Separation of Slip- and High-Frequency Flux Densities and its Application in Rotor Iron Loss Fine Analysis of Induction Motors Haisen Zhao, Bing Li, Wang Yilong Yang Zhan*, Guorui Xu
Dong Dong Zhang School of Electrical Engineering Xi’an Jiaotong University Xi’an, Shanxi, China
State Key Laboratory of Alternate Electrical Supply System with Renewable Energy Sources (North China Electric Supply University), Beijing, China
[email protected],
[email protected],
[email protected],
[email protected] [email protected] Abstract˖The fine analysis of rotor losses of induction motors requires separation of the slip- and high-frequency electromagnetic quantities under load conditions. However, the conventional method involves a full slip-cycle finite element simulation, which is very expensive in terms of CPU time and memory usage. To obtain a precise iron loss prediction with reduced time consumption, this paper proposes three methods for fast separation of the slip- and high-frequency electromagnetic quantities at rotor side of an induction motor. The first method (Method A) eliminates the influence of the slip-frequency component by calculating the derivative of the rotor flux density that is obtained by Time-Stepping Finite Element Method (T-S FEM). The second method (Method B) extracts the magnitude of slip-frequency flux density from the spatial distribution of the flux density wave within one supply cycle, and the major highfrequency components are extracted from the wave after subtraction of the full slip-frequency wave determined by Least Square Fitting (LST) the curve in time domain. The third method (Method C) reproduces the full slip-cycle waves by incorporating the wave segments at different positions in rotor teeth, and then separates the slip- and high- frequency components by Discrete Fourier Transform (DFT). The three methods are implemented to separate the slip- and high- frequency rotor flux densities in a 5.5kW induction motor. The accuracy of the results and the CPU time of the three methods are compared with those by the conventional method. The predicted rotor iron losses are also validated by experiments to verify the effectiveness of the presented methods. Index terms - Induction motors, rotor loss, slip frequency, Discrete Fourier Transform, Finite Element Method
I. INTRODUCTION The iron loss distribution characteristic of the stator and rotor in induction motors, particularly the iron losses caused by the harmonic fields are complex in induction motors, especially for the inverter-fed induction motor that are widely used nowadays [1-2]. A fine analysis of the different frequency portions of the total loss is critical to in-depth understanding of the induction motor loss mechanism and the development of design strategy for premium efficiency or high-speed motors. How to reduce the iron loss is the key to promoting the efficiency of induction motors. A precise prediction and fine analysis of iron loss are therefore required during the design of induction motors. This work is supported in part by the National Natural Science Foundation of China under Grant No. 51307050 and the Fundamental Research Funds for the Central Universities 2015ZD003 and 2016MS18.
978-1-5090-2998-3/17/$31.00 ©2017 IEEE
Finite Element Method (FEM) is widely used in electric engineering for its advantages over the analytical electromagnetic calculation method: 1) it can model the actual structure of induction motors with more detail; 2) more detailed distribution of flux density in the iron core can be obtained; 3) the iron loss component related to rotary magnetization can be included; 4) some manufacturing factors can also be taken into account effectively [3]-[7]. Time-stepped finite-element Method (T-S FEM) provides an approach to predicting more detailed flux density distribution. In turn, this data may enable the accurate prediction of iron loss. Many researchers have proposed complex 3-D finite element model for more accurate prediction of hysteresis loss and eddy current loss [8-9]. However, these complex finite element models have not yet made their way into the mainstream design and analysis of induction machines due to the large cost in computation time and memory. Although the 2-D FEM including the multi-slice FEM [10] that deals with the skewed induction motors can also calculate the iron loss with acceptable accuracy at apparently cheaper expense in CPU time than 3-D FEM, a prediction of rotor iron loss in induction motors still requires greater effort. Under no-load condition, the calculation of a supply cycle would be sufficient to take into account the flux harmonics as the slip is nearly zero. Under load condition, a full slip cycle which is typically a few dozen times the supply cycle must be simulated in order to include the high-frequency as well as the slip-frequency components in the rotor flux densities [11]. This calculation would usually be very time consuming even if 2-D FEM is used [12]. Considering inverterfed induction motors this calculation would be even extremely expensive due to the very small time steps and large consumption of memory. In order to obtain a precise rotor iron loss with shorter time consumption, this paper proposes three methods for fast rotor iron loss prediction. Method A is based on derivative calculation of flux density, which eliminates the effect of fundamental component. Method B utilizes the spatial distribution of the flux density obtained from the periodic elements in rotor teeth and Least Square Fitting (LSF) of the
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one-supply-cycle curve for the expression of fundamental flux density, and then extracts the high-frequency flux density harmonics by Discrete Fourier Transform (DFT) in time domain. Method C reproduces the flux density wave of a full slip cycle by incorporating the wave segments obtained from the data of a few supply cycles in the periodic rotor teeth elements, and directly applies DFT to the incorporated wave for the fundamental and the harmonics. The effectiveness and the accuracy of the rotor iron loss calculated by the proposed methods are validated with the test results from a 5.5 kW induction motor.
