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Simple Analytical Expressions for the Force and Torque
of Axial Magnetic Couplings
Thierry Lubin, Mezani Smail, Abderrezak Rezzoug
To cite this version:
Thierry Lubin, Mezani Smail, Abderrezak Rezzoug. Simple Analytical Expressions for the Force and
Torque of Axial Magnetic Couplings. IEEE Transactions on Energy Conversion, Institute of Electrical
and Electronics Engineers, 2012, 11 p. �10.1109/TEC.2012.2183372�. �hal-00673920�
1
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 two-dimensional (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.
Index Terms— Torque transmission, axial magnetic coupling,
analytical model, axial force.
I. I
NTRODUCTION
AGNETIC
couplings are of great interest in many
industrial applications. They can transmit a torque from
a primary driver to a follower without mechanical contact. As
the torque could be transmitted across a separation wall, axial
field magnetic couplings are well suited for use in isolated
systems such as vacuum or high pressure vessels. Moreover,
they present a maximum transmissible torque (pull-out torque)
giving an intrinsic overload protection.
Axial magnetic couplings consist of two opposing discs
equipped with rare earth permanent magnets as shown in Fig.
1. The magnets are magnetized in the axial direction. They are
arranged to obtain alternately north and south poles. The flux
is closed by soft-iron yokes. The torque applied to one disc is
transferred through an air-gap to the other disc. The angular
shift between the two discs depends on the transmitted torque
value. The main drawback of axial-type magnetic couplings is
the significant value of the axial attractive force between the
two discs.
An accurate knowledge of the magnetic field distribution is
necessary for predicting the torque and the axial force. The
magnetic field can be evaluated by analytical methods [1-22]
or by numerical techniques like finite elements [23-26].
Finite elements simulations give accurate results considering
three dimensional (3-D) effects and nonlinearity of magnetic
materials. However, this method is computer time consuming
Manuscript received October 31, 2011.
T. Lubin, S. Mezani and A. Rezzoug are with the Groupe de Recherche en
Electrotechnique et Electronique de Nancy (GREEN), Université Henri
Poincaré, 54506 Nancy, France (e-mail: thierry.lubin@green.uhp-nancy.fr).
Fig. 1. Geometry of the studied axial-type magnetic coupling (p = 6)
and poorly flexible for the first step of design stage.
Analytical methods are, in general, less computational time
consuming than numerical ones and can provide closed-form
solutions giving physical insight for designers. So, they are
useful tools for first evaluations of magnetic couplings
performances and for the first step of design optimization.
Three-dimensional analytical models for ironless permanent
magnet couplings have been proposed in the literature [1-16].
The proposed models are developed for axial magnetic
couplings with parallelepiped magnets or cylindrical tile
magnets. As the magnets are in free space (with no other
magnetic materials present), analysis is based either on the
amperian model with Biot-Savart law or on the coulombian
method with equivalent surface charges. Although these
methods give very accurate results, they are not suitable for the
study of magnetic couplings with iron-core structures.
An alternative analytical method to compute the torque for
magnetic couplings with iron yokes is based on boundary
value problems with Fourier analysis. This method consists in
solving directly the Maxwell’s equations in the different
regions (air-gap, magnets....) by the separation of variables
method [17], [18]. The magnetic field distribution is obtained
in each region by using boundary and interface conditions. The
torque and the force are then computed by using the Maxwell
stress tensor. In [19] and [20], two-dimensional (2-D)
analytical models for radial-type magnetic couplings were
developed and closed-form expressions for the torque was
given and used for design optimization. In [21] and [22], quasi
3-D analytical models are proposed to compute the
performances of axial-flux permanent magnets machines. A
Simple Analytical Expressions for the Force and
Torque of Axial Magnetic Couplings
Thierry Lubin, Smail Mezani, and Abderrezak Rezzoug
M
2
modulation function is defined to take into account the radial
dependence of the magnetic field.
