Estimation of Lorentz Force From Dimensional Analysis

IEEE TRANSACTIONS ON MAGNETICS, VOL. 52, NO. 8, AUGUST 2016 7004813

Estimation of Lorentz Force From Dimensional Analysis:Similarity Solutions and Scaling Laws

Matthias Carlstedt1, KonstantinWeise (née Porzig)1, Marek Ziolkowski1,2,Reinhard Schmidt1, and Hartmut Brauer1

1Advanced Electromagnetics Group, Technische Universität Ilmenau, Ilmenau 98693, Germany2Applied Informatics Group, West Pomeranian University of Technology, Szczecin 70313, Poland

In this paper, we consider the use of dimensional analysis for modeling electromagnetic levitation and braking problems, whichare described by the Lorentz force law. Based on Maxwell’s equations, to illustrate the underlying field problem, we formulate acomplete mathematical model of a simple academic example, where a permanent magnet is moving over an infinite plate at constantvelocity. The step-by-step procedure employed for dimensional analysis is described in detail for the given problem. A dimensionlessmodel with a reduced number of parameters is obtained, which highlights the dominant dependences, and it is invariant to thedimensional system employed. Using the dimensionless model, a concise parametric study is conducted to illustrate the advantagesof the dimensionless representation for displaying complex data in an efficient manner. We provide an exhaustive study of thedependences of the Lorentz force on the dimensionless parameters to complete the analysis, and we give results for a generalizedrepresentation of the problem. Finally, scaling laws are derived and illustrated based on practical examples.

Index Terms— Dimensional analysis, eddy currents, Lorentz force, magnetic levitation, permanent magnet (PM), scaling laws,similarity.

I. INTRODUCTION

WHEN an electrically conducting object moves througha magnetic field, such as that produced by a permanent

magnet (PM) or a current-carrying coil, motion-induced eddycurrents appear inside the conductor, which lead to electro-magnetic levitation and braking forces. The phenomenon iswell described by the Lorentz force law, and the correspondingtechnical developments have been made in this area for over100 years.

At present, this phenomenon had numerous different appli-cations, including magnetic bearing [1]–[3], coupling [4],precision actuation [5], [6], magnetic suspension [7]–[11],and energy harvesters [12], [13]. However, the best knowntechnical applications are magnetically levitated trains, whichuse electromagnets, PMs, or superconducting magnets forlevitation and guidance. Magnetically levitated trains werefirst proposed by Bachelet [14] in 1912, but high-speed trans-portation systems became popular in the early 1970s and theyhave evolved into a recognized form of modern transportationin the 21st century. There has been great success in thedevelopment of sufficient expressions of the underlying fieldproblem [15]–[22]. During the 1990s, great efforts weremade [23]–[28], which laid the foundations for the currentworld speed record in a test run of 603 km/h [29].

More recently, the Lorentz force phenomenon has beenextensively investigated in the context of non-destructivetesting and in the evaluation of electrically conductivematerials using the PMs. Two different approaches areemployed for material characterization to use the sec-

Manuscript received August 26, 2015; revised October 31, 2015 andJanuary 5, 2016; accepted March 4, 2016. Date of publication March 9, 2016;date of current version July 18, 2016. Corresponding author: M. Carlstedt(e-mail: [email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TMAG.2016.2539927

ondary magnetic field obtained from the motion-induced eddycurrents: measuring the secondary field using magnetic fieldprobes [30], [31] or measuring the Lorentz force acting on thePM [32]–[34].

In educational applications, the slowing down of a magnetfalling in a non-ferromagnetic, electrically conducting pipe isemployed as a popular demonstration to introduce engineeringand physical science students to the basics of electromagneticinduction phenomena. This problem has been extensivelystudied by focusing on experimental, analytical, or numericalsolutions [35]–[43].

Notable contributions to the falling magnet problem usingdimensional analysis were made in [44]–[46], where thesestudies briefly demonstrated the possibility of estimating theterminal velocity of the magnet based on the dimensionalanalysis supported by laboratory experiments.

The phenomenon itself is well understood and fullydescribed by the magnetic field transport equation [47], whichcan describe many different problems, and it is an integral partof engineering design for numerous devices with electromag-netic interactions. However, even seemingly simple tasks areactually difficult to solve and they require modern simulationsoftware, as well as specially trained professionals in the fieldof electromagnetics. In most cases, the dependences of thedifferent variables are not obvious and their complex interac-tions make it difficult to gain familiarity with the phenomenon.Another difficulty is caused by the high number of differentparameters in the applications of electromagnetic levitationand braking. Thus, many optimization tasks are extremelytime-consuming and often uneconomical.

In this paper, we contribute to the solution of technicallevitation problems by considering the idea of Lorentz forceestimation with the help of dimensional analysis for a rathersimple academic problem to illustrate the principal steps,thereby aiding the understanding of how to handle morecomplex applications. In Section II, we briefly describe a

0018-9464 © 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

7004813 IEEE TRANSACTIONS ON MAGNETICS, VOL. 52, NO. 8, AUGUST 2016

Fig. 1. Geometry and parameters of the problem under investigation. A vast,non-magnetic, electrically conducting plate moves rectilinear with a constantvelocity under the axially magnetized cylindrical PM at rest.

seemingly simple example of magnetic levitation to illustratethe underlying field problem. In Section III, we performa dimensional analysis of this problem, which highlightsthe dominant dependences, and thus reduces the number ofparameters. Based on this representation of the problem, wepresent a more general and efficient method for displayingthe complex data obtained from the numerical solution ofan exact expression of the Lorentz force in Section IV.In Section V, we derive scaling laws for the particular problemas a basic example to allow more complex technical applica-tions. Section VI concludes this paper.

II. PROBLEM DEFINITION

In the problem considered in this paper, a PM and an infiniteplate of a non-ferromagnetic, electrically conducting materialare in constant rectilinear motion with respect to each other(Fig. 1).

We define two frames of reference as S′ and S for the plateand the PM, respectively. The motion of both the parts isdescribed by the relative velocity v = vex of the frame S′in the fixed frame S. In our example, the PM has a cylindricalshape with diameter D and height H . It is assumed that thePM material is homogeneous and magnetized in the axialdirection with the magnetization M = Mez and M = Br/μ0,where Br is the remanent magnetization of the magneticmaterial. The base of the cylinder (z = h) is parallel tothe surface of the plate. The plate of thickness t is madeof a non-magnetic material with a homogeneous electricalconductivity σ . The PM is described by means of the surfacecurrent model with Js = M×n [48]. In addition, displacementcurrents are neglected because we are only interested in thecases where the plate is traveling at velocities much lower thanthe speed of light (v � c). In order to determine the forceacting on the PM, first, the problem of finding the magneticfield H of an infinitely thin current loop with Jsdz, locatedin the vicinity of the moving plate, is formulated and lateron extended to the case of the PM of finite height. In thequasi-static case (v = const), the governing equations derivedfrom Maxwell’s equations take the following form in theframe S :

∇2H − σμ0(v · ∇)H = 0, in the conductor (1)

∇2H = 0, outside the conductor (2)

∇ · H = 0, everywhere. (3)

The equations with the appropriate boundary conditionscan be analytically solved using the 2-D Fourier transformapproach [17], [20], [49].