harmonic wave can be separated by subtracting the fundamental wave as shown in Fig. 2 (b), and a time-domain DFT can be applied for the harmonic flux densities. A frequency-domain iron loss model can be used with the extracted fundamental and harmonic flux densities as the inputs to predict the rotor iron losses.
II. CHARACTERISTICS OF FLUX DENSITY UNDER LOAD CONDITION It is well known that the rotor structure of an induction motor is periodic in space, with respect to a pole pair pitch, as shown in Fig.1 (a). The elements at the same position in the rotor teeth are exposed to the flux densities waves with equal magnitude and different time lag depending on the angles between the elements. The rotor flux density is formulated as follows [12]. B(θ R , t ) = Mˆ cos{npθ R + ϕ + [ n(1 − s) − 1]ω1t} ⋅ (1) ª º 1− s Pˆ cos « qN S (θ R + ω1t ) » p ¬ ¼ ˆ where M is the peak of a mmf harmonic, Pˆ is the peak of a permeance harmonic, Ȧ1 is the supply frequency, s is the slip, șR is the mechanical angle referred to the rotor, p is the number of pole pairs, NS is the number of stator slots, ij is the phase angle, n=1, 5, 7, 11, 13, ... is the mmf harmonic order, and q=0, 1, 2, ... is the permeance harmonic order. Equation (1) indicates that a period of simulated flux density at a certain position in different rotor teeth reflect different phase of the travelling wave. The fundamental (slip-frequency) magnitude of the timevarying flux density at a certain position is equal to the fundamental magnitude of the spatial flux density waveform constituted by a series of flux densities at the different sample positions, at a certain moment, as shown in Fig. 1 (b). The spatial waveform constituted by the flux densities at these counterpart positions (1, 2, 3, …,14) over a pole pair pitch has equal magnitude as the slip-frequency waveform in time domain. With the spatial sample flux densities, B(i), as the inputs, the spatial harmonic components are given by the DFT as follows N −1
B(i ) = ¦ B(k ) e
−j
2π ki N
, k = 0,1,...N − 1
(2)
k =0
where N is the number of samples over one pole pair pitch. By making use of the spatial information, it is possible to calculate the fundamental flux density with the data from a shorter period of simulation. One supply cycle of flux density wave is the sum of the fundamental F(t) and harmonic components H(t). After the above processing the only unknown information about F(t) is the phase angle, and it can be obtained by LSF to the one supply cycle of simulated wave shown in Fig. 2 (a). Once F(t) is determined by the DFT and LSF, the
(a)
1.5 0.5
1 N
2 N
3 N
4 N
5 N
6 N
7 N S 8
0.5
S 9
S 10
S S S S 11 12 13 14
-1.5 (b) Fig.1 Flux density waveform of special position (a) spatially periodic rotor structure with 28 rotor slots and 4 poles, as an example. (b) schematic diagram of spatial waveform.