In this paper, we propose new formulas for the torque and
the axial force of an axial-type magnetic coupling with iron
yokes (fig. 1). The analytical study is based on the solution of
2-D Laplace’s and Poisson’s equations in air-gap and
permanent magnets regions by using the separation of
variables method. The torque expression is used to study the
influence of geometrical parameters (number of pole pairs and
air-gap length). In order to study the accuracy of the proposed
formulas, the results are compared with those obtained from 3-
D finite elements simulations and experimental results.
II. P
ROBLEM DESCRIPTION AND ASSUMPTIONS
As shown in Fig. 1, the geometrical parameters of the
studied magnetic coupling are the inner and outer radii of the
magnets R
1
and R
2
, the air gap length e, and the magnets
thickness h. The pole-arc to pole-pitch ratio of the permanent
magnets is α. The number of pole-pairs is p.
Analytical study of axial magnetic couplings is complicated
because of the three-dimensional nature of the magnetic field
distribution. However, in order to simplify the analysis and to
carry out closed-form expressions for the axial force and
torque, the 3-D problem is reduced to a 2-D one by
introducing a cylindrical cutting surface at the mean radius of
the magnets R
e
=(R
1
+R
2
)/2 at which the magnetic field will be
computed [21], [22].
Fig. 2 shows the resulting 2-D model by considering the
unrolled cylindrical cutting surface. With this approach, we
neglect the radial component of the magnetic field and we
consider that the axial and tangential components do not
depend on the r-coordinate. Moreover, for simplicity, we
adopt the following assumptions:
1) The iron yokes have infinite magnetic permeability,
2) The magnets are axially magnetized with relative recoil
permeability
1=
r
µ
.
As shown in Fig.2, the whole domain is divided into three
regions: the PMs regions (regions I and III) and the air-gap
region (region II). The magnets of region III are shifted by an
angle δ (torque angle) from the magnets of region I. Due to the
periodicity of the magnetic field distribution, the studied
domain is limited by 0 ≤ θ ≤ 2π/p.
A magnetic vector potential formulation is used in 2D
cylindrical coordinates to describe the problem. According to
the adopted assumptions, the magnetic vector potential in each
region has only one component along the r-direction and only
depends on the θ and z-coordinates. The electromagnetic
equations in each region expressed in term of the magnetic
vector potential are
2
0
2
in Regions I and III (PMs)
0 in Region II (air-gap)
µ
∇ = − ∇×
∇ =
A M
A
(1)
Fig. 2. 2-D model of the axial magnetic coupling at the mean radius of the
magnets R
e
=(R
1
+R
2
)/2.
with
0
r
z
B
M
µ
= = ±
z z
M e e
(2)
where M is the magnetization vector, B
r
the remanence of the
magnets, e
z
the unit vector along the axial direction and ±
indicates the magnetization direction.
III. 2-D
A
NALYTICAL MODEL
By using the separation of variables method, we now
consider the solution of Poisson’s equations for PMs regions
and Laplace’s equation for the air-gap region.
A. Solution of Poisson’s Equation in the PMs Regions
(Regions I and III)
Poisson’s equation in the magnets region (region III) can be
written in a cylindrical coordinates system as
2 2
0
2 2 2
1
III III z
e
e
A A M
R
R z
µ
θ
θ
∂ ∂ ∂
+ = −
∂
∂ ∂
for
2
0 2 /
h e z h e
p
θ π
+ ≤ ≤ +
≤ ≤
(3)
where
µ
0
is the permeability of the vacuum and M
z
is the axial
magnetization of the magnets.
Knowing that the tangential component of magnetic field at
2
z h e
= +
is null (soft-iron yoke with infinite permeability)
and considering the continuity of the axial component of the
flux density at
z h e
= +
, we obtain the following boundary
conditions
2
0
III
z h e
A
z
= +
∂
=
∂
(4)
( , ) ( , )
III II
A h e A h e
θ θ
+ = +
(5)
where
( , )
II
A z
θ
is the magnetic vector potential in the air-gap
region.
The distribution of the axial magnetization M
z
is plotted in
Fig.3, δ is the relative angular position between the magnets of
region I and region III. The axial magnetization can be
expressed in Fourier’s series and replaced in (3)
3
Fig. 3. Magnetization distribution along
θ
-direction (region III).
