2005 Analytical Modeling of an Axial Flux Permanent Magnet Synchronous Generator for Wind Energy...

8/14/2019 2005 Analytical Modeling of an Axial Flux Permanent Magnet Synchronous Generator for Wind Energy Application

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AbstractIn this paper, an analytical modeling of an axial flux

permanent magnet synchronous generator (AFPMSG) is

investigated. The proposed model is based on an exact twodimensional (2D) solution of the magnetic field in the generator.

Then, the generators quantities such as the phase electromotive

force (EMF), the cogging torque and electromagnetic torque are

written based on the developed exact 2D solution. To validate the

proposed modeling, a 2D finite elements analysis (FEA) is

performed, and results issued from the proposed model are

compared with those stemming from a 2D FEA simulations.

According to the simulation results, it is possible to evaluate the

performance of the AFPMSG with reasonable accuracy via the

developed analytical model.

I. INTRODUCTION

In recent years, wind energy conversion (WEC)

conventional systems with gearbox have been more and more

replaced by a WEC systems without gearbox. Since the WEC

systems with gearbox are subject to vibration, noise and

fatigue. The novel WEC systems which are achieved by

generators, directly driven from the wind turbine, offer other

advantages, namely higher overall efficiency and reliability,

reduced weight and maintenance cost [1]. The direct-drive

generators must have a large number of poles since the

rotational speed of the wind turbine is very low. Generators

with a large number of poles are easily achieved using

permanent magnet synchronous generator (PMSG). In the

PMSG family, AFPMSG can be an attractive solution if the

number of poles is large. The length of AFPMSG is short

compared to the radial flux PMSG [2-4]. The PM which are

used in the AFPMSG can be of a flat shape which is easy to

manufacture. There are many alternatives for the design of the

AFPMSG: N rotors and N+1 stators for internal rotor disc

machine type, or N stators and N+1 rotors for internal stator

disc machine type. They could be either slotted or slotless

(with a toroidal airgap windings) depending on the stator

structure.

The AFPMSM being investigated in this paper has a

double-sided topology (fig.1) and is dedicated to wind energy

applications. In order to develop this relatively new type of

generators, one needs an analytical model helping the designer

to predict the performances of this type of AFPMSM. So, in

this paper, an analytical modeling of AFPMSG is investigated.

Firstly, an exact 2D analytical model of the magnetic field in

the AFPMSG is developed. This exact 2D solution of themagnetic field is obtained using the separation of variables

method to solve the magnetic vector potential formulation

resulting from Maxwell equations. Secondly, the generator's

quantities are written using the obtained exact 2D solution of

the magnetic field. Thirdly, the three phase voltage of

generator is rectified and then connected to a resistive load.

The voltage equations are used to calculate the armature

current waveforms. Results issued from the proposed model

are compared with those stemming from a 2D FEA

simulations.

II. EXACT 2D ANALYTIC MAGNETIC FIELD SOLUTION

The geometry of the studied AFPMSG is presented in fig.

1. The rotor PMs are surface mounted, axially magnetized and

separated by nonmagnetized zones. The cross sectional view

in Fig. 1 shows the arrangement of airgap, rotor PMs and

stator along the axial direction. Taking into account the

geometry of this generator, its main flux is obviously three

dimensional. Since, the boundary conditions in the radial

direction (inner and outer radii) are very hard to choose in the

case of analytical approach. The first task is to reduce the 3D

problem to a 2D boundary-value problem which can be solved

using the separation of variables technique. This goal is

Analytical modeling of an axial flux permanent

magnet synchronous generator for wind energy

application

J. Azzouzi, G. Barakat and B. Dakyo

Electrical Engineering Department, GREAH, University of Le Havre

25, rue Philippe Lebon, 76600 Le Havre Cedex France, phone: (+332) 32744331, fax: (+332) 32744348,

e-mail: [email protected], [email protected], [email protected].

