Spatial Discretization Methods for Air Gap Permeance ... - IEEE Xplore

IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 48, NO. 6, NOVEMBER/DECEMBER 2012

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Spatial Discretization Methods for Air Gap Permeance Calculations in Double Salient Traction Motors Esin Ilhan, Student Member, IEEE, Maarten F. J. Kremers, Student Member, IEEE, Emilia T. Motoasca, Member, IEEE, Johan J. H. Paulides, Member, IEEE, and Elena A. Lomonova, Senior Member, IEEE

Abstract—Weight limitations in electric/hybrid cars demand the highest possible power-to-weight ratio from the traction motor, as in double salient permanent magnet (PM) machines. Their high flux densities in the air gap result in nonlinear analytical models, which need to be time optimized. The incorporated reluctance networks are sensitive to the correctness of air gap permeances. Conventionally, in these networks, air gap permeances are calculated by approximating flux paths; however, it is time inefficient. For an improved simulation time, spatial discretization techniques are presented to calculate air gap permeances in double salient PM machines. The spatial techniques discussed here cover the tooth contour method and Schwarz–Christoffel (SC) mapping as semidiscrete methods, which are used to discretize only the air gap region. Their results are verified by the spatial discrete method and finite element method, which discretizes the whole machine geometry. For consistency in this paper, all methods are explained on a three-phase 12/10 flux-switching PM motor. Obtained air gap permeances show a very good agreement with only 0.8% difference. Further on, machine characteristics such as phase flux linkage and cogging torque are also compared to show the impact of the modeling techniques. The total machine simulation time is improved by 20% using the SC method. Although methods are explained particularly for double salient PM machines, formulas are generalizable for other machine types as well. Index Terms—Double salient machines, doubly salient machines, flux-switching permanent magnet (PM) machines, Schwarz–Christoffel (SC) mapping, spatial discretization, tooth contour (TC) method (TCM), traction motors.

N OMENCLATURE F Φ P B Nr Ns Rg La

Magnetomotive force. Magnetic flux. Permeance. Magnetic flux density. Rotor pole number. Stator pole number. Center of air gap radius. Axial length.

Manuscript received December 28, 2011; revised March 14, 2012 and May 1, 2012; accepted May 31, 2012. Date of publication October 26, 2012; date of current version December 31, 2012. Paper 2011-EMC-654.R2, approved for publication in the IEEE T RANSACTIONS ON I NDUSTRY A PPLICA TIONS by the Electric Machines Committee of the IEEE Industry Applications Society. The authors are with Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands (e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [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/TIA.2012.2226692

Fig. 1. poles.

g θrt wr Δθ Δe

Three-dimensional illustration of an FSPM with ten rotor and 12 stator

Air gap. Angular rotor tooth width. Rotor tooth width. Mechanical displacement. Electrical displacement. I. I NTRODUCTION

D

UE TO THEIR high energy density, double salient machines such as flux-switching permanent magnet (PM) (FSPM) machines are very suitable as traction motors for electric/plug-in hybrid vehicles. FSPMs come with many theoretical advantages such as robust rotor design, low demagnetization probability, and wide operation range [1]–[6]. In its rather unconventional structure, as shown in Fig. 1, FSPM embodies all its energy sources in stator, similar to a parallel magnetic circuit motor. With the shortened stator flux paths, due to the alternating magnets, the magnetic operating point of the machine is pushed to the knee section of the BH curve, resulting in a high starting torque. This torque is demanded for regular start–stop actions, because electric vehicles are mainly used for commuting due to their limited drive range [7], [8]. Additionally, for a safe drive, a traction motor should be controlled in a fast and robust way. Analytical models can easily be coupled with such controllers for the desired fast response. Double salient PM machines with numerous advantages for vehicle applications require still answers in a number of important research questions. In these machines, whereas nonuniform air gap results in more complicated analytical models, high air gap flux density levels create the necessity for nonlinear models. In nonlinear models, the permeability μr of the soft

