Fig. 8. Example of FE calculated losses for different harmonic orders.
The losses at the various interesting frequencies can besummed to obtain the overall power consumption. This isconceptually correct, because in any system excited bymultiple sources with different frequencies, the totalactive power equals the sum of the active power due to allthe sources taken independently. This result will beformally proven to hold (Section IV) also for the assumedhysteresis loss model (II.B-2).
One could wander if the use of FE analysis tools could be skipped once an analytical model of magnetic corelosses, as per (1) and (9), has been established and
parametrically identified. The answer is that thementioned analytical expressions for the magnetic power loss have a local meaning, i.e. p Fe and p H , defined by (1)and (9) respectively, indicate specific losses within aninfinitesimal machine volume (dV ) where the magneticflux density and field intensity peak values ( B1 and H 1)can be assumed constant. To compute the overall core
losses P Fe, though, the specific losses need to beintegrated over the entire stator and rotor core volumes:
∫∫ +=
corerotor
Fe Fe
core stator
Fe Fe Fe dV pdV p P δ δ (12)
where δ Fe indicates the core material density. In the ringtest, the flux density inside the motor yoke is almostuniform (Fig. 4), which allows for (12) to be simplifiedinto:
( )2211 f Bk f Bk M p M
dV pdV p
E H yoke Fe yoke
corerotor
Fe Fe
core stator
Fe Fe
+==
=+
∫∫α
δ δ
(13)
as done in writing (2). The same, of course, does notapply to normal motor operation, when the flux densityamplitude strongly varies from point to point (Fig. 7),especially when harmonic exciting fields are present. Theneed is then explained for a FE tool capability of numerically computing integrals (12) over the entirevolume of stator and rotor cores.
IV. I NVESTIGATION INTO THE HYSTERESIS MODEL
SUITABILITY FOR HARMONIC FIELDS
In II-B it has been shown how the fact of modeling the
hysteresis process through a fictitious phase shift φ between B and H sine waves is reasonably realistic as far as the fundamental magnetizing component is concerned.
In this section the implications of applying the samehysteresis loss model when the magnetization has timeharmonics will be investigated.
Let us suppose that B in each point of the motor corecontains not only the fundamental B1 but also some timeharmonics, of amplitudes B3, B5, B7 , etc. and phase angles
ψ 3, ψ 5, ψ 7 , etc. with respect to the fundamental, as itactually happens under PWM supply. Neglectingsaturation, this means that, in any given point, themagnetic field intensity can be written as:
...)5cos(
)3cos()cos()(
55
331
+++
+++=
ψ ω µ
ψ ω µ
ω µ
t B
t B
t B
t H
(14)
If hysteresis is modeled through (6)-(7) and the
fictitious shift angle φ is held the same for all harmonicsimulations (Section III), the equivalent flux density will
be:
...)5cos(
)3cos()cos()(
55
331
++++
+++++=
φ ψ ω
φ ψ ω φ ω
t B
t Bt Bt B(15)
The resulting trajectory of the ( H , B) point looks likethe plots of Fig. 9, where a 23
rdorder harmonic, of 5%
amplitude, is superimposed to the fundamental as an
example, for different possible angles φ .
1− 104
× 0 1 104
×
2−
1−
0
1
2
1− 104
× 0 1 104
×2−
1−
0
1
2
1− 104
× 0 1 104
×2−
1−
0
1
2
Loop due to the fundamental
Loop due to the fundamental with a 23rd harmonic
Fig. 9. Example of trajectory of the ( H , B) point in presence of a 23rd order harmonic, of 5% amplitude, in the flux density.
It is remarked that the B- H trajectory around the main
ellipsis is not affected at all by the phase angles ψ 3, ψ 5,ψ 7 , etc.
