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Journal ArticleDOI

A New Analytical Torque Formula for Axial Field Permanent Magnets Coupling

Bastien Dolisy1, Smail Mezani1, Thierry Lubin1, Jean Lévêque1 
07 May 2015-IEEE Transactions on Energy Conversion (IEEE)-Vol. 30, Iss: 3, pp 892-899

AbstractIn this paper, we present a simple and accurate analytical expression to compute the torque of axial-field magnetic couplings. The torque expression is obtained by solving the three-dimensional (3-D) Maxwell equations by the method of separation of variables. Here, we adopt the assumption of linearization at the mean radius, the problem is then solved in 3-D Cartesian coordinate (we neglect the curvature effects). To show the accuracy of the torque formula, the results are compared with those obtained from 3-D finite-element simulations and from experimental tests. As the proposed formula needs very low computational time and depends directly on the geometrical parameters, it is used for a design optimization using multiobjective genetic algorithms.

Topics: Coupling (63%), Linearization (53%), Torque (53%), Curvature (52%), Maxwell's equations (52%)

Summary (2 min read)

Introduction

  • Analytical expression to compute the torque of axial-field magnetic couplings.
  • It consists of two similar rotors facing each other.
  • The electromagnetic computation of magnetic couplings is carried out using several methods.
  • The main drawback of FE methods is the long computation time and the lack of flexibility.
  • A new and purely analytical expression for the torque evaluation is then derived from the 3D solution.

A. Magnetic force from the electrostatic-magnetostatic analogy

  • Unlike s, the magnetic charge m doesn’t have any physical meaning.
  • It is introduced for modelling purposes in which it usefully replaces some magnetic field sources (magnets, current carrying solenoids,…).
  • Furthermore, expression (2) which uses Lorentz force in free space gives, for their coupler, the right values of the force along the x and y directions only (no other material than air in these two direction).
  • Since ferromagnetic materials are present in the z-direction, (2) will not give the right value of the force and the authors have to use Maxwell stress tensor or virtual work methods.

B. Magnetic field due the magnets of one rotor

  • Fig.3 shows the problem to solve after linearization.
  • The main dimensions of the linearized coupler are Due to the alternate polarity along the x-direction, only one pole is considered with anti-periodic boundary conditions along x.
  • These conditions state that the tangential magnetic field components Hx and Hy are zero (iron boundaries).

C. Equivalent surface charge density of the second PM rotor

  • This dot product has to be performed on all the external surfaces of the magnet volume.
  • Fig.6 shows a rectangular permanent magnet with a uniform magnetization in the z-direction.
  • From (13), the magnet is then represented by two surface charge densities + and -.

D. Torque expression

  • The force is computed using (2) where the integration is performed on the surfaces carrying 𝜎+ and 𝜎−.
  • According to the boundary condition (3), the tangential components 𝐵𝑥 and 𝐵𝑦 of the flux density are null on the charged surface 𝜎+ (at 𝑧 = ℎ𝑡), so the forces that contribute to torque (Fx and Fy) also vanishes.
  • 𝑋𝑖 −𝑙𝑚𝑥+𝑋𝑖 𝑑𝑥 𝑑𝑦 (14) The variable 𝑋𝑖 in (14) corresponds, in cartesian coordinates, to the angular lag (load angle) 𝜑 between the two rotors of the coupling.
  • Another useful quantity to compute is the flux over a pole surface.
  • The thickness of iron yokes can be determined using flux conservation law.

III. EVALUATION OF THE TORQUE FORMULA

  • The authors analyze the accuracy of the developed torque formula whose results are compared to those issued from 3D FE computations (Comsol multiphysics®) carried out on the actual cylindrical coupling.
  • The yoke thickness is choosen to avoid magnetic saturation.
  • Hence, a linear model with a relative permeability value equals to 1000 is used for the ferromagnetic material so the computation time is reduced without any loose of accuracy.
  • The corresponding experimental and analytical results also serve to evaluate the torque formula (17).
  • If the authors add a supplementary harmonic (𝑁 = 3), the computation provides very accurate result with an error estimate lower than 2%.

