This paper presented an improved analytical method for calculating armaturereaction field in the surfacemounted permanent magnet machines accounting for opening slots. The analytical model is divided into two types of subdomains. The current of the armature is centralized in the center of the slots. The field solution of each subdomain is obtained by applying the interface and boundary conditions of the model. Two 30pole/36slot prototype machines with different slotopening width are used for validation. The FE (finite element) results confirm the validity of the analytical results with the proposed model. The investigation shows that the wider the slotopening width is, the smaller the peak value of radial and circumferential components of flux density, and the analytical armaturereaction field produced by centralized current in the slots is similar with the armaturereaction field produced by distributed current in the slots in the FE.
1. Introduction
Permanentmagnet machines have become more and more popular in the commercial, industrial and military products benefiting from higher power ratio to mass, torque ratio to volume, efficiency and lower vibration and noise over conventional electrically excited synchronous machines and asynchronous machines
[1

3]
. The magnetic field is one of the most important issues in the permanentmagnet machines. It influences the performance of the motor such as torque ripple
[4]
, armature winding inductances
[5
,
6]
, acoustic noise and vibration spectra
[7]
, radial force distribution
[8
,
9]
, etc. Its accurate prediction can significantly facilitate and expedite their optimal design in terms of efficiency, compactness, cost and reliability. At the present time, the numerical methods for magnetic field calculation, such as finiteelements method, provide accurate results concerning all kinds of magnetic sizes of permanent magnet machines, taking into account the saturation and without making any simplification of the geometry. But the numerical methods are very timeconsuming, not suit to the initial design and optimization of the machines. Usually, the numerical methods are very good for the adjustment and validation of the design. Furthermore, the results which are obtained by numerical methods may be not accurate to calculate cogging torque and unbalanced magnetic force
[10
,
11]
since it is sensitive to the FE meshes. Indeed, the motor performance can be obtained by the analytical methods of field computation based on the sufficient hypotheses
[12]
.
Carter’s coefficient has been widely used to consider the slotting effects in conventional electrically excited synchronous machines and asynchronous machines
[3
,
13]
. Its accuracy is not enough sometimes. The current of armature winding is equivalent to the distributed current sheet in the inner surface of the stator by Zhu et al.
[14]
. Then he proposed the analytical method accounting for stator slot openings by the application of the conformal transformation method and a “2d” relative permeance function
[15]
. Zarko
[16]
introduced the notion of complex relative airgap permeance, calculated from the conformal transformation of the slot geometry, to take into account the effect of slotting. This solution follows directly from the SchwarzChristoffel transformation, which is a complex function by nature.
Zhu
[5]
proposed distributed current area, and divided the field domain into two regions: the region with airgap and magnet and the region with slotless winding. The accuracy of the 2D model depended on the assumed radial location of the equivalent current sheet. Wu
[17]
calculated armaturereaction field of surfacemounted permanentmagnet machines accounting for toothtips. The field domain is divided into three types of subdomains. The subdomain model predicted similar flux density in the air gap and magnets, but exhibited much higher accuracy for the flux density in the slots. Its solution is complicated. Rahideh
[18]
presented analytical armaturereaction field distribution of slotted brushless machines with surface inset permanent magnets. Overlapping and nonoverlapping windings with all teeth or alternate teeth were also studied.
In this paper, an improved analytical method accounting for opening slots is derived for calculating the armaturereaction field distribution of machine. In the derivation, the current in the slot is centralized in the center of the slot. The field domain is divided into two types of subdomains: (1) permanent magnet and airgap; (2) Slots. The analytical field expressions of two subdomains excited by armature windings are obtained by the variable separation method. The coefficients in the expressions are determined by applying the interface and boundary conditions. The investigation shows the developed model has high accuracy to calculate the armaturereaction field of surfacemounted permanent magnet machines. And the wider the slotopening width is, the smaller the peak value of radial and circumferential components of flux density. The FE results verify the validity of the analytical model.
2. Analytical Field Modeling
In this paper, the analytical modeling is based on the following assumptions:

(1) Linear properties of permanent magnet;

(2) Infinite permeable iron materials;

(3) The relative permeability in the PM is equal to air;

(4) Negligible end effect;

(5) Simplified slot as shown inFig. 1;