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magnetic material depends on the applied field intensity H in accordance with the BH curve of the material. Therefore, to include nonlinearity in analytical models, researchers usually rely on reluctance network models or hybrid models, which are, to some extent, always verified with measurement or finite element method (FEM) simulation data [9]–[12]. An important model parameter is the correct calculation of air gap permeances. By definition, the air gap permeances are not directly affected by μr (H) dependence of the soft magnetic material. As a part of the whole magnetic circuit, flux paths in the air gap are subject to change; however, in double salient PM machines, this change is negligible next to the dominant change of permeance in soft magnetic material [11]. To calculate the air gap permeances in double salient machines, most commonly, flux paths through the air gap are geometrically approximated [13], [14]. The prediction of these paths is difficult and time consuming due to the nonuniformity of the air gap. There are very few publications focused on alternative analytical techniques for air gap permeance calculations, such as Fourier analysis and curve fitting [9], [15]–[17], but proposed techniques are very difficult to incorporate nonlinearities. Therefore, this paper focuses on spatial discretization techniques to efficiently predict the air gap permeances in double salient machines. Schwarz–Christoffel (SC) mapping and tooth contour (TC) method (TCM) are explained in detail on the double salient air gap of the three-phase 12/10 fluxswitching PM machine. The results for the air gap permeances are compared to FEM, for which flux software from the Cedrat company is used. Further on, the models are also used to calculate permeancedependent machine parameters such as phase flux linkage and cogging torque. Obtained results show very good agreement among the discussed models. Although the techniques presented here can also be used for nonlinear modeling of double salient machines, as discussed in [11], [13], [18], nonlinear modeling is not in the scope of this paper. II. M ETHODS The spatial discretization techniques discussed in this paper are categorized as semidiscrete and discrete spatial techniques. As semidiscrete spatial methods, SC mapping and TCM are evaluated, which discretize only the double salient air gap. As the spatial discrete method, FEM is evaluated, which discretizes the whole machine geometry. The software used for FEM calculations enables the user to define the density of mesh elements, as given in Fig. 2(a). For an optimum simulation time during the rotor movement, remeshing is performed only in 1/3 of the air gap, as in Fig. 2(b). FEM is introduced here only for verification of the results. For consistency in this paper, all methods are explained on the 12/10 FSPM shown in Fig. 1. A. Spatial Semidiscrete Method: SC Mapping Technique The SC mapping technique is a conformal mapping used to transform a shape into a simpler polygon to decrease computational complexity. SC mapping is used in several electromagnetic problems to simplify the nonuniform air gap of an electrical machine [19]. In these papers, SC technique is not used to calculate air gap permeances but, rather, to analytically

Fig. 2. (a) FEM model with mesh elements. (b) Enlarged view of air gap with mesh elements.

solve the Maxwell equations in the equivalent (mapped) air gap by means of charge modeling or Fourier series expansions. However, these analytical models cannot account for iron nonlinearity, which can be only incorporated with reluctance network models. SC mapping is a mathematical function mapping a complex w domain to another complex z domain. The mapping function f (w) = z is given in closed form as f (w) = A + C

w n−1

(w − wk )αk −1 dw

(1)

k=1

where A and C are complex constants, αk ’s are the interior angles among the w points, and zk = f (wk ) for k = 1, . . . , n − 1 [20]. As an example of mapping of the w domain into the z domain, an L-shaped polygon in the complex w domain is chosen, as shown in Fig. 3. This polygon is mapped with the SC integral in (1) to another complex z domain. The corners of the L-shaped polygon are called vertices, denoted as encircled numbers in Fig. 3. Sometimes, SC mapping can be defined around certain vertices; for example, in Fig. 3, (1) is defined around vertices 1, 2, 4, and 5. The number of vertices is proportional to the complexity of the mapping function f (w). For polygons with a maximum of three vertices, the mapping function can be calculated explicitly. For more complicated polygons, determining the mapping function f (w) becomes a parameter problem, which can only be solved numerically [20]. To determine the SC mapping function numerically, a MATLAB toolbox is available, which automatically calculates the mapping function of a given polygon. In the MATLAB SC mapping toolbox, several mapping methods are available for different polygons in the z domain, e.g., disk, half plane, strip, rectangle, Riemann surfaces, etc. [21]. Regardless of its shape, there are two general requirements to describe a polygon in the w domain. 1) The polygon has to have a quadrilateral polygon, i.e., defined in polar coordinates that have to be converted to the Cartesian coordinate system. 2) The corner points (vertices) of the polygon have to be defined in the complex plane in counterclockwise direction.

ILHAN et al.: DISCRETIZATION METHODS FOR AIR GAP PERMEANCE CALCULATIONS IN TRACTION MOTORS

Fig. 3.

