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Faculty of Engineering Papers

The University of Auckland Year 

Magnetic Field Analysis of an Ironless Brushless DC Machine

Udaya Mandawala

^∗

John T. Boys

^†

∗University of Auckland,

†University of Auckland, New Zealand, [email protected] This paper is posted at ResearchSpace@Auckland.

http://researchspace.auckland.ac.nz/engpapers/5

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Magnetic Field Analysis of an Ironless Brushless DC Machine

Udaya K. Madawala, Member IEEE,and John T. Boys

Department of Electrical and Computer Engineering, The University of Auckland, Auckland, New Zealand

This paper presents an analytical solution for the magnetic field inside an unconventional ironless brushless (ILBL) dc motor design.

We discuss the unique characteristics of the design, which adopts an “inside-out” construction with an internal ironless stator. With the intention of obtaining an analytical solution for the magnetic field, we propose a model based on a magnetic pole concept for rep- resenting the magnetic circuit. The magnetic field inside the motor, which has no iron for guiding magnetic flux paths, is obtained by solving Laplace’s equation for magnetic scalar potential. We present both analytical and numerical solutions for the magnetic field and compare them with measured results of a 20-pole prototype IL design to show the validity of the model. The IL design is simple, easy to manufacture and, as indicated by the results, has an optimum number of magnets, for which its performance becomes maximum.

Index Terms—Brushless machines, Laplace equation, magnetic analysis, permanent magnets.

I. INTRODUCTION

C

ONVENTIONAL dc machines, excited by field windings, are largely being replaced by low-cost permanent-magnet dc (PMDC) machines [1]. This trend is mainly due to the availability of low cost high-energy permanent magnets as they can be used for replacing the bulky wound field excitation and its associated losses in conventional dc machines. Today, the continuation of this trend coupled with the advances in semiconductor technology has transformed conventional dc machines into efficient, compact, and high power/weight ratio brushless dc (BLDC) machines with electronic commutators, thereby eliminating the necessity for a troublesome mechanical commutator. Moreover, BLDC machines are often equipped with relatively inexpensive yet sophisticated drive systems to cater for a wide range of load conditions. As a consequence, they have today become the obvious and most popular choice for a variety of industrial applications. Nevertheless, even with these very attractive characteristics, many attempts are still being made to further improve or optimize both the design and operation of BLDC machines as opportunities are always created for improvement by the continual advances in the technological world. The improvements to the motor design, in particular the magnetic circuit, are generally carried out by making use of commercially available finite-element software packages. This is mainly because magnetic field analysis is complex and nonlinear in nature, and is best solved by numerical techniques, as analytical solutions are often difficult to arrive at and may still require numerical integration. In addition, there are many software packages currently available in the market, which are relatively inexpensive, accurate, easy to use, and suitable for such analysis. Naturally, this has led most authors to focus largely on numerical techniques when obtaining solutions for magnetic field problems [2]–[4]. Nevertheless, attempts are still being made to obtain analytical solutions for magnetic field problems associated with electric machines in order to facilitate efficient and optimum designs through the change of motor parameters [5], [6].

Digital Object Identifier 10.1109/TMAG.2005.852952

Fig. 1. Rotor and stator of an ILBL motor.

This paper describes a nontraditional design of a BLDC machine that uses an ironless stator structure. Although the magnetic field inside this ironless (IL) design can be analyzed by a finite-element software package, the authors have instead fo- cused mainly on obtaining an analytical solution, which is more difficult to achieve for this design without a clearly defined magnetic circuit for guiding the magnetic flux, in comparison to conventional designs. To facilitate an analytical solution, a model based on magnetic pole concept with magnetic scalar potential is proposed for representing the magnetic circuit. The magnetic field inside the design is obtained by solving Laplace’s equation for the magnetic scalar potential using conformal transformation and Schwarz–Christoffel mapping, as described in [7].

The validity of the proposed model and the analytical solution is verified by experimental results and a numerical solution.

II. ARCHITECTURE OFIRONLESS-BRUSHLESS(ILBL) MACHINE

As the name implies, the unique feature of this proposed ILBL machine is its ironless stator. The ILBL motor, shown in Fig. 1, adopts an internal stator structure that can be viewed as an inside-out construction of a conventional BLDC motor design.

