This paper presents design procedure of the magnetic bearings used for highspeed electric machines and flywheel energy storage systems. Magnetic bearing can be categorized by innerrotor type and outerrotor type according to the position of the rotary disc. These two types are applicable based on application environments such as application space, required attraction force, and controllability. Magnetic bearing is generally designed based on the ratio (geometrical coefficient or geometrical efficiency) of pole width to rotor journal radius but proper ratio is only decided by the analysis. This is the difficulty of the magnetic bearing design. In this paper, proper design technology of the innerrotor type and outerrotortype eight pole magnetic bearings is introduced and compared with the FEM analysis results, which verifies the proposed design procedure is suitable to be applied to the design of the magnetic bearings for the industrial applications and flywheel energy storage system.
1. Introduction
As the magnetic bearing is a kind of ring support without any contact between rotor and stator, unlike conventional ball bearings which usually need lubricant, it has some merits such as very high speed RPM, low noise level, semipermanent life expectancy, low maintenance cost, and environmentfriendly. These advantages, especially, enable to maximize the ripple effect on very high speed rotor application industries
[1

2]
.
For last three decades, researches on the magnetic bearing have been under active progress in USA, Europe and Japan such as ETH (Eidgenössische Technische Hochschule, Swiss Federal Institute of Technology(Swiss)), S2M(France), University of Virginia(USA), Texas A&M University(USA), NASA(USA), Chiba university(Japan) and National Defense Academy of Japan(Japan)
[3

9]
. From the researches, registration of patents are 183,192 in USA, 32,008 in Europe, 12,872 in Japan and 1,900 in China, respectively.
Magnetic bearing technology can be classified into three great divisions. One of them is pure active magnetic bearing method which is suitable to small rating applications because it should manage the load by itself without any support. Second method is hybrid bidirectional control which has some merits for controllability and power consumption points of view. This hybrid method is to support the load by the repulsive force of permanent magnet and the attraction force of magnetic bearing. Third method is hybrid unidirectional control which uses only unidirectional magnetic bearing for support. In general, first method is widely used because of excellent damping effect to external force but second method is also used in case power consumption is critical.
Therefore, proper magnetic bearing should be designed out of regard for various applicability and system requirement. As magnetic bearing is a noncontact bearing, it should have various uniform airgap according to applicable systems and this makes it difficult to standardize the design procedure of the magnetic bearing.
Magnetic bearing design has been researched by several researchers. Pang D dealt with the optimal design of the pancake type magnetic bearing for fly wheel energy storage devices. Pang focused on the pancake shape and suggested proper design variables. He focused on not the optimized geometrical dimensions but the comparison between different design procedures
[10]
. J Imlach et al dealt with the geometrical dimensions of the poles and electrical characteristics of the coils based on the seven design constraints for four radial magnetic bearings
[11]
. NanChyuan Tsai et al dealt with the optimal bearing shape in order to minimize electrical energy consumption for fly wheel energy storage devices
[12

14]
. However, these design researches were limited in the innerrotor type magnetic bearings and put forth a multilateral effort into solving a problem of design optimization which is very complicated.
This paper presents a design procedure of both innerand outerrotor type magnetic bearings and verifies usefulness of the proposed design procedure by comparing with the simulation results of numerical analysis and finite element method.
2. Fundamental Formula for Magnetic Bearing
Fig. 1
shows a basic schematic diagram of an innerrotor type 8 pole magnetic bearing. Magnetic bearing basically controls the airgap by using the attraction force of the electromagnets and this attraction force is decided by the geometry of magnetic bearing. Magnet width
W_{p}
of a magnetic bearing as shown in
Fig. 1
, is related to the attraction force, and diameter D of the inner rotor of a magnetic bearing is related to the surface area which affect to the attraction force. Therefore, the attraction force of a magnetic bearing can be represented by the geometrical coefficient (geometrical efficiency), σ as in (1).