The specific power consumption (W/kg) associated tothe distorted hysteresis loop (including harmonics) is:
[ ]
[ ]...53)sin(
...)3sin(3)sin(
...)3cos()cos(
)()(
25
23
21
331
0
331
0
+++=
=+++++×
×
+++=
=∂
∂=
∫
∫
B B B f
dt t Bt B
t B
t B f
dt t
t Bt H
f p
Fe
T
Fe
T
Fe H
φ δ µ
π
φ ψ ω ω φ ω ω
ψ ω µ
ω µ δ
δ
(16)
0
200
400
600
800
1000
1200
1400
1600
1800
2000
5 0 0
1 0 0 0
1 5 0 0
2 0 0 0
2 5 0 0
3 0 0 0
3 5 0 0
4 0 0 0
4 5 0 0
5 0 0 0
5 5 0 0
6 0 0 0
6 5 0 0
7 0 0 0
W
Harmonic order
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is exactly the hysteresis loss computed from the harmonic
analysis at frequency h f (Section III), in accordance with(9). This confirms that the superposition principle for thetotal power due to several sources at different frequenciesholds also with the adopted hysteresis loss model.
The problem is that, to the authors’ knowledge, nosignificant evidence can be produced on the likelihood of the assumed hysteresis model in presence of high order harmonics. References have been found in the literature[10], [11] demonstrating how, in presence of a distortedexcitation current waveform, the hysteresis loop shapemay significantly vary with respect to the one recordedunder sinusoidal supply; in [10], [11] authors observe the
possible occurrence of “minor loops” around the mainloop, but do not provide any practical or theoreticalcriteria to predict the way in which the hysteresis loopchanges with the harmonic content of the excitationcurrents. Therefore, the hypothesis of deformedhysteresis loops as shown in Fig. 9 may constitute asomewhat arbitrary assumption.
Nevertheless, one should consider that what is of importance for the subject study is the power losscomputation only. To this end, a reasonable assumption isthat, due to the very small amplitude of motor harmoniccurrents in the case of PWM supply, the shape of thehysteresis loop does not change importantly compared tothe case of sinusoidal supply. In particular, the possible“minor loops” originating from high order harmonics [10]
are assumed of negligible area compared to the area of the main loop.
Fortunately, the above assumption is consistent withthe fact that hysteresis power loss, as (9) shows, weightsless and less compared to the eddy-current losscomponent, as the harmonic order h increases. In fact, if we compute the hysteresis to eddy-current loss ratio for the h
thorder harmonic we have from (1) and (9):
h f k f h Bk
B f h
p
p
Fe E h E
h Fe
E
H 1)sin()sin(
222
2
δ µ
φ π φ
δ µ
π
== (18)
Assuming the same values as in II.B-1 for the parameters in (18), the diagram of Fig. 10 is obtained.
10 20 30 40 500
1
2
3
4
h
E
H
p
p
Fig. 10. Hysteresis to eddy-current harmonic loss ratio versus harmonic
order.
It can be seen that, while for the fundamental (h=1)hysteresis losses prevail over the eddy-current ones by a
factor around 3.5, the situation is reversed for high order harmonics. In particular, the current harmonics produced
by the PWM, which are high-order ones, will cause largeeddy-current losses and small hysteresis losses. This factmakes the computation results very little sensitive to the
possible inaccuracies in the model used for harmonichysteresis loss computation.
V. EXPERIMENTAL RESULTS
The method proposed in this paper was applied to predict the inverter-related extra-losses in a 5 MWinduction motor (4 poles, 900 rpm, 2400 V) designed to
be supplied by a MV three-level inverter of knownvoltage spectrum (Fig. 6). After the machine test datawere available, the design spectrum was adjustedaccording to measurements so as to match the realvoltage distortion, which proved better than predictedfrom the design data (Table I).
TABLE IVOLTAGE DISTORSION FROM MEASUREMENTS
Voltage RMS Voltage fundamental THD Current
V V % A
2461 2343 32,1 426
2442 2323 32,4 369
2631 2533 28,0 592
2229 2094 36,5 277
2441 2344 29,1 417
Because no ring test data were available on themachine under study, the motor model was tuned, asdescribed in Section II, based on ring-test data from
previously-built similar machines.
The FE harmonic analysis, repeated for all harmoniccomponents as illustrated in Section III, led to the power loss spectrum displayed in Fig. 8. Summing the power losses over all the frequencies, the total power consumption due to PWM harmonics was obtained equalto 6638 W.