Lcs

Lw

Ri

La

Ro

e

Fig. 1. Cross sectional view of the studied AFPMSG

0-7803-8987-5/05/$20.00 2005 IEEE. 1255


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achieved by the use of cylindrical cutting plane (fig. 2) at the

main flux region. This approach consists in considering that

the radial component of the magnetic field is neglected and

the axial and circumferential components do not depend on

the radial direction. Consequently, the magnetic vector

potential has only a radial component which depends on the

circumferential and axial coordinates. The magnetic quantities

are then expressed as follows in cylindrical coordinates:

r

eA ),( zA= (1)

zeeB z),(Bz),(B z += (2)

The permeability of the magnet yoke and the armature are

assumed to be infinite. The permeability of PM is assumed to

be equal to that of air. Also, the remanence of PM is

considered to be ideal and oriented in the axial direction

yielding to the following Fourier series:

( ) ( ) ( ) kFkEB Mkk

MkR sincos += (3)

where is the circumferential. So, the magnetic field created

by the permanent magnets in the generators satisfies the basic

equations of magnetostatics

0. = B (4)

0= H (5)

and the constitutive relation between Band H:

Rrec BHB += (6)

where recis the recoil permeability and BRis the remanence

of PMs. Combining (5) and (6) yields to:

RBB = (7)

Using the fact that the magnetic flux density Bis equal to the

curl of a vector potential Aand choosing the Coulomb gauge

(div A= 0), equation (7) becomes:

RBA = (8)

Therefore, equation (8) is reduced to a scalar formulation and

can be written in the following form:

RB

Rz

AA

R

112

2

2

2

2 =+ (9)

Using the separation of variables technique, the vector

potential solution form in the PMs region can be written as:

( ) ( )( ) ( )( ) ( )( ) ( )

++

+++=

0

0

kMkkk

Mkkk

kERkzshFRkzchE

kFRkzshDRkzchC

k

raA

sin

cos(10)

and in the airgap as well as in the slots region:

( ) ( )( ) ( )( ) ( )( ) ( )

++

+

+=0

0

k kk

kk

kRkzshFRkzchE

kRkzshDRkzchC

k

raA

sin

cos(11)

The slotted AFPMSG is assumed to have Qs slots which are

numbered l=0,..., Qs-1. The generator is divided into three

regions: PM region, airgap and slots region (fig. 2). The origin

of the circumferential direction is arbitrarily chosen to be in

the center of the first slot.

A. Slots areaIn the slots, the magnetic field satisfies the following

boundary conditions:

( )( )( )

( )

+=

+==

sl

s

l

lz

QlwQlwforzB

22220,

2

1

(12)

wsl LzforzB == 0),(),( (13)

With these boundary conditions, the general solution for the

magnetic filed in the slots area can be expressed as:

( ) ( )

( ) ( ) ( )( )( )lm

wl

ml

wmwRzm

LwR

mz

wR

mfB

1

0

11

+

=

cossinh

tanhtanh(14)

( ) ( )

( ) ( ) ( )( )( )lm

wl

ml

z

wmwRzm

LwRmz

wRmfB

1

0

1

+

=

coscosh

tanhtanh(15)

B. Airgap: area IIIn this region, the general solution of magnetic field is:

( )( ) ( ) ( ) ( )( ) ( )

( ) ( ) ( ) ( )( ) ( )

++

+=

022

22

2

k kk

kk

kRkzchFRkzshE

kRkzchDRkzshCB

sin

cos(16)

( )( ) ( ) ( ) ( )( ) ( )

( ) ( ) ( ) ( )( ) ( )

+

+

= 022

22

2

k kk

kk

zkRkzshFRkzchE

kRkzshDRkzchC

B

cos

sin

(17)

C. PM region: area IIIThe boundary condition for this region is:

( ) ( ) ezB +== alzfor0,3 (18)

The general solution for the magnetic field in this region

where this boundary condition has already been accounted for

is:

( ) ( ) ( ) ( ) ( )( )

( ) ( )( ) ( )( )rkzrelkrkz

kEkCB

a

k

kk

coshtanhsinh

sincos

0

333

+

+=

(19)

III

II

I3I2I1

z

Fig. 2. Unrolled surface of a cylindrical cutting plane placed in the

main flux region.

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( ) ( ) ( ) ( ) ( ) ( )( )

( ) ( )( ) ( )( )RkzRelkRkz

kEkCBB

a

k

kkRz

sinhtanhcosh

cossin

+

+= 0

333 (20)

D. Boundary condition between areasBoundary conditions between regions are required to

determine the Fourier series coefficients of the magnetic field.