B. Comparison to experimental and 3D FE results

  • The authors compare experimental measurements for the static torque with numerical and analytical computations.
  • An analytical formula which is derived in [11] using a 2D analytical model (mean radius model and first harmonic approximation) is given by (21).
  • This formula doesn’t take into account the edge effects.
  • 9 show the results of the torque calculation for two airgap values (e=4mm and e=9.5mm) obtained by all the methods in use.
  • This clearly shows the necessity to consider the radial fringing effects in axial field couplings.

C. Curvature effects

  • In order to address the limits of the analytical formula regarding the curvature effects, the analytical computation (linearized coupling) are compared to 3D finite element simulations (actual cylindrical topology) for several dimensions of the magnet.
  • For a given air gap, the error introduced by the linearization assumption depends on the radial excursion Rout-Rin and on the mean pole pitch which is equal to (𝑅𝑜𝑢𝑡 + 𝑅𝑖𝑛)/2𝑝. (see Fig.2).
  • But finally, the authors found that this single parameter λ led to the same conclusions.
  • The first one uses 3D FEM to obtain the torque of the cylindrical coupling, also known as Two computations are performed.
  • This corresponds to 504 combinations (252 for each topology).

D. Optimization of the coupler using genetic algorithms

  • Genetic Algorithms (GA) are widely used as a robust and effective tool in optimization problems.
  • The objective T2 (Nm) is the torque for an air gap e=10 mm and the objective MPM corresponds to the total mass of the PMs.
  • The optimization procedure uses 100 individuals evolving during 100 generations, but 50 generations are enough to reach a stable solution.
  • Notice that many of the solutions of Fig.11 have checked by the 3D FE model and the error in the worst case is less than 3%.

IV. CONCLUSION

  • A new analytical expression to compute the torque of a PM axial field magnetic coupling has been derived.
  • This expression has been obtained thanks to 3D magnetostatic analytical.
  • By introducing a surface charge density, the torque computation used the electrostatic-magnetostatic analogy to evaluate the Lorentz force.
  • The authors have shown that the proposed torque formula is very accurate and computationally very efficient.
  • Thus, it has been used to optimize the studied coupler by a multiobjective genetic algorithm.

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Submitted on 16 Jun 2015
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A new analytical torque formula for axial eld
permanent magnets coupling
Bastien Dolisy, Smail Mezani, Thierry Lubin, Jean Lévêque
To cite this version:
Bastien Dolisy, Smail Mezani, Thierry Lubin, Jean Lévêque. A new analytical torque formula for axial
eld permanent magnets coupling. IEEE Transactions on Energy Conversion, Institute of Electrical
and Electronics Engineers, 2015, 8 p. �10.1109/TEC.2015.2424159�. �hal-01164388�

