1
Evaluation of Iron Loss Models in Electrical Machines Z. Q. Zhu1, Shaoshen Xue1, Jianghua Feng2, Shuying Guo2, Zhichu Chen2, Jun Peng2, W. Q. Chu2 1
Department of Electronic and Electrical Engineering, University of Sheffield, Mappin Street, Sheffield S1 3JD, UK 2 CRRC Zhuzhou Institute Co. Ltd, Shidai Road, Shifeng District, Zhuzhou, Hunan, 412001, China Email:
[email protected]
Abstract--In this paper, more than ten different iron loss models are experimentally evaluated, which covers alternating and rotating fields, influence of temperature, DC bias flux density and distorted flux density due to PWM inverter. Iron loss models considering alternating fields are evaluated by the measured results of a lamination ring specimen. The iron loss model considering rotating field and nonsinusoidal field are evaluated by the measured results of an electrical machine under different conditions. Based on these comprehensive investigations, the iron loss models having the best prediction accuracy for each case are identified. Index terms--Iron loss, Electrical machine.
I. INTRODUCTION Iron loss is one of the major losses in electrical machines. The accurate prediction of the iron loss is essential for the electromagnetic and thermal design of electrical machines. Many different iron loss models have been developed in [1]-[4]. These iron loss models are widely used since they have solid physical basis while very easy to implement [5]-[7]. The iron loss models in [1]-[4] are also evolved to many different forms to adapt conditions where the flux density alternates nonsinusoidally, such as flux density with DC bias [8]-[10], flux density when supplied by pulse width modulation (PWM) inverter [11] [12], etc. Furthermore, the temperature can influence iron loss significantly [13]. Different iron loss models are developed to consider the temperature dependencies of iron loss in [14] [15]. All the iron loss models in [1]-[15] are based on alternating flux density. However, in electrical machines, the flux density can be rotational. Different models have been developed in [16]-[20] to consider the influence of rotational field. Although many different iron loss models have been developed, they have not been evaluated and compared with each other, which can provide useful guidelines on the iron loss prediction. In this paper, more than ten different iron loss models are evaluated against the measured results. This paper is organized as follows. In Section II, the iron loss test method in steel lamination ring specimen is demonstrated. Based on the ring specimen test, different iron loss models considering alternating flux density are evaluated in Sections III and IV. In Section III, the temperature is constant. In Section IV, the temperature dependency of iron loss is considered. By selecting the most accurate iron loss model considering alternating flux density, the rotational field influence on the iron loss is further discussed in Section V. Different iron loss models for rotational flux density are evaluated against the testing results of an electrical machine. In Section VI, different iron loss models considering nonsinusoidal flux density such as DC bias or 978-1-5386-3246-8/17/$31.00 © 2017 IEEE
fluctuation caused by PWM inverter are also evaluated against the testing results of an electrical machine. II. IRON LOSS TEST IN STEEL LAMINATION SPECIMEN The ring specimen iron loss test has been widely used to measure the iron loss of steel laminations under alternating flux density [21]. The measurement system is shown in Fig. 1. Table I shows the parameters of the ring specimen. The ring specimen is wounded by the excitation coil and the measuring coil. The excitation coil is supplied by an AC power source which is the California Instrument 4500iL in this paper. The measuring coil having the same number of turns with excitation coil is closely wounded together with the excitation coil and connected to the oscilloscope to measure the induced voltage. Thus, the voltage drop on the excitation coil’s resistance is excluded in the measured induced voltage. The current in the excitation coil is measured by the Tektronix A622 current probe. The iron loss , the field strength and the flux density density can be calculated as: 1
(1) (2) (3)
where is the iron loss density. is the instant induced voltage of the measuring coil. is the instant current in the excitation coil. is the time period of the current and the voltage. and are the mass density and the volume of the ring specimen, respectively. is the number of turns of the is the effective excitation coil and the measuring coil. length of the ring specimen. is the cross sectional area of the ring specimen. In order to investigate the influence of temperature on the iron loss, the ring specimen can be heated by its own iron loss to the designate temperature. A K-type thermal couple is also installed to measure the temperature. Thus, iron loss under different temperatures can be obtained. The test range of the measuring system is summarized in Table II. Fig. 2 shows the measured iron loss in the ring specimen at different frequencies, flux densities and temperatures. It can be seen that the iron loss varies with the frequency and flux density in an identical pattern when the lamination temperature is 40°C and 100°C. This pattern has been investigated and modelled widely in [1]-[4]. However, the iron losses at 40°C and 100°C are different when the flux density and frequency are the same. In other word, the temperature influences the iron loss
2 significantly. The temperature dependency of iron loss has been investigated and modelled in [13]-[15]. The iron loss models will be discussed later in this paper.
curve fitting of the test results. The iron loss model can be expressed as: .
