IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 6, JUNE 2007
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Electromechanical Parameters Calculation of Permanent Magnet Synchronous Motor Using the Transfer Relations Theorem Seok-Myeong Jang, Kyoung-Jin Ko, Han-Wook Cho, and Jang-Young Choi Department of Electrical Engineering, Chungnam National University, Daejeon, 305-764, Korea This paper presents an analytical solution to predict magnetic field distribution and to calculate parameters of permanent magnet (PM) synchronous motor equipped with surface-mounted magnet using an electromagnetic transfer relations theorem (TRT) in terms of 2-D model in polar co-ordinates system. The analytical results are validated by comparison with finite element (FE) analyses and experimental results. Index Terms—Analytical solution, permanent magnet synchronous motor, transfer relations theorem.
I. INTRODUCTION
E
LECTROMAGNETIC analyses of electric machines are widely classified into two methods; one is numerical method, the other is analytical method such as space harmonic method. The former such as FE analysis provide an accurate means of determining the field distribution, with due account of saturation, etc., they remain time-consuming. The latter provide as much insight as analytical solutions into the influence of the design parameters on the machine behavior. In particular, analytical technique using the electromagnetic transfer relations theorem (TRT) that proposed by Melcher is more useful than the existing space harmonic method because it reduces analytical burden such as derivation of the governing equation [1], [2]. This paper presents an analytical solution to predict magnetic field distribution and to calculate parameters of permanent magnet (PM) synchronous motor equipped with surface-mounted magnets by using TRT in terms of 2-D model in polar coordinates system. The analytical results are validated by comparison with finite element analyses and experimental results.
Fig. 1. Two-dimensional model in polar coordinates system. TABLE I DESIGN SPECIFICATIONS OF PM SYNCHRONOUS MOTOR
II. TRANSFER RELATIONS AND MAGNETIC FIELD SOLUTION IN PM SYNCHRONOUS MOTOR A. Model and Assumption Fig. 1 shows the 2-D model of a typical 4-pole, 3-phase PM synchronous motor equipped with a parallel magnetized PM rotor and a slotted stator core. The main design specifications of PM synchronous motor are given in Table I. In order to establish analytical solutions for the magnetic field distribution, this paper assumes that the relative permeability of the stator core and rotor shaft is infinite and the . current is distributed in an infinitesimal thin sheet at B. Derivation of Transfer Relations in PM Region Fig. 2 shows the simplified analytical model with only PM field for deriving transfer relations at the magnet surface [(d) and (e) in Fig. 2(A)]. Since there is no free current in the PM region [4] (1)
Fig. 2. Simplified analytical model with only PM field (A) when stator and rotor shaft are air and (B) considering the stator and rotor shaft core.
The magnetic vector potential A is defined as . By the geometry of the cylindrical structure, the magnetic vector which is independent z. In the polar coorpotential has only dinate system, the magnetization M is given by (2) In nonconducting regions, the magnetic vector potential is and assumed to have Coulomb gauge dependence satisfies the Poisson equation (3) (4)
Digital Object Identifier 10.1109/TMAG.2007.893803 0018-9464/$25.00 © 2007 IEEE
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IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 6, JUNE 2007
where denotes the free space permeability, is the spatial wave number of the th harmonic. Here, is the complex Fourier coefficient of th order parallel magnetization component
Fig. 3. Normal and tangential flux density distribution by PM at the imaginary boundary Y .
(5) , flux density of normal and tangential compoBy nent could be expressed as (6) Using general solution of (4), the transfer relation between the and the flux density which relate the vector potential field evaluated at the magnet region—(d), (e) identified in the model of Fig. 2 are
Fig. 4. (A) Analytical model for predicting armature-reaction field. (B) Normal and tangential flux density by armature-reaction field at the imaginary boundary Y .
