Stray Load and Iron Losses in Small Induction Machines ...download.xuebalib.com/3xySJXAB7vu.pdf · represent stray load losses estimated according to (8). Simple and fairly accurate

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TEC-00491-2017.R1 1

Abstract—In this paper, an engineering approach to estimation of the fundamental stray load and iron losses in small induction machines (IMs) is proposed. This approach consists of experimental testing carried on several IMs of different efficiency class and rotor cage material, all rated 1.5 kW, followed by the derivation of simple empirical formulas for estimation of these two types of losses. The iron losses are evaluated as variable with respect to the operating frequency and flux, whereas the stray load losses are evaluated as variable with respect to the operating frequency, flux and torque. Application of the proposed formulas does not require any additional tests other than those imposed by international standards, and, equally important, they do not include any constructional details of the IM. The validity of the formulas is verified by comparison of the estimated and measured losses in wide ranges of the operating frequency, flux, and torque. The proposed methods of estimation offer possibilities for application in the analysis of small IMs under various operating conditions with sinusoidal supply, as well as in developing advanced IM models and vector control algorithms.

Index Terms—Energy efficiency, estimation, induction machines, iron losses, stray load losses.

I. INTRODUCTION

TRAY load and iron losses of an induction machine (IM) are often omitted from the equivalent circuit analysis,

mostly because they are assumed to be negligible. It is habitual to consider the iron losses taking about 1% – 4% of the IM rated power and the stray load losses (SLLs) taking about 0.5% – 3% of the IM rated power at full load [1]. While it may be argued that these percentages are already far from negligible, it can also be suspected that they are yet higher in small IMs (rated up to 15 kW), which are known to have reduced efficiency compared to large IMs. In addition to reducing the efficiency, losses generally increase the temperature within the IM and reduce its loading capacity. Hence, it is of critical importance to accurately determine and, if possible, reduce the IM losses.

In [2]–[4], it was reported that both the iron and stray load

Manuscript submitted June 29, 2017. This work has been fully supported by Croatian Science Foundation under

the project (IP-2016-06-3319). M. Bašić, D. Vukadinović, and M. Polić are with the Faculty of Electrical

Engineering, Mechanical Engineering and Naval Architecture, University of Split, 21000 Split, Croatia (e-mail: [email protected]).

losses, when ignored, can have a significant impact on the IM vector control in terms of the induced disagreement between the reference and the actual flux vector. This disagreement is induced by the fundamental loss component since any vector controller operates as the fundamental-harmonic controller. While vector control is not a subject of this paper, this still presents a valuable piece of information about the influence of these losses and adds to the importance of their assessment.

According to the widely used international standards IEEE 112-B [5] and IEC 60034-2-1 [6], the iron losses are determined by subtracting the mechanical and stator copper losses from the measured input power at no load. However, these standards do not deal with the iron loss variation with the operating frequency. Furthermore, the rotor component of the iron losses is commonly neglected due to low rated slip of regular IMs. Other approaches to iron loss assessment include measurements on magnetic samples [7]–[10], thermometric methods [11] and finite element analysis (FEA) [8], [9], [12], but they all involve extensive experimental testing and/or intensive computations, or lack the generality.

According to the same standards, the SLLs are indirectly determined from experimental testing by subtracting the sum of the mechanical losses, iron losses, and stator and rotor copper losses from the total measured losses, at load torques up to 50 % above the rated. The IEEE and IEC standards consider the SLLs to be dependent solely on the load torque, which is not quite true. Namely, in [13] and [14], the authors have demonstrated that the SLLs’ vary with both the amplitude and frequency of the supply voltage. Still, they have missed to quantitatively relate the SLLs with these variables by means of some sort of empirical formula. In [15] and [16], theoretical expressions are given for the calculation of the SLLs, but these expressions are rather complex, require knowledge of a number of constructional details of the IM, and are more intended to provide an insight into the possibilities of reducing the SLLs already at the IM design stage. A simple empirical formula proposed in [17] assigns a fixed percentage of the IM input power to the SLLs – a correction of the similar formulas given in [5] and [6] – but it neglects the dependency of the SLLs on the IM operating conditions, as well as the variability of the SLLs within IMs of the same ratings. Other attempts to assess the SLLs include those based on FEA, as in [18], which are extremely time consuming, especially in the preparation phase.