The external rotor of ILBL dc machines is a thin cup-shaped cylindrical steel drum, inside which a series of low cost Arnox

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H-8 ceramic permanent magnets (PMs) are firmly glued in place to provide the field excitation. It is mounted on bearings located inside the nonmagnetic stator, made out of plastic. PMs of flat pole surfaces, instead of preferable arc-shaped, are used to lower the manufacturing cost and they are assembled as close as prac- tically possible to each other to make the rotor interior circular and flux linkages maximum. As a result of the inside-out design, ILBL motors invariably have a high number of pole pairs and high rotor inertia. Both these characteristics are advantageous for lowering speed fluctuations as torque pulsations occur at a high frequency and they are further damped out by the high rotor inertia. However, with a high number of poles, the commutator invariably has high switching losses and demands precision po- sition sensors to meet the required high degree of switching accuracy for PM operation.

The plastic stator of the ILBL design contains a single-layer, full-pitched three-phase concentrated winding. It is skein- wound in a thin slotless configuration and glued on to the stator core to keep the manufacturing cost low. This winding arrangement results in back-electromotive force (back-EMF) voltages of 10%–15% of third harmonic distortion in typical ILBL motors, which are ideal for low-cost BLDC operation in a unipolar or bipolar switching arrangement to keep the torque ripple low. With the adoption of a plastic core with no iron, the ILBL design inevitably has a low operating flux density.

This is further compounded by the use of Arnox H8 magnets as they appear as an additional air gap in the magnetic circuit of the design. Typical air gap flux density of an ILBL motor is in the range of 0.14–0.18 T and, therefore, the motor does not appear to be capable of producing a comparable torque.

The drawback of this low operating flux density is, however, largely offset by the “inside-out” construction of the design as it makes the split ratio of ILBL machines as high as 0.85. This is because the specific output of a machine is largely governed by its rotor diameter since an increase in rotor diameter causes a corresponding increase in both the magnetic flux and current in the machine whereas an increase in axial length results in only an increase in the magnetic flux and not in the current. As a consequence, the air-gap radius of ILBL machines is larger than that of a similar sized conventional design and, hence, mostly compensates for the specific power reduced by the lower air-gap flux density. In comparison to similar sized motors, ILBL motors with no iron are considerably lighter and quieter in operation as there are no ferrostriction forces. The design is low in cost, robust, and suitable for applications such as blowers, small pumps, conveyors, etc., where its characteristics are advantageous.

III. MODELINGPMSUSINGPOLEMODEL

A. Pole Model

From Maxwell’s equations under time invariant conditions, the magnetic flux density and the magnetic field intensity in a medium can be given by

(1) and

(2) where is the conduction current density in the medium. The relationship between and is given by

(3) where is the permeability of free space and magnetization is due to the induced and residual magnetizations of the medium.

By substituting for in (1)

(4) If is zero, then is a curl free vector and, hence, may be derived from the gradient of a scalar given by

(5) where is the magnetic scalar potential of . Substituting (5) in (4) gives the Poisson’s equation for

(6) where is the volume charge density of the medium. For mediums where divergence of is zero, Poisson’s equation reduces to Laplace’s equation

(7) According to (6), is zero for a uniformly magnetized magnet since all spatial derivatives inside the magnet are essentially zero. Therefore, the magnetic charges are confined only to the pole surfaces of the magnet and can be represented by a surface charge density using a pole model [8]. This implies that the alternate north and south pole surfaces of the magnets in ILBL machines can be represented by equivalent potential surfaces in direct analogy to electrostatics. If the magnets exhibit linear demagnetization characteristics with rigid magnetization [9], then the potential difference between the surfaces can be determined by modeling the magnet as an intrinsic magnetomotive force given by

(8) in series with internal and external magnetic re- luctances, respectively.

B. A Model for the Magnetic Circuit

Because of the axisymmetric nature of the design, the magnetic field analysis is restricted only to a sector containing a single magnet with the following assumptions.

• The effect of the armature reaction and the presence of stator winding on the magnetic field distribution is negligibly small.

• Arnox-H8 magnets are uniformly and rigidly magnetized perpendicular to the pole surface.