The Lorentz force exerted on the PM is calculated usingParseval’s theorem [50] as

F = μ0

4π2 �e

{∫ h+H

h

∫ ∞

−∞

∫ ∞

−∞J∗

s × H(e)dkxdkydz

}(4)

where kx and ky are the transform variables and H(e) isthe Fourier transform of the magnetic field associated withthe eddy currents induced in the conductor. The 2-D Fouriertransform Js of the source current density is given by

Js = [ Jsx , Jsy]T = jπ Js0D J1

(k

D

2

) [ky

k,−kx

k

]T

(5)

where k2 = k2x + k2

y , Js0 = Br/μ0 is the magnitude of thesource current density, and J1(·) denotes the first-order Besselfunction of the first kind. The components of the force F aregiven by the formulas

Fx = μ0

2π2

∫ ∞

0

∫ ∞

0�m[T (k, β)](kx | Jsy|2 − ky J ∗

sy Jsx)

× (1 − e−k H )2

k3 e−2khdkxdky (6)

Fy = 0 (7)

Fz = μ0

2π2

∫ ∞

0

∫ ∞

0�e[T (k, β)](| Jsx |2 + | Jsy|2)

× (1 − e−k H )2

k2 e−2khdkxdky (8)

with T (k, β) obtained as

T (k, β) = (β2 − 1) tanh βkt

2β + (1 + β2) tanh βkt(9)

where β = α/k and α2 = jμ0σvkx + k2.The set of equations (6)–(8) builds the mathematical model

for the problem under investigation by describing the relationbetween the dependent parameter of interest F and eight rel-evant physical parameters v, μ0, σ , Br , D, H , h, and t .

We use the presented approach to provide reliable datafor the subsequent analysis. Clearly, other methods, such asfinite-element analysis or laboratory experiments, would alsobe suitable for obtaining valid solutions for the force, but thelatter would require an additional effort in terms of uncertaintyanalysis.

III. DIMENSIONAL ANALYSIS

In this section, we present the main contribution of thispaper. We apply the dimensional analysis procedure to thedefined problem using the mathematical reformulation givenby Price [51]. We assume that any complete physical relationmust be dimensionally consistent, which is also known asthe statement of dimensional homogeneity [52]–[54]. Further-more, we acknowledge that any physical relationship that isexpressed by a complete equation must be invariant to theapplied dimensional system [53]–[55].

CARLSTEDT et al.: ESTIMATION OF LORENTZ FORCE FROM DIMENSIONAL ANALYSIS 7004813

A. Definition of the Physical Model

The first step in the dimensional analysis procedure is thepreliminary physical analysis of the system and the definitionof the problem, which is described in detail in Section II.The next step is to create a list of the physical parame-ters xi of x = {x1, x2, . . . , xI }, which are expected to berelevant to the features of the phenomena of interest. Theseparameters should be described using a consistent system ofunits [G] = {[G1], [G2], . . . , [GK ]}, which comprise funda-mental units [Gk] that are sufficient to define the magnitudeof any physical quantity [56]. It should be mentioned thatit is customary (as suggested by Maxwell) to denote thedimensions of a quantity φ by [φ] [57]. The dimension [xi ]of any physical parameter xi can be written as the product ofthe powers of the fundamental units

[xi ] =K∏

k=1

[Gk]dki (10)

where dki equals to the power to which the kth fundamentalunit is raised in the i th physical parameter of x. To improvethe clarity of the description, the dimensional analysis employsthe International System of Units (SI), but the reader is freeto choose any other appropriate system [58].

We start the list with the first important parameter in thisproblem, which is the force F acting on the PM. We want toperform the dimensional analysis in the scalar form, so theforce as a vector quantity has to be decomposed into orthogo-nal components. Due to the symmetry of the problem, only Fx

and Fz are of interest because Fy ≡ 0. Additional parameterscomprise the magnitude of the relative velocity v, the electricalconductivity of the plate σ , and the PMs remanence Br , whichare assumed to be relevant to the acting force. The nextparameter in the list is the magnetic permeability μ = μ0μr ,where μ0 is the vacuum permeability and μr is the relativepermeability of the plate. We want to restrict the investigationto non-ferromagnetic materials (μr ≡ 1), so we only have toconsider the vacuum permeability μ0 in our list. It should bementioned that the vacuum permeability μ0 appears in ourlist, because we describe all the parameters in SI units. Othersystems of units would also lead to other constants, such as theelectromagnetic velocity in Gaussian and Heaviside-Lorentzunits. Finally, we employ a group of geometrical parametersto describe all the lengths and distances in our problem,i.e., the cylinder’s diameter D and height H , the distancebetween the PM and plate h, and the plate’s thickness t .

The full list contains ten (I = 10) parameters

x = {Fx , Fz, v, σ, Br , μ0, D, H, h, t} (11)

which comprise the physical model x of our problem. Theresult of the second step is summarized in Table I, where thedimensions are in fundamental units.

Clearly, all the elements of the physical model x can bedescribed using a reduced base of K = 4 fundamental unitsexpressed in terms of mass [G1] = M, length [G2] = L,time [G3] = T, and electric current [G4] = I.

A comprehensive form to represent all the elements of x andtheir corresponding dimensions is the dimensional matrix D,

TABLE I

LIST OF THE PHYSICAL PARAMETERS AND CONSTANTS

where the elements dki are given in (10) such that

D

=

Fx Fz v σ Br μ0 D H h t⎡⎢⎣

⎤⎥⎦

M 1 1 0 −1 1 1 0 0 0 0L 1 1 1 −3 0 1 1 1 1 1T −2 −2 −1 3 −2 −2 0 0 0 0I 0 0 0 2 −1 −2 0 0 0 0

(12)

for example, the dimension of the velocity v is the product ofthe fundamental unit [L] raised to the power of 1 and [T] tothe power of −1.

To continue the analysis, we must assert that the physicalmodel is complete, i.e., it includes all the parameters requiredto build a correct mathematical model. This model mustbe dimensionally homogeneous and, thus, invariant to thedimensional system used. This is evident from the previousproblem definition (Section II), but in general, the complete-ness would only be a hypothesis without any knowledge ofthe mathematical model.

B. Calculation of a Dimensionless Basis Set

Using the complete physical model, we can define a func-tional relationship that includes all the previously identifiedparameters. Without loss of generality, this relationship canbe written as

g(x) = g(Fx , Fz, v, σ, Br , μ0, D, H, h, t) = 0 (13)

where g is an unknown function. From Buckingham’s�-theorem [55], we know that a dimensionally homogeneousequation can be reduced to a relation of independent dimen-sionless parameters �j for a basis set � = {�1, . . . ,�J },such that

G(�) = G(�1,�2, . . . ,�J ) = 0 (14)

where G still is an unknown function, but J ≤ I . Eachdimensionless parameter �j is a product of the powers of thegoverning parameters xi with independent dimensions [57]

� j =I∏

i=1

xsi ji , j = 1 . . . J (15)


where si j denotes the power to which the i th physical para-meter is raised in the j th dimensionless element of �.

