HAL Id: hal-01062497
https://hal.archives-ouvertes.fr/hal-01062497
Submitted on 9 Sep 2014
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entic research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diusion de documents
scientiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
Steady-State and Transient Performance of Axial-Field
Eddy-Current Coupling
Thierry Lubin, Abderrezak Rezzoug
To cite this version:
Thierry Lubin, Abderrezak Rezzoug. Steady-State and Transient Performance of Axial-Field Eddy-
Current Coupling. IEEE Transactions on Industrial Electronics, Institute of Electrical and Electronics
Engineers, 2015, 62 (4), pp.2287-2296. �10.1109/TIE.2014.2351785�. �hal-01062497�
Abstract –This paper presents an approach for quick
calculation of steady-state and transient performances of an
axial-field eddy-current coupling. Based on a two-dimensional
(2D) approximation of the magnetic field distribution, a simple
analytical expression for the transmitted torque is first
developed. This expression is valid for low slip values which
correspond to the normal working area of such devices (high
efficiency). The proposed torque formula is then used to study
the steady-state (constant speed operations) and the transient
performances of the coupling (small variations of the slip
speed). The results are compared with those obtained from 3-D
finite elements simulations and tests. It is shown that good
agreements are obtained.
Index Terms-- Analytical model, magnetic transmission,
eddy-current, torque, transient performance.
NOMENCLATURE
R
1
Inner radius of the magnets
R
2
Outer radius of the magnets
R
m
Mean radius of the magnets (R
m
=(R
1
+R
2
)/2)
R
0
Inner radius of the copper
R
3
Outer radius of the copper
L Radial length of the magnets (L=R
2
-R
1
)
H Radial length of the copper (H=R
3
-R
0
)
Copper end-lengths to magnets length
= (H-L)/L
a Thickness of the back-iron (magnets side)
b Magnets thickness
c Air-gap thickness
d Copper thickness
e Thickness of the back-iron (copper side)
α PMs pole-arc to pole-pitch ratio
p Pole-pairs number
B
r
Remanence of the permanent magnets
Conductivity of the conducting plate (copper)
= p/R
m
I. INTRODUCTION
AGNETIC couplings can transmit a torque from a
primary driver to a follower without mechanical
contact. Therefore, they are well suited for use in isolated
systems such as vacuums, high-pressure or cryogenic
vessels. Compared to mechanical couplings, they present
great advantages such as self protection against overload
condition and great tolerance to shaft misalignment [1]-[4].
There are two main types of magnetic coupling:
synchronous [1]-[13] and induction type (eddy-current)
[14]-[30], both with radial or axial flux configuration. In
this paper, we focus on single-sided axial field eddy-current
coupling as shown in Fig. 1. It consists of two disks: one is
composed of rare earth permanent magnets (axially
magnetized), and the other one is equipped with a
conducting plate (generally copper). To improve the
performances, the flux is closed by two back iron yokes.
The working principle of eddy-current couplings is well
known [14]. The eddy currents, which are induced in the
conducting plate by the slip speed (
=
1
-
2
), interact with
the magnetic field and generate a torque. The normal
working range area of such devices corresponds to low slip
values where the temperature rise, due to induced- current,
is limited. In this condition, the torque is proportional to the
slip [21]. As shown in [14], the efficiency of eddy-current
couplings is given by
=1-s where s is the slip (s =
/
1
).
Fig. 2 shows the prototype of the studied eddy-current
coupling placed on the test bench. The physical and
geometrical parameters are given in Table I. These
parameters have been optimized but the optimization
procedure is not developed in this paper. Design
optimization of eddy-current couplings can be found in
[22], [29] and [30]. With these parameters and considering
c = 3 mm for the air-gap, we obtain a torque of about 10
Nm for a slip speed of 150 rpm (11). A slip speed of 150
rpm corresponds to an efficiency of 95% (
=1-s) when the
eddy-current coupling is used with a motor running at a
nominal speed of 3000 rpm [14].
Fig. 1. Geometry of the studied axial-field eddy-current coupling (p = 5)
with its geometrical parameters.
Steady-State and Transient Performance of
Axial-Field Eddy-Current Coupling
Thierry Lubin, and Abderrezak Rezzoug
Université de Lorraine, Groupe de Recherche en Electrotechnique et Electronique de Nancy, GREEN, F-54500
Vandœuvre-lès-Nancy, France
M
Fig. 2. Eddy-current coupling prototype (air-gap c=3mm).
TABLE I
PARAMETERS OF THE STUDIED EDDY-CURRENT COUPLING
Symbol
Quantity
value
R
1
Inner radius of the magnets
30 mm
R
2
R
0
R
3
a
b
c
d
e
α
Outer radius of the magnets
Inner radius of the conducting plate
Outer radius of the conducting plate
Thickness of the back-iron (magnets side)
Magnets thickness
Air-gap length
Conducting plate thickness
Thickness of the back-iron (copper side)
PMs pole-arc to pole-pitch ratio
60 mm
15 mm
75 mm
10 mm
10 mm
variable
5 mm
8 mm
0.9
p
Pole-pairs number
5
B
r
Remanence of the permanent magnets (NdFeB)
1.25 T
Conductivity of the conducting plate (copper)
57 MS/m
Eddy-current couplings have been studied for a long time
and many papers can be found in the literature [14]-[23].