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SC mapping of an L-shaped polygon in the w domain.

The first requirement states that the path between all vertices has to be linear, since an interior angle α has to be defined around each vertice k. The second requirement is due to the generated MATLAB code. Note that all vertices both in original w and mapped z domains in Fig. 3 are in counterclockwise direction. The relative position of the vertices is not affected by the mapping function. It should be noted that, in some air gap geometries, convergence problems may occur in the toolbox. This is also referred to as crowding, which is a common phenomenon in computational conformal mapping [20]. It is a result of the large ratios between polygon sides. For motors, this situation may occur in the case of large-air-gap-to-slot-depth ratios in radial direction. One solution to overcome this problem is to reach a suitable ratio of polygon sides by extending the boundaries in the tangential direction. SC Mapping on Double Salient Air Gap of FSPM: In this part, how to calculate the air gap permeance of FSPM with SC mapping is explained by modeling a periodic part of the double salient structure. The modeled partial air gap of FSPM is surrounded by one stator period together with one rotor tooth, as shown in Fig. 4(a). Since SC mapping only applies to quadrilateral polygons, all curved boundaries have to be transformed into linear ones. The curved air gap boundary can be linearly discretized and implemented in the mapping function (1); however, it will considerably increase computational time. Therefore, to obtain the w-domain coordinates of each relevant (x, y) point of the air gap boundary, the following function is used: w = ln(s) = ln (|s|) + i · arg(s) y √ = ln (x2 + y 2 ) + i · arctan x

(2)

with s = x + iy. Using (3), double salient air gap in the s domain is transformed into a polygon in the w domain as seen in Fig. 4(b). Next, the desired mapping polygon in the z domain is chosen in MATLAB SC toolbox. Since a rectangle is the basic 2-D shape of a flux path as shown in Fig. 5, crrectmap mapping function is used to obtain the rectangle polygon in the z domain. By applying the mapping function, air gap flux paths in the w domain are projected into rectangular flux tubes in the z domain, as seen in Fig. 4(d). The crrectmap mapping function is defined around four vertices, which are points of the rectangle in the z domain, as shown in Fig. 5. The air gap

Fig. 4. Sequential steps in SC mapping: (a) FSPM elementary cell in the s domain. (b) Double salient air gap in the w domain. (c) Vertices of (b). (d) Rectangular flux tube in the z domain.

Fig. 5. Rectangular flux tube in the z domain.

flux is assumed to flow from the surface z1 −z2 to the surface z3 −z4 . To calculate the permeance of this flux tube, these four vertices have to be chosen carefully, because z1 −z2 has to satisfy Dirichlet and z3 −z4 homogeneous Neumann boundary conditions [20]. In FSPM, because the air gap flux is assumed to flow from stator tooth to rotor tooth as shown in Fig. 5, the crrectmap function is applied on the corresponding stator vertices 11–14 as z1 −z2 and rotor vertices 1–2, 2–3, and 3–4 as z3 −z4 . Three parallel air gap permeances between one stator tooth surface and three rotor tooth surfaces are calculated with μ0 S μ0 (|z1 − z2 |) La μ0 dS = = (3) Pgap = l l |z1 − z4 | where μ0 is the air permeability, S is the cross section of the considered air gap flux tube, and l is the length of the air gap flux tube. The complex coordinates z1 −z4 of the rectangle are

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Fig. 6.


Air gap permeance Pgap in SC frame: (a) Half period. (b) Full period.

used to calculate S and l in (3). Finally, three parallel air gap permeances are summed up to one air gap permeance. The aforementioned SC mapping method can be summarized as follows. 1) Draw a single stator and rotor tooth section and denote it in the s domain [Fig. 4(a)]. 2) Transform the geometry from polar to Cartesian coordinates using complex ln(s) function in the w domain [Fig. 4(b)]. 3) Determine the vertices for the SC mapping in counterclockwise direction [Fig. 4(c)]. 4) Apply the crrectmap function on the corresponding stator and rotor vertices [Fig. 4(d)]. 5) Calculate air gap permeance Pgap for one rotor position. Following these steps, air gap permeance in one rotor position is calculated. To calculate quantities like phase flux linkage or torque, the air gap permeance needs to be varied with position. Rotor is moved for one electrical cycle, since it is the period of the permeance function Pgap (θ). To reduce the simulation time, the symmetry of FSPM in Fig. 4(a) is used, where the stator teeth are located on both sides of the PMs. Only one stator tooth side for half electrical cycle is required for simulations as shown in Fig. 6(a). Whereas the x-axis in Fig. 6(a) shows the linear displacement for the rectangular SC frame in the half permeance period, air gap permeance depending on the angular displacement for one electrical cycle is shown in Fig. 6(b).

