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Instituto Superior Técnico, Lisboa – Portugal 1
Characterization and Optimization of a Spherical Induction Machine for Motor Applications
Rui Jorge Viegas Calçada
November 2016
Abstract – Multiple degrees of freedom spherical rotor
electric machines have some advantages over classical
machines for some applications. This dissertation studies
the characteristics of a spherical induction machine and
attempts to optimize it through analytical, numerical and
experimental analysis. Special focus is given to the
machines’ copper windings, which represent a significant
portion of the equivalent air-gap thickness, having a direct
influence on the magnetic circuit. An analytical model is
presented and validated with Finite Element Analysis
(FEA) simulations. Planar and spherical single-layer
copper stators geometries are studied and their
electromagnetic characteristics compared to the double-
layer stator geometry. Planar and spherical prototypes are
analysed. The acquired data validated the models
developed and showed significant potential for the
improvement of the machines’ electromagnetic and
thermal characteristics for a single-layer configuration
versus a double-layer one.
Index Terms – Copper windings, double-layer, equivalent
air-gap, FEA, single-layer, spherical induction machine.
I. INTRODUCTION
Electrical machines with multiple degrees of freedom are a
target of research for many years. More specifically,
synchronous and magnetic reluctance machines [1,2] have
showed that significant torque and power can be produced at
the expense of very complex rotor design and resorting to
permanent magnets. The spherical induction machine [3,4]
allows for a very simple rotor construction, with low inertia and
without the use of permanent magnets. It is the aim of this
document to provide a study of the characteristics of this
machine and to optimize its design for torque production and
efficiency. A great deal of attention is given to the stator
configuration. The bulky double-layer of copper windings
necessary to ensure multiple degrees of freedom make the
machines’ magnetic circuit inefficient through its influence on
the equivalent air-gap thickness. Single-layer configurations
are studied to find if they provide advantages over double-layer
geometries.
Electromagnetic and thermal analytical models are
developed. These models allow us to have a simple and fast
numerical program to study the sensitivities of the machines’
characteristics to its parameters. Special focus is on torque
production, although other electromagnetic quantities, such as
the induced current density in the rotor and the radial magnetic
flux density are analysed.
FEA simulations are performed in order to study different
copper windings geometries and to validate the analytical
models. Due to the computational power and the long
processing times required by these simulations, planar
geometry simulations are studied before spherical geometry
ones. Slotting is also approached using FEA simulations.
In order to validate the aforementioned work, a planar and a
spherical prototype are studied. Constructive details are
approached and the results are compared to the analytical and
FEA simulations results.
II. ANALYTICAL MODELS
a) Electromagnetic model
The electromagnetic analytical model is based on Fig. 1.
Some assumptions are made to allow for the mathematical
solution:
A homogeneous zone of constant thickness and low
magnetic permeability (air-gap);
An infinitesimal thickness current density in the
internal stator surface (copper conductive layer);
An infinitesimal thickness conductive layer in the
external rotor surface (aluminium conductive layer);
Stator and rotor made of high magnetic permeability
material;
Air-gap thickness much smaller than stator’s length
(no border effects).
Fig. 1 - Analytical model schematic.
A magnetomotive traveling wave originates as a result of the
stator current density (Eq. 1). This also induces a current
density in the rotor (Eq. 2).
Js⃗⃗ = Re{Js ej(ωt−kφ)}uθ⃗⃗⃗⃗ (1)
Jr⃗⃗ = Re{Jr̅ej(ωt−kφ)}uθ⃗⃗⃗⃗ (2)
where ω is the copper currents angular frequency and k is the
spatial wavelength in the uφ⃗⃗⃗⃗ ⃗ direction (equivalent to twice the
number of the machines’ pole pairs).
Being of infinitesimal thickness, the stator equivalent current
density and the rotor equivalent electrical conductivity are given
by Equations 3 and 4.
Jseq = JsδCu (3)
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σAleq = σAl ∙ δAl (4)
where δCu and δAl are the real stator and rotor thicknesses. The
magnetic vector potential will have a radial component and a
spatial/time dependent component, similar to the
magnetomotive force source. Using cylindrical coordinates its
solution can be calculated, as in Equations 5 and 6 [5].
