Design and Analysis of an Air-Cored Axial Flux Permanent Magnet Generator

for Small Wind Power Application B. Xia, M. J. Jin, J. X. Shen and A. G. Zhang

College of Electrical Engineering, Zhejiang University, Hangzhou, 310027, China E-mail: M_J_Jin@zju.edu.cn

Abstract-An axial flux permanent magnet (AFPM) machine with dual rotors and single air-cored stator is studied in this paper. The machine is applied for vertical-shaft small power off-grid wind generating system. A 2-dimensional (2D) finite element analysis (FEA) method is proposed to approximately solve to the 3-dimensional (3D) magnetic field inside the AFPM machine. And a 2D analytical method is also presented to compute the air gap flux density for this dual-rotor single-air-cored-stator axial-flux surface-mounted permanent magnet machine structure. Then, a 3D FEA model is developed to validate the accuracy of these two proposed 2D methods. Finally, a prototype machine is fabricated, and experiments are carried out to test its performances and electromagnetic features. Experimental results confirm that the 2D FEA and analytical approaches are both efficient and simple with sufficient accuracy.

I. INTRODUCTION

Wind power, considered as one of the cleanest renewable energies, is now receiving more and more attention. In some developing countries like China, with the supportive policies of the government, the utility of wind power is growing fast. Many wind power stations with large scale wind turbines have been built to provide electricity to the grid in places with good wind resources. However, in some remote but windy areas where grid is not available, small low-speed stand-alone high-efficiency wind generators can be very attractive for household electrical appliance as well as outdoor monitor equipments [1]. Axial flux permanent magnet (AFPM) machine, with its compact structure, flat shape, and high torque density, fits perfectly for these applications. In this paper, a 1kW, 300rpm, air-cored outer-rotor surface-mounted AFPM synchronous machine is studied as a direct-drive vertical-shaft wind generator (VSWG).

Due to its instinct structure, the flux inside an AFPM machine flows in both axial and circumferential directions. Therefore, 3-dimensional (3D) field analysis is necessary for precise resolving [2]. Though finite element analysis (FEA) software has been extensively used nowadays with development of computer technologies, 3D FEA modeling and simulation are still too complex and quite time-consuming, and even unworkable for low-grade computers. Thus, they are not very practical in optimizing designs. To overcome this problem, a 2-dimensional (2D) FEA method with sufficient accuracy is proposed in Section III, which simplifies the modeling and reduces the computing time.

Besides, analytical methods are also solutions to the computation of magnetic field. Some analytical methods for AFPM machines have been presented, in both 2D and 3D models for different AFPM machine topologies [3-5]. These solutions enable quick parametric analysis, yet have certain limitations since simplifications are made. A 2D analytical method is developed for double-rotor, single-air-cored-stator AFPM machine in Section IV.

3D FEA model is also set up to validate the proposed 2D FEA and analytical methods. Finally, a prototype machine is tested to validate both the 2D FEA and analytical methods and confirm the machine performance.

II. MACHINE STRUCTURE

The AFPM machine structure is showed in Fig. 1. It consists of two external rotors and one internal stator sandwiched between the two rotors. Air-cored stator structure is adopted to reduce the machine weight. Concentrated trapezoidal coils are assembled with non-magnetic non-conducting material epoxy resin. As a result, cogging torque is eliminated, which improves the dynamic characteristics [6] and gives the generator a much lower cut-in speed. Besides, stator core loss does not exist due to the absence of iron core, and the magnetic force between the stator and the rotor is minimized. Sector-shaped high energy sintered Nd-Fe-B magnets are glued onto the rotor back iron surfaces, positioned with aluminum frames which are not illustrated in Fig. 1. Magnets in red make the air gap flux flow downwards, whilst those in blue make the flux go upwards. Vertical-shaft

Fig. 1. Machine structure.

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wind impellers can be directly installed on the top rotor. This direct-drive vertical-shaft configuration captures wind energy from all directions, and the construction is simplified because the yaw, tower and gear box are unnecessary [7]. Consequently, efficiency and reliability are further improved.