III. SEPARATION METHODS OF SLIP- AND HIGH-FREQUENCY WITH LOAD CONDITIONS The above analysis indicates that the magnitude of the spatial waveform of the flux density in one pole-pair pitch is equal to the magnitude of the time waveform of the flux density in a slip cycle. A periodic flux density waveform can be obtained with periodic points in teeth at an arbitrary moment. This waveform is approximately equivalent to the slip-frequency waveform in time domain. The rotor fundamental flux density are obtained by the DFT applied to this spatial wave. This paper proposes three methods for separating the slip- and high-frequency components. Method A is based on derivative calculation of flux density, which eliminates the effect of the rotor fundamental (slip-frequency) component. Method B and Method C are both utilize the flux densities from periodic elements in rotor teeth, specifically, Method B uses only one supply cycle of flux density data to find the major harmonics, while Method C uses a few supply cycles of flux density data for full harmonics solution. The detail of these methods is given as follows. A. Method A In Method A, the rotor fundamental flux density is ignored due to its low frequency, and the rotor iron loss is assumed to
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be mainly caused by harmonic fields. This method is used to calculate the rotor iron loss in a 5.5kW induction motor at the rated speed of 1450 rpm. The specification of this motor is shown in Table I.
Number of phases Number of poles Number of stator Slot Number of rotor Slot Winding connection
-0.6
3 4 36 28 Delta
dt
d k +1 g ( t ) dt k +1
H k +1 (n) = ( −1)
= ( −1)
( k +1) / 2
( k +1) / 2
¦n n
k +1
w1k +1 Bn cos ( nw1t + ϕn )
0
t (s)
0.02
(b)
B. Method B
(7)
¦ nk +1w1k +1 H (n) n
The magnitude of the nth harmonic H(n) of g(t) can be expressed as H(n)=(-1)(k+1)/2 Hk+1(n)/(nw1)k+1
0.02
t (s)
Fig. 2. Waveform of a supply cycle by Method A. (a) the original flux density of a supply cycle. (b) the flux density after the derivative operation.
n
=
0
(a)
p(t) = ak tk + ak-1 tk-1 + Ă + a1 t + a0 (6) where k= 1, 3, 5Ă. By taking the (k+1)th derivative of (4), the influence of p(t) can be eliminated, hence k +1
-0.4
-0.6
-0.8
As shown in Fig. 2 (a), the rotor slip-frequency component is regarded as a polynomial under load conditions. When using the data of a supply cycle to solve the flux density harmonics, the flux density wave is viewed as the harmonics superimposed with the polynomial. Due to the partial wave of the fundamental flux density included in the data of one supply cycle, DFT cannot be used directly to accurately separate the rotor harmonic components. It is therefore necessary to eliminate the influence of the polynomial function, the supply cycle of the rotor flux density, B(t), is decomposed into harmonic part, g(t), and the polynomial part, p(t). B(t) = g(t) + p(t) (4) (5) g ( t ) = ¦ Bn cos ( nω1t + ϕn )
d k +1 B ( t )
B (T)
B (T)
5.5 50 380 11.7 1450
-0.2
-0.4
TABLE I SPECIFICATIONS OF A 5.5KW INDUCTION MOTOR Rated power (kW) Rated frequency(Hz) Rated voltage (V) Rated current (A) Rated Speed (rpm)
% W
-0.2
With the spatial DFT described in section II, the accuracy of the calculated fundamental at an arbitrary moment depends on the number of spatial samples chosen for the spatial DFT. As the spatial flux density contains numerous harmonics as well as the fundamental, the number of sample elements should be carefully selected to accurately separate the fundamental. The spatial sampling frequency in the rotor teeth are supposed to be at least twice the slot-harmonic order, following Nyquist Sampling Theorem. Considering the large harmonic flux densities on the rotor teeth surface that may interfere with the extraction of the fundamental, relatively more samples are chosen on the rotor teeth surface as shown in Fig. 3, and the related selection criterion is given by ே ʹሺ ೞ േ ͳሻ ۓ ۖʹ ሺ െ ͳሻ ൌ ݈ ௧௧ k=1,2,3… (9) ʹ ݈௦௧ ൌ ܾ ۔ ே ۖ ܰൌ݇ כೝ ە where N is the number of samples in the rotor teeth within a pole pair pitch; k is the number of samples on a single tooth; a is the arc length from the tooth tip to the first sample; b is the arc length between adjacent samples; Nr is the number of rotor slots, and lteeth and lslot are the width of rotor teeth and slots, respectively; a, b and k satisfy the following equations and inequality that are derived from (9).
where n is the harmonic order, ¹1 is the angular frequency of the supply. From the above derivation, after taking the (k+1)th derivative of the rotor flux density of one supply cycle, the rotor high order harmonics can be found by implementing DFT and the transformation formulated as (8). The high-order harmonics extracted by this method is shown in Fig. 2 (b). With this proposed method, the magnitudes of flux density harmonics are calculated for the rotor iron loss prediction. The method only uses a supply cycle of flux density data to obtain the harmonic components by just simply ignoring the fundamental component. The waveform after the derivative operation obviously shows difference with the original in a supply cycle. How to find a precise polynomial, p(t), is the key to good accuracy.