Fig. 7.

TCM model of FSPM.

Fig. 8. TCM network model of the FSPM with black permeances used for air gap permeance calculations. TABLE I E LEMENTS OF THE TCM M ODEL

B. Spatial Semidiscrete Method: TCM The second spatial semidiscrete technique, discussed in this paper, is called TCM. This method can be classified as a reluctance network model [22], where a whole electrical machine [23], [24] or a part of it [12], [25], [26] can be modeled. Unlike the SC mapping technique, which transforms the air gap geometry into a simpler polygon, TCM discretizes the air gap boundary at the soft magnetic material (iron). Discretization can be done in both curved and linear boundaries. The iron boundaries are called TCs, which are shown in Fig. 7 for 12/10 FSPM. In Fig. 8, the complete TCM network is given with the explanations in Table I; this network can be used to calculate quantities like flux linkage, magnetomotive force, torque, or flux density. Since the focus of this paper is air gap permeances, only the black air gap permeances in Fig. 8 are relevant, which interconnect all TCs around the air gap. Using TCM, Ps∗∗ , Pc , and Pml are calculated. Among the three, Ps∗∗ has a consider-

ably higher value compared to the other two permeances, which leads to Pgap = Ps∗∗ .

(4)

The permeance values change, depending on the relative position of the rotor to stator. The sum of the local fields, which are represented over these permeances, gives the total magnetic field in air gap [27]. This feature allows a wide application area including an easy representation of the double salient air gap by TCM permeance network. Numerical methods can easily be integrated into TCM’s reluctance network to determine air gap permeances. Although


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it is generally explained in a Cartesian coordinate system on polygons, it can also be implemented on complex structures such as arc-shaped magnets [25]. Without any need for transformation of the curved boundaries, as it is required in SC mapping, FEM is used to predict the air gap permeances. Considering simulation time, using only FEM for the machine simulation would yield a shorter calculation time compared to the proposed method, which uses both FEM and TCM. However, for the TCM method, since the shortest simulation time is aimed, only in the double salient air gap, the scalar magnetic potential distribution is calculated using Pgap =

Φ F

· dS B F

= La Bn dx

=

S

.

(5)

F=1

Fig. 9. (a) FEM model of FSPM used in TCM calculations with the TCs in rotor and stator.

Note that (5) holds only for F = 1A turns. In FEM simulations, the 1A scalar magnetic potential is set at TCs, i.e., TC is activated. To calculate Pgap , the normal component of magnetic flux density Bn along the length of the nonactive TC is calculated in FEM. As indicated in the SC method, these calculations give only one value of air gap permeance. Position-dependent air gap permeance Pgap (θ) follows a normal (Gaussian) distribution used in the probability theory and statistics [28] Pgap (θ) =

1 2 2 √ e−(θ−mean) /(2σ ) σ 2π

(6)

where θ stands for relative rotor position, σ stands for standard deviation, and mean stands for the mean value of Pgap (θ). The maximum permeance is reached when the considered rotor and stator TCs come at minimum distance, i.e., if they are aligned. Depending on the position of the TCs, the shape of permeance function changes in its variance, skewness, and kurtosis. In the next part, it will be discussed how these air gap permeances and the FEM model of the TCM are created for the double salient air gap of 12/10 FSPM. TCM on Double Salient Air Gap of FSPM: To calculate the air gap permeances of FSPM in Fig. 1 using the TCM, the first step is to create the air gap model for FEM calculations with a minimum number of TCs. Since no material properties are involved in the magnetostatic FEM simulations of TCM, the double salient air gap of FSPM can be modeled as in Fig. 9. Because the PM relative permeability is very similar to that of air, PMs can be included in the TCM as well. Considering the geometrical symmetries and periodicities, it is sufficient to represent a part of the air gap by TCs, which consists of three rotor teeth and one stator cell with one phase coil around one PM. Three rotor teeth are modeled to cover all air gap permeances in one electrical cycle. Because these TCs are activated on a small area only, it should be ensured that all the other TCs and boundaries are of zero potential. Therefore, it is generally recommended to consider at least three rotor pole pitches while creating the TCM model [29]. Because all energy sources of FSPM are located in the stator, it is sufficient to set the stator TCs at 1A magnetic