1
(5)
12
where and are the electrical conductivity and the thickness of the lamination. is the time period. It is concluded in [3] that the contributions of classical loss and excess loss cannot be separated. Alternatively, a two-term iron loss model is developed, where the classical loss and excess loss in the three-term model are combined into a global eddy current loss. This model can be expressed as:
Fig. 1. Schematic diagram of ring specimen iron loss measuring. TABLE I PARAMETERS OF RING SPECIMEN Type of silicon steel lamination Thickness of single lamination Outer diameter of ring specimen Inner diameter of ring specimen Effective thickness of ring specimen No. of turns for excitation and measuring coils N
(6) mm mm mm mm
V300-35A 0.35 150 125 14 102
TABLE II TEST RANGE OF RING SPECIMEN IRON LOSS MEASURING SYSTEM Maximum output voltage in RMS value V 150 Maximum output current in RMS value A 30 Frequency range Hz 50-1000 Temperature range °C 40-100 Iron loss density (W/kg)
350
Bm=0.60T, 40°C Bm=0.60T, 100°C Bm=1.00T, 40°C Bm=1.00T, 100°C Bm=1.40T, 40°C Bm=1.40T, 100°C Bm=1.73T, 40°C Bm=1.73T, 100°C
300 250 200 150 100 50 0 0
200
400 600 Frequency (Hz)
800
1000
Fig. 2 Measured iron loss in the ring specimen at different frequency, flux density and temperature.
According to the investigation on the iron loss, the hysteresis loss and eddy current loss coefficients vary with frequency and flux density. Therefore, an iron loss model with variable coefficients is presented in [4]: ,
.
(4)
where is the amplitude of the flux density. and are the hysteresis loss coefficients. is the classical loss is the excess loss coefficient. The coefficients coefficient. can be obtained by the curve fitting of the test results at different flux density and frequency. In [2], the hysteresis loss coefficient is obtained by fitting the hysteresis loss test result under a DC hysteresis test. The classical loss is calculated by the properties of the steel laminations. The excess loss coefficient can be obtained by the
TABLE III COEFFICIENTS OF IRON LOSS MODELS (4), (5) AND (6) Parameters for iron loss model (4) 3.25×10-2
3.6×10-2
6.67×10-5 5.95×10-4 Parameters for iron loss model (5) (mm) (S/m) 3.5×10-4 2.0×106 5.95×10-4 Parameters for iron loss model (6) 3.76×10-2
12 khyst (×10-2 W/kg/T2/Hz)
.
(7)
It can be seen from the iron loss models (4)-(7) that the coefficients are very important for the prediction accuracy. The iron loss coefficients are constants in models (4), (5) and (6) while the coefficients are variables in model (7). Table III lists the coefficients for iron loss models (4), (5) and (6). Fig. 3 shows the variable coefficients of iron loss model (7). Fig. 4 shows the relative prediction errors of the iron loss models. Table IV shows the numerical results. It can be seen that the prediction accuracies of iron loss models (4), (5) and (6) are close to each other while the prediction accuracy of (7) is much better. This is due to the fact that the iron loss models (4), (5) and (6) are based on constant coefficients, which cannot consider the coefficient variation with flux density and frequency. On the other hand, the prediction accuracy keeps good in the whole test range of this paper with the help of variable coefficients.
III. IRON LOSS MODELS UNDER CONSTANT TEMPERATURE It should be noted that none of the iron loss models in [1]-[4] considers the influence of temperature on the iron loss. Iron loss coefficients are obtained based on the measured iron loss, generally at a constant temperature. In this section, all the iron loss coefficients are obtained based on the measured results at 40°C. The iron loss models presented in [1]-[4] are the most widely used models for alternating flux density. In [1], iron loss is calculated by the three-term formula including the hysteresis loss, the classical loss and the excess loss.