In the studied generator, the permeability of stator and rotor

core is assumed to be infinite. Then, the magnetic field

strength in iron is zero. Also, it is assumed that stator and

rotor iron as well as PM are non conductive materials. So, the

surface current density is equal to zero everywhere. Then, the

boundary conditions between airgap (region II) and PM area

(regionIII) are:

( ) ( ) ( ) ( )3232, HHandBBezFor zz === (21)

This boundary condition results in the following relations

between coefficients:

( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )

++=

=

+=

+=

RkeERelkEF

RkeEEE

RkeFRelkCD

RrkeFCC

Mkakk

Mkkk

Mkakk

Mkkk

sinhtanh

cosh

sinhtanh

cosh

32

32

32

32

(22)

Boundary conditions between regions I and II can be

expressed as follows:

( ) ( ) ( ) ( ) ( ) ( )llllzz ;forHHandBB 21

22 == (23)

and between regionII and stator core adjacent to it is:

( ) ( ) ( )112

2 0 += ll ;forH (24)

The treatment of boundary conditions (22), (23) and (24)

yields to a system of equations made up of relations between

the coefficients of Fourier of the magnetic field in the region

III.

( ) ( ) ( )( )

( )( ) ( ) ( ) ( )( )

( )

( ) ( )( )

( )( ) ( ) ( ) ( )( )

++

+=

++

+=

=

=

RneEEknRelak

RelakRkeEE

RneFCknRelak

RelakRkeFC

Mnn

n

M

kk

Mnn

n

Mkk

cosh,,tanh

tanhsinh

cosh,,tanh

tanhsinh

3

1

1

13

3

1

1

13

(25)

where

( )( )( )

( )( ) ( )( )[ ] ( )( )[ ]( )2211

2

122222222

2

wnkwkn

LwR

m

wnmwkm

Qkmwnk

m

m

ws

+

=

=

coscos

tanh,,

m

(26)

when knis multiple of 2pand zero otherwise.The Fourier series coefficients of the magnetic field in the

PMs region can be easily obtained by solving numerically theprevious system (25) and the airgap magnetic field

coefficients are deduced by means of (22).

III. COMPUTATION OF THE GENERATOR'S QUANTITIES

A. Computation of electromotive forceThe permanent magnet flux linking a coil j having nsturns

in series can be expressed in the cylindrical coordinates by

[5]:

( ) ( )

( ) ( ) ==

2

0

222

2dBF

RRSdBn zD

io

S

sc jj,. )(

rr(27)

where Sis the surface of the coil, is the angular position of

the rotor with respect to the stator frame, is the angular

position in the stator frame. FDj() designates the distribution

function of coil j. If b and es designate the coil openingand the slot opening respectively, the distribution function of

this coil is illustrated in fig. 3 and its expression is given by

[6]

( )

( )[ [

( ) ( )( )( ) ( )[ [

( ) ( )[ [

+

++

+

=

2;for

;for

;for

bb

bb

bse

b

esesssp

esess

esssp

ess

sD

n

n

F

2

22

2

20

(28)

The distribution function of the coil j can be written in the

form of Fourier series as follows:

( ) ( ) ( ) kFkEEFDk

k

Dk

DDj sincos ++=

0 (29)

Replacing Bz(2) by its expression (17), the airgap flux

embraced by the coiljin the linear case is then given by

( ) ( )

( ) ( ) ( )( ) ( ) ( )( )( ) ( ) ( )( ) ( ) ( )( )

++

+

=

022

22

22

2

kDk

Dkkk

Dk

Dkkk

ioj

kFkERkzshFRkzchE

kEkFRkzshDRkzchC

RR

sin.cos

sincos

.

)(

)((30)

So, the induced electromotive force in the coil j is obtained

by Faraday's law:

( ) ( )

( )( ) ( ) ( )( )( )( ) ( ) ( )( )

++

+

=

022

22

22

2

kDk

Dkkk

Dk

Dkkk

ioj

kEkFRkzshDRkzchC

kFkERkzshFRkzchE

RRe

cossin)()(

cos.sin)()(k

.

)(

)((31

-

FDj(S)

nspes

s

b

Fig. 3. Distribution function of coil 'j'.

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)

where is the angular speed of the rotor. If the stator phase is

constituted ofNccoils in series, the distribution function of the

stator phase is obtained by summing all the distribution

functions of the Nc coils. Also, the stator phase EMF is

obtained by summing the EMFs of theNccoils.

