Estimation of Iron Losses in Induction Motors - Semantic Scholar

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 57, NO. 1, JANUARY 2010

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Estimation of Iron Losses in Induction Motors: Calculation Method, Results, and Analysis Zbigniew Gmyrek, Aldo Boglietti, Senior Member, IEEE, and Andrea Cavagnino, Member, IEEE

Abstract—This paper intends to develop a more accurate approach for determining the no-load iron losses in pulse width modulation (PWM) inverter fed induction motors. The proposed method is validated by means of a prototype motor with a plastic rotor cage. The iron losses have been computed by the timestepping finite element method, both with sinusoidal and PWM supply. The iron losses have then been estimated by adding up the contribution generated by orthogonal components of the flux density, as if the iron losses generated by these components were independent phenomena. The rotational hysteresis losses, as well as excess ones, have been calculated applying a correction factor based on experimental data. These factors are a function of the peak flux density and ellipticity of the B vector loci. Experimental validations are provided for several frequency and magnetic saturation values. In addition, this paper demonstrates the necessity to consider the harmonics initial phase in order to increase the accuracy in the iron loss prediction. Index Terms—Finite element method (FEM), induction motors, iron losses, pulsewidth modulation (PWM) supply.

I. I NTRODUCTION

T

HE INDUCTION motor design requires a correct estimation of the iron losses, in particular in the presence of deformed magnetic flux in the motor core. As a consequence, utilization of simple dependence of iron losses on the frequency and the peak value of the flux density does not fulfil designer expectations. Many authors propose utilization of these concepts, combined with careful calibration by experimental data [1]. Alternatively, it is possible to estimate the iron losses in the frequency domain or in the time domain [2]–[4]. In the frequency domain, the loss separation among the static hysteresis, classical eddy current, and excess loss is widely used. These iron loss components are calculated in terms of the Bm peak flux density, the f frequency, and the material dependent coefficients kh , kc , and ke , respectively. The iron loss calculation in the frequency domain does not consider the initial phases of each individual flux harmonic. In particular, the harmonic initial phase is very significant for lower order harmonics with relatively high amplitude, because the peak value of the flux changes and the static hysteresis iron

Manuscript received October 30, 2008; revised May 19, 2009. First published June 5, 2009; current version published December 11, 2009. Z. Gmyrek is with the Institute of Mechatronics and Information Systems, Technical University of Lodz, 90-924 Lodz, Poland (e-mail: gmyrek@ p.lodz.pl). A. Boglietti and A. Cavagnino are with the Department of Electrical Engineering, Politecnico di Torino, 10129 Torino, Italy (e-mail: aldo.boglietti@ polito.it; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2009.2024095

loss also changes. This problem is discussed in Section III, where the harmonic initial phases are considered. From this point of view, the iron loss estimation should be executed in the time domain. Several time-domain static or dynamic hysteresis models have been proposed for the iron loss calculation [5], [6]. Unfortunately, these models require the execution of many experiments in order to identify precise parameters. The failure to fulfill all requirements leads to incorrect iron loss results [7]. This paper presents the time-domain iron loss calculation method using time-stepping finite element method (FEM). In this method, the iron loss was separated on three components. These components were computed in postprocessor mode using the flux density waveforms, calculated in elementary regions of the model of the induction motor. Computations have been performed for sinusoidal as well as pulsewidth modulation (PWM) supply. In order to verify the computation results, experimental tests have been carried out on a motor prototype for some fundamental harmonic frequencies and peaks of the flux density. II. E XPERIMENTAL T EST B ENCH Before describing the adopted experimental approach, it is important to highlight the following concepts linked to electrical machine loss segregation. In papers concerning the comparison between measured and computed iron losses, a still unsolved problem is as follows: “What is the meaning of the measured iron losses?” It is well known that the iron losses in induction motors are measured by no-load test, and the iron losses determination is defined by the International Standards. These iron losses must be considered as convectional iron losses, and they cannot be considered as the actual ones. In particular, the additional iron losses in no-load conditions due to the stator space harmonics give a contribution up to 10% of the conventional measured iron losses. This was demonstrated by the experimental point of view in [15] using the same induction motor prototype described in the following. In addition, the determination of the mechanical losses is questionable, and their value determines the value of the measured iron losses. Other examples of inaccuracy in the measurement of the iron losses can be found. As a consequence, speaking of the competence of a method for predicting the iron losses, it is very important the measured data are accurate. Therefore, the iron losses as defined by the International Standards are not the best ones; it is well known that several loss contributions are active in an induction motor, i.e., stator and rotor Joule losses, iron losses, mechanical and additional losses. As the aim of this paper is the validation of the proposed iron loss computation method, an ad hoc motor prototype has been built. In particular, an original aluminum squirrel cage rotor has been replaced with