Eight pole magnetic bearing
here,
(
n_{p}
is number of pole).
(hence, sin45˚ is ideal for the 8 pole magnetic bearing)
The attraction force can be represented by Maxwell equation as in (2).
here,
B
is the magnetic flux density,
A_{g}
is the crosssectional area of the airgap,
μ_{o}
is the permeability of the air.
In (2),
A_{g}
is the product of
W_{p}
and rotor journal length,
L_{j}
.
From (1) and (2), the attraction force can be represented by (3).
By inserting (4) into (3), the attraction force of a magnetic bearing including the geometrical coefficient (geometrical efficiency) can be represented in (5).
Maximum attraction force per 1 pole is obtained by the maximum available current,
 2.1 Innerrotor type magnetic bearing
Design parameters for an innerrotor type magnetic bearing is shown in
Fig. 2
. Decision of the winding areas is very important factor to secure the performance because the attraction force is proportional to the square of the currents and square of the number of turns. And decision of the outer diameters of the magnetic bearing and rotor which are determined by the system constraint, should be done at the very first beginning. In case a magnetic bearing is applied to a highspeed rotating machine, the outer diameter of the rotor can be decided by the standard of the backup bearing which is to prevent from any contact between rotor and stator.
Innerrotor type magnetic bearing
In this paper, 70mm of inner diameter angular contact from NKS is chosen as the backup bearing (ball bearing) and 69.4mm of outer diameter for inner rotor is decided by considering the 0.6mm of airgap. The ratio of the sum of pole width to circumference of the inner rotor is defined as Iron ratio,
r_{i}
that is,
which is 0.9 (40˚) in this paper. The outer radius,
R_{1}
should be decided by considering space constraint and outer radius of the rotor, which will be explained in the next section. From these constraints and (1),
W_{p}
is as in (6).
In the
Fig. 2
, section shaded is the winding area which is product of
t_{c}
and
l_{c}
.
Here,
Number of winding can be calculated as in (9) by the winding area from (7) and (8).
here,
β
is the packing factor.
 2.2 Outerrotor type magnetic bearing
Outerrotor type magnetic bearing is shown in
Fig. 3
. Design procedure of the outerrotor type is same as the innerrotor type except winding area from different airgap location. That is,
R_{4}
of the outerrotor type is the inner radius of the rotor (
R_{4}
=
D
/2), while
R_{3i}
of the innerrotor type is the inner radius of the rotor and airgap (
R_{3i}
=
D
/2+
g
_{o}
). This difference affects to the calculation of
t_{c}
and
l_{c}
. Therefore, in case innerrotor type and outerrotor type have the same outer radius (
R_{1}
), followings are obtained.
R_{2} = R_{1} −W_{p} , R_{3o} = R_{1} −W_{p} − g_{o} ,
Outerrotor type magnetic bearing
As the final outcome, the performance of the innerrotor type and outerrotor type magnetic bearings becomes different from the different winding areas.
3. Simulation Results of the Finite Element Method
In this section the innerrotor type and outerrotor type magnetic bearing design is dealt with based on the design procedures mentioned in the previous section by using finite element method and also the comparison for the load capacity between the innerrotor type and outerrotor type is performed. Commercial magnetic field analysis program, Maxwell was employed for the finite element analysis. For the design of the two different types of the magnetic bearings, authors assume that the inner diameter of the rotor,
D
is 69.4mm, the air gap,
g
_{o}
is 0.6mm, the journal length in the axial direction,
L_{j}
is 35mm. Three different kinds of dimensions for the outer radius (
R_{1}
=75mm, 88mm, 100mm) are selected arbitrary. Also authors assume that the geometrical efficiency, σ which presents the relation between the width of the magnetic bearing pole and the rotor inner radius has 0.01 ~ 0.7.
Table 1
shows the design parameters.
