An Analytical 3-D Model for Calculating Eddy Current Damping Force for a Magnetic Levitation System...

2472 IEEE TRANSACTIONS ON MAGNETICS, VOL. 48, NO. 9, SEPTEMBER 2012

An Analytical 3-D Model for Calculating Eddy-Current Damping Force fora Magnetic Levitation System With Permanent MagnetMehran Ebrahimian , Mohammad Khodabakhsh , and Golamreza Vossoughi

Center of Excellence in Design, Robotics and Automation, Mechanical Engineering Department, Sharif University ofTechnology, Tehran, Iran

Physics Department, Sharif University of Technology, Tehran, IranAzadi street, Tehran 11365-9567, Iran

An analytical solution for obtaining steady-state eddy-current-based force on a levitated permanent magnet above a plate with linearconductivity in the field of an electromagnet having cylindrical symmetry is presented in this paper. In literature, the force due to eddycurrent in this levitation system have been used for high precision positioning of a levitated permanent magnet without providing anexplicit analytical model. In this system, a change in the coils current and also the motion of the levitated permanent magnet in 3-Dspace generate eddy current in the plate. A novel explicit solution for obtaining damping forces due to these eddy currents is obtainedas a function of geometric and material properties of the system. The analytical solution is derived by solving the quasi-static Maxwellequations using separation of variables method. Comparisons show that the analytical solution closely matches with the results of thefinite element analysis.

Index TermsAnalytical solution, eddy current, eddy-current damping, magnetic levitation.

I. INTRODUCTION

E DDY CURRENT has wide usage in some electricalsystems such as contactless manipulation [1], magneticbearing [2], [3], and nondestructive testing [4]. The contactlessmanipulation systems manipulate an object in the absence offrictional losses stemming from mechanical contacts.Modeling the eddy current in different systems has al-

ready been investigated by using analytical and finite elementmethods. In [5][7], eddy current in a conductive plate underinfluence of a field source has been analyzed by numerical andfinite element methods.Analytical models are developed to study the eddy-current

effects. In [8], an analytical model is presented for a thin con-ductive plate exposed to a time-varying magnetic field. In thiswork, only magnetic fields in the direction perpendicular to theplate can be modeled. This can be a limitation in most problems.Also, an analytical model for obtaining eddy-current effects in aconductive plate with a source field above the plate is presentedin [9], [10]. In this work, the source magnetic field is symmetricin one of the directions parallel to plate and Maxwell equationshave been solved in 2-D space.Magnetic levitation system is a contactless manipulation

system that uses the eddy currents in stable positioning of thelevitated object [11][13]. An important advantage of using theeddy current in magnetic levitation systems is higher precisionpositioning of the levitated object stemming from the resultantdamping forces [1], [14]. In [1], a semi-analytical model forobtaining real time eddy-current damping forces when themagnetic field has a cylindrical symmetry is studied withoutpresenting an explicit analytical relation for damping forces.

Manuscript received January 18, 2012; revised March 23, 2012; acceptedMarch 29, 2012. Date of publication April 16, 2012; date of current versionAugust 21, 2012. Corresponding author: M. Khodabakhsh (e-mail: [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TMAG.2012.2194791

In this paper an analytical model for obtaining steady statedamping force on a permanent magnet, levitated in a field ofelectromagnet above a conductive plate is presented. So an ex-plicit analytical relation for eddy forces is obtained as a functionof geometric andmaterial properties of the system. The dampingforce can be generated by changing electromagnets current andalso the motion of levitated object. The motion of permagnet(permanent magnet) is studied in all directions in 3-D space.The magnetic field is considered to have cylindrical symmetrywhich is reasonable in many applications.In the presented analytical solution, quasi-static Maxwell

equations are solved by separation of variables method in eachregion and the global solution is derived using boundary con-ditions between the regions. In obtaining eddy forces the effectof eddy current generated in the permagnet is ignored, becausethe conductivity and size of the plate are much larger thanthose of the permagnet. Also, the permagnet is modeled as amagnetic dipole. By integration over magnetic dipole elementsto find the force on a real permagnet, it is shown that the simplemagnetic dipole model has sufficient precision. By comparingthe results from this analytical solution with the results fromthe finite element analysis, the analytical solution is validated.