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a plastic squirrel cage. In this way, the rotor Joule losses and the additional rotor losses due to the air-gap spatial MMF harmonics are completely absent. Consequently, this prototype can be considered as a test bench, with the rotor cage losses equal to zero. The iron losses can be measured as the difference between the absorbed electrical power and the stator Joule losses only. Another advantage of this prototype is that the locked rotor test allows accurate measuring of the actual iron losses including those iron losses in the rotor lamination. In fact, in the locked rotor test, the rotor iron losses are active and measured, but this is not a problem because the simulation can be done in the same conditions. Obviously, in locked rotor condition, there are no stator and rotor teeth iron loss contributions due to slotting effects. It is important to remark that the locked rotor tests are straightforward to perform and simulate. In fact, in the no-load synchronous test, the rotor of the prototype has to be driven by a synchronous motor with the same pole number. In order to avoid the presence of slip (and related iron rotor losses), the supply frequency must be the same both for the motor under test and the synchronous motor. This is easy to do in the tests with sinusoidal supply, while for the inverter supply, this condition cannot be obtained due to the impossibility of synchronizing the fundamental frequency of the inverter with the sinusoidal power supply of the synchronous motor. It is important to underline that very small value of slip for the prototype leads to consistent variation of the absorbed power of this machine. As any torque and speed transducer are positioned between the two machines, the mechanical power exchanges cannot be measured, and they are wrongly attributed to the core losses of the motor under test. The prototype nameplate data are the following: 11 kW, 4 poles, 50 Hz, 230 V, delta connection. All specifications of this machine are known, such as stator and rotor lamination dimensions, winding specification, used magnetic material characteristics, and so on. Sinusoidal supply locked rotor tests have been performed at several frequencies, ranging from 10 Hz up to 100 Hz. In addition, PWM supply locked rotor tests have been performed, with a fundamental frequency of 50 Hz and switching frequency from 1300 Hz up to 7300 Hz. During the tests, the electrical quantities have been measured by a power meter with 800-kHz bandwidth. As requested for the simulations, the phase voltage and phase current waveforms have been acquired by a digital oscilloscope. III. I RON L OSS E STIMATION M ETHOD For the considered prototype, a 2-D FEM model has been developed, as shown in Fig. 1. The model was divided in elementary regions where the normal and tangential components of the flux density were computed. These regions can be arbitrarily small, so it was possible to consider that both components of the flux density are constant in the region. Calculation results, made with utilization of FEM model, were stored and next used in postprocessing mode, where iron loss components have been computed in all elementary regions. In estimating process of iron loss components, it was assumed that the iron loss under rotating field is the sum of the iron loss under alternating conditions in radial (r) and tangential (Θ) directions. The static hysteresis loss Ph considering the effect of

Fig. 1.

Mesh of the investigated model.

rotating magnetic field (without minor loops), can be calculated according to formula Ph =

N 1 α α ∗ (B + BmΘi ) · khi . T i=1 mri

(1)

∗ is the coefficient depending on material parameters In (1), khi and elementary volume as well as ellipticity of the B vector loci in ith region, Bmri , BmΘi are the maximum value of the normal and tangential component of the flux density in ith region, N is the number of regions, α is the coefficient of the hysteresis loss, and T is the fundamental harmonic period. As a consequence, the solution to the problem depends on the assumption, confirmed by many researchers, that without minor loops, the static hysteresis loss depends only on the peak of flux density. In the case of the minor loop existence, the problem is more complicated. Many authors suggest using a simple formula and adding up the static hysteresis losses, originating from fundamental and high-order harmonics [9], [10]. However, the same amplitude of harmonics and different initial phases will change the maximum value of the deformed flux, and in this way, the total static hysteresis loss. Moreover, these can lead to the occurrence of the minor loops, as shown in Fig. 2. The analysis of the static hysteresis loss error shows the necessity of the consideration of harmonic initial phase. The static hysteresis error was defined as absolute value of the difference between Phx − Ph0 over Ph0 . The Ph0 symbol represents the static hysteresis loss when the initial phase of higher harmonic is zero, whereas Phx symbol represents the static hysteresis loss in the case of nonzero initial phase. Some calculated errors are shown in Fig. 3. In this figure, the initial phase of the harmonic was chosen as 180◦ . The largest errors occurred for this phase value. The estimations were executed by utilization of the static Jiles–Atherton model (c = 0.1, α = 0.001, k = 1500 A/m) with voltage excitation [14]. The voltage contained the fundamental and single high-order harmonic, with changeable initial phase. The higher amplitude of harmonic, with respect to the fundamental one, or/and the lower order of harmonic, will cause stronger dependence of the static hysteresis loss on the initial phase of harmonic, which is clearly visible in Fig. 3. Investigations executed by authors and described in [8] present the dependence of the iron losses originating from the higher harmonic on the amplitude of fundamental harmonic. These observations lead to the conclusion that, only in special


