Download Static Airgap Magnetic Field of Axial Flux Permanent Magnet Disk

Static Airgap Magnetic Field of Axial Flux Permanent Magnet Disk Motor
Xuanfeng Shangguan, Kai Zhang
Static Airgap Magnetic Field of Axial Flux Permanent Magnet Disk Motor
1
Xuanfeng Shangguan, 2Kai Zhang
School of Electrical Engineering and Automation, Henan Polytechnic University, Jiaozuo,
China, [email protected]
*2,
School of Electrical Engineering and Automation, Henan Polytechnic University, Jiaozuo,
China. [email protected]
1,
Abstract
Due to flat structure of the axial flux permanent magnet disc motor (AFPMDM), the airgap
magnetic field distribution is more complex. To study the static airgap magnetic field of AFPMDM, the
conventional magnetic circuit method and the method of taking slice in finite element software Magnet
are used respectively. Then axial airgap flux density distribution characteristics along circumferential
direction and radial direction are analyzed. The accurate three-dimensional airgap magnetic field
distribution can be gotten with the above methods. At last, analytical method, finite element method
and average radius method are used to calculate the airgap magnetic flux of per pole respectively.
Three results are approximate and can reflect per pole magnetic flux of AFPMDM generally. The
research for airgap magnetic field of AFPMDM provides theoretical basis for its wide application.
Keywords: Airgap Magnetic Field, AFPMDM, Slice, Per Pole Magnetic Flux
1. Introduction
The disc axial flux permanent magnet motor, with advantages of axial compact structure, easy to
heat dissipation, high efficiency, obviously energy saving effect, high torque - inertia ratio and power
density and so on [1, 2], especially the size and weight of which is about 50% of the ordinary
permanent magnet motor, is especially suitable for occasions demanding small size, low weight the
low-speed drive system [3].
With flat structure, the airgap magnetic field distribution is along the axial direction, so the cross
section of this motor can not be selected to create a 2D model simply the same as common radial motor
is dealt with [4]. The axial airgap flux density of AFPMDM along the circumferential direction at
different radius is different, and the axial airgap flux density along the radius direction in the same
electrical degree is also different [5]. Therefore, in order to calculate airgap magnetic field distribution
of AFPMM accurately, 3D finite element analysis is asked to use [6, 7]. In this paper, airgap magnetic
field distribution only under permanent magnet excitation is studied. In order to save computing time,
according to the symmetry of the magnetic field distribution, only a pair of poles 3D motor model [810] is built.
2. No-load airgap magnetic field calculation of AFPMDM by magnetic circuit
method
The magnetic field analysis of axial flux permanent magnet disc motor is very complex. The main
magnetic circuit contains two closed magnetic circuits shown in Figure 1: one magnetic circuit is
starting from N pole, passing airgap and the magnetic yoke, through the airgap to reach S pole, at last,
through the magnetic yoke returning to N pole; the other one is closed through the airgap, magnetic
yoke and end cap [11, 12]. Due to the particularity of the permanent magnet magnetic circuit
distribution, the length of the magnetic circuit at different radius is not the same, thus increasing the
computational complexity of the magnetic circuit.
However, because the airgap length of AFPMDM is longer, and the main magnetic circuit is
unsaturated, so for engineering, we often take the magnetic circuit of the average radius as the total
magnetic circuit of AFPMDM to calculate.
International Journal of Digital Content Technology and its Applications(JDCTA)
Volume7,Number7,April 2013
doi:10.4156/jdcta.vol7.issue7.139
1175
Static Airgap Magnetic Field of Axial Flux Permanent Magnet Disk Motor
Xuanfeng Shangguan, Kai Zhang
(a)
(b)
Figure 1. Main magnetic circuit of AFPMDM (a) radial direction (b) circumferential direction
1-shaft 2-yoke 3-permanent magnet 4-cover
It’s assumed that the magnetic circuit is unsaturated, the magnetic potential drop of iron core and
the armature reaction are ignored. Thus,
H    H m hM
(1)
  1   2
(2)
Where  is the total airgap length of motor, 1 is the distance between the surface of the
permanent magnet and armature plate surface namely the main airgap length of the motor,  2 is the
bond length between the permanent magnet and rotor disk, hM is the length of the magnetization
direction of permanent magnet, H  is airgap magnetic field strength, H m is the magnetic field strength
of PM.