Fig. 10. Air gap permeance functions [Pgap (θ)] between TCc on the stator tooth and the rotor TCs.

potential for the permeance calculations using the relationships given in (5). The activated stator TCs, eight TCs labeled a−h, are located on stator tooth faces, and rotor TCs, labeled 1–3, are located on rotor tooth faces, as shown in Fig. 8. Due to the considered three pole pitches in the model, all three rotor teeth in Fig. 8 are numbered in the same order. Consequently, TCs with the same numbering contribute to the same permeance, just with a difference in the relative position θ. In double salient machines, the dominant air gap permeances are located on the outer tooth sides, which are TC2 for the rotor tooth side and TCc and TCf for the stator tooth side. Results in Fig. 10 also verify that, compared to other air gap permeances, the permeance between TC2 −TCc or TC2 −TCf determines the general shape of Pgap (θ). The dominant permeance shape has a standard deviation of σ = 0.5. If more permeances are required in addition to Pgap , such as Pc or Pml , these can be also calculated by the same TCM model. III. R ESULTS A. SC Mapping Preliminary results in Fig. 11(a) show that air gap permeances calculated by SC mapping have higher values at certain rotor positions compared to the values calculated by FEM. To

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SC frame. The value 0.1309 mm has no physical meaning, but it is only used to calculate the ratios. The correction factor varies with the rotor position, i.e., depends on the y-axis position in Cartesian coordinates and on the θ-axis position in the polar coordinate system. For consistency in the formulas, the distance calculations for both Dcar and Dpol are given in the Cartesian coordinate system Dcar,pol =

(Δx)2 + (Δy)2

Δxcar,pol = g Fig. 11. Comparison between SC and FEM for air gap permeance Ps12 calculation. (a) Before correction factor. (b) After correction factor.

Δypol

π Nr

π |z3 − z4 | · 0 : = 2π Nr 4Ns

π Rg sin 0 : = . 2 2Nr

Δycar =

wr · θrt

0:

(8)

All angular displacements presented in (8) are in mechanical degrees. Both Dcar (SC mapping) and Dpol (FEM) are distances calculated for the half electrical period, which corresponds to [0 : π/Nr ] = [0◦ : 18◦ ] in mechanical degrees. Using the correction factor, the results from SC mapping and FEM show a very good agreement in Fig. 11(b) with an error of only 0.8%, compared to the previous results in Fig. 11(a). B. TCM

Fig. 12. Calculation of the correction factor for the SC method.

investigate the reason and solve the discrepancy, a correction factor is introduced. Correction Factor: Fig. 11(a) results of SC mapping and FEM are identical only at the stator-tooth–rotor-tooth alignment. At the remaining rotor positions, the error percentage increases with increasing distance between stator–rotor teeth. This result is due to the differently increasing distances between stator and rotor teeth in two different coordinate systems. The distance in the polar coordinate system of FEM does not linearly increase compared to the distance in the Cartesian coordinate system of the SC mapping. To minimize the difference, a correction factor corrfac is implemented, which has to depend on rotor position, as corrfac =

Dcar Dpol

(7)

where Dcar is the distance between the midpoints of rotor tooth and stator tooth in the Cartesian coordinate system and Dpol is the corresponding distance in the polar coordinate system, as shown in Fig. 12. For the Cartesian coordinate system in Fig. 12, the rotor tooth width wr corresponds to 7.5◦ mechanical and 0.1309 mm in the

