IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 1, JANUARY 2004 59

Modeling Maglev Passenger Compartment StaticMagnetic Fields From Linear Halbach

Permanent-Magnet ArraysJames F. Hoburg, Fellow, IEEE

Abstract—Passenger compartment magnetic field levels in alow-speed magnetic levitation (maglev) vehicle that uses linearHalbach permanent-magnet arrays for both levitation and propul-sion are computed through superposition of fields due to patchesof magnetization charge at surfaces where the magnetization isdiscontinuous. End effects due to the finite lengths of the arrayslead to fields that decay much less rapidly with distance fromthe arrays than the near-field exponential decay based on arraywavelength, and do not have the strong side/weak side characterof the near fields. End effects dominate the maglev passengercompartment fields. Contour plots of computed fields due to themagnet arrays for a specific maglev design show field magnitudesof about 1.5 G at floor level, 0.5 G at seat level, and 0.2 G at headlevel.

Index Terms—Halbach array, human field exposure, maglev,magnetic levitation.

I. INTRODUCTION

MODERN high strength (Nd–Fe–B) permanent magnetsprovide a lightweight alternative to electromagnets as

on-board magnetic field sources on a magnetic levitation (ma-glev) vehicle [1], [2]. In a system that is under development asa low-speed urban maglev alternative to light rail transporta-tion, permanent-magnet arrays on the moving vehicle are usedfor both levitation (through interaction with passive conductingloops in track coils) and propulsion [through interaction withground-based linear synchronous motor (LSM) traveling wavecoils]. The magnets are configured as “Halbach” arrays, whereinthe magnetizations rotate from one magnet to the next in the di-rection of travel of the vehicle, so as to accomplish two separatepurposes:

1) to create fields that vary periodically with space in thedirection of travel of the vehicle, with a dominant firstFourier component;

2) to put nearly all of the field either above or below thearray, so as to maximize the strength of the field that in-teracts with the track coil or LSM.

In the context of evaluating magnetic field levels in thevehicle passenger compartment due to the Halbach perma-

Manuscript received June 5, 2003; revised September 17, 2003. This workwas supported in part by the U.S. Department of Transportation, Federal TransitAdministration Office of Technology, through General Atomics, under Cooper-ative Agreement CA-26-7025.

The author is with the Department of Electrical and Computer Engi-neering, Carnegie Mellon University, Pittsburgh, PA 15213 USA (e-mail:hoburg@ece.cmu.edu).

Digital Object Identifier 10.1109/TMAG.2003.821559

nent-magnet arrays, the fact that the arrays have finite lengthsbecomes a significant issue—in fact, the dominant field con-tributions in the passenger compartments are quite differentfrom the near fields that interact with the track coils and LSM.The strong side/weak side character of Halbach array fields isaltered in the far fields, where end effects become dominant.The terms “near” and “far” here refer to evaluation distancesfrom the array that are respectively small and large in compar-ison with the spatial period of the magnet arrays. This usageis not to be confused with near and far antenna fields, wheredistances are compared with an electromagnetic wavelength.Similarly, the term “wavelength” in the text to follow refersto the spatial period of the arrays, and has nothing to do withan electromagnetic wavelength. This paper documents thedifferences between near and far fields from linear Halbacharrays, and shows the significance of these differences withregard to passenger compartment magnetic field levels.

II. BACKGROUND: HALBACH ARRAY NEAR FIELDS

The Halbach array was conceived by Klaus Halbach as acircular array of permanent-magnet cubes, with magnetizationsrotating from one magnet to the next, for use in particle accel-eration experiments [3].

A linear version of the Halbach array, wherein the circle is“laid out flat” and repeated for two wavelengths with eightmagnets per wavelength, is portrayed in Fig. 1. One extramagnet (beyond the two wavelengths) is included so as tocreate a symmetric configuration, with vertical magnetizationsin the magnets at each end. The individual magnets in Fig. 1actually represent several magnet cubes, all with the samemagnetization direction, situated adjacent to one another in thedirection perpendicular to the plane of view.