1
Abstract In this paper, we present a simple and accurate
analytical expression to compute the torque of axial-field
magnetic couplings. The torque expression is obtained by solving
the three-dimensional (3D) Maxwell equations by the method of
separation of variables. Here we adopt the assumption of
linearization at the mean radius, the problem is then solved in 3D
Cartesian coordinate (we neglect the curvature effects). To show
the accuracy of the torque formula, the results are compared
with those obtained with 3D finite-element simulations, and
experimental tests. As the proposed formula needs very low
computational time and depends directly on the geometrical
parameters, it is used for a design optimization using
multiobjective genetic algorithms.
Index Terms Genetic algorithms, magnetic coupling,
permanent magnets, torque transmission, 3D analytical model.
I. INTRODUCTION
agnetic couplings (or couplers) can transmit a torque
without mechanical contact. This is very interesting for
applications requiring isolation between two different
atmospheres. They can be used in the naval propulsion [1] for
torque transmission between motors and propellers, or in
chemical industry for health constraints. In addition, the
absence of mechanical contact increases the lifetime of the
system and reduces noise, vibrations and mechanical friction
losses. Moreover, it provides natural protection against
overloads.
Magnetic couplings can have axial or radial flux topologies
(Fig. 1). They both consist of two rotors, each of which is
composed of an array of permanent magnets alternately
magnetized along the θ-direction. The two rotors present the
same number of pole pairs (p=6 in Fig.1).
The axial-flux topology is studied in this paper. It consists of
two similar rotors facing each other. As shown in Fig. 2, the
air-gap between the two rotors is noted e.
The magnets are sector shaped with a thickness (along z-
direction) noted h for both rotors. The inner and outer radii are
respectively noted R
in
and R
out
.
The authors are with the Groupe de Recherche en Electrotechnique et
Electronique de Nancy, Université de Lorraine, 54506 Nancy, France (e-mail:
´ b-dolisy@hotmail.fr).
(a) Radial flux (b) Axial flux
Fig. 1 Topologies of rotating magnetic couplings (p=6).
The magnet angular opening to pole opening ratio is noted α
and varies between 0 and 1. The angular lag (load angle)
between the two rotors is noted φ.
The electromagnetic computation of magnetic couplings is
carried out using several methods. The Finite-element (FE)
method is so far the preferred method of analysis. Indeed, it
leads to accurate results taking into account the non-linearity
of magnetic materials and the actual coupler geometry [2-4].
The main drawback of FE methods is the long computation
time and the lack of flexibility. It is therefore unsuitable for
optimization purposes which require many repetitive
computations. In order to reduce the computation time,
analytical models can be developed by solving the partial
differential equations (PDE) arising from Maxwell’s
equations. It is necessary to make some assumptions regarding
the linearity of magnetic materials and geometry
simplification [5-9]. Usually, the problem is solved under a
two-dimensional (2D) approximation which, in some situation
like in axial field couplers, results in a 30% overestimation of
the torque compared to 3D FE prediction [10-12]. Three-
dimensional analytical models for magnetic couplings have
been proposed in the literature [6],[13] and [14]. Biot-Savart
like formulas are used to determine the magnetic field
distribution in ironless structures (magnets in free space).
The method of images could be used to consider infinitely
permeable iron walls but the computation time increases.
Recently, it has been shown that Fourier analysis can be used
to solve 3D problems with ferromagnetic parts [7].
A new analytical torque formula for axial field
permanent magnets coupling
B. Dolisy, S. Mezani, T. Lubin, J. Lévêque
M

2
Fig. 2 Dimension of the permanent magnets of an axial flux coupler topology
(iron yokes not shown).
In [5], the authors developed a 3D analytical model to
compute the no load flux in axial-field permanent magnet
synchronous machine. In this method (also called sub-domain
method), it is necessary to numerically solve an algebraic
system of linear equations to calculate the Fourier coefficients.
Hence, even keeping its analytical formalism, the “fully
analytical” meaning of the sub-domain method is somewhat
lost. Nevertheless, in terms of computation time, such a
method remains more efficient than a FE analysis.
In this paper, the sub-domain method is used to analytically
determine the magnetic field distribution in the axial magnetic
coupling shown in Fig. 1b. A new and purely analytical
expression for the torque evaluation is then derived from the
3D solution. The proposed torque formula, which depends
directly on the physical and geometrical parameters, is
obtained by solving the PDEs in 3D Cartesian coordinates by
assuming a linearized geometry at the mean radius (we neglect
the curvature effects). We also consider an infinite
permeability of the iron yokes.
The torque expression is obtained in two steps:
- Firstly, we only consider the magnets on one side of the
coupling (the magnets on the other rotor are turned off). Then,
we compute the magnetic field by using a magnetic scalar
potential formulation.
- Secondly, using the analogy between the electrostatic and the
magnetostatic fields, the magnetic force acting on the magnets
placed on the opposite side is obtained by using the equivalent
electrostatic Lorentz force.
To analyze the accuracy of the proposed torque formula, the
results are compared to those obtained from 3D finite-element
simulations and from experimental investigations. Finally, the
analytical formula is used for a genetic algorithm
multiobjective optimization of the coupler.
II. ANALYTICAL MODEL
As stated above, the torque expression will be derived using
the analogy that exists between the electrostatic and the
magnetostatic fields.
A. Magnetic force from the electrostatic-magnetostatic
analogy
For simplicity, let us consider, in free space, an electrostatic
uniform surface charge density
s
(C/m
2
), subjected to an
electric field
󰇍
[16]. The Lorentz force (N) exerted on
s
is
󰇍
󰇍
󰇍
󰇍
(1)
where S is the surface which carries
s
.
From the magnetostatic point of view, it is usual to use an
equivalent magnetic surface charge
m
in A/m
[7],[11],[14],[15],[16]. Unlike
s
, the magnetic charge
m
doesn’t have any physical meaning. However, it is introduced
for modelling purposes in which it usefully replaces some
magnetic field sources (magnets, current carrying
solenoids,…).
The magnetic force (N) which is analogous to the electrosatic
one, given by (1), is then obtained by
󰇍
󰇍
󰇍
󰇍
󰇍
(2)
Here, S is the surface which carries
m
.
The force expressions (1) and (2) show that the electrostatic-
magnetostatic analogy links the electric field
󰇍
to the
󰇍
field
(called flux density).
Concerning the studied magnetic coupler, all what we need to
compute the force is the magnet’s magnetic surface charge on
one rotor and the magnetic field created by the magnets of the
second rotor (the magnets on the first rotor are turned off).
Furthermore, expression (2) which uses Lorentz force in free
space gives, for our coupler, the right values of the force along
the x and y directions only (no other material than air in these
two direction). However, since ferromagnetic materials are
present in the z-direction, (2) will not give the right value of
the force and we have to use Maxwell stress tensor or virtual
work methods.
To deal with the presence of iron media an equivalent surface
charge of the ferromagnetic material could be introduced [15].
B. Magnetic field due the magnets of one rotor
The iron-yokes have an infinite permability. Hence, the
magnetic field is null in the iron parts.
The boundary condition on the iron interface is then
󰇍
󰇍
󰇍
(3)
where
󰇍
󰇍
is the outward normal to the considered surface and
󰇍
󰇍
󰇍
the magnetic field strength.
Rare-earth permanent magnets have a relative permeability
close to that of air (
).