ା
ሺଵିሻ
ܽ ൌ ೞ ଶ ቐ ା ܾ ൌ ೞ
(8)
݇
ଶሺேೞ േሻ ேೝ
(10)
The number of samples chosen for spatial DFT depends on the radius of the concerned area to the center. As highfrequency flux density harmonics are more concentrated on the rotor teeth surface, rather than the middle of the rotor teeth area or rotor yoke area. Relatively sparse samples can be applied to the inner area of the rotor teeth. It is noted that periodic rotor mesh should be made intentionally as the proposed technique requires that every element in one slot pitch has a counterpart with the same shape and position in another slot pitch. Fig. 4 (a) and (b) show the time-varying radial and tangential flux densities of the samples shown in Fig. 3, for this 5.5kW induction motor. Fig. 4 (c) and (d) show the spatial waveform of the flux density from the sample elements.
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% W
-0.2
0.2
B (T)
B (T)
-0.4
0
-0.6 -0.2
) W -0.8
0
+ W
0.02
t (s)
0
0.02
t (s)
(a)
(b)
Fig. 5. Flux density waveform in a supply cycle. (a) Flux density within a supply cycle. (b) Flux density calculated by method B.
C. Method C
Fig. 3. Selection of samples on the top of teeth
Due to the influence of the slip-frequency component, a direct DFT in time domain is unable to accurately separate the flux density harmonics using the data of one supply cycle. With the presented method, the magnitude of the fundamental component included in the flux density, as shown in (11), is separated by spatial DFT. Prior to the spatial DFT, the angular slip frequency, Ȧs, is already known from the load condition. Once the fundamental magnitude, Bf, is obtained by the spatial DFT, the only unknown information about the fundamental flux density wave is the phase angle, ij, as shown in (12), which can be found by LSF as shown in Fig. 5 (a). With F(t) retrieved by the above technique the harmonic wave H(t) can be separated by subtracting F(t) from B(t), as shown in Fig. 5 (b). Then the magnitude of each harmonic as given by (13) can be obtained by time-domain DFT. B(t) = H(t)+F(t) (11) ܨሺݐሻ ൌ ܤ ሺ ωs ݐ ϕ ሻ
ωh = 1 −
(12)
1− s ( np ± qN s ) ω1 p
(13)
Due to the periodic rotor structure in a cage induction motor an identical position in different rotor teeth and bars are exposed to flux density and eddy current waveforms with the same shape and different phase angle, respectively. A certain period of simulated flux densities at the identical positions in the rotor teeth can be viewed as samples of different phases of the flux density waves, including the fundamentals and the harmonics. An example of these identical positions over one pole-pair pitch is shown in Fig.1, with the red dots for the rotor teeth (flux densities). The fundamental flux densities in these positions present phase shifts of 2pʌ/Nr, as shown in Fig.6. With the simulation data of p/(sfNr) seconds, the whole waves of the flux density in these positions can be reproduced by combining the Nr/p wave segments of p/(sfNr) seconds obtained from the different teeth, respectively. The harmonics are also included in these incorporated waves. These incorporated waves can be made for limited number of such positions (i.e. the finite elements) in the whole cross-sections of rotor teeth. By applying DFT to all the incorporated waves corresponding to different elements in the rotor teeth, the fundamental and the harmonic flux densities of all the element in the rotor teeth can be extracted for rotor loss calculation. B
B 3 … …
2 (a)
1
(b)
1.5
t
…
ts =
1.5
… 0.5
… -0.5
1 0
Spatial position (degree)
360
-1.5
0
Spatial position (degree)
2
șR Nr/p
0.5
-0.5
-1.5
p sfN r …
… … 3 … … … … … … … …
Nr/p
360
Fig. 6. Reproducing the waves of rotor tooth flux density (c)
(d)
Fig. 4. Time and spatial waveform of flux density on rotor surface. (a)radial flux density of one supply cycle. (b)tangential flux density of one supply cycle (c)Radial flux density of space waveform. (d)tangential flux density of space waveform.