GMYREK et al.: ESTIMATION OF IRON LOSSES IN INDUCTION MOTORS

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Fig. 3. Static hysteresis loss estimation error versus the harmonic order. The flux density amplitude of the harmonics: (a)—50 mT, (b)—100 mT, (c)—200 mT. The amplitude of the fundamental harmonic—1 T. The initial phase of the harmonics—180◦ .

Fig. 2. Examples of the static loop (Bmax 1 = 1.6 T, Bmax 5 = 0.3 T, Bmax 7 = 0.1 T; initial phase of the fifth harmonic—90◦ ; initial phases of the seventh harmonic: (a)—0◦ , (b)—60◦ , (c)—90◦ , (d)—120◦ , (e)—150◦ , (f)—180◦ ).

cases, it is possible to estimate the static hysteresis loss with reference to the amplitude of higher harmonics only. Therefore, authors have used the static Jiles–Atherton model to estimate the static hysteresis loss in analyzed models for both axial and rotational magnetization [14]. By the postprocessing computation in individual regions, the waveforms of the flux density components were obtained. These waveforms were used to determine, in all regions, the following quantities: the el ellipticity and the peak of the flux density. The el ellipticity was defined as the relation of the major axis over the minor axis of the B vector loci. It is well known that, under rotational magnetization, the hysteresis losses can be greater than under longitudinal magnetization, which is very well described in literature and confirmed by measurements [11], [12]. Taking this into account, the static hysteresis loss was estimated according to formula Ph =

N kh [kel (eli , Bmaxi ) · P (Bmaxi )] . T i=1

(2)

In (2), kh is the coefficient depending on material parameters, kel is the coefficient depending on eli ellipticity and

Fig. 4. Measured trend of the kel coefficient versus ellipticity and peak of the flux density (BMAX ). (a)—BMAX = 1.6 T, (b)—BMAX = 1.4 T, (c)—BMAX =< 1.0 T.

Bmaxi maximum value of the flux density in ith region (see Fig. 4), P (Bmaxi ) is the static hysteresis loss in ith region, determined with the help of the Jiles–Atherton model, N is the number of regions, and T is the fundamental harmonic period. The P (Bmaxi ) losses (for ith region) were determined in the following way: First, the direction of the long axis of the vector B loci was set; next, the flux density waveform, related to this direction was computed. This waveform was used as excitation in the Jiles–Atherton model, and the static hysteresis loop was the result of executed calculations. The last step of this process is the calculation of the area of the static


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hysteresis loop including minor loops. This area was multiplied by the iron volume, and in this way, the static hysteresis loss referred to ith region was obtained. The proposed method of the static hysteresis loss estimation has the following advantages: it considers the initial phases of harmonics, it considers the influence of fundamental harmonic on the static hysteresis losses originating from higher harmonics (this problem was described in [8]), and it enables the static hysteresis loss estimation in case of rotational magnetization. The longer estimation time, with respect to analytical models, is the flaw in this method. The classical eddy current loss Pecl , was evaluated by ⎡ 2 E N i i − Br,j Br,j+1 1 ⎣ Pecl = kc E j=1 i=1 Δt 2 ⎤ i i − BΘ,j BΘ,j+1 ⎦ · vi (3) + Δt where kc is the coefficient depending on material characteri i , BΘj are the radial and tangential components of istics, Brj the flux density in ith region and jth moment, E is a number of moments in fundamental harmonic period (E = T /Δt), Δt is the step time, and vi is the volume of the iron related to ith region. The excess loss Pex was estimated by formula (4), where ke is the material coefficient for this loss contribution ∗ (eli , Bmaxi ) is the coefficient depending on el ellipticity and kex and Bmaxi maximum value of the flux density in ith region (see Fig. 5). Fiorillo et al. [13] suggest the slight dependence of this factor on frequency

∗ coefficient versus ellipticity and peak Fig. 5. Measured trend of the kex of the flux density (BMAX ). (a)—BMAX = 1.6 T, (b)—BMAX = 1.4 T, (c)—BMAX =< 1.0 T.