Stray Load and Iron Losses in Small Induction Machines under Variable Operating Frequency

and Flux: A Simple Estimation Method

Mateo Bašić, Member, IEEE, Dinko Vukadinović, Member, IEEE, Miljenko Polić

S



TEC-00491-2017.R1 2

What the previous studies have ultimately failed to provide, in our opinion, are simple, yet accurate, expressions for the iron loss and SLL estimation, applicable for various IMs and for various operating conditions. Variations of the operating frequency and flux are in this view particularly interesting since they are often encountered in IM control systems (e.g., variable-speed drives, wind-based power generating systems, flux-weakening operation, loss minimization strategies, etc.), but are not covered by the standards. Hence, the main goals of this study are to fill some of the remaining methodological gaps and to provide the sought expressions.

II. THEORETICAL BACKGROUND

The iron losses in IMs are commonly divided into two principal contributors, namely the hysteresis losses, Ph, and the eddy-current losses, Pec. In some cases, the excess (or anomalous) loss component is also separated from the classical eddy-current loss component in order to achieve a better agreement between the measured and predicted results, as in the Bertotti’s model [19]. However, such separation has not been made in this paper in order to preserve the simplicity.

Both Ph and Pec increase with the frequency and the magnetic flux density of the applied magnetic field. For sinusoidal supply, this dependency can be modeled as [20]

xphh BfkP (1)

22peec BfkP (2)

where kh and ke are the coefficients related to hysteresis and eddy-current losses, respectively, f is the frequency of the magnetic field, Bp is the peak value of the flux density, and x is the Steinmetz coefficient.

The coefficients kh and ke are commonly considered constant, though there are studies which point to their variation with both the frequency and flux density of the applied magnetic field [8]–[10], [20]. As a trade-off between accuracy and simplicity, the dependency of the coefficients on the flux density is here taken into account, whereas the dependency on the frequency is neglected. This is justified for sinusoidal magnetic fields of frequencies up to 400 Hz and flux densities up to 2 T [8]–[10]. The coefficient x has also been shown to vary with the flux density and frequency [8], [9]. However, given the encompassed flux densities and frequencies, here it is assumed x = 2. In an IM, Bp value is proportional to the magnitude of the stator flux-linkage vector, Ψs (in the following: stator flux). So, summed up, the IM’s iron losses under sinusoidal supply can be expressed as a function of the stator flux and stator frequency, fs, as

222ssesshFe fKfKP (3)

The rotor iron losses are neglected in this study due to the

reasons previously discussed. In the following, the coefficients Kh and Ke from (3) are referred to as the hysteresis loss coefficient and the eddy-current loss coefficient, respectively.

Ultimately, by assuming hysteresis losses dominance in the frequency range fs ≤ 50 Hz [7], [12], it follows:

2sshFe fKP (4)

The SLLs in the IM can be viewed as the sum total of many

different loss components which are not encompassed by the conventional losses and whose theoretical background is yet not entirely understood. They occur both in the stator and rotor of the IM simply due to non-ideal construction of a real machine. Some of the factors affecting the SLLs are the number and shape of slots, distribution of windings, rotor bar skewing, magnetic saturation, etc. Magnetic flux pulsations in the air gap of the IM due to stator and rotor slotting and their interaction could be singled out as a key source of the SLLs. An overview of the stray loss components and their origins is thoroughly discussed in [15] and [16]. Moreover, complex theoretical expressions are given for calculation of the particular SLL components, but they require knowledge of a number of constructional details of the IM. So, the expression which describes the SLLs with the shared simplicity and generality of (3) or (4) is lacking in the literature.