∇2Az⃗⃗⃗⃗ =
∂2Az⃗⃗⃗⃗
∂r2+
1
r2∂2Az⃗⃗⃗⃗
∂φ2+∂2Az⃗⃗⃗⃗
∂θ2+1
r
∂Az⃗⃗⃗⃗
∂r= 0 (5)
A(r) = C1rk + C2r
−k (6)
C1 and C2 coefficients can be calculated by analysing the
boundary conditions shown in Fig. 2 (Equations 7, 8 and 9).
Fig. 2 - Boundary conditions.
{
Stator: ∫ Hdl = ∫ Jseqn⃗ dS
S1
d
c
Rotor:∫ Hdl = ∫ Jreqn⃗ dSS2
f
e
(7)
C1 =Jseqμ0rs
k+1
k
1 + jSμ0σAleq
krr
rs2k − rr
2k + jSμ0σAleq
krr(rs
2k + rr2k)
(8)
C2 =Jseqμ0rs
k+1rr2k
k
1 − jSμ0σAleq
krr
rs2k − rr
2k + jSμ0σAleq
krr(rs
2k + rr2k)
(9)
where μ0 is the vacuum magnetic permeability, rs is the stator
internal radius, rr is the external rotor radius and S = ω − kωr
is the machines’ slip parameter.
The magnetic flux density can be calculated by Eq. 10 and its
radial component in the rotor surface by Eq. 11. The rotor
induced current density is given by Eq. 12, the force density is
given by Eq. 13 and the machines’ electromagnetic torque by
Eq. 14.
B⃗⃗ = ∇ × A⃗⃗ (10)
Br⃗⃗⃗⃗ = −j
k
r(C1r
k + C2r−k)ej(ωt−kφ)ur⃗⃗ ⃗ (11)
Jreq⃗⃗ ⃗⃗ ⃗⃗ = σAleq[E⃗⃗
+ v⃗ × B⃗⃗ ] ⇔
⇔ Jreq⃗⃗ ⃗⃗ ⃗⃗ = −jσAleqS(C1r
k + C2r−k)ej(ωt−kφ)uθ⃗⃗⃗⃗
(12)
f = Jreq⃗⃗ ⃗⃗ ⃗⃗ × B⃗⃗ = fr⃗⃗ + fφ⃗⃗ ⃗ (13)
Tφ = rr∫ < fφ⃗⃗ ⃗ > dS
SAl
=< fφ⃗⃗ ⃗ > 2πrr3 (14)
An equivalent air-gap is considered including all the low
magnetic permeability zones (Eq. 15).
δair−gapeq = δAl + δair−gap + 2δCu (15)
For the machines’ parameters in Table 1, Fig. 3 shows the
torque sensitivity to changes in frequency f, pole pairs k, and
aluminium thickness δAl.
Table 1 - Electromagnetic analytical model machines' parameters.
Rotor radius rr 50 [mm]
Stator current density Js 3x106 [A/m2]
Frequency f 50 [Hz]
Number of pole pairs k 2
Aluminium thickness δAl 2 [mm]
Air-gap thickness δair−gap 1 [mm]
Copper thickness δCu 2 x 5 [mm]
Fig. 3 - Torque sensitivity to frequency, number of pole pairs and aluminum thickness.
We can conclude:
An increase in frequency shifts the maximum torque
point to the left;
Increasing the number k decreases the torque
significantly;
For the same machine dimensions, an increase in the
aluminium thickness increases the maximum torque
produced and shifts the torque curve to the left.
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For some given machines’ dimensions, there will be an
optimum aluminium thickness that maximizes torque. The
starting torque can then be maximized through frequency
control.
b) Thermal model
The thermal model is based on a layer approach in which the
machine is composed of spherical and semi-spherical layers.
A geometry illustration is presented in Fig. 4. In Fig. 5 an
amplified view of the machines’ boundaries is presented, with
the thermal elements evidenced.
Fig. 4 - Thermal model geometry.
Fig. 5 - Amplified view of the stator-rotor boundary region with the thermal model elements evidenced.
