IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 6, JUNE 2011
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Slotless Permanent-Magnet Machines: General Analytical
Magnetic Field Calculation
Pierre-Daniel Pfister1;2 and Yves Perriard2
Moving Magnet Technologies SA (MMT), 25000 Besançon, France
Laboratory of Integrated Actuator (LAI), Ecole Polytechnique Fédérale de Lausanne (EPFL), 1015 Lausanne, Switzerland
This paper presents a general analytical model for predicting the magnetic field of slotless permanent-magnet machines. The model
takes into account the effect of eddy currents in conductive regions and notably in conductive permanent magnets without neglecting
their remanent field. The modeling of this effect is important for the design of very high speed slotless permanent-magnet machines, as
the power losses are linked with the frequency of the field. The model takes into account any number of layers. It implies that, for one,
the fields can easily be calculated in a design including a permanent magnet and a conductive retaining sleeve. The model is applicable
both to internal rotor and external rotor permanent-magnet machines. The effect of the relative permeability and of the conductivity of
the permanent magnet or of the yoke on the magnetic fields is also taken into account. Any magnetization can be taken into account, in
particular a Halbach type permanent magnet, or a radially magnetized permanent magnet can be considered.
Index Terms—Analytical magnetic fields solutions, permanent-magnet machines, slotless motors.
I. INTRODUCTION
S
LOTLESS permanent-magnet (PM) machines are increasingly used for very high speed (VHS) applications. The
analytical modeling of the magnetic field is important for the
design and optimization of such applications. Also, analytical
models of the fields calculated for slotless structures are widely
used in the design of slotted structures using conformal mapping or Carter coefficients.
Many papers (see Section I-C) have already been written
about the analytical calculation of magnetic fields in several
particular cases. The objective of the present paper is to show
a general model (see Table I), applicable to a very large family
of slotless machines.
A. Considered Structure
The structure considered in this paper is the following. It
is made of -concentric contiguous hollow cylinders. Every
cylinder is characterized by:
• its inner and outer radius;
• its permeability and conductivity;
• its spatial and temporal applied current density harmonics;
• its remanent field spatial harmonics;
• its rotational speed.
At the interface between two contiguous hollow cylinders,
the interface is characterized by its spatial and temporal applied
surface current density harmonics.
Each hollow cylinder is called a “layer.” The model is called
an -layer model. Every layer in which the magnetic field is
calculated, and every boundary is numbered starting with the
innermost one as shown in Fig. 1.
In the presented model any spatial magnetization harmonics
can be taken into consideration. Allowing models of different
kinds of magnetization and in particular the ideal Halbach magnetization as it is defined in [1] and [2] to be created. Radial
magnetization can be approximated by this model.
Fig. 1. Section of the contiguous hollow cylinders: the example of a 3-layer
structure, with infinite permeability in the center and in the exterior. The structure is defined to be general. The model is able to deal with any number of layers.
It is able to deal with PM motors, PM generators, eddy-current brake, and other
kind of electromechanical structures.
B. Which Differential Equation?
As it is shown in Section II, starting from Maxwell’s equations and in the constitutive equations of the materials, a diffusion equation can be derived in its generalized form: it includes
the effect of magnetic remanent fields, eddy currents, and applied currents. In this paper, it is called: a generalized diffusion
equation (GDE).
Depending on the assumptions which govern the physics of
the structure, the governing equation can take different forms
that are named differently (Fig. 2). These different forms can be
deduced from the GDE.
Different authors have used these different forms of the GDE:
Laplace equation [3], Poisson equation [4], the diffusion equation [5], and the generalized form of the diffusion equation [6].
C. Literature Review
Manuscript received June 25, 2010; revised September 15, 2010; accepted
December 06, 2010. Date of publication February 17, 2011; date of current version May 25, 2011. Corresponding author: P.-D. Pfister (e-mail: pierredaniel.
pfister.public@gmail.com).