The study of such devices is usually based on a two-
dimensional approximation for the magnetic field
distribution. The induced currents in the conducting plate
are computed by solving a 2D diffusion equation leading to
a complex expression for the transmitted torque [16]-[18].
More recently [20], [21], the magnetic equivalent circuit
(MEC) method has been used to predict the performance of
eddy-current couplings by taking into account the magnetic
saturation effects and all material properties. With the MEC
method, it is possible to study eddy-current couplings with
complex geometries [23]. In such cases, analytical models
have significant issues. The main drawback of the MEC
method is that the flux paths must be a priori known in
order to define the reluctance expressions which appear in
the torque expression [20], [21], and [23].
For an engineering purpose, it is important to have a
simple and accurate torque formula in order to quickly
study the effects of geometrical parameters on the coupling
performances. As indicated previously, the working area of
eddy-current couplings corresponds to low slip values.
Therefore, analysis of such device can be limited to low slip
values, which simplify greatly the analytical development.
The goal of this paper is to develop the simplest model
which gives acceptable results in the design area (low slips).
To achieve this outcome, we have to do some assumptions
that will be justified in the next developments. The 3D
effects (actual paths of the induced currents) are taken into
account by using the Russell correction factor [31]. The
proposed torque formula, which depends directly on the
physical and geometrical parameters, is then used for the
analysis of both steady-state and transient operations. It
should be noted that transient performance analysis of eddy-
current couplings is rarely treated in the literature [24]-[26],
and very few experimental results are given [27], [28].
II. TORQUE EXPRESSION FOR LOW SLIP VALUES
Axial flux eddy-current coupling is an inherent 3D
geometry from the modeling point of view. However, in
order to simplify the analysis, the 3D problem of Fig. 1 can
be reduced to a 2D problem by introducing a cutting plane
at the average radius of the magnets R
m
=(R
1
+R
2
)/2. Fig. 3
shows the resulting 2D model in Cartesian coordinate (y, z)
with an infinite length along the x-direction. The 3D effects
(actual path of the induced currents) cannot be overlooked
for such device and will be taken into account by using the
classical Russell correction factor [31]. Moreover, for
reasons of simplicity, we adopt the following assumptions:
1) The reaction field due to induced currents is neglected
because of the low slips assumption,
2) The iron-yokes are considered with infinite magnetic
permeability and zero conductivity,
3) Only the first harmonic is considered for the magnetic
field distribution,
4) Magnets are axially magnetized with relative recoil
permeability
1
r
.
As the iron-yokes present an infinite permeability, the
tangential component of the magnetic field is zero at z=0
and z=b+c+d (boundary conditions). This limits the number
of region where the magnetic field has to be computed. This
hypothesis can be regarded as accurate because the
thicknesses a and e of the back-iron (see Fig. 3) must be
determined to avoid magnetic saturation.
As it can be seen in Fig. 3, the whole domain of the field
problem is divided into two regions: the PMs region (region
I) and the air-gap with copper region (region II). As the
reaction field is neglected, the air-gap and copper can be
connected because we have the same governing equation for
this region, i.e. Laplace’s equation.
A magnetic vector potential formulation (A
i
=A
i
(y,z)e
x
) is
used to solve the problem. From the Maxwell equations and
considering the adopted assumptions, we have to solve the
following equations in each region
22
0
22
22
22
in Regions I (PMs)
0 in Region II (air -gap and copper)
I I z
II II
A A M
y
yz
AA
yz
(1)
Fig. 3. 2D model of the eddy-current coupling at the mean radius of the
magnets R
m
=(R
1
+R
2
)/2.
where
0
is the vacuum permeability and M
z
is the axial
magnetization of the magnets. The distribution of the axial
magnetization M
z
(y) is plotted in Fig. 4. The axial
magnetization can be expressed in Fourier’s series. It has
been shown in [18] that the first harmonic component plays
a dominant role in the torque transmission for eddy-current
couplings (around 98% of the total torque). So, in the
following development, we will consider only the first
harmonic of the magnetization.
