Analytical Modeling of the Magnetic Field in Axial Field Flux-Switching

Permanent Magnet Machines at No-Load

Mouheb Dhifli*, Habiba Bali**, Yanis Laoubi*, Georges Barakat*, Yacine Amara*

*GROUPE DE RECHERCHE EN ELECTROTECHNIQUE ET AUTOMATIQUE DU

HAVRE. 75, rue bellot 76058

Le Havre, France

Tel.: +33 / (0) – 2.32.85.99.53

E-Mail: mouheb.dhifli@etud.univ-lehavre.fr

** UNIVERSITE DE JIJEL, FACULTE DES SCIENCES ET DE LA TECHNOLOGIE,

DEPARTEMENT DE GENIE ELECTRIQUE. ALGERIE

BP 98, Ouled Aissa 18000

Jijel Algeria

Keywords

« axial field flux-switching machines », « permanent magnets », « analytical solution », « axial flux-

switching », « finite element analysis », « electromagnetic analysis ».

Abstract

In this paper an analytical model for prediction of the no-load magnetic field in axial field flux

switching permanent magnet machines (FSPM) is presented. The proposed method is based on an

exact 2D analytical solution of Maxwell‟s equations in low permeability regions (stator/rotor slots,

air gap, permanent magnets (PMs) and surrounding air regions), using the separation of variables

technique. For each region, the magnetic vector potential solution is expressed as a sum of Fourier

series. Combining obtained solutions in each region and boundary conditions leads to a set of linear

equations which allows calculation of the coefficients of the Fourier series in the different regions. The

developed model is validated by comparing magnetic field components for a machine modeled with

this analytical model and the finite element method (FEM).

I. Introduction

In the automotive industry , demand for electromechanical machines which have a high torque density

and a high and variable speed capability is increasing [3]. The flux switching permanent magnet

(FSPM) machine is a good candidate since it combines the advantages of a switched reluctance

machine (high speed and robust rotor structure) and a brushless permanent magnet machine as shown

in Fig. 1 (high torque density) [1][2].

Fig. 1 The advantages of the SRM (a) and the PMSM (b) are embodied in a FSPM (c).

In the literature, this machine is often modeled by FEM or with the magnetic equivalent circuit (MEC)

model due to the nonlinear behavior and the doubly salient topology [1][3][4][5]. A hybrid model

which combines the MEC and the Fourier analysis exist too [3][4][6].

In fact MEC model allows the prediction of the electromagnetic performance of a AFSPM machine

but it suffers from coarse discretization of the resulting field solution, whereas the FEM has the

disadvantage of a long computational time [3].

In this paper, analytical modeling of AFSPM machine is presented as a solution to solve the

disadvantages of the MEC model and to reduce computation time compared to FEM. This technique

is based on an exact 2D solution of Maxwell equations using the separation variables method and

Fourier analysis. The 2D configuration is set by unrolling the curved surface obtained by using a

cylindrical cutting plane at the main flux region of the studied machine [7]-[9]. The topology of the

machine is given together with a brief definition of the working principle in section II.

The magnetic field distribution is calculated accurately in the different regions of the AFSPM machine

(stator/rotor, slots, air gap, PMs and the surrounding air regions) with a reasonable computation time.

Results issued from the proposed model, for both local and global quantities, are compared and

validated with those stemming from a 2D FEM. Furthermore, this model could be an effective tool to

explore a set of solutions in the early stages of the AFSPM machine design procedure.

II. AFSPM MACHINE

A. Topology

The geometry of the studied AFSPM machine 12/10-pole is presented in Fig. 2, which is a double-

sided structure [1][10]. The stator includes armature coils and twelve permanent inductor magnets, the

rotor is completely passive which allows to operate at very high speed. This three-phase machine

comprises four magnets and four concentrated coils per phase, while the rotor contains ten teeth

(Nr = 10). For a better understanding of the FSPM, it is mandatory to understand the underlying

physical phenomena inside the machine.

Fig. 2: Topology of the 12/10 AFSPM machine Fig. 3: Working principle of the AFSPM machine

B. Flux switching principle

The stator part of the FSPM machine consists of 12 modular U-shaped laminated segments between

which are placed permanent magnets (PMs). The magnetization is reversed in polarity from one

magnet to the next [10][11]. The principle of flux-switching has been shown in Fig.3. According to the

position of the mobile part, the magnetic flux-linkage in the armature winding switches direction and

can be counted as either positive or negative, and is then alternative. The working principle is similar

to the conventional radial field FSPM [3][4][12][13].