• The steel rim is infinitely permeable.

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Fig. 2. A model representing a sector containing a magnet.

• The end effects are negligible and, hence, a two-dimensional solution is adequate.

With the absence of iron, the magnetic flux lines in ILBL machines are in air. As a consequence, the magnets have an operating range on the – curve instead of a static operating point, causing on the surface of the magnet to vary from a minimum, where the magnets are virtually in contact, to a maximum at the center of the magnet where the external magnetic reluctance is maximum. The potential distribution on the surfaces of the magnet is, therefore, not deterministic for this particular design.

However, on all interpolar axes is essentially zero because of the alternate north and south pole configuration of the design.

As a consequence, along the innermost steel rotor surface of infinite permeability can also be considered as zero. With these boundary conditions, a sector containing a magnet can be represented by a magnetic circuit model in Fig. 2.

If there are no conduction currents, inside the machine may be derived from using (5). The fact that is constant inside the magnet and zero elsewhere in the sector implies that inside the sector should satisfy Laplace’s equation. Thus, the magnetic field distribution inside an ILBL machine may be determined by formulating the sector as a boundary value problem and solving Laplace’s equation for [10].

IV. ANALYTICALSOLUTION FOR THE MAGNETIC

FIELDDISTRIBUTION

As described, the magnetic potential distribution over pole surfaces is indeterminate in this design. Therefore, in order to obtain an analytical solution with an acceptable accuracy, a con- straint of the model was slightly changed as outlined below.

With the absence of iron, the radial air-gap magnetic field around the corners of the magnet is considerably small and varying, whereas it is significant and reasonably constant over the rest of the pole surface. This implies that the change in surface potential is significant only around the corners of the magnet, which is only a small part of the pole surface. In this respect, it is reasonable to specify a constant surface potential distribution in the model by (8), as the error introduced at the corners is insignificant and has only a little impact on the flux linkage, as justified by experimental results.

With the above change to the model, the problem now reduces to solving Laplace’s equation subject to only Dirichlet boundary condition [11] and not to both Newmann and Dirichlet boundary conditions. This can be achieved by formulating the problem in the complex domain and using the method of conformal transformation [12].

Fig. 3. A sector with a flat magnet in (a)Z-plane and (b)t-plane.

Noting that any function that satisfies Laplace’s equation is a harmonic function, is represented by a real part of a complex function defined by

(9) Since is an analytic function that satisfies the Cauchy–Rie- mann equation, the magnetic flux paths in the machine can then easily be obtained by plotting the curves corresponding to constant values.

Fig. 3(a) shows a single sector, containing a flat surface magnet in the complex -plane. The surface potential of the magnet is represented by a constant . If and are the polar coordinates of a point in -plane, then a point in any of the sectors may be represented by

for (10)

where is an integer representing each sector with a magnet, is the number of magnets in the machine, and is the sector angle defined by

(11) Since all sectors are identical, a solution can be obtained by considering a single sector in Fig. 3(a) with . The dif- ficulty in obtaining an analytical solution with the two singularities at points A and B of this problem, can be overcome by mapping the isosceles triangle onto a half-space in the -plane using the Schwarz–Christoffel theorem with the transformation

(12) where is a constant. As shown in Fig. 3(b), the vertices A and B of the triangle are mapped onto 1 and 0 on the real axis of the -plane, respectively, while the remaining vertex C is chosen to be a point at infinity in the -plane. Substituting for in (12)

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Solving for constant from set boundary conditions

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Since these two integrals are functions [13], (14) can be represented by

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where and represent

the incomplete and complete functions, which can be evalu- ated by a series expansion. From Abramowitz and Stegun [14], an incomplete function can be described by a hypergeometric function as

(16) which may be expanded to an absolutely convergent function

(17) Therefore, the transformation, which maps the isosceles triangle in the -plane onto a half-space in -plane, is

(18) Thus, the solution for Laplace’s equation for complex potential

in -plane is given by

(19) By substituting for and in equation

(20) As evident from (19), even an analytical solution for magnets with flat pole surface is achievable, it is not still possible to obtain an explicit expression for in terms of because of the complexity of the solution. A workable analytical solution can, however, be obtained if the magnets of flat pole surfaces used in the prototype design, are considered to be as curved pole surface magnets in the model, resembling to a motor with

a ring magnet. This is justifiable because the rotor interior of 70.75 mm radius with flat surface magnets, is very much similar (with only a 0.85 mm deviation at the center) to that with curved pole surfaces.