E N 1 ∗ Pex = ke k (eli , Bmaxi ) · vi E j=1 i=1 ex ⎡ 1.5 1.5 ⎤ i i i i BΘ,j+1 − BΘ,j Br,j+1 − Br,j ⎦ . (4) ·⎣ + Δt Δt

The coefficients kh , kc , and ke were measured for longitudinal magnetization under sinusoidal excitation whereas ∗ for rotational magnetization under coefficients kel and kex sinusoidal excitation and some ellipticities of vector B loci. The flux density waveform was controlled by means of a feedback included in the test bench. The obtained results confirm that ignoring relative phase angle of the harmonics, the total iron loss estimation error increases up to 14%, when compared to 10% when initial phase angles were taken into account. These values are concerning the rated voltage, whereas for lower voltage the error increase is smaller due to a lower magnetic saturation level. IV. M ODEL D ESCRIPTION The model of the induction motor was analyzed using an FEM software. A part of the model mesh is shown in Fig. 1. The problem was solved as nonlinear, isotropic, and magnetostatic. Then, in the postprocessor mode, the model was divided into arcs. Each arc was divided into segments, creating in this way elementary regions (see Fig. 6). The number of elementary

Fig. 6. Stator (a) and rotor (b) pitches with marketed elementary arcs, elementary region and points shown in Figs. 7–9.

regions, placed on arcs, can be unrestricted. The stator and rotor models were divided into 40 arcs. Each stator arc was divided into 3600 segments and each rotor arc into 2800 segments. The arc number and its segments were chosen experimentally, with respect to the accepted estimation error of total iron losses and minimum calculation time. The proposed model was used in the iron loss estimation process, both under sinusoidal and PWM supply. Figs. 7 and 8 show the calculated loci of the B vectors in two stator and rotor core regions (marked as points 1 and 2 in Fig. 6), with sinusoidal and PWM supply, respectively. The obtained results show that in several parts of the motor core, rotational magnetization occurs, confirming the necessity for equations which take this phenomenon into account. Even with sinusoidal supply, due to the magnetic nonlinearity, the



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Fig. 7. B vector loci under sinusoidal supply [(a)—for the rotor area—point 1 in Fig. 6, (b)—for the stator area—point 2]. Sinusoidal supply: 1—230 V/50 Hz, 2—200 V/50 Hz, 3—160 V/50 Hz, 4—100 V/50 Hz.

Fig. 8. B vector loci under PWM supply [(a)—for the rotor area—point 1 in Fig. 6, (b)—for the stator area—point 2]. The rms value of the fundamental harmonic: 1—230 V/50 Hz, 2—200 V/50 Hz, 3—160 V/50 Hz. The PWM supply—switching frequency 2 kHz, constant modulation index.

waveforms of the flux density components in individual regions of the model are not sinusoidal. As a consequence, even under reduced voltage, formulations based on the fundamental harmonic only cannot be used. Some calculation results are shown in Fig. 9. The problem becomes more complicated when the rotor rotates, because in the stator and rotor teeth close to the airgap zone, the magnetization is more complicated. In this paper, this problem has not been analyzed and taken into account. For

all regions of the postprocessor model, the harmonic analysis of the radial and tangential flux components was performed. In the stator back iron, peaks of the radial and tangential components are different, but the presence of higher harmonics in the radial component waveform leads to iron losses comparable with those coming from the tangential components. In this region, in the tangential component, the greatest harmonic amplitude does not exceed 3% of the fundamental


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Fig. 9. Calculated waveforms of the radial component of the flux density [(a)—for a point in stator tooth—point 3 in Fig. 6, (b)—for a point in stator back iron—point 2]. Sinusoidal supply: 1—230 V/50 Hz, 2—200 V/50 Hz, 3—160 V/50 Hz, 4—100 V/50 Hz.