II. DAMPING FORCE IN MAGNETIC LEVITATION SYSTEM

In Fig. 1, the schematic of the magnetic levitation systemis shown. An electromagnet, a levitated object which is a per-magnet and a conductive plate are parts of this levitation system.The levitated object can be moved in direction by exerting themagnetic field of the electromagnet, and in and directionsby moving the electromagnet.The eddy-current-based force may be generated by the vari-

ation of the position of the levitated object which creates eddycurrent in the conductive plate. This eddy current generatesa magnetic field which opposes the motion of the permagnet(Lenzs law). Also, the variation of the electromagnet currentgenerates eddy current in the plate which subsequently gener-ates eddy force on the object. It is shown in the Appendix that

0018-9464/$31.00 2012 IEEE

EBRAHIMIAN et al.: ANALYTICAL 3-D MODEL FOR CALCULATING EDDY-CURRENT DAMPING FORCE 2473

Fig. 1. Schematic of the levitated object above the plate.

in the first order of perturbation, the eddy current in the plateis linearly proportional to the velocity of the levitated objectand the rate of change of current in the electromagnet. Becauseour object is a permagnet, the eddy force is proportional toeddy current in the plate. Hence, the eddy force is a linearcombination of objects velocity and rate of electromagnetscurrent change. Thus, we can obtain the eddy-current-basedforce generated by each effect separately and obtain the totaleddy force by summing these two eddy forces coming fromdifferent sources.Also, to compute eddy forces the permagnet is considered as

a magnetic dipole. If the size of the permagnet is small withrespect to its distance from the plate, the analytical solution ob-tained in this paper can be used approximately by assuming thelevitated object as a magnetic dipole placed right at its center.Generally we can consider the levitated object as the sum ofmagnetic dipoles at different coordinates and obtain the eddyforces by integrating over these small magnetic dipole elements.But this increases the complexities. By calculating these inte-grals, it is seen that when , the distance between the center ofthe object and conductive plate (see Fig. 1), is such thatand , the magnetic dipole approximation is valid with5% relative error. In these equations is the objects diameterand is its height. In most situations is in the above ranges,so we can avoid these complexities in our calculations.

III. EDDY-CURRENT-BASED FORCE DUE TO THE MOTION OFLEVITATED PERMAGNET

A. Introduction

In this section, an analytical solution is presented for ob-taining eddy force due to the motion of the small permagnet.If , the relative permeability of the plate, is larger than 1,

the plate will pull-in the levitated object toward itself, which isundesirable. Also, it is shown that using a plate with high con-ductivity produces a bigger eddy-current-based force. Hence, it

is better to use a high conductive plate (i.e., many of Aluminumalloys) with a relative permeability of .

B. Systems Model

From the quasi-static Maxwells equations for linear conduc-tors we have the following equation for magnetic vector poten-tial in the conductive plate [15]

(1)

where is the magnetic vector potential, is the permeabilityof free space, and is the conductivity of the plate.In the nonconductive places the right side of this equation

vanishes. Using this equation and boundary conditions the mag-netic vector potential can be obtained in order to find the eddyforces on the object.First we analyze the motion of the permagnet in z direction.

According to Fig. 1, we write the position of the permagnet as, where and are the initial position and

velocity of the permagnet, respectively.It is known that a magnetic dipole with a moment is

equivalent to a coil with a small radius , and a large currentwhere . Hence, using this conversion we can makethis problem similar to the electromagnet based eddy-currentproblem presented in the next section.The space current density, , in the cylindrical coordinate is

(2)

where is Diracs delta function and is the angular unit vectorin cylindrical coordinate. By the Taylor series expansion aroundthe point , the current density can be written as

(3)

In the first order of perturbation to find nonzero eddy force,the higher order terms of the series expansion can be neglectedbecause of the small permagnets velocity.The first term of (3) which is due to a static dipole cannot

generate eddy current, because it is not time dependent. Thusfor analyzing the eddy-current-based force only the second termis necessary.Since vector potentials and fields are linear functions of cur-

rent sources, a virtual current density, , can be defined as

(4)

Hence, the magnetic field generated by is derivative of thevirtual fields, generated by , with respect to . Hence, wemust first derive the magnetic fields due to current density .This current density is obtained from a coil with radius posi-tioned at a distance (in parallel to the conductive plate) car-rying a current .


We can write this time dependent current as the sum of har-monic currents with amplitude in different angular fre-quencies using Fourier transform below

(5)

One can write

(6)

The derivation of the magnetic vector potential with respectto this harmonic current is shown in the Appendix. Thus, the am-plitude of vector potential in angular frequency can be writtenas

(7)where is the first-order of the first-kind Bessel function andquantities and are defined in (47) and (48) of Appendix.The second term of (7) is the vector potential due to the direct

effect of the coil current (see (48)) which is independent ofand . Also, the first term, which decreases by increasing , isthe vector potential due to eddy current in the plate (see (47)).Hence, to obtain the eddy force of the plate on the permagnet,only the first term should be considered.Taking inverse Fourier transform and differentiating with re-

spect to and limiting the coil radius to zero (in the magnetdipole limit) lead to an equation for magnetic vector potentialin the following form:

(8)

To obtain the eddy-current-based force we have [15]

(9)

Substituting into (9) and using (8), theeddy-current force due to the permagnets motion can be writtenas

(10)

where is

(11)

Equation (10) represents the eddy force due to the motion ofthe permagnet perpendicular to the plate. This idea can be usedto find the eddy force due to the motion of permagnet parallelto the plate by replacing by , in (8). Hence,we have

(12)

where is

(13)

Fig. 2. Component of of eddy-current density A/m in the plate due to themotion of the levitated object resulted from finite element method.

