It is difficult to calculate the magnetic force of an object of magnetic material in contact with other objects using the existing methods, such as Maxwell stress tensor method, magnetic charge method, or magnetizing current method. These methods are applicable for force computation only when the object is surrounded by air. The virtual airgap concept has been proposed for calculating the contact force. However, its application is limited to magnetostatic system. In this paper, we present the virtual airgap concept for contact surface force in the eddycurrent system. Its validity and usefulness are shown by comparison between numerical and experimental examples.
1. Introduction
The magnetic force is an important design factor for the mechanical structural design of electric power apparatus. Large magnetic force on a part of the system can cause mechanical problems such as deformation, vibration, noise, and even fracture, either in the part, or in neighboring parts. To avoid such mechanical problems, accurate analysis of the magnetic contact force will help in designing and manufacturing of the systems
[1

4]
.
Maxwell stress tensor method, magnetic charge method and magnetizing current method are typical ways to calculate the magnetic force. These methods are effective only when calculating the total magnetic force of an isolated object that is surrounded by air. Since the integration path for calculating the force should be taken in the air, it is difficult to compute the force between contacting objects using such existing methods. To solve this problem, the virtual airgap concept was introduced, and it has only been used for magnetic systems that have large flux leakage without eddycurrent
[5]
.
This paper presents an analysis method for the contact force in two different electromagnet models, the Imodel and the Emodel, in both magnetostatic and eddycurrent systems; and its effectiveness is verified by experiment.
2. Maxwell Stress with the Virtual Airgap
In
Fig. 1
, there is no airgap between material 1 and material 2. However, we can imagine an airgap, which exists on the contact surface with distance 0[mm]. That is the basic concept of the virtual airgap.
The magnetic field with virtual airgap concept continuity, respectively.
The field intensity inside the virtual airgap is derived as
In the above expression,
H
_{vg _ from1}
and
H
_{vg _ from2}
are, respectively,
and
where, 1 and 2 are the material number’s,
n
and
t
are a normal and a tangential unit vector, and
μ
_{0}
is the permeability in the air.
B
_{1n}
and
H
_{1t}
are the normal component of the magnetic flux density and tangential component of magnetic field intensity in the material 1, respectively.
B
_{2n}
and
H
_{2t}
are the normal component of the flux density and tangential component of the field intensity in material 2. So, the field intensity in the virtual airgap
H
_{vg}
is
The virtual airgap gap field that satisfies the boundary conditions comes from each field in the material 1 and the material 2. In other words,
B_{n}
and
H_{t}
should satisfy the continuity, respectively.
where,
B
_{r}
is the residual flux density, and
M
_{0}
is the magnetization of permanent magnet materials
[5

7]
.
Therefore, the Maxwell stress on the contact surface in magnetic system is
The above equation is the same form as normal Maxwell stress. However, the magnetic field that is used to calculate stress tensor is not a magnetic field in the materials, but the virtual airgap field. When the integration surface is with the air, the virtual airgap field is the same with as the magnetic field at that surface.
3. Numerical Analysis and Experiment of Electromagnet Models
Electromagnet models are divided into Imodel and Emodel according to flux leakage. In contrast to Emodel, the large flux leakage occurs in Imodel. Two models are designed, and its numerical analysis and experiment are carried out in two systems. One is magnetostatic system and the other is eddycurrent system.
 3.1 Numerical analysis
In magnetostatic system, the input current, 4[A]×907[turns], is applied to Imodel, and the force is calculated in the range of 0~5[mm]. In Emodel analysis, the input current is 0.3[A]×83[turns] and the calculation range is 0~1[mm].
In eddycurrent system, the peak value of input current is 2.4[A]×907[turns], and the force is calculated with airgap from 0 to 5[mm] in Imodel. The peak value of input current is 0.42[A]×83[turns] and the force is calculated with airgap from 0 to 1[mm] in Emodel. The frequency of eddycurrent system is 1[Hz], which is enough to observe the eddycurrent.