IV. COMPARISON WITH THE THREE METHODS With the methods proposed above, the waveform under slipcycle are restored by a supply-cycle which are supposed to be obtained by a full slip-cycle, the effectiveness of the proposed methods is validated by comparing the waveform given by the conventional full slip cycle simulation, as shown in Fig. 7. The slip-cycle waveform is obtained by a full simulation of 32 supply cycle which is shown in Fig. 7 (a), and the magnitude of
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fundamental and harmonics by DFT are shown in Fig. 7 (b). Fig. 7 (c) and (d) are waveforms which are calculated by Method A and magnitudes of harmonics calculated by DFT. It should be noted that, as the main consideration is high-frequency harmonic during rotor iron loss prediction, the fundamental component is ignored, hence the slip-frequency waveform cannot be restored by a supply cycle. but we can still validate the effectiveness by the magnitudes of major harmonics. Fig. 7 (e) and (f) present the slip-frequency waveform which is restored by Method B and magnitudes of harmonic calculated by DFT. In this method, the high-frequency harmonic and fundamental components are both considered, it therefore can reach a precise prediction with only a supply cycle simulation. It should be noted that this waveform is reproduced by the major harmonics only, hence the waveform shows the difference compared to the conventional one. Fig. 7 (g) and (h) present the slip-frequency waveform reproduced by Method C and the magnitudes of harmonics, respectively. As shown in Fig. 7 (g) and (h), this method shows the best accuracy. The slipfrequency waveform and magnitudes of fundamental and harmonic are nearly the same with the original waveform. However, more supply cycles are supposed to be calculated compared to the Method B. As can be seen from Fig. 7, the effectiveness of the presented methods is validated by comparing the slip-frequency waveform and magnitudes of the major harmonics to those by the conventional method, which simulates a full-slip cycle, i.e. 32 supply cycles. The magnitudes of the major harmonics calculated by these three methods are shown in Table. II. 1.5
1.2
0.5
0.8
B (T)
B (T) -0.5
-1.5
0.4
0
0.16
0.48
0.32
0
0.64
t (s)
500
(a)
1000
1500 f (Hz)
2000
(c) 0.5
0.8 B (T)
B (T) -0.5
-1.5
0.4
0
0.16
0.48
0.32
0
0.64
500
1000
t (s)
(e) 1.2
0.5
0.8
V. FINE ANALYSIS ON ROTOR IRON LOSSES A. Iron Losses Prediction Model
The iron loss can be calculated with the harmonic flux densities as inputs to the numerous frequency-domain iron loss models, e.g. [13]-[16], The conventional iron loss model divides the iron loss into eddy current loss and hysteresis loss, as follows. (14) PFe = PH + PE where PFe is total iron loss, PH is hysteresis loss, and PE is eddy current loss, in W/kg. PH and PE are given as follows. (15) PH = kh fBˆ α PE = k e
2000
2500
¦
0.16
0.32
t (s)
0.48
0.64
0
(16)
(
¦
)
ª PE = ke ¦ A j «¦ Bˆ m 2 f m 2 1 + k2 m Bˆ m β ¬m j
(
0.4
0
2
§ dB · ¨ ¸ dt ³ 2 2π T © d t ¹ 1
(17) is not accurate for Bˆ greater than 1.2T or f greater than 400 Hz [14], the variable-parameter models are proposed for the cases with larger variation of magnitude and frequency of flux density [17]. This paper uses the variable-parameter model that formulates the hysteresis and eddy current loss as (18) and (19), respectively. ª º (18) PH = kh A j « kr Bˆ mα f m k1m Bˆ m β1m » j ¬ m ¼
B (T)
B (T) -0.5
-1.5
1500 f (Hz)
(f)
1.5
Table II
where Bˆ is the magnitude of the flux density at the frequency of f, T is the period corresponding to f, and kh, Į and ke are the parameters determined by fitting the measured loss in magnetic material. When B is sinusoidal, the formulation of PE can be simplified as follows. (17) PE = k e Bˆ 2 f 2 Because the classical iron loss model formulated as (14) to
(d) 1.2
(h)
