IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 11, NOVEMBER 2014
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Equivalent Circuit Considering the Harmonics of Core Loss in the Squirrel-Cage Induction Motor for Electrical Power Steering Application Su-Jin Lee, Ji-Min Kim, Dong-Kyun An, and Jung-Pyo Hong Department of Automotive Engineering, Hanyang University, Seoul 133-791, Korea Consideration of the saturation in the core is essential in designing motors for automotive application, because of the stringent size limitations in automotive electric motors, such as those equipped with electrical power steering. In the design of squirrel-cage induction motors, equivalent circuit analysis using lumped parameters is often used to investigate the motor performance. However, estimation of the motor characteristics has been limited because the standard equivalent circuit that considers space harmonics does not consider the core loss. To improve the reliability of the characteristic analysis, a new method that can solve these problems is required. Thus, study of a new equivalent circuit that considers the saturation in the core and the core loss harmonics is needed. Therefore, to obtain the reliable characteristics of induction motors, this paper presents a modified space harmonic equivalent circuit that considers the harmonics of core loss resistance. The modified equivalent circuit is verified by comparing the analysis results of the suggested method and the experimental results of the test model. Index Terms— Core loss, electrical power steering (EPS), equivalent circuit, harmonics, induction motor.
I. I NTRODUCTION NDUCTION motors are used in many different fields to operate the small fans of large automobile engines. Recently, electric vehicles and fuel cell electric vehicles have required different operating regions for better exploitation of induction motors [1]–[6]. Electrical power steering (EPS) has increasingly been favored as alternative to hydraulic power steering (HPS) owing to the advances in electrical machines, sensors, and control electronics. EPS offers several advantages over conventional HPS, such as improved fuel economy, ability to operate even when the engine is OFF, and elimination of hydraulic fluid. These benefits result in significant energy savings [7]–[9]. Increase in the saturation in the core cannot be remedied, because of the stringent size limitations in automotive electric motors. Therefore, considering saturation in the core is essential in designing motors for automotive application. The most convenient method of analyzing the characteristics of a squirrel-cage induction motor is finite element analysis (FEA). However, this method requires much computational time and large memory capacity. In squirrel-cage motor design, equivalent circuit analysis using lumped parameters is often used to investigate the induction motor performance [10]. Nevertheless, estimation of the motor characteristics is limited because the standard equivalent circuit that considers the space harmonics does not consider the core loss. To improve the reliability of the characteristic analysis, a new method that can resolve these problems is required. Thus, study of a new equivalent circuit that considers the saturation in the core and the core loss harmonics is needed. Therefore, to derive the reliable characteristics of induction motors, this paper presents a modified space harmonic equivalent circuit that considers the harmonics of the core loss resistance. The relative permeability
I
Manuscript received March 7, 2014; revised April 26, 2014, May 12, 2014, and May 20, 2014; accepted June 1, 2014. Date of current version November 18, 2014. Corresponding author: J.-P. Hong (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMAG.2014.2329316
of the stator and rotor core is modified using the B–H data from the steel manufacturer. In addition, the magnetic flux distribution in the load is investigated, and core saturation is considered. The modified equivalent circuit is verified by comparing it with the analysis results of the suggested method and the experimental results of the test model. II. M ODIFIED E QUIVALENT C IRCUIT A. Conventional Equivalent Circuit The input voltage and current in the conventional equivalent circuit of an induction motor has been assumed to have a sinusoidal waveform. However, because a motor with a teeth-andslot structure essentially includes slot harmonics, as suggested in [10], an equivalent circuit analysis that considers space harmonics is needed. It is occasionally useful to visualize the electromagnetic behavior of various space harmonics as similar to the behavior of separate motors with a common stator winding and common shaft but with magnetizing reactance and secondary impedance that correspond to the air gap flux wave of each specific harmonic. The various space harmonics can be viewed from the above perspective as the harmonic torque in the fundamental speed-torque curve. Nevertheless, estimation of the motor characteristics is limited because the standard equivalent circuit that considers the space harmonics does not consider the core loss. To improve the reliability of the characteristic analysis, a new method that can solve these problems is required. B. Modified Equivalent Circuit The modified equivalent circuit is shown in Fig. 1. In this circuit, phase quantities r1 , x 1 , r2 , x 2 , rc , and x m are identical to those used in the standard equivalent circuit. The magnetizing reactance and core loss resistance of each harmonic are based on the component of the air gap flux in that particular harmonic. Because a motor with a teeth-and-slot structure essentially includes slot harmonics, the flux density in the air gap contains the slot harmonics of the stator and rotor. The air gap flux density of stator and rotor magnetomotive forces (MMFs) is shown in Fig. 2. The ratio of the area per pole
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Fig. 1.