According to the magnetic flux continuity principle:
(3)
Am Bm   A B
Where A and Am are respectively the effective area of per pole airgap and the area of one pole
magnetic flux provided by the PM, B and Bm are the airgap magnetic flux density and the magnetic
flux density of the PM at the operating point,  is the leakage coefficient.
Assuming p is the number of pole pairs, Dmi and Dmo are the inner and outer diameter of PM,
 p and  i are the pole arc coefficient and the calculation pole arc coefficient. Thus,
Am 
1
2
 p ( Dmo
 Dmi2 )
8p
(4)
A 
1
2
 K F  i ( Dmo
 Dmi2 )
8p
(5)
Where K F is the airgap density distribution coefficient, is defined as the ratio of flux density
amplitude mean and flux density amplitude maximum in a group of airgap flux density curves
distributed along with the circumference.
Permanent magnetic material response curve is
(6)
Bm   r 0 H m  Br
According to the formula (1) ~ (6), assuming  i   p , the magnetic flux density of the PM at the
operating point Bm and the airgap magnetic flux density B can be obtained .
Bm 
 K F Br
 K F  r

(7)
hM
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Static Airgap Magnetic Field of Axial Flux Permanent Magnet Disk Motor
Xuanfeng Shangguan, Kai Zhang
B 
Where
r
and
Br
 K F  r

(8)
hM
Br are respectively relative magnetic permeability and remnant magnetization.
3. 3D airgap magnetic field analysis of single-side AFPMDM
Single-sided axial magnetic flux motor shown in Figure 2a is the simplest disc motor. It has only
one rotor side and one stator side. The advantages of this motor are compact structure, short shaft and
high torque but existing the great single-sided magnetic force. In this paper, the airgap magnetic field
distribution of a pair of poles is analyzed by the 3D static magnetic field solver of Magnet. Figure 2b
shows the flux density vector distribution of the motor excited by permanent magnet.
(a)
(b)
Figure 2. Single-sided AFPMDM (a)structure (b)magnetic flux density vector distribution
Taking a slice which is perpendicular to the shaft at the center of airgap, the airgap magnetic field is
reflected in this slice, and the airgap flux density contour map can be got as shown in Figure 3.
Figure 3. Airgap flux density contour map
Axial airgap flux density distribution curve of one pole shown in Figure 4 can be obtained through
taking the axial airgap flux density values along the circumferential direction respectively in the
average radius, inner diameter and outer diameter from the slice as shown in Figure 3. Besides, the
axial airgap flux density distribution curve along the radial direction can be got by taking a straight line
in the center of the pole (electrical angle is 90°). It’s shown in Figure 5.
Comparison analysis between Figure 4 and 5 shows that the amplitudes of airgap flux density in
different radius are not the same, that’s because the magnetic path length in different radius is different.
Airgap flux density distribution in a certain radius is close to flat-top wave, the flux density amplitude
in average radius is maximum, but due to the influence of the edge effect and the end flux leakage, the
amplitude of airgap magnetic flux density near the inner and outer diameter of the magnetic poles
decreases obviously.
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Static Airgap Magnetic Field of Axial Flux Permanent Magnet Disk Motor
Xuanfeng Shangguan, Kai Zhang
1 average radius
2 inner diameter
3 outer diameter
air gap magnetic flux density Bz(T)
0.8
1
0.7
0.6
0.5
3
0.4
2
0.3
0.2
0.1
0.0
0
20
40
60
80
100
120
140
160
180
electrical degree(°)
air gap magnetic flux density Bz(T)
Figure 4. Axial airgap flux density distribution curve along the circumferential direction
0.70
0.65
0.60
0.55
0.50
0.45
0.40
30
35
40
45
50
radius(mm)
Figure 5. Axial airgap flux density distribution curve along the radial direction
According to the above method, the axial airgap flux density is taken from the slice along the
circumferential direction and radial direction. The corresponding axial airgap flux density curve will be
got. Then the cross-cutting airgap flux density curve nets can be obtained, namely, the accurate 3D
airgap flux density space distribution graph in this airgap plane is shown in Figure 6.