Some assumptions were made to allow for the analytical
solution of this model:
It is assumed that the ambient is capable of absorbing
all the heat generated;
The heat follows a radial path towards the
environment;
It is considered that the heat dissipation to the
environment occurs mostly through convective
thermal resistances as opposed to conductive
resistances;
Since we are only interested in the steady-state
temperatures, the thermal capacities are not
considered, becoming open circuits.
An electrical circuit equivalent of the thermal model [6] is
shown in Fig. 6.
Fig. 6 - Electrical circuit equivalent of the thermal model.
The conductive thermal resistances can be calculated
thorough Equations 16-18 [8].
Rcond =
ΔT
P0 (16)
P0 = ∫ q(r)⃗⃗⃗⃗ ⃗⃗ ⃗⃗
S
× n⃗ dS (17)
q(r) = −KdT ⟺ ΔT = ∫ −
q(r)
Kdr
re
ri
(18)
where q(r) is the heat flux density, K is the thermal conductivity
and S, ri and re are the external spherical surface, the internal
radius and the external radius of the considered layer. The
conductive thermal resistances are given by Equations 19-22.
RAl =1
4πKAl(
1
rr − δAl−1
rr) (19)
Rcondair−gap =1
2πKair(1
rr−1
rs) (20)
RCu =1
2πKCu(
1
rs + δCu−
1
rs + 2δCu) (21)
Rstator =1
2πKstator(
1
rs + 2δCu−
1
rs + 2δCu + δs) (22)
Heat transfer through convective thermal resistances occurs
from the rotor and stator to the environment and in the air-gap.
Assuming a laminar flux in the air-gap [7] and an infinite
external radius environment layer, the convective thermal
resistances can be calculated through Equations 23-27 [9,10].
Rconv =
ΔT
Pconv (23)
Pconv = h ∙ A ∙ ΔT (24)
h =
KarNu∗
2ri (25)
𝑃𝐴𝑙
𝑇𝐶𝑢
𝑇𝐴𝑙
𝑅𝐴𝑙 𝑅𝑟𝑜𝑡𝑜𝑟
𝑅𝑟𝑜𝑡𝑜𝑟 −𝑒𝑥𝑡
𝑇𝑎𝑚𝑏
𝐶𝐶𝑢 𝐶𝐶𝑢 𝐶𝐴𝑙 𝐶𝑟𝑜𝑡𝑜𝑟
𝑅𝐶𝑢2
𝑅𝐶𝑢2
𝑅𝐶𝑢2
𝑅𝐶𝑢2
𝑃𝐶𝑢1 𝑃𝐶𝑢2
𝑅𝑐𝑜𝑛𝑣𝑎𝑖𝑟 −𝑔𝑎𝑝
𝑅𝑐𝑜𝑛𝑑𝑎𝑖𝑟−𝑔𝑎𝑝
𝑅𝑠𝑡𝑎𝑡𝑜𝑟 −𝑒𝑥𝑡 𝑅𝑠𝑡𝑎𝑡𝑜𝑟
𝐶𝑠𝑡𝑎𝑡𝑜𝑟
𝑃𝐴𝑙
𝑅𝐶𝑢2
𝑅𝐶𝑢2
𝑅𝐶𝑢2
𝑅𝐶𝑢2
𝑇𝑎𝑚𝑏
𝑅𝑒𝑠𝑡𝑎𝑡𝑜𝑟 −𝑒𝑥𝑡 𝑅𝑒𝑠𝑡𝑎𝑡𝑜𝑟
𝑅𝑟𝑜𝑡𝑜𝑟 −𝑒𝑥𝑡
𝑅𝑐𝑜𝑛𝑣 𝑒𝑛𝑡𝑟𝑒𝑓𝑒𝑟𝑟𝑜
𝑅𝑐𝑜𝑛𝑑 𝑒𝑛𝑡𝑟𝑒𝑓𝑒𝑟𝑟𝑜
𝑅𝐴𝑙
𝑃𝐶𝑢1 𝑃𝐶𝑢2
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Nuair−gap∗ = 2 + 0.14(Ra∗)1/3 (26)
Nuambient
∗ = 2 +0,589(PrGr)
1/4
[1 + (0,469Pr
)
916]
4/9
(27)
where Pconv is the convective heat transfer across a given layer,
h is the convective heat transfer coefficient, A is the surface
orthogonal to the heat transfer and Nu∗ is the Nusselt number.
