Abstract —In this paper, a nonlinear magnetic circuit network model for the analysis of a three-phase 12-stator-slot/10-rotor-pole flux-switching permanent-magnet (FSPM) motor is proposed considering localized saturation effect, which enables the predictions of open-circuit air-gap field distributions, phase flux-linkage, phase back-electro-motive-force (back-EMF) and winding inductances waveforms. Further, a developed model with bypass-bridges is also proposed, which enables the winding inductances to be predicted with more accuracy. The predicted results are confirmed by both finite element analysis and experimental measurements.

I. INTRODUCTION

Flux-switching permanent magnet (FSPM) machines as shown in Fig. 1 have been extensively investigated due to the high torque (power) density, essentially sinusoidal back-EMF waveforms, together with compact and robust structure since both the armature windings and magnets are located in stator instead of rotor as those in the conventional rotor-PM machines [1]-[13].

Conventionally, finite-element method (FEM) is widely used for analyzing the electromagnetic performances of electrical machines, especially for switched-reluctance machines and PM brushless machines due to the nonlinear saturation in iron, with the development of commercial software package, e.g., ANSYS, ANSOFT and FLUX, etc. However, compared with FEM-based analysis, traditional magnetic circuit-based analysis, such as nonlinear lumped parameter magnetic circuit (LPMC) [3] and nonlinear magnetic network model (NMNM) [14] are often preferred at the primary design stage due to the acceptable accuracy and considerable computational time saving. In [3], a nonlinear LPMC model was proposed for a 12-stator-slot/10-rotor-pole FSPM machine, in which the stator outer diameter and active stack length is 45mmand of 25mm, respectively. In addition, the rotor pole arc equal to that of the stator tooth arc. Recently, a hybrid model combining LPMC and the Fourier analysis is introduced in [12]-[13]. However, both the LPMC and hybrid models did not take the localized saturation effect into account and the inductances obtained by LPMC model slightly differ from those by FE analysis. From Fig. 2, it can be seen that the flux densities in the teeth of both stator and rotor irons are usually heavily saturated due to flux focusing effect.

Hence, in this paper a nonlinear magnetic network model which takes localized saturation effect into account is

proposed for a 12-stator-slot/10-rotor-pole prototyped FSPM motor [4], in which the stator outer diameter is 128mm and the stack length is 75mm. Additionally, the rotor pole arc is 1.4 times of that of �the stator tooth arc for more sinusoidal back-EMF and less cogging torque [4]. The detailed design dimensions and parameters are listed in Table I. Furthermore, since the leakage flux between stator tooth and back iron are significant when armature current is loaded as shown in Fig. 3, an improved model is proposed, in which the back iron and stator tooth in each U-shaped segment is divided into two serial ones between which a bypass-bridge is embedded to taking into account the leakage flux around. The predicted electromagnetic performances from both models are compared with FE analysis and validated by experimental measurements.

TABLE I DESIGN SPECIFICATIONS OF 12/10 FSPM MACHINE

Items Dimensions and Parameters Stator outer diameter (mm) 128Active stack length (mm) 75

Rotor inner diameter (mm) 22Stator tooth number 12Rotor pole number 10

Air-gap length (mm) 0.35 Split ratio (Dsi/Dso) 0.55 Split ratio (Dsi/Dso) 7.5

Slot opening (degree) 7.5Magnet thickness (degree) 7.5

Rotor pole arc (degree) 10.5 Rotor pole yoke width (degree) 21

(a)

Analysis of Flux-Switching Permanent-Magnet Machine by Nonlinear Magnetic Network Model

with Bypass-Bridges Gan Zhang1, Ming Cheng1, Wei Hua1

1School of Electrical Engineering, Southeast University, China E-mail: huawei1978@seu.edu.cn

(b)Fig. 1. A 3-phase 12-stator-slot/10-rotor-pole FSPM machine.

(a) cross-section, (b) topology.

1

.348E-04.385946

.7718571.158

1.5441.93

2.3162.701

3.0873.473

Fig. 2. Open-circuit flux density distribution.

Fig. 3. Flux distribution with phase-A loaded.