harmonic amplitude. A different conclusion can be formulated for the normal flux component, where the greatest harmonic amplitude can reach 30% of the fundamental harmonic amplitude. In addition, many other significant harmonics are present. In other regions, for example in the regions close to the tooth symmetry axis, the third harmonic dominates, while in the regions close to the slot symmetry axis, the fifth harmonic dominates. This flux distortion leads to significant errors in the iron loss estimation, in particular when the adopted formulations use information concerning the fundamental harmonic only, because in some regions, the sum between classical and excess iron losses is three times higher than the results obtained with fundamental harmonic only. The analysis of the loss estimation error leads to the conclusion regarding the necessity of harmonic consideration in the loss estimation process. The loss estimation error was defined as the relation of the difference in the iron losses calculated by using the fundamental harmonic only and those calculated with the harmonics, over the iron losses calculated by using the fundamental harmonic only. For example, in the region marked by point 2 in Fig. 6, the waveform of the tangential flux component is not very deformed. Taking the different amplitudes of normal and tangential flux density components into account, the total error of the iron loss estimation, in the analyzed region, does not exceed 15% (presented results concern the sinusoidal supply 230 V/50 Hz). In teeth, the average estimation errors reach 20%. In the stator back iron, the following error estimation results were obtained: 34%–39%—for 100 V/50 Hz, 22%—for 160 V/50 Hz, 16%—for 200 V/50 Hz, and 15%—for 230 V/50 Hz. Another analysis was carried out for the static hysteresis loss. Estimation errors caused by keeping the initial phases

was discussed in Section III. As previously mentioned, in some regions, the third and fifth harmonic having significant amplitude, occur. In the investigated model, the initial phases of these harmonics were in the range from 0◦ to 40◦ . Thus, in these regions, the static hysteresis loss differs about 15% from those without initial phase utilization. Moreover, analysis of the magnetic field distribution was extended into the B vector loci ellipticity analysis. The results referring to the stator and rotor pitches are shown in Figs. 10 and 11. In the stator pitch, the regions with el ellipticity less than 0.1 occupies 32% of the iron area in the stator pitch (sinusoidal supply 230 V/50 Hz). For other ellipticities, the percentage contribution is as follows: 0.1 < el < 0.3%−42.4%, 0.3 < el < 0.5%−14%, 0.5 < el < 0.7%−6.2%, 0.7 < el < 0.9%−3.8%, el > 0.9%−1.6%. In the case of a nonsaturated magnetic circuit (100 V/50 Hz), the percentage contributions of regions with specified ellipticity are as follows: el < 0.1%−38.6%, 0.1 < el < 0.3%−42.5%, 0.3 < el < 0.5%−10%, 0.5 < el < 0.7%−4.6%, 0.7 < el < 0.9%−3.2%, el > 0.9%−1.1%. In the rotor pitch, the percentage contributions of regions are as follows: el < 0.1%− 26.4% of the iron area in the rotor pitch, 0.1 < el < 0.3%− 22.9%, 0.3 < el < 0.5%−16.6%, 0.5 < el < 0.7%−20.2%, 0.7 < el < 0.9%−10.1%, el > 0.9%−3.8%. The waveforms of the flux density components, calculated for all elementary regions of the model, were the basis for determination of the iron losses. In each elementary region, the iron loss components (static hysteresis, classical eddy current and excess losses) were estimated and stored. Then, they were added up. In this way, the total iron loss in the stator and rotor cores were estimated and the results are reported in the next section.



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Fig. 10. Regions of (black color) stator pitch with specified ellipticity of the B vector loci. Ellipticity: (a) el < 0.1, (b) 0.1 < el < 0.3, (c) 0.3 < el < 0.5, (d) 0.5 < el < 0.7, (e) 0.7 < el < 0.9, (f) el > 0.9. Sinusoidal supply 230 V/50 Hz.