Ra∗, Pr and Gr are the Rayleigh, Prandtl, and Grashof numbers,
respectively, given by Equations 28-30.
Ra∗ =
gβarΔT
(μarρar
)αar(2ri)
3 (28)
Pr =μarcarKar
(29)
Gr =
g(2ri)3ρar
2ΔTβarμar
2 (30)
where g is the earth’s gravitational acceleration, βar, μar, ρar,
αar and car are the coefficient of thermal expansion, absolute
viscosity, density thermal diffusivity and thermal capacity of air,
respectively. The thermal convective resistances can be
computed by Equations 31-33.
Rconvair−gap =2rr
Kar ∙ Nuair−gap∗ ∙ 2π rr
2 (31)
Rrotor−ext =2rr
Kar ∙ Nuambient∗ ∙ 2π rr
2 (32)
Rstator−ext =2(rs + 2δCu + δs)
KairNuambient∗ ∙ 2π(rs + 2δCu + δs)
2 (33)
The heat losses in the electrically conductive layers of the
machine are given by Equations 34-36.
PCu1 =
1
σCu(Js
√2)2
∙2π
3[(rs + δCu)
3 − rs3] (34)
PCu2 =1
σCu(Js
√2)2 2π
3[(rs + 2δCu)
3 − (rs + δCu)3] (35)
PAl =
1
σAl(Jr
√2)2
∙2π
3[rr
3 − (rr − δAl)3] (36)
c) Double-layer vs single-layer copper windings
The models developed can be combined and adapted to the
single-layer copper geometry. In Fig. 7 and Fig. 8 the windings
distribution and the electromotive force production is clarified.
A two-phase system [A/-A] and [B/-B] is shown, displaced 90º
electrically from each other. To ensure 3 DOF movement, note
that we need two layers of copper windings or, in the case of
the single-layer geometry, it is assumed that this can be
achieved with only one layer of windings.
Fig. 7 - Copper windings distribution per phase for a machine with 4 pole pairs.
Fig. 8 - Electromotive force production for Fig. 7 geometry.
Equations 37-43 allow for the study of the single-layer
geometry.
Tφ = rr∫ < fφ⃗⃗ ⃗ > dS
SAl
=< fφ⃗⃗ ⃗ > πrr3 (37)
δair−gapeq = δAl + δair−gap + δCu (38)
RCu =
1
2πKCu(1
rs−
1
rs + δCu) (39)
Rstator =
1
2πKstator(
1
rs + δCu−
1
rs + δCu + δs) (40)
Rstator−ext =
2(rs + δCu + δs)
KairNuambient∗ 2π(rs + δCu + δs)
2 (41)
PCu =
1
σCu(Js
√2)2
∙π
3[(rs + δCu)
3 − rs3] (42)
PAl =
1
σAl(Jr
√2)2
∙π
3[rr
3 − (rr − δAl)3] (43)
−𝐴
−𝐴
−𝐴
−𝐴 −𝐵
−𝐵
−𝐵
−𝐵
𝑓𝑚𝑚
𝑓𝑚𝑚
𝑓𝑚𝑚
𝑓𝑚𝑚
𝜑
𝜑
𝜑
𝜑
𝜔𝑡 = 0
𝜔𝑡 = 𝜋2
𝜔𝑡 = 3𝜋2
𝜔𝑡 = 2𝜋
𝜑 = 𝜋2 𝜑 = 𝜋
4 𝜑 = 𝜋8 𝜑 = 3𝜋
8 𝜑 = 5𝜋8 𝜑 = 3𝜋
4 𝜑 = 7𝜋8 𝜑 = 𝜋 𝜑 = 0
Instituto Superior Técnico, Lisboa – Portugal 5
The copper insulation is usually the limiting factor in the
machines’ durability and its integrity depends strongly in the
machines’ temperature. Considering a NEMA 180 H insulation
class, a 10 ºC margin is typically given to account for hotspots
and an additional 20 ºC margin is given to ensure the machines’
longevity. Thus, limiting the copper temperature to 150 ºC, the
electromagnetic properties are compared and plotted in Fig. 9,
for the following cases.