3
The studied coupler doesn’t contain any current source. To
solve the magnetostatic problem, it is then more convenient to
use a magnetic scalar potential 󰇛
󰇜 formulation
(
󰇍
󰇍

). In the different media, the flux density
󰇍
is given by
Air region:
󰇍


󰇍
󰇍
󰇍
(4)
(5)
where
󰇍
󰇍
is the magnetization of the magnet.
To simplify the analysis, we make the assumption of
linearization at the mean radius, so the curvature effect is not
considered. However, this allows to solve the problem in a
Cartesian coordinates system, which is simpler than to solve
the problem in cylindrical coordinates where special functions
appear [5]. The validity of the linearized model is discussed
later in the paper.
Fig.3 shows the problem to solve after linearization. The main
dimensions of the linearized coupler are
Due to the alternate polarity along the x-direction, only one
pole is considered with anti-periodic boundary conditions
along x.
A second anti-periodic condition is applied at the external
boundaries on the y-coordinate. This condition is a fictitious
but a necessary one in order to get a solution. Nevertheless, by
setting


, this anti-periodic condition leads to
󰇍
at  
, which is of course a more realistic physical
condition (Usually,


allows to obtain accurate
results).
As stated above, the whole resolution domain contains magnet
and air regions, Fig.4.
Domain I () corresponds to the magnet region of
height h. The magnetization vector is noted
󰇍
󰇍
󰇍
󰇍
󰇛
󰇜
󰇍
󰇍
󰇍
and it is obtained by expanding the magnetization into a
double Fourier series along x and y-directions (Fig. 5).
󰇛
󰇜


󰇛

󰇜

󰇛

󰇜


(7)
Fig. 3 Dimensions of one magnet pole after linearization (axis of rotation,
located at 

not shown).
where n, m are odd integers and B
r
is the residual flux density
of the permanent magnets. Notice that the magnetization given
by (7) is divergence free
󰇍
󰇍
.
Domain II (
) is composed of the actual air-gap
and the second magnet (whose magnetization is turned off).
The magnetic scalar potential is noted
in domain I and

in domain II.
and
are the solution of Laplace
equation
󰇍
󰇍
(8)
Fig. 4 Domains and equations in the plan (x,z).
Fig. 5 Magnetization Mz as a function of x and y (domain I).




