This paper presented an improved analytical method for calculating the opencircuit magnetic field in the surfacemounted permanent magnet machines accounting for slots and eccentric magnet pole. Magnetic field produced by radial and parallel permanent magnet is equivalent to that produced by surface current according to equivalent surfacecurrent method of permanent magnet. The model is divided into two types of subdomains. The field solution of each subdomain is obtained by applying the interface and boundary conditions. The magnet field produced by equivalent surface current is superposed according to superposition principle of vector potential. The investigation shows harmonic contents of radial flux density can be reduced a lot by changing eccentric distance of eccentric magnet poles compared with conventional surfacemounted permanentmagnet machines with concentric magnet poles. The FE(finite element) results confirm the validity of the analytical results with the proposed model.
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 produced synchronous machines and asynchronous machines
[1

3]
. The magnetic field distribution in the airgap is one of the most important issues in the permanentmagnet machines. It is foundation to other issues. 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
[4
,
5]
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
[7]
.
Zhu
[8]
worked for both internal and external rotor motor topologies, and either radial or parallel magnetized magnets by accounting for the effects of both the magnets and the stator windings. B. Ackermann
[9]
proposed a twostep method, solving slotless field firstly and predicting the slotting effect based on slotless field, to reduce the computational complexity for each rotor position, although slotting effect still needs to be evaluated for all rotor positions. The method is very quick but is an approximate method. P. Kumar
[10]
developed an analytical model for the instantaneous airgap field density with the assumption that the iron (both stator and rotor yoke) has finite permeability and the thickness of the stator yoke is finite. The slotting effect can be accounted for by the conformal mapping method which transforms the field solution in the corresponding slotless domain into the slotted domain. One typical example is the Carter’s factor which compensates the main flux for the slotting effect. F.W. Carter applied the conformal transformation to calculate magnetic field of electrically produced machines accounting for the slots
[11]
firstly. Cogging torque, acoustic noise and vibration spectra can not be analyzed by use of Carter’s coefficient.
It is very important to take into account slots in the analytical methods. Liu
[11
,
12]
presents an analytical model suitable for analyzing permanent magnet motors with slotted stator core which calculate the airgap flux density taking into account the effect of the interaction between the pole transitions and slot openings and solving the governing functions. In
[7]
, the interface of magnet field between slots and air gap was obtained in form of Fourier series. Its high accuracy for the flux density distributions in both airgap and magnets of the machines with different slot opening widths was confirmed by FE. Zhu
[14]
extended
[7]
to account for any pole and slot combinations, and an accurate analytical subdomain model with stator slotting effects was presented for computation of the opencircuit magnetic field in surfacemounted permanent magnet machines. Some mistakes were clarified at the same time. Wu
[15]
developed an improved analytical subdomain model for calculating the opencircuit magnetic field in surfacemounted permanent magnet machines accounting for the toothtips in the slots based on 2D polar coordinate.
In
[7]
, and
[11

14]
, the field domain was divided into some types of subdomains, and calculated according to interface and boundary condition. The radial and circumferential components of magnet magnetization were expressed in the form of Fourier series to solve Possion equation in the concentric permanent magnet. But the interface between eccentric permanent magnet and airgap is very difficult to be obtained. K. Boughrara
[16]
used twodimensional field theory in polar coordinates to determine the flux density distribution, cogging torque, back EMF and electromagnetic torque in the slotted air gap of permanentmagnet motors with surface mounted magnet bars which are magnetized in shifting direction, but not like Halbach array magnetization. The sinusoidal waveform of the flux density distribution was obtained, but installation process of the permanent magnet bars is not easy to achieve.