Digital Object Identifier 10.1109/TMAG.2011.2113396
The resolution of a differential equation in order to obtain
the magnetic field in an electrical machine is not recent. With
the exception of some papers where the 3-D analytic magnetic
field solution is presented [7], [8], most publications involve
0018-9464/$26.00 © 2011 IEEE
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Fig. 2. The differential equation used for the calculation of the magnetic vector
potential.
2-D solutions. Here are some important contributions to this
theory.
1) Already in 1929, B. Hague wrote a book about 2-D solutions of Poisson’s equation [9]. He calculated the fields due
to currents in a cylindrical geometry.
2) In 1977, A. Hughes and T. J. E. Miller [10] presented a
model of the field created by a conducting sheet in a 5-layer
structure.
3) Based on the work of B. Hague, N. Boules wrote a paper in
1985 [11] entitled “Prediction of no-load flux density distribution in permanent magnet machines,” where the PM’s
magnetization is replaced by equivalent currents.
4) In 1993, Z. Q. Zhu and D. Howe wrote an excellent series
of four papers [12]–[15] on the calculation of the magnetic
field in electrical machines based again on the magnetic
scalar potential. The fields due to the PM are directly calculated using the PM magnetization.
5) In 1995, Z. J. Liu et al. used the magnetic vector potential to
calculate the fields and the eddy currents in the stator yoke
[16]. They divided the space into three concentric layers
and expressed the analytical field solution in the layers.
6) In 1997 and 1998, after discussion with N. Boules, F. Deng
wrote two papers [17], [18] about eddy-current power losses
due to the commutation of the PM machine. She was able to
solve the differential equation not only in different layers,
but also in different sectors in some layers. In these calculations, the remanent field of the PMs is neglected. The
applied current in the coils is approximated by a surface
current density between two layers. This model enables the
calculation of the eddy-current power losses in the motor
teeth due to the pulsewidth modulation [19].
Several improvements and contributions have been made on
this topic in the last 10 years. Amongst these, the following publications can be cited.
1) In 2003, S. R. Holm calculated the magnetic fields in a
cylindrical structure. He included in his calculations the
fields due to an applied current density in a layer [4].
2) In 2007, Shah et al. solved a 6-layer structure in cartesian
coordinates [20], based on a previous work [21].
3) In 2007, A. Chebak determined the solution of the differential equation for a 4-layer structure, -pole-pair PM, with
magnetization harmonics and calculated the eddy currents
in the stator [22].
4) In 2008, M. Markovic calculated the eddy-current
power losses in a one-pole-pair PM [5], considering
its magnetization.
This timeline is not exhaustive, but shows some important
milestones on the path to the resolution of the differential equations for obtaining the fields. Many other papers could be cited:
[6], [23], [24], [27], [28]. A brief overview of some papers which
IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 6, JUNE 2011
solve the differential equation is presented in Table I. Some remarks about these two tables are as follows.
• A “Yes” in the “Eddy currents” row means that the eddy currents, if they are calculated, are a direct solution of the differential equation. If the geometry is more complex than a concentric contiguous hollow cylinder, it may be better to make
simplifying assumptions than to directly solve the general
differential equation in order to derive eddy currents.
• Concerning the “Innermost boundary” row:
” means that the material inside the boundary
—“
1 is considered to be of infinite permeability. No magnetic field is calculated in this region. Since the layer
is defined as a region in which the magnetic fields are
calculated, in this case the center is not considered as a
layer, as in Fig. 1.
” means that the radius of the boundary 1 tends
—“
to zero.
• Concerning the “Outermost boundary” row:
” means that the exterior of the boundary
—“
is considered to be of infinite permeability.
” means that the outermost layer covers the
—“
tends to
whole space: the radius of the boundary
infinity.
For more information about the “Innermost boundary” and
the “Outermost boundary,” see Section IV-B.
• If a “(1)” stands in the “Magnetic potential” row, it means
that instead of solving a differential equation involving a
magnetic scalar or vector potential, the authors solved a
differential equation directly involving the current density.
• If a “(2)” stands in the “Eddy currents in the PM(s)” row,
it means that the PM remanent field and its harmonics are
neglected in the calculation of the eddy currents. In these
cases the eddy currents are due to the excitation current.