The rotation of magnetization from one magnet to the next(Fig. 1, left to right) results in constructive superposition on oneside (Fig. 1, bottom) and destructive superposition on the otherside (Fig. 1, top) [4]. In general, for a magnetization directionthat is clockwise from one magnet to the next moving from leftto right, the near field below the array is much stronger than thenear field above the array, and varies sinusoidally in the hori-zontal direction with the same fundamental wavelength as thearray. (This configuration is used for vehicle-mounted arraysthat induce currents in ground-based conducting “track” loopsso as to cause a net levitation force on the vehicle.) A reversal inthe direction of magnetization rotation yields a correspondingreversal in the strong side/weak side polarity—counterclock-

0018-9464/04$20.00 © 2004 IEEE

60 IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 1, JANUARY 2004

Fig. 1. Two-wavelength linear Halbach array, with 45 clockwise rotation from one magnet to the next. The array is several magnet cubes wide in the directionperpendicular to the plane of view. The resultant near field is much stronger below than above the array.

wise rotation from left to right produces a strong side field aboveand a weak side field below the array. (This configuration is usedfor additional vehicle-mounted arrays to increase the lift/dragratio and to interact with linear synchronous motor propulsioncoils. The various Halbach arrays on the vehicle are describedin detail in Section IV.)

For an array that consists of several spatial periods, the nearfields on the strong side can be approximated by assuming anarray of infinite extent and by retaining only the fundamentalFourier component, which decays exponentially with verticaldistance from the array less rapidly than the higher Fourier com-ponents. The resulting closed-form expressions are simple andaccurate for use in levitation and drag force descriptions. Witharray wavelength , the near fields are expressed in terms ofa wavenumber that is defined by . For a remanentfield at the surface of each magnet cube, the strong sidefields have a fundamental Fourier component with magnitude

, where the relationship between and is based on thewavenumber , the magnet thickness , and the number of mag-nets per wavelength

(1)

The maglev vehicle arrays that are described here have wave-length m with magnets per wavelength.Thus, each magnet has thickness m. For a typicalNd–Fe–B magnet with remanent field T, (1) yields

T. The near field for the array shown in Fig. 1 thenhas strong side fundamental Fourier components in the regionunder the array, with measured horizontally to the right andwith (which is negative) measured vertically from the bottomsurface, that take the forms

(2)

The exponential decay length is m.At a distance 1 m under the surface of the array (or 1 m above

the surface of an upward facing array), the exponential decay in(2) yields fields with magnitude T mG. However,an essential point, as documented in this paper, is that this cal-culation does not correctly capture very significant end-effectfields for a finite length array. These fields decay less rapidlythan exponentially, and thus dominate the far fields.

III. HALBACH ARRAY FIELDS BASED ON

MAGNETIZATION CHARGES

Since each permanent-magnet cube in a Halbach array has aknown uniform magnetization, its contribution to the total fieldcan be calculated on the basis of square patches of magnetiza-tion surface charge density at the faces of the cube where the

magnetization is discontinuous. Each patch consists of a uni-form magnetization surface charge density ,where is the discontinuity in normal component of mag-netization across the magnet face, and serves as a source forthe magnetic flux density in the free space region outside themagnet cube at observation position vector , as described byintegration of elemental contributions over the surface , withsource position vector and unit vector pointing fromsource to observation position

(3)

Equation (3) is integrated to yield closed-form expressions forfield components from each magnet face. For example, a singlehorizontal patch at height that extends between and

and between and contributes field compo-nents at observation point

(4)

(5)

(6)

where

(7)

(8)

(9)

(10)

Magnet cubes that have 45 rotations from horizontal and ver-tical have patches of magnetization surface charge density onboth horizontal and vertical faces.