The computed value was then compared to the datacollected during the motor no-load test. This test wasrepeated, at the same speed, in two different conditions,i.e. with the motor supplied from the grid (sinusoidalvoltage source) and with the motor supplied from theinverter, obtaining the no-load current diagrams reportedin Fig. 11.
Fig. 11. Diagrams (interpolation lines) of the stator no-load current(fundamental rms values) versus the stator voltage (fundamental per unit value) with sinusoidal and inverter supply at no-load and at the
same speed.
It is remarked that, for the data processing to makesense, the fundamental components (and not the total rms
200
250
300
350
400
450
500
550
600
650
0,85 0,9 0,95 1 1,05 1,1 1,15
A
V
Sinusoidal supply
Inverter supply
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values) of both motor currents and voltages had to berecorded by means of a spectrum analyzer during the test.
The procedure followed to compute the experimentalcore losses under inverter supply along with their increment with respect to the sinusoidal supply isanalytically illustrated in Table II.
TABLE IICORE LOSS EVALUATION UNDER GRID AND INVERTER SUPPLY
# Description ote
(1) No-load current, sinusoidal supply 383 A At rated V
(2) No-load current, inverter supply 455 A At rated V
(3) Total no-load losses, sinusoidal supply 45 kW At rated V
(4) Total no-load losses, sinusoidal supply 53,4 kW At rated V
(5) Windage and friction losses 7,1 kW At rated rpm
(10) Core loss increase due to inverter (kW) 7,38 kW (9)-(8)
(11) Core loss increase due to inverter (%) 20,9 % (10)/(8)×100
The value of windage and friction losses was inferredfrom the no-load test under sinusoidal supply inaccordance with [9]. It can be seen that a good agreementis obtained in terms of experimental and computed valuesfor the core loss increase due to inverter supply: the valueinferred from measurements is 7380 W (Table II), whilethe calculated value is 6638 W. According tomeasurements the increase equals to 20.9%, according tocalculations to 18.8%.
VI. CONCLUSIONS
A finite element (FE) approach has been discussed inthis paper to compute the magnetic core losses caused inan induction motor by PWM inverter supply. The FEmodel tuning procedure resorts to eddy-current andhysteresis loss characterization from available ring-testdata. The magnetic hysteresis phenomenon has beenmodeled introducing a frequency-independent phase lag
between the flux density and field intensity local values.The appropriateness of this hysteresis model in presenceof time harmonics in the magnetic field has been
critically investigated. It has been concluded that possibleinaccuracies of this modeling approach are beneficiallymitigated by the prevalence of the eddy-currentcomponent over the hysteresis one in the losses due tohigh order harmonics.
The application of the proposed method to a real 5MW induction motor drive based a three-level MVinverter has been finally presented. A good accordancehas been found between test data processing and the
proposed calculation method. In both ways, the extra corelosses caused by the PWM supply are evaluated around20% of the rated core losses under sinusoidal supply.
R EFERENCES
[1] C.J. Melhorn, L. Tang, “Transient effects of PWMdrives on induction motors”, IEEE Transactions on
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[2] L.T. Mthombeni, P. Pillay, “Core losses in motor laminations exposed to high-frequency or nonsinusoidal excitation”, IEEE Transactions on
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[3] Y. Kawase, T. Yamaguchi, Y. Mizuno, “3-D Eddy
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Industry Applications, 2003, vol. 123, pp. 323-329.[4] J. Saitz, “Computation of the core loss in an
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[5] G.B. Kliman, Sang Bin Lee, M.R. Shah, R.M.Lusted, N.K. Nair, “A new method for synchronousgenerator core quality evaluation”, IEEE Transactions on Energy Conversion, vol. 19, Sept.2004, pp. 576-582.
[6]
A.E. Fitzgerald, C. Kingsley, A. Kusko, Electric Machinery, 1978, FrancoAngeli s.r.l., Milan, Italy, p.438.
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[8] B.K. Bose, Modern Power Electronics and AC Drives, 2001, Pearson Education.
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