Eccentric magnet pole is a typical example of eccentric magnet pole, and performance of motors can be optimized by changing eccentric distance of the eccentric magnet poles. Zhang
[17]
deduced expressions of Fourier transform coefficient for magnetomotive force of eccentric magnet pole. The main exciting force wave can be reduced through suitable selection of the eccentric distance. Xu
[18]
analyzed the influence of the eccentric magnet poles on the waveform of airgap flux density and the motor performances, proposed a novel optimal designing method for the eccentric magnet poles with analytical expression. In
[19]
, based on the magnetic field which was produced by a pair of windings on the airgap, the expressions of the flux density produced by parallelmagnetized permanent magnet with different shapes were deduced with surfacecurrent method. But slots were not accounted. At the present time, the analytical model of permanentmagnet machines accounting for eccentric magnet poles and slots has not been analyzed comprehensively.
In this paper, an improved analytical method accounting for slots and eccentric magnet pole is derived for calculating the magnetic field distribution of machine. In the derivation, magnetic field produced by radial and parallel eccentric permanent magnet is equivalent to that produced by surface current according to surfacecurrent method of permanent magnet. The field domain is divided into two types of subdomains. The analytical field expressions of two subdomains produced by a pair of windings are obtained by the variable separation method. The coefficients in the field expressions are determined by applying the interface and boundary conditions. The magnet field produced by equivalent surface current is superposed according to superposition principle of vector potential. Compared with conventional surfacemounted permanentmagnet machines with concentric magnet poles, harmonic content of radial flux density can be reduced a lot by changing eccentric distance of eccentric magnet poles. The investigation shows the developed model has high accuracy to calculate the flux density of surfacemounted permanent magnet machines with eccentric magnet poles. The finite element (FE) results verify the validity of the analytical model.
2. Analytical Field Modeling
 2.1 Equivalent surface current of magnet pole
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 to1;

(4) Negligible end effect;

(5) Simplified slot as shown inFig. 1.
Symbols and types of subdomains.
The twodimensional conventional subdomain model 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 characterised by a
R_{s}
radius); (2) subdomain of slots.
The permanent magnet with eccentric structure is shown in
Fig. 2
. The distance between point E and point O can be given by
The eccentric structure of permanent magnet
where
H
is the eccentric distance,
R_{r}
is the radius of rotor,
h_{max}
is the maximum thickness of permanent magnet and
ζ
is the radian between OE and the center line of permanent magnet.
The radian between O
_{1}
E and the center line of permanent magnet can be given by
where
R
_{2}
is the radius of arc BC.
The radius of arc BC can be given by
The eccentric distance of the permanent magnet can be given by
where
h
_{min}
is the minimum thickness of PM,
α_{p}
is polearc to polepitch ratio and
P
is pole pairs.
The equivalent surface current is equal to the circumferential component of coercivity on the surface of magnet pole
[19]
. And the equivalent surface current of eccentric magnet pole is shown in
Fig. 3(a)
for parallel magnetization.
Equivalent surface current of eccentric magnet pole. (a) Parallel magnetization; (b) Radial magnetization.
The current density of AB and CD can be given by
where
H_{cj}
is coercivity of permanent magnet.
The surface current density of side BC can be given by
The surface current density of side AD can be given by
The equivalent surface current of eccentric magnet pole is shown in
Fig. 3(b)
for radial magnetization.
The surface current density of side AB and CD can be given by
The surface current density of side BC can be given by
The surface current density of side AD is zero.
 2.2 Magnet field produced by a pair of windings
Equivalent surfacecurrent method is based on magnet field produced by a pair of windings. The current of windings can be given by
where Δ
l
is the length infinitesimal in the side AB, CD, AD and BC of magnet pole.
In this section, magnet field produced by a pair of windings is analyzed. Subdomain model with a pair of windings is shown in
Fig. 4
.
Subdomain model with a pair of windings
 2.2.1 Magnet field in the first type of subdomains
Since in the 2D field, the vector potential has only zaxis component which satisfies:
α
and
β
are the labels of winding.
a
and
ζ
present the position of the windings in the polar coordinate system.
i_{c}
is current. The vector potential of point
Q
(
r
,
θ
) produced by
α
and
β
is given by
and
respectively.