It is difficult to consider fringing effect in 2D analysis for Imodel. So, the 3D analysis should be carried out. However, there is limitation to analyze Imodel when the airgap becomes smaller due to mesh problem. To solve this problem, 3D analysis results under 1[mm] airgap is obtained using 2D analysis results because the difference between 2D and 3D model depends on only geometry with the same material properties. The ratio coefficient is calculated using 2D and 3D analysis result near 1[mm] airgap.
In Emodel, the field fringing in z direction is much smaller than the field fringing in x direction. So, we only analyze 2D model.
 3.1.1 Numerical analysis method
The force is calculated using Maxwell stress in numerical analysis. When the iron block is contact with the electromagnet, the contact force is calculated using two ways, which are only Maxwell stress and Maxwell stress with the virtual airgap concept, respectively.
Maxwell stress with the virtual airgap concept in magnetostatic model is written as
where
σ_{svg}
is Maxwell stress tensor with the virtual airgap concept.
B_{vg}
and
H_{vg}
are
B
and
H
in the virtual airgap.
n
is normal unit vector.
x
and
y
in subscript is a direction of vector.
The maximum intensity of magnetic flux density in eddycurrent system is less than 0.3[T]. The magnetic permeability of ferrite is almost linear in this resign. So, timeharmonic analysis is used for the eddycurrent system. Time average Maxwell stress with virtual airgap concept in the eddycurrent system is as follows:
where 𝕽 is real function and ‘ * ’ is conjugate operator.
 3.1.2 Geometry and material properties of the numerical models
The core in electromagnet is ferrite and the magnetic material, the iron block, is S18. Size and shape of analysis models are shown in
Fig. 2
. The conductivities of ferrite and S18 are 0[S/m] and 6.17 × 10
^{6}
[S/m] and its BH curve are shown in
Fig. 3
.
The (a) Imodel for 2D analysis, (b) Imodel for 3D analysis and (c) Emodel for 2D analysis
(a) BH curve for ferrite and (b) BH curve for S18
 3.2 Experiment
In the experiment of the magnetostatic system, the minimum distance to measure the magnetic force exists between the iron block and the electromagnet. When the distance is smaller than 0.05[mm], the iron block contacts partially with the electromagnet due to the inhomogeneity of material surface. So, it is difficult to measure the magnetic force acting on the iron block using the pushpull gauge.
In the eddycurrent system, the minimum distance is changed to 0.06[mm] due to the vibration of the iron block.
The input current in each experiment is the same with the numerical analysis.
 3.2.1 Measuring equipment for magnetic force
The magnetic force is measured using a pushpull gauge, which has a −50~50[N] measuring range, 0.01[N] accuracy, and 0.02[s] sampling time. The jig to hold the electromagnet and the magnetic material is MCnylon, which is an inelastic, dielectric and nonmagnetic material. The motor controlled gauge stand can move 30~300 [mm/min]. All bolts and nuts are nonmagnetic material. The power supply is controllable 0~150[V] with 0.1[V] step and 0~60[Hz] with 1[Hz] step.
 3.2.2 Geometry and material properties of the experimental models
In
Fig. 5(a)
, Imodel, the iron (S18) block size is 20 × 30 × 15[mm], the magnetic core (Ferrite) size is 20 × 30 × 80[mm] and the electromagnet is bounded with 907[turns] coil.
Measuring equipment for magnetic force
The (a) Imodel and (b) Emodel
In
Fig. 5(b)
, Emodel, the iron block size is 42 × 40 × 10[mm], the magnetic core size is 42 × 40 × 21.2[mm] and the electromagnet is bounded with 83[turns] of coil.
All materials used in experiments and numerical analysis are the same.
 3.2.3 Experimental procedures
The experiment in magnetostatic system is carried out. The iron block and the electromagnet are fixed at the pushpull gauge and the stand using jig. The pushpull gauge is set zero and enter DC input. The iron block goes down slowly using the electric stand. Regarding the iron block is contact with the electromagnet, when the push force shown in the pushpull gauge. While the iron block rise and lower 0.5[mm/s] velocity, the pull force is recorded in every 0.02[s]. This process is repeated and calculate the average value at each distance.
