2044 IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 4, JULY 2004
Analytical Prediction of the Incremental Inductance
of the Permanent Magnet Synchronous Motors
Shyh-Jier Wang and Shir-Kuan Lin
Abstract—An analytical method for calculating the incremental
line inductance of a permanent-magnet synchronous motor is pro-posed in this paper. First, the incremental inductance is simplified as a linear combination of the slot inductance and the air-gap in-ductance, which are both estimated by the equivalent magnetic cir-cuits. The reluctance of the rotor back irons and the stator back irons are ignored when the slot inductance is calculated. However, for the calculation of the air-gap inductance, the iron saturations and the perturbations of the permanent magnet are considered. The proposed method is verified by a finite-element method.
Index Terms—Analytical method, finite-element method, incremental inductance, iron saturation, permanent magnet synchronous.
I. INTRODUCTION
I
T IS KNOWN that the accurate calculation of the winding inductance is essential for the design of a permanent-magnet synchronous motor (PMSM). Several works have been pre-sented to deal with the calculation of the winding inductance of a PMSM. The two-dimensional (2-D) analytical method proposed by Zhu [1] ignored the reluctance of the rotor back irons and the stator irons, although it was simple. A numer-ical method based on the finite-element method (FEM) was presented by Lowther [2]. He ignored the perturbations of the permanent magnet (PM). On the other hand, Demerdash [3] and Gyimesi [4] proposed FEM methods to deal with the so-called incremental inductance. These methods took into account the effects of the iron saturation, so that the variation of the incremental inductance with the rotor position and the winding current can be predicted. However, little attention has been paid to the development of an analytical method for the computation of the incremental inductance of a PMSM.This paper is aimed at the development of an analytical method for the incremental inductance of a PMSM, which can enormously reduce computations in comparison with FEM.
II. MAINRESULT
It is known that the line inductance of a PMSM, denoted by , can be described as follows [5]: where is the 2-D equivalent inductance, is the end-turn induc-tance of windings. is brought by three-dimensional (3-D) magnetic flux in a PMSM, and is usually much less than . Manuscript received October 15, 2003. This paper was supported in part by the National Science Council of Taiwan under Grant NSC 92-2213-E-009-057. The authors are with the Department of Electrical and Control Engineering, National Chiao Tung University, Hsinchu 300, Taiwan, R.O.C. (e-mail: [email protected]; [email protected]).
Digital Object Identifier 10.1109/TMAG.2004.830629
Fig. 1. Simplified shoes, teeth, and slots structure.
Fig. 2. Flux pattern of winding current.
Thus, this paper is concerned with only . Some assumptions listed here are made to simplify the analysis.
A1) The shoes, the teeth and the slots of the stator are sim-plified to rectangular shapes with the dimensions , ,
, , , and as shown in Fig. 1.
A2) For the calculation of the air-gap permeance, the field pattern is assumed to be that the flux lines are always orthogonal to the intersection surface of the air-gap and the steel (see Fig. 2).
A3) The B-H curve of the PM is linear with the constants of the relative permeability and the remanence . A4) The reluctance of the rotor and the stator back iron are usually seen as zero, so that the magnetic field within the motor can be assumed the linear superposition of the PM and the winding current. The nonlinear interac-tion of these two fields is considered only for the cal-culation of the air-gap inductance. This means that the relative permeability of the stator back iron is still regarded as a nonlinear function of the flux density in the shoes and the teeth of the stator.
WANG AND LIN: ANALYTICAL PREDICTION OF THE INCREMENTAL INDUCTANCE OF THE PMSMs 2045
Fig. 3. (a) Equivalent magnetic circuit of . (b) Equivalent magnetic circuit of . (c) Equivalent magnetic circuit of air-gap flux.
Suppose that only two of the three phases are excited by the same winding current . These two phases are phases b and c in Fig. 1. It is known [5] that is the sum of the air-gap and the slot inductances
(1) where are the total numbers of slots, the divider 3 is the number of the phases of a PMSM. The inductances and are functions of the flux linkage and , respectively. The flux linkages of the windings are shown in Fig. 2, where they are divided into five parts: , , , , and . is the flux crossing the air-gap, while consists of , , , and
. Since there is no current in phase a, the flux linkages and cross the shoes and the teeth of phase a, respectively. Thus, can also be divided into four parts as
(2) The equivalent magnet circuits of and are, respec-tively, shown in Fig. 3(a) and 3(b). Note that is the number of turns of each stator tooth, is the magnetic reluctance of
, and is that of . It follows from Fig. 3 that and . According to the definition of the inductance [6], we obtain
(3)
It is well known that , where
and are defined in Fig. 1, is the motor axial active length, and is the permeability of the air. Consider the rectangular slot of flux linkage , in Fig. 2, i.e., the slot between the teeth of phases b and c. The magnetic motive force (MMF) at the top of the slot is equal to , while it is zero at the bottom. This is because the turns are uniformly distributed in the slot, and the magnetic field crossing the slot increases linearly from the bottom of the slot to the top. Since and
, we have ,
where is the distance from the bottom of the slot (see Fig. 2). This allows us to express from the energy viewpoint [7] as (4) where the integral volume is over the slot volume between
phases b and c, i.e., . Similarly, can
be related to
(5) The equivalent magnetic circuit in Fig. 3(c) describes the air-gap flux that goes from the shoe of phase b through the
PM yoke to the shoe of phase c and then comes back to the tooth of phase b from the tooth of phase c (see Fig. 2). According to assumption A4), in addition to the reluctance of , only the reluctance of the stator back irons of phases b and c are taken into account, which are denoted by and , respectively.