As previously discussed, the IEEE and IEC standards provide somewhat simplified interpretation of the SLLs by regarding them as solely dependent on the load torque [5], [6]. However, it was clearly shown in [13] and [14] that, apart from load torque, the SLLs increase with increasing frequency and decreasing flux (i.e., in the flux-weakening region).

III. MEASUREMENT AND ESTIMATION PROCEDURES

A. Measurement of Iron Losses

The iron losses were determined by the classical no-load test in accordance with the standards IEEE 112-B and IEC 60034-2-1. In order to determine the dependency PFe (fs), the tests were extended to five supply frequencies between 10 Hz and 50 Hz. The open-loop U/f control strategy was utilized, which allowed the same range of the stator flux to be covered at each frequency. Variable sinusoidal supply was provided by means of a three-phase synchronous generator, driven by a controlled dc motor. The fundamental phase voltages, currents and real power were measured with high accuracy by means of the power analyzer Fluke Norma 4000.

A total of four squirrel-cage IMs (4-pole, star-connected, rated 1.5 kW) were included in the testing. The first three IMs (in the following: IM-1, IM-2, and IM-3) are die-cast aluminum rotor IMs of the efficiency classes IE1, IE2, and IE3, respectively, according to IEC 60034-30-1 [21]. The fourth is a die-cast copper rotor IM (in the following: IM-4), classified as IE3-efficiency IM according to IEC 60034-30-1. Such selection of IMs allows establishing a correlation between the iron losses and both the efficiency and rotor cage material of the IM, as well as to check how well the proposed methods of iron loss estimation cope with such diverse IMs.

Fig. 1 presents the iron losses determined for the IM-1. This machine was chosen as a model example here because it has the most pronounced iron losses of all four analyzed IMs.



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Fig. 1. Measured no-load iron losses vs. stator flux for different supply frequencies (IM-1). Circles denote the measured values.

The stator flux was calculated from the stator angular

frequency and stator induced EMF, which was in turn calculated from the measured stator phase voltage and current, with previously determined stator winding resistance. Note that the stator flux range in Fig. 1 corresponds to the range 0.45 Ψsr – 1.1 Ψsr, where Ψsr = 0.89 Wb is the rated stator flux of the considered IM. This pretty much corresponds to the expected operating range; higher values of the stator flux imply significant magnetic saturation of the IM, leading to distorted flux waveforms and increased losses, whereas lower values of the stator flux imply significantly reduced loading capacity of the IM, leading to increased danger of short circuit. Note also that in the field-weakening region, Ψs = 0.45 Ψsr corresponds to more than double rated speed.

B. Estimation of Iron Losses – Method A

It is well known that the iron losses of an IM are most commonly modeled by a constant resistor, which is in fact tantamount to neglecting the hysteresis loss component in (3). However, it was shown in [7] and [12] that it is this component which is dominant in the range fs ≤ 50 Hz. The first proposed method of iron loss estimation – Method A – relies upon this observation. The hysteresis loss coefficient Kh is calculated from (4) for each value of PFe and Ψs obtained from the standard no-load test at 50 Hz (Fig. 1). Thus obtained values of Kh are then plotted against Ψs and interpolated by a polynomial in order to find the dependency Kh (Ψs).

Fig. 2 shows the example of the 4th order polynomial fit for the IM-1. While it is evident that Kh varies with the stator flux, it can be argued that for the expected operating range of Ψs (Fig. 1 and shaded area in Fig. 2) Kh could be approximated by a constant value (e.g., the value corresponding to Ψsr) without introducing significant error. This holds even more true for other tested IMs.

Since Kh is in this study considered frequency invariant, the iron losses for any other supply frequency fs < 50 Hz can then be evaluated as

2)( ssshFe fKP (5)

Fig. 2. Hysteresis loss coefficient vs. stator flux (IM-1). Dashed line represents the fourth order polynomial, whereas circles denote values obtained by measurement.

The estimation of the iron losses according to the Method A

is thus a matter of simple linear scaling with respect to the frequency. Complete results obtained by this method for all the analyzed IMs are presented in Section IV.