The interaction of the PM’s magnetization with the eddy
currents is also neglected.
• As is shown in Section III, the form of the diffusion equation in a PM where eddy currents are considered is different
and implies a much simpler solution when the PM is parallelly magnetized than when higher harmonics of remanent
field are considered. If a “(3)” stands in the “Eddy currents
in the PM(s)” row, it means that the calculation is valid only
for a 1-pole-pair parallelly magnetized PM.
There are some limitations to the different resolutions of
the differential equation in cylindrical coordinates presented in
Table I.
• The number of layers is limited.
• The number of simplifying hypotheses is high.
• The models of eddy currents in the PMs neglect the effect
of the remanent field. The only known exception is one
case examined in the literature which is a one-pole-pair
central PM diametrically magnetized [5].
• Each model is the result of laborious calculations.
D. Objective
The aim of the present paper is to show a model which has
the following advantages.
1) The procedure which gives the analytical solution is fast.
2) The procedure gives a model for any number of layers.
3) As the magnetization of the PM is defined by its harmonics,
any magnetization can be set, including the ideal Halbach
magnetization.
PFISTER AND PERRIARD: SLOTLESS PERMANENT-MAGNET MACHINES
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TABLE I
COMPARISON OF THE SOLUTIONS OF THE DIFFERENTIAL EQUATIONS. THE REMARKS ARE IN THE TEXT
4) The eddy currents in the PM are determined, with any
number of pole pairs, magnetization harmonics, applied
current density harmonics and applied surface current density harmonics.
5) The model can handle for the innermost boundary: zero
radius or nonzero radius with infinite permeability inside
the boundary.
6) The model can handle for the outermost boundary: infinite radius or finite radius with a material of infinite permeability outside it.
7) The model can handle at the same time applied surface
current densities and applied current densities.
E. Outline of the Paper
Maxwell’s equations and two equations which describe materials are used to obtain the GDE. The GDE is considered in the
Fourier space and solutions are found. Boundary conditions are
described and finally a general solution is found for a multilayer
system.
F. Assumptions
1) The permeability and the conductivity are isotropic.
2) Until (22), the system is considered to be 3-D. Afterwards,
3-D effects are neglected and the system is considered to
be 2-D.
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3) For the 2-D, no lamination of any materials is considered.
4) The materials are considered to be linear with no
saturation.
5) The PMs do not demagnetize.
6) Each layer is cylindrical.
7) The wavelengths of all time-varying fields are large compared with the physical dimensions of the device.
is not uniquely defined. Let be an arbitrary scalar function, and
and
be defined as
(10)
(11)
It implies that
II. FROM THE MAXWELL’S EQUATIONS TO THE GENERALIZED
DIFFUSION EQUATION
(12)
A. Maxwell’s Equations
Maxwell’s equations are written in the following form:
(13)
(2)
Therefore, the potentials defined by (10) and (11) give the same
field. To define uniquely, the Coulomb gauge [30] has been
chosen:
(3)
(14)
(1)
(4)
where
is the electric displacement, is the resistivity,
the electric field strength, is the magnetic flux density,
the magnetic field strength, and is the current density.
is
is
D. Generalized Diffusion Equation
Since the permeability is assumed to be isotropic and constant, (4), (5), (6), and (7) are combined to obtain
B. The Constitutive Materials Equations
The constitutive equation of a PM is
(15)
(5)
where
is the remanent field and is the permeability. Since
the permeability is isotropic, is scalar. Since the materials are
assumed to be linear without saturation, is constant. In soft
ferromagnetic materials, the same equation is used, but with
.
The current density in a moving conductor with relative velocity is generated by the Lorentz force and is given by [29]:
(6)
is the conductivity which is assumed to be constant.
C. Vector Potential
As the divergence of is equal to zero
netic vector potential such that
(15) is rewritten as
(17)
Equation (7) is the most general form of the GDE presented
in this paper. It can be simplified, depending on material’s properties and other hypotheses:
• It can be reduced in PMs with no applied currents, where
the eddy currents are considered. A GDE type equation is
obtained:
(18)
is taken into consideration. Equations (2) and (7) are combined
to obtain
(8)
• It can be reduced in a conductive media, where no current
is applied where the eddy currents are considered. A diffusion type equation is obtained:
(19)
• It can be reduced in the air. A Laplace type equation is
obtained:
(20)
which gives after integration
(9)
is a electric scalar potential.