0.6
air gap flux desityBz （T）
0.8
0.5
0.6
0.4
0.4
0.3
0.2
0.2
0
240
180
45
120
0.1
40
60
electrical degree（ °）
35
0
30
radius（ mm）
Figure 6. The space distribution diagram of airgap magnetic field
Similarly, one slice is taken from the surface of permanent magnet; the airgap magnetic field
distribution of permanent magnet surfaces is shown in Figure 7.
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Static Airgap Magnetic Field of Axial Flux Permanent Magnet Disk Motor
Xuanfeng Shangguan, Kai Zhang
0.7
0.6
air gap flux densityBz(T)
0.8
0.5
0.6
0.4
0.4
0.2
0.3
0
0.2
-0.2
240
0.1
180
45
120
40
60
electrical degree（ °）
0
0
35
radius （ mm）
30
Figure 7. The space distribution diagram of airgap magnetic field of critical the PM surface
Figure 7 shows that the magnetic field distribution of PM surface is flat-top shape. It’s impacted by
the shape of PM, and the permanent magnet is magnetized along the axis, therefore, the flux leakage is
relatively small when the slice is taken from the PM surface.
4. Study for the variation regular of axial airgap flux density amplitude
In order to research the relationship between flux density amplitude and pole arc coefficient, the
motor model is built by selecting the pole arc coefficient  p =0.8,0.7,0.6,0.5, respectively. And the
slice is taken from the center plane of airgap to solve the 3D static magnetic field, and then the axial
airgap flux density distribution curve of one pole along the circumferential direction at the average
radius corresponding to different pole arc coefficient can be got that is shown in Figure 8.
pole arc coefficient 0.8
pole arc coefficient 0.7
pole arc coefficient 0.6
pole arc coefficient 0.5
0.8
air gap flux density Bz(T)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0
20
40
60
80
100
120
140
160
180
200
electrical degree(°)
Figure 8. Airgap flux density change curves corresponding to different pole arc coefficient
It can be seen from Figure 8 that the airgap flux density amplitude at the average radius keeps a
constant. It is nothing to do with the pole arc coefficient.
Next, the factors that affect the airgap flux density amplitude at the average radius will be studied.
Firstly, the magnetization length of the permanent magnet is changed while keeping the airgap length
that is 4mm unchanged. Secondly, the length of the airgap is changed while keeping the magnetization
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Static Airgap Magnetic Field of Axial Flux Permanent Magnet Disk Motor
Xuanfeng Shangguan, Kai Zhang
length of the PM a constant hM  6mm . According to the above two methods, the airgap flux density
amplitude curve at the average radius is shown in Figure 9.
 =4mm
hM=6mm
air gap flux density amplitudeB(T)
0.8
0.7
0.6
0.5
0.4
0.3
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
hM/
Figure 9. The change curve of airgap flux density amplitude at the average radius
It can be seen from the above figure that the airgap flux density amplitude is related to the size
of hM  , but the flux density amplitude corresponding to the same value of hM  in different
situations is not the same. As can be seen from the formula (8), considering the flux leakage, magnetic
saturation and so on, the flux leakage will become larger with the increase of airgap length when the
value of hM  is a constant. That is to say, the flux leakage coefficient  becomes larger and the
amplitude of airgap flux density will decrease. Overall, the amplitude of airgap flux density is mainly
determined by the size of hM  .
5. Calculation of per pole airgap magnetic flux with different methods
Whether static characteristic or dynamic characteristic is considered, the airgap magnetic flux is an
important parameter for motor, and it affects electromagnetic torque and back electromotive force
directly. So it is very necessary to calculate per pole airgap magnetic flux.