( )
( )
1
( ) sin
z k
k
M M kp
θ θ δ
∞
=
= −
∑
(6)
( )
0
4
cos 1 with 1,3,5,7....
2
r
k
B
M k k
k
π
α
πµ
= − =
(7)
Taking into account the boundary conditions (4) and (5), the
general solution of the magnetic vector potential in Region III
can be written as
( )
( )
( )
( )
1
1
( , )
2
( cos( ))cos
2
( sin( ))sin
III
e
III
k k
k
e
e
III
k k
k
e
A z
kp
ch z h e
R
a K kp kp
kp
ch h
R
kp
ch z h e
R
c K kp kp
kp
ch h
R
θ
δ θ
δ θ
∞
=
∞
=
=
− −
+
− −
+ +
∑
∑
(8)
with
0
2
e
k k
R
K M
kp
µ
=
(9)
The integration constants
III
k
a
and
III
k
c
are determined
using a Fourier series expansion of
( , )
II
A h e
θ
+
over the
interval [0, 2π/p]
2 /
0
2
cos( ) ( , )cos( )
2
p
III
k k II
p
a K kp A h e kp d
π
δ θ θ θ
π
+ = +
∫
(10)
2 /
0
2
sin( ) ( , )sin( )
2
p
III
k k II
p
c K kp A h e kp d
π
δ θ θ θ
π
+ = +
∫
(11)
The expressions of the coefficients
III
k
a
and
III
k
c
are given
in the appendix.
One can apply the same procedure for region
I
by
considering a zero value for δ. This leads to the following
expression for the magnetic vector potential
( )
( )
1
1
( , ) ( )cos
sin
e
I
I k k
k
e
e
I
k
k
e
kp
ch z
R
A z a K kp
kp
ch h
R
kp
ch z
R
c kp
kp
ch h
R
θ θ
θ
∞
=
∞
=
= +
+
∑
∑
(12)
The integration constants
I
k
a
and
I
k
c
in (12) are determined
using a Fourier series expansion of
( , )
II
A h
θ
over the interval
[0, 2π/
p
]
2 /
0
2
( , )cos( )
2
p
I
k k II
p
a K A h kp d
π
θ θ θ
π
+ =
∫
(13)
2 /
0
2
( , )sin( )
2
p
I
k II
p
c A h kp d
π
θ θ θ
π
=
∫
(14)
The expressions of the coefficients
I
k
a
and
I
k
c
are given in
the appendix.
B. Solution of Laplace’s Equation in the Air-Gap Region
(Region II)
Laplace’s equation in the air-gap region can be written in a
cylindrical coordinates system as
2 2
2 2 2
1
0
II II
e
A A
R z
θ
∂ ∂
+ =
∂ ∂
for
0 2 /
h z h e
p
θ π
≤ ≤ +
≤ ≤
(15)
The continuity of the tangential component of the magnetic
field at
z h
=
and at
z h e
= +
leads to the following
boundary conditions
II I
z h z h
A A
z z
= =
∂ ∂
=
∂ ∂
and
II III
z h e z h e
A A
z z
= + = +
∂ ∂
=
∂ ∂
(16)
By taking into account the boundary conditions (16), the
general solution of the magnetic vector potential in the air-gap
can be written as
( ) ( )
( )
( ) ( )
( )
1
1
( , )
( )cos
( )sin
II
e e
II II
e e
k k
k
e e
e e
II II
e e
k k
k
e e
A z
kp kp
ch z h e ch z h
R R
R R
a b kp
kp kp
kp kp
sh e sh e
R R
kp kp
ch z h e ch z h
R R
R R
c d kp
kp kp
kp kp
sh e sh e
R R
θ
θ
θ
∞
=
∞
=
=
− − −
− +
− − −
+ − +
∑
∑
(17)
4
The integration constants
II
k
a
,
II
k
b
,
II
k
c
and
II
k
d
are
determined using Fourier series expansions of
I
h
A z
∂ ∂
and
III
h e
A z
+
∂ ∂
over the air-gap interval [0, 2π/
p
]
2 /
0
2
cos( )
2
p
II
I
k
h
A
p
a kp d
z
π
θ θ
π
∂
=
∂
∫
(18)
2 /
0
2
cos( )
2
p
II
III
k
h e
A
p
b kp d
z
π
θ θ
π
+
∂
=
∂
∫
(19)
2 /
0
2
sin( )
2
p
II
I
k
h
A
p
c kp d
z
π
θ θ
π
∂
=
∂
∫
(20)
2 /
0
2
sin( )
2
p
II
III
k
h e
A
p
d kp d
z
π
θ θ
π
+
∂
=
∂
∫
(21)
The expressions of these coefficients are developed in the
appendix.