(16)
, a mag(7)
where
Using the Coulomb gauge and the identity
• It can be reduced in a static media, where the eddy currents
are not considered, but where current is applied. A Poisson
type equation is obtained:
(21)
PFISTER AND PERRIARD: SLOTLESS PERMANENT-MAGNET MACHINES
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where
is the applied current density. For (21), the
applied current density is deduced from the scalar potential
using (6):
(22)
By hypothesis, the complete system is 2-D. The cylinis used. The magnetic
drical coordinate system
vector potential and the current density are hence along
.
and
are in the
plane. The angular velocity of the material is so
.
Equation (17) is expressed along :
(23)
III. THE GENERALIZED DIFFUSION EQUATION SOLUTIONS
Now that the GDE is formulated, its solutions need to be
found. Since the structure is made of cylinders, there is a spatial periodicity of any quantity, but in particular of the magnetic
vector potential. Moreover, as the transient states are not considered, there is a time periodicity of period which corresponds
to the period of the applied current. The angular frequency is
defined by
(24)
For any quantity
applies:
, the following periodicity condition
(25)
(26)
The easiest way to find solutions is to consider the different
variables in the Fourier space.
A. Complex Fourier Series
The complex Fourier series of any quantity
is defined as
(27)
is the real part of .
where
The GDE (23) is expressed for the
with
(30)
The solution of (29) depends on the material properties and
on many parameters. Two cases need to be considered.
1) The case where eddy currents can possibly occur, if the
layer of the system is defined by the right harmonics, rotation speed and material conductivity. This case is called
ECPO.
2) The case where the hypothesis is made that no eddy
currents occur or where they are neglected. This case is
called NECO.
Some examples of both cases follow.
1) The ECPO case:
• A PM layer with a given conductivity rotating in a field
created by different applied current harmonics.
• A cylinder of copper rotating in a synchronous field. No
eddy currents occur, nevertheless, if the harmonic content of the field is enriched, eddy currents would occur.
2) The NECO case:
• Any nonconductive media.
• Laminated iron in which eddy currents are neglected. If
the eddy currents are not neglected in laminated iron,
the problem is intrinsically 3-D and cannot be solved
by the present model. For laminated iron, the simplest
approach is to solve the equation without eddy currents
first, and then use an a posteriori model of the power
losses due to the eddy currents as a function of the field.
• A coil layer. The insulation between the wires implies
that the NECO case is a better approximation than the
ECPO case.
B. The Generalized Diffusion Equation in the “Eddy Currents
Can Possibly Occur” Case
Good examples of conductive media would be: titanium or
copper cylinders, a conductive PM, and iron. In the ECPO case,
the following hypotheses are made.
1)
and
are constant. This assumption is always true for any ideal Halbach PM. This assumption is a
good approximation in PMs used for electrical machines.
2) There is no external current:
.
3) The material is conductive:
.
4) The field is synchronous:
, with the number of
poles pairs. In the moving part:
. In the standstill
part:
.
In that case (29) becomes
th harmonic:
(31)
(28)
In the resolution of (31), two cases need to be separated.
1) The first case is implied by the following conditions:
• For a part which belongs to the rotor:
or
.
• For a part which belongs to the stator:
or
.
If one of the above conditions is fulfilled, the solution is
(29)
(32)
which can be simplified as
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IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 6, JUNE 2011
with the particular solution of the differential equation:
(33)
It is important to notice that in a material which has no
. Also if the remanent field is
remanent field
parallel
.
2) The second case is implied by the following conditions:
and
• For a part which belongs to the rotor:
.
• For a part which belongs to the stator:
and
.
The solution is
(34)
is the Bessel function of the first kind (see
where
Appendix E) and
is the Bessel function of the second
kind (see Appendix F).