Design variables of the magnetic bearings
Design variables of the magnetic bearings
Fig. 4
presents the relation between geometrical efficiency and the width of the magnetic bearing pole when the rotor journal radius is predecided based on the design requirement. As seen in the figure when the geometrical efficiency is 0.5, the rotor inner radius and pole width of the magnetic bearing is same, which means that
R_{5}
and
W_{p}
are decided based on the σ.
The relations between geometrical efficiency and rotor inner radius and pole width of the magnetic bearings
Fig. 5
shows the magnetic force produced from the magnetic bearings of the innerrotor type and outerrotor type. As seen in the figure if the magnetic saturation of the magnetic core is not considered the maximum magnetic force is around σ = 0.2.
Maximum magnetic force (Without considering the magnetic saturation of the core)
Considering the magnetic saturation of the magnetic core, as seen in
Fig. 6

7
, the maximum magnetic force is at 0.33, 0.45, 0.55 of σ.
Magnetic force for the innerrotor type (Considering the saturation of the magnetic core)
Magnetic force for the outerrotor type (Considering the saturation of the magnetic core)
These results mean that when the magnetic saturation is not considered the narrower the pole width of the magnetic bearing the lower σ, instead the wider coil area the greater magnetic force.
However it is impossible that such magnetic force can be produced. When the magnetic saturation is considered the σ moves to around 0.5 (in this paper
B_{max}
=1.2T). This means that σ moves in the direction for
W_{p}
and
R_{5}
to be same length, and means that in order to suppress the magnetic saturation
W_{p}
should be wider.
Fig. 8
indicates the magnetic force of the two rotor types when the magnetic saturation is considered. The magnetic force moves in the direction that pole width
W_{p}
becomes wider when the outer radius becomes greater, and the magnetic force of the innerrotor type is greater than that of the outerrotor type. This result presents that the load capacity of the innerrotor type magnetic bearing is superior to that of the outerrotor type magnetic bearing.
Magnetic force considering the saturation
Figs. 9

11
show that the magnetic flux calculated by finite element method and numerical analysis of the magnetic force for the innerrotor type magnetic bearing. In the FEM analysis maximum flux density is 1.2 Tesla for the silicon iron (SiFe).
FEM analysis (Inner OR 75mm)
FEM analysis (Inner OR 88mm)
FEM analysis (Inner OR 100mm)
The greater OR(outer radius) the greater maximum magnetic force as shown in those figures.
Fig. 12
shows the maximum magnetic force of the magnetic bearing that has been designed through the design procedures proposed in this paper and the maximum magnetic force calculated by the FEM, and also indicates that the maximum magnetic force calculated by those two methods is almost same.
Comparison between numerical and FEM analysis (Innerrotor type)
Figs. 13

15
show that the magnetic flux calculated by finite element method and numerical analysis of the magnetic force for the outerrotor type magnetic bearing. The greater OR(outer radius) the greater maximum magnetic force as shown in the innerrotor type.
Fig. 16
also shows that the maximum magnetic force calculated by FEM and the maximum magnetic force calculated by numerical analysis proposed in this paper are almost same.
FEM analysis (Outer OR 75mm)
FEM analysis (Outer OR 88mm)
FEM analysis (Outer OR 100mm)
Comparison between numerical and FEM analysis (Outerrotor type)
4. Conclusion
This paper presented a design procedure of both innerand outerrotor type magnetic bearings and verified usefulness of the proposed design procedure by comparing with the simulation results of numerical analysis and finite element method. From the results, it was proved that the load capacity depended on the geometrical coefficient (geometrical efficiency), and relation between magnetic force and according to the various outer radius. The results of the maximum attractive magnetic force calculated by numerical analysis also were almost same with that of the finite element method analysis, which means that it is possible for the proposed design procedures to be employed for the design of the magnetic bearing system and also to be applied for the industrial applications and flywheel energy storage system.