Case 0: Double-layer geometry, with 5 mm per layer;
Case 1: Single-layer geometry with a 5 mm layer;
Case 2: Single-layer geometry with a 10 mm layer.
Fig. 9 - Electromagnetic properties for the three cases.
We can observe that single-layer geometry cases 1 and 2
perform better than double-layer geometry, showing a 18,46 %
and 29,74 % increase in starting torque, respectively. Case 1
has the highest efficiency and performs slightly better than
case 2 for lower slip values.
III. Finite Element Analysis
a) Planar geometry FEA simulations
Due to the large amount of computational resources
necessary for the FEA, the machine is first simulated using a
simple planar geometry. In Fig. 10 a schematic of the planar
configuration is shown. In Fig. 11 different copper layer
geometries and their current distributions are showed.
Fig. 10 - Planar configuration used for FEA simulations. a) Sectional plane; b) General view.
Fig. 11 - Copper layer geometries configurations and electric current directions (in red). a, b) current for the double-layer
geometry; c, d, e, f) geometry and currents for planar configurations “Duo”, “Quad”, “Star” and “Diamond”,
respectively.
In Table 2 the electromagnetic force is compared for each of
the copper layer. Even though only one of the geometries
showed an increase in force, it is enough to suggest that the
force production is strongly dependent in the windings
geometry. A spherical geometry FEA is then required to assess
the impact of a single-layer copper configuration.
Fmmx
a) b)
c) d)
f) e)
Fmmx
Fmmy
Instituto Superior Técnico, Lisboa – Portugal 6
Table 2 - Electromagnetic force comparison between the different planar copper layer geometries.
Geometry Força total 𝐅𝐓 [N] 𝐅𝐓 − 𝐅𝐓𝐝𝐨𝐮𝐛𝐥𝐞−𝐥𝐚𝐲𝐞𝐫
𝐅𝐓𝐝𝐨𝐮𝐛𝐥𝐞−𝐥𝐚𝐲𝐞𝐫
Double-layer 0,192 0 %
“Duo” 0,188 - 2,08 %
“Quad” 0,062 - 67,71 %
“Star” 0,062 - 67,71 %
“Diamond” 0,260 35,42 %
b) Spherical geometry FEA simulations
Three spherical configurations are simulated: double-layer,
single-layer “Star” and single-layer “Diamond”. In Table 3, a
summary of the machines’ parameters is presented. An
exploded view of the simulated geometries are shown in Fig.
12. Both the electromagnetic and thermal properties are
simulated. In Table 4 the average values of the rotor induced
current density module |Jr|, the radial magnetic flux density Br
and the electromagnetic torque Tφ are compared for the
different geometries. In Tables 5 and 6 the electromagnetic
double and single layer configurations FEA results are
compared to the analytical ones.
Table 3 - Spherical simulations machines' parameters.
Parameter Double layer “Star” “Diamond”
Rotor thickness
𝛅𝐫 [mm] 5 5 5
Rotor radius
𝐫𝐫 [mm] 50 50 50
Aluminium
thickness
𝛅𝐀𝐥 [mm]
1 1 1
Air-gap thickness
𝛅𝐚𝐢𝐫−𝐠𝐚𝐩 [mm] 1 1 1
Copper layer
thickness
𝛅𝐂𝐮 [mm]
2x5 10 10
Stator thickness 𝛅𝐬
[mm] 10 10 10
Copper current
density 𝐉𝐬 [A/m2] 3x106 3x106 3x106
Frequency 𝐟 [Hz] 10 10 10
Table 4 – Electromagnetic properties average values
comparison between the three spherical configurations FEA
results.
Double
layer "Star" "Diamond"
|𝐉𝐫| [𝐌𝐀/𝐦𝟐] 1,94 3,97 4,04
𝐁𝐫 [𝐦𝐓] 35,5 58,4 59,1
𝐓𝛗 [𝐦𝐍.𝐦] 36,1 66,6 60,0
|𝐓𝛗 − 𝐓𝛗𝐝𝐨𝐮𝐛𝐥𝐞−𝐥𝐚𝐲𝐞𝐫|
𝐓𝛗𝐝𝐨𝐮𝐛𝐥𝐞−𝐥𝐚𝐲𝐞𝐫
∙ 𝟏𝟎𝟎 0 % 84,49 % 66,21 %
Fig. 12 - Exploded views of the spherical configurations used in the FEA simulations. a) Double-layer; b) spherical "Star"; c)
spherical "Diamond".