The particular solution is the following for even :
Fig. 3. Diagram of an electromagnetic s-layer structure. Every layer is characterized by its constitutive material properties, by its rotational speed and by the
external current applied. This structure is general, it can represent, for example,
eddy-current brakes and PM slotless motors. In this representation, the layer i
contains a coil defined by the angles and .
C. The Generalized Diffusion Equation Solutions in the “No
Eddy Currents Occur” Case
With respect to the solution of the GDE in the domain of electrical machines, magnets with low conductivity, air, litz wire,
can be related to or approximated by the NECO case. In a nonconductive media, the following assumptions are made.
and
are constant.
1) As in the ECPO case:
.
2) The field is synchronous
The following expression of the GDE can be solved when the
material is not conductive, or when the eddy currents can be
neglected:
(37)
(35)
In the resolution of (37), different cases are separated depending on
is the Struve function (see Appendix G),
is a generalized hypergeometric function (see
Appendix H), and
where
(38)
is the generalized Meijer G function (see Appendix I).
With odd, no general formula for
was found. Here
is the formula for
:
with
(39)
In the case of a balanced three-phase machine, the harmonics
are given by
(36)
(40)
From (31), it can be simply deduced that when the magnetization of the part is parallel or when there is no remanent magne.
tization:
where
and
are the boundary angle of the coil,
is
the time harmonics of the current density. The two angles
and
are shown in Fig. 3.
PFISTER AND PERRIARD: SLOTLESS PERMANENT-MAGNET MACHINES
1745
Using
, the following expression is obtained:
(44)
which gives
Fig. 4. Boundary conditions.
IV. THE -LAYER PROBLEM
The vector potential obtained by solving the GDE is determined in the previous sections. The constants
and
remain to be determined. These two constants are defined by
the boundary conditions. The considered structure is an -layer
concentric structure as shown in Fig. 3. The two boundary conditions for each interface are deduced in Section IV-A. As there
are layers, there are
interfaces and hence
conditions. In Section IV-B, the innermost and outermost boundaries
give two more conditions. The total number of conditions is .
The
conditions give the
equations that are needed to deconstants.
fine the
The interior boundary of the th layer is called . Its permeability is called .
is the coefficient associated with
the th layer for the harmonic
and .
represents the
coefficient
of the th layer for
.
represents the function
of the layer for
.
is defined to be true for layer if:
The condition
• For a layer which belongs to the rotor:
and
.
• For a layer which belongs to the stator:
and
.
Otherwise it is defined to be false.
represents the condition which expresses the fact
that the harmonic
creates eddy currents in the layer .
A. Two Kinds of Boundary Conditions
between layer
The boundary defined by
and layer is taken into consideration.
implies
that
. By the definition of the magnetic
vector potential, in the 2-D case it follows:
(45)
in the 2-D approximation. The second boundary condition for
the vector potential is obtained:
(46)
The last expression can be expressed as a Fourier series:
(47)
B. Three Kinds of Innermost and Outermost Boundary
Conditions
The innermost and outermost boundary conditions are at the
inner side of the innermost layer and at the outer side of the outermost layer of the -layer structure. The boundary conditions
are deduced in the following three cases:
1) Center: If the innermost boundary is defined in
,
this boundary is called “center.”
needs to be defined in
. It implies that if
is true, (34) gives that
(48)
(49)
(41)
and if
which can be expressed as a Fourier series:
is false, (32) and (38) imply that
(50)
(51)
(42)
2) Infinite Permeability: If inside from the innermost
boundary the permeability is considered to be infinite:
,
(47) gives
This is the first boundary condition.
implies that
(52)
(43)
If the exterior of the outermost boundary is considered to be
infinite, (47) gives
where
is by definition the surface current density at the
boundary between layer
and layer . Fig. 4 shows the different vectors at the boundary.
(53)
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IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 6, JUNE 2011
3) Infinite Radius: In this case, for the layer , only a nonconductive material is considered. The assumption is made that
. This implies that the magnetic
there is no flux in
vector potential needs to be zero in
. Equation (38)
gives
(54)
(55)
C. Matrix
The magnetic vector potential given by a single wire is not
taken into consideration. In the problem, for any current density
which flows in one direction, there is always a current density
which flows in the other direction. Therefore, the amplitude of
is assumed to be equal to
the vector potential’s harmonic
zero.