According to (11) and (12), the sum of vector potential can be given by
where
A
_{m1}
,
B
_{m1}
,
C
_{m1}
,
D
_{m1}
are coefficients to be determined,
μ _{0}
is the permeability of the air,
r
is the radial of point Q,
θ
is the degree between point Q and center line of the windings,
ρ_{α}
and
ρ_{β}
are the coordinates when the origins are
α
and
β
respectively.
If the origin is point O, ln
ρ_{α}
and ln
ρ_{β}
can be expanded into infinite series about
θ
and
r
.
The radial and circumferential components of flux density can be obtained from the vector potential distribution by
While
r
<
a
, the flux density in the first subdomain can be given by
for the circumferential component.
In the outer surface of rotor, the circumferential component of the flux density is zero
Substituting (16) into (17),
B
_{m1}
and
D
_{m1}
can be given by
While
r
>
a
, substituting (18) and (19) into (13), the general solution of vector field in the first subdomain can be given by
where
While
r
>
a
, the flux density in the first subdomain can be given by
for the radial component, and
for the circumferential component, where
The vector potential produced by the equivalent surface current of
j
th magnet pole can be given by:
where
A_{mj}
and
C_{mj}
are coefficients to be determined, and (
j
− 1)
π
/
P
is the angle between center line of first pole and that of
j
th pole.
 2.2.2 Magnet field in the second type of subdomains
The governing function in the slots is:
The vector potential in the subdomain
2i
is
[15]
where
R_{s}
is the inner radial of the stator,
R_{sb}
is the radial of the slot bottom,
θ_{i}
is the angle between center line of the ith slot and center line of the windings as shown in
Fig. 4
,
b_{sa}
is the slot opening width angle,
D_{n2i}
is coefficient to be determined, and
So the flux density in the second subdomain can be given by
for the radial component, and
for the circumferential component.
 2.2.3 Interface condition between two types of subdomain
 (a) 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.
By evaluating (34) at the
r
=
R_{s}
interface, (34) simplifies down to
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 the vector potential distribution in the first subdomain, the circumferential component of the flux density in the inner surface of stator can be given as
According to (24), (37) and (42):
Combining (36), (38), (39) and (43), the following equations can be obtained:
 (b) The Second Interface Condition
The second interface condition is that the vector potential of the ith slot opening is equal in the two types of the subdomains.
According to (20), the vector potential in the inner surface of stator can be given as
where
The equation (45) can be expanded into Fourier series along the stator inner surface of the ith slot:
Where
According to (30), the vector potential in the inner surface of stator can be obtained:
The vector potential in the inner surface of stator is equal in two subdomains.
Substituting (48) and (52) into (53), the following equation can be obtained:
Then
Substituting (49) into (55), the following equation can be obtained:
while
n
=1, 2, 3….
Combining (46), (47) and (53), the following equation can be obtained:
According to (44) and (57), the matrix format can be given as
Then the coefficients A1, C1 and D2i can be obtained according to (58).
 2.2.4 The superposition principle of vector potential
The superposition principle can be applied to the vector potential by equivalent surface current in the surfacemounted permanent magnet machines.
In the
Fig. 3
, the vector potential produced by equivalent surface current of side AB and CD can be superposed:
where Δ
r
is the length infinitesimal in the side AB and CD of PM and
The vector potential produced by equivalent surface current of side BC can be superposed:
where Δ
γ
_{1}
is the angle infinitesimal in the side BC of PM, and the origin is point O
_{1}
,
The vector potential produced by equivalent surface current of side AD can be superposed:
where Δ
γ
_{2}
is the angle infinitesimal in the side AD of PM, and the origin is point O,
According to (15), (59), (61) and (64), the radial and circumferential components can be obtained.
3. FiniteElement Validation
The major parameters of two 30pole/36slot prototype machines which are used for validation are shown in
Table 1
. The minimum thickness of permanent magnet is 7mm in the prototype machine with eccentric magnet poles. And it is 12mm in the prototype machine with concentric magnet poles. The analytical prediction is compared with the linear FE prediction.