III. COMPUTATION OF THE EXACT 2D ANALYTICAL MAGNETIC

FIELD DISTRIBUTION

A. Assumptions

Stator 1

Stator 2

Windings

PMs

Stator and rotor

configuration

Stator 1

Direction of movement

Stator 2

Rotor tooth

Knowing the 3D nature of axial flux machine, the first geometrical assumption is to reduce the 3D

problem to a 2D one. The 2D analytical model is less time consuming and allows to take into account

the 3D intrinsic nature of the magnetic field in the axial flux machine by the multi-slice technique. The

main assumption of the analytical model is that the permeability of the soft-magnetic material is

considered infinite. The permeability of PMs is assumed to be equal to that of air and their resistivity

is infinite. Also, eddy current effects are neglected [7][8]. This goal is achieved by the use of a

cylindrical cutting plane at the mean radius and then the derived plane is unrolled to obtain the surface.

According to the AFSPM topology, half of the machine needs to be modeled [3][12].

In this approach the magnetic vector potential has only a radial component which depends on the axial

and circumferential coordinates. Therefore, in cylindrical coordinates the magnetic quantities can be

considered as follows:

rezAA , (1) zz ezBezBB ,, (2)

The second part is to model the machine in a Cartesian plane (x, y) as shown in Fig. 4, without unduly

compromising accuracy [6]. The studied machine is symmetrical over a period noted per and have a

period length given by (4), where moyR is the average radius:

sper Np,gcd2 (3) permoyper RL (4)

Fig. 4: FSPM machine model in Cartesian coordinates.

The magnetic quantities are then expressed as follows:

and (5)

B. Exact 2D analytical magnetic field solution

The FSPM machine model in Cartesian coordinates has different field regions (rotor slots (I), air gap

(II), PMs (III), stator slots (IV) and surrounding air (V)). The governing field equations, in terms of

the Coulomb Gauge, , is given by:

(6)

Therefore, equation (6) is reduced to a scalar formulation and can be written in the following form:

x

B

y

B

y

A

x

A yRxR

2

2

2

2

(7)

The vector potential solution for every region can be written by using the separation of variables

technique [9][14] as:

xx e

y

AB

yy e

x

AB

0 A

IIIregioninBA

VandIVIIIregionsinA

R2

2 ,,0

wes wa

wer hdr

hes

ha

e

(0,0) x

y

I

II

III

IV

V

0,3,3

,3,3,3

0,3

0,3

sinhsincos

coshsincos1,

kl

kl

k

lk

lklll

kykxFkxD

kykxEkxC

kyDayxA (8)

0

0 sincossinhcosh1

,

n

ni

nni

nni

nni

nn

ii xkFxkEykDykCk

ayxA (9)

note that i represents the regions I, II, IV and V. The potential vector is the derived to find field

expressions.

To facilitate post processing of the machine performance (local and global quantities), the following

change of variable is made: *rx and zy , note that 𝑟 = moyR is the average radius.

Therefore, equation (7) can be written in the scalar form in cylindrical coordinates as follows:

RB

rz

AA

r

112

2

2

2

2 (10)

The rotor coordinate system that refers to stator coordinate system is considered and defined by

relation (11), where 𝑟 = moyR is the average radius and d is the relative displacement of the rotor:

ezzR and dR (11)

Therefore, the vector potential solution for each region can be written as:

0,3,3

,3,3,3

0,3

0,3

sin

cos,

ml

ml

m

lm

lmlll

mprmpzshFrmpzchE

mprmpzshDrmpzchC

mp

rzDazA

(12)

0

0sin

cos,

ni

ni

n

in

inii

nprnpzshFrnpzchE

nprnpzshDrnpzchC

np

razA

(13)

where i represents the regions I, II, IV and V.

Magnetic field vector components are deduced from the magnetic vector potential zA i , as

shown:

A

rBz

1and

z

AB

(14)

Field components expressions will be derived for each region in the following subsections.