III. 2D FEA MODELING

Since flux travels circumferentially in the rotor back iron and goes axially in the air gap and magnets, by employing radial cross section at different cutting radius (r) between outer radius (Ro) and inner radius (Ri) of the magnets, an entire flux path is obtained in the cross section. Then, by pulling the cylindrical cross section straight, a 2D FEA planar model is obtained. As the flux distribution is periodic in the circumferential direction and symmetrical in the axial direction, half of the machine in the axial direction and one electrical cycle (i.e., two pole-pitches) in the circumferential direction are required for the field resolving. Thus, the 2D FEA model can be simplified as that shown in Fig. 2. It should be pointed out that the model can be further simplified if only a quarter of electric cycle (viz., half a pole-pitch) is required by using symmetrical boundaries. Moreover, due to the air-cored topology, the air gap is relatively large, and armature reaction is low and negligible. As a result, coils are simplified. The 2D FEA model illustrated in Fig. 2 indicates its simplicity, and it is quite convenient to be modified and remodeled. With just one variable (the cutting radius of the cross section is chosen here), parametric models can be built. As the cutting radius (r) changing from Ri to Ro in a proper step, 2D FEA models are constructed to simulate the AFPM machine at different radius.

After the parametric analysis, the results obtained from 2D FEA are mapped back to the 3D model and a grid is formed by lines at different radius and electrical degree (θ), as illustrated in Fig. 3. The grid consists of elements and nodes and every element contains 4 nodes. The flux density at different nodes of the grid represents field distribution in the machine air gap, and can be easily computed from the 2D parametric FEA model. Assuming that the flux through an element is determined evenly by the 4 nodes, magnetic flux of each element is calculated by:

2/)(),( 221 θΔ⋅−= + ii rrjis (1)

4/),()]1,1()1,(),1(),([),(

jisjiBjiBjiBjiBji

⋅+++++++=ΔΦ

(2)

Concentrated trapezoidal armature coil windings are adopted in this machine. And for this slotless structure, the width of the coil band has to be considered when calculating the generated electromotive force (EMF). However, it will be too complicated to consider each conductor as the flux through each turn of coil is different. For convenience, each coil band is assumed to be a thin conductor at the middle position of the coil band. Then, the flux of the elements covered by the assumed coil is added up. And the back EMF can be calculated by:

dt

jide ji ⎥⎦

⎤⎢⎣⎡ ΔΦΣΣ

=),(

(3)

where s, r, Δθ, ΔΦ, B and e represent the area and the cutting radius of the element-i,j, and the angular step, the flux through element-i,j, and the back EMF in a single turn, respectively.

IV. 2D ANALYTICAL METHOD

Fig. 4 shows the 2D analytical solution model. Duo to the ironless stator structure, the effective air gap including the mechanical air gap and the height of coil winding is much

Fig. 2. 2D FEA model.

Fig. 3. Mapping relation of 3D and 2D models.

Fig. 4. 2D analytical model.

larger than that in slotted machines. Thus the rotor back iron has little magnetic saturation. Consequently, the relative permeability is assumed to be infinitely large for simplicity. Rectangular coordinate system is adopted here, and x-axis represents the distance in the circumferential direction, whilst y-axis represents the axial direction.

The scalar magnetic potential φ in all regions follows Laplace’s equation [8]:

02

2

2

2

=∂∂+

∂∂

yxϕϕ

(4)

The general solution to (4) is given by:

))(()]sinhcosh)(sin

cos[()]sin

cos()sinhcosh[(

0000

222

121

11

11

yDCxBAymDxmCxmB

xmAymD

ymCxmBxmA

nnnnnn

nnnnn

nnn

nnnn

+++++

++

+=

∑

∑∞

=

∞

=

ϕ

(5)

Equation (5) must satisfy the boundary conditions below: )2()( pxx τϕϕ += (6)

)(,0 LyH xIII == (7)

)0(,0 == yH xII (8)

)(, mxIIxI

yIIyI hyHHBB

=⎩⎨⎧

=

= (9)

)(, mxIIIxI

yIIIyI hLyHHBB

−=⎩⎨⎧

=

= (10)

where τp, hm and L denotes the pole-pitch, thickness of magnets and distance between the two rotor back irons.