II. NONLINEAR MAGNETIC NETWORK MODEL

The basic equation which governs each element of the magnetic network model is given by,

FG

�� (1)

where, �, G, and F are the flux, permeance, and magneto-motive-force (MMF), respectively, and,

lSG r 0��� (2)

where, �r, �0, S, l are the relative permeability, the permeability of free space, cross-section area, and effective length, respectively. �r is determined by iteration from the B-H curve of the lamination material.

As shown in Fig. 3, the fluxes produced by magnets and armature currents are in parallel, and the variation of the flux

density in the magnets is very small. Thus, the permanent magnets can be simply modeled as an equivalent MMF [3],

mr

rm hBF

0��� (3)

and permeance,

m

amrm h

llG 0��� (4)

where, hm and lm are the magnet thickness and width, respectively, and Br is the magnet remanence.

Although the permeances of the ferromagnetic regions and the permanent magnets can be determined by equations (1)-(4), the permeances of the air-gap region are much more complicated, which is also the key to predict the air-gap flux density. As illustrated in Fig. 4, the flux paths in the air-gap region around each U-shaped stator segment are simplified, and the stator and rotor surfaces are assumed to be equipotentials [3]. Fig. 5 shows the typical simplified air-gap flux paths between the stator and the rotor, and equations for each corresponding permeance.

Thus, a complete nonlinear magnetic network model of the FSPM machine can be obtained. However, the model needs to be modified adaptively according to the rotor position. Fig. 6(a) shows the half initial original nonlinear magnetic network model at a typical rotor position. Furthermore, an improved nonlinear bypass-bridge magnetic network model is proposed as shown in Fig. 6(b), in which bypass-bridges are embedded in stator circuits. Consequently, the improved model enables the self- and mutual inductances to be calculated with more accuracy.

Compared with the flux paths in Fig. 7(a), in which the air-gap flux paths between stator and rotor poles are simplified as one branch, another parallel branch is added to take the localized saturation effect in iron into account when the rotor pole overlaps stator tooth as shown in Fig. 7(b). It should be emphasized that a localized saturation depth coefficient ksat is proposed in equation (5) to represent the deepness of localized saturation effect,

hhk sat

sat � (5)

where, hsat and h are the localized saturation depth and stator slot depth, respectively, as illustrated in Fig. 7(b). Since the localized saturation varies as the rotor rotates, ksat needs to be modified in accordance with localized saturation.

Furthermore, to improve predicting accuracy, localized saturation effect is accounted for in both nonlinear magnetic network models proposed previously. Hence considering localized saturation effect, two improved models termed as original nonlinear model (ONM) and bypass-bridge model (BBM) as shown in Fig. 8(a) and (b), which are derived from initial nonlinear magnetic network model and initial nonlinear bypass-bridge model respectively. The results are calculated by BBM and ONM considering localized saturation effect.

Fig. 4. Simplified flux paths in air-gap region.

Fig. 5. Typical air-gap permeances. (a) G1 = (�0LaX1)/( g), (b) G2 = (2�0La)/ (�)ln(1+(�X1)/ (�R1+2g)), (c) G3 =(�0La)/ (�)ln(1+(2�X1)/ (�(R1+ R2)+2g)), (d) G4 = (2�0La X1)/ (�(R1+ R2+

X1)+2g)), (e) G5 = 0.26�0La..

+-

+-

+ -

+-

+ -

+-

+-

(a)

+-

+-

+-

+ -

+-

+-

+-

Bypass-bridge

(b)Fig. 6. Initial nonlinear magnetic network models.

(a) Initial original-nonlinear model, (b) Initial bypass-bridge model

Fig. 7. Air-gap permenaces between stator and rotor pole. (a) Simplified condition, (b) Considering localized saturation effect.

(a)

Bypass-bridge

(b)Fig. 8. Nonlinear magnetic network models.