Fig. 2.
IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 11, NOVEMBER 2014
Fig. 3.
DC saturation curve of steel.
Fig. 4.
Concept of the single tooth equivalent.
Fig. 5.
Magnetic circuit model.
Modified space harmonic equivalent circuits of an induction motor.
Air gap flux density of the stator and rotor windings.
of the k harmonic to the area per pole of the fundamental wave is 1/k. Then, because the flux is equal to flux density B multiplied by the area per pole, the ratio of the flux per pole for the k harmonic to the flux per pole of the fundamental harmonic is equal to 1/k times. Because the voltage induced in the winding is proportional to the magnetizing reactance, rotor parameters [r2 ]k , and [x 2 ]k are based on the impedance of the effective rotor winding. We note that the slip function of [r2 ]k , and [x 2 ]k for each harmonic is set up for the rotor slip for that particular harmonic and is dependent on the harmonic order and on whether the harmonic field is forward-rotating (sk+ ) or backward-rotating (sk− ) [11] sk+
n
= =
sk = =
s −n nk − n n s − kn = k ns = nk ns k 1 − k (1 − s1 ) = (1 − k) + ks1 ns +n nk + n n s + kn = k ns = nk ns k 1 + k (1 − s1 ) = (1 + k) − ks1 .
(1)
(2)
Several harmonic-torque dips on the resultant motor torque will occur, and the space harmonics can largely affect the induction motor starting. III. E FFECT OF C ORE S ATURATION Fig. 3 shows a plot of the corresponding B and H values of magnetic core steel. This plot shows the familiar normal saturation curve of the material, which shows the important magnetic characteristic of that material. The single tooth equivalent concept is shown in Fig. 4. This method is introduced to calculate the reluctance of the teeth, yoke, air gap, and rotor using the lumped constant method. Fig. 5 shows the magnetic circuit model. The reluctance is expressed solely
on the basis of the geometric data of the motor, material information, and winding distributions. For the calculation, the flux (i ) in each tooth is calculated using the tridiagonal matrix algorithm (input: MMF, output: flux). The flux density in the core is obtained by single-tooth analysis and using a magnetic circuit model, which considers the nonlinearity of non-oriented silicon steel. Here, we input the stator and rotor MMFs to consider the change in the flux path due to the stator and rotor MMFs. The validity of the analysis results is verified by comparing them with the FEA results. Fig. 6 shows the comparison of the flux density of the stator teeth using the modified equivalent circuit results with that using the FEA results under load at a 0.2 slip. The profiles of the flux density distribution of the stator and rotor MMFs are similar. To consider the saturation in the core, this paper introduces saturation factor ksat . The saturation factor accounts for the sum of the MMF losses in the core and air gap divided by the MMF loss in the air gap. Consequently, the presence of the core may be considered as increased air gap g [9] ksat =
ATg + ATc ATg
(3)
where ATg , and ATc are the MMF losses in the air gap and the core, respectively. The saturation factor is calculated using the convergence of the direct iteration method. IV. C ALCULATION OF THE H ARMONICS C ORE L OSS Motors are becoming increasingly favored over high-speed machine owing to their high efficiency, miniaturization, and
LEE et al.: EQUIVALENT CIRCUIT CONSIDERING HARMONICS OF CORE LOSS IN SQUIRREL-CAGE INDUCTION MOTOR FOR EPS APPLICATION
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TABLE I F UNCTION E XPRESSION OF THE C ORE L OSS C OEFFICIENTS
Fig. 6.
Distribution of the flux density in the stator teeth at a 0.2 slip.
Fig. 8.
Cross section of half of the analyzed motor and the test. TABLE II S PECIFICATIONS OF THE M ODEL
Fig. 7.
Core loss data.
