Electrical generator or motor structure – Dynamoelectric – Rotary
Reexamination Certificate
2002-03-28
2004-08-31
Mullins, Burton (Department: 2834)
Electrical generator or motor structure
Dynamoelectric
Rotary
C310S156010
Reexamination Certificate
active
06784590
ABSTRACT:
BACKGROUND OF THE INVENTION
1. Field of the Invention
The present invention relates to a permanent magnet motor, and more particularly to an electric motor having a permanent magnet and an armature facing the permanent magnet with a gap therebetween, capable of reducing cogging torque.
2. Description of the Prior Art
A conventionally implemented permanent magnet motor, wherein a cogging torque is reduced by providing auxiliary grooves is disclosed in the Japanese Patent Publication No. 63048147.
According to such permanent magnet motor, however, the cogging torque cannot be reduced sufficiently, because a normal magnetizing yoke is used, so that the magnetic wave form thereof is not suitable.
Minimization of cogging torque generated in the grooves in the iron core
The torque generated in the general electromagnetic machine system can be expressed by Formula 1 under the condition of constant magnetic flux according to the principle of virtual work.
T
=
-
∂
W
m
∂
θ
(
1
)
Here, Wm denotes a total magnetic energy, and &thgr; denotes a rotation angle.
The cogging torque will now be considered. The magnetic energy W
m
due to the permanent magnet is stored in the magnet, the iron core and the air gap portion. The magnetic energy in the magnet is almost constant, and the energy in the iron core is very small because the iron core has a high permeability. Accordingly, a cogging torque T
c
can be expressed by Formula 2 by the angular differentiation of only a magnetic energy W
g
in the air gap portion.
T
c
=
-
∂
W
g
∂
θ
(
2
)
In order to simplify, it is assumed that the iron core is rotated, and the magnetic energy is stored in a cylindrical air gap portion entirely, and that a magnetic energy is W
g
(&thgr;) when the relative angle of the stator and the rotor is &thgr;. The W
g
(&thgr;) can be expressed by Formula 3 by integration by rotation at the air gap portion.
W
g
(
θ
)
=
l
g
L
S
r
g
2
μ
0
∮
C
B
g
2
(
θ
+
γ
)
ⅆ
γ
(
3
)
Here, l
g
denotes an air gap length, L
s
denotes an effective thickness of iron core, &mgr;
0
denotes a vacuum permeability, r
g
denotes a mean radius of air gap portion, and B
g
(&thgr;+&ggr;) denotes a distribution of the magnetic flux density in the air gap with respect to an angle &ggr; in the iron core rotated by an angle &thgr;.
In a smoothed iron core
1
having no winding grooves as shown in
FIG. 1
, no cogging torque due to the rotation is generated because there are no winding grooves. Accordingly, the magnetic energy W
g
(&thgr;) in the Formula 3 is constant having no relation to the rotation angle (&thgr;). On the contrary thereto, it is considered that if the winding grooves exist, B
g
(&xgr;) or B
g
2
(&xgr;) lacks substantially at the angle of &ggr;, so that the cogging torque is generated. Here, &xgr;=&thgr;+&ggr;. The W
g
can be expressed by Formulas 4-6, if the lacked magnetic energy due to the winding grooves is &dgr; W
g
.
δ
W
g
=
∑
k
=
1
3
w
g
(
θ
+
γ
k
)
(
5
)
w
g
(
θ
,
γ
k
)
=
l
g
L
S
2
μ
o
k
sk
B
g
2
(
θ
+
γ
k
)
(
6
)
Here, W
g
denotes a magnetic energy in the air gap portion of the smoothed iron core, s denotes a number of grooves, &ggr;k denotes an angle of a No. k winding groove, k
sk
denotes a coefficient determined by a figure of the No. k winding groove, and B
g
(&thgr;+&ggr;k) is a magnetic flux density in the air gap at a position of No. k groove.
By putting the Formulas 4 to 5 in the Formula 2, the cogging torque can be expressed by Formula 7.