We are interested in finding this basis set, so we considerthe dimensional equations of this statement

[� j ] =I∏

i=1

[xi ]si j . (16)

Analogous to (10), we describe the dimension of eachelement � j as a product of the powers of the fundamentalunits

[� j ] =K∏

k=1

[Gk]ckj . (17)

If they are dimensionless, each [� j ] must be equal to one, sowe can conclude that ckj ≡ 0,∀k, j . Furthermore, we write thedimensional formula for the right-hand side of (16) using (10),such that

I∏i=1

[xi ]si j =I∏

i=1

(K∏

k=1

[Gk]dki

)si j

. (18)

By combining (18) and (17), the dimensional equation (16)can be written as

K∏k=1

[Gk]ckj =I∏

i=1

(K∏

k=1

[Gk]dki

)si j

. (19)

For the sake of simplicity, we rewrite (19) by taking thelogarithm of both the sides as

K∑k=1

ckj log[Gk] =I∑

i=1

si j

K∑k=1

dki log[Gk] ∀ j (20)

which holds in the non-trivial case for [Gk] �= 1 only if

ckj =I∑

i=1

si j dki ∀ j, k. (21)

Using the matrix notation, it is evident that the unknown basisset of dimensionless parameters is equal to the non-trivialsolutions of the homogeneous system of linear equations

c j = Ds j , c j ≡ 0 ∀ j (22)

with the dimensional matrix D given by (12) and the basis setvectors s j . D is underdetermined, so infinitely many solutionsform a vector space. The vector space dimension is equalto J , the number of dimensionless products in a completeset of parameters. This so-called nullity of D is stated by therank-nullity theorem of linear algebra

J = nul(D) = I − rk(D) (23)

where I is the number of columns and rk is the rank ofthe dimensional matrix D. Consequently, from the physicalmodel (11) with I = 10 parameters and the dimensionalmatrix (12) of rank rk(D) = 4, the problem is fully describedby a set of J = 6 independent dimensionless products � j .

The solutions set of (22) represents the kernel (null space)of D, which can be calculated using the Gaussian elimination.Given that D is initially built using an arbitrary ordering of

the physical parameters, and then the row echelon form of theunderdetermined system mainly depends on the arrangementselected. Therefore, we are free to reorder the columns ofthe dimensional matrix in any form desired. In the following,we use a slightly modified matrix D, where the physicalparameter h is moved to the fourth place of x, to obtain asparse null space basis. This yields a clearly arranged resultand the distance h is set to the characteristic length of theproblem.

The calculation of a rational null space of this reorderedmatrix yields six vectors that correspond to dimension-less parameters � j , which are combined into the solutionmatrix S = [s1; s2; . . . ; sJ ] as follows:

S =

s1 s2 s3 s4 s5 s6⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Fx −1 −1/2 0 0 0 0Fz 1 0 0 0 0 0v 0 1/2 1 0 0 0h 0 3/2 1 −1 −1 −1σ 0 1/2 1 0 0 0Br 0 1 0 0 0 0μ0 0 0 1 0 0 0D 0 0 0 1 0 0H 0 0 0 0 1 0t 0 0 0 0 0 1

. (24)

The corresponding set of dimensionless parameters � j isconstructed using (15) for each solution vector s j , where xs j

is employed as a shorthand notation for this computation, assuggested in [51]. The calculated set of independent dimen-sionless parameters �

�∗1 = xs1 = Fz/Fx (25a)

�∗2 = xs2 =

√σv B2

r h3/Fx (25b)

�∗3 = xs3 = μ0σvh (25c)

�∗4 = xs4 = D/h (25d)

�∗5 = xs5 = H/h (25e)

�∗6 = xs6 = t/h (25f)

comprises a dimensionless model of our problem, and this isthe result of the third step. In contrast to the previously definedphysical model x, the new formulation is invariant to thedimensional system used. More importantly, it fully describesthe phenomenon of interest using only six parameters insteadof the original ten parameters.

C. Discussion and Reformulation of the Dimensionless Basis

The last step of the dimensional analysis is to evaluate andinterpret the derived dimensionless basis set in the light ofobservations or confirmed mathematical models. To be able tointerpret these results it is useful to discuss (25) in the givenform.

The first dimensionless parameter �∗1 is the ratio of both

the force components, Fz and Fx , which can be intuitivelyinterpreted as the direction of the force in the xz-plane. Thesecond parameter �∗

2 illustrates the remarkable features of a


dimensional analysis by indicating the dominant dependencesof the parameters. This expression clearly agrees with thestatements in [59] about the Lorentz force acting on a magneticdipole located at a distance L above a semi-infinite electricallyconducting fluid. An estimate is given by the proportionalityF ∝ μ2

0σvm2 L−3, where m is the magnetic dipole momentm ∝ Br L3/μ0. The third dimensionless parameter �∗

3 iscalled the magnetic Reynolds number Rm , which is wellknown in magnetohydrodynamics, where it indicates the ratioof magnetic advection relative to magnetic diffusion [47].The last three dimensionless parameters indicate the shape orgeometric similarity [52], and they describe the relative sizesof the bodies involved.

This particular set of dimensionless parameters might betoo abstract for a convenient description of the Lorentz forceexerted on the PM in our problem. Therefore, we are interestedin a form that clearly separates independent and dependentvariables, as well as their parameters. The basis set vectors s j

are orthogonal and they span the null space, so any linearcombination of these vectors is a solution of the homogeneoussystem (22). Thus, we are free to transform the initial basis setby multiplying the dimensionless parameters with each otherto any desired power.

In the reformulated basis set, we define the independentvariable �1 as the magnetic Reynolds number Rm . Thevariables Fx and Fz , where tilde indicates a dimensionlessforce component, are the linear combinations of (25a)–(25c).Furthermore, we define the dimensionless geometric parame-ters δ, ξ , and τ by rearranging (25d)–(25f).

After some simple calculations and reordering, we obtain areformulated basis set

Rm := �1 = μ0σvh (26a)

δ := �2 = D/h (26b)

ξ := �3 = D/H (26c)

τ := �4 = t/h (26d)

Fx := �5 = μ0 Fx/(Br h)2 (26e)

Fz := �6 = μ0 Fz/(Br h)2 (26f)

which provides a more suitable representation for thefollowing discussion.

IV. RESULTS AND DISCUSSION

A. Dimensionless Representation of Complex Data

In this section, we present the calculated force componentsthat act on the PM to evaluate the derived dimensionless basisset.

The procedure of the force evaluation implemented inMATLAB is shown schematically in Fig. 2. To demonstrate theadvantages of the dimensionless representation, we discuss theconcise parametric study depicted in Table II for four differentsettings. We consider the PMs of various sizes with typicalmagnetic remanences for neodymium–iron–boron (NdFeB)PMs. The electrical conductivity of each plate is in the rangefor aluminum and copper alloys. All the settings define thesystems that are similar in shape to each other, and thusthey have identical dimensionless parameters δ, ξ , and τ .

Fig. 2. Flowchart of dimensionless Lorentz force calculation. Box L F: forcecalculation described by (6) and (8). Box � → x: conversion of dimensionlessto dimensional parameters. Box x → �: conversion of Fx,z to Fx,z .

TABLE II

EXAMPLE SETTINGS

The geometric similarity is a necessary condition for thecomplete similarity in this paper.