0
4
( ) sin sin
2
r
z
B
M y y
with
m
pR
(2)
According to the symmetry of the magnetic field
distribution and considering the classical boundary and
interface conditions for such problem, solutions of (1) is
sinh ( ) cosh
( , ) 1 cos
sinh
I
c d z
A y z K y
b c d
(3)
sinh cosh ( )
( , ) cos
sinh
II
b z b c d
A y z K y
b c d
(4)
with
4
sin 2
rm
BR
K
p
(5)
The induced current density in the conducting plate can
be deduced from Lorentz’s equation
II
II m
A
R
y
x
J v B e
(6)
where
=
1
-
2
is the slip speed (rd/s) between the two
discs. The transmitted torque can be evaluate by the power
losses dissipated in the conducting plate
2
2
0
( , )
m
R
b c d
bc
PL
T J y z dydz
(7)
From (4), (6) and (7), we can derive a closed-form
expression for the electromagnetic torque which depends
directly on the physical and geometrical parameters
12TT
T K K
(8)
where K
T
is given by
Fig. 4. Magnetization distribution along y-direction (region I)
2 3 2
2
2
8
sin
2
sinh ( ) sinh(2 )
1
2
sinh ( ( ))
T r m
K B R Ld
bd
d
b c d
(9)
As expected for low slip values, the torque is
proportional to the slip speed
(rd/s). The expression has
been arranged in order to isolate its dimensional part.
It has been shown in [18], [20] and [22] that the 3D end-
effects cannot be neglected for such devices. Fig. 5 shows
the actual paths of the induced currents in the conducting
plate (3D effects).The induced currents flow along a closed
path and not only in the x-direction as it was obtained with
the 2D model (6). To take into account the end-effects, an
efficient correction factor (10) has been given by Russel and
Norsworthy [31]
(2 / )tanh( / 2)
1
1 tanh( / 2)tanh( / 2)
Russel
LL
k
LL
(10)
The torque expression with the 3D effects can be finally
expressed as
'
T
TK
with
'
T Russel T
K k K
(11)
where K’
T
is the torque coefficient. Fig. 6 shows the
variation of K’
T
versus the air-gap length (the other
parameters are those given in table I). As expected, K’
T
decreases with the air-gap length.
Fig. 5. Actual paths for the induced-currents (3D effects)
Fig. 6. Torque coefficient K
T
’ (Nm.s/rad) versus air-gap length for the
studied magnetic coupling (Table I).
III. STEADY-STATE ANALYSIS
In this section, we use the torque formula (11) to study
the steady-state performances of the coupling (constant
speed operation). In order to show the effectiveness of the
proposed model, the results obtained with the analytical
formula (11) are compared with those obtained from 3D
nonlinear finite element analysis (FEA). The finite element
simulations (COMSOL Multiphysics®) are conducted by
considering the reaction field due to the induced currents
and the real geometry of the coupling (i.e. cylindrical
structure as shown in Fig. 1). Moreover, the nonlinear
magnetic property for the back-irons (B-H curve) has been
considered in the finite-element simulations and is shown in
Fig. 7.
Fig. 7. B(H) curve of the back-irons ( M-27 steel) used in 3D FEA
A. Torque-Speed Characteristic
The torque-speed characteristic of the studied coupling is
shown in Fig. 8. As indicated previously, only the low slip
values are considered here (high efficiency area). The
geometrical parameters are the ones given in Table I with
c=3mm. Fig. 8 shows that the torque is well predicted by
the analytical formula if we consider the Russel correction
factor. In this case, the deviations between the proposed
formula and the 3D nonlinear FEA is always below 10%.
Fig. 8. Torque-speed characteristic for the studied coupling (with
c=3mm): comparison between analytical results and 3D FEA.
The flux density (z-component) and the eddy-current
distributions at the surface of the copper obtained with 3D
FEA are respectively shown in Fig. 9a and Fig. 9b. These
results correspond to a slip speed of 150 rpm. The
maximum value of the flux density on the surface of the
copper is around 0.55T (with alternate polarity).
(a)
(b)
Fig. 9. 3D FEM analysis: (a) flux density distribution (z-component) on
the surface of the copper, (b) eddy-current distribution at 150 rpm.
B. Influence of Geometrical Parameters
The torque formula is now used to study the influence of
several geometrical parameters (copper thickness, pole-
pairs number, and the radial length of the copper). Fig. 10
shows the torque versus copper thickness. We have fixed
the air-gap length c=3mm and the slip speed at 150 rpm (the
other parameters are those given in Table I). We can
observe that an optimal value exists for the copper thickness
(T ≈ 10 Nm for d ≈ 5mm). This optimal value is well
predicted by the analytical formula (11) when compared to
3D FEA.
The torque versus the pole-pairs number is shown in Fig.
11. For the analysis, we have considered c=3mm, the other
geometrical parameters are kept constant and are given in
Table I. Once again, we can observe that an optimum value
exists for the pole-pairs number (p≃4). This optimum value
is well predicted by using the torque formula. The error is
never greater than 15% compared to 3D FEA.
As indicated in the nomenclature,
is the ratio of the
radial copper end-lengths to the radial magnet length (active
length):
=0 corresponds to the same radial dimensions for
the copper and the magnets (H=L) whereas
=1
corresponds to H=2L. Fig. 12 shows the torque versus the
ratio
obtained with the analytical formula and with 3D
FEA. We can observe that the torque increases with the
copper end-lengths and then becomes constant for
≥ 1 (an