T
c
=
∂
(
δ
W
g
)
∂
θ
=
l
g
L
S
2
μ
o
∂
∂
θ
(
∑
k
=
1
s
k
sk
B
g
2
(
θ
+
γ
k
)
)
(
7
)
The right side of the Formula 7 is the sum of magnetic energy portions lost by the winding grooves. It can be said that it is similar to the function of the hole in the semiconductor engineering. Specifically, it can be said that the cogging torque is generated by the reduction of the magnetic energy due to the grooves. Accordingly, a manner for reducing the cogging torque is now studied under the point of view as follows.
FIG. 2
shows results of the distribution of the magnetic flux density in the air gap measured by providing a hole element on the surface of the iron core and rotating the iron core, in order to know a figure of B
g
(&xgr;). The analysis is proceeded on the assumption that a figure of the distribution of the magnetic flux density in the air gap is shown in
FIG. 3
with respect to the electrical angle p &xgr;. &bgr; denotes a ratio of an inclined portion. It is supposed that the magnetic flux density is varied as a figure of a fourth part of a sine wave in a section corresponding to &bgr; shown in Formula 8.
(0<&bgr;≦1) (8)
The B
g
(&xgr;) can be expressed by Formula 9.
B
g
(
ξ
)
{
=
-
1
for
-
π
2
p
ξ
<
-
β
π
2
=
sin
p
ξ
β
for
-
β
π
2
≤
p
ξ
≤
β
π
2
=
1
for
β
π
2
<
p
ξ
≤
π
2
(
9
)
The Formula 9 can be expressed by Fourier series in the form of Formula 10 consisting of terms of odd number order.
B
g
(
Ϛ
)
=
∑
n
=
o
∞
b
2
n
-
1
sin
(
(
2
n
-
1
)
p
ξ
)
(
10
)
The coefficient can be expressed by Formula 11 in case of &bgr;=0 and by Formula 12 in case of 0<b<1.
b
2
n
-
1
=
4
(
2
n
-
1
)
π
(
11
)
b
2
n
-
1
=
4
(
2
n
-
1
)
π
(
β
2
(
2
n
-
1
)
2
-
1
)
cos
(
2
n
-
1
)
β
π
2
(
12
)
In case of &bgr;=1, only the fundamental wave is presented.
B
g
2
(&xgr;) can be expressed by Formula 13 which is a even function consisting of terms of even number order.
B
g
2
(
ξ
)
=
a
o
+
∑
n
=
1
∞
a
2
n
cos
2
np
ξ
(
13
)
FIG. 4
shows the change of each harmonic coefficient a
2n
of B
g
2
with respect to &bgr;. When &bgr; is zero, it becomes a square wave, and when &bgr; is 1, it becomes a pure sine wave. The second order component corresponds to the fundamental wave, and becomes larger in value when the order number is smaller in value. The maximum value thereof exists in the middle portion of the change of &bgr;.
By putting Formula 13 in Formula 7, Formula 14 can be obtained.
T
c
=
l
g
L
S
2
μ
o
∑
n
=
1
∞
[
∂
∂
θ
∑
k
=
1
s
k
sk
a
2
n
cos
2
np
(
θ
+
γ
k
)
]
=
l
g
L
S
μ
o
∑
n
=
1
∞
[
∑
k
=
1
s
npk
sk
a
2
n
sin
2
np
(
θ
+
γ
k
)
]
(
14
)
In order to minimize the cogging torque, it is understood that a sum of components due to the winding grooves should be set to zero as shown in Formula 15 in the most of the harmonics of low order (n=1, 2, 3. . . ) which affect largely on the cogging torque.
∑
k
=
1
s
npk
sk
a
2
n
sin
2
np
(
θ
+
γ
k
)
=
0
(n: natural numeral)
(
15
)
This is the principle of minimization of the cogging torque due to the iron core grooves. A manner for reducing the cogging torque with respect to the three-phase permanent magnet motor on the basis of the principle is now considered.
Minimization of the cogging torque in the three-phase winding grooves
A recent conventional small motor of non-lap concentra
Abukawa Toshimi
Ohnishi Kazuo
Sekiguchi Tokio
Japan Servo Co. Ltd.
Patterson Thuente Skaar & Christensen P.A.
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