The results of the parametric study S1–S4 are shown inFig. 3. The dimensional representations of the calculated forcecomponents on the PM are shown in Fig. 3(a) versus thevelocity up to v = 35 m/s. As expected, all four settingsdiffer in terms of the magnitude of the forces generated.In addition, the characteristic points P1–P4 of intersection forthe force components differ in terms of their velocity andforce. Nevertheless, it is clear that all the four settings sharea common characteristic shape for the resulting forces. Thisis clearer when we look at the results for the dimensionlessrepresentation shown in Fig. 3(b).

As predicted, all the four configurations yield dimension-less representations with identical results in terms of boththeir magnitude and shape. The resulting force componentsFx and Fz merge in the dimensionless representation atidentical magnetic Reynolds numbers on two separate curves,and thus they share one point of intersection P0. Clearly,identical magnetic Reynolds numbers do not imply that thevelocities are the same, but they do indicate electromagneticsimilarity. This is why it is not possible to replace the abscissain Fig. 3(b) by a dimensional representation of the velocitywithout defining the remaining parameters of Rm . Further-more, it is clear that the magnet’s remanence Br does notaffect the electromagnetic similarity, and it merely comprisesa scaling factor of the second power for the generated forcecomponents.

B. Dependence on Rm

In the following study, we illustrate that a dimensionlessrepresentation remains valid without further constraints andit can be used to highlight the dominant dependences in theproblem under consideration. For this reason, we extend the


Fig. 3. Comparison of (a) dimensional and (b) dimensionless representationsof the numerical results for the parametric study (Table II).

study presented in Fig. 3 to a larger range for the magneticReynolds number. Furthermore, we examine the hypothesisthat the Lorentz force components can be described over awide range using simple power-law dependences based onthe magnetic Reynolds number. Therefore, we plot the resultsin double logarithmic scale, where the underlying power-lawdependence is indicated by a straight line. This is also usefulfor the subsequent estimation of the scaling laws.

In this paper, we assume that the mathematical model usedto calculate the force components only holds for the velocities

that are much smaller than the speed of light. Therefore,we restrict our investigation to Rm ≤ 103, which is expectedto give reliable results.

Fig. 4 shows the force components calculated for fixed δ,ξ , and τ depending on the magnetic Reynolds number Rm .It is clear that for both the Lorentz force components, tworegions exist where the problem is dominantly described by aparticular power of Rm . We can clearly distinguish one regionwith low and another with high Rm . In the region whereRm ≤ 10−1, the dimensionless force components Fx and Fz

are proportional to Rm and R2m , respectively. In the region

where Rm ≥ 101, Fx is proportional to R−1/2m , whereas Fz

goes to saturation. This observation confirms our hypothesisregarding the dependence on Rm .

We recall that the magnetic Reynolds number Rm is ameasure of the relative strength of advection and diffusion, andthus their respective characteristics can be attributed directly tothe corresponding phenomena. Between the regions of low andhigh Rm (moderate Rm), both the phenomena occur side byside, thereby preventing further characterization using power-law dependences.

C. Dependence on τ

Next, we investigate how the dimensionless Lorentz forcecomponents depend on the dimensionless geometric parame-ters δ, ξ , and τ . Therefore, we expand our previous studybased on the considerations of the influence of the platethickness τ on the force components. This analysis shouldhave a significant impact, because τ determines the availableregion where the eddy currents are induced. For this purpose,we express a similar power-law hypothesis for the dependenceon τ .

Fig. 5 shows the dependences of the Lorentz force compo-nents on the dimensionless plate thickness τ , for fixed Rm ,δ, and ξ . It is clear that for both the force components,two regions exist where the problem is dominantly describedby a particular power of τ . We can clearly distinguish oneregion with low and another with high τ . In the region whereτ ≤ 10−1, Fx ∝ τ and Fz ∝ τ 2, whereas in the region whereτ ≥ 102, Fx and Fz have no further dependence on τ . Thisobservation also confirms our power-law hypothesis about thedependence on τ .

In the following, we refer to the region that has a constantpower-law dependence on τ with thin plate behavior andthe region without dependence on τ with infinite half-spacebehavior. Between these two regions, where a moderate τdominates, no further characterization is useful without theadditional investigation of the explicit eddy current densitydistribution.

The next step of our investigation is to analyze how thedependences of the force components on the dimensionlessplate thickness τ change for different values of the magneticReynolds number Rm . Fig. 6 shows filled contour plots for thecommon logarithm of the force components as functions of themagnetic Reynolds number Rm and dimensionless plate thick-ness τ for δ = 10 and ξ = 1. The force component generatedis constant along each contour line. Each value is depicted as


Fig. 4. Dimensionless Lorentz force components Fx (blue curve) and Fz (magenta curve) as functions of the magnetic Reynolds number Rm forfixed δ, ξ , and τ . Blue region: transition zone for the mixed dominance of advection and diffusion, which separates low and high Rm .

Fig. 5. Dimensionless Lorentz force components Fx (blue curve) and Fz (magenta curve) as functions of the dimensionless plate thickness τ forfixed Rm , δ, and ξ . Blue region: transition zone that separates small and large τ .

the power to base 10 in the color bar. The contour intervalemployed, i.e., the difference in elevation between successivecontour lines, is constant in each graph. Thus, the distancebetween two lines is a measure of the gradient for a forcecomponent at a certain point, which is always perpendicularto the contour lines. At a dimensionless thickness τ ≥ 101, weagain observe two regions where both the force componentsare proportional to the powers of Rm , similar to that givenin Fig. 4, but they are invariant to τ . In the region whereτ ≤ 10−1, Fx is proportional to τ and Fz to τ 2 until acharacteristic Reynolds number, which is proportional to τ .

In addition, the maximum force Fx exceeds that for largedimensionless thickness. After the force maximum is reached,Fx is inversely proportional to Rm and τ , whereas Fz againgoes to saturation, and thus it has no further dependence oneither Rm or τ . This observation also confirms our hypothesisregarding the power-law dependence on τ .

D. Generalized Representation

Furthermore, we are interested in generalizing these state-ments of proportionality. We have seen that it is possible to


Fig. 6. (a) Dimensionless drag force Fx and (b) dimensionless lift force Fzas functions of the magnetic Reynolds number Rm and the dimensionlessplate thickness τ for fixed δ = 10 and ξ = 1.

distinguish between the regions that are dominantly describedby either diffusion (low Rm ) or advection (high Rm), and theregions with and without dependences on the plate thickness,so we can define characteristic values RmC and τC thatapproximately differ between each of these regions. We defineRmC x,z as the value of the maximum curvature of log (Fx,z) forτ = 103 (approximation of infinite half-space) and with fixed δand ξ . Analogously, we define τC x,z for Rm = 10−3 (low Rm ).As a result, we obtain the estimates of the equilibriums for thedifferent phenomena. To avoid the need for multiple partialderivations to calculate the maximum curvature, we use asimple geometric estimate. The basic idea is to transformthe dimensionless force components obtained into an almost

Fig. 7. Transformation of the dimensionless force components into asymmetric representation (a) Fx , (b) Fz .

symmetric representation

Fsym = F1

τ 1/2 R1/2m

(27a)

Fsymx = Fx

1

τ 1/2 R1/4m

(27b)

Fsymz = Fz

1

τ Rm(27c)

shown in Fig. 7.Using this transformation, the characteristic values

RmC x,z and τC x,z are defined as

RmC x,z(δ, ξ) = arg maxτ0=103,Rm∈R

Fsymx,z (Rm , τ0, δ, ξ) (28a)

and

τC x,z(δ, ξ) = arg maxRm0=10−3,τ∈R

Fsymx,z (Rm0, τ, δ, ξ) (28b)

whereas RmC x,z and τC x,z are the only functions of the dimen-sionless geometric parameters δ and ξ . The term symmetrichelps to clarify that the absolute values of the exponentsof proportionality in the separated regions are equal to eachother after transformation. This also ensures that only onecharacteristic point exists for all the possible sets of para-meters. Equations (28a) and (28b) are numerically evaluatedusing a derivative-free minimization of the negative symmetricrepresentation −Fsym

x,z defined by (27). The minimization isbased on golden section search and parabolic interpolationprovided by the MATLAB function fminbnd [60].