The results in Fig. 13(a) show examples of the air gap permeances P calculated by TCM; these permeances denoted as Ps12 , Ps21 , Ps22 , and Ps31 are located between the five rotor teeth P 1−P 5 and stator teeth c1, f 1, c2, and f 2, respectively. Results in Fig. 13(b) show how different located air gap permeances are changing with the same rotor position. Permeance values are distributed over one electrical cycle, which corresponds to 36◦ mechanical due to Nr = 10. Since FEM is used in the calculation of air gap permeances, TCM–FEM results for air gap permeance P are identical, but the next-part machine models are compared. All values in Fig. 13 are extracted from the same TCM model. Although only a part of the air gap is modeled, permeance values can easily be used in other parts of the machine as well, i.e., the machine periodicity does not affect the TCM periodicity in Pgap calculations. The periodic character of Pgap (θ) is seen in Fig. 13(b) as a traveling wave. Semidiscrete Versus Discrete: Air gap permeances play an important role to determine the phase flux linkage and torque capabilities of the machine. These machine characteristics are highly sensitive to the correctness of air gap permeances. Using the results obtained for Pgap (θ) by the SC, TCM, and FEM methods, phase flux linkages are compared with each other. A good agreement is achieved in Fig. 14 among these spatial discretization methods. The 1.8% difference between SC and TCM is caused by different Pgap (θ) values. The higher difference between SC and FEM (also between TCM and FEM) is due to the reluctance network model previously shown in Fig. 8. At the lower values of phase flux linkage, there is a very good agreement between SC and FEM; however, the values deviate


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calculation between SC and FEM is 2.9%, whereas the error for cogging torque calculation between TCM and FEM is 5.1%. Note that the error values given here are only observed due to the reluctance network model of the FSPM but not due to the differences in the air gap permeance calculations Pgap (θ). Results of Pgap (θ) calculation are, for TCM and FEM, identical, whereas the difference between SC and FEM is only 0.8%. Among three methods, to calculate cogging torque and phase flux linkage, FEM requires the longest computational time of 15 min, since it requires remeshing of the double salient air gap for each rotor step and FSPM is only half periodical. TCM has a shorter computational time with 13.2 min compared to FEM, because its calculations partially rely on FEM. The fastest method is with 12-min SC mapping, which solely runs in MATLAB. With the proposed methods, engineers do not have to rely, for each nonlinear step, on FEM simulations in the early design phase. For optimization, these methods can be used up to a certain level; however, it should be noted that, for motors with a high saliency ratio, discretization of SC mapping may lead to a longer than expected simulation time.

IV. C ONCLUSION

Fig. 13. (a) Air gap permeances between rotor teeth P1–P5 and stator tooth f 1. (b) Permeance functions Pgap (θ) calculated by TCM.

Double salient PM machines show a nonlinear magnetic behavior combined with the slotted air gap structure, which restricts the use of known analytical methods. Instead of aiming machine modeling techniques, this paper has focused on the issue of air gap permeances. As spatial discretization techniques, SC mapping, TCM, and FEM have been presented in this paper. Detailed explanations are given on how to use the techniques to calculate air gap permeances on an example double salient machine. A three-phase 12/10 flux-switching PM (FSPM) motor is chosen as the example due to its numerous advantages in vehicle applications. Results have shown that the SC method is the fastest, with a minor error of 0.8%. For the SC mapping, a geometrical correction factor formula is presented, which can easily be adapted for other electrical machines. Although the spatial discretization technique FEM is time optimized, both TCM and SC have shown an improvement in simulation time by 20%. R EFERENCES

Fig. 14. Comparison of discretization methods calculated with SC, TCM, and FEM for (a) phase flux linkage and (b) cogging torque calculation.

at some rotor positions. Whereas the mean error is considerably lower, for the phase flux linkage, TCM deviates maximum by 9.7% from FEM results and SC by 7.3%. The cogging torque calculations show a better agreement in Fig. 14(b), where the deviations are mainly observed in the second peak of each period. The error for cogging torque

[1] A. Zulu, B. Mecrow, and M. Armstrong, “A wound-field three-phase fluxswitching synchronous motor with all excitation sources on the stator,” IEEE Trans. Ind. Appl., vol. 46, no. 6, pp. 2363–2371, Nov./Dec. 2010. [2] C. Pollock, H. Pollock, R. Barron, J. Coles, D. Moule, A. Court, and R. Sutton, “Flux-switching motors for automotive applications,” IEEE Trans. Ind. Appl., vol. 42, no. 5, pp. 1177–1184, Sep./Oct. 2006. [3] J. Tapia, F. Leonardi, and T. Lipo, “Consequent-pole permanent-magnet machine with extended field-weakening capability,” IEEE Trans. Ind. Appl., vol. 39, no. 6, pp. 1704–1709, Nov./Dec. 2003. [4] E. Sulaiman, T. Kosaka, and N. Matsui, “Parameter optimization study and performance analysis of 6s−8p permanent magnet flux switching machine with field excitation for high speed hybrid electric vehicles,” in Proc. 14th EPE, Sep. 1, 2011, pp. 1–9. [5] H. Pollock, C. Pollock, R. Walter, and B. Gorti, “Low cost, high power density, flux switching machines and drives for power tools,” in Conf. Rec. 38th IEEE IAS Annu. Meeting, Oct. 2003, vol. 3, pp. 1451–1457. [6] N. Lobo, E. Swint, and R. Krishnan, “ M -phase N -segment flux-reversalfree stator switched reluctance machines,” in Conf. Rec. IEEE IAS Annu. Meeting, 2008, pp. 1–8.