The coefficients
, with
and
need
to be calculated to obtain the magnetic vector potential. In order
to calculate them, the following vector is defined:
Fig. 5. Representation of the magnetic field calculated analytically considering
eddy currents in the outer yoke. The different layers from the center to the exterior are: the rotor (iron yoke, 2-pole-pair ideal Halbach PM), the air gap, the
stator (iron yoke), air.
D. Solution
Equation (57) is inverted to obtain the formula for each
constant:
(59)
(56)
..
.
V. FIELD REPRESENTATION
A. Magnetic Flux Density
The magnetic flux density
is given by
(60)
Using the boundary conditions of Sections IV-A and IV-B,
the following system is found:
which gives in polar coordinates
(57)
(61)
can be expressed as
Xmn =
B1mn 0 0 0 0
L2mn R2mn 00 00
0 0 L3mn R3mn
0 0
..
.
..
.
..
.
..
.
..
0 111 0
0 111 0
0
0
0 111 0
0
0
.
..
.
..
.
..
.
0
0
0
0
0
..
.
..
.
0
0
0
0
0
..
.
..
.
0
0
0
0
0
Now can be represented as a function of the different harmonics of the magnetic vector potential:
..
.
..
.
(62)
0 0 0 0 0 0 0 1 1 1 Lsmn Rsmn
0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 1 1 0 0 B(s+1)mn
with
and
being (1 2) matrices, and
and
being (2 2) matrices. The expression of these matrices
is given in Appendix A.
The vector
has
elements:
..
.
The variables
(58)
are defined in Appendix B.
B. Some Illustrations
The magnetic field in different configurations is represented,
as an illustration of the power of the model. The following representations are the result of the fully analytical model.
• Fig. 5 represents a five-layer model of the following structure, from the center to the exterior: the rotor (iron yoke,
2-pole-pair ideal Halbach PM), the air gap, the stator (iron
yoke), air. The innermost boundary: “center,” and the outermost boundary: “infinite radius.” The figure shows the
deformation of the magnetic flux density field lines due to
the eddy currents in the stator.
• Fig. 6 represents a two-layer model. The inner layer is an
ideal Halbach 3-pole-pair PM, and the exterior layer is air.
PFISTER AND PERRIARD: SLOTLESS PERMANENT-MAGNET MACHINES
Fig. 6. Representation of the magnetic field calculated analytically due to a
three-pole-pair ideal Halbach PM. The innermost and outermost boundaries
have the condition of “infinite permeability.” The two layers from the center
to the exterior are: the PM and air.
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Fig. 8. Representation of the magnetic field calculated analytically due to a
two-pole-pair ideal Halbach PM (left), and due to a radially magnetized PM
(right).
TABLE II
PROTOTYPE SPECIFICATIONS
Fig. 7. Representation of the magnetic field calculated analytically due to a
one-pole-pair PM. The PM is magnetized with a first and second harmonic. It
makes it asymmetric. The five different layers from the center to the exterior
are: a rotor yoke, a PM, air, a stator yoke, and air.
The innermost and outermost boundaries have the condition of “infinite permeability.”
• Fig. 7 represents the possibilities of considering different
magnetization harmonics.
• Fig. 8 represents the difference between an ideal Halbach
PM and a radially magnetized PM. The radially magnetized PM is created taking into consideration the harmonics
1,3,5,7 of the radial remanent field, and the tangential remanent field harmonics are equal to zero.
VI. MODEL VALIDATION
A. Very High Speed Permanent-Magnet Machine
As it is not possible to validate the model in its generality
using finite-element methods, we present here different cases
which show the model’s validity. The first illustration is a
very high speed PM machine whose specifications are given
in Table II. A section of the machine is shown in Fig. 10.
This machine is slotless and reaches more than 200 000 rpm
and more than 2 kW of output power [35]. As the machine is
Fig. 9. Geometry of a coil of the VHS PM machine.
slotless, the coils shown in Fig. 9 are in the air gap between the
stator yoke and the rotor.