The experiment in eddycurrent system is as follows. The geometrical setting is the same as for magnetostatic experiment. After the zero point is set, the AC input is entered.
It is difficult to measure the force varying with the distance when the force also varies with time. So, the force is measured during 7[s] when the iron block is held at each position. The time average force at each position is calculated using the measured data except for the first and the last second one.
4. Results and Discussions
Compare experimental results and numerical results, we show the numerical models are reflected the experimental models well. And compare the contact force computed by the Maxwell stress with the virtual airgap concept
F_{vg}
, the contact force computed by Maxwell stress only
F_{c}
and the force at numerical minimum distance
F_{mn}
. Through this, we show the validity of virtual airgap concept.
The distance between the iron block and the electromagnet is
d
.
 4.1 The Imodel
The Imodel has large flux leakage. So, in this model, the magnetic force slowly increases when the iron block approaches the electromagnet.
 4.1.1 The Imodel in magnetostatic system
In
Fig. 6
, the magnetic force is 9.30[N] when
d
is 1[mm] in the experiment. The magnetic force is 2.85[N] in 2D model and 11.56[N] in 3D model at the same
d
. The force tendency between 3D nuamerical results and experimental results is similar.
The magnetic force varying with distance in magnetostatic system for the Imodel
When the distance is near 1[mm], the ratio between model and 3D model is chosen as 4, which is based on the analysis results.
The 3D analysis results calculated using the 2D analysis results are
F_{vg}
=16.17[N],
F_{c}
=4.41[N] and
F_{mn}
=16.32 [N]. The force,
F_{me}
, at minimum distance is 16.33[N] in the experiment.
 4.1.2 The Imodel in eddycurrent system
In
Fig.7
, the time average magnetic force is 2.03[N] in the experiments when
d
is 1[mm]. The force is 0.58[N] in the 2D model and 2.15[N] in the 3D model at same distance.
The magnetic force varying with distance in eddycurrent system for the Imodel
The ratio between 2D and 3D analysis is also 4 in eddycurrent system. Considering the ratio, the 3D analysis results are
F_{vg}
=3.28[N],
F_{c}
=0.79[N], and
F_{mn}
=3.28[N]. The magnetic force,
F_{me}
, at minimum distance is 3.15[N].
 4.2 The Emodel
The small airgap makes large field change in the Emodel. So, the magnetic force rapidly increases when the iron block approaches the electromagnet.
 4.2.1 The Emodel in magnetostatic system
In
Fig. 8
, the force at experimental minimum distance, 0.05[mm], is 17.55[N] in the experiments. At the same distance, the magnetic force is 16.39[N] in the numerical analysis result. The magnetic force
F_{mn}
is 68.76[N] at the numerical minimum distance, 0.01[mm]. The virtual airgap force,
F_{vg}
, is 85.50[N] and the contact force,
F_{c}
, without the virtual airgap is 0.19[N].
The magnetic force varying with distance in magnetostatic system for the Emodel
 4.2.2 The Emodel in eddycurrent system
In
Fig. 9
, the force at experimental minimum distance, 0.06[mm], is 9.11[N] in the experiments. At same distance, the magnetic force is 9.28[N] by numerical analysis. Where the numerical minimum distance, the magnetic force
F_{mn}
is 66.8[N]. The virtual airgap force,
F_{vg}
, is 84.5[N] and the contact force,
F_{c}
, is 0.11[N].
The magnetic force varying with distance in eddycurrent system for the Emodel
 4.3 Discussions
The numerical analysis results agree with the experimental ones. The virtual airgap concept is verified comparing the three forces, which are the magnetic contact force with the virtual airgap, the magnetic contact force without the virtual airgap and the magnetic force at numerical minimum distance.
 4.3.1 The system energy and the continuity of the force
Force can be defined as the differential of the system energy. If the system energy is differentiable, the force should be continuous. In
Fig. 10
, the system energy in eddycurrent system, Emodel, is shown at the gap. The system energy changes smoothly until the iron block is in contact with the electromagnet. The magnetic force is continuous in this model. However, the Maxwell stress without the virtual airgap concept yields discontinuous results.