The relation of magnetic field strength and scalar magnetic potential is given by:

⎪⎪⎩

⎪⎪⎨

⎧

∂∂−=

∂∂−=

yH

xH

y

x

ϕ

ϕ

(11)

And, the flux distribution density in region I is obtained as:

⎩⎨⎧

==

yIyI

xIxI

HBHB

0

0

μμ

(12)

Applying boundary conditions (6)-(10) to (11) and (12), using the general solution (5), the air gap flux density in axial direction is given by:

p

Ln

pn p

mryI

xne

yLnhnn

BB

p

τπ

τπ

τπ

ππτ

cos

)2/(coshsinh1[8

2/

⋅Λ

⋅

−⋅= ∑∞

(13)

p

Ln

pn p

mrxI

xne

yLnhnn

BB

p

τπ

τπ

τπ

ππτ

cos

)2/(sinhsinh1[8

2/

⋅Λ

⋅

−⋅= ∑∞

(14)

where

p

m

p

m

hn

p

mr

p

m

hLn

p

mr

p

m

ehnhn

ehnhn

τπ

τπ

τπμ

τπ

τπμ

τπ

)cosh(sinh

)cosh(sinh)(

−+

+=Λ−

(15)

n=1, 3, 5, … μ0 is the permeability of free space and μr is the relative recoil permeability of magnet.

V. 3D FEA VALIDATION

Since 2D FEA and analytical methods have not taken the end effect into account, a 3D FEA model is developed to validate these 2D methods. The 3D model and flux density of rotor back iron, magnets and coils are showed in Fig. 5. The maximum flux density is 1.6 T in the back iron. It proves the assumption that there is little saturation in the back iron.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

90 115 140 165 190

Cutting Radius(mm)

Flux

den

sity

(T)

θ=90 (3D FEA)θ=11.6 (3D FEA)θ=90 (2D FEA)θ=11.6(2D FEA)θ=90 (2D Anal.)θ=11.6 (2D Anal.)

Fig. 6. Air gap field distribution along radius direction, solved with 3D FEA, 2D FEA and 2D analytical model.

Fig. 5. Flux density distribution from 3D FEA.

The 3D FEA method is compared with the two proposed 2D methods in Fig. 6 and Fig. 7. Results of 2D FEA and 2D analytical method show great agreement with each other. Fig. 6 reveals that as the cutting radius of the cross section increases, the flux density in the air gap from 2D analysis increases. 3D FEA method validates the magnetic field distribution from 2D methods, except that the flux density actually decreases when getting very close to the inner and outer radius of the magnet. That is because the end effect becomes significant on the inner and outer edges of the magnets, but it cannot be reflected in the 2D models. Flux density from 3D FEA and 2D FEA, 3D FEA and 2D

analytical method are compared along circumferential direction at different radius in Fig. 7 (a) and (b) respectively. The results show that both 2D methods match the 3D FEA method. Overall, the magnet inner and outer regions are quite small, and we believe that magnetic field distribution computed from 2D methods can simulate the actual 3D flux distribution with small error due to the inner and outer regions of the magnets.

Since the 3D FEA method has validated the 2D FEA flux distribution, according to (1)-(3), the phase back EMF can be calculated based on the results from 2D FEA method. Waveforms of one-phase back EMF from the 3D and 2D FEA results are compared in Fig. 8, and it confirms that the 2D FEA method totally agrees with the 3D FEA method with an error less than 2%.

VI. PROTOTYPE MACHINE EXPERIMENTS

The prototype AFPM generator is built, Fig. 9. Design parameters of the machine are illustrated in TABLE I.

A test rig is then set up, where the generator is put in a horizontal position and driven with the geared induction motor, Fig. 10. For laboratory experiments, the induction motor is fed with a commercial programmable frequency converter, thus, its speed can be adjusted to simulate various wind condition.

To test the machine and verify the analysis methods proposed in this paper, the machine back EMF at rated speed (300 rpm) is measured in the experiment. Fig. 11 shows the line-line voltage of 2D FEA, 3D FEA and the prototype. The experimental waveform is similar to the 2D and 3D FEA results. The measured peak value is a little smaller than the FEA-predicted data, and the error is about 6%. This error is caused possibly by the model inaccuracy, the characteristic inconsistency of the permanent magnets, or the manufacturing imperfection. However, the error is acceptable. Thus, the 2D FEA method is validated to be simple, time-saving and sufficiently accurate. Moreover, as proved previously, the 2D analytical model agrees well with the 2D FEA, therefore, both 2D approaches, viz., 2D FEA and 2D analytical model, are verified by both 3D FEA and experiments.