Table 5 - Double-layer electromagnetic comparison between
the analytical and FEA results.
Double-layer Analytical FEA Error [%]
|𝐉𝐫| [𝐌𝐀/𝐦𝟐] 1,82 1,94 6,59
𝐁𝐫 [𝐦𝐓] 33,5 35,5 5,97
𝐓𝛗 [𝐦𝐍.𝐦] 33,4 36,1 8,08
Table 6 - Single-layer electromagnetic properties comparison between the analytical and FEA results.
Single-layer “Star” Analytical FEA Error [%]
|𝐉𝐫| [𝐌𝐀/𝐦𝟐] 3,63 3,97 9,37
𝐁𝐫 [𝐦𝐓] 67,1 58,4 12,91
𝐓𝛗 [𝐦𝐍.𝐦] 66,7 66,6 0,15
We can conclude:
Both single-layer geometries perform better than the
double-layer one, with the spherical “Star”
configuration achieving an increase in
electromagnetic torque of 84,49%;
Even though the spherical “Star” geometry has lower
values of both rotor induced current density module
and radial magnetic flux density, the interaction
between these is more efficient, generating more
torque than the spherical “Diamond” geometry;
The FEA simulations validate the electromagnetic
analytical model, with deviations always below the
13%.
a) b) c)
Instituto Superior Técnico, Lisboa – Portugal 7
Thermal FEA simulations were done for the single-layer
geometry. Four situations were considered: blocked rotor
(slip=100%, worst case) or with a 10% slip value for a stator
current density of 1,5 MA/m2 and 3,0 MA/m2. The results are
presented in Table 7.
Table 7 - Comparison between thermal analytical and FEA results for the single-layer geometry machine.
Single-layer
geometry
Analytical FEA
TCu [C°] TAl [C°] TCu [C°] TAl [C°]
JS = 1,5 MA/m2
slip = 100% 72.42 87.59 72.87 84.45
JS = 1,5 MA/m2
slip = 10% 32.87 34.66 32.96 31.58
JS = 3,0 MA/m2
slip = 100% 165.22 217.72 169.01 202.19
JS = 3,0 MA/m2
slip = 10% 62.17 69.27 62.46 56.78
Fig. 13 - Average copper windings temperature as a function
of input power for the double and single-layer geometries.
From the temperature values obtained, it may seem that the
double-layer geometry has a better thermal performance than
the single-layer one. However, we can see from Fig. 13 that the
copper temperature per unit of input power for the single-layer
geometry is slightly lower than for the double-layer one, mainly
due to the increase in rotor induced current density. One of the
reasons for the temperature discrepancy between analytical
and FEA is the assumption that the heat always follows a radial
path when in reality there is a significant portion of the rotor
generated heat that travels up through conduction where it is
dissipated to the ambient. Despite this, the analytical and FEA
copper temperature results are very similar, validating the
thermal analytical model.
IV. Experimental results
a) Planar prototype
A planar prototype based on Fig. 14 was built. Four sections
make up the stator, each of these with two copper windings
around it. The windings were wound using a mould and their
resistances were measured to ensure that the number of turns
was similar in all the phases. In Fig. 15 the completed stator
assembly is shown. The stator is supported in a wooden board,
underneath which the electrical connections are made. As a
rotor, a square aluminium sheet was used with some silicon
steel transformer core plates laid on its top. The rotor is
supported by five ball bearings, allowing low friction movement
in all directions. The prototype ready for experimental analysis
is shown in Fig. 16.
Fig. 14 – Model of the planar prototype built.
Fig. 15 – Completed stator assembly.
Fig. 16 – Planar prototype ready for experimental analysis.