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[7] E. Sulaiman, T. Kosaka, and N. Matsui, “Design and performance of 6-slot 5-pole PMFSM with hybrid excitation for hybrid electric vehicle applications,” in Proc. IPEC, Jun. 2010, pp. 1962–1968. [8] K. Chau, C. Chan, and C. Liu, “Overview of permanent-magnet brushless drives for electric and hybrid electric vehicles,” IEEE Trans. Ind. Electron., vol. 55, no. 6, pp. 2246–2257, Jun. 2008. [9] B. Gysen, E. Ilhan, K. Meessen, J. Paulides, and E. Lomonova, “Modeling of flux switching permanent magnet machines with Fourier analysis,” IEEE Trans. Magn., vol. 46, no. 6, pp. 1499–1502, Jun. 2010. [10] L. Huang, H. Yu, M. Hu, J. Zhao, and Z. Cheng, “A novel flux-switching permanent-magnet linear generator for wave energy extraction application,” IEEE Trans. Magn., vol. 47, no. 5, pp. 1034–1037, May 2011. [11] E. Ilhan, B. Gysen, J. Paulides, and E. Lomonova, “Analytical hybrid model for flux switching permanent magnet machines,” IEEE Trans. Magn., vol. 46, no. 6, pp. 1762–1765, Jun. 2010. [12] E. Ilhan, J. Paulides, L. Encica, and E. Lomonova, “Tooth contour method implementation for the flux-switching PM machines,” in Proc. 19th ICEM, Sep. 2010, pp. 1–6. [13] Z. Zhu, Y. Pang, D. Howe, S. Iwasaki, R. Deodhar, and A. Pride, “Analysis of electromagnetic performance of flux-switching permanentmagnet machines by nonlinear adaptive lumped parameter magnetic circuit model,” IEEE Trans. Magn., vol. 41, no. 11, pp. 4277–4287, Nov. 2005. [14] C. Stuebig and B. Ponick, “Determination of air gap permeances of hybrid stepping motors for calculation of motor behaviour,” in Proc. 18th ICEM, Sep. 2008, pp. 1–5. [15] M. Hesse, “Air gap permeance in doubly-slotted asynchronous machines,” IEEE Trans. Energy Convers., vol. 7, no. 3, pp. 491–499, Sep. 1992. [16] V. I. A. , L. Strete, and I. F. Soran, “Analytical flux linkage model of switched reluctance motor,” Rev. Roum. Sci. Tech.—Electrotech. Energy Bucarest, vol. 54, no. 2, pp. 139–146, Sep. 2009. [17] E. Panddurariu, L. Somesan, I.-A. Viorel, C. Martis, and O. Cornea, “Switched reluctance motor analytical models, comparative analysis,” in Proc. 12th OPTIM, May 2010, pp. 285–290. [18] K. Chau, M. Cheng, and C. Chan, “Nonlinear magnetic circuit analysis for a novel stator doubly fed doubly salient machine,” IEEE Trans. Magn., vol. 38, no. 5, pp. 2382–2384, Sep. 2002. [19] D. Krop, E. Lomonova, and A. Vandenput, “Application of Schwarzchristoffel mapping to permanent-magnet linear motor analysis,” IEEE Trans. Magn., vol. 44, no. 3, pp. 352–359, Mar. 2008. [20] T. Driscoll and L. Trefethen, Schwarz–Christoffel Mapping: Cambridge Monogrpahs on Applied and Computational Mathematics. Cambridge, U.K.: Cambridge Univ. Press, 2002. [21] [Online]. Available: http://www.math.udel.edu/~driscoll/SC/ [22] C. Gerada, K. Bradley, M. Sumner, P. Wheeler, S. Pickering, J. Clare, C. Whitley, and G. Towers, “The mesh do the result,” IEEE Ind. Appl. Mag., vol. 13, no. 2, pp. 62–72, Mar./Apr. 2007. [23] D. Petrichenko, M. Hecquet, P. Brochet, V. Kuznetsov, and D. Laloy, “Design and simulation of turbo-alternators using a coupled permeance network model,” IEEE Trans. Magn., vol. 42, no. 4, pp. 1259–1262, Apr. 2006. [24] T. Ramonosoa, J. A. Farooq, A. Djerdir, and A. Miraoui, “Reluctance network modelling of surface permanent magnet motor considering iron nonlinearities,” Energy Convers. Manage., vol. 50, no. 5, pp. 1356–1361, May 2009. [25] M. Kremers, E. Ilhan, D. Krop, J. Paulides, and E. Lomonova, “Reluctance network model for the in-wheel motor of a series-hybrid truck using tooth contour method,” in Proc. 14th Biennial IEEE Conf. Electromagn. Field Comput., Chicago, IL, 2010, p. 1. [26] D. Krop, L. Encica, and E. Lomonova, “Analysis of a novel double sided flux switching linear motor topology,” in Proc. 19th ICEM, Sep. 2010, pp. 1–5. [27] V. Kuznetsov, “General method of calculation of magnetic fields and processes in electric machines having discrete space-distribution windings,” Ph.D. dissertation, Moscow University, Moscow, Russia, 1990. [28] J. Stewart, Calculus Early Transcendentals, 4th ed. Pacific Grove, CA: Brooks/Cole, 1999. [29] V. A. Vyacheslav and P. Brochet, “Numerical modeling of electromagnetic process in electromechanical systems,” Compel, vol. 22, no. 4, pp. 1142–1154, 2003.