Fig. 8 shows the determination of the characteristic valuesfor an arbitrary set of δ and ξ . This description of characteristicpoints is equally valid for our generalization, such as that ofthe maximum curvature. However, it has particular advantagesin the case of real measurements where multiple derivationswould lead to incorrect results due to the amplified sensornoise.

When these two characteristic values are determined for aspecific set of δ and ξ , we can normalize τ and Rm fromFig. 6. Furthermore, we can normalize the dimensionless forcecomponents against a characteristic value of interest, e.g., themaximum force in a specific parameter range or the force atone of the two characteristic points of the maximum curvature.


Fig. 8. Determination of the characteristic values (a) RmCx,z along τ = 103

and (b) τCx,z along Rm = 10−3 after the transformation of the dimensionlessforce components Fx,z into a symmetric representation F

symx,z .

Figures 9 and 10 show filled contour plots for the commonlogarithm of the force components normalized to their max-ima as the functions of the generalized magnetic Reynoldsnumber Rm/RmC x,z and the generalized dimensionless platethickness τ/τC x,z for arbitrary δ and ξ . The normalized forcecomponent generated is constant along each contour line. Eachvalue is depicted as the power to base 10 in the color bar, andthe order of magnitude is given relative to the maximum forcein the range considered. The contour intervals employed areagain constant in each graph.

In Fig. 9, the parameter space for the normalized forcecomponent Fx/Fx,max is divided into four regions. The redcurve separates an infinite half-space and thin-plate behavior.The blue curve distinguishes dominantly diffusive (low Rm )and advective (high Rm) regions. Each region has specificproportionality to the powers of Rm and τ . The slope of thered curve for the values of Rm/RmC x ≥ 1 can be obtainedusing the proportionalities in the two regions with high Rm as

τ

τC=

(Rm

RmC

)−1/2

,Rm

RmC≥ 1. (29)

The slope of the blue curve for the values of τ/τC ≤ 1 canbe obtained in a similar manner using the two regions withthin-plate behavior as

Rm

RmC=

(τ

τC

)−1

,τ

τC≤ 1. (30)

Fig. 9. Normalized drag force Fx/Fx,max as function of the generalizedmagnetic Reynolds number Rm/RmCx and the generalized platethickness τ/τCx for arbitrary δ and ξ . Four regions with high or low Rmand thin plate or infinite half-space behavior are distinguished.

Fig. 10. Normalized lift force Fz/Fz,max as function of the generalizedmagnetic Reynolds number Rm/RmCz and the generalized platethickness τ/τCz for arbitrary δ and ξ . Two regions with low Rm andthin plate or infinite half-space behavior are distinguished from a third regionwith high Rm .

Of particular importance are the regions where both advec-tive behavior (high Rm) prevails and thin-plate behavior(low τ ) dominates. For the dominantly diffusive regions(low Rm), the dependences on Rm are retained for both theforce components. However, in the transition between domi-nantly diffusive (low Rm ) and advective (high Rm ) behavior,the dependences on τ are inversely proportional.

In a similar manner, we can proceed with the normalizedforce component Fz/Fz,max (Fig. 10). In contrast to Fig. 9,the parameter space is divided into only three regions in


this case. The blue curve separates dominantly diffusive andadvective behavior with the same estimate as that given by (30)but with squared proportionalities compared with the normal-ized force component Fx/Fx,max. The distinction between aninfinite half-space and thin-plate behavior is only valid forRm/RmCz ≤ 1, whereas for Rm/RmCz ≥ 1, Fz/Fz,max goes tosaturation and is almost invariant to the changes in Rm and τ(dashed line).

During extensive parametric studies, we observed that therepresentations, as shown in Figs. 9 and 10, are completelyinvariant to the changes in the remaining geometrical para-meters δ and ξ . Thus, the selected representation includes allthe similarity solutions for the defined problem, which areindependent of the input parameters selected in x.

The characteristic variables RmC x,z and τC x,z, as well asthe value Fx,z at the specific parameter point, must be knownto denormalize the standard dimensionless representations.Furthermore, this type of representation is less suitable forcalculating the actual values of the force components, but ithelps us to better understand the phenomenon itself.

Furthermore, during our investigations, we observed thatthe statement of invariance remained valid for the PMs withdifferent base areas, e.g., quadratic or regular octagonal (notexplicitly shown here). This observation is rather surprisingand it leads to the hypothesis that electromagnetic similar-ity also exists between the PMs with different geometries.However, this does not necessarily mean that different PMswill induce an identical eddy current distribution inside theconductor or that the dimensionless force components willbe the same, but it does imply the existence of identicalnormalized representations for PMs with different shapes.

Clearly, the specific values of the characteristic variablesRmC x,z and τC x,z depend on the base area and the geometricparameters of the PM. The dependences for the cylindrical PMare shown in Fig. 11.

It can be seen that the shapes of the curves are similar toRmC x,z and τC x,z for both the force components Fx,z , as wellas for the absolute value of the force F (not shown). In partic-ular, RmC x,z and τC x,z appear to be inversely proportional toeach other over the whole range of δ and ξ . This is confirmedby calculating the product of the factors RmC x,z and τC x,z , andthe related standard deviation σD (Table III) for the parameterrange investigated in Fig. 11.

The mean relative standard deviations σD of ∼2.5% areprobably the results of truncation errors during the numericalintegration required to calculate the force components.

The constancy of the products can be used to reduce theeffort required to determine RmC and τC by calculating onefrom the other, which simplifies their subsequent application tomodel experiments. Furthermore, this supports our hypothesisthat electromagnetic similarity also exists between the PMswith different geometries.

V. SCALING LAWS

We applied dimensional analysis to a simple problem thatcould be solved by an exact analytical formulation. However,many engineering problems are so complex that no analyticalsolution can be obtained. In many of these problems, model

Fig. 11. (a, b) Characteristic magnetic Reynolds number RmCx,z alongτ = 103, (c,d) characteristic dimensionless plate thickness τCx,z alongRm = 10−3, as the functions of δ and ξ .

TABLE III

PRODUCT OF THE CHARACTERISTIC VALUES RmC AND τC CALCULATED

FOR δ = 10−1.4 . . . 101.4 AND ξ = 10−3 . . . 103

experiments are the only way to avoid expensive and time-consuming experiments with wide variation in the governingparameters.