,
2
2
8.03×10-5
Low frequency High frequency
10 8 6 4 2 0 0
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Flux density (T) (a)
2
3
keddy (×10-5 W/kg/T2/Hz2)
12 10 8 6 4
Low frequency High frequency
2 0 0
0.2
0.4
0.6
0.8 1 1.2 1.4 Flux density (T)
(b) Fig. 3 Coefficients of iron loss model (7). (a)
1.6
1.8
2
methods for modelling the temperature dependency will be carried out based on iron loss model (7). As presented in [14], in order to take the temperature influence into account, one method is to consider the temperature dependency of eddy current loss while assuming hysteresis loss is constant. The eddy current loss coefficient is related to the electrical resistivity. The relationship between resistivity and temperature can be expressed as: 1
(b)
.
(8)
where is the resistivity at temperature . is the resistivity at the base temperature . is the temperature coefficient provided by manufacturers. The iron loss model can be expressed as: (9)
1
In [15], an iron loss model is developed to take temperature dependency of both hysteresis and eddy current losses into account. Temperature dependent coefficients are introduced, the iron loss model considering temperature influence can be expressed as:
(a)
,
(b) Fig. 4 Prediction relative errors of iron loss models at 40°C (a) Model (4) and Model (5). (b) Model (6) and Model (7). TABLE IV AVERAGE PREDICTION RELATIVE ERROR OF IRON LOSS MODELS AT 40°C
Iron loss model Relative error (%)
(4) 11.3
(5) 10.3
(6) 10.7
(7) 2.4
IV. IRON LOSS MODELS CONSIDERING TEMPERATURE INFLUENCE According to the comparison in Section III, iron loss model (7) is the most accurate iron loss model with the help of variable coefficients. However, it should be noted that the iron loss model (7) cannot consider the influence of temperature on iron loss while the temperature affects iron loss significantly. One engineering way to solve this problem is to measure the coefficients of the iron loss model at different temperatures. The iron loss at different temperatures can be then predicted by using these coefficients obtained at different temperatures. However, the coefficients have to be measured at many different temperatures, which is time consuming. Therefore, different methods for modelling the temperature dependency of iron loss have been developed in [14] and [15]. In these iron loss models, temperature dependent coefficients are introduced to the existing iron loss models in order to reflect the temperature influence on iron loss. As illustrated in Section III, the iron loss model (7) has the best accuracy when the temperature is constant. Therefore, in this section, different
,
,
,
(10)
Fig. 5 shows the prediction relative error of different iron loss models with temperature variation. Table V shows the average prediction relative errors at 100°C. It can be seen that the relative prediction errors of model (7) increase significantly when the temperature increases. This is due to the coefficients of model (7) are obtained at 40ºC. When the temperature rises, the prediction value is fixed, while the actual iron loss changes. The accuracy of model (9) has shown a slight improvement over that of model (7), with the help of the temperature dependent resistivity. However, the prediction errors of model (9) still vary with the temperature significantly. Iron loss is overestimated or underestimated dramatically when the temperature is high. This is caused by the temperature dependency of hysteresis loss, which is not taken into account in model (9). On the other hand, iron loss model (10) can predict the iron loss with a very low and stable relative prediction error when the temperature changes significantly.
Fig. 5 Prediction relative error of different iron loss models with temperature variation. TABLE V AVERAGE PREDICTION RELATIVE ERROR OF IRON LOSS MODELS AT 100°C
Iron loss model Relative error (%)
(7) 7.5
(9) 5.2
(10) 2.8
4 V. INFLUENCE OF ROTATIONAL FIELD ON THE IRON LOSS A. Methods to consider rotational field In electrical machines, the flux density in stator or rotor cores can be not only alternating but also rotational. More importantly, it is widely reported that the rotational flux density influences the iron loss significantly [16]-[18] [20]. In [16], it is concluded that the rotational flux density can be decomposed into two alternating directions, i.e. the x-axis and the y-axis as shown in Fig. 6. The total iron loss under rotational flux density can be then obtained by the sum of iron losses at these two directions. The iron loss can be expressed as: ,
,
,
(11)
where is the total iron loss under rotational flux , density. and , , are the iron loss calculated by the alternating flux density on x-axis and y-axis, respectively. In [17] and [19], the rotational flux density is decomposed into the major-axis and minor-axis as shown in Fig. 6 instead of the x-axis and the y-axis. The iron loss can be then expressed as: ,
,
,
(12)
where and are the iron losses calculated by , , the alternating flux density along the major-axis and the minor-axis, respectively.