In order to compare near and far fields, with end effects in-cluded, the components of magnetic flux density are calculatedby using the magnetization charge methodology to superimposecontributions from all magnet cubes in a linear Halbach arraythat is like the one portrayed in Fig. 1, but with nine wavelengthsplus one extra magnet, with eight magnets per wavelength,and five magnet cubes deep in the direction perpendicular to

HOBURG: MODELING MAGLEV PASSENGER COMPARTMENT STATIC MAGNETIC FIELDS 61

Fig. 2. Fields components and field magnitude evaluated 0.1 m above thetop surface and below the bottom surface of a nine-wavelength linear Halbacharray, with � = 0:4 m. The field components vary sinusoidally with the samewavelength as the array, and are much stronger on the strong (lower) side.Field magnitude in the region under the array is slightly less than a near-fielddescription based on (2) due to the finite array length.

the plane of view. The total number of magnetization surfacecharge patches is .The calculation of fields through superposition of magnetiza-tion charge contributions is exact and is computationally veryefficient in comparison with a three-dimensional finite-elementdescription, which has been used as an alternative to verifythe results shown below. A finite-element description for asource with this level of complexity requires tens of thousandsof tetrahedra over the solution region to obtain results that arecorrect to within 1%.

In order to characterize fields due a finite length linear Hal-bach array, the magnetization charge methodology is used tocalculate field components and , and field magnitude

, for the nine-wavelength array along linesabove and below the array at three different distances from itssurfaces. In each case, the line is centered with respect to thedepth of array and extends 2 m beyond each end of the array inthe direction. The lines are positioned vertically (in the di-rection) at distances of 0.1, 0.5, and 2.0 m from the top (“weakside”) and bottom (“strong side”) surfaces of the array, and theresultant field components are plotted in Figs. 2–4, respectively.The near-field magnitude based on (2), , is shownfor comparison on the strong side plots.

In Fig. 2, at distances above and below the surfaces of thearray, the fields are nearly as would be expected on the basis ofa near-field Halbach array analysis—the field components vary

Fig. 3. Field components and field magnitude evaluated 0.5 m above andbelow the respective surfaces of the nine-wavelength Halbach array. At adistance a little larger than the wavelength, end-effect fields become strongerthan the more rapidly decaying near fields.

sinusoidally in the direction with a wavelength correspondingto the array wavelength, the weak side fields are small in ampli-tude, and the strong side fields are nearly in agreement with thesimple description of (2) over the nine-wavelength length of thearray, decaying rapidly beyond the ends of the array. The factthat the array has a finite length reduces the strong side fieldmagnitude only slightly from the near-field description.

In Fig. 3, at distances a little more than one wavelength fromthe surfaces of the array, the sinusoidal variation in the direc-tion is still apparent over most of the length of the array on thestrong side, but two effects associated with the finite length ofthe array become apparent.

1) In addition to the field components that vary sinusoidallywith according to (2), each end of the array contributesa much larger “end-effect” field. The end-effect fieldsdecay in the vertical direction much less rapidly than theexponential decay of the near fields. Thus, their contribu-tion begins to dominate at larger distances from the array.

2) The end-effect fields are completely different with regardto the “strong side/weak side” issue. At this distance, the“weak side” field components have peak values near theends that are more than half those of the “strong side”field components.

In Fig. 4, at distances five times the wavelength from the sur-faces of the array, the sinusoidal variation in the direction iscompletely masked by the end-effect fields, which decay much

62 IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 1, JANUARY 2004

Fig. 4. Field components and field magnitude evaluated 2.0 m above andbelow the respective surfaces of the nine-wavelength Halbach array. Thenear-field sinusoidal variation is completely masked by the end-effect fields.

less rapidly than the near fields and which extend out over theentire length of the array. The “weak side” fields are now largerin peak magnitude than the “strong side” fields.

In order to show the transition from near field to far fielddecay, the field magnitude, , for the nine-wavelength linear Halbach array is plotted on a semilog scaleas a function of vertical distance from the array surfaces inFig. 5. (Since varies with at any one vertical distance ,the maximum value over all is used in this plot.) For rela-tively small distances, the field below the array is much strongerthan the field above the array, and both lines have a slope thatcorresponds to the exponential decay based on wavelength. Inthis region, the strong side field magnitude is nearly identical tothe field magnitude based on (2), . For relativelylarge distances, the decays become much less rapid, and the fieldabove the array has a larger magnitude than that below the array.