C. Estimation of Iron Losses – Method B

The second proposed method of iron loss estimation – Method B – allows separation of the hysteresis and eddy-current loss components and relies on the assumption of constant ratio α = Kh /Ke. A similar approach was adopted in [20], but by implying constant values of both Kh and Ke.

By dividing (3) with Ke and substituting Kh /Ke with α, after some mathematical manipulations, the following expressions for Kh and Ke are obtained:

sss

Feh

ff

PK

2,

sss

Fee

ff

PK

12

(6)

For any supposed value of α, Kh and Ke can be calculated

from (6), with PFe and Ψs obtained from the standard no-load test at 50 Hz. Then by including the calculated values of Kh and Ke in (3), PFe can be calculated for any other frequency. To determine how well the calculated iron losses agree with the measured iron losses at this other frequency, the comparison of the two is necessary. This implies performing the no-load test at a supply frequency different from 50 Hz, which is the inherent drawback of the Method B. Nevertheless, if the test is performed, obtaining the value of α is then a matter of simple fitting. From experience, a satisfactory value of α is reached by trial-and-error in a matter of a few tries. In the example shown in Fig. 3 for the IM-1, α is determined to be 150 by following the described procedure, with the no-load tests performed at 50 Hz and 30 Hz. Note that the described method is not oversensitive to the value of α since the error introduced for α = 50 or α = 600 is only marginal.

Subsequently, the iron losses can be separated into hysteresis and eddy-current losses. Fig. 4 shows the corresponding characteristics for fs = 50 Hz and fs = 10 Hz. Presented results confirm the hypothesis about the dominance of hysteresis losses, especially at lower frequencies. Similar conclusions were reached for other analyzed IMs.



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Fig. 3. Determination of α by fitting procedure according to the Method B (IM-1). Circles denote measured values, whereas dashed lines represent characteristics obtained by scaling the 50 Hz characteristic (solid line) with different values of α.

Fig. 4. Separation of iron losses into hysteresis and eddy-current losses at fs = 50 Hz and fs = 10 Hz according to Method B (IM-1).

Finally, regarding the proposed Method B, the question of

the need for estimation as such may be raised since the method anyway imposes measurements at different supply frequencies. Namely, it may be argued that if the iron losses can be measured at two different frequencies, then they could be measured at all the frequencies of interest – of course, provided that the measuring equipment is adequate. Still, if the proposed method is viewed in the context of its potential application in developing advanced IM mathematical models and/or corresponding control algorithms, it is desirable to have a simple and accurate expression for the iron loss estimation in the entire operating range, thus avoiding the use of look-up tables and storing data in the processor memory. Needless to say, in this way extensive experimental testing is also avoided.

D. Measurement and Estimation of Stray Load Losses

In order to determine the SLLs, the standards IEEE 112-B and IEC 60034-2-1 require performing three separate tests, namely thermal test at the rated load, no-load test, and variable-load test at rated thermal conditions [5], [6]. These tests are performed using a sinusoidal supply of rated voltage and frequency values. Both standards, in fact, impose very similar methods for the SLL determination. The main differences are that the IEC standard accounts for the iron loss dependency on load and imposes slightly higher correlation coefficient (0.95 instead of 0.9), whereas the IEEE standard imposes final correction of the measured mechanical power.

In any case, the measurement of the SLLs is considered highly susceptible to errors, not only because of the difficulties related to accurate torque measurement, but also due to the complex nature and diverse causes of the SLLs. The level of accuracy with which the SLLs are determined also depends on the level of accuracy with which all the other losses are determined. The task gets more difficult if the SLLs are to be determined for various values of frequency and flux. The iron losses, in particular, are known to vary with frequency and flux, so this variation should be accurately evaluated, by means of measurement or estimation, prior to the determination of the SLLs.

In this study, the SLLs’ dependency on the load torque was determined with minor modifications of the methods imposed by the standards [5], [6]. Namely, the iron loss dependency on load has been taken into account, as required by the IEC standard, and the final correction of the measured mechanical power has also been done, as required by the IEEE standard. These modifications are aimed to achieving better overall accuracy of the SLL determination and are recommended by the authors. With the same aim in mind, measurements were repeated three times and then averaged, while making sure that the correlation coefficient of 0.95 is satisfied for each set of measurements.