The field calculated using the analytical model (10 space
harmonics) and using finite-element methods is represented in
Fig. 10. In each figure, a current density of 10.4 10 A/m is
applied to the coil which is on the upper right side. No current
is applied to the two other coils. The value of permeability and
remanence are the ones given in Table III. The permeability of
. Fig. 10 shows
the outer yoke is assumed to be
that the agreement is excellent.
Another good physical property that can be calculated to
compare the model to finite-element methods, is the total flux
passing through one coil. The same hypotheses are considered
as the one used for the calculation of Fig. 10 except that no
applied current is inside the coils. The total flux passing through
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Fig. 11. Total flux passing through one coil as a function of the angular position
of the rotor. The dots represent calculation using the finite-element methods, the
continuous line represents the calculation using the analytical model.
Fig. 10. Comparison of the magnetic field given by the analytical model (top)
and the finite-element methods (bottom) during the calculation of the torque
created by one phase. The current in set in coil 1. In a+, the current density
is 10.4 A/mm and in a- the current density is 10:4 A/mm . In b and c the
current density is equal to 0.
0
TABLE III
PROTOTYPE MATERIALS
a coil is calculated as a function of the angle. The dots in Fig. 11
represent the finite-element method calculations, the continuous line represents the analytical model. As for Fig. 10, the
agreement is excellent. The comparison between the analytical
model and the finite-element method gives a difference of less
%.
than
B. Eddy-Current Brake
The dynamic electromagnetic model is validated using a
2-pole-pair eddy-current brake structure. The structure made of
concentric cylinders is the following, starting from the center.
Fig. 12. Magnetic field in the eddy-current brake obtained by finite-element
methods.
1) A 2-pole-pair ideal Halbach type PM in the center: the reT and the relative permeability
manent field is
is 1.03. The outer radius is 5 mm. The PM is rotating at a
rad/s.
speed of
2) A layer of air.
3) A yoke: the inner radius is 5.5 mm, the outer radius is
7 mm, its relative permeability is 2000, is conductivity
2.4 10 S.
4) A layer of air.
The magnetic fields calculated using the analytical model is
represented in Fig. 13 and the one obtained using the finiteelements methods is represented in Fig. 12. Fig. 14 represents
the radial magnetic field in the conductive yoke at different radii.
We see a good agreement between the model and the finiteelement methods.
VII. CONCLUSION
Starting with Maxwell’s equations, the formalism developed
in this paper allows the obtention of the analytical expression
of the vector potential and the magnetic field at any point of a
-layer cylindrical system, whereas in the literature only special
cases of this model have been found.
The presented model is successful in the following aspects:
each layer can rotate, be conductive, have a remanent magnetic
PFISTER AND PERRIARD: SLOTLESS PERMANENT-MAGNET MACHINES
1749
with
and
being (1
being (2 2) matrices.
:
The first column of
2) matrices, and
and
(63)
:
The second column of
Fig. 13. Magnetic field in the eddy-current brake calculated using the analytical
model.
(64)
The first column of
:
(65)
The second column of
:
Fig. 14. Comparison between the radial magnetic field obtained by finite-element methods or calculated analytically in the conductive part of the eddy-current brake.
field and be subject to an applied current density, each boundary
can be characterized by the presence of a current surface density.
The calculation of the fields in multipolar conductive magnets
is also a contribution of this paper.
The study of different designs shows a good agreement between the general analytical model and finite-element methods.
APPENDIX A
(66)
and
depend on the side boundary conditions, they
are defined using considerations in Section IV-B.
is considered first. If the interior side boundary is a
“center”:
A. The Expression of
can be expressed as
Xmn =
B1mn 0 0 0 0
L2mn R2mn 00 00
0 0 L3mn R3mn
0 0
..
.
..
.
..
.
..
.
..
0 111 0
0 111 0
0
0
0 111 0
0
0
.
..
.
..
.
..
.
0
0
0
0
0
..
.
..
.
0
0
0
0
0
..