V. M EASURED AND C OMPUTED R ESULTS : C OMPARISON AND D ISCUSSION Estimation of the iron loss components in all regions of the model allows a deep analysis of distribution of the specific iron losses. The specific iron loss was defined as relation of the total iron loss in a specified region over the iron mass related to that region. Figs. 12 and 13 show examples of the specific iron loss distributions in the stator and rotor pitches, under different

magnetic saturation. The stator and rotor slots position is clearly visible in these figures. In the case of high magnetic saturation, the largest iron losses occur in the teeth, as was expected. In high saturation, the iron loss distribution in the stator teeth is nearly uniform [see Fig. 12(a)], whereas in the rotor teeth, it is irregular (due to complex shape of slot). In low saturation, the largest iron losses occur at edges of slots. It is important to highlight the relatively high iron losses in the rotor yoke, both


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Fig. 11. Regions of (black color) rotor pitch with specified ellipticity of the B vector loci. Ellipticity: (a) el < 0.1, (b) 0.1 < el < 0.3, (c) 0.3 < el < 0.5, (d) 0.5 < el < 0.7, (e) 0.7 < el < 0.9, (f) el > 0.9. Sinusoidal supply 230 V/50 Hz. Rotor was locked under no-load test.

under high and low saturation level. The iron loss occurs in the rotor due to the locked rotor conditions. Moreover, for elementary regions laying on specified stator or rotor arc, the average iron loss densities were determined according to

pk =

N 1 pi . mN Vk i=1

(5)

In (5), pk is the average iron loss density, related to the kth stator or rotor arc, m is the specific iron mass, N is the number of elementary regions on the kth arc, Vk is the elementary iron volume, and pi are the iron losses in the current elementary region. The results, obtained for a 230 V/50 Hz supply, are shown in Figs. 14 and 15, where the curves represent the iron losses originating from the flux components. As expected, the largest average iron loss density is present in the teeth near the yoke.



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Fig. 12. Distribution of the specific iron loss in the stator pitch. (a) For 230 V/50 Hz. (b) For 100 V/50 Hz.

Fig. 14. Trend of the iron loss density in the stator core versus the number of elementary stator arc. (a) Losses due to tangential flux component. (b) Losses due to radial flux component. (c) Total loss (solid lines: sinusoidal supply, dashed lines: PWM supply with switching frequency 1.3 kHz, saturation as for 230 V/50 Hz).

Fig. 13. Distributions of the specific iron loss in the rotor pitch. (a) For 230 V/50 Hz. (b) For 100 V/50 Hz. Rotor was locked under no-load tests.

Fig. 15. Trend of the iron loss density in the rotor core versus the number of elementary rotor arc. (a) Losses due to tangential flux component. (b) Losses due to radial flux component. (c) Total loss (solid lines: sinusoidal supply, dashed lines: PWM supply with switching frequency 1.3 kHz, saturation as for 230 V/50 Hz).


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TABLE I M EASURED AND C ALCULATED R ESULTS OF I RON L OSSES U NDER PWM S UPPLY C ONDITIONS . F REQUENCY OF F UNDAMENTAL H ARMONIC —50 H Z . ROTOR WAS L OCKED U NDER N O -L OAD T ESTS

Fig. 16. Measured (points) and calculated (continuous line) iron losses versus frequency for sinusoidal supply (1,2,3,4—description as in Fig. 7).

Fig. 17. (Continuous line) Calculated and (points) measured iron losses versus line voltage for the PWM supply. The switching frequency 1300 Hz, constant modulation index.

In teeth near the air gap, the average iron loss density is smaller than at the end of the back iron. In the rotor, a different trend was observed. In regions close to the air gap, the average iron loss density reached significant values. As a consequence, complex rotor slot shapes lead to large inequality of the average iron loss density, as shown in Fig. 15. The comparison between the calculated and measured iron losses, under sinusoidal supply and different magnetic saturations, is shown in Fig. 16. The comparison under the PWM supply is shown in Fig. 17, where ±10% error bars with respect to measured values are marked. The good agreement can be

seen, both for sinusoidal and PWM supply at constant and variable modulation index as reported in Table I. The symbols used in Table I are as follows: CMI constant modulation index; VMI variable modulation index; line-to-line voltage; Vlin Pmeas measured iron losses; Pstat calculated stator iron losses; calculated rotor iron losses; Prot Pcalc sum of calculated iron losses. It is an authors’ opinion that for PWM inverter fed induction motors, an iron loss prediction error lower than 10% confirms the goodness of the proposed procedure when compared with the results obtained using other methodologies reported in literature [16]–[21]. VI. C ONCLUSION In this paper, an improved time-domain iron loss estimation method has been presented. The proposed approach is based on the knowledge of the flux density waveforms, both in time and space. In order to validate the predicted results, an induction motor prototype with plastic squirrel cage has been built. The experimental campaign has highlighted the difficulty in obtaining accurate and reliable iron loss data from the synchronous no-load tests. Thanks to the presence of a nonconductive rotor