Esin Ilhan (S’09) was born in Istanbul, Turkey. She received the B. degree in electrical engineering from Istanbul Technical University, Istanbul, in 2007 and the M.Sc. degree in electrical engineering from Eindhoven University of Technology (TU/e), Eindhoven, The Netherlands, in 2009, with a full scholarship. She is currently with the Electromechanics and Power Electronics Group, TU/e. Her research interests include double salient PM machines for automotive applications.

Maarten F. J. Kremers (S’05) was born in Venray, The Netherlands, in 1984. He received the B.Sc. and M.Sc. degrees in electrical engineering from Eindhoven University of Technology (TU/e), Eindhoven, The Netherlands, in 2006 and 2010, respectively, where he is currently working toward the Ph.D. degree in the Electromechanics and Power Electronics Group. Since 2010, he has been with TU/e as a Researcher. His research activities are focused on permanent-magnetmachines.

Emilia T. Motoasca (M’06) was born in Romania. She received the M.Sc. degree in electrical engineering from Transilvania University of Brasov, Brasov, Romania, in 1996 and the Ph.D. degree from the Laboratory of Electromagnetic Research, Faculty of Information Technology and Systems, Delft University of Technology, Delft, The Netherlands, in 2003. She is currently an Assistant Professor with the Electromechanics and Power Electronics Group, Eindhoven University of Technology, Eindhoven, The Netherlands. Her research interests include numerical and analytical methods for electromagnetic field calculations, electrodynamics of deformable solids, biomedical applications of (micro)sensors, and actuators.

Johan J. H. Paulides (M’03) was born in Waalwijk, The Netherlands, in 1976. He received the B.Eng. degree from the Technische Hogeschool’sHertogenbosch, ’s-Hertogenbosch, The Netherlands, in 1998 and the M.Phil. and Ph.D. degrees in electrical and electronic engineering from The University of Sheffield, Sheffield, U.K., in 2000 and 2005, respectively. Since 2005, he has been a Research Associate with Eindhoven University of Technology, Eindhoven, The Netherlands. His research activities span all facets of electrical machines, particularly linear and rotating permanent-magnet excited machines for automotive and high-precision applications.

Elena A. Lomonova (SM’07) was born in Moscow, Russia. She received the M.Sc. (cum laude) and Ph.D. (cum laude) degrees in electromechanical engineering from Moscow Aviation Institute, Moscow, Russia in 1982 and 1993, respectively. She is currently a Professor with Eindhoven University of Technology, Eindhoven, The Netherlands. She has worked on the electromechanical actuators design, optimization, and development of the advanced mechatronics systems.