The rotor overhang is used to enhance the airgap flux and improve the power density. Due to the asymmetry in the axial direction caused by the overhang, a time consuming 3D analysis is necessary when designing a motor with overhang. To solve this problem, this paper proposes an equivalent magnetic circuit model (EMCM) which takes account overhang effects without a 3D analysis by using effective airgap length. The analysis time can be reduced significantly via the proposed EMCM. A reduction in the analysis time is essential for a preliminary design of a motor. In order to verify the proposed model, a 3D finiteelement method (FEM) analysis is adopted. 3D FEM results confirm the validity of the proposed EMCM.
1. Introduction
Permanentmagnet (PM) brushless machines are increasingly being used in various applications, such as electric vehicles, industrial servos and in wind power generation systems
[9]
. This increased popularity is due to their high torque, high power density, high efficiency, and low maintenance requirements as a consequence of the use of PM materials in the rotor. Surfacemounted permanentmagnet (SPM) motors have the advantages of low torque ripple and low cogging torque as compared with interior permanentmagnet (IPM) machines.
The overhang is defined in this paper as a configuration that the rotor length is longer than the stator length in the axial direction. In general, the overhang structure is used to enhance the airgap flux and improve the power density while utilizing the free space caused by the stator end winding
[29]
. A 3D finiteelement method (FEM) is necessary to analyze the magnetic fields in the axial direction for a proper consideration of the overhang effects. Although FEM can precisely obtain the magnetic flux distribution and electromagnetic performances of electrical machines
[22

24]
, it is time consuming and computationally expensive, especially at predesign stages.
To reduce computational time for the motor design process, analytical methods are essential during the preliminary design of electric machines. Two types of analytical methods are usually used. The first one is based on the formal solution of Maxwell’s equations. It can calculate parameters and performances of electric machines with high accuracy
[1

5
,
8
,
13
,
15
,
19

21
,
27
,
28
,
30]
. Also, this method can be used for analyzing practical electrical machines with the geometric complexity, such as synchronous reluctance motors
[15]
, switched reluctance machines (SRM)
[27]
, SPM machines
[2
,
4
,
5
,
30]
, slotless motors
[19
,
20]
, and PM actuator
[13
,
21]
. The second method is an equivalent magnetic circuit model (EMCM). As an EMCM has the advantage of simplicity compared to the aforementioned method, it is widely used as an analytical method for various types of machines
[6
,
10

12
,
14
,
16

18
,
25
,
26
,
31
,
32]
, such as SPM machines
[10
,
12
,
18]
, IPM machines
[16
,
31]
, fluxswitching PM machines
[32]
, SRM
[25]
, axial flux PM motors
[14
,
17]
, and induction machines
[26]
.
In the case of SPM motors, an EMCM required to predict the airgap and magnet flux density analytically has been developed while taking into account the leakage flux around the magnets in the rotor
[18]
. However, overhang effects in SPM machines with rotor overhang are presently not included in the EMCM. In this paper, we propose an EMCM considering not only the leakage flux in the rotor but also the overhang effects for SPM machines with rotor overhang. A 3D FEM analysis is used to verify the proposed model.
2. Analytical Model of SPM Machines with Overhang
Although the effects of the stator slots have been taken into account
[10

12
,
14
,
16

18
,
25
,
26
,
32]
, an EMCM which disregards slotting effects was developed in order to focus on the overhang effects and simplify the analytical model in this paper. It is assumed that the effects of saturation within the core are negligible, as the magnetic flux density within the core is low in general and because normally there is no significant saturation.
With the motor topology generalized as the linear translation type shown in
Fig. 1
, it is possible to derive an EMCM applicable to any specific topology. In order to enhance the airgap flux and improve the power density, the overhang structure is used in the linear translation motor here, as shown in
Fig. 2
.
Linear transfer motor topology
Configuration of a motor with overhang
Qu
et al
. developed an EMCM for SPM machines that takes into account the airgap leakage fluxes around the magnets
[18]
. The airgap leakage fluxes consist of the magnettomagnet leakage flux and the magnettorotor leakage flux. With the circulararc straightline permeance model
[7]
, these leakage fluxes were modeled. However, this analytical model cannot be applied to SPM motors with overhang because the overhang results in asymmetry of the magnetic fields in the axial direction.
An EMCM based on the analytical model developed by Qu
et al
. considering overhang effects as well as airgap leakage fluxes is proposed.
Fig. 3
shows the proposed EMCM. The variables shown in
Fig. 3
are defined as follows:
Equivalent magnetic circuit for Fig. 1 and Fig. 2

