Electromagnetic Parameters of aThree Phase Transverse Flux Machine

with an External Rotor for Wheel Hub DrivesErich Schmidt1, Daniel Brunnschweiler, Stefan Berchten2

1Institute of Electrical Drives and MachinesVienna University of Technology, Vienna, Austria

2MagnetDrives AG, Zug, Switzerland

Abstract – The transverse flux machine with apermanent magnet excited rotor is one of the mostimportant machine topology for the application withelectric driven vehicles. In particular with wheelhub drive systems, the design of such a machine withan external rotor is favourably against conventionalinduction machines due to the high output torquein conjunction with low heat losses. The paper dis-cusses 3D finite element analyses of a novel designof a three phase transverse flux machine with anexternal rotor. In order to setup an environmentsuitable for a prototype design of the single sidedtransverse flux machine, special attention has to begiven to modelling as well as analysis methods. Theresults presented here are focussed on the most im-portant electromagnetic parameters such as no-loadvoltages, short-circuit currents and electromagnetictorque.

Index Terms – Transverse flux machine, Perma-nent magnet machine, Finite element analysis

I. Introduction

In general, two different topologies of a transverseflux machine exist [1]–[5]. Thereby, double-sided ar-rangements can be used only in either single phaseor two phase mode. On the other hand, single-sidedarrangements can be designed for three phase mode,too.

In contrast to conventional induction machines withmore than one phase, there is no common rotat-ing field with all designs of transverse flux machines[1]–[6]. In order to produce a resulting shaft torqueas comparatively as smooth as with conventional in-duction machines, either stator or rotor parts haveto be mechanically shifted according to the numberof phases. Nevertheless, the alternating fields withan electrical angular shift according to the number ofphases cause higher harmonics with the electromag-netic torque as well as noticeable shear stresses in allcarrier parts and the shaft [6].

The novel single-sided design with its main datalisted in Table I consists of a three phase arrangementwith an external rotor. Fig. 1 depicts two poles of the

basic arrangement of one single phase whereas Fig. 2illustrates two poles of the complete three phase ar-rangement. The rotor parts of the three phases arearranged in line and carry the permanent magnetswith an alternating magnetization in circumferentialdirection. The stator parts of the three phases carrythe ring windings of the three phases and have an ap-propriate mechanical angular shift necessary for thethree phase operation.

TABLE I

Main Data of the Transverse Flux Machine

Number of poles 120

Rated speed 400 rpm

Maximum speed 1200 rpm

Outer stator diameter 380 mm

Outer rotor diameter 410 mm

Airgap length 1 mm

Total axial length 100 mm

Rated stator current 120 A

Remanent flux density 1.30 T

Coercive field strength 985.25 kA/m

Fig. 1: Basic arrangement of two poles of a single sided trans-verse flux machine with flux concentration

Fig. 2: Three phase transverse flux machine with in line rotorsegments and inserted permanent magnets as well asshifted stator segments and ring windings, finite ele-ment model of two poles

II. Finite Element Modelling

The finite element model as shown in Fig. 2 con-sists of completely independent stator as well as rotorparts and includes only two poles of the machine asthe smallest necessary part. In order to reflect the pe-riodicity of the magnetic field, appropriate repeatingperiodic boundary conditions for the unknown degreesof freedom of the 3D magnetic vector potential are ap-plied at the boundaries being two poles pitches apart.

The various nonlinear finite element analyses uti-lize a 3D vector potential formulation with an incor-porated Coulomb gauge

curl(ν∼· curlA

)= J , divA = 0 (1)

with appropriate Neumann and Dirichlet boundaryconditions (

ν∼· curlA

)× n = K on ΓH , (2a)

A × n = 0 on ΓB , (2b)

where ν∼

denotes an anisotropic reluctivity tensor, ΓH

the boundary where n × H is specified and ΓB theboundary where n ·B is specified [7]–[9].

Regarding the arrangement as depicted in Fig. 2,the nonlinear anisotropic material properties of thelaminated stator and rotor iron regions are describedby

ν∼rϕz

(B

)=

⎡⎣ νrr

(B

)0 0

0 νϕϕ

(B

)0

0 0 νzz

(B

)⎤⎦ , (3)

νϕϕ

(B

)=

(1 − kF

)ν0 + kF

√νrr

(B

)νzz

(B

),

where kF denotes the stacking factor of the lamina-tions.

With an intent of investigating cross-coupling effectsbetween the three phases, Dirichlet boundary condi-tions of the magnetic vector potential on the sym-metry planes between the phases in axial directioncan be used optionally. With these internal bound-ary conditions, the magnetic flux densities are onlytangential on these symmetry planes yielding fully de-coupled phases. Without any coupling of the threephases, they will act independently in particular onthe electromagnetic torque. On the other hand, thereexist additional axial stray flux portions between thethree phases. In dependence on the axial distance ofthe three phases, these flux portions will influence theevolved electromagnetic torque with both no-load andload cases.

The injection of the three phase currents with thering windings of the stator follows

i1(ϕ) = I cos(β + ϕ + ϕ0

), (4a)

i2(ϕ) = I cos(β + ϕ + ϕ0 −

2π

3

), (4b)

i3(ϕ) = I cos(β + ϕ + ϕ0 −

4π

3

), (4c)

where ϕ0 denotes the initial angular rotor position ofthe first phase. As shown in Fig. 2, stator and rotor ofthe first phase are arranged in an unaligned positionwhich is represented by ϕ0 = π/2. Moreover, β de-notes the current angle with respect to the rotor fixedcoordinate system. Thereby, current angles of β = 0or β = ±π produce no resultant torque. A maximumresultant torque is generated with a current angle ofβ = ±π/2 [6].

The various angular rotor positions are now calcu-lated by utilizing the sliding surface approach in con-junction with a domain decomposition algorithm [6],[10]–[12]. Thereby, the independent stator and rotorparts Ωst and Ωrt have an equidistant mesh discretiza-tion in circumferential direction on a cylindrical slid-ing surface Γsl within the air-gap while the mesh dis-cretization in axial direction coincides between bothparts.

The two independent model parts are now describedwith their own symmetric stiffness matrices Kst,Krt

and by the matrix equations

Km Um + Gm = Pm , m = {st, rt} , (5)

wherein Gst,Grt summarize the boundary terms re-lated to unknown tangential components of the mag-netic field strength at the sliding surface boundary Γsl.Afterwards, the stiffness matrices of the model partsare decomposed according to two exclusive subsets ofinterior (i) and exterior (e) unknown degrees of free-

dom,[Km,ii Km,ie

KTm,ie Km,ee

][Um,i

Um,e

]+

[0

Gm,e

]=

[Pm,i

Pm,e

].(6)

The coupling of the two model parts is now providedby a direct mapping of the degrees of freedom alongthe sliding surface boundary Γsl between stator androtor parts depending on the angular rotor position,

Urt,e = Ek Ust,e , Gst,e = −ETk Grt,e . (7)

According to the used discretization in circumferentialdirection, this locked-step coupling allows for angularpositions with an electrical increment of Δϕ = π/24.

Therefore, the finite element mesh remains com-pletely unchanged for all angular rotor positions with-out any remeshing. Consequently, different numericalerrors with respect to the angular rotor position areavoided. Additionally, distinct decompositions of thesmaller stiffness matrices Km,ii of the two indepen-dent model parts significantly reduce the calculationtimes for various rotor positions even in a nonlinearanalysis.

III. Analysis Results

A. No-Load Voltages and Short-Circuit Currents

Fig. 3 and Fig. 4 show the no-load voltages of thethree phases with a Y-connected stator for the ratedspeed of n = 400 rpm, respectively. Further, Fig. 5shows the symmetrical short-circuit currents of thethree phases with a Y-connected stator for the ratedspeed of n = 400 rpm.

Caused by the high level of saturation, the no-loadvoltages contain a significant 3rd harmonic componentresulting in a non-vanishing sum of the three phasevoltages as drawn additionally. In comparison of thecoupling effects, the fundamental harmonic decreasesfrom 206.4 V to 203.2 V for decoupled and coupledphases. On the other hand due to the low level ofsaturation, the symmetrical three phase short-circuitcurrents are nearly sinusoidal with respect to time.Thereby, the short-circuit currents are smaller withcoupled against decoupled phases.

B. Electromagnetic Torque

As described in [6], the torque calculation requiresonly the portions along a cylindrical surface within theair-gap. Hence, the electromagnetic torque is evalu-ated from the analysis results by summarizing the cir-cumferential component of the Maxwell stress vector

pϕ,mn = ν0 Br(rT , ϕm, zn)Bϕ(rT , ϕm, zn) (8)

0 π/4 2π/4 3π/4 π 5π/4 6π/4 7π/4 2π

Angular position ϕ (rad)

−300

−250

−200

−150

−100

−50

0

50

100

150

200

250

300

Voltage Ui (V)

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Fig. 3: No-load voltages of the three phases, rated speed ofn = 400rpm, decoupled phases

0 π/4 2π/4 3π/4 π 5π/4 6π/4 7π/4 2π

Angular position ϕ (rad)

−300

−250

−200

−150

−100

−50

0

50

100

150

200

250

300

Voltage Ui (V)

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Fig. 4: No-load voltages of the three phases, rated speed ofn = 400rpm, coupled phases

0 π/4 2π/4 3π/4 π 5π/4 6π/4 7π/4 2π

Angular position ϕ (rad)

−120

−100

−80

−60

−40

−20

0

20

40

60

80

100

120

Current Isc (A)

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Fig. 5: Short-circuit currents of the three phases, rated speedof n = 400 rpm, decoupled phases (dashed lines) andcoupled phases (solid lines)

along a cylindrical surface within the air-gap as

Tz = p

Nz∑n=1

Nϕ∑m=1

rT Δvmn

ΔrTpϕ,mn , (9)

where p is the number of pole pairs, Nϕ and Nz denotethe numbers of air-gap elements in circumferential andaxial direction, rT and ΔrT are radius of center ofgravity and radial thickness of each hexahedral air-gap element along the cylindrical surface, and Δvmn

denote the finite element volume, respectively.

Fig. 6 and Fig. 7 show the cogging torque of thethree phases and the entire machine with decoupledand coupled phases. Fig. 8 and Fig. 9 show the loadtorque of the three phases and the entire machine withthe rated stator current of I = 120 A in the quadratureaxis according to a maximum electromagnetic torquewith decoupled and coupled phases.

0 π/4 2π/4 3π/4 π 5π/4 6π/4 7π/4 2π

Angular position ϕ (rad)

−20

−15

−10

−5

0

5

10

15

20

Torque Tz (Nm)

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Fig. 6: Cogging torque of the three phases and the entiremachine, decoupled phases

0 π/4 2π/4 3π/4 π 5π/4 6π/4 7π/4 2π

Angular position ϕ (rad)

−20

−15

−10

−5

0

5

10

15

20

Torque Tz (Nm)

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Fig. 7: Cogging torque of the three phases and the entiremachine, coupled phases

0 π/4 2π/4 3π/4 π 5π/4 6π/4 7π/4 2π

Angular position ϕ (rad)

0

50

100

150

200

250

300

350

400

450

500

Torque Tz (Nm)

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Fig. 8: Load torque of the three phases and the entire machinewith rated stator current of I = 120A, decoupled phases

0 π/4 2π/4 3π/4 π 5π/4 6π/4 7π/4 2π

Angular position ϕ (rad)

0

50

100

150

200

250

300

350

400

450

500

Torque Tz (Nm)

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Fig. 9: Load torque of the three phases and the entire machinewith rated stator current of I = 120A, coupled phases

Fig. 10 and Fig. 11 show the load torque of the en-tire machine with various quadrature axis stator cur-rents of I = 30 A . . .150 A with decoupled and coupledphases, respectively. Corresponding, Table II lists av-erage value and distortion factor

ΔTz =Tz,max − Tz,min

2 Tz,av(10)

as well as measurement data from an initial prototype.

The three phases contribute to the electromagnetictorque independently with only little interaction. Dueto the appropriate electrical shift of the three phasecurrents, the resulting shaft torque is comparativelyas smooth as for conventional three phase inductionmachines. With coupled against decoupled phases,the cogging torque of the three phases is slightlylarger while with the entire machine both coggingtorque and average value of the load torque are in-significantly smaller. Nevertheless, the electromag-

netic torque shows higher harmonics with in partic-ular a significant 6th harmonic component which gainin significance with the mechanical construction.

0 π/4 2π/4 3π/4 π 5π/4 6π/4 7π/4 2π

Angular position ϕ (rad)

0

50

100

150

200

250

300

350

400

450

500

Torque Tz (Nm)

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Fig. 10: Load torque of the entire machine with statorcurrents I = 30 A . . . 150A, decoupled phases

0 π/4 2π/4 3π/4 π 5π/4 6π/4 7π/4 2π

Angular position ϕ (rad)

0

50

100

150

200

250

300

350

400

450

500

Torque Tz (Nm)

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Fig. 11: Load torque of the entire machine with statorcurrents I = 30 A . . . 150A, coupled phases

C. Field Results

Fig. 12 and Fig. 13 show the distribution of theradial magnetic flux density within the air-gap of onephase for the aligned and unaligned position of rotorand stator poles for the no-load condition.

With the aligned position, the magnetic flux crossesfrom stator to rotor along adjacent pole regions of sta-tor and rotor causing a concentration of the radialmagnetic flux density within the air-gap along thesepoles faces. On the other hand with the unaligned po-sition, the magnetic flux crosses from stator to rotoronly via small overlapping regions of the pole faces ofstator and rotor.

TABLE II

Load Torque, Average Value, Distortion Factor and

Measurement Data

Decoupled phases Coupled phases

Current Average Distortion Average DistortionMeasure-load value factor value factor ment data

I (A) Tz,av (Nm) ΔTz (1) Tz,av (Nm) ΔTz (1) Tz,m (Nm)

30 109 0.0550 107 0.0555 104

60 209 0.0520 206 0.0535 196

90 295 0.0530 291 0.0540 269

120 366 0.0550 361 0.0565 325

150 425 0.0585 419 0.0600 367

Rotor position: AlignedNonlinear Magnetostatic AnalysisMagneticFluxDensity RComponentNo-load Condition

-1.50000 --1.20000

-1.20000 --0.90000

-0.90000 --0.60000

-0.60000 --0.30000

-0.30000 - 0.00000

0.00000 - 0.30000

0.30000 - 0.60000

0.60000 - 0.90000

0.90000 - 1.20000

1.20000 - 1.50000

Fig. 12: Distribution of the radial magnetic flux densitywithin the air-gap of one phase, aligned positionof rotor and stator poles, no-load condition

Rotor position: UnalignedNonlinear Magnetostatic AnalysisMagneticFluxDensity RComponentNo-load Condition

-1.50000 --1.20000

-1.20000 --0.90000

-0.90000 --0.60000

-0.60000 --0.30000

-0.30000 - 0.00000

0.00000 - 0.30000

0.30000 - 0.60000

0.60000 - 0.90000

0.90000 - 1.20000

1.20000 - 1.50000

Fig. 13: Distribution of the radial magnetic flux densitywithin the air-gap of one phase, unaligned posi-tion of rotor and stator poles, no-load condition

IV. Conclusion

The transverse flux machine with a permanent mag-net excited rotor is well suited in particular for wheelhub drive applications of electric driven vehicles. Thepresented novel concept of such a machine with an ex-ternal rotor consists of a three phase arrangement with

an appropriate mechanical angular shift of the statorparts whereas the rotor parts are arranged in line. Allphases contribute to the electromagnetic torque inde-pendently with only less interaction. Due to an ap-propriate electrical shift of the three phase currents inthe stator ring windings, the resulting shaft torque iscomparatively as smooth as with conventional induc-tion machines.

The 3D finite element analyses of the single sidedtransverse flux machine use only one single finite el-ement model with completely independent rotor andstator parts for all angular rotor positions. On thesliding surface between both parts, they are modelledwith an equidistant finite element discretization withrespect to the movement direction of the rotor. Thevarious angular rotor positions are analyzed withoutany remeshing of air-gap regions by using a domain de-composition algorithm in the nonlinear calculations.

The presented results are focussed on the most im-portant parameters such as no-load voltages, short-circuit currents and electromagnetic torque of thisnovel prototype design. First measurement results ob-tained from an initial design show a good agreementwith the numerical results discussed herein.

References

[1] Lange A., Canders W.R., Laube F., Mosebach H.: ”Com-parison of Different Drive Systems for a 75kW Electri-cal Vehicle Drive”. Proceedings of the International Con-ference on Electrical Machines, ICEM, Espoo (Finland),2000.

[2] Hackmann W.: Systemvergleich unterschiedlicher Rad-nabenantriebe fur den Schienenverkehr: Asynchronma-schine, permanenterregte Synchronmaschine, Transversal-flussmaschine. Doctoral Thesis (in German), TU Darm-stadt, 2003.

[3] Blissenbach R.: Entwicklung von permanenterregtenTransversalflussmaschinen hoher Drehmomentdichte furTraktionsantriebe. Doctoral Thesis (in German), RWTHAachen, 2002.

[4] Anpalahan P.: Design of Transverse Flux Machines us-ing Analytical Calculations and Finite Element Analysis.Licentiate of Technology Thesis, Royal Institute of Tech-nology, Stockholm, 2001.

[5] Njeh A., Masmoudi A., Elantably A.: ”3D Finite Ele-ment Analysis Based Investigation of the Cogging Torqueof a Claw Pole Transverse Flux Permanent Magnet Ma-chine”. Proceedings of the IEEE International Conferenceon Electric Machines and Drives, IEMDC, Madison (WI,USA), 2003.

[6] Schmidt E.: ”3D Finite Element Analysis of the CoggingTorque of a Transverse Flux Machine”. IEEE Transactionson Magnetics, Vol. 41, No. 5, May 2005.

[7] Biro O., Preis K., Richter K.R.: ”On the Use of theMagnetic Vector Potential in the Nodal and Edge FiniteElement Analysis of 3D Magnetostatic Problems”. IEEETransactions on Magnetics, Vol. 32, No. 3, May 1996.

[8] Hameyer K., Belmans R.: Numerical Modelling and De-sign of Electrical Machines and Devices. WIT Press,Southampton, 1999.

[9] Jin J.: The Finite Element Method in Electromagnetics.John Wiley & Sons, New York (USA), 2002.

[10] Schmidt E.: ”Application of a Domain Decomposition Al-gorithm in the 3D Finite Element Analysis of a Trans-verse Flux Machine”. Proceedings of the IEEE Cana-dian Conference on Electrical and Computer Engineering(CCECE), Winnipeg (MB, Canada), 2002.

[11] De Gersem H., Gyselinck J., Dular P., Hameyer K., Wei-land T.: ”Comparison of Sliding Surface and Moving BandTechniques in Frequency-Domain Finite Element Modelsof Rotating Machines”. Proceedings of the 6th Interna-tional Workshop on Electric and Magnetic Fields, EMF,Aachen (Germany), 2003.

[12] De Gersem H., Weiland T.: ”Harmonic Weighting Func-tions at the Sliding Surface Interface of a Finite El-ement Machine Model Incorporating Angular Displace-ment”. IEEE Transactions on Magnetics, Vol. 40, No. 2,March 2004.