Using (10), the magnetic vector potential at an imaginary boundary can be obtained as (11)
(7) where the geometric parameters
are
Fig. 3 shows the comparison of between TRT and FE method results for normal and tangential flux density by PMs at the imaginary boundary . D. Magnetic Field Solution by Winding Current
(8)
C. Magnetic Field Solutions by PM Regions Fig. 2(B) shows the simplified analytical model for predicting generalized magnetic vector potential due to PM. Since there is no source term in the airgap region, the transfer relations can be easily expressed as follows [elimination only the source term of (7)]:
Fig. 4(A) shows the analytical model for predicting armature-reaction field distribution. Since this paper assumes that the current is distributed in an infinitesimal thin sheet and the relative recoil permeability of the PM in unity, both air/stator wind. ings and magnet regions remain characterized by curl , Since there is no source terms in the airgap regionthe transfer relations can be easily expressed as follows: (12) The boundary conditions used in analytical prediction of the magnetic vector potential as follows:
(9) The boundary conditions used in analytical prediction of the magnetic vector potential due to the PMs are as follows:
Following an analysis similar to (12), the transfer relations at can be expressed as an imaginary boundary (13)
and Following an analysis similar to (9), the transfer relations at in airgap regions can be exan imaginary boundary pressed as (10)
Using (13), the vector potential at an imaginary boundary can be obtained as (14) Therefore, the magnetic fields at substituting (14) for (6).
can be obtained by
JANG et al.: ELECTROMECHANICAL PARAMETERS CALCULATION OF PM SYNCHRONOUS MOTOR
Fig. 5. TRT, FE analysis, and experimental result for back EMF at 1000(rpm).
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Fig. 6. TRT and FE analysis result for Torque [I = 1(A)].
Fig. 4(B) shows the comparison of between TRT and FE analysis results for flux density due to armature reaction field at the imaginary boundary . III. BACK EMF AND TORQUE A. Back EMF The back EMF is given by the product of angular velocity and the rate of change in the flux linkage with respect to angular and area , the flux linkage can position. From flux density be express as
Fig. 7. Comparison of between TRT, FE analysis and experimental result for (A) peak back EMF with angular speed and (B) peak torque with the current. TABLE II PARAMETERS OF PM SYNCHRONOUS MOTOR
(15) By (15), the flux linkages due to PMs for one slot pitch can be expressed as (16) where and are the pitch, stator inner radius, and stack length, respectively. Therefore, the back EMF can be expressed as (17) where and are the slots per phase per pole and the turns per phase, respectively. When the analysis model work at a 1000 (rpm), the analytical result of back EMF waveform by TRT in phase A is given against FE method and experimental result, shown in Fig. 5. B. Electromagnetic Torque This paper used the Maxwell stress tensor for calculation of for magnetithe electromagnetic torque. The stress tensor cally linear materials associated with the Korteweg–Helmholtz force density is (18) where is the Kronecker delta. The net electromagnetic torque is the integral over the encan be exclosing the surface of (18). Therefore, the torque pressed as follows:
C. Back EMF and Torque Constant Fig. 7 shows the peak back EMF with the angular speed and the peak torque with the current. Table II shows the parameters such as torque constant and back EMF constant of PM synchronous motor. The analytical results by TRT are shown in good agreement with those obtained from FE analyses and experimental results. IV. CONCLUSION The TRT has been outlined and applied to the study of surface-mounted PM synchronous motor. The magnetic field distribution, back EMF and torque constant of the PM machine with parallel magnetized PM rotor are given. The analytical results have been verified by FE analysis and experiment. Further, the results we derived herein can be easily used to analyze other type of rotary machines. ACKNOWLEDGMENT This work was supported in part by the Ministry of Commerce, Industry and Energy (MOCIE) through the IERC program, Korea. REFERENCES
(19) where is the area of the surface (b), is the torque density, and are the magnetic field intensity at the surface and ( ), respectively. Fig. 6 shows the analysis result of the electromagnetic torque at a current (A) by the analytical technique together with those by the FE analyses.
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[1] X. Wang, Q. Li, S. Wang, and Q. Li, IEEE Trans. Energy Convers., vol. 18, pp. 424–432, Sep. 2003. [2] J. R. Melcher, Continuum Electromechanics. Cambridge, MA: MIT Press, 1981. [3] Z. Q. Zhu and D. Howe, IEEE Trans. Magn., vol. 29, no. 1, pp. 124–135, Jan. 1993. [4] N. Boules, IEEE Trans. Ind. Applicant, vol. IA-21, no. , pp. 633–643, Jul.–Aug. 1985
Manuscript received October 31, 2006; revised January 29, 2007 (e-mail:
[email protected]).
