction causes an asymmetry with respect to the tooth axis. Fig. 6 compares radial and circumferential flux components on full-load (normal commutation) at the back of a tooth (search coils E and F), and also shows the resulting flux locus, which exhibits pronounced rotation. Table 1 compares the iron losses on open-circuit with those at rated load under normal commutation. The effect of advancing (i.e. field-weakening) and retarding the commutation by 20 °, whilst maintaining the rated current, is also included. However, it will be seen that on load there is a considerable discrepancy between measured and computed results. The most likely cause is attributable to the measurement technique
employed, viz. the torque-speed/input power method, which inherently limits the achievable accuracy, particularly with distorted voltage and current waveforms. The alternative would be to employ a calibrated calorimetric technique [7]. In addition, the computed results do not account for stator end-effects or rotor losses. Also, because of the high quality of the lamination
Table 1 Iron loss at 2000 rpm Operating condition Open-circuit Normal commutation 20° advanced commutation 20° retarded commutation
Iron loss (pu) Computed
Measured
1.0 1.63 1.12 1.69
0.98
3.64 3.31 3.38
582
K. Atallah, D. Howe/Journal of Magnetism and Magnetic Materials 133 (1994) 578-582
material, the iron loss on full load normal commutation represents only about 15% of the total loss, the copper loss accounting for 75% and friction and windage 10%. Nevertheless, the operating condition can be seen to have a considerable effect on the total iron loss.
4. Conclusions It has been shown that the iron loss in brushless dc motors can be markedly affected by the load condition. It has also been shown that a calculation procedure based on coupling a drive system simulation to magnetostatic finite element analyses, and incorporating single-sheet test results, enables the prediction of the flux density waveforms and the total iron loss under any
operating condition of the drive. However, further experimental validation is required.
References [1] K. Atallah, Z.Q. Zhu and D. Howe, IEEE Trans. Magn. 28 (1992) 2997. [2] K. Atallah, Z.Q. Zhu and D. Howe, Proc. ICEM, Manchester, 1992, p. 1025. [3] J.D. Lavers, P.P. Biringer and H. Hollitscher, IEEE Trans. Magn. 14 (1978) 386. [4] G. Bertotti, IEEE Trans. Magn. 24 (1988) 621. [5] F. Fiorillo and A. Novikov, IEEE Trans. Magn. 26 (1990) 2094. [6] T. Yamaguchi and K. Narita, Elec. Eng. Jpn 96 (1976) 15. [7] B. Baholo, P.H. Mellor, D. Howe and T.S. Birch, J. Magn. Magn. Mater. 133 (1994) 433 (this volume).