C. Region I (rotor slots regions)

Boundary conditions to be satisfied by the magnetic field components in region I in slot „i‟, are:

erdlld

derdl

Rl

zw

NlwforzB

1,

,122

0, (15)

(16)

Where 1,,0 dNl , pNNN srd ,gcd , and p is the pole pair number.

The general expressions of the magnetic field components in region I are given by:

ehzforzB drRl 0,,1

0

,1,1

0

,1,1

sin,

cos,

m

dlR

erdr

er

er

drR

lmRR

lz

m

dlR

erdr

er

er

drR

lmRR

l

w

m

hwr

mch

wr

hzmch

zB

w

m

hwr

mch

wr

hzmsh

zB

(17)

D. Region II (air gap region)

In this region, the general solution of the magnetic field is:

npr

npzshF

r

npzchEnp

r

npzshD

r

npzchCzB

npr

npzchF

r

npzshEnp

r

npzchD

r

npzshCzB

nn

n

nnz

nn

n

nn

cossin,

sincos,

22

0

222

22

0

222

(18)

E. Region III (permanent magnet region)

The boundary condition for this region is given by, where and :

(19)

The general solution for the magnetic field components in this region is:

0

,3,3,3

0

,3,3,30

,3

sin,

cos,

m

alaa

lm

a

lm

lz

m

alaa

lm

a

lm

ll

w

mz

rw

mshDz

rw

mchCzB

w

mz

rw

mchDz

rw

mshCDzB

(20)

F. Region IV (stator slots regions)

Regions IV are composed of Nm regions (which correspond to half of stator slots). Boundary

conditions to be satisfied by the magnetic field components are listed in (21), where and

:

esl hzforzB 0,,4 (21)

General expressions of magnetic field components can be written by taking into account these

boundary conditions:

1,,0 mNl pNNN ssm ,gcd

adlla

maallz

w

NlwforzB

1,

,322

0,

1,,0 mNl

esdsas www 2

eselle

sdsa

ellz

w

lww

forzB

1,

,420,

0

,4,4

0

,4,4

sin,

cos,

m

el

es

es

es

es

es

lm

lz

m

el

es

es

es

es

es

lm

l

w

m

hwr

mch

wr

hzmch

GzB

w

m

hwr

mch

wr

hzmsh

GzB

(22)

G. Region V (surrounding air)

The boundary condition for this region can be expressed by setting the magnetic field components to

zero at z . These magnetic field components can be written as:

0

555

0

555

cossinexp,

sincosexp,

n

nnz

n

nn

npnFnpEzr

npzB

npFnpEzr

npzB

(23)

H. Boundary condition between areas

In order to compute the Fourier series coefficients of the magnetic field, boundary conditions between

regions (interface conditions) are required. In the studied machine, the magnetic field strength in iron

is zero since the permeability of stator and rotor core is assumed to be infinite. Also, it is assumed that

the stator and the rotor iron as well as PMs are not conductive materials. So, the surface current

density is equal to zero everywhere. Then, the boundary conditions between region I (rotor slots

regions) and region II (air gap region) are:

1,,0,0,

,0, 1,

2,1

2,1

d

ldddl

zdl

z

dl

Nlfor

eBB

eHH

(24)

12 0, dlddlforeH (25)

The use of the last boundary conditions yields to a system of equations made up of relations between

Fourier coefficients of the magnetic field in the air gap region. The next boundary conditions that will

be exploited between regions II, III (stator slots regions) and IV (PMs regions), can be expressed as

follows:

1,,00,0,

0,0, 1,

,32

,32

m

laal

lzz

l

Nlfor

BB

HH

(26)

1,,00,0,

0,0, 1,

,42

,42

m

leel

lzz

l

Nlfor

BB

HH

(27)

11,

1,2 00,alle

ellaforH

(28)

This step helps establish relations between coefficients of region II with those of region III and region

IV. Final step consists in the exploitation of the interface conditions between regions III and V and can

be written as follows:

1,,0,,

,, 1,

,35

,35

m

laal

al

zaz

al

a

Nlfor

hBhB

hHhH

(29)

11,5 0, allaa forhH (30)

The treatment of the aforementioned boundary conditions leads to a set of unknown variables for the

magnetic field expressions. However, to evaluate the unknown coefficients additional boundary

conditions are necessary . These can be obtained by using the continuity of the magnetic vector

potential property firstly between region II and region III, then between region III and region V [9].

This approach can be obtained by setting one of the arbitrary coefficients found in Eq. 9 to zero.

Consequently, system of linear equations is obtained and the unknowns are coefficients of the

magnetic field in regions II and V. Solving this linear equation and using interface conditions give

coefficients of magnetic vector potential in other regions.

IV. GLOBAL QUANTITIES COMPUTATION

Computation of the cogging torque, flux linkage, EMF allows the evaluation of the machine's

performance. In the next subsections global quantities will be evaluated.

A. Electromotive force calculation

The EMF is computed using Faraday‟s law based on the time variation of magnetic flux as given by

[14]-[16]:

t

cjje

(31)

The cj is the PM flux linking a coil “j” with tn turns in series and can be expressed in the

cylindrical coordinates by:

2

0

222

,2

dBFRR

ndsBn dzDjio

t

s

tcj (32)

where S is the surface of the coil, d is the angular position of the rotor with respect to the stator

frame and is the angular position in the stator frame.

DjF designates the distribution function of a coil “j”. It can be expressed using Fourier series as

follows [16]:

0

0 coscosn

jDn

jDn

jDDj npFnpEEF (33)

B. Cogging torque computation

The cogging torque is calculated by applying Maxwell stress tensor method and can be expressed as

follows:

ia0

2

0

222

0

,,1

o

i

R

R

zC ddreBeBrT (34)

Where 2

zB and 2

B are respectively the normal and tangential magnetic field components in region II

Ri and Ro are respectively the outer and the inner radius.

After integration, the cogging torque can be given by:

(35)

In the next section, the electromagnetic performance of the 12/10-pole FSPM machine will be

predicted by using the foregoing model and 2D FEA.

V. Results and discussion

The magnetic field distribution is calculated for a disc type axial flux switching permanent magnet

machine which main parameters are given in Table I and verified with 2D FEA. This analysis has been

done with a relative permeability of 1∗ 𝑒5 for ferromagnetic cores.

Therefore the derived analytical expressions of the field distribution have been validated by finite-

element calculations of the tangential and normal magnetic field components in various regions of the

machine.

The solution in the centre of the air gap for a given position of the rotor relatively to the stator is

shown in Fig.5 where excellent agreement is obtained. The magnetic field components are also

compared for both methods in the external air region as shown in Fig. 6. For the rotor slots and the

stator slots a comparison between analytical model and FEM results shows small differences as

presented in Fig. 7 and Fig.8. This can be explained by meshing irregularities and poor accuracy of

the model inside slots. Despite these differences, it can be seen that the analytical prediction agrees

well with the finite-element solution in the slot opening, air gap and external air regions. the

comparison between the analytically predicted and finite-element calculated magnetic field

components in PMs region is presented in Fig.9. As seen, fairly good agreement is again achieved.

0

2222

0

22

3n

nnnnio

C EDFCRR

T

Fig.5: Comparison of the magnetic field components in

the air gap region (z = - e/2) (region II)

TABLE I – Machine Parameters

Parameter Value

Pole pair number 10

Slot number 12

Air gap e (mm) 1

Relative permeability of PMs

r 1.045

Remanent flux density BR (T) 1.16

Active length Lpp (mm) 60

hdr, hes, ha (mm)

20, 10 and

21

wa, wer and wes (mm) 6, 15 and 4

VI. Conclusion

A general analytical model of a disc type switching flux permanent magnet machine is discussed,

where the periodic geometry is divided into regions. The proposed model is based on an exact 2D

solution of the magnetic field. The mathematical approach leading to the exact solution of the

Maxwell‟s equations using the separation of variables method in the different regions was briefly

exposed. Then, the global quantities expressions were derived from the 2D solution of the magnetic

field. Finally, the analytical results for the magnetic field were compared with those obtained by FEA,

and good agreement was obtained. The developed model can then be used in the early stage of the

design process of such type of electrical machines.

Fig. 6: Comparison of the magnetic field components

in the external air region (z = ha + 1/2) (region V)

(region II)

Fig. 7: Comparison of the magnetic field components in

the rotor slot region (z = hes /2) (region VI)

(region II)

Fig. 9: Comparison of the magnetic field components

in the permanent magnet region (z = ha /2) (region III)

Fig. 8: Comparison of the magnetic field components

in the stator slot region (z = -hdr /2) (region I)

(region II)

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