(a) Original-nonlinear model (ONM), (b) Bypass-bridge model (BBM)

III. PREDICTED ELECTROMAGNETIC PERFORMANCE

A. Air-gap filed distribution Fig. 9 shows the open-circuit air-gap field distributions at

two typical rotor position , namely (a) phase flux is zero @ �r=0o, (b) phase flux is maximum @ �r=9o from FE analysis. Obviously, the flux distribution is far from sinusoidal and exhibits significant harmonics, similar to that of switched reluctance machines due to doubly salient structure.

Fig. 10 compares the predicted air-gap flux density distributions at the two positions in Fig. 9. It can be seen that good agreements with FE analysis are achieved by both ONM and BBM, due to the considering of localized saturation effect.

(a)

(b)Fig. 9. Open-circuit field distribution.

(a) �r=0o, (b) �r=9o

-6

-4

-2

0

2

4

6

0 30 60 90 120 150 180Cita along gap (mechanical degree )

Flux

den

sity

( T

)

FEM ONM BBM

(a)

-6

-4

-2

0

2

4

6

0 30 60 90 120 150 180Cita along gap (mechanical degree )

Flux

den

sity

( T

)

FEM ONM BBM

(b)Fig. 10. Open-circuit air-gap field distributions by FEM, ONM and BBM.

(a) �r=0o, (b) �r=9o

B. Phase flux and back-EMF per turn The phase flux can be obtained directly predicted from

ONM and BBM, and the phase back-EMF determined from

dtde �

� (6)

Fig. 11 compares the predicted open-circuit coil fluxes in a complete rotor period, i.e., 36o in mechanical degrees by ONM, BBM and FEM analysis, respectively. It can be seen that the predicted results from both ONM and BBM are in good agreement with FE analysis.

Fig. 12 compares the predicted and measured back-EMF per turn waveforms at 1500 rpm on the prototyped FSPM motor in [4]. As it is seen, good agreements are achieved, verifying the effectiveness of the proposed magnetic network models.

-3

-2

-1

0

1

2

3

0 60 120 180 240 300 360Rotor position (electrical degree)

Flux

(mW

b)

Phase_A FEM

Phase_A ONM

Phase_A BBM

Fig. 11. Open-circuit phase flux waveforms.

-3

-2

-1

0

1

2

3

0 60 120 180 240 300 360Rotor position (electrical degree)

Back

-EM

F (V

) @

1500

rpm

Phase_A FEM

Phase_A ONM

Phase_A BBMMeasured

Fig. 12. Open-circuit phase back-EMF waveforms.

C. Inductances Fig. 12 compares predicted self- and mutual inductances

per turn from ONM, BBM and FE. As will be seen, the inductances calculated by ONM differ significantly from those by FE analysis as shown in Fig. 13(a), due to the simplification of leakage flux paths between stator tooth and back iron. However, as shown in Fig. 13(b), inductances predicted by BBM agree well with those by FE analysis since the implementation of bypass-bridge enables leakage flux between stator tooth and back iron to be accounted for especially when armature current loaded, indicating that the BBM is more suitable for predicting the electromagnetic performance of FSPM machines.

-2

-1

0

1

2

3

4

0 60 120 180 240 300 360Rotor position (electrical degree)

Indu

ctan

ce p

er-tu

rn (�

H)

Laa FEM Laa ONMMab FEM Mab ONMMac FEM Mac ONMLaa un-saturated FEM Laa un-saturated ONM

(a)

-2

-1

0

1

2

3

4

0 60 120 180 240 300 360Rotor position (electrical degree)

Indu

ctan

ce p

er-tu

rn (�

H)

Laa FEM Laa BBMMab FEM Mab BBMMac FEM Mac BBMLaa un-saturated FEM Laa un-saturated BBM

(b)Fig. 13. Predicted inductances waveforms.

(a) Inductances from FEM and ONM, (b) Inductances from FEM and BBM

IV. CONCLUSION

In this paper, n original and an improved nonlinear magnetic network model with bypass-bridge have been proposed to predict the electromagnetic performance, i.e., the air-gap distribution, the phase flux, the back-EMF and inductances for a FSPM machine. To enhance the accuracy of the predicted electromagnetic performance, localized saturation effect is taken into account in both nonlinear magnetic models. The predictions have been validated by both FEM calculations and experimental measurements.