; 


(6)

4
By considering the anti-periodic boundary conditions along
the x and y coordinates, the use of the method of separation of
variables leads to the following form solutions for
and

󰇛
󰇜
󰇟




󰇠



󰇛

󰇜

󰇛

󰇜
(9)
The coefficients A
I
, B
I
, A
II
and B
II
are obtained using the
interface and boundary conditions.
The Boundary condition in domains I and II are set at z=0 and
z=h
t
, respectively. These conditions state that the tangential
magnetic field components H
x
and H
y
are zero (iron
boundaries). This leads to
(10)
Interface conditions between domains I and II are set at z=h.
Domain I and II have the same magnetic permeability
r
=1),
so the normal flux density (B
z
) and the tangential magnetic
fields (H
x
and H
y
) of the domain I and II will be equal at z=h.
The two following expressions arise






















(11)
Finally, the coefficients A
I
, B
I
, A
II
and B
II
are calculated by
solving an algebraic system of linear equations arising from
(10) and (11). They are given by


󰇛



󰇜


󰇛


󰇜
(12)
C. Equivalent surface charge density of the second PM rotor
The equivalent surface charge density of a magnet
(Coulombian model) with uniform magnetization is given by
󰇍
󰇍
󰇍
󰇍
󰇍
(13)
Where
󰇍
󰇍
󰇍
󰇍
󰇍
󰇍
󰇍

󰇍
󰇍
󰇍
is the magnetization vector and
󰇍
represents the outward normal to the considered surface.
This dot product has to be performed on all the external
surfaces of the magnet volume.
Fig.6 shows a rectangular permanent magnet with a uniform
magnetization in the z-direction. From (13), the magnet is
then represented by two surface charge densities
+
and
-
.
In our problem, the surface charge density
is located
at
and the surface charge density
 at
.
D. Torque expression
The force is computed using (2) where the integration is
performed on the surfaces carrying
and
. However,
according to the boundary condition (3), the tangential
components
and
of the flux density are null on the
charged surface
(at
), so the forces that contribute to
torque (F
x
and F
y
) also vanishes. Hence, the integation is only
performed on the charged surface
(at ).
The axis of rotation (the shaft axis) is parallel to the Oz axis.
This axis has constant coordinates noted (x
0
,y
0
) in the (Oxyz)
reference frame. The z-component of the torque is then
obtained by
󰇛
󰇜
󰇛
󰇜









(14)
The variable
in (14) corresponds, in cartesian coordinates,
to the angular lag (load angle) between the two rotors of the
coupling.
and are related by


.
Notice that the maximum (pull-out) torque is obtained for a
position
Fig. 6 Equivalent surface charge density for a rectangular permanent magnet.

Citations
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Journal ArticleDOI
TL;DR: The present review explores various enabling technologies used to implement the vectorial thrusters (VT), based on water-jet or propellers, and proposes ideas for the improvement of this new generation of underwater thrusters as extending the magnetic coupling usage to obtain a fully contactless vector thrust transmission.
Abstract: Manoeuvrability is one of the essential keys in the development of improved autonomous underwater vehicles for challenging missions In the last years, more researches were dedicated to the development of new hulls shapes and thrusters to assure more manoeuvrability The present review explores various enabling technologies used to implement the vectorial thrusters (VT), based on water-jet or propellers The proposals are analysed in terms of added degrees of freedom, mechanisms, number of necessary actuators, water-tightness, electromagnetomechanical complexity, feasibility, etc The usage of magnetic coupling thrusters (conventional or reconfigurable) is analysed in details since they can assure the development of competitive full waterproof reconfigurable thrusters, which is a frictionless, flexible, safe, and low-maintenance solution The current limitations (as for instance the use of non conductive hull) are discussed and ideas are proposed for the improvement of this new generation of underwater thrusters, as extending the magnetic coupling usage to obtain a fully contactless vector thrust transmission

6 citations


Cites background or methods from "A New Analytical Torque Formula for..."

  • ...However, most of the time, MCs are classified regarding their magnetic flux geometries, which are usually classified as axial or radial [78]....

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  • ...proposed what is called as “ideal” AMC and RMC, with rotors made from arc-shaped magnets in Halbach arrays, and their torque density analyses are based on a new analytical formulation, including the curvature effects that were neglected in [78]....

