IEEE TRANSACTIONS ON MAGNETICS, VOL. 34, NO. 6, NOVEMBER 1998 3897

Three-Dimensional Analytic Model ofPermanent Magnet Axial Flux Machine

Yuriy N. Zhilichev

Abstract—A three-dimensional (3-D) magnetic field problem inan axial flux machine with a toroidal winding is considered. Theprecise solutions generated by the integral transform and Fouriermethods in elementary subregions are joined using the iterativeSchwartz algorithm. The comparison between two-dimensionaland 3-D models is performed and a correction factor taking intoaccount the radial variation of the magnetic flux is given.

Index Terms—Axial flux, electrical machine, magnetic field,permanent magnet.

I. INTRODUCTION

A XIAL-flux, permanent-magnet machines with a toroidalslotless winding (Fig. 1) are used as portable generators

providing low voltage dc output [1], [2]. They are now beingconsidered for drives of electric and hybrid road vehicles andfor generators in wind mills.

A set of cylindrical two-dimensional (2-D) models areused in [3] as the basis for a quasi-three-dimensional (3-D)representation of the magnetic field. Although the quasi–3-D model gives rapid and accurate predications of the mainterminal parameters of the machine, it needs some additionalparameters (obtained experimentally) such as effective outerand inner radii of the airgap. This model cannot predicatethe magnetic field reduction near the ends of the air gap dueto the edge effects. A 3-D analysis is needed for a rigoroustreatment of this device.

This paper presents a 3-D solution which combines the inte-gral Hankel transform method [4] and discrete Fourier methodin elementary subregions with the Schwartz iterative routinefor joining the sections of the solution. The model presentedhere is compared with the 2-D and one-dimensional (1-D)models reported earlier. From the comparison, a correctionfactor for the 2-D solution is obtained. It can be used toincrease the accuracy of formulas for the motor characteristicsderived from the 2-D analysis of the magnetic field [3].

II. PRINCIPLES OF THEMAGNETIC FIELD CALCULATIONS

Fig. 2 shows the simulated model of the axial flux machine.We assume that the magnetic field in the airgap is generatedonly by the permanent magnet. The magnetic field of thestator and armature reaction in slotless machines is normallyweaker by two orders of the magnitude than that of the fieldenergized by the permanent magnet. Due to the symmetry with

Manuscript received July 9, 1997; revised June 25, 1998.The author is with the Institute of Physical Energetics, Latvia Academy of

Sciences, Riga LV-1006, Latvia (e-mail: fei@edzi.lza.lv).Publisher Item Identifier S 0018-9464(98)07209-4.

Fig. 1. Layout of axial flux machine.

Fig. 2. Three-dimensional model for simulation.

respect to the plane , we can consider only a half of theproblem region, i.e., . A scalar magnetic potential of thefield energized by the permanent magnet satisfies the Poissonequation

inside the permanent magnetoutside the permanent magnet

(1)

where is the magnetization vector. For anisotropic rare-earth and ferrite magnets, which are normally used in themotors, the components of the magnetization vector dependon the magnetic field strength as follows:

along the easy (or preferred) magnetization axis

along the transverse magnetization axes, where arethe reversible susceptibilities along the easy and transversemagnetization axes, respectively.

0018–9464/98$10.00 1998 IEEE

3898 IEEE TRANSACTIONS ON MAGNETICS, VOL. 34, NO. 6, NOVEMBER 1998

Substituting these relationships in (1) we obtain those in [4]

(2)

where for

are the inner and outer radii of the magnet,respectively,

is the pulse unit function:is the delta function

.

The function gives the distribution of the permanentmagnet magnetization over the azimuthal angle. This func-tion depends on the type of the magnetization fixture andmay be different for square-, trapezoidal-, or cosine-wavemagnetization [4].

For all types of magnetization the function can beexpanded in Fourier series as

where is the number of the pole pairs. Assuming the scalarpotential at infinity is zero and if the back iron and stator coreare unsaturated, the boundary conditions are

The outer radius of the rotor iron disk (back iron) is equal tothe outer radius of the permanent magnet . Because of themotor symmetry with respect to the plane , we can write

According to the field symmetry and periodicity with respectto the polar angle, we have

The problem formulated can be solved by a combination ofthe integral transformation method and Schwartz method foroverlapping subregions. We assume that the permanent magnetand stator core have the same inner and outer radii

Fig. 3. Elementary sections of the field region.

We subdivide the region occupied by the magnetic fieldinto three rectangular overlapping subregions with rectangularcross sections (Fig. 3): subregion I (cross section ABKO)

subregion II (cross section AFJG)

subregion III (cross section DEML)

We have restricted the height of the region III with

where is the height of the back iron. This is done in orderto obtain the solution in the regions I and III in the similarforms. We assume that the boundary conditions on the top ofthe region III are

As region II overlaps with regions I and III, some parts of itsboundary surfaces are inside these regions and vice versa. Theconditions at these parts (indicated by broken lines in Fig. 3)of the interior boundaries (IB) are unknown. We denote thepotentials at these boundaries as

III. SOLUTION OF THE MAGNETIC FIELD PROBLEM

Due to the field periodicity over the polar angle the potentialin each region can be written in the form of Fourier series

I, II, III (3)

The functions , , , , and can be alsoexpanded into similar series

IB CF, GH, IJ, HB, IC

ZHILICHEV: PERMANENT MAGNET AXIAL FLUX MACHINE 3899

The solution of the Poisson equation (2) in the region II canbe obtained using the discrete Fourier transformation over thepolar angle and the Hankel transformation over the radialcoordinate. The procedure of the solution is exactly the sameas that used in [4]. The result is

(4)

where

IB GH, IJ, CF.

The solutions of Laplace’s equation in the regions I and IIIcan be presented in the form of Fourier series

(5)

where are the modified Bessel functions of the first order

The similar solution of Laplace’s equation can be written forthe region III

(6)

where are the modified Bessel functions of the secondorder

The general solutions in the regions I, II, and III contain theunknown distribution of the potential on the interior bound-aries CF, GH, IJ, HB, IC. To determine these distributions theiterative procedure called Schwartz’s method for overlappingregions is applied. First we assume that

and we find the solution in the region II using (4) with

This solution is exactly the same as in [4]. By applicationof this solution, the functions and coefficients

can be easily calculated. For example

A similar expression can be also written for the first approx-imation of the coefficients . Substituting these functionsinto (4) we obtain the potential in the regions I and III as wellas distribution of the potential over the surfaces GH, IJ, CF.This is the first iteration of the Schwartz method. Two or threeiterations are normally sufficient to achieve the convergenceof the solution in all three regions.

IV. COMPARISON BETWEEN 1-D, 2-D, AND 3-D MODELS

The performance of the motor [i.e., torque and back elec-tromotive force (EMF)] depend only on the axial componentof the magnetic field density in the mechanical airgap

. According to the quasi-3-D model thiscomponent is calculated as [3]

cosh

(7)

Although (7) contains the radial coordinate, it does not depictany weakening of the magnetic flux near edges ( )and, therefore, it is 2-D. From (4) we can now derive thenormal component of the real 3-D magnetic field as shownat the bottom of the next page in (8). It is clear that thedistribution of along the radial coordinate depends on thedimensions ( ) as well as on the number of poles2 . There are some dimensionless motor parameters whichmainly affect the distribution . The total magnetic gapto mean radius ratio and the polepitch to radius ratio are the most significant parametersthat determine the magnetic field topology of the machine.

3900 IEEE TRANSACTIONS ON MAGNETICS, VOL. 34, NO. 6, NOVEMBER 1998

Fig. 4. Axial flux density versus radial coordinate: broken line with “x” =2-D model (7), solid line = 3-D model (8).

When , the magnetic flux density inside the airgap( ) tends to the 1-D distribution

(9)

This is an asymptotic expression which can be obtained from(7) as well as from (8).

From (9) the maximum flux density in the airgap is foundto be

(10)

This flux density (or even more simple expressions) are usedin one dimensional models [1], [2]. To compare different for-mulas for magnetic field calculation, consider the experimentalmodel of the “TORUS” generator, which has been used in [3]to verify the quasi-3-D calculation model. A ceramic ferritering with 110 mm ID 220 mm OD, 19 mm thick, remanence

, is used in a permanent magnet field system.The other parameters of the generator are mm,

mm, mm, and mm. The fluxdensity in the airgap calculated in accordance with (8), (7)is given in Figs. 4 and 5 for . As we haveexpected, the 2-D model gives precise values of the magneticflux density only around the mean radius of the generator butnot near the edges where the flux density drops down rapidly.This is a typical plot of in the axial flux machine.The axial flux density versus radial coordinate is given inFig. 6 for some combinations of geometrical parameters when

.It should be noted that (8) of the real 3-D model is

much more complex than (7) of the quasi-3-D model. Thecomputational time (PC 486 DX2) of the axial component

( const const) is less than asecond when (7) is used. It takes about five minutes to calculate

Fig. 5. Axial flux density versus angle: “x” = 2-D model (7), solid line =3-D model (8).

Fig. 6. Normalized flux density versus radial coordinate forhm=R1 = 0:15; g=R1 = 0:1; p = 2 and for different magnetic gapto mean radius ratios� = 1:4 (1), 1.6(2), 1.8 (3), 2.0(4), 2.2(5).

the same distribution using (8) and the iteration proceduregiven in Section III.

To evaluate the magnetic flux weakening due to the edgeeffects in the disk-type machines, we can use the followingratio:

(11)

where is the axial component of the real 3-D magneticfield (8).

The integration parameter is normally in the range from0.85–0.95 (Fig. 7). The parameter can be used as thecorrection factor for formulas of the output characteristicsderived in [3] on the base of the 2-D magnetic field analysis.For example, a total magnetic flux generated by the permanentmagnet in the airgap (over a pole pitch) is defined as

(12)

(8)

ZHILICHEV: PERMANENT MAGNET AXIAL FLUX MACHINE 3901

Fig. 7. Correction factor forp = 2 wheng=R1 = 0:1 andhm=R1 = 0:15(1); g=R1 = 0:05 and hm=R1 = 0:15 (2); g=R1 = 0:05 andhm=R1 = 0:10 (3).

Replacing the integral over by andcomputing from (7) with a fixed mean radius

, we have

Similar formulas containing a correction factor can beeasily derived also for the back EMF, magnetic flux in thestator core, and for other output parameters.

V. CONCLUSIONS

The magnetic field density in the airgap of the axial fluxmachine depends on the radial coordinate. Due to the edgeeffect the total useful magnetic flux typically is less by 5–15%.The simple correction factor can be used to take into accountthe abatement of the axial flux density near the inner andouter radii of the machine. This factor improves the accuracyof formulas for the output characteristics derived earlier onthe base of the 2-D approach.

REFERENCES

[1] E. Spooner and B. J. Chalmers, “‘TORUS’: A slotless, toroidal-stator,permanent-magnet generator,”Proc. IEE B,vol. 139, no. 6, pp. 497–507,1992.

[2] C. C. Jensen, F. Profumo, and T. A. Lipo, “A low loss permanent magnetbrushless DC motor utilizing tape wound amorphous iron,”IEEE Trans.Ind. Applicat.,vol. 28, no. 3, pp. 646–651, 1992.

[3] Y. Zhilichev and E. Spooner, “Quasi-3-D analysis of permanent-magnetaxial-flux machines,” submitted for publication.

[4] Y. N. Zhilichev, “Calculation of 3-D magnetic field of disk-typemicromotors by integral transformation method,”IEEE Trans Magn.,vol. 32, pp. 248–253, Jan. 1996.

Yuriy N. Zhilichev was born in Riga, Latvia, in 1954. He received theDiploma in physics from the Latvian University, Riga, in 1976, the Cand.Sc.degree in electrical engineering from the Institute of Electrodynamics of theUkrainian Academy of Sciences, Kiev, in 1987, and the Dr.-Eng. degree inelectrical engineering from the Institute of Physical Energetics of the LatvianAcademy of Sciences, Riga, in 1992.

Since 1976, he has been working as a researcher at the Institute of PhysicalEnergetics of the Latvian Academy of Sciences. His currrent title is LeadResearcher. His major scientific interests are computational electromagnetics,electrical machines, permanent magnets, and applied superconductivity.