IV. MAGLEV VEHICLE PASSENGER COMPARTMENT FIELDS

The descriptions of Section III are directly relevant to a low-speed urban maglev system that is currently being developedand tested by General Atomics, with support from the FederalTransit Administration. Fig. 6 shows an end view of three pairsof linear Halbach arrays, labeled “propulsion,” “upper levita-tion,” and “lower levitation” on opposite sides of the vehiclecenterline in a maglev vehicle design. The propulsion arraysare positioned just below ground-based LSM coils that serveas sources for the propulsion traveling wave. The upper lev-itation and lower levitation arrays are positioned respectively

Fig. 5. Maximum (over y) magnitude of field as a function of distance z

from the array surfaces. At relatively small distances, the strong side fieldmagnitude is nearly identical to the description based on (2). The transitionfrom exponential decay based on the wavelength to far fields with a much lessrapid decay is apparent.

above and below passive levitation coils, so as to induce currentsand yield a net levitation force that is based on vehicle motion.A horizontal line in Fig. 6 indicates the level of the floor in thepassenger compartment.

Each array in Fig. 6 is nine wavelengths plus one magnetlong in the direction perpendicular to the plane of view; the two“upper levitation” arrays each are five magnets wide and are“downward facing,” so as to produce fields that interact withground-based passive levitation coils just beneath them. The“lower levitation” arrays each are three magnets wide and are“upward facing” toward the passive levitation coils—their roleis to reduce the net drag force on the vehicle. The propulsionmagnets are each two magnets wide, with a center “blank” be-tween the two magnets, and are “upward facing” so as to interactwith the LSM coils just above them.

For purposes of analyzing the electromechanical couplingsof both the LSM to the propulsion arrays and of the levitationmagnet arrays to the passive levitation coils, a near-field Hal-bach array description is quite accurate. However, for purposesof assessing passenger compartment magnetic field levels, thefar-field effects associated with finite length arrays that aredescribed in Section III become dominant. The magnetizationcharge methodology provides a simple means to superimposecontributions from the relatively complex configuration ofsources so as to determine these fields.

Fig. 7 shows contour plots of the magnitude of magnetic fluxdensity (in tesla) due to superimposed fields from the propul-sion, upper levitation, and lower levitation magnets at floorlevel, seat level (0.45 m above the floor), and head level (1.2 mabove the floor) over the area of the passenger compartment. Atfloor level, peak fields occur over each end of each array, withvalues just under 0.0003 T 3 G. A calculation based on theHalbach array near field (2), using the nearest (upward facingpropulsion) array, whose upper surface is 0.7 m below floorlevel, would produce floor level fields of only 0.12 G—thus,

HOBURG: MODELING MAGLEV PASSENGER COMPARTMENT STATIC MAGNETIC FIELDS 63

Fig. 6. End view of propulsion, upper levitation, and lower levitation linear Halbach arrays, LSM support structure, track coils, and floor elevation within thepassenger compartment.

Fig. 7. Contour plots of magnitude of magnetic flux density in tesla due tothe superimposed fields of the six arrays shown in Fig. 6 at floor, seat, andhead levels. At floor level, individual peaks from the ends of the arrays areapparent, with largest values of about 3 G. At head level, only a broad singlemound remains, with largest value just under 0.4 G.

the far-field calculation is essential for distances correspondingto elevations in the passenger compartment. At seat level, theend-effect fields still have maxima over the ends of the arrays,but the local peaks over the individual arrays on each side of thevehicle are smoothed out into a broad single peak, with a valuejust under 0.0001 T 1 G. At head level, the fields are broadlydistributed over the entire passenger compartment, with peakvalues near the center just under 0.00004 T 0.4 G. The decayin peak field magnitude from floor to seat to head level is, asexpected, much less rapid than the exponential decay that ischaracteristic of Halbach array near fields.

Through a simple 180 phase shift of the propulsion magnetorientations with respect to the levitation magnets, the super-position of far-field contributions from the various arrays andresultant passenger compartment fields can be significantly re-duced. If the end magnets in the propulsion arrays are orientedwith downward instead of upward magnetizations (but the ro-tation from magnet to magnet remains counterclockwise so asto yield a near field that is upward facing to interact with the

Fig. 8. Contour plots with a 180 phase shift of the propulsion magnetorientations with respect to the levitation magnets. The end-effect fields tendto cancel with this configuration, with a resultant significant reduction inpassenger compartment fields in comparison with those shown in Fig. 7. Thelargest floor level field is about 1.5 G.