Emodel system energy in eddycurrent system according to distance
 4.3.2 Magnetic field and virtual airgap field
Fig. 11
shows the Emodel in eddycurrent system, and the red lines in each figure is the surface of the iron block. There is a 0.01[mm] airgap between the iron block and the electromagnet in
Fig. 10(a)
, while the iron block is in contact with the electromagnet in
Fig. 10(b)
.
The (a) surface of the iron block when the airgap is 0.01[mm] and (b) the contact surface
Fig. 12(a)
shows the normal component of the magnetic field when the airgap is 0.01[mm]. And
Fig. 12(b)
shows the magnetic field when the iron block is in contact with the electromagnet. The maximum value of magnetic field in each states are 9 × 10
^{5}
[A/m] and 1.7 × 10
^{4}
[A/m]. The former is about sixty times stronger.
Magnetic field on (a) surface of the iron block when the airgap is 0.01[mm] and (b) the contact surface
The magnetic force is normally proportional to the square of the field intensity. Therefore, this field differences make a discontinuous force. This result shows that the magnetic field is not continuous at the boundary of different materials. However, the magnetic force should be continuous in this system.
Fig. 13
shows the virtual airgap field on the contact surface. Its field tendency is similar to the field when the airgap is 0.01[mm]. Concretely, the maximum value of magnetic field, 6 × 10
^{6}
[A/m], is sixth times stronger than the magnetic field at 0.01[mm] airgap. However, when the edge effect is ignored, the virtual airgap field is only 1.2 times stronger than the magnetic field with minimum gap.
Virtual airgap field on contact surface
5. Conclusion
In the analysis models, the magnetic force increases as the gap between the electromagnet and the iron block becomes smaller. The force varies continuously when the system energy change smooth. However, the discontinuity occurs in the analysis results obtained using only the existing Maxwell stress tensor when the contact force is compared with the force with the minimum gap. It comes from the Maxwell stress which is only defined in air.
In contrasts, the contact force obtained using the Maxwell stress with virtual airgap concept increases continuously as the gap becomes smaller.
Therefore, the virtual airgap concept is a tool that should be used to calculate the contact force for not only magnetostatic systems but also eddycurrent systems in various shapes.
Acknowledgements
This work is supported by the Human Resources Development program(No. S20141172000) of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korea government Ministry of Trade, Industry and Energy and by Korea Ministry of Environment as EcoInnovation R&D Project(No. S20140712000).
BIO
ByungSu Park He received B.S, and M.S. degrees in engineering from Sungkyunkwan University. His research interests are electromagnetic contact force and numerical analysis of eddy current system.
HwiDae Kim He received B.S degree in electronic and electrical engineering from Sungkyunkwan University. His research interests are Electromagnetics, Magnetic Material and Numerical analysis.
HongSoon Choi received the B.S., M.S., and Ph.D. degrees in electrical engineering from Seoul National University, Seoul, Korea, in 1986, 1988, and 2000, respectively. From 1988 to 1994, he was a Senior Research Engineer with Samsung Electro Mechanics Company. From 1995 to 1997, he was a Senior Researcher with Korea Electrical Engineering and Science Research Institute. From 1997 to 2003, he was a cofounder and a Research Director of KOMOTEK Company which develops and produces precision motors. From 2003 to 2006, he was a Research Professor with Sungkyunkwan University. Since 2007, he has been a Professor of the Department of Electrical Engineering, Kyungpook National University. His current research interests are design of electric machines, theory of electromagnetic force density, and multiphysics of electrics and mechanics.
Il Han Park He received MS and PhD degrees in Electrical Engineering from Seoul National University in 1986 and 1990, respectively. He is a Professor at Sungkyunkwan University and his present academic interests are in electromagnetic force and its multiphysics problems coupled with fluid, microparticle, mechanical dynamics, heat transfer and electric discharges.
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