.
..
.
0
0
0
0
0
..
.
..
.
0 0 0 0 0 0 0 1 1 1 Lsmn Rsmn
0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 1 1 0 0 B(s+1)mn
(67)
If the interior side boundary is “infinite permeability”, the first
is
column of
(68)
1750
IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 6, JUNE 2011
The second column of
:
:
For
(69)
is now taken into consideration. By hypothesis, in
the case of an exterior side boundary which is an “infinite radius” only a nonconductive material is taken into account:
If the exterior side boundary is “infinite permeability,” the first
is
column of
and
(70)
:
The second column of
C. Gamma Function
The
function [36] is defined by
(76)
(71)
This implies that for any
:
B. The expression of
The vector
has
(77)
elements:
D. Pochhammer Function
..
.
(72)
The general definition of the Pochhammer function [36] is
depends on the innermost boundary condition:
(78)
It can be simplified when
is a positive integer:
(79)
(73)
and
depends on the outermost boundary condition:
E. Bessel Function of the First Kind
The Bessel function of the first kind [36],
following differential equation:
, satisfies the
(80)
(74)
For
It is defined as
:
(75)
(81)
PFISTER AND PERRIARD: SLOTLESS PERMANENT-MAGNET MACHINES
1751
F. Bessel Function of the Second Kind
REFERENCES
The Bessel function of the second kind [36],
isfies the following differential equation:
, also sat(82)
If
, it is defined as
(83)
, it is defined as
If
(84)
G. Struve Function
The Struve function [36] is defined as
(85)
H. Generalized Hypergeometric Function
For positive
Generalized hypergeometric function [37],
, is defined as
(86)
where
is the Pochhammer function.
I. Generalized Meijer G Function
The Generalized Meijer G function can be defined in terms
of the Fox H function:
a1 ; ; ak ; ak+1 ; ; au
Gj;k
u;v z; r
b1 ; ; bj ; bj+1 ; ; bv
; au ; r
+1 ; r ;
j;k
rHu;v z ab11;;rr ;; ;; abkj ;; rr ;; abjk+1
; r ; ; bv ; r
k
0 ad 0 rs jd=1 bd rs z0s s
r
d=1
u
{ L d=k+1 ad rs vd=j+1 0 bd 0 rs
...
...
...
(
=
=
(
...
) ...
) ...
(
(
0(1
2
0(
) (
) ...
(
) (
) ...
(
)
+
)
0(
0(1
+
)
)
)
)
d
(87)
with
and
. The infinite contour
of integration separates the poles of
at
from the poles of
at
. Such a contour always exists in the cases
.
Any good mathematical software can calculate directly such
a function. Many books and papers give more detail about this
function [37].
ACKNOWLEDGMENT
The authors want to thank Moving Magnet Technologies SA
and Sonceboz SA for their support for the research accomplished
on very high-speed machines [26]. The present paper is resulting
from this research.
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Pierre-Daniel Pfister was born in Bienne, Switzerland, in 1980. He received the
M.Sc. degree in physics in 2005 and the Ph.D. degree in 2010 from the Swiss
Federal Institute of Technology-Lausanne (EPFL). He studied for one year at
the University of Waterloo, Canada.
After receiving the Ph.D. degree, he continued to serve as a development
engineer for Sonceboz SA (Switzerland)/Moving Magnet Technologies SA
(France). His research interests are in the field of permanent magnet machines,
very high speed machines, and analytical optimization.
Yves Perriard was born in Lausanne, Switzerland, in 1965. He received the
M.Sc. degree in microengineering from the Swiss Federal Institute of Technology-Lausanne (EPFL) in 1989 and the Ph.D. degree in 1992.
Co-founder of Micro-Beam SA, Yverdon, Switzerland, he was CEO of this
company involved in high precision electric drives. He was a Senior Lecturer
from 1998 and has been a Professor since 2003. He is currently director of the
Integrated Actuator Laboratory and vice-director of the Microengineering Institute at EPFL. His research interests are in the field of new actuator design
and associated electronic devices. He is author and co-author of more than 80
publications and patents.