Following the described procedure, the standard tests were extended to three supply frequencies, namely 30 Hz, 40 Hz, and 50 Hz, in order to determine the SLLs’ variation with fs. This was done by utilizing the open-loop U/f control strategy. As in the iron loss measurements, variable sinusoidal supply was provided by means of a three-phase synchronous generator, whereas high accuracy of electrical measurements was again ensured by utilizing the power analyzer Fluke Norma 4000. During the tests, special care was taken not to exceed the maximum allowed RMS value of the stator phase current, set to 50 % above the rated. For some of the tested IMs, this resulted in lower load torques achievable at frequencies lower than the rated. The tested IMs were loaded by means of a dc generator, with the load torque measured by means of the torque transducer TMB 308 (Magtrol) mounted on the motor shaft. In addition, in order to accurately determine the SLLs, the iron losses were previously evaluated according to the Method B, which is assumed to be more accurate than the Method A.

Fig. 5 shows the SLLs versus the load torque squared, TL2,

for the IM-1 of rated torque 10.3 Nm. The measured SLLs were corrected according to the standards to obtain zero SLLs at zero load torque. Once more, the IM-1 was chosen as a model example because it has the most pronounced SLLs of all the tested IMs.

Due to the constant U/f ratio, the stator flux remained close to the rated value for all measured points. Hence, the changes in the slope of the characteristics in Fig. 5 reflect the SLLs’ dependency on the supply frequency. Since the slopes in Fig. 5 decrease almost linearly with the frequency, the implication is that the SLLs also decrease almost linearly with the frequency. Note that similar conclusions were reached for other tested IMs, as well as by other authors in [14].



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Fig. 5. Corrected measured stray load losses vs. torque squared for different supply frequencies and Ψs ≈ Ψsr (IM-1). Solid lines denote characteristics obtained by linear fitting for different supply frequencies.

From the above considerations it follows that the SLLs for

the rated flux can be, in a simpler variant, approximated as

sr

s

r

LSLLSLL

f

f

T

TKP

2

(7)

where KSLL is the slope of the 50 Hz characteristic from the variable-load test – also representing the SLL value at rated operating conditions – whereas Tr and fsr are the rated torque and rated stator frequency of the IM, respectively.

Expression (7) allows quick estimation of the SLLs as a function of the torque and frequency without the need for any additional tests other than those imposed by the standards. More accurate scaling with respect to fs can be achieved based on the tests performed at different frequencies (Fig. 5), but this involves an elaborate test rig. It also cancels the supposed generality of (7) since in this case the dependency PSLL (fs) would have to be individually determined for each IM.

Still, (7) does not account for the SLLs’ variation with the stator flux. In order to determine this variation, the IM was supplied with the voltage of variable amplitude and constant frequency (fs = 50 Hz) while loaded with constant torque, thus achieving different values of the stator flux. Such variations of the stator voltage are easily achieved, e.g., by supplying the IM from the grid, through a variable transformer.

The range of the stator flux values encompassed by the variable-flux test should be as wide as possible, but there are some practical limitations to it. Namely, for the load torque values around the rated, the achievable flux range is rather narrow since even a small reduction in magnetization leads to stopping of the IM. On the other hand, the relative measurement error increases with reduction of the load torque and, after a certain point, dispersion of the measured values becomes too large to properly identify the flux dependency. Therefore, we recommend performing the variable-flux test with the load torque of about half the rated value.

Fig. 6 shows the SLLs versus the stator flux, determined by the variable-flux test for four different load torques (IM-1). It can be seen that the minimum stator flux achieved for the load torque of 8 Nm is about 65 % of the rated value, which cannot be considered sufficient.

Fig. 6. Corrected measured stray load losses vs. stator flux for different load torques and fs = fsr (IM-1). Solid line represents a polynomial fit of the SLLs measured at TL = 6 Nm, whereas dashed-line characteristics are obtained by scaling the solid-line characteristic.