COMPARISON BETWEEN CONVENTIONAL AND PROPOSED METHODS B (T) Conventional Method A Method Method B C 1.575Hz (slip 1.0605 1.0634 1.0633 frequency) 291Hz (n=7, q=0) 0.0484 0.0297 0.0194 0.0254 (300Hz) (300Hz) (298Hz) 584Hz(n=7, q=1) 0.0358 0.0233 0.0243 0.0259 (600Hz) (600Hz) (575Hz) 880Hz (1st slot 0.147 0.115 0.195 harmonic, 0.1897 (850Hz) (850Hz) (877Hz) n=1,q=1) 1756Hz (2nd slot 0.0391 0.0661 0.0718 harmonic, n=1, 0.0755 (1750Hz) (1750Hz) (1753Hz) q=2)
2500
(b)
1.5
(g)
Fig. 7. (a) Calculated flux density waveform under a slip cycle. (b) Harmonics of flux density under a slip-cycle. (c) Flux density waveform in one supply cycle by Method A. (d) Harmonics spectrum of flux density by Method A. (e) Reproducing the flux density waveform by Method B (f) Harmonic spectrum of flux density by Method B. (g) Reproducing the flux density waveform by Method C. (h) Harmonics spectrum of flux density by Method C.
500
1000
1500 f (Hz)
2000
2500
2m
º
)» ¼
(19)
where j denotes the jth element in the motor finite element model, m denotes the mth harmonic in an element, Aj is the area
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of the jth element, Bˆm and fm are the magnitude and frequency
B. Loss Comparison with the Measured
of the mth harmonic flux density, respectively, k1m, ȕ1m, k2m and ȕ1m vary piecewisely with Bˆm and fm, and kr is a factor
The losses of the 5.5 kW induction motor are measured to further validate the calculated rotor losses by the proposed methods and by the conventional method. Fig. 8 shows the test rig of this motor. The losses are measured by Method B specified in IEEE Std. 112-2004 [18]. Table V shows the measured total iron loss, Pfe. As it is actually impossible to experimentally separate the rotor iron loss from the total iron loss, the rotor iron loss, Pfer, shown in Table V, is obtained by subtracting the stator iron loss, Pfes, calculated by the FEM, from the measured total iron loss. It is noted that the iron loss model used in this paper does not take into account the interbar current loss that accounts for a significant portion of the additional load loss, meanwhile, the tested iron losses generally includes the harmonic copper loss. Therefore, the calculated rotor iron loss is lower than the tested one. However, through the different method comparison, the effectiveness of the presented method can also be verified well. In next step, the proposed technique will be used with more advanced rotor electromagnetic models in the future for detailed rotor loss calculation.
accounting for the rotational flux. B. Rotor Iron Loss Analysis with the Proposed Method
With the T-S FEM, the above model is utilized to calculate the iron losses of the 5.5kW induction motor, and the fine analysis of iron losses is also performed. The calculated hysteresis, eddy current and total iron losses in the rotor by the conventional method are 6.53 W, 32.89 W and 39.43 W, respectively. The calculated rotor iron loss by the proposed methods are shown in Table. III. a) It apparently shows that the Method A presents a coarse accuracy due to the ignorance of the slip-frequency component, but this method can be implemented just by adding a derivative operation during the post-processing. It is the easiest and fastest method. b) Methods B and C both show a good accuracy by utilizing the spatial periodic elements in rotor teeth considering the slip-frequency waveforms. The rotor iron losses given by Methods B and C are 38.92 W and 39.43 W, respectively, and the rotor iron loss calculated by the conventional method is 39.43 W. c) Method B can obtain the major harmonics as well as fundamental component with one supply cycle, while Method C is capable of reproducing every harmonic by using the data of 3 supply cycles for this 5.5kW motor. Method C shows the best accuracy. TABLE III LOSS ANALYSIS WITH THE THREE METHODS Methods Conventional Method A Method B Method C