It is seen that the damping coefficient for radial direction ishalf of the damping coefficient for axial direction.

C. Results

In this section, we validate the analytical solution obtained inthe last section. The results are validated with the numerical re-sults obtained from finite element method. In obtaining the finiteelement method results, the Mesh size of each subdomain hasbeen refined to get more accurate data. We have used COMSOLMultiphysics software version 4.1 with a free triangular mesh.The number of the elements is about 100 000 and the number ofDOF is about 200 000 in our simulation. In Fig. 2, a symmetricdiagram of the eddy-current density due to the motion of thepermagnet with A m and upward velocity ofmm/s is shown. The plate with the thickness of 5 mm has theconductivity of m . Also, the distance betweenthe permagnet and the plate is 10 mm.According to Fig. 2, the maximum value of the current den-

sity is generated 5 mm away from the center of the plate. Theeddy-current density vanishes after 5 cm from the center of theplate. Therefore, the plates with a radius larger than 5 cm can beconsidered as infinite plates in and directions and the calcu-lations are valid for these plates.Figs. 3 and 4 show , the damping coefficient in direc-

tion [see (11)], as a function of and . In all of these figures,the plate conductivity is set to be 30 m .These figures show the precision of our results and can be

considered as a validation for our solution. The error of analyt-ical results with respect to the finite element method (FEM) re-sults is less than 3%. Also, from (11) it is seen that by increasing

increases until it reaches to a maximum value whichis proportional to ; this effect can be seen in Fig. 4. Equation(11) indicates that the damping coefficient for is greaterthan 85% of its value for semi-infinite plate . Soa plate with a thickness of the order of the permagnet distancecan be a good choice for practical considerations.


Fig. 3. The damping coefficient in (11), as a function of the permagnet heightfrom the plate. The plate thickness is assumed to be 1 cm.

Fig. 4. The damping coefficient in (11), as a function of plate thickness. Thedistance between permagnet and plate is assumed to be 1 cm.

IV. EDDY-CURRENT-BASED FORCE DUE TO CHANGE OFELECTROMAGNET CURRENT

A. Eddy-Current-Based Force Due to Change of Coil CurrentIn this section, the goal is to obtain an analytical model of

eddy-current-based force due to change of electromagnet cur-rent. An electromagnet consists of many coils and cores. If wesolve the problem for one coil, we can solve the problem for acomplex electromagnet by summing the effect of each coil.The geometry of problem is shown in Fig. 5.The coil current is and is a function of time. To find nonzero

eddy force in the first order of perturbation, we use Taylor seriesexpansion about the initial time value and disregard the higherorder terms:

(14)

Using Fourier transform, we can write this current as sumof harmonic currents with amplitude in different angularfrequencies:

(15)

Fig. 5. Position of the coil and permagnet with respect to plate.

, the amplitude of magnetic vector potential hasbeen obtained in the Appendix as

(16)

where quantities and are defined in the Appendix [see(47) and (48)].The second term of (16) is the vector potential due to the di-

rect effect of the coil current [see (48)]. Also, the first term is thevector potential due to the eddy current of the plate. Therefore,to obtain the eddy force, only the first term should be consid-ered. The eddy force on the object is obtained from (9) and iswritten in the following form in two directions of and :

(17)(18)

where and are

(19)

(20)

where index in refers to electromagnet-based eddyforce.Since increasing the distance from the symmetry axis results

in a considerable reduction of the magnetic field, for effectivedamping in the radial direction, , the distance of the objectfrom the symmetry axis, should be small compared to and .Therefore, we can replace the zero order of the first-kind Besselfunction by its Taylor series to obtain the first nonzero term ofeddy force in each direction as follows:

(21)


Fig. 6. The damping coefficient in (22), as a function of the permagnet heightfrom the plate. The coil radius is 2.5 cm and the plate thickness is 1 cm.