On the other hand, by reducing the load torque, proper

identification of the SLLs’ variation with the stator flux becomes increasingly difficult. Hence, the variation of PSLL with Ψs was determined by a polynomial fit for the load torque of 6 Nm (solid-line curve), which is about half the rated torque. For correction, the curve was shifted along the y-axis, together with the corresponding measurement points, in order to make sure that the SLL value at the rated stator flux corresponds to that obtained during the variable-load test for the same load torque (Fig. 5).

Dashed-line characteristics in Fig. 6 were actually obtained by scaling the solid-line characteristic with respect to the square of the torque ratio, with the load torque of 6 Nm used as the base value. The obtained results once more confirm the SLLs’ linear dependency on the torque squared.

It is evident from Fig. 6 that the SLLs increase with decreasing stator flux, as in [13] and [14]. There, a significant increase of the SLLs has been found in the flux weakening region, but the dependency PSLL (Ψs) was not clearly isolated. However odd such relation between PSLL and Ψs may seem, one has to be aware that during the variable-flux test, the IM slip increases with decreasing stator flux. Thus, the obtained results indirectly relate the SLLs with the IM slip as well.

In this paper, the impact of the stator flux (and indirectly the slip) on the SLLs is expressed by expanding (7) as follows:

s

sr

sr

s

r

LSLLSLL

f

f

T

TKP

2

(8)

It is important to point out that the calculation of the SLLs

according to (8) does not require any additional tests other than those imposed by the standards [5], [6]. Again, yet more accurate scaling can be achieved by expanding the standard tests to other frequencies and by including the variable-flux test, but only at the expense of an elaborate testing procedure and loss of generality.

The accuracy of the proposed SLL estimation was checked through comparison of the results obtained from (8) with those obtained by measurement for the same four IMs as encompassed in the iron loss analysis. Complete results are presented in the following section.



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IV. RESULTS AND DISCUSSION

A. Measured and Estimated Iron Losses

The results from measurement and estimation of the iron losses are shown in Fig. 7, plotted against the stator flux in the range 0.5Ψsr – 1.1Ψsr. The measurements have been made at five different frequencies in the range 10 Hz – 50 Hz. The values obtained by measurement are compared to the values estimated by the two proposed methods for IM-1 – IM-4.

The greatest iron losses are expectedly obtained for the IM-1; somewhat lower values are obtained for the IM-2 and the IM-3; the lowest iron losses are obtained for the IM-4. For example, at the rated stator flux and frequency, they amount to 111 W, 60 W, 54 W, and 37 W, respectively, which points to evident differences in the iron core build (note that the above values range from 2.5 % to 7.4 % of the IMs’ rated power).

The estimated iron loss values agree very well with the measurement data, to an extent that the usage of the both proposed methods of estimation is justified. As expected, the values estimated according to the Method B have slightly better agreement with the measurement data in the whole considered range. Moreover, the error introduced by the Method A seems to increase with increasing stator flux and decreasing efficiency of the tested IMs, for all the encompassed frequencies. The only exceptions are the characteristics obtained at 10 Hz for the IM-2 and IM-3, which can rather be assigned to the measurement error considering low absolute values of iron losses.

B. Measured and Estimated Stray Load Losses

The results from measurement and estimation of the SLLs are shown in Fig. 8, plotted against the torque squared in the range 0 – 1.5Tr. The measurements have been made at three different frequencies in the range 30 Hz – 50 Hz. The values obtained by measurement are compared to the values estimated by the proposed method, again for all four IMs.

As it can be seen, the SLLs of the IM-1 have the greatest value; they are somewhat lower for the IM-4; the lowest SLLs are obtained for the IM-2 and the IM-3s. For example, at the rated operating conditions, they amount to 124 W, 67 W, 34 W, and 40 W, respectively (note that these values range from 2.3 % to 8.3 % of the IMs’ rated power). In addition, the SLLs notably decrease with frequency. Interestingly, the SLLs of the IM-4 are nearly 50% greater compared to the IM-2 and IM-3, despite (or because of) the copper cage. Furthermore, the latter two IMs have similar values of the SLLs for all tested frequencies, which might indicate that they share the geometry of both the air gap and the iron core.