Φg: the airgap flux excited by one magnet pole

Φr: the flux source of one magnet pole

Rg: the reluctance corresponding to Φg

Rmo: the reluctance corresponding to Φr

Rr: the reluctance of the rotor yoke

Rs: the reluctance of the stator yoke

Rmm: the reluctance corresponding to the magnettomagnet leakage flux

Rmr: the reluctance corresponding to the magnettorotor leakage flux.
R_{r}
and
R_{s}
can be ignored due to the assumption mentioned earlier.
Fig. 3
can be simplified into
Fig. 4
through the use of symmetry. In
Fig. 4
,
R_{m}
is calculated from
Fig. 3
as
Simplified circuit of Fig. 3
where
and
For the EMCM considering the overhang effect, the effective airgap length should be defined. Flux lines between overhang structure and stator core are modeled with consideration for the distribution of magnetic flux density in the air gap.
Fig. 5
shows modeling of flux lines in the air gap with a circular arc and a straight line. The magnetic flux distribution in the air gap which is calculated through FEM is shown in
Fig. 6
. The effective airgap length is derived via the modeling of flux lines with a circular arc and a straight line as
Modeling of the flux lines in the air gap with a circular arc and a straight line: (a) A straightline permeance model; (b) A circulararc permeance model
Distribution of magnetic flux density by FEM
where
z
is the position in the overhang, as shown in
Fig. 5
, and
g
is the airgap length.
The permeances in the nonoverhang region is derived in
[18]
as
where
μ
_{0}
is the permeability of air,
μ_{r}
is the magnet relative recoil permeability,
L_{st}
is the stator stack length,
g
is the airgap length,
w_{f}
is the width between two adjacent magnets,
w_{m}
is the magnet width,
h_{m}
is the magnet length, and min(·) is the minimum function.
With the effective airgap length
g_{e}
, the permeances of the infinitesimal stack length
d_{z}
in the overhang can be derived. The permeances can be calculated by integrating the permeances of the infinitesimal stack length over the overhang length, as (9)(12), where
L_{oh}
is the rotor overhang length.
With the permeances in the nonoverhang region and the overhang region, the permeances for the entire motor with overhang are derived as follows:
Using the equations above and the reciprocal relationship between the permeance and reluctance, the reluctances are calculated as follows:
According to the flux division, the airgap flux and the flux from the magnet can be derived as
and
The airgap flux density and the magnet flux density can be induced by
and
where
and
3. Results and Verification
For the verification of the proposed analytical model, the results from a FEM and the analytical results calculated from (23) and (24) are investigated. The investigations are conducted in various cases with different values of
g
,
w_{f}
and
L_{oh}
.
Table 1
and
Table 2
show the results for a motor employing ferrite magnets with
B_{r}
= 0.4 T and rare earth magnets with
B_{r}
= 1.07 T, respectively.
B_{m}
and
B_{g}
of FEM results is average value.
B_{m}
is calculated by averaging flux density in the middle of a magnet though rotor stack length.
B_{g}
is calculated by averaging flux density in the middle of the air gap though rotor stack length.
Comparison between the analytical results and the FEM results of the motor employing ferrite magnets
At L_{st} =100.0 mm, h_{m} = 4.0 mm, w_{m} = 20.0 mm, μ_{r} = 1.05, and B_{r} = 0.40. *: Difference = (3D FEM results − Analytical results) / 3D FEM results × 100.
Comparison between the analytical results and the FEM results of the motor employing rare earth magnets
At L_{st} =100.0 mm, h_{m} = 4.0 mm, w_{m} = 20.0 mm, μ_{r} = 1.0384, and B_{r} = 1.07. *: Difference = (3D FEM results − Analytical results) / 3D FEM results × 100.
As shown in
Table 1
and
Table 2