Firstly, using the analytical method to calculate the per pole airgap magnetic flux. For nonsinusoidal magnetic flux density waveforms, the per pole airgap magnetic flux formula is as follows
[9]:
Rout
2

2
 f    i Bmg
rdr   i Bmg
 Rin2 )
(9)
( Rout
Rin
2p
2p
Where, Bmg is the amplitude of airgap flux density, Rout is the outer radius of PM, Rin is the inner
radius of PM.
Per pole airgap magnetic flux can be calculated combining the equations (8) and (9) with motor
parameters,  and K F are respectively approach to 1.4 and 0.88 according to their characteristic curves.
Secondly, the magnetic field integrator of finite element software is used to calculate per pole airgap
magnetic flux. Taking a slice at the airgap center of one pole and completing the static simulation of
3D magnetic field, airgap flux density distribution of one pole can be got as shown in Figure 10. Then
the airgap magnetic flux of per pole can be calculated by magnetic field integrator on this slice.
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Static Airgap Magnetic Field of Axial Flux Permanent Magnet Disk Motor
Xuanfeng Shangguan, Kai Zhang
Figure 10. Magnetic flux density distribution of one pole
Finally, using the average radius method to calculate the per pole airgap magnetic flux. Taking the
axial flux density values of one pole in the average radius along the circumferential direction from the
slice in Figure 10 and the default interpolation in the software is 1001 points, so per pole magnetic flux
can be calculated by putting these flux density values into formula (10).
f  
Rout 1001
Rin
1001
 B ldr   B l ( R
i 1
i
i 1
i
out
 Rin )
(10)
The calculation results of the above three methods are listed in Table 1:
Table 1. The calculation results of the above three methods
Calculation method
Analytic method
FEM
Average radius method
4.83×10-4Wb
Per pole magnetic flux
5.21×10-4Wb
5.35×10-4Wb
It can be seen that calculation results of per pole magnetic flux are close to each other with the
above three methods. The result calculated by the finite element method is the most accurate in the
three methods, because the 3D static solver of Magnet is used. This method will take long computing
time. Besides, the calculation gap between analytic method and FEM is 0.38×10-4Wb. So the error of
analytic method is relatively large because the leakage flux coefficient is from experience curve.
However, the gap between average radius method and FEM is only 0.14×10-4Wb, the error of average
radius method is very small. Therefore the AFPMDM can be equivalent to linear motor to model and
analyzed by using the average radius method.
6. The main parameters of the motor in this paper
Motor parameters
Table 2. The main parameters of the motor
Value
Motor parameters
Rated voltage
Rated power
Number of phases
Number of pole pairs
PM material
Remnant magnetization density
one pole angle
48V
168W
3
3
NdFeB
1.2T
48°
Inner diameter of PM
Outer diameter of PM
Thickness of PM
thickness of stator core
air gap length
Thickness of magnetic yoke
Value
60mm
105mm
6mm
15mm
4mm
5mm
7. Conclusion
By taking slice in airgap center, airgap magnetic field spatial distribution of AFPMDM is obtained.
In this way, the complex axial airgap magnetic field distribution can be displayed simply, conveniently,
and accurately. Axial airgap magnetic field distribution is close to flat-top wave. Due to the influence
of the edge effect and the end flux leakage, the airgap flux density decrease gradually near the inner
and outer diameter of the magnetic poles. The airgap flux density amplitude is maximum at average
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Static Airgap Magnetic Field of Axial Flux Permanent Magnet Disk Motor
Xuanfeng Shangguan, Kai Zhang
radius and it is determined by the size of hM  . Then, the analytic method, finite element method and
average radius method are adopted to calculate per pole magnetic flux of AFPMDM. In general, the
above three methods can calculate per pole magnetic flux of AFPMDM with certain accuracy. The
above research has achieved anticipated effect and provided a basis for the research of AFPMDM
airgap magnetic field.
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