It can be seen from Fig. 8 that there is a very good agreement between the estimated and measured SLLs for up to the rated torque, for all the tested IMs; for the IM-4 (Fig. 8d), the close agreement stretches to the whole tested range.

Fig. 9 shows the results from measurement and estimation of the SLLs, plotted against the stator flux in the range 0.5Ψsr – 1.1Ψsr. Note that there is no solid-line characteristic in Fig. 9, unlike in Figs. 7 and 8, since obtaining such

characteristic would require measurements not envisaged by the international standards (Section III-D).

It can be seen that the SLLs increase with decreasing stator flux – this is especially emphasized for the IM-1 – whereas the best agreement between the measured and estimated SLLs is achieved for the IM-4.

(a)

(b)

(c)

(d)

Fig. 7. Measured and estimated iron losses vs. stator flux for different supply frequencies: (a) IM-1, (b) IM-2, (c) IM-3, and (d) IM-4. Solid line represents a polynomial fit of the iron losses measured at fs = 50 Hz, dotted lines denote iron losses estimated by Method A, whereas dashed lines denote iron losses estimated by Method B.



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(a)

(b)

(c)

(d)

Fig. 8. Measured and estimated stray load losses vs. torque squared for different supply frequencies (Ψs ≈ Ψsr): (a) IM-1, (b) IM-2, (c) IM-3, and (d) IM-4. Solid line represents a linear fit of the SLLs measured at fs = 50 Hz, whereas dashed lines represent the SLLs scaled according to (8).

V. CONCLUSION

In this paper, a novel approach to estimation of the IM’s fundamental iron and stray load losses is presented. The experimental testing was carried on various small IMs, from which the following conclusions were drawn: The SLLs were shown to vary with both the stator frequency

and magnetic flux, apart from the load torque. Hence, it is strongly advised to take these factors into consideration in order to accurately assess the SLLs under variable operating conditions.

(a)

(b)

(c)

(d)

Fig. 9. Measured and estimated stray load losses vs. stator flux for different load torques (fs = fsr): (a) IM-1, (b) IM-2, (c) IM-3, and (d) IM-4. Dashed lines represent stray load losses estimated according to (8).

Simple and fairly accurate formulas for evaluation of the

iron and stray load losses were obtained. They include dependencies on the stator frequency and flux, and, in case of the SLLs, the load torque, some of which are not dealt with by the international standards. In this paper, the dependence of the SLLs on the stator flux has been established for the first time.

The fundamental iron and stray load losses were shown to be greater than commonly assumed, especially for the IM-1. Hence, for more accurate IM analysis, it is advisable to take these losses into account. Although the fundamental



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component is relevant only for sinusoidal supply and vector control schemes, it may be argued that it is generally better to take into account only this component than to neglect these losses altogether.

The assumption of the hysteresis loss dominance within the range fs ≤ 50 Hz was proven correct, especially for the IMs of higher efficiency class. Hence, to simplify the analysis, the eddy-current losses of an IM may be neglected in the iron loss model at least up to the rated speed. A constant iron loss resistance thus may not represent the iron losses very well in the IM equivalent circuit.

The proposed formulas for estimation of the iron and stray load losses do not require knowledge of constructional details and contain only variables that can be determined by measurement, either directly or indirectly. This should enable easy implementation of the proposed estimation methods as part of the advanced IM model or control algorithm, which is the subject of future research.

Note that the above conclusions are derived from tests carried on a limited sample of IMs, so it would be incautious to assume they are generally valid. However, their generality was to some extent established by including IMs of different efficiency class and rotor cage material in the test sample.

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[10] M. Popescu, D. M. Ionel, A. Boglietti, A. Cavagnino, C. Cossar, and M. I. McGilp, “A general model for estimating the laminated steel losses under PWM voltage supply,” IEEE Trans. Ind. Appl., vol. 46, no. 4, pp. 1389–1396, July–Aug. 2010.