(6) The current in the slot is centralized in the center of the slot.
The twodimensional subdomain model for calculating armaturereaction field is shown in
Fig. 1
. The magnet field is divided into two types of subdomains for the convenience of analysis: (1) subdomain of permanent magnet and airgap (The first subdomain is limited by a circle characterized by a
R
_{s}
radius); (2) subdomain of slots.
Symbols and types of subdomains with windings.
 2.1 Armaturereaction field in the first type of subdomains
Since in the 2D field, the vector potential has only zaxis component which satisfies Laplace equation in the first type of the subdomains:
where
A
_{z1}
is vector potential in the air,
r
and
θ
are the radius and he angle between OP and the line of
0
°, which is shown in
Fig. 1
.
The radial and circumferential components of flux density can be obtained from the vector potential distribution by
where
B
_{r}
and
B
_{θ}
are radial and circumferential components of flux density respectively.
In the first type of subdomains, the general solution of (1) can be given by
According to the boundary condition in the first type of subdomain, the circumferential component of flux density is zero in the outer surface of rotor. So
where
m
is the harmonics of the armaturereaction field in the airgap, and
R
_{r}
is the radius of the outer surface of rotor.
Substituting (4) into (3), the vector potential in first type of subdomains can be given by
where
C
_{m}
and
D
_{m}
are coefficients to be determined, and
According to (2) and (5), the radial and circumferential components of flux density can be given by
where
B
_{r1}
and
B
_{θ1}
are radial and circumferential components of flux density respectively in the airgap, and
 2.2 Armaturereaction field in the second type of subdomains
The vector potential in the
i
th slot produced by
I
(the current of the winding) can be given by
where
μ
_{0}
is permeability of the air,
n
is the harmonics of the armaturereaction field in the slots, and
where
a
is the distance between windings and the center of the machine.
Boundary conditions in the
i
th slots: ①
B
_{r}
= 0 while
θ
=
θ
_{i}
±
b
_{sa}
/ 2 and
R
_{s}
≤
r
≤
R
_{sb}
. ②
B
_{θ}
= 0 while
r
=
R
_{sb}
and
θ
_{i}
−
b
_{sa}
/ 2 ≤
θ
≤
θ
_{i}
+
b
_{sa}
/ 2 . Where
R
_{s}
is the radius of the inner surface of stator.
R
_{sb}
is the bottom radius of the slot.
b_{sa}
is the radian of the each slot.
Then according to boundary conditions, the armaturereaction field produced by the single slot can be solved in the slot. Since the armaturereaction field is symmetrical about the line
θ
=
θ
_{i}
in the polar coordinates, according to the general solution of (11), the vector potential in the slot can be given as
where
A
_{ni}
and
B
_{ni}
are coefficients to be determined,
μ_{0}
is the permeability of the air,
r
is the radius of point P,
θ
_{i}
is the angle between center line of the
i
th slot and the line of
0
º.
Then
for
r
<
a
, and
for
r
≥
a
, where
 2.3 Interface conditions between two types of subdomains
 2.3.1 The First Interface Condition
The first interface condition is that the circumferential component of the flux density in the inner surface of stator
r
=
R
_{s}
is equal.
According to the vector potential distribution in the
i
th slot, the circumferential component of the flux density along the stator bore can be obtained:
where
where
The circumferential component of the flux density along the stator bore outside the slot is zero since the stator core material is infinitely permeable. So Fourier series of the circumferential component of the flux density in the inner surface of stator can be given by
where
where
According to (8), (20) and the first interface condition, the following equations can be obtained:
Substituting (8), (20) into (25), the following equations can be obtained:
Combining (17), (21), (22) and (26), the following equations with matrix format can be obtained:
where
where
C
_{1}
,
D
_{1}
and
A
_{2i}
are the column vectors for coefficients
C
_{m}
,
D
_{m}
and
A
_{ni}
, e.g.
C
_{1}
=[
C
_{1}
,
C
_{2}
, ...,
C
_{M}
]
^{T}
,
M
is the maximum harmonic order in the air gap and magnet regions,
N
is the maximum harmonic order in the slot regions and
N
_{s}
is the number of slots.
 2.3.2 The second interface condition
The second interface condition is that the vector potential of the
i
th slot opening is equal in the two types of the subdomains.
According to (5), the vector potential in the inner surface of stator can be given as
where
The equation (44) can be expanded into Fourier series along the stator inner surface of the
i
th slot:
for
θ
_{i}
−
b
_{sa}
/ 2 ≤
θ
≤
θ
_{i}
+
b
_{sa}
/ 2 , where
where
According to (14), the vector potential in the inner surface of stator can be obtained:
where
where
According to (47), (53) and the second interface condition, the following equation can be obtained:
Combining (45), (46), (52) and (55), the following equations with matrix format can be obtained:
where
According to (27) and (56), the final equations with matrix format are shown in (67).
Then the coefficients
C
_{1}
,
D
_{1}
and
A
_{2i}
can be obtained according to (67).
3. FiniteElement Validation
The major parameters of two 30pole/36slot prototype machines with different slotopening width which are used for validation are shown in
Table 1
. The analytical prediction is compared with the linear FE prediction. The Finite element mesh of the geometry is shown in
Fig. 2
. The current with 700
A
