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Modeling of Core Loss Resistance for - Equivalent Circuit Analysis of IPMSM considering Harmonic Linkage Flux Byeong-Hwa Lee1 , Soon-O Kwon1 , Tao Sun1 , Jung-Pyo Hong2 , Geun-Ho Lee3 , and Jin Hur4 Department of Automotive Engineering, Hanyang University, Seoul 133-791, Korea Graduate School of Automotive Engineering, Kookmin University, Seoul 136-702, Korea Department of Electricity, Namhae College, Gyeongnam 668-801, Korea Department of Electrical Engineering, Ulsan University, Ulsan 680-749, Korea This paper presents modeling of core loss resistance for equivalent circuit analysis of IPMSM considering harmonic linkage flux. High efficiency is one of the major advantages of IPMSM, therefore precise loss analysis is required for high efficiency design. equivalent circuit analysis in the calculation of core loss due to harmonic components However, there is practical limitation of of flux especially in high speed region where field weakening is generally applied. Harmonic components of linkage flux leads to the underestimation of core loss. Especially operation with high constant speed power range results in severe underestimate of core loss. axis equivalent circuits in this paper, therefore core losses due to Harmonic components of linkage flux are added to the general harmonic components are considered by increased voltage drop across core loss resistance. From the presented modeling of core loss resistance, precise core loss can be obtained. Verification with FEA and experiments are presented. Index Terms—Core loss, core loss resistance, harmonic inductance, interior permanent magnet synchronous motor.
I. INTRODUCTION
I
NTERIOR permanent magnet synchronous motors (IPMSMS) have higher torque density and efficiency per unit rotor volume than induction motor and reluctance motor by utilizing magnetic and reluctance torque, and wide operating speed range can be obtained with the help of field weakening control. Therefore, their applications are getting widened from small power to large power systems. In addition, manufacturing can be simplified comparing to surface mounted permanent magnet motor due to non-existence of sleeve [1]–[3]. - axis equivalent circuit analysis (ECA) generally used for IPMSM, since it provides precise analysis results with fast calculation time. Finite element analysis (FEA) can also be used for high precision of analysis result however, it requires large computation. Especially for the various operation conditions, the ECA is more effective [4]. Generally, motor parameters are calculated by FEA and steady state characteristics are calculated by ECA. In the loss calculation in ECA, core loss is modeled as a resistance and proportional to the square of voltage. However that model results in the underestimation of core loss at field weakening region since it does not consider harmonic components. Fig. 1 shows the core loss comparison between FEA and ECA dealt with in this paper. Maximum torque per ampere (MTPA) control is applied in the constant torque region, and field weakening control is applied to the last region [5]. In the constant torque region, the difference of core loss from FEA and equivalent circuit analysis is not significant however in the field weakening region, there are large difference. The main cause of the
Fig. 1. Core loss comparison.
difference is that harmonic components of core loss are not included in the ECA [6]. II. ANALYSIS THEORY A. Core Loss Resistance Core loss resistance can be expressed by core loss, flux linkage, and speed (1). By using FEA in no-load case, core loss is calculated and core loss resistance is approximated by speed, coefficients of and . This approximation has problems when the harmonic components of flux density exists because core loss resistance is calculated by no-load core loss and no-load back EMF, especially field weakening, which induces distortion of flux density distribution in stator region [7]. This model is improved and used in this paper to apply ECA in this paper. When the speed or load varies from no load condition, core loss can be modeled as shown in (2), (3) [3]. Because core loss resistance corresponding to speed is modeled in (1), coefficient of frequency in (2) is omitted, therefore, core loss resistance according to load condition can be expressed as (4)
Manuscript received June 01, 2010; revised November 02, 2010; accepted December 01, 2010. Date of current version April 22, 2011. 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.2010.2099647 0018-9464/$26.00 © 2011 IEEE
(1) (2)
LEE et al.: MODELING OF CORE LOSS RESISTANCE FOR - EQUIVALENT CIRCUIT ANALYSIS OF IPMSM
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Fig. 3. Inductance calculation process. TABLE I SPECIFICATION OF ANALYZED AND TESTED IPMSM FOR IN-WHEEL APPLICATION
Fig. 2. d-q axis equivalent circuit of PMSM. (a) d-axis equivalent circuit; (b) q -axis equivalent circuit; (c) vector diagram.