, , and can be approximated as
(6)
where , , , , , and are defined in Fig. 1, and are the relative permeability of phase b in the shoe and the tooth, respectively, and, and are those of phase c. It then
fol-lows from Fig. 3(c) that . Thus
(7) However, the relative permeability of the stator back iron , , and are nonlinear functions of the flux densi-ties in the shoes and the teeth of the stator by assumption A4),
i.e., , , , and are nonlinear
functions, where and are the flux densities of phase b in the shoes and the teeth, respectively, and and are those of phase c. These flux densities are defined as follows:
(8) where and are, respectively, the fluxes of the PM flowing through phases b and c, and , and are, respec-tively, the equivalent slot linkage fluxes of phase b and c. , , and can be calculated in advance, but the method for and is complicated and will be described later at the end of this section. It is easy to calculate and with the following equations [6]:
(9) Note that , since phases b and c are in series as shown in Fig. 1. It is apparent that , , and are functions of , so that the flux should make the following error function zero from the equivalent circuit shown in Fig. 3(c):
(10) in (10) is a nonlinear equation with a single variable . Applying the Newton–Raphson iteration method to (10), we can easily obtain that makes , provided that and are known. After is obtained, can be calculated
2046 IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 4, JULY 2004
Fig. 4. (a)B 0 curve of the stator back iron. (b) and with respect to .
by (7). is the PM flux flowing through the coils of phase b and is in the form of
(11) where is the PM flux density in the air-gap, and the integral path is along the edges of stator shoes of phase b, i.e., the length of the path is (see Fig. 1). is a function of revolution angle , and can be predicted by a conventional 2-D analytical method [8]. can be obtained in a similar method.
III. EXAMPLE
A 3-phase, 12-pole, and 9-slot dc brushless motor used as a spindle motor of a 50x CD-ROM is taken as an example. The motor has a Y-connected windings, surface-mounted NdFeB
magnets with and , and nonlinear
iron steels with the curve as shown in Fig. 4(a), where is the relative permeability of the stator iron. It shows that the stator iron will be saturated when , magnetic flux den-sity, is about 1.3 T. The geometrical parameters of the motor are
, , , ,
, , , , and
.
It is assumed that only two phases b and c are excited by the same current as shown in Fig. 1. It is revealed in (7) that is proportional to , and is a nonlinear function of , so varies with . In the example, of the motor is computed for several current levels ( , 0.4 A, 0.6 A, 0.8 A, and 1 A) and different rotor positions over the entire 360 electrical cycle.
We can calculate out the slot inductances, ,
, , and
from (3)–(5), respectively. The total slot inductance is then by (2). Furthermore, we use (9) to obtain
and . They are both
proportional to . Note that and are functions of , and are calculated using (11). The results of them are shown in Fig. 4(b). Substituting , , , and into (8), we
obtain , , , and , and then , ,
, and from Fig. 4. Note that and
vary with the rotor position , so do , ,
, and . Finally, is calculated out by solving in (10) with the Newton–Raphson method. It then allows us to compute by (7) and by (1) for several distinct exciting currents and different . The results
Fig. 5. Results ofL . (a) Proposed analytical method. (b) An FEM.
of are shown in Fig. 5(a). The counterpart results of obtained by a FEM is also shown in Fig. 5(b). The profile of
for negative currents is the same but with the 180 phase shift, so is not shown in the paper.
The results of both methods are consistent except that the W shapes in the FEM have larger depths. The possible reason for this difference is that the assumption A4) used to develop (3)–(5) ignores the perturbation of the PM fluxes for the slot inductance. The slot inductance may diminish rapidly in the same way as the air-gap inductance, as the flux density of the iron reaches the saturation state. However, the analytical method does not consider this effect.
IV. CONCLUSION
This paper proposes an analytical method to calculate the incremental inductance of a PMSM, which is computationally much more efficient than FEM. An example is used to illustrate the proposed analytical method. The results of the analytical method is found very close to those of a FEM with the excep-tion of the difference in the lower peak values of the incremental inductance. Fortunately, this difference does not affect the use-fulness of the proposed method.
REFERENCES
[1] Z. Q. Zhu, D. Howe, and J. K. Mitchell, “Magnetic field analysis and inductances of brushless DC machines with surface-mounted magnets and nonoverlapping stator windings,” IEEE Trans. Magn., vol. 31, pp. 2115–2118, May 1995.
[2] D. A. Lowther and P. P. Silvester, Computer-Aided Design in
Mag-netics. New York: Springer-Verlag, 1986.
[3] N. A. Demerdash, F. A. Fouad, and T. W. Nehl, “Determination of winding inductances in ferrite type permanent magnet electric machinery by Finite Elements,” IEEE Trans. Magn., vol. MAG-18, pp. 1052–1054, Nov. 1982.
[4] M. Gyimesi and D. Ostergaard, “Inductance computation by Incremental Finite Element analysis,” IEEE Trans. Magn., vol. 35, pp. 1119–1122, May 1999.
[5] T. J. E. Miller, M. I. Mcgilp, D. A. Staton, and J. J. Bremner, “Calcula-tion of inductance in permanent-magnet DC motors,” Proc. Inst. Elect.
Eng.—Elect. Power Applicat., vol. 146, no. 2, pp. 129–137, 1999.
[6] F. W. Sears and M. W. Zemansky, University Physics. Reading, MA: Addison-Wesley, 1979.
[7] D. C. Hanselman, Brushless Permanent Magnet Motor Design. New York: McGraw-Hill, 1993.
[8] Z. Q. Zhu and D. Howe, “Analytical prediction of the cogging torque in radial-field permanent magnet brushless motors,” IEEE Trans. Magn., vol. 28, pp. 1371–1374, Mar. 1992.