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  • ...In [78], it is proposed a new analytical formulation using a subdomain method and the torque density optimised by Genetic Algorithms (GA)....

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References
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Journal ArticleDOI
Abstract: A quasi-three-dimensional (3-D) analytical model of the magnetic field in an axial flux permanent-magnet synchronous machine is presented. This model is derived from an exact two-dimensional analytical solution of the magnetic field extended to the 3-D case by a simple and effective radial dependence modeling of the magnetic field. The obtained quasi-3-D solution allows rapid parametric studies of the air-gap magnetic field. Then, analytical modeling of the cogging torque is presented. It is based on the obtained quasi-3-D analytical solution. Results issued from the proposed model in the air gap are compared with those stemming from a 3-D finite-element method simulation as well as with prototype measured values.

143 citations


Additional excerpts

  • ...Usually, the problem is solved under a two-dimensional (2-D) approximation which, in some situation like in axial field couplers, results in a 30% overestimation of the torque compared to three-dimensional (3-D) FE prediction [10]–[12]....

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01 Jan 1953

121 citations


"A New Analytical Torque Formula for..." refers methods in this paper

  • ...For simplicity, let us consider, in free space, an electrostatic uniform surface charge density σs (C/m2), subjected to an electric field E [16]....

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  • ...From the magnetostatic point of view, it is usual to use an equivalent magnetic surface charge σm in A/m [7], [11], [14]– [16]....

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Journal ArticleDOI
Abstract: In this paper, a theoretical analysis of an axial magnetic coupling is presented, leading to new closed-form expressions for the magnetic axial force and torque. These expressions are obtained by using a 2-D approximation of the magnetic coupling geometry (mean radius model). The analytical method is based on the solution of Laplace's and Poisson's equations by the separation of variables method. The influence of geometrical parameters such as number of pole pairs and air-gap length is studied. Magnetic field distribution, axial force, and torque computed with the proposed 2-D analytical model are compared with those obtained from 3-D finite elements simulations and experimental results.

84 citations


"A New Analytical Torque Formula for..." refers background or methods in this paper

  • ...Axial flux magnetic coupling prototype [11], [12]....

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  • ...An analytical formula which is derived in [11] using a 2-D analytical model (mean radius model and first harmonic approximation) is given by (21)....

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  • ...7) (more details can be found in [11] and [12])....

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  • ...In [11] and [12], the authors have constructed a prototype axial field coupling and developed a 2-D formula to evaluate the torque....

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  • ...From the magnetostatic point of view, it is usual to use an equivalent magnetic surface charge σm in A/m [7], [11], [14]– [16]....

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Journal ArticleDOI
13 Apr 1993
Abstract: An analytical method adapted to calculating the forces between permanent magnets is developed, and applied to the torque calculation of synchronous couplings. The method is based on the forces exerted between two elementary barshaped magnets. The curvature effect is taken into account by a corrective coefficient, and the yokes by magnetic images. The method is relatively simple and gives accurate results which are very useful for optimization of the coupling shape. >

77 citations


"A New Analytical Torque Formula for..." refers background in this paper

  • ...3-D analytical models for magnetic couplings have been proposed in the literature [6], [13] and [14]....

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Journal ArticleDOI
Abstract: A formula is presented for computing the field in axial-field permanent-magnet motors. The formula is based on a three-dimensional (3-D) analytical analysis and is expressed in terms of a finite sum of elementary functions. It is readily programmed and ideal for parametric studies of field strength. It is also well suited for parallel processing and could be developed into a motor model for real-time performance simulations. It is applied here to a practical motor geometry and verified by the use of 3-D finite element analysis (FEA).

66 citations


"A New Analytical Torque Formula for..." refers background in this paper

  • ...3-D analytical models for magnetic couplings have been proposed in the literature [6], [13] and [14]....

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Frequently Asked Questions (1)
Q1. What are the contributions in "A new analytical torque formula for axial field permanent magnets coupling" ?

In this paper, a new analytical torque formula for axial field permanent magnets coupling has been derived using 3D magnetostatic analytical.