temperatures of different parts of the machine become stable after 120 minutes heating since the thermal transfer process is completed and the thermal balance is achieved. Also, the temperatures of different parts of the machine are closed to each other and the average temperature is 101°C. In this case, the temperature of the electrical machine can be approximately considered as 101°C. The iron losses are then measured by applying the pre-tuned input current. Since the measuring process will only take a few seconds, the temperature variation during the measurement can be neglected. According to the steel lamination test result in Section IV, the iron loss model (10) has the best accuracy with the help of variable coefficients and the consideration of temperature influence. Therefore, in this section, different methods for consideration of rotational flux density are applied to model (10). The different models are then evaluated by the comparison between the predicted and measured results. Fig. 10 shows the measured and predicted iron losses at different temperatures. On one hand, the iron losses at different temperatures vary significantly and the iron loss model can track this variation. On the other hand, the iron loss model (12) considering the rotational flux density on major-minor axis has a better accuracy.
Fig. 7. Electrical machine iron loss test system. Fig. 6. Rotational flux density loci in electrical machines.
Fig. 8. Thermal couples in the electrical machine. 140
Temperature (ºC)
B. Evaluation in an electrical machine In order to evaluate different methods for considering rotational flux density, an electrical machine test is carried out in this section. Fig. 7 shows the schematic diagram of the electrical machine test system. A 12-slot/10-pole IPM machine is powered by a three-phase AC power source with sinusoidal current. In order to exclude the influence of eddy current loss in magnets and the mechanical loss, the rotor is locked and there is no magnet in the rotor. The iron loss is then obtained by subtracting the copper loss from the total loss. Furthermore, in order to measure temperature at different parts of the electrical machine, six thermal couples are equipped at the stator yoke, the coil, the stator tooth, the tooth tip, the rotor magnet slot and the rotor yoke as shown in Fig. 8. The evaluation process is as follows. Firstly, the measured iron loss and phase current waveform can be obtained by the electrical machine test. Secondly, the electrical machine is modelled in the finite element analysis (FEA) software with the measured phase current waveforms. The flux density of every finite element can be then recorded and extracted. The iron loss is then calculated from the flux density in each element using the iron loss models. In order to measure the iron loss at different temperatures, the iron loss is measured at the room temperature (24°C) at first. Then the ring specimen is heated to the designate temperature by the iron loss. Fig. 9 shows the temperature variation when the input phase current of the electrical machine is 3.11A, 1000Hz. It can be seen that the
Stator yoke Stator tooth Rotor magnet slot
120 100
Coil Tooth tip Rotor yoke
80 60 40 20 0
20
40
60 Time (min)
80
100
120
Fig. 9. Evaluation process of iron loss models.
Fig. 10. Measured and predicted results using different methods.
5 VI. IRON LOSS MODEL FOR NONSINUSOIDAL FLUX DENSITY The iron loss models discussed in Sections III, IV and V are based on the assumption that the flux density waveform is sinusoidal. However, in actual electrical machines, the flux density can be different from sinusoidal. On one hand, the DC bias flux density exists in many types of electrical machines such as permanent-magnet synchronous machine (PMSM), brushless DC machine (BLDC) and switched reluctance motor (SRM), etc. More importantly, DC bias flux density can significantly influence the iron loss. On the other hand, electrical machines are usually powered by PWM inverters. The current waveform of electrical machines can be significantly distorted from sinusoidal. This can result in significant fluctuation of flux density in electrical machines and then additional iron losses. In this section, different iron loss models for consideration of the flux density with DC bias and for electrical machine fed by PWM inverters are evaluated against the measured iron loss in the electrical machine.
to its negligible eddy current loss. Fig. 12 shows the measured and predicted iron losses in the electrical machine. It can be seen that the iron loss at 19°C and 101°C varies significantly while the predicted results of iron loss model (13) keep constant. On the other hand, the iron loss model (15) can track this variation with the help of temperature dependent coefficients.
Fig.11. DC bias coefficients variation with temperature.