ACKNOWLEDGMENT

This work was supported by National Natural Science Foundation of China (Project 50807007), the Specialized Research Fund for the Doctoral Program of Higher Education of China (Project 200802861038), the Fund Program of Southeast University for Excellent Youth Teachers, Southeast University Science Fund for Distinguished Young Scholars and a grant from the Key Technology R&D Program of Jiangsu Province, China (BE2009085).

REFERENCES

[1] S. E. Rauch and L. J. Johnson, “Design principles of flux-switching alternators,” AIEE Trans. 74III, vol. 74, no. 3, pp. 1261-1268, Jan. 1955.

[2] E. Hoang, A. H. Ben-Ahmed, J. Lucidarme, “Switching flux permanent magnet polyphased synchronous machines,” Proc. 7th Europe Conf. Power Electron and Applicat., vol. 3, pp. 903-908, 1997.

[3] Z.Q. Zhu, Y. Pang, D. Howe, S. Iwasaki, R. Deodhar, and A. Pride, “Analysis of electromagnetic performance of flux-switching permanent magnet machines by non-linear adaptive lumped parameter magnetic circuit model,” IEEE Trans. Magn., vol. 41, no. 11, pp. 4277-4287, Nov. 2005.

[4] Wei Hua, Ming Cheng, Z.Q. Zhu, D. Howe, “Analysis and Optimization of Back-EMF waveform of a Flux-Switching Permanent Magnet Motor,” IEEE Trans. Energy Conversion, vol. 23, no. 3, pp. 727-733, Sep. 2008.

[5] Wei Hua, Ming Cheng, “Static Characteristics of Doubly-Salient Brushless Machines Having Magnets in the Stator Considering End-Effect,” Electric Power Components and Systems, vol. 36, no. 7, pp. 754-770. July 2008.

[6] Z.Q. Zhu, J.T. Chen, Y. Pang, D. Howe, S. Iwasaki, and R. Deodhar, “Analysis of a novel multi-tooth flux-switching pm brushless ac machine for high torque direct-drive applications,” IEEE Trans. Magn., vol. 44, no. 11, pp. 4313-4316, Nov. 2008.

[7] J.T. Chen, Z.Q. Zhu, D. Howe, “Stator and rotor pole combinations for multi-tooth flux-switching permanent-magnet brushless ac machines,” IEEE Trans. Magn., vol. 44, no. 12, pp. 4659-4667, Dec. 2008.

[8] Wei Hua, Ming Cheng, Z.Q. Zhu, “Comparison of electromagnetic performance of brushless motors having magnets in stator and rotor,” Journal of Applied Physics, 2008, 103(7): 07F124.

[9] Z.Q. Zhu, D. Howe, “Electrical machines and drives for electric, hybrid, and fuel cell vehicles,” Proc. IEEE, vol. 95, no. 4, 2007, pp. 746-765.

[10] K.T. Chau, C.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.

[11] Y. Amara, E. Hoang, M. Gabsi, and et al, “Design and comparison of different flux-switch synchronous machines for an aircraft oil breather application,” Eur. Trans. Electr. Power, vol. 15, no. 6, pp. 497-511, Nov./Dec. 2009.

[12] E. Ilhan, B.L.J. Gysen, J.J.H. Paulides, and E.A. Lomonova. “Analytical hybrid model for flux switching permanent magnet machines,” IEEE Transactions on Magnetics, 46(6), pp: 1762-1765, 2010.

[13] B.L.J. Gysen, E. Ilhan, K.J. Meessen, J.J.H. Paulides, and E.A. Lomonova. “Modeling of flux switching permanent magnet machines with Fourier analysis, IEEE Transactions on Magnetics,” 46(6), pp: 1499-1502, 2010.

[14] Ming Cheng; Chau, K.T.; Chan, C.C.; Zhou, E.; Huang, X. “Nonlinear Varying-Network Magnetic Circuit Analysis for Doubly Salient Permanent-Magnet Motors, IEEE Transactions on magnetics,” vol. 36, no. 1, pp: 339-348, 2000.