cage, the experimental and computation result comparison was possible through locked rotor tests. The problems concerning the static hysteresis loss estimation process have been highlighted. Taking these into account, the static Jiles–Atherton model was used to estimate the static hysteresis losses. Simultaneously, it enabled the calculation of estimation errors caused by considering just the initial phase. For the analyzed induction motor, the total computed iron loss differs about 10% with respect to the measured iron losses. When estimation algorithm left the initial phases, this difference increased up to 14% (for 230 V/50 Hz). For lower saturation of the magnetic circuit (e.g., 160 V/50 Hz), the total iron loss results (these including and leaving initial phases) were comparable. Results show the importance of considering the harmonics initial phase. The initial phases are often omitted in other approaches, with a resultant reduction in accuracy of iron loss prediction. The comparison was made both in sinusoidal and PWM supply. The agreement between the measured and computed iron losses was excellent, with a percentage error lower than 10%, confirming the value of the proposed approach in the presence of highly distorted voltage conditions such as in the case of PWM supply. R EFERENCES [1] J. R. Hendershot and T. J. E. Miller, Design of Brushless Permanent Magnet Motors. Mentor, OH: Magna Physics, 1994. [2] G. Bertotti, “General properties of power losses in soft ferromagnetic materials,” IEEE Trans. Magn., vol. 24, no. 1, pp. 621–630, Jan. 1988. [3] A. Boglietti, A. Cavagnino, M. Lazarri, and M. Pastorelli, “Predicting iron losses in soft magnetic materials with arbitrary voltage supply,” IEEE Trans. Magn., vol. 39, no. 2, pp. 981–989, Mar. 2003. [4] F. Fiorillo and A. Novikov, “An improved approach to power losses in magnetic laminations under nonsinusoidal induction waveform,” IEEE Trans. Magn., vol. 26, no. 5, pp. 2904–2910, Sep. 1990. [5] I. D. Mayergoyz, Mathematical Models of Hysteresis. New York: Springer-Verlag, 1991. [6] D. C. Jiles and D. L. Atherton, “Theory of ferromagnetic hysteresis,” J. Magn. Magn. Mater., vol. 61, no. 1/2, pp. 48–60, Sep. 1986. [7] L. R. Dupre, O. Bottauscio, M. Chiampi, M. Repetto, and J. A. A. Melkebeek, “Modeling of electromagnetic phenomena in soft magnetic materials under unidirectional time periodic flux excitations,” IEEE Trans. Magn., vol. 35, no. 5, pp. 4171–4184, Sep. 1999. [8] Z. Gmyrek, A. Boglietti, and A. Cavagnino, “Iron loss prediction with PWM supply using low- and high-frequency measurements: Analysis and results comparison,” IEEE Trans. Ind. Electron., vol. 55, no. 4, pp. 1722– 1728, Apr. 2008. [9] H. Domeki, Y. Ishihara, C. Kaido, Y. Kawase, S. Kitamura, T. Shimomura, N. Takahasi, T. Yamada, and K. Yamazaki, “Investigation of benchmark model for estimating iron loss in rotating machine,” IEEE Trans. Magn., vol. 40, no. 2, pp. 794–797, Mar. 2004. [10] C. A. Hernandez-Aramburo, T. C. Green, and A. C. Smith, “Estimating rotational iron losses in an induction machine,” IEEE Trans. Magn., vol. 39, no. 6, pp. 3527–3533, Nov. 2003. [11] W. Salz and K.-A. Hempel, “Power loss in elliptical steel under elliptically rotating flux conditions,” IEEE Trans. Magn., vol. 32, no. 2, pp. 567–571, Mar. 1996. [12] Z. Gmyrek and J. Anuszczyk, “The rotational power loss calculation in the square sample,” in Computer Engineering in Applied Electromagnetism. New York: Springer-Verlag, 2005, pp. 245–250. [13] F. Fiorillo, C. Ragusa, and C. Appino, “Magnetic losses under twodimensional flux loci in Fe–Si laminations,” J. Magn. Magn. Mater., vol. 316, no. 2, pp. 454–457, Sep. 2007. [14] D. C. Jiles, “A self consistent generalized model for the calculation of minor loop excursions in the theory of hysteresis,” IEEE Trans. Magn., vol. 28, no. 5, pp. 2602–2604, Sep. 1992.