B. Computation of cogging torque:The cogging torque is calculated by means of the moment

of the Maxwell stress tensor applied to the rotor and can be

written in the following form:

( ) ( ) ( )( ) =S

zc dSeBeBrT ,, 33

0 (32)

where Sis the PMs surface and dS=rdrd .After the integration on the radial direction between the inner

and outer radii, the cogging torque expression becomes:

( ) ( ) ( ) ( )( ) =

2

03333

031 deBeBRRT zaiaec ,, (33)

Incorporating (19) and (20) in (30) and integrating on the

tangential direction yields to the final expression of the

cogging torque in terms of Fourier series coefficients of

regionIII :

( ) ( ) ( )

( ) ( )

kEEFC

kFEECRR

T

Mkk

Mkk

k

Mkk

Mkk

aiaec

sin

cos

++

+

= 00

33

3 (34)

where:

( ) ( ) ( )( ) ( )( )( ) ( ) ( )( ) ( )( )

+=

+=

rkerelkrkeEE

rkerelkrkeCC

akk

akk

coshtanhsinh

coshtanhsinh

3

3

C. Stator phases voltages equationsThe three phases voltages of generator are rectified and then

connected to a resistive load R. The rectifier is a three-phase

diode bridge. The diodes are considered ideals. Consequently,

the conduction of the diodes corresponds to a short circuit and

the blocking corresponds to an open circuit. With these

conditions, for each sequence, the two diodes which conduct

correspond to the phase having the most positive voltage for

the diode of the higher half-bridge and to the phase having the

most negative voltage for the diode of the lower half-bridge.

The rectifier voltage and the rectifier current are connected to

the phase voltages and the phase currents by means of a

commutation vector (whose elements are exposed in Table I)

written as follows :

[ ] [ ]TrrrRS = (35)

where the T subscript is the transpose of a vector. The

relations between the rectifier voltage, the rectifier current ,

the stator voltage and the rotor current are expressed as :

[ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ]Rt

RRR

tRR

tRR

VSVISi

VSViSI

;

;

2

1

2

1

==

==(37)

where [i ] = [ ia, ib, ic ]Tis the stator currents vector and [V]

=[ va , vb , vc]T is the stator voltages vector. So, the stator

phases voltages can be expressed in a vector-matrix form as

follows

[ ] [ ] [ ] [ ] [ ] [ ]idt

dLiReV s = (38)

where:

[ ]

=

c

b

a

e

e

e

e ; [ ]

=

s

s

s

LMM

MLM

MML

L ; [ ]

=

s

s

s

s

R

R

R

R

00

00

00

[e] the EMF vector, Rs the per-phase stator resistance, L the

main inductance of a stator phase and M is the mutual

inductance between two stator phases.

Combining (37) and (38), the differential equation governing

the generator-rectifier system can be written as [7]:

[ ] [ ] [ ]

[ ] [ ] [ ] [ ][ ]( )( ) RRstRtsR

tR

R

IRSRSeR

SLSdt

dI

+

=1

(39)

A. Computation of electromagnetic torque:The general electromagnetic torque equation is written as:

( ) ( ) ( ) cm

jjjem TtitEt += =1

1.

(40)

VR

R

iR

ea

eb

ec

RsLs ia

ib

ic

Fig. 4. Circuit diagram for the generator with rectifier model

Table I : Values of the commutation vector components

V+ V- r r rVa Vc 1 0 -1

Vb Vc 0 1 -1

Vb Va -1 1 0Vc Va -1 0 1

Vc Vb 0 -1 1

Va Vb 1 -1 0

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where Tcis the cogging torque, m is the number of phases, Ejand ijare respectively the EMF and the current of the phasej.

IV. RESULTS AND DISCUTION

The proposed analytical model is now used to study a 10

kW / 130 rpm, 28-poles, three-phase armature windingsaccommodated in 84 stator slots AFPMSG. The rotor PMs are

surface mounted, axially magnetized and the remanence is

1.25 T. The geometrical dimensions of the studied AFPMSG

are:Ri=0.31m andRo= 0.35m,La= 5.1mm, e = 1.3mm

and Lw = 65 mm. These values are issued from an

optimization design procedure based on the global equations

and using the genetic algorithm [8].

Figure 5 reports the comparison between the magnetic field

components in the airgap calculated by (16) and (17) and the

2D FEA simulation results. These magnetic field components

are calculated in the medium of airgap (z = e/2) and for an

arbitrary angular position (= 0) of the rotor with respect tothe stator. The effect of the slots is clearly visible. In figures 6

and 7, the computed values of the axial flux density are

compared to the FEA simulation results respectively on the

surface between the airgap and the PMs area (z= e) and in the

medium of PMs area (z = e+la/2). The effect of the slots

decreases progressively aszincreases.

In the three figures 5, 6 and 7, the computed axial flux values

are slightly lower than the corresponding FEA simulation

results in front of the slots. In spite of the small differences,

the results obtained from the presented analytical model can

be considered as largely satisfactory.