We must stress that in the current problem, we are interestedin the influence of the parameters v, σ , and Br and thegeometric parameters D, H , h, and t on the force componentsFx and Fz that act on the PM. Based on the dimensionalanalysis, we know the form in which all the parameters mustappear in the unknown functions that determine the actingforce. From the discussion in Section IV, we know thatwe can distinguish between different regions of dependencesfrom Rm and τ for the force components. These regions areseparated by the areas of transition, which include the definedcharacteristic points RmC and τC . These characteristic valuesare the only functions of the dimensionless diameter of thePM δ and the aspect ratio ξ .


In order to clearly formulate the scaling laws for ourproblem, we distinguish between a prototype, which is theobject of interest, and a model, which we employ to performexperiments under controlled conditions. For the prototype andthe model, we can consider three different cases.

In the first case, we perform experiments based on amodel with electrodynamic similarity. Electrodynamic simi-larity includes geometric similarity and it occurs if and onlyif each dimensionless parameter (Rm , δ, ξ , and τ ) has thesame value in the model and the prototype. When we designthe model experiment, we must consider that not only all thegeometric parameters need to be scaled linearly. The magneticReynolds number Rm itself also changes with the geometricscale and it must be adapted by changing the product of σv.For example, if we use the same material for the plate in ann-time larger model, then the relative velocity between theplate and the PM must be decreased by 1/n for Rm to beconstant. The forces obtained from the model experimentsshould then be rescaled using (26e) and (26f) to obtain acorrect evaluation. Using the n-time larger model, we knowthat the measured forces are n2 times larger than those forthe prototype. Furthermore, we know that if we use a PMwith an m-time higher magnetic remanence Br , then themeasured forces are also m2 times larger. Again, it is clearthat we do not necessarily have to use the same grade of PMmaterial to obtain a similar electrodynamic model. We onlyneed to consider these differences when the experimentalresults are evaluated. All these statements about scaling inthe case of electrodynamic similarity are a direct consequenceof the dimensional analysis, and an additional discussion ofdependences in different regions is not required.

The second and more general case occurs when we allowRm and τ in the model to differ within a certain range, butwhere δ and ξ are equal in the model and the prototype. This isachieved by identifying which of the different regions containsthe prototype and ensuring that the model experiment occurs inthe same region. If the model is closer to the transition zonesthan the prototype, but it is still outside, then the results can bescaled to those of the prototype. To identify the region of themodel, we need to slightly vary Rm and τ , and then observethe changes in the measured force components. If the changefits the proportionality of one of the characteristic regions, thenthe current region is identified and the scaling parameters areknown.

In the third case, we allow Rm and τ to differ over thewhole range in the model, where the phenomenon is stillmainly described by the same physical effects. It is necessaryto perform four steps to ensure a correct estimation of theLorentz force. First, either RmC or τC must be found asdescribed in (28) by varying the free parameters τ or Rm ,respectively. Second, the other characteristic value must becalculated using the corresponding factor from Table III.Third, the axes in Fig. 9 or 10 should be denormalized bymultiplying the axis scale with the respective characteristicvalue. Finally, the complete graph is denormalized using thedimensionless force component measured by the model ata single arbitrary point. In the result, for a specific con-figuration of δ and ξ , we can estimate the forces for a

very large range of settings, but without the need for directexploration.

As an example of the third case, we take a specific setof δ and ξ , i.e., a fixed distance for a specific cylindrical PM.In the first step, the plate thickness t is changed gradually ata velocity that corresponds to a magnetic Reynolds numberRm = 10−3. The measured values of the force in thex-direction Fxn are multiplied by the associated t−1/2

n to obtainthe symmetric representation given by (27). The characteristicvalue τC x is estimated at the maximum of Fsym

xn and storedwith the measured force Fx . Next, RmC x is estimated usingTable III for a circular base shape as RmC x = 1.94 × τC x .Using these three values, Fig. 9 can be denormalized.

VI. CONCLUSION

In this paper, we contributed to the process of modeling andscaling in the Lorentz force applications using dimensionalanalysis. For this particular problem, we defined a physicalmodel, a list of relevant parameters x, and their individualdimensions [x]. Using this list, we set up a dimensionalmatrix D to calculate a dimensionless basis set � comprisinga dimensionless model of the same problem with a reducednumber of parameters independent of the dimensional systemused. We transformed this basis set to obtain a representationthat is easy to handle. We conducted a concise paramet-ric study to illustrate the advantages of the dimensionlessrepresentation for displaying complex data in an efficientmanner.

In particular, we showed the influence of the magneticReynolds number Rm and the dimensionless plate thickness τfor one arbitrary pair of remaining dimensionless geometricparameters. Using a power-law hypothesis for both depen-dences, we defined four readily distinguished regions, whereeach can be described by a simple power law.

The positions of the transition zones between separatedregions greatly depend on the geometric parameters δ and ξ .Therefore, the results were normalized against the character-istic values RmC and τC , which are defined as the pointswith the maximum curvature of the dimensionless force com-ponents Fx and Fz . This normalization yields a generalizedrepresentation of the dimensionless force components, whichis completely invariant to the changes in the geometric parame-ters δ and ξ . The apparent inversely proportional relationshipbetween the characteristic parameters RmC and τC for differentPM shapes was a surprising result, which was shown tobe an additional simplification that facilitates the subsequentformulation of scaling laws.

Finally, we discussed scaling laws for three relevant scenar-ios, which were illustrated with practical examples.

In conclusion, we must stress that the success of dimen-sional analysis depends on the absolute requirement that thephysical model is complete. This is an unattainable goal formany technical problems, so a satisfactory compromise isrequired to focus on the relevant aspects of the phenomenaincluded. This compromise depends on the judgments thatonly come with experience and a continual validation based onobservations and numerical implementations of the developedmathematical models.


ACKNOWLEDGMENT

This work was supported by the Deutsche Forschungs-gemeinschaft in the framework of the Research TrainingGroup at the Technische Universitaet Ilmenau, Germany underGrant 1567.

REFERENCES

[1] H. Bleuler, “A survey of magnetic levitation and magnetic bearingtypes,” JSME Int. J. III, Vibrat., Control Eng., Eng. Ind., vol. 35, no. 3,pp. 335–342, 1992.

[2] H. Bangcheng, Z. Shiqiang, W. Xi, and Y. Qian, “Integral design andanalysis of passive magnetic bearing and active radial magnetic bearingfor agile satellite application,” IEEE Trans. Magn., vol. 48, no. 6,pp. 1959–1966, Jun. 2012.

[3] J. Sun, D. Chen, and Y. Ren, “Stiffness measurement method of repulsivepassive magnetic bearing in SGMSCMG,” IEEE Trans. Instrum. Meas.,vol. 62, no. 11, pp. 2960–2965, Nov. 2013.

[4] A. Canova and B. Vusini, “Analytical modeling of rotating eddy-currentcouplers,” IEEE Trans. Magn., vol. 41, no. 1, pp. 24–35, Jan. 2005.

[5] W.-J. Kim, D. L. Trumper, and J. H. Lang, “Modeling and vector controlof a planar magnetic levitator,” IEEE Trans. Ind. Appl., vol. 34, no. 6,pp. 1254–1262, Nov. 1998.