Parameters of Prototype Machines (Unit: mm)
Parameters of Prototype Machines (Unit: mm)
Fig. 5
show the results between analytical and FE predictions of flux density in the airgap at
r
=198.5mm of motor with eccentric magnet poles.
Fig. 6
show the results between analytical and FE predictions of flux density in the airgap at
r
=198.5mm of motor with concentric magnet poles. As can be seen, the predicted flux density by subdomain model with equivalent surfacecurrent method almost completely matches FE results.
FE and analytically predicted flux density waveforms in the airgap at r =198.5mm of motor with eccentric magnet poles: (a) Radial component; (b) circumferential component.
FE and analytically predicted flux density waveforms in the airgap at r = 198.5mm of motor with concentric magnet poles: (a) Radial component; (b) circumferential component.
Harmonic analysis of radial component of flux density with five pair of magnet poles in airgap is shown in
Fig. 7
. Because five pair of magnet poles are one cycle to radial component of flux density in the airgap. Then the 5th order spatial harmonic is fundamental harmonic. The same harmonic orders are 7th, 17th, 19th, 29th 31th et al. because of the influence of slots. The main different harmonic orders between Eccentric magnet poles and concentric magnet poles are the 15th and 25th order spatial harmonic. The harmonic content of radial component of flux density is 10.59% in the motor with eccentric magnet poles. The harmonic content of radial component of flux density is 23.17% in the motor with concentric magnet poles.
Harmonic analysis of radial component of flux density at r = 198.5mm of motor. (a) Eccentric magnet poles; (b) Concentric magnet poles.
4. Conclusion
This paper presented an improved method for calculating the magnetic field in the surfacemounted permanent magnet machines accounting for slots and eccentric magnet pole. Magnetic field produced by radial and parallel eccentric permanent magnet is equivalent to that produced by surface current according to surfacecurrent method of permanent magnet. The model is divided into two types of subdomains. The field solution of each subdomain is obtained by applying the interface and boundary conditions. The magnet field produced by equivalent surface current is superposed according to superposition principle of vector potential. The investigation shows harmonic contents of radial flux density can be reduced a lot by changing eccentric distance of eccentric magnet poles compared with conventional surfacemounted permanentmagnet machines with concentric magnet poles. The FE results confirm the validity of the analytical results with the proposed model.
BIO
Yu Zhou 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 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.
Wei Wang She received the B.S. degree in harbor, waterway and coastal engineering, from the School of Hydraulic Engineering, Changsha University of Science and Technology, China in 2009 and the M.S. degree in the school of Water Resources and Hydropower Engineering, Wuhan University, China in 2015. Currently, she is working in Power China Zhongnan Engineering Corporation limited.
Shi Zhou 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 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.
Zhu Z. Q.
,
Howe D.
2007
“Electrical machines and drives for electric, hybrid, and fuel cell vehicles.”
Proc. IEEE
95
(4)
746 
765
DOI : 10.1109/JPROC.2006.892482
Zhu Z. Q.
,
Chan C. C.
2008
“Electrical machine topologies and technologies for electric, hybrid, and fuel cell vehicles,”
IEEE Vehicle Power and Propulsion Conf.
1 
6
ElRefaie A. M.
2010
“Fractionalslot concentratedwindings synchronous permanent magnet machines: Opportunities and challenges,”
IEEE Trans. Ind. Electron.
57
(1)
107 
121
DOI : 10.1109/TIE.2009.2030211
Zhou Y.
,
Li H.
,
Meng G.
,
Zhou S.
,
Cao. Q.
“Analytical Calculation of Magnetic Field and Cogging Torque in SurfaceMounted Permanent Magnet Machines Accounting for Any Eccentric Rotor Shape,”
IEEE Trans. Ind. Electron.
published online
DOI : 10.1109/TIE.2014.2369458
Lin D.
,
Ho S. L.
,
Fu W. N.
2009
“Analytical Prediction of Cogging Torque in SurfaceMounted PermanentMagnet Motors,”
IEEE Trans. Magn.