LSM field), the calculated magnetic flux density contour plotsare shown in Fig. 8. The maximum value at floor level for thisconfiguration is reduced to about 1.5 G.

The results for passenger compartment field levels that areshown here describe only contributions from the permanent-magnet arrays. Additional contributors to the fields include:

1) currents in the LSM traveling wave coils;2) induced currents in the levitation coils.

Each of these sources produces fields that are static in the refer-ence frame of the passenger compartment, with spatial phaseswith respect to the permanent-magnet array fields that are de-pendent on vehicle speed. For all speeds, both contributions arerelatively small in comparison with the permanent-magnet arrayfields. An additional effect that is ignored here is the inherentshielding that is provided by a steel support plate for the LSMcoils above the propulsion magnet arrays.

Relevant standards with regard to static magnetic fieldlevels for human exposure safety include those of The Inter-national Commission on Nonionizing Radiation Protection

64 IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 1, JANUARY 2004

(ICNIRP)—(400 G for the general public and 5 G for peoplewith electronic implants or metallic metal implants) andthose of the American Conference of Governmental IndustrialHygienists (ACGIH)—(600 G for the whole body for 8 h, and5 G for people with electronic implants, metallic implants, orpain device users) [5]. The maximum values of 3 G at floorlevel in Fig. 7 and 1.5 G at floor level in Fig. 8 are significantlyunder the 5 G standard.

V. CONCLUSION

The work that is shown in this paper is motivated by the ques-tion of magnetic field levels in the passenger compartment of alow-speed maglev vehicle that uses Halbach arrays for both lev-itation and propulsion. Results are based on the use of a compu-tational methodology wherein field contributions from patchesof magnetization charge at the surfaces of the individual mag-nets of the arrays are superimposed to find total fields at anyobservation point.

The following conclusions are evident from this work.

1) End effects become dominant at large distances froma linear Halbach array because their fields decay muchless rapidly with distance than the near-field exponentialdecay that is based on the array wavelength. End effectsare the dominant contributors to passenger compartmentfields.

2) The strong side/weak side character of Halbach array nearfields is not present in the end-effect fields. Configurationof the relative spatial phases of multiple arrays providesa way to obtain cancellation, and thus to reduce net farfields.

3) Calculated passenger compartment field levels are signif-icantly below relevant standards with regard to static mag-netic field levels for human exposure safety.

An important point with regard to passenger compartmentfields in a maglev design that uses permanent magnets for itson-board excitation is that the fields are static. This is true forthe LSM traveling wave as seen in the frame of the vehicle, andfor the fields that are produced by the induced currents in thetrack, as well as the stronger fields of the permanent-magnetarrays.

ACKNOWLEDGMENT

The interest, insight, and advice of Dr. R. F. Post ofLawrence Livermore National Laboratory, Dr. H. Gurol ofGeneral Atomics, and V. Pindiprolu of the Federal TransitAdministration are gratefully acknowledged.

REFERENCES

[1] “Final report,” Low Speed Maglev Technology Development Pro-gram, U.S. Dept. Transportation (Federal Transit Administration),FTA-CA-26-7025-02.1, 2002.

[2] R. F. Post, “Maglev: A new approach,” Sci. Amer., vol. 282, pp. 82–87,Jan. 2000.

[3] K. Halbach, “Application of permanent magnets in accelerators andstorage rings,” J. Appl. Phys., vol. 57, pp. 3605–3608, Apr. 1985.

[4] J. Ofori-Tenkorrang and J. H. Lang, “A comparative analysis of torqueproduction in Halbach and conventional surface-mounted permanent-magnet synchronous motors,” in Conf. Rec. 1995 IEEE Industry Appli-cations Conf., vol. 1, Oct. 8–12, 1995, pp. 657–663.

[5] “Final programmatic environmental impact statement,” Maglev Deploy-ment Program, U.S. Dept. Transportation (Federal Railroad Administra-tion), DOT/FRA/RDV-00/02, DOT-VNTSC-FRA-00-04, 2001.