Rotor iron loss (W) 39.43 26.87 38.92 39.43
Hysteresis loss (W) 6.53 4.33 4.14 6.75
Eddy-current loss (W) 32.89 22.53 34.78 32.67
Fig. 8. Test rig of the 5.5 kW induction motor TABLE V LOSS MEASUREMENT Total iron loss Pfe (W) Calculated stator iron loss Pfes (W) Rotor iron loss Pfer = Pfe - Pfes (W) Rotor iron loss by conventional (W) Rotor iron loss by Method A (W) Rotor iron loss by Method B (W) Rotor iron loss by Method C (W)
VI. VALIDATION OF THE PRESENTED METHODS A. Time Consumption with a Full Slip Cycle
The simulation of the 5.5 kW induction motor shown in section III is performed by using a computer with an Intel i74790K processor and 16GB RAM. The total number of elements is 41,456 and the time step is set to 0.2 ms. The full slip-cycle simulation consumes 160 min, while Methods A and B reduce the CPU time to 4 min which is only 1/40 of that by the conventional method. Method C reduces the CPU time to 11 min, which is almost 1/16 of that by the conventional method, as shown in Table IV. TABLE. IV CPU TIME COST OF THE DIFFERENT METHODS Method Number of supply cycles Time cost (min)
Conventional 32 160
Method A 1 4
Method B Method C 1 3 4 11
143.4 88.4 55.0 39.4 26.9 38.9 39.4
VII. CONCLUSION This paper proposes three methods to shorten the CPU time for separation of the slip- and high-frequency flux density of induction motors under load conditions. The accuracies and time cost of these methods are also compared with those of the conventional method. Experimental validation of the rotor iron loss prediction is also performed, and the effectiveness of the presented methods is verified. The proposed methods can significantly reduce the calculation overhead in terms of CPU time and memory usage. In addition, by utilizing both the timevarying and the spatial information of the rotor flux densities, Methods B and C achieve better accuracy, making them suitable for fine analysis of rotor iron losses in induction motors.
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ACKNOWLEDGMENT The authors would like to thank Yang Xiujun and Li Yachun, of Hebei Electric Motor company, Ltd., for their technical support in manufacturing the prototypes. REFERENCES [1] V. Ruuskanen, J. Nerg, M. Rilla, and J. Pyrhonen, “Iron Loss Analysis of the Permanent-Magnet Synchronous Machine Based on Finite-Element Analysis Over the Electrical Vehicle Drive Cycle,” IEEE Trans. on Ind. Electr., vol.63, No.7, pp.4129-4136, Jul. 2016. [2] P. Rasilo, A. Salem, A. Abdallh, F. D. Belie, L. Dupre and J. A. Melkebeek. “Effect of Multilevel Inverter Supply on Core Losses in Magnetic Materials and Electrical Machines,” IEEE Trans. Ener. Conv., vol.30, no.4, pp.736744, Jun. 2015. [3] H. S. Zhao, D. D. Zhang, Y. L. Wang, G. R. Xu, Y. Zhan, X. F. Liu, and Y. L. Luo, “No-load iron loss distribution characteristics and its fine analysis for inverter-fed induction motors,” Proceeding of CSEE, vol.36, No.8, pp. 2260-2269, Aug. 2016. [4] K. K. Nallamekala and K. Sivakumar, “A Fault-Tolerant Dual Three-Level Inverter Configuration for Multipole Induction Motor Drive With Reduced Torque Ripple”, IEEE Trans. on Ind. Electr., vol.63, No.3, pp.1450-1457, Oct. 2016. [5] T. J. Brauhn, M. H. Sheng, B. A. Dow, H. Nogawa, and R. D. Lorenz, “Module-Integrated GMR-Based Current Sensing for Closed Loop Control of a Motor Drive,” IEEE Trans. Ind. Appl., pp.342-349, Sep./Oct. 2015. [6] Pries, Jason, and H. Hofmann, “Steady-State Algorithms for Nonlinear Time-Periodic Magnetic Diffusion Problems Using Diagonally Implicit Runge–Kutta Methods,” IEEE Trans. Magn., vol.51, No.4, pp.1-4, Apr. 2015. [7] J. B. Padilha, P. K. Peng, N. Sadowski and N. J. Batistela, “Vector Hysteresis Model Associated With FEM in a Self-Excited Induction Generator Modeling,” IEEE Trans. Magn., vol.52, No.3, pp.1-4, Mar. 2016.
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