is distributed in the two slots in FE, while the current with 700
A
is concentrated in the two slots in the analytical method.
Parameters of prototype machines (Unit: mm)
Parameters of prototype machines (Unit: mm)
Finite element mesh of the geometry
Fig. 3
shows the results between analytical and FE predictions of armaturereaction flux density in the airgap of machine with slotopening width=8 mm. The peak value of radial component of flux density are 0.083T airgap r =198mm of motor. And the peak value of circumferential component of flux density are 0.07T in the airgap r =198mm of motor.
FE and analytically predicted armaturereaction flux density waveforms in the airgap r =198mm of motor having slotopening width =8 mm: (a) Radial component; (b) circumferential component.
Fig. 4
shows the results between analytical and FE predictions of armaturereaction flux density in the airgap and slots of machine with slotopening width =16 mm. The peak value of radial component of flux density are 0.077T in the airgap r =198mm of motor. And the peak value of circumferential component of flux density are 0.048T in the airgap r =198mm of motor.
FE and analytically predicted armaturereaction flux density waveforms in the airgap r =198mm of motor having slotopening width =16 mm: (a) Radial component; (b) circumferential component.
As can be seen from
Fig. 3
and
Fig. 4
, the predicted armaturereaction flux density by subdomain model almost completely matches FE results, and the error is less than 2%. The results show that the wider the slotopening width is, the smaller the peak value of radial and circumferential components of flux density. At the same time, the analytical armaturereaction field excited by centralized current in the slots is similar with the armaturereaction field excited by distributed current in the slots in the FE.
4. Conclusion
This paper presented an improved method for calculating the armaturereaction field in the surfacemounted permanent magnet machines accounting for slots. The analytical model is divided into two types of subdomains. The current in the slot is centralized in the center of the slot. The field solution of each subdomain is obtained by applying the interface and boundary conditions. The analytical armaturereaction field excited by centralized current in the slots is similar with the armaturereaction field excited by distributed current in the slots with FE method. And the FE results confirm the validity of the analytical results with the proposed model. The investigation shows that the wider the slotopening width is, the smaller the peak value of radial and circumferential components of flux density.
BIO
Yu Zhou was born in China, 1983. He received the B.S. degree and the M.S. degree in electrical engineering from the College of Electrical Engineering, Naval University of Engineering, Wuhan, China in 2005 and 2009 respectively. From 2012, he is working toward the Ph.D. degree in the College of Electrical Engineering, Naval University of Engineering, Wuhan, China. He is one of IEEE Members now. His major research interests include power electronics, and design and control of permanentmagnet machines.
Huaishu Li was born in China, 1965. He received the B.S. degree and the M.S. degree in electrical engineering from the College of Electrical Engineering, Naval University of Engineering, Wuhan, China in 1986 and 1991 respectively, and the Ph.D. degree from Huazhong University of Science & Technology, Wuhan, China in 2001. From 1991, he lectures in the College of Electrical Engineering, Navy University of Engineering, Wuhan, China. His major research interests include power electronics, and design and control of permanentmagnet machines.
Qingyu Wang was born in China, 1986. He received the B.S. degree in manufacturing and its automation from Tsinghua University, Beijing, China in 2009 and the M. S. degree in navigation from Dalian Naval Academy, Dalian, China in 2011. From 2012, he is working toward the Ph.D. degree in the College of Electrical Engineering, Naval University of Engineering, Wuhan, China. His major research interests include power electronics, and design and control of linear motor.
Zhiqiang Xue was born in China, 1979. He received the B.S. degree in electrical engineering and the automatization specialty from the Ordnance Engineering College, Shijiazhuang, China in 2003, and the M.S. degree in electrical engineering from the College of Electrical Engineering, Naval University of Engineering, Wuhan, China in 2009. From 2015, he is working toward the Ph.D. degree in the College of Electrical Engineering, Naval University of Engineering, Wuhan, China. His major research interests include power electronics, and design and control of permanentmagnet machines.
Shi Zhou was born in China, 1988. He received the B.S. degree in physics from Nanjing University, Nanjing, China, in 2011. From 2013, he is working toward the M.S. degree in the College of Electrical Engineering, Naval University of Engineering, Wuhan, China. His research interest is multiphase permanentmagnetic synchronous generator.
Qing Cao was born in China, 1991. She received the B.S. degree in electric engineering from Hunan Institute of Engineering, Xiangtan, china, in 2013. She is working toward the M.S. in the College of Electrical Engineering, Naval University of Engineering, Wuhan, China. Her research interest is linestart permanentmagnetic synchronous motor.
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