(3) (4)
: core loss resistance calculated by no load analysis; where : reference angular velocity; : changed angular velocity; : reference frequency; : changed frequency; : reference : core loss resistance flux density; : changed flux density; : no load linkage flux; : load linkage flux; calculated; : no load core loss; : load core loss; : coefficients related to flux variation; : coefficient related to frequency variation. B. Equivalent Circuit Considering Harmonics Harmonic components of linkage flux exist even with sinusoidal current inputs. Even though it is impossible to add harmonic components of flux to - axis equivalent circuit considering physical aspects, however it is inevitable to add such components in order to explain additional voltage drop and core loss due to harmonic components of flux. By using Fourier analysis, harmonic components of flux at load condition are identified, then, each components are divided by fundamental components
of current. Finally, harmonic inductances are added together. At this point, harmonic order is not considered since it is impossible to consider their phase angle. By adding all harmonic inductance, their representative value is expressed as (5). Corresponding voltage equations are (6) and (7). In (7), polarity of voltage drop due to harmonic components should be determined to increase total input voltage. Because harmonic inductance does not contributes to field weakening or reluctance torque. Output torque can be expressed as (8) without harmonic inductance (5) (6) (7) (8) where , and : phase current, - and -axis current; and : - and -axis voltages; and : linkage flux by permanent magnet and -th harmonic component of linkage flux; : armature winding resistance; and : -and -axis inductances; : harmonic linkage inductance; : output torque; : number of pole pairs. Fig. 2(a) and (b) show IPM equivalent circuit using harmonic is parallel connected and adinductance, core loss resistance ditional voltage drop increases core loss. Fig. 2(c) shows vector diagram considering harmonic inductance. A process to calculate inductance considering harmonic inductance is shown in Fig. 3. In order to calculate harmonic components of linkage flux, harmonic components are extracted
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Fig. 7. The linkage flux and harmonic components at no-load.
Fig. 4. Output performance.
Fig. 8. The linkage flux and harmonic components at base speed (120 A, ).
:
Fig. 9. The linkage flux and harmonic components at max speed (76 A, 78 )
:
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Fig. 5. Structure of IPMSM for in-wheel type EV.
Fig. 6. No-load core loss and core loss resistance.
simply in the general inductance calculation process using fundamental components of no-load linkage, load flux linkage, , , and . III. ANALYSIS AND EXPERIMENTAL VERIFICATION A. Analysis Model Model in the article model is IPMSM for in-wheel type electric vehicle. Table I shows specifications. Output characteristics are shown in Fig. 4. In the constant torque region, MTPA control is applied to and field weakening control is applied in constant power region. Rotor and stator structure are shown in Fig. 5. In Fig. 6, no-load core loss and corresponding are shown. is expressed by function of speed using (1). Fig. 7–9 show no load linkage flux, linkage flux at max. torque, and max. speed. Linkage flux at no-load contain small harmonics. In Fig. 8, MTPA control is applied and field weakening control is applied at Fig. 9. As shown in the figures, linkage flux at MTPA region contains similar harmonics with no load condition, however, huge harmonics exist in field weakening region, and this harmonic components require additional consideration for calculating input voltage. , , and calculated by FEA acFig. 10 shows the cording to current and current phase angle. is roughly 10% of at large current and current phase angle.
Fig. 10. L , L , and L according to current and current phase angle.
Fig. 11. Fabricated rotor and stator.
B. Experimental Verification Analysis is conducted by presented core loss consideration (ECA2) and compared with conventional consideration of core loss (ECA1) and experiments. Fig. 11 shows the fabricated rotor and stator. In order to reduce eddy current loss in PMs, PMs are segmented along circumferential direction. Fig. 12 shows the comparison of back EMF waveforms from FEA and experiments at 1000 rpm. No-load back EMF contain
LEE et al.: MODELING OF CORE LOSS RESISTANCE FOR - EQUIVALENT CIRCUIT ANALYSIS OF IPMSM
Fig. 12. Comparison of no-load back EMF waveforms.
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Fig. 15. Comparison of core loss.
ECA2 shows higher voltage in field weakening region, and agrees with experimental results. Fig. 15 shows the core loss comparison; eddy current loss in the PMs are calculated by 3-D FEA with currents from experiments and subtracted from measured loss. There is a small difference in the MTPA region between results, however in the field weakening region differences becomes significant as the speed increases, this leads to significant over estimation of efficiency. Fig. 13. Test setup for load test.
IV. CONCLUSION Modeling of core loss resistance and consideration of harmonic components of core loss are made by harmonic linkage flux in this paper. Voltage equations and procedures are presented. Presented methods are verified experimentally. Some difference exists with harmonic considerations, however, the presented model can prevent significant underestimation of core loss. In the next study, method for the PWM carrier frequency will be considered for core loss analysis. REFERENCES
Fig. 14. Comparison of current and voltage. (a) Input current. (b) Line to line voltage.
small harmonic components. Fig. 13 shows the experimental setup for load tests. Fig. 14 shows the comparison of current and voltage satisfying load conditions shown in Fig. 4. Comparing with ECA1,
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