VII. CONCLUSIONS

AFPM machines are among the most suitable candidates for direct drive small and medium wind power generators. In this paper, a dual-rotors and single-air-cored-stator AFPM generator for low-speed small wind power application is analyzed, using proposed 2D FEA and 2D analytical methods. 3D FEA model is also developed, validating both proposed 2D approaches. The machine back EMF is also calculated based on the 2D FEA methods. A prototype machine is tested. Experimental results are compared with the FEA results, confirming that the 2D approaches are simple, efficient and sufficiently accurate.

The proposed 2D FEA method can also be applied to other structures of AFPM machines with slotted stators and different shape magnets.

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 120 240 360

Angular distance(elect. deg)

Flux

den

sity

(T)

r=122.5(3D FEA)r=140(3D FEA)r=157.5(3D FEA)r=122.5(2D FEA)r=140(2D FEA)r=157.5(2D FEA)

(a)

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 60 120 180 240 300 360

Angular distance(elect. deg)

Flux

den

sity

(T)

r=122.5(3D FEA)r=140(3D FEA)r=157.5(3D FEA)r=122.5(2D Anal.)r=140(2D Anal.)r=157.5(2D Anal.)

(b) Fig. 7. Air gap field distribution along circumferential direction.

(a) 3D FEA and 2D FEA methods. (b) 3D FEA and 2D analytical method.

-250-200-150-100-50050100150200250

0 60 120 180 240 300 360

Electrical Degree

Phas

e ba

ck E

MF

(V)

2D FEA3D FEA

Fig. 8. One-phase back EMF from 3D FEA and 2D FEA.

REFERENCES [1] T. F. Chan and L. L. Lai, “An axial-flux permanent magnet

synchronous generator for a direct-coupled wind-turbine system”, IEEE TRANSACTIONS ON ENERGY CONVERSION, vol. 22, No.1, pp. 86-94, March 2007.

[2] Luca Del Ferraro, Roberto Terrigi and Fabio Giulii Capponi, “Coil and Magnet Shape Optimization of an Ironless AFPM Machine by Means of 3D FEA”. IEMDC '07. IEEE International, 2007.

[3] T. F. Chan, L. L. Lai and Shuming Xie. “Field Computation for an Axial Flux Permanent-Magnet Synchronous Generator”, IEEE TRANSACTIONS ON ENERGY CONVERSION, vol. 24 No.1, pp. 1-11, March 2009.

[4] Virtic, P., Pisek, P., Marcic, T., Hadziselimovic, M., and Stumberger, B., “Analytical Analysis of Magnetic Field and Back Electromotive Force Calculation of an Axial-Flux Permanent Magnet Synchronous Generator With Coreless Stator” IEEE Transactions on Magnetics, vol. 44, pp. 4333-4336, Nov. 2008.

[5] Yuriy N. Zhilichev, “Three-dimensional analytic model of permanent magnet axial flux machine”, IEEE Transactions on Magnetics, vol. 34, No. 6, pp. 3897-3901, November 1998.

[6] W.Z. Fei and P.C.K. Luk, “Design of a lkW High Speed Axial Flux Permanent-Magnet machine”, Electric Machines and Drives Conference, 2009.

[7] Ahmad M. Eid, Mazen Abdel-Salam and M. Tharwat Abel-Rahman, “Vertical-Axis Wind Turbine Modeling And Performance With Axial-Flux Permanent Magnet Synchronous Generator For Battery Charging Applications,” MEPCON 2006. Eleventh International Middle East, vol. 1, pp. 162-166, Dec. 2006.

[8] Z. Q. Zhu, David Howe, Ekkehard Bolte, and Bernd Ackermann, “InstaInstantaneous Magnetic Field Distribution in Brushless Permanent Magnet DC Motors Part I: Open-Circuit Field”, IEEE Transactions on Magnetics, vol. 29, pp. 124-135, Jan. 1993.

TABLE I MAIN DESIGN PARAMETERS

Parameter Value Unit Stator coil number 21

Rotor pole number 28

Pole embrace 0.89

Magnet outer radius 175 mm

Magnet inner radius 105 mm

Magnet thickness 7.5 mm

Thickness of back iron 6 mm

-400

-300

-200

-100

0

100

200

300

400

0 60 120 180 240 300 360

Electrical Degree

No-

load

Lin

e V

olta

ge (V

)

2D FEA3D FEAExperimental

Fig. 11. Line-line back EMF of 3D FEA, 2D FEA and prototype. Fig. 9. Prototype AFPM Machine.

Fig. 10. Measurement Setup for Prototype Machine.