[6] J. W. Jansen, E. A. Lomonova, and J. M. M. Rovers, “Effects of eddycurrents due to a vacuum chamber wall in the airgap of a moving-magnetlinear actuator,” J. Appl. Phys., vol. 105, no. 7, pp. 07F111-1–07F111-3,2009.

[7] M. Kanamori and Y. Ishihara, “Finite-element analysis of an elec-tromagnetic damper taking into account the reaction of the magneticfield,” JSME Int. J. III, Vibrat. Control Eng., Eng. Ind., vol. 32, no. 1,pp. 36–43, 1989.

[8] H. A. Sodano and J.-S. Bae, “Eddy current damping in structures,” ShockVibrat. Dig., vol. 36, no. 6, pp. 469–478, 2004.

[9] H. A. Sodano, D. J. Inman, and W. K. Belvin, “Development of a newpassive-active magnetic damper for vibration suppression,” J. Vibrat.Acoust., vol. 128, no. 3, pp. 318–327, 2006.

[10] C. Elbuken, M. B. Khamesee, and M. Yavuz, “Eddy current dampingfor magnetic levitation: Downscaling from macro- to micro-levitation,”J. Phys. D, Appl. Phys., vol. 39, no. 18, pp. 3932–3938, 2006.

[11] B. Ebrahimi, M. B. Khamesee, and F. Golnaraghi, “Permanent magnetconfiguration in design of an eddy current damper,” Microsyst. Technol.,vol. 16, nos. 1–2, pp. 19–24, Jan. 2010.

[12] L. Zuo, B. Scully, J. Shestani, and Y. Zhou, “Design and characterizationof an electromagnetic energy harvester for vehicle suspensions,” SmartMater. Struct., vol. 19, no. 4, pp. 045003-1–045003-10, 2010.

[13] L. Zuo and X. Tang, “Large-scale vibration energy harvesting,” J. Intell.Mater. Syst. Struct., vol. 24, no. 11, pp. 1405–1430, 2013.

[14] E. Bachelet, “Foucault and eddy currents put to service,” Engineer,vol. 114, pp. 420–421, Oct. 1912.

[15] C. A. Guderjahn et al., “Magnetic suspension and guidance for highspeed rockets by superconducting magnets,” J. Appl. Phys., vol. 40,no. 5, p. 2133, 1969.

[16] J. R. Reitz, “Forces on moving magnets due to eddy currents,” Appl.Phys., vol. 41, no. 5, pp. 2067–2071, 1970.

[17] J. R. Reitz and L. C. Davis, “Force on a rectangular coil moving abovea conducting slab,” Appl. Phys., vol. 43, no. 4, pp. 1547–1553, 1972.

[18] P. L. Richards and M. Tinkham, “Magnetic suspension and propulsionsystems for high-speed transportation,” Appl. Phys., vol. 43, no. 6,pp. 2680–2691, 1972.

[19] R. H. Borcherts, L. C. Davis, J. R. Reitz, and D. F. Wilkie, “Baselinespecifications for a magnetically suspended high-speed vehicle,” Proc.IEEE, vol. 61, no. 5, pp. 569–578, May 1973.

[20] S.-W. Lee and R. C. Menendez, “Force on current coils moving over aconducting sheet with application to magnetic levitation,” Proc. IEEE,vol. 62, no. 5, pp. 567–577, May 1974.

[21] D. De Zutter, “Levitation force acting on a three-dimensional staticcurrent source moving over a stratified medium,” Appl. Phys., vol. 58,no. 7, pp. 2751–2758, 1985.

[22] J. Meins, L. Miller, and W. J. Mayer, “The high speed Maglevtransport system TRANSRAPID,” IEEE Trans. Magn., vol. 24, no. 2,pp. 808–811, Mar. 1988.

[23] W. M. Saslow, “Maxwell’s theory of eddy currents in thin conductingsheets, and applications to electromagnetic shielding and MAGLEV,”Amer. Phys., vol. 60, no. 8, pp. 693–711, 1992.

[24] Final Report on the National Maglev Initiative (NMI), Washington, DC,USA, Tech. Rep. DOT/FRA/NMI-93/03, 1993.

[25] J. H. Lever, “Technical assessment of Maglev system concepts: Finalreport by the government Maglev system assessment team,” New York,NY, USA, Tech. Rep. SR 98-12, 1998.

[26] R. F. Post, “Maglev: A new approach,” Sci. Amer., vol. 282, no. 1,pp. 82–87, 2000.

[27] R. F. Post and D. D. Ryutov, “The inductrack: A simpler approachto magnetic levitation,” IEEE Trans. Appl. Supercond., vol. 10, no. 1,pp. 901–904, Mar. 2000.

[28] H.-W. Lee, K.-C. Kim, and J. Lee, “Review of Maglev train technolo-gies,” IEEE Trans. Magn., vol. 42, no. 7, pp. 1917–1925, Jul. 2006.

[29] J. McCurry. (Apr. 24, 2015). Japan’s Maglev train breaks world speedrecord with 600 km/h test run. The Guardian. [Online]. Available:http://www.theguardian.com/world/2015/apr/21/japans-maglev-train-notches-up-new-world-speed-record-in-test-run

[30] H. M. G. Ramos, T. Rocha, D. Pasadas, and A. Ribeiro, “Velocityinduced eddy currents technique to inspect cracks in moving conductingmedia,” in Proc. IEEE Int. Instrum. Meas. Technol. Conf. (I2MTC),May 2013, pp. 931–934.

[31] T. J. Rocha, H. G. Ramos, A. L. Ribeiro, and D. J. Pasadas, “Magneticsensors assessment in velocity induced eddy current testing,” Sens.Actuators A, Phys., vol. 228, pp. 55–61, Jun. 2015.

[32] H. Brauer and M. Ziolkowski, “Eddy current testing of metallic sheetswith defects using force measurements,” Serbian J. Elect. Eng., vol. 5,no. 1, pp. 11–20, 2008.

[33] B. Petkovic, J. Haueisen, M. Zec, R. P. Uhlig, H. Brauer, andM. Ziolkowski, “Lorentz force evaluation: A new approximation methodfor defect reconstruction,” NDT E Int., vol. 59, pp. 57–67, Oct. 2013.

[34] M. Carlstedt et al., “Application of Lorentz force eddy current testingand eddy current testing on moving nonmagnetic conductors,” Int. J.Appl. Electromagn., vol. 45, no. 1, pp. 519–526, 2014.

[35] C. S. MacLatchy, P. Backman, and L. Bogan, “A quantitative magneticbraking experiment,” Amer. Phys., vol. 61, no. 12, pp. 1096–1101, 1993.

[36] K. D. Hahn, E. M. Johnson, A. Brokken, and S. Baldwin, “Eddy currentdamping of a magnet moving through a pipe,” Amer. J. Phys., vol. 66,no. 12, p. 1066, 1998.

[37] M. H. Partovi and E. J. Morris, “Electrodynamics of a magnet movingthrough a conducting pipe,” Can. Phys., vol. 84, no. 4, pp. 253–271,2006.

[38] Y. Levin, F. L. da Silveira, and F. B. Rizzato, “Electromagnetic braking:A simple quantitative model,” Amer. Phys., vol. 74, no. 9, pp. 815–817,2006.