[11] A. Bousbaine, “A thermometric approach to the determination of iron losses in single phase induction motors,” IEEE Trans. Energy Convers., vol. 14, no. 3, pp. 277–283, Sep. 1999.

[12] M. Ranta, M. Hinkkanen, E. Dlala, A. K. Repo, and J. Luomi, “Inclusion of hysteresis and eddy current losses in dynamic induction machine models,” in Proc. IEMDC, Miami (FL), 2009, pp. 1387–1392.

[13] A. Boglietti, A. Cavagnino, L. Ferraris, and M. Lazzari, “Impact of the supply voltage on the stray-load losses in induction motors,” IEEE Trans. Ind. Appl., vol. 46, no. 4, pp. 1374–1380, July-Aug. 2010.

[14] A. Boglietti, R. Bojoi, A. Cavagnino, and S. Vaschetto, “Influence of the sinusoidal supply frequency on the induction motor stray load losses,” in Proc. IECON, Montreal (QC), 2012, pp. 1847–1851

[15] P. L. Alger, G. Angst, and E. J. Davies, “Stray load losses in polyphase induction machines,” Trans. AIEE, vol. 78, no. 3, pp. 349–355, April 1959.

[16] A. M. Odok, “Stray-load losses and stray torques in induction machines,” Trans. AIEE, vol. 77, no. 3, pp. 43–53, April 1958.

[17] P. Pillay, M. Al-Badri, P. Angers, and C. Desai, “A new stray-load loss formula for small and medium sized induction motors,” IEEE Trans. Energy Convers., vol. 31, no. 3, pp. 1221–1227, Mar. 2016.

[18] K. Yamazaki, A. Suzuki, M. Ohto, and T. Takakura, “Circuit parameters determination involving stray load loss and harmonic torques for high-speed induction motors fed by inverters,” IEEE Trans. Energy Convers., vol. 28, no. 1, pp. 154–163, Mar. 2013.

[19] G. Bertotti, “Physical interpretation of eddy current losses in ferromagnetic materials. I. Theoretical considerations,” J. Appl. Phys., vol. 57, no. 6, pp. 2110–2117, Mar. 1985.

[20] A. Boglietti, A. Cavagnino, D. M. Ionel, M. Popescu, D. A. Staton, and S. Vaschetto, “A general model to predict the iron losses in PWM inverter-fed induction motors,” IEEE Trans. Ind. Appl., vol. 46, no. 5, pp. 1882–1890, Sept.–Oct. 2010.

[21] Rotating Electrical Machines – Part 30-1: Efficiency Classes of Line Operated AC Motors, IEC 60034-30-1, 2014.

Mateo Bašić (M’12) received the B.S. degree and the Ph.D. degree in electrical engineering from the University of Split, Croatia, in 2006 and 2013, respectively. Since 2008, he has been with the University of Split, Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, where he is currently an Assistant Professor. His

research interests include modelling and control of electrical machines, power electronics, and artificial intelligence.

Dinko Vukadinović (M’17) received the B.S. degree from the University of Split, the M.S. degree from the University of Zagreb and the Ph.D. degree from the University of Split, Croatia, in 1997, 2002 and 2005, respectively, all in electrical engineering. In 1998, he joined the University of Split, Faculty of Electrical Engineering,

Mechanical Engineering and Naval Architecture, where he is currently a Full Professor and the Chair of Power Electronics and Control. His research interests include automation in electric power engineering, power electronics, electrical machines, and artificial intelligence.

Miljenko Polić received the B.S. degree in electrical engineering from the University of Split, Croatia, in 2011, where he is currently pursuing the PhD degree. In 2013, he joined the University of Split, Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture as an assistant. His research

interests include induction machine control systems, electrical machines, and power electronics.

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Stray Load and Iron Losses in Small Induction Machines ...download.xuebalib.com/3xySJXAB7vu.pdf · represent stray load losses estimated according to (8). Simple and fairly accurate