Hence, the integrals can be written as Laplace transform offirst-kind Bessel function and can be solved analytically as

(22)

where and

(23)

Also, at small distances , the second term of (16) which leadsto the direct magnetic force on the permagnet, , is ob-tained as follows:

(24)

B. Results

In this section, we are going to validate the analytical solu-tion obtained in the last section. The results are compared withthe numerical results obtained from the finite element method.In obtaining the finite element method results, the Mesh sizeof each subdomain has been refined until the results converge.Figs. 6, 7, and 8 show , the damping coefficient in thedirection in (22), as a function of and . In all of thesefigures, the plate conductivity is set to be 40 m and thedistance between the plate and the coil is set to be 10 cm. Also,

is in SI units.Precision of the analytical results has been shown in compar-

ison with finite element method results. The error of analyticalresults is less than 4%. Since we have only considered the ef-fect of one coil, has an insignificant value (about ).But for an electromagnet with a large number of coils and ironcores, the amount of is in the order of milli-Newton.This amount is in the order of for a dipole with ve-locity 1 cm/s in direction (see Fig. 3).

Fig. 7. The damping coefficient in (22), as a function of coil radius. The per-magnet height from the plate is 5 cm and the plate thickness is 1 cm.

Fig. 8. The damping coefficient in (22), as a function of plate thickness. Thepermagnet height from the plate is 5 cm and the coil radius is 2.5 cm.

C. Eddy-Current-Based Force Due to Change of Current for aComplex Electromagnetic System

Since the motive force in the levitation system is producedby several electromagnets in different positions, the modelingof such systems can be complex. As a solution, we present amethod which states that by using the coil model stated in theprevious section, any electromagnet can be modeled. To thisend, (24) from the previous section can be written as

(25)

where is

(26)

It is seen that is only a function of coil parameters anddoes not depend on plate parameters.


Thus, having the direct electromagnet force acting on the per-magnet from experimental data or electromagnet catalog, byusing inverse Laplace transform from (25), can be ob-tained as follows:

(27)

where . Notice that is the distance of permagnetfrom the plate. Also, from (19) and (26) for small we have

(28)

Substituting (27) into the (28), the following relation can beobtained:

(29)

Also, substituting (29) in (23) a similar equation can be ob-tained for .In the presented model, parameter has not been deter-

mined. It can be obtained from the geometrical properties ofthe electromagnet. Notice that in this model we compare themagnetic field of the electromagnet with the field of a coil.Since the magnitude of the field of a coil is maximum in itscenter at a distance of from the plate, and considering thefact that the magnetic field due to the magnetic field generationsystem has got a maximum point (vs. distance from the plate),the distance between the plate and the point with maximummagnetic field generated by the electromagnet can be usedas an approximation for . In most cases, the geometry ofelectromagnet is symmetric, so the maximum value of themagnetic field is in the center of the electromagnet.Thus, the eddy force due to the change of electromagnets cur-

rent for a complex electromagnet can be obtained using this ap-proximate method. To obtain an exact relation for the dampingcoefficient we can decompose the electromagnet into parts withdistance from the plate and calculate the damping coefficientof each part, , then find the total damping coefficient by sum-ming them.

V. CONCLUSION

An explicit analytical solution has been presented to calcu-late eddy-current-based forces due to the motion of the per-manent magnet and the change of the electromagnet current inthe magnetic levitation systems described above. The perma-nent magnet has been modeled as a magnetic dipole; it has beenshown that this model has a maximum error of 5% in a widerange of workspace. Close results of the analytical solution andFEM modeling validates the analytical solution.These analytical relations can be used to calculate the eddy

force in modeling some electrical systems. The amount of eddyforce near the plate is large enough to act as a damping force onthe levitated object, and it can be useful in position control ofthe levitated object with high precision.

APPENDIX

In this section, we want to obtain the vector potential due to acoil with radius and in a height from the plate with current

.Since the current source in the space is harmonic function of

time with angular frequency , considering space symmetry, wecan write

(30)

where index is the number of regions. The space under theplate is region 1, inside the plate is region 2, the space betweenthe upper surface of the plate and the plan is region 3,and above this plane is region 4. Using (1) and (30), we have

(31)

(32)

where in (1) is replaced by .Using separation of variables method and Bessel transform,

the following results for the four regions are obtained:

(33)

(34)

(35)

(36)

where is the first order of the first-kind Bessel functions and.

and can be obtained according to boundary condi-tions in the boundary surfaces between different regions. Thus,six equations are obtained that can be written as

(37)(38)(39)(40)(41)(42)

where is

(43)


Solving (37) through (43), the following results are obtained:

(44)

(45)

Therefore, having a coil with radius and in a height fromthe plate with current , the vector potential in theregion 3 will be in the form of

(46)

where and are obtained as follows:

(47)

(48)

and is

(49)

where .

ACKNOWLEDGMENT

The authors would like to gratefully thank M. Abedinnasabfor his help on the finite-element analysis in the sabbatical pe-riod in Nanyang Technological University.

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