The axial and tangential components of the magnetic flux
density in the air-gap can be deduced from the magnetic vector
potential by
1
II
IIz
e
A
B
R
θ
∂
= −
∂
II
II
A
B
z
θ
∂
=
∂
(22)
IV. A
XIAL
-
FORCE AND TORQUE EXPRESSIONS
A. Electromagnetic torque
The electromagnetic torque is obtained using the Maxwell
stress tensor. A line at
[
]
,
z h h e
ζ
= ∈ +
in the air-gap region
is taken as the integration path so the electromagnetic torque is
expressed as follows
2
3 3
2 1
0
0
( , ) ( , )
3
e II IIz
R R
T B B d
π
θ
θ ζ θ ζ θ
µ
−
=
∫
(23)
Incorporating (22) into (23), the analytical expression for
the electromagnetic torque becomes
(
)
3 3
2 1
0
1
( )
3
e k k k k
k
R R
T W X Y Z
π
µ
∞
=
−
= +
∑
(24)
where the coefficients
W
k
,
X
k
,
Y
k
and
Z
k
are given in the
appendix.
The torque can be computed with a good precision by
considering only the fundamental components of the the flux
density distribution in the air-gap (
k = 1
). This is especially
true for large number of PM pole-pairs and/or large air-gap.
Considering the first harmonic approximation, we can derive a
closed-form expression for the electromagnetic torque which
depends directly on the geometrical parameters.
( )
( )
( )
3
2
2
3 2
1
2
0
2
16
1 sin
3 2
2 1
r
e
B R
sh a
T R sin p
sh a
R
π
α δ
π µ
ν
= −
+
(25)
with
e
h
a p
R
= and
2
e
h
ν
=
(26)
As expected, the torque presents a sinusoidal characteristic
with the relative angular position
δ. Its maximum value (pull-
out torque) is obtained at the angle δ=π/2p.
B. Axial-Force
Axial magnetic force is an important parameter for the
design of an axial magnetic coupling. This attractive force
must be known because it affects directly the rotor structure
and bearings. Indeed, the bearing lifetime depends on the
bearing load. By using the Maxwell stress tensor, the axial
force expression is
( )
2
2 2
2 2
2 1
0
0
( , ) ( , )
4
IIz II
R R
F B B d
π
θ
θ ζ θ ζ θ
µ
−
= −
∫
(27)
Substituting (22) into (27), the analytical expression for the
axial force becomes
(
)
( ) ( )
( )
2 2
2 1
2 2
0
1
4
k k k k
k
R R
F Z X W Y
π
µ
∞
=
−
= + − +
∑
(28)
Considering only the fundamental component of the
magnetic field in the air-gap (
k
= 1), we can derive a closed-
form expression for the axial force
( )
( )
( ) ( )
( )
2
2
2
2 2
1
2
2
0
2
8
1 sin
2
2(1 )
cos 2(1 ) 1
r
sh a
B R
F R
R sh a
p ch a
π
α
π µ
ν
δ ν
= −
+
× + +
(29)
From (25) and (29), we can see that the torque and the axial
force dependence on the design parameters are explicit. For
engineering purpose, it is important to have simple relations to
study rapidly the effects of the geometrical parameters on the
coupling performances. This is developed in the following
subsection.