light weight. Thus, precise prediction of the core loss, which accounts for a large percentage of the total energy loss in induction motors is very important to improve the performance capability of the motor under all operating conditions. The three typical prediction models are the empirical formula, FEA, and equivalent circuit models. These models have both advantages and disadvantages. In the empirical formula model, a set of approximation models is used to predict the core loss of the induction motor. This method has been proven to have high precision in predicting the core loss in transformers where alternating field is dominant. Nevertheless, in rotating machines where the flux density variation is very complex, the results are less satisfactory. On the other hand, FEA considers the rotational core loss as well as the core loss caused by the flux density harmonics in rotating machines. However, this method is cumbersome and requires much iteration for every operating condition. In the equivalent circuit model, however, if the modeling of the core loss resistance inserted in parallel to the circuit is correct, prediction of the core loss according to the load condition is feasible without requiring a large number of iterations, in contrast to the FEA. In this paper, the core loss density that considers the excess or anomalous loss is expressed as follows [9]: Pck = Ph + Pe + Pa = kh f B n + ke f 2 B 2 + ka f 1.5 B 1.5 (4) where B is the peak flux density value, f is the frequency, kh is the hysteresis loss coefficient, ke is the eddy current loss coefficient, ka is the anomalous loss coefficient, and n is the Steinmetz constant, the decision number in this paper is two. Fig. 7 shows the core loss data from the steel manufacturer. As you can be seen, only the data at 50 Hz are given. Therefore, in this paper, the core loss data expressed by different functions are obtained as follows. 1) The core loss data provided by the manufacturer are rearranged and plotted as a function of Pcore / f versus B. 2) The frequency-versus-core loss data are plotted through the curve fitting of Pcore / f versus B.
3) The derivation of the non-linear curve fitting functions for the core loss coefficients is shown in Table I. The fundamental core loss and harmonic core loss are calculated using the frequency and flux density of the core, respectively, such as at each tooth, yoke, and rotor calculated in Section IV. The electromotive force (EMF) is estimated using the ratio of the k harmonic content, and each harmonic core loss resistance is then calculated. Thus, we can calculate the core loss resistance in the harmonic equivalent circuit using [rc ]k =
E k2 Pck /3
(5)
where E k is the kth back-EMF and Pck is the total core loss. The model is developed solely on the basis of the geometric data, material information, and winding distributions. The model considers the local saturation in the individual stator and rotor teeth as well as the back yoke sections of the motor. V. V ERIFICATION OF THE H ARMONIC E QUIVALENT C IRCUIT The described models were applied to analyze a three-phase, four-pole, and squirrel-cage induction motor. The parameters of the machine equivalent circuit were calculated to operate at a fundamental frequency of 50 Hz. The cross-sectional view of the analysis model and the test motor for EPS application are shown in Fig. 8. The detailed specifications of this motor are listed in Table II. Fig. 9 shows the testing apparatus used to measure the motor characteristics. Table III shows the comparison of the test model and equivalent circuit results. As previously stated, the conventional equivalent circuit analysis assumes that the input voltage and current had a sinusoidal waveform, and slot harmonics were ignored. Therefore, these results had a big error. Because the existing standard equivalent circuit that considers the space harmonics did not consider the
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IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 11, NOVEMBER 2014
accurate result. The obtained results illustrate the suitability of the proposed model for non-sinusoidal wave supply analysis that considers harmonic core losses.
Fig. 9.
Testing apparatus for measuring the characteristics. TABLE III
C OMPARISON OF THE T EST AND E QUIVALENT C IRCUIT R ESULTS
VI. C ONCLUSION To obtain more reliable characteristics of induction motors, this paper has presented a modified space harmonic equivalent circuit that considers the harmonics of core loss resistance and the core saturation. The modified space harmonic equivalent circuit solely based on the geometric data, material information, and winding distributions has the following advantages. 1) The modified harmonic equivalent circuit is established by the proposed approach, which improves the estimation of the characteristic of induction motors compared with the existing method. The analysis result in this paper was validated by experiment. 2) The modified harmonic equivalent circuit can immediately offer guidance for the initial design of induction motors. In other words, the proposed method can reduce the design time and cost. Thus, the proposed method is very useful in the initial design of induction motors. Additionally, improvement in quality and reduction in cost at the initial design stage are the expected advantages of this paper. ACKNOWLEDGMENT This work was supported by the Ministry of Science, ICT and Future Planning, Korea, through the Convergence Information Technology Research Center Support Program under Grant NIPA-2013-H0401-13-1008 supervised by the National IT Industry Promotion Agency. R EFERENCES
Fig. 10. (a) Current spectrum under load. (b) Power factor under load. (c) Efficiency under load. Comparison of the conventional and modified equivalent circuit results with the measurements result under load.
core loss, input current efficiency, and power factor, it had a larger error than the modified equivalent circuit. Fig. 10 shows the comparison of the conventional and modified equivalent circuit results with the measurement results under load at a 0.2 slip. The figure shows that the conventional equivalent circuit underestimated the space harmonics, whereas the modified equivalent circuit achieved a more accurate result. Additionally, the standard equivalent circuit that considers the space harmonics underestimated the harmonic core loss, whereas the modified equivalent circuit provided a more
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