[39] G. Ireson and J. Twidle, “Magnetic braking revisited: Activities for theundergraduate laboratory,” Eur. Phys., vol. 29, no. 4, pp. 745–751, 2008.

[40] N. Derby and S. Olbert, “Cylindrical magnets and ideal solenoids,” Amer.Phys., vol. 78, no. 3, pp. 229–235, 2010.

[41] G. Donoso, C. L. Ladera, and P. Martín, “Damped fall of magnets insidea conducting pipe,” Amer. J. Phys., vol. 79, no. 2, pp. 193–200, 2011.

[42] R. P. Uhlig, M. Zec, H. Brauer, and A. Thess, “Lorentz force eddycurrent testing: A prototype model,” J. Nondestruct. Eval., vol. 31, no. 4,pp. 357–372, Dec. 2012.

[43] B. Irvine, M. Kemnetz, A. Gangopadhyaya, and T. Ruubel, “Magnettraveling through a conducting pipe: A variation on the analyticalapproach,” Amer. Phys., vol. 82, no. 4, pp. 273–279, 2014.

[44] J. A. Pelesko, M. Cesky, and S. Huertas, “Lenz’s law and dimensionalanalysis,” Amer. Phys., vol. 73, no. 1, pp. 37–39, 2005.

[45] M. K. Roy, M. K. Harbola, and H. C. Verma, “Demonstration of Lenz’slaw: Analysis of a magnet falling through a conducting tube,” Amer. J.Phys., vol. 75, no. 8, p. 728, 2007.

[46] S. R. Bistafa, “On the derivation of the terminal velocity for the fallingmagnet from dimensional analysis,” Revista Brasileira Ensino Física,vol. 34, no. 2, pp. 1–4, 2012.

[47] P. A. Davidson, An Introduction to Magnetohydrodynamics (CambridgeTexts in Applied Mathematics). New York, NY, USA: Cambridge Univ.Press, 2001.

[48] E. P. Furlani, Permanent Magnet and Electromechanical Devices: Mate-rials, Analysis, and Applications. San Diego, CA, USA: Academic,2001.

[49] M. Ziółkowski, Modern Methods for Selected Electromagnetic FieldProblems. Szczecin, Poland: West Pomeranian Univ. Technol. Press,2015.

[50] R. N. Bracewell, The Fourier Transform and Its Applications(McGraw-Hill Series in Electrical and Computer Engineering), 3rd ed.New York, NY, USA: McGraw-Hill, 1999.

[51] J. F. Price, “Dimensional analysis of models and data sets,” Amer. J.Phys., vol. 71, no. 5, p. 437, 2003.


[52] W. E. Baker, P. S. Westine, and F. T. Dodge, Similarity Methodsin Engineering Dynamics: Theory and Practice of Scale Modeling(Fundamental Studies in Engineering), vol. 12. New York, NY, USA:Elsevier, 1991.

[53] P. W. Bridgman, Dimensional Analysis. New Haven, CT, USA:Yale Univ. Press, 1963.

[54] K. Hutter and K. Jöhnk, Continuum Methods of Physical Modeling: Con-tinuum Mechanics, Dimensional Analysis, Turbulence. Berlin, Germany:Springer, 2004.

[55] E. Buckingham, “On physically similar systems; illustrations of the useof dimensional equations,” Phys. Rev., vol. 4, no. 4, pp. 345–376, 1914.

[56] T. Szirtes and P. Rózsa, Applied Dimensional Analysis and Modeling,2nd ed. New York, NY, USA: Elsevier, 2007.

[57] G. I. Barenblatt, Dimensional Analysis. New York, NY, USA:Gordon and Breach, 1987.

[58] S. V. Gupta, Units of Measurement: Past, Present and Future. Interna-tional System of Units (Springer Series in Materials Science), vol. 122.Heidelberg, Germany: Springer, 2010.

[59] A. Thess, E. Votyakov, and Y. Kolesnikov, “Lorentz force velocimetry,”Phys. Rev. Lett., vol. 96, no. 16, p. 164501, 2006.

[60] MATLAB Version 8.2.0.29 (R2013b), The MathWorks Inc., Natick, MA,USA, 2013.

Matthias Carlstedt was born in Nordhausen, Germany, in 1984. He receivedthe M.Sc. degree in mechanical engineering from the Technische UniversitätIlmenau, Ilmenau, Germany, in 2012, where he is currently pursuing thePh.D. degree with the Advanced Electromagnetics Group.

His current research interests include system identification, structuraldynamics, and model validation for nondestructive testing applications.

Konstantin Weise (née Porzig) (S’12) was born in Leipzig in 1986.He received the B.Eng. degree in electrical engineering from the Universityof Applied Science Leipzig, Leipzig, Germany, in 2009, and the M.Sc. degreein electrical engineering from the Technische Universität Ilmenau, Ilmenau,Germany, in 2012, where he is currently pursuing the Ph.D. degree.

His current research interests include biomedical engineering problems andnondestructive testing applications.

Marek Ziolkowski was born in Szczecin, Poland, in 1954. He received theM.Sc. and Ph.D. degrees in electrical engineering from the Technical Univer-sity of Szczecin, Szczecin, in 1978 and 1984, respectively, and the Habilitationdegree from the Faculty of Electrical Engineering, West Pomeranian Univer-sity of Technology, Szczecin, in 2015.

He has been with the Department of Electrical and Computer Engineering,Technical University of Szczecin, since 1978. Since 2001, he has alsobeen with the Advanced Electromagnetics Group, Technische UniversitätIlmenau, Ilmenau, Germany. His current research interests include numericalsimulations and visualization of electromagnetic fields with applications toforward/inverse problems in nondestructive evaluation, bioelectromagnetics,small electrical machines, and magnetic fluid dynamics.

Reinhard Schmidt was born in Erfurt, Germany, in 1981. He received theDiploma Engineering degree in electrical engineering from the TechnischeUniversität Ilmenau, Ilmenau, Germany, in 2013.

He has been with the Advanced Electromagnetics Group, TechnischeUniversität Ilmenau, since 2013, where he is currently a Scientific Assistant.His current research interests include nondestructive testing of metal injectionmolding specimens in the framework of Lorentz force eddy current testing.

Hartmut Brauer was born in Berlin, Germany, in 1953. He received theDiploma and Dr.-Ing. (Ph.D. equivalent) degrees in electrical engineering fromthe Technische Universität Ilmenau, Ilmenau, Germany, in 1975 and 1982,respectively.

He has been with the Advanced Electromagnetics Group, TechnischeUniversität Ilmenau, since 1975, where he is currently a Senior Lecturer andSenior Researcher. He has authored or co-authored over 100 papers in journalsand books. His current research interests include the theory and computationof electromagnetic fields, and inverse field problems/optimization in electricalengineering with applications to nondestructive evaluation, bioelectromagnet-ics, magnetic fluid dynamics, and educational aspects.

Dr. Brauer is a Founder Member of the International Compumag Societyin 1994 and has been a member of the International Steering Committeeof the International Workshops on Optimization and Inverse Problems inElectromagnetism since 2008.

Documents

Estimation of Lorentz Force From Dimensional Analysis