45
(9)
3296 
3302
DOI : 10.1109/TMAG.2009.2022398
Wang D. H.
,
Wang X. H.
,
Qiao D. W.
,
Pei Y.
,
Jung S. Y.
2011
“Reducing Cogging Torque in SurfaceMounted PermanentMagnet Motors by Nonuniformly Distributed Teeth Method,”
IEEE Trans. Magn.
47
(9)
2231 
2239
DOI : 10.1109/TMAG.2011.2144612
Dubas F.
,
Espanet C.
2009
“Analytical solution of the magnetic field in permanentmagnet motors taking into account slotting effect: Noload vector potential and flux density calculation,”
IEEE Trans. Magn.
45
(5)
2097 
2109
DOI : 10.1109/TMAG.2009.2013245
Zhu Z. Q.
,
Howe D.
,
Chan C. C.
2002
“Improved analytical model for predicting the magnetic field distribution in brushless permanentmagnet machines,”
IEEE Trans. Magn.
38
(1)
229 
238
DOI : 10.1109/20.990112
Ackermann B.
,
Sottek R.
1995
“Analytical modeling of the cogging torque in permanent magnet motors,”
Elect. Eng.
78
(2)
117 
125
DOI : 10.1007/BF01245643
Kumar P.
,
Bauer P.
2008
“Improved analytical model of a permanentmagnet brushless DC motor,”
IEEE Trans. Magn.
44
(10)
2299 
2309
DOI : 10.1109/TMAG.2008.2001450
Heller B.
,
Hamata V.
1977
Harmonic Field Effects in Induction Machines
Elsevier Scientific
New York
Liu Z. J.
,
Li J. T.
2008
“Accurate prediction of magnetic field and magnetic forces in permanent magnet motors using an analytical solution,”
IEEE Trans. Energy Convers.
23
(3)
717 
726
DOI : 10.1109/TEC.2008.926034
Liu Z. J.
,
Li J. T.
2007
“Analytical Solution of AirGap Field in PermanentMagnet Motors Taking Into Account the Effect of Pole Transition Over Slots,”
IEEE Trans. Magn.
43
(10)
3872 
3883
DOI : 10.1109/TMAG.2007.903417
Zhu Z. Q.
,
Wu L. J.
,
Xia Z. P.
2010
“An accurate subdomain model for magnetic field computation in slotted surfacemounted permanentmagnet machines,”
IEEE Trans. Magn.
46
(4)
1100 
1115
DOI : 10.1109/TMAG.2009.2038153
Wu L. J.
,
Zhu Z. Q.
,
Staton D.
,
Popescu M.
,
Hawkins D.
2011
“An improved subdomain model for predicting magnetic field of surfacemounted permanent magnet machines accounting for toothtips,”
IEEE Trans. Magn.
47
(6)
1693 
1704
DOI : 10.1109/TMAG.2011.2116031
Boughrara K.
,
Chikouche B. L.
,
Ibtiouen R.
,
Zarko D.
,
Touhami O.
2009
“Analytical model of slotted airgap surface mounted permanentmagnet synchronous motor with magnet bars magnetized in the shifting direction,”
IEEE Trans. Magn.
45
(2)
747 
758
DOI : 10.1109/TMAG.2008.2008751
Zhang R.
,
Wang X. H.
,
Qiao D. W.
2010
“Reduction of exciting force wave for permanent magnet motors by eccentric magnet pole,”
Proceedings of the CSEE
30
(27)
20 
25
Xu Y. Y.
,
Ge H. J.
,
Jing Y.
2013
“Optimal design of eccentric magnet pole for permanentmagnet synchronous motors,”
Journal of Harbin Engineering University
34
(7)
873 
877
Zhou Y.
,
Li H. S.
,
Huang K. F.
2013
“Analytical calculation of airgap magnetic field of the trapezoidal surface permanent magnet,”
Guangzhou, China
Proc. IEEE CSAE
450 
454