, there is little difference between results of the motor employing the ferrite magnet and the rare earth magnet. The accuracy of the EMCM is affected by
L_{oh}
. As the length of the overhang increase, the leakage flux at the end of the overhang is increased and the estimation of the magnetic flux path becomes difficult. This increases the difference between the analytical results and the FEM data. In cases in which
L_{oh}
is less than 7 mm, the differences of
B_{m}
and
B_{g}
are less than 3% and 2%, respectively. The differences of
B_{m}
and
B_{g}
are respectively less than 5% and 3% when
L_{oh}
is 10mm.
In the condition with same stator stack length, the total flux passing through airgap increases when
L_{oh}
increases. However, the average flux densities
B_{m}
and
B_{g}
decline as the overhang increases, as shown in
Table 1
and
Table 2
, this is because the area is more increased compared to the flux. In cases in which
L_{st}
is 100 mm, the total flux passing through the air gap increases by about 11% and 15% when
L_{oh}
is 7 mm and 10 mm, respectively.
4. Conclusion
This paper is noteworthy in that the time for design of SPM machines can be reduced remarkably via the reduction of time for initial design by using the proposed EMCM, which is analytical method considering overhang effects.
BIO
HanKyeol Yeo He received B.S. degree in electronic and electrical engineering from Sungkyunkwan University, Suwon, Korea, in 2012. He is currently working toward the Ph.D. degree in electrical engineering from the Seoul National University, Seoul, Korea. His current research interests include numerical analysis and design of electrical machines.
DongKyun Woo He received B.S. degree in electrical engineering from Yonsei University, Seoul, Korea, in 2007. In 2014, he received a Ph.D. in electrical engineering from Seoul National University, Seoul, Korea, through the Combined Master’s and Doctorate Program. Currently, he is carrying out research at Power & Industrial Systems R&D Center of Hyosung from 2014. His current research interests include numerical analysis and design of electrical machines.
DongKuk Lim He received B.S. degree in the electrical engineering from Dongguk University, Seoul, Korea, in 2010. He is currently working toward the Ph.D. degree in electrical engineering and computer science from the Seoul National University, Seoul, Korea. His current research interests include design of electrical machines.
JongSuk Ro He received B.S. degree in mechanical engineering from Han Yang University, Seoul, Korea, in 2001. In 2008, he earned a Ph.D. in electrical engineering from Seoul National University, Seoul, Korea, through the Combined Master’s and Doctorate Program. He conducted research on cellular phone modules at R&D center of Samsung Electronics as a Senior Engineer from 2008 to 2012. From 2012 to 2013, he was at Brain Korea 21 Information Technology of Seoul National University as a Post Doctoral Fellow. Currently, he is carrying out research at Electrical Energy Conversion System Research Division of Smart Grid Team at Korea Electrical Engineering & Science Research Institute as a Researcher. His research interests are numerical analysis and optimal design of electric machines.
Hyun Kyo Jung (S’82M’90SM’99) He received the B.S., M.S., and Ph.D. degree in Electrical engineering from the Seoul National University, Seoul, Korea, in 1979, 1981, and 1984, respectively. From 1985 to 1994, he was a member of the faculty with Kangwon National University. From 1987 to 1989, he was with the Polytechnic University of Brooklyn, Brooklyn, NY. From 1999 to 2000, he was a Visiting Professor with the University of California at Berkeley. He is currently a Professor at the School of Electrical Engineering and Computer Science/Electrical Engineering, Seoul National University. His research interests are the analysis and design of the electric machine.
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