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[15] A. Boglietti, A. Cavagnino, M. Lazzari, and M. Pastorelli, “A critical approach to the iron losses in induction motors,” in Energy Efficiency in Motor Driven System. New York: Springer-Verlag, 2003, pp. 71–77. [16] A. Boglietti, O. Bottauscio, M. Chiampi, M. Pastorelli, and M. Repetto, “Computational and measurement of iron losses under PWM supply conditions,” IEEE Trans. Magn., vol. 32, no. 5, pp. 4302–4304, Sep. 1996. [17] A. Boglietti, P. Ferraris, M. Lazzari, and M. Pastorelli, “Influence of the inverter characteristics on the iron losses in PWM inverter fed induction motors,” IEEE Trans. Ind. Appl., vol. 32, no. 5, pp. 1190–1194, Sep. 1996. [18] A. Boglietti, A. Cavagnino, and A. M. Knight, “Isolating the impact of PWM modulation on motor iron losses,” in Conf. Rec. IEEE IAS Annu. Meeting, Edmonton, AB, Canada, Oct. 5–9, 2008, pp. 1–7. [19] D. M. Ionel, M. Popescu, S. J. Dellinger, T. J. Miller, R. J. Heideman, and M. I. McGilp, “On the variation with flux and frequency of the core loss coefficients in electrical machines,” IEEE Trans. Ind. Appl., vol. 42, no. 3, pp. 658–667, May/Jun. 2006. [20] T. L. Mthombeni and P. Pillay, “Core losses in motor laminations exposed to high-frequency or non-sinusoidal excitation,” IEEE Trans. Ind. Appl., vol. 40, no. 5, pp. 1325–1332, Sep./Oct. 2004. [21] C. Cester, A. Kedous-Lebouc, and B. Cornut, “Iron loss under practical working conditions of a PWM powered induction motor,” IEEE Trans. Magn., vol. 33, no. 5, pp. 3766–3768, Sep. 1997.

Zbigniew Gmyrek was born in Bialogard, Poland, in 1961. He received the M.Sc. degree in electrical engineering and the Ph.D. and D.Sc. degrees from the Technical University of Lodz, Lodz, Poland, in 1987, 1995, and 2006, respectively. He joined the Department of Electrical Machines and Transformers, Technical University of Lodz, in 1987 and the Institute of Mechatronics and Information Systems in 2003, where he is currently a Researcher. He is the author of more than 83 papers and conference proceedings. His research interests include the energetic behavior of electrical machines, magnetic materials in the electrical machines, methods of the iron loss calculation, particularly under deformed flux conditions, electromagnetic compatibility, and numerical models of ferromagnetics as well as electrical machines.

Aldo Boglietti (M’04–SM’06) was born in Rome, Italy, in 1957. He received the Laurea degree in electrical engineering from Politecnico di Torino, Turin, Italy, in 1981. Since 1984, he has been with the Department of Electrical Engineering, Politecnico di Torino, first as a Researcher in electrical machines; then, he became an Associate Professor of electrical machines in 1992. He has been a Full Professor since November 2000 and is currently the Head of the Department of Electrical Engineering. He is the author of about 100 papers. His research interests include energetic problems in electrical machines and drives, high-efficiency industrial motors, magnetic materials, and their applications in electrical machines, electrical machine and drive models, and thermal problems in electrical machines. Prof. Bogliettei is an Associate Editor for the IEEE T RANSACTIONS ON I NDUSTRIAL E LECTRONICS, the Secretary of the Electric Machines Committee of the IEEE Industry Applications Society and the Chair of the Electrical Machine Committee of the IEEE Industrial Electronics Society.

Andrea Cavagnino (M’04) was born in Asti, Italy, in 1970. He received the M.Sc. and Ph.D. degrees in electrical engineering from Politecnico di Torino, Turin, Italy, in 1995 and 1999, respectively. Since 1997, he has been with the Electrical Machines Laboratory, Department of Electrical Engineering, Politecnico di Torino, where he is currently an Assistant Professor. His research interests include electromagnetic design, thermal design, and energetic behaviors of electric machines. He is the author of more than 70 papers published in technical journals and conference proceedings. Dr. Cavagnino is an Associate Editor for the Electric Machines Committee of the IEEE T RANSACTIONS ON I NDUSTRY A PPLICATIONS.