Experimental values of the magnetic flux density at the
surface of the stators windings were gathered using a Hall
effect sensor, for linearly spaced points along the direction of
the traveling magnetic wave. The force was also measured for
each of the two traveling waves and for both of these active at
the same time. The results are presented in Fig. 17 and Table
8.
a)
b) c)
𝐴𝑥 −𝐴𝑥 −𝐵𝑥 −𝐵𝑥
Instituto Superior Técnico, Lisboa – Portugal 8
Fig. 17 – Graph showing the experimental (“x” points) and
FEA (blue curve) magnetic flux density results.
Table 8 – Experimental force in the x direction, y direction and total force produced.
Experimental
value [N] FEA [N] Error [%]
𝐅𝐱 0,822 0,901 8,82
𝐅𝐲 0,826 0,907 9,00
𝐅𝐓 1,165 1,279 8,91
We can see that the experimental data is very consistent with
the FEA results, further validating the theoretical study done to
this point.
b) Spherical prototype
A spherical double-layer prototype was analysed. The use of
silicon steel plates is not effective in the mitigation of eddy
currents due to the anisotropic property of the material. As we
can see in Fig. 18, there is always areas in which the magnetic
field is perpendicular to the silicon steel plates, generating eddy
currents. The solution was to use a soft magnetic composite
material. The chosen material was the 3P SOMALOY® [11]
commercialized in wafers by the Swedish company Hӧganӓs
AB. The material’s properties are presented in Table 9.
Table 9 – Soft magnetic composite material 3P SOMALOY® properties.
Magnetic saturation 1.8 T
Maximum relative magnetic
permeability 850
Losses at B=1 T and f=50 Hz 5 W/kg
Density 7630 kg/m3
Fig. 18 – Magnetic field lines (in white) perpendicular to the silicon steel plates direction, generating eddy current losses
(in dashed blacked).
The prototype is based on the model in Fig. 19. The 3P
SOMALOY® wafers were milled and glued together to create
the stator (Fig. 20) and each of the two rotor parts (Fig. 21).
Fig. 19 – Model of the spherical double-layer machine.
Fig. 20 – Spherical machine stator.
Fig. 21 – Spherical machine rotor.
Due to the constructive complexity of the copper windings,
only one copper layer is fitted to the stator. In Fig. 22 the
finished prototype is presented.
z z
Instituto Superior Técnico, Lisboa – Portugal 9
Fig. 22 – Finished spherical machine prototype.
Radial magnetic flux density values at the rotor surface were
collected. The results are compared with the FEA simulation in
Fig. 23. In Fig. 24 the machines’ electromagnetic torque is
measured for different values of copper current density.
Fig. 23 – Comparison between experimental and FEA results.
Fig. 24 – Electromagnetic torque comparison between analytical and experimental data, for three different copper
current density.
The experimental data is consistent with both the analytical
and FEA study, validating the models used.
V. Slotting
Slotting can be used in the spherical induction machine to
enhance stator-rotor magnetic linkage and to physically
accommodate the copper windings. Slotting also has an impact
on the thermal characteristics of the machine by reducing its
volume of copper. In a double-layer geometry, the orthogonal
arrangement of the copper windings only allows for slotting in
the inner most layer. Otherwise, shorting of the magnetic circuit
is likely to occur, negatively affecting performance. In the
single-layer geometry, slotting covers the whole volume of
copper. In Fig. 25 and Fig. 26 slotting for the double and single-
layer geometries is shown, respectively.
Fig. 25 – Transverse cut of the double-layer machine
geometry with discriminated phases and slotting in the inner
most copper layer.
Fig. 26 – Transverse cut of the single-layer machine geometry
with discriminated phases and slotting covering the whole
copper volume.
Various number of slots per phase, 𝑛, and total slot thickness
per phase, 𝛿𝑐, combinations were analysed using FEA
simulations: 𝑛 = 1, 2, 4, 8 and 12; 𝛿𝑐 =1°, 2°, 3°, 4° and 5°. 𝛿𝑐
is measured as an angle of the stator section. In Fig. 27 we
have an example of one of these configurations.
𝐴𝑥
𝐴𝑥
𝐵𝑥
𝐵𝑥
𝐴𝑦 𝐵𝑦
𝐴𝑥
𝐴𝑦
𝐵𝑦
𝐵