100
A. Iron loss model for flux density with DC bias
(13) 1
(14)
where is the iron loss. is the ratio between and hysteresis losses with and without DC bias flux density. are the hysteresis and eddy current losses coefficients, is the respectively. is the frequency of AC flux density. and are the DC bias amplitude of AC flux density. coefficients and can be obtained by steel lamination tests. is According to the study in [10], the coefficient not only the DC bias flux density but also the temperature dependent. Furthermore, hysteresis and eddy current loss are also flux density, frequency and coefficients and temperature dependent as illustrated in Section IV. Therefore, an iron loss model considering the temperature dependence of flux density, frequency and DC bias flux density is developed in [10] as: ,
,
, ,
, 1
,
,
(15) (16)
Fig. 11 shows measured DC bias coefficients and at and vary almost linearly different temperatures. Both with the temperature for the investigated temperature range. and obtained based on measured iron losses Thus, with at two different temperatures, they can be used to predict the iron loss at the other temperatures. In order to evaluate the iron loss models for flux density with DC bias, tests on the electrical machine are carried out. The electrical machine test is carried out by using the electrical machine and process introduced in section V. In order to provide DC bias flux density in the machine, permanent magnets are installed in the rotor. Ferrite magnets are used due
Iron loss (W)
80
The iron loss model developed in [8] is one of the most widely used iron loss models considering the influence of DC bias flux density as:
Measured-19ºC Measured-101ºC Existing model (13) Improved model (15)-19ºC Improved model (15)-101ºC
60 40 20 0 0
200
400 600 800 1000 1200 Frequency (Hz) Fig.12. Measured and predicted iron losses by different models.
B. Iron loss model for electrical machine fed by PWM inverters According to the study in Section IV, the iron loss model (10) is one of the most accurate iron loss models when the flux density is sinusoidal. However, when the electrical machine is fed by PWM inverter, the phase current waveform can be significantly distorted from sinusoidal. This distorted current will cause flux density fluctuations in the electrical machine. On one hand, the hysteresis minor loops occur as shown in Fig. 13. These minor loops result in additional hysteresis loss. On the other hand, the harmonics of the distorted flux density also cause additional eddy current iron loss. Therefore, in [12], an iron loss model for electrical machine fed by PWM inverter is developed: ,
, 1
, 1
∆
,
(17) (18)
where is the coefficient considering the influence of hysteresis minor loops. is the coefficient depending on lamination properties. ∆ is the amplitude of the flux density fluctuations shown in Fig. 13. In order to evaluate the iron loss models for electrical machines fed by PWM inverter. Two sets of tests are carried out in this section. One is conducted when the IPM machine is powered by an inverter controlled by a dSPACE controller at different switching frequencies. The other one is carried out
6 when the machine is powered with sinusoidal current by the three-phase current source California Instrument 4500il. For each iron loss test, the RMS value of the phase current is kept the same. Fig. 14 shows the measured and predicted results at different switching frequencies. It can be seen that when the electrical machine is powered by sinusoidal current, both of the models (10) and (17) can accurately predict the iron loss. When the switching frequency decreases, the iron loss increases significantly. The iron loss model (10) cannot reflect this variation due to the lack of the consideration of minor loops and eddy current harmonics. On the other hand, the iron loss model (17) keeps good accuracy even when the switching frequency is very low. B Bm
B
∆B2 ∆B1
REFERENCES [1] [2]
[3]
[4]
[5] [6]
∆B3 t ∆B6
∆B4
H
∆B5 Fig.13. Flux density distortion in electrical machine when supplied by PWM inverter.
[7]
[8]
[9]
[10]
[11] [12]
Fig.14. Comparison of measured and predicted results.
[13]
VII. CONCLUSIONS In this paper, more than ten iron loss models are comprehensively evaluated against the measured iron losses considering alternating and rotating fields, temperature influence, flux density with DC bias and flux density fluctuation caused by PWM inverter. Some conclusions can be highlighted as below: 1) The iron loss model developed in [4] has the best accuracy when the flux density is alternating sinusoidally at constant temperature with the help of variable coefficients. 2) The iron loss model developed in [10] has the best accuracy for considering the temperature variation 3) It is more accurate to calculate the iron loss under rotational flux density by decomposing the flux density into major-minor axis than into radial-tangential axis. 4) The iron loss models developed in [10] can accurately predict iron loss under flux density with DC bias at different temperatures while taking the temperature influence into account. 5) The iron loss models developed in [12] can effectively consider the additional iron loss caused by PWM inverter in the electrical machine.
[14] [15]
[16]
[17] [18] [19]
[20] [21]
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