Fig. 4(a) shows a single sector of rotor radius , containing a magnet with a curved pole surface in the complex -plane.

This problem has two singularities at points B and D where changes from 0 to on the interpolar axis when the magnets are virtually in contact. A solution for this problem can be obtained by mapping the two singularities in the -plane onto a half-space in an intermediate -plane, as in Fig. 3(b) with a single singularity, by the transformation from Kober [14]

(21) Substituting for and solving for gives

(22) Therefore, the solution for Laplace’s equation for in -plane [15] is given by

(23) Substituting for yields

(24) or

(25) This is the solution for Laplace’s equation for complex potential . By substituting for from (9) and considering the real part, at any point inside the sector can be found by

(26) Since is derived from the gradient of , the corresponding radial and tangential , components of are given by

(27) and

(28) Equations (27) and (28) can thus directly be used for determining the magnetic field at any point inside a sector.

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Fig. 4. A sector with a curved magnet in (a)Z-plane and (b)t-plane.

Fig. 5. Computed flux patterns due to magnets. (a) Analytical. (b) Numerical.

V. RESULTS ANDDISCUSSION

All theoretical calculations and experimental measurements discussed in this section have been based on a 20-pole prototype ILBL machine and its details are given in the Appendix. Both the validity of the proposed magnetic model and the accuracy of the analytical solution were verified by solving Laplace’s equation numerically by computer simulation with no modification to the model, and comparing the results with those obtained ex- perimentally on the prototype.

Fig. 5 is a comparison of the flux patterns computed from the analytical and numerical solutions. The flux patterns are similar but all flux paths from the analytical solution emanate at right angles to the pole surface. This was expected from the analytical solution because of the imposed constant surface potential (calculated from (8) with an estimated average reluctance) and curved pole surface. In this situation, the crowded flux paths, where the magnets are virtually in contact, indicate a strong magnetic field with a significantly large radial component, es- pecially in close proximity to the pole surface. However, at the center of the magnet, the flux lines are sparse and, hence, the magnetic field is weak. But they tend to follow the natural circular path, resulting in a large radial component as evident from the results. In contrast, the numerically computed flux paths of a real machine neither emanate from nor enter into the magnet at right angles as a consequence of estimating the actual surface potential iteratively using Dirlichlet boundary condition. They tend to bend more toward the adjacent magnet at the corners and are more radial in the middle of the magnet. This suggests that in both cases the flux paths with a larger radial penetration around the center have a larger contribution to the flux linkage in a practical winding than those around the corners, as indicated by the winding location with dotted lines. An important

Fig. 6. Computed radial flux density distribution. (a) Analytical. (b) Numerical.

conclusion that can be arrived at from these patterns is that the erroneous flux paths at the corners of the magnets of the analytical solution have only a little contribution to the flux linkage and, hence, the analytical solution can be considered to be of acceptable accuracy when determining the performance.

The radial component of the air-gap magnetic field is a very good indication of the shape of the back-EMF voltages gener- ated in the motor terminals. Therefore, as shown in Fig. 6, it was computed analytically at different radial distances and com- pared with that obtained numerically to investigate the shape of back-EMF voltages and also to justify the modifications made to the model in obtaining the analytical solution.

As evident from Fig. 6, the magnetic flux paths are more radial and stronger closer to the surface, than further away from the magnet where flux paths are weak. The results are in good agreement except those around the corners of the magnets at large radii, where the analytically computed radial component is noticeably large. This in fact was identified when formulating the analytical solution, and is also obvious from the flux patterns of Fig. 5(a) as they emanate at right angles to the pole surface. However, the air-gap field distributions are very much similar in both cases, a little further away from the magnet surface where a practical winding with typical dimensions would be located. This can be attributed to rapidly decaying radial component of the magnetic field around the corners of the magnets, where magnetic flux paths tend to bend naturally more toward the adjacent magnets. These results, therefore, suggest that the

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Fig. 7. Measured versus computed radial flux density distribution.