BIO
JunHo Lee He was born in Seoul, Korea on July 3, 1964. He received the B.E. degree, M.E. degree from KwangWoon University, Seoul, Korea, and doctoral degree in electrical engineeering from Kanazawa National University, Kanazawa, Japan, in 1987, 1989 and 1998, respectively. He worked at the Department of Mechanical and Aerospace Engineering, University of Virginia, USA from 1998 to 2004 as a senior scientist and joined to KRRI(Korea Railroad Research Institute) from 2005. He is currently doing research for the development of the flywheel energy storage system to be applied for the railway system. His research areas of interest are application of modern control theory, design of robust controller, in vibration control of magnetic bearing system, in control type magnetic levitation system and the design of the wireless power transfer system. Fig. 16 Comparison between numerical and FEM analysis (Outerrotor type)
ChanBae Park He received the M.S. degree in electrical engineering from Seoul National University, Seoul, Korea, in 2003 and the Ph.D. degree in electrical engineering from Hanyang University, Seoul, Korea, in 2013. From 2003 to 2006, he worked as Senior Engineer in Digital Appliance R&D Center at Samsung Electronics. Since 2007, he has been a Senior Researcher at the Korea Railroad Research Institute, Uiwang, Korea. His research interests include design and analysis of motor/generator, transformer, and superconducting devices for energy conversion systems. Currently, he is working for development of propulsion systems for railway transit.
ByungSong Lee He received the M.S. degree and Ph. D degree in electrical engineering from ChungAng University, Seoul, Korea, in 1991 and 1995, respectively. From 1996 to 1997, he worked as Senior Researcher in Railway Vehicle R&D Center at KHRC(Korea High speed Rail Construction Authority)., where he developed propulsion system for high speed train. Since 1998, he has been a Senior and Principal Researcher at the Korea Railroad Research Institute, Uiwang, Korea. His research interests include IPT(Inductive Power Transfer) and energy conversion systems for railway train. Currently, he is working for development of IPT system for railway vehicles.
SuGil Lee He received the M.S. and Ph.D. degree in electrical engineering from SoongSil University, Seoul, Korea, in 1997 and 2010. Dr. Lee is currently a principal researcher at the Department of Propulsion System at the Korea Railroad Research Institute in Uiwang. His main research interest is propulsion control system design and hybrid tilting control of electrical railway vehicles.
JaeHee Kim He received the B.S. degree in electrical engineering from Korea University, Seoul, Korea, in 2005 and the Ph.D. degree in electrical engineering from POSTECH, Pohang, Korea, in 2010. His research interests include design and analysis of array antennas, mobile phone antennas and metamaterials. Currently, he is working for development of wireless power transfer systems for railway.
ShinMyung Jung He was born in Korea, in 1981. He received the B.S., M.S. and Ph.D. degrees in electrical engineering from the Korea Advanced Institute of Science and Technology, Daejeon, Korea, in 2004, 2006, and 2013, respectively. Since 2013, he has been a senior researcher at Korea Railroad Research Institute, Uiwang, Gyeonggido, Korea. His research interests include power electronics and control, which include motor drives, fault tolerant control, and highperformance switching regulators.
HyungWoo Lee He received the B.S. and M.S. degrees from Hanyang University, Seoul, Korea, in 1998 and 2000, respectively, and the Ph.D. degree from Texas A&M University, College Station, TX, in 2003, all in electrical engineeering. In 2004, he was a Postdoctoral Research Assistant in the Department of Theoretical and Applied Mechanics, Cornell University, Ithaca, NY. In 2005, he was a contract Professor at the BK division of Hanyang University, Seoul, Korea. From 2006 to 2012, he worked as a senior researcher at the Korea Railroad Research Institute, Uiwang, Korea. He joined Korea National University of Transportation as professor of the department of Railway Vehicle System Engineering in March 2013. His research interests include design, analysis and control of motor/generator, power conversion systems, and applications of motor drives such as Maglev trains, conventional railway propulsion systems, and modern renewable energy systems.
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