Figure 8 shows the three phases EMF waveforms obtained

using the analytical expression (31). Figure 9 gives the phase

stator current waveform when the three phase voltage of

generator is rectified and then connected to a resistive load.

Figure 10 shows the rectified current wave form. The

temporal variation of active power is presented in Fig. 11. The

fluctuation of in the power is due to the effect of

commutation.

V. CONCLUSION

In this paper, an analytical modeling of an AFPMSG is

investigated. The proposed model is based on an exact 2D

solution of the magnetic field in the generator. . This exact 2D

Fig. 8. Simulated EMF per phase for a 130 rpm versus the angular

position of the rotor with respect the stator frame

0 5 10 15 20 25-1.5

-1

-0.5

0

0.5

1

1.5

Stator surface

2D FEAAnalytical model

PMs remanence

Bz

Bphi

Fig. 5. Comparison between FEA results and analytical magnetic model

in the airgap for arbitrary angular position.

0 5 10 15 20 25-1.5

-1

-0.5

0

0.5

1

1.5

Analytical model

2D FEM

Fig. 6. Comparison between FEA results and analytical magnetic modelfor z = e and arbitrary angular position .

0 5 10 15 20 25-1.5

-1

-0.5

0

0.5

1

1.5

2D FEA

Analytical model

Fig. 7. Comparison between FEA results and analytical magnetic model

for z = e+la/2 and arbitrary angular position.

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solution of the magnetic field is obtained using the separation

of variables method to solve the magnetic vector potential

formulation resulting from Maxwell equations. Then, the

generators quantities such as the phase EMF, the cogging

torque and electromagnetic torque were written based on the

developed exact 2D solution. Finally, the proposed model was

used to study an AFPMSG dedicated to 10 kW / 130 rpmWEC system. the obtained results were compared with those

stemming from 2D FEA. According to the simulation results,

it is possible to evaluate the performance of the AFPMSG

with reasonable accuracy via the developed analytical model.

REFERENCES

[1] Anders Grauers, " Design of direct-driven Permanent Magnet

Generators for Wind Turbines" Technical Report No. 292 Chalmers

University of Technology, Sweden.[2] E. Spooner and B. J. Chalmers, "TORUS: A slotless, toroidal-stator,

permanent-magnet generator," IEE proceedings-B, vol. 139, no. 6,

November 1992, pp. 497-506.

[3] B. J. Chalmers, W. Wu, E. Spooner, "An axial-flux permanent magnet

generator for a gearless wind energy system," IEEE Trans. On energy

conversion, vol. 14, no. 2, June 1999, pp. 251-257.

[4] F. Caricchi, F. Crescimbini, F. Mezzetti, and E. Santini, "Design ant

testing of a small-size wind-photovoltaic system prototype,"European

community wind energy conference, Lbeck-Travemnde Germany, 8-

12 March 1993, pp. 740-743.

[5] J. Azzouzi, G. Barakat and B. Dakyo, "Analytical model for a design

approach of an axial flux permanent magnet synchronous machine for

wind energy application," inProc. 10thEuropean conference on Power

Electronic and applications (EPE 03), Toulouse, France, September

2nd- 4th, 2003.

[6] G. Houdouin, G. Barakat, B. Dakyo and E. Destobbeleer, A winding

function theory based global method for the simulation of faulty

induction machines, Proc. of the IEEE IEMDC'03, Madison,

Wisconsin, USA, June 2003.

[7] E. J. R. Sambatra, G. Barakat, B. Dakyo and X. Roboam, " Safety

Operation Locations of Permanent Magnets Synchronous Machine for

Stand Alone Wind Energy Converter," EPE 03, Toulouse, France,

September 2nd-4th, 2003.

[8] J. Azzouzi, G. Barakat and B. Dakyo, " Design optimization of axial

flux permanent magnet synchronous generator for direct-drive wind

energy application," 16th

international conference of electricalmachines (ICEM 04), Cracow, Poland, September 5-8, 2004.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160

5

10

15

20

25

Fig. 10. Rectified current waveform

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160

2000

4000

6000

8000

10000

12000

Fig. 11. Temporal variation of active power.

0.04 0.045 0.05 0.055 0.06 0.065-30

-20

-10

0

10

20

30

Fig. 9. Phase current waveform

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2005 Analytical Modeling of an Axial Flux Permanent Magnet Synchronous Generator for Wind Energy...

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