ISSN: 1693-6930
TELKOMNIKA Vol. 11, No. 2, June 2013: 241 – 248
244
δ
R 4
mm
R
m
R 2
m
R 2
mo
R 2
mo
R 2
mi
R 2
mi
R 2
mr
R
mr
R 2
r
2
m
2
δ
2
r
Figure 4. Equivalent LPMC of AFPM
mm
R
m
R 4
mo
R 4
mi
R 4
mr
R 2
2
δ
δ
R 4
2
m
2
r
Figure 5. A reduced form of Figure 4
3. Calculation of Mensioned the Reluctances 3.1 Reluctances corresponding to air-gap and magnet
The reluctances corresponding to air-gap and magnet can be expressed as
eff e
A R
µ δ
δ
4
m r
m m
A h
R µ
µ
5 where, h
m
is the magnetizing height of magnet, μ and μ
r
are permeability of air and relative permeability of magnet respectively, δ
e
is effective length of the air gap taking into account the stator slotting, and A
m
, A
eff
are the flux passing area of magnet and that of the air-gap respectively and can be expressed as
p D
D A
mi mo
p m
8
2 2
πα
6
p D
D A
e mi
mo p
eff
8 ]
[
2 2
δ δ
πα
7 Where α
p
is pole embrace coefficient, p is pole-pair number and D
mo
, D
mi
are the outer and inner diameter of magnet respectively.
3.2 Reluctances corresponding to magnet-to-rotor leakage flux from region 4 and 5
The expressions of the two reluctances can be obtained by calculating their permeances. The circular-arc straight-line permeance model
[6]
is one of the most satisfactory techniques for modeling flux flowing in the air gap, and that to model magnet-to-pole leakage
flux from region 4 as depicted in Figure 6, so the resultant permeance P
mi
is an infinite sum of differential permeances and can be obtained by the integral as:
m m
m mi
p p
D D
p m
mi mi
h h
h D
p p
h D
r rdrd
P
mi mi
p
πδ π
α µ
δ α
µ π
θ µ
δ α
π
ln 2
1 2
1
2 1
2 1
9
In a same way the leakage flux permeance corresponding to magnet-to-magnet leakage flux from region 5 can be obtained as
TELKOMNIKA ISSN: 1693-6930
Formula Expression of Airgap Leakage flux Coefficient of Axial-Flux Permanent Magnet Motor Xiao Gong
245
m m
m mo
p p
D D
p m
mo mo
h h
h D
p p
h r
D rdrd
P
mo mo
p
πδ π
α µ
δ α
µ π
θ µ
δ α
π
ln 2
1 2
1
2 1
2 1
10
Equation 10 is solved under the condition that the difference between the outer radius of magnet and that of rotor is more than or equal to δ
e
, if this difference is less than δ
e
, replace δ
e
in equation 10 with this difference. Equation 9 is solved under the condition that the difference between the inner radius of magnet and that of rotor is far larger than δ
e
. Using the above equations, and noting that the reciprocal relationship between
reluctance and permeances, the needed reluctances can be expressed as:
mi mi
P R
1
11
mo mo
P R
1
12
Figure 6. Permeance model of magnet-to-rotor leakage flux form region 4
3.3 Reluctances corresponding to magnet-to-rotor leakage flux from region 3 and magnet-to-magnet leakage flux
Based on Figure 3, the circular-arc straight-line permeance models describing the magnet-to-rotor leakage flux from region 3 and magnet-to-magnet one can be depicted as in
Figure 7 and Figure 8 respectively, and the resultant permeances can be easily achieved as
m m
mi mo
D D
r m
mr
h h
D D
h r
rdrd P
mo mi
πδ δ
π µ
θ π
θ µ
δ δ
δ
ln 2
2 1
2 1
2 1
2 1
13
] 1
2 1
1 2
1 ln
1 2
1 1
2 1
1 2
1 ln
1 2
1 1
2 1
1 2
1 ln
[ 1
1
2 1
2 1
p mi
p mi
p mi
p mo
p mo
mo p
mi p
mo p
D D
r p
mm
D p
D D
D D
p D
D p
D p
p p
r r
rdrd P
mo mi
α δ
δ α
δ α
δ α
δ α
δ δ
α δ
α δ
δ α
δ δ
δ α
π µ
α π
θ π
θ µ
δ δ
δ
14
ISSN: 1693-6930
TELKOMNIKA Vol. 11, No. 2, June 2013: 241 – 248
246 If the distance between two adjacent magnet is less than δ
e
2, replace δ
e
with πD
mi
1- α
p
4p in Equation 13.
Figure 7. Permeance model of magnet-to- rotor leakage flux form region 3
Figure 8. Permeance model of magnet-to- magnet leakage flux
3.4 The formula expression of airgap leakage flux coefficient