Fig. 8. Comparison of indirect versus direct flux density.

proposed approximate analytical solution can be used for analyzing the magnetic field inside which a practical winding is located, with a reasonable degree of accuracy.

The validity of the developed magnetic model was verified directly by measuring the flux density inside the machine by a flux meter and indirectly by locating search coils inside the machine.

The comparison of the measured radial flux density distribution of a typical ILBLDC machine by search coils and that computed from the numerical solution, is shown in Fig. 7. As apparent, the measured results are in good agreement with those computed numerically at different radial distances and the small discrep- ancy can be attributed to the assumed infinitely permeable back iron and the variation of during magnetization.

The comparison of both indirectly and directly measured flux density and the numerically computed flux density is shown in Fig. 8. Since it is not practicable to use the flux meter to mea- sure the flux density at different radial distances over the pole surface, flux densities along the pole axis instead were measured for comparison. Results show that directly measured flux densities, which can be considered as the closest to the actual values, are in very good agreement with the values computed numerically, verifying the validity of the proposed magnetic model.

Flux linkages in a typical ILBL machine largely depend on several design parameters—particularly, the number of magnets

Fig. 9. Flux linkage for different magnet thicknesses.

Fig. 10. Flux linkage for different rotor radii.

used, the thickness of the winding and the magnets , and the dimensions of the machine. Thus, for a given winding thickness and machine dimensions, there should be an optimum number of magnets for which the flux linkages become maximum. This effect is illustrated in Fig. 9 by numerically calculating the flux linkages for different winding thicknesses (represented as a per- centage of rotor radius ) for a rotor radius of 70.75 mm. As evident, the thinner the magnet the higher the optimum number of magnets that gives the maximum flux linkages as weaker magnetic flux produced by thin magnets cannot penetrate deep into the stator core.

The effect of the rotor radius on the optimum pole number is shown in Fig. 10 for magnets of 9.5 mm thickness. In this situation, the optimum pole number increases with the rotor radius since larger rotors results in larger effective air gaps and winding thicknesses.

VI. CONCLUSION

An unconventional ironless BLDC machine design has been described. The magnetic field inside the design that has no iron for guiding flux paths was obtained by using a model based on the magnetic pole concept. To facilitate an analytical solution, the magnetic scalar potential instead the magnetic vector potential, has been used. An approximate analytical solution has been presented and the results are in good agreement except those at the corners of the magnets which have only a little contribution to the flux linkages, as verified by both a numerical solution and

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experimental results. Results indicate that for a given set of dimensions the IL design has an optimum number of poles for which its flux linkages are maximum.

APPENDIX

DETAILS OF200W PROTOTYPEILBLDC MACHINE

Rotor (pressed cup)

Outer radius mm

Inner radius mm

Length mm

Stator (plastic)

Radius mm

Length mm

Winding resistance Ohms Winding inductance H

Back EMF constant V rpm

Arnox-H8 Permanent Magnets

Thickness mm

Width mm

Length mm

Remanent flux density T

Coercivity kA m

Recoil permeability Number of magnets

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Manuscript received March 22, 2005; revised May 17, 2005.

Udaya K. Madawala(M’96) was born in Sri Lanka in 1962. He received the B.Sc. degree with honors in electrical engineering from the University of Moratuwa, Sri Lanka, in 1987 and the Ph.D. degree in power electronics from the University of Auckland, Auckland, New Zealand, in 1992.

He was employed by Fisher & Paykel Ltd., New Zealand, in 1992, where he worked as a research and development engineer in the area of motor design and control. In 1997, he joined the department of Electrical and Computer En- gineering at the University of Auckland as a Research Fellow and became a Lecturer in 2002. His research interests are in the fields of BLDC motor design and control, resonant converters, supercapacitor applications, and unbalanced operation of three-phase induction machines.

John T. Boysgraduated from the University of Auckland, Auckland, New Zealand, in 1962.

After receiving the Ph.D. degree, he worked for SPS Technologies (USA) for five years before returning to academia where he is currently Professor of Elec- tronics at The University of Auckland. His fields of interests are motor control and inductive power transfer. He has a particular interest in the transfer of technology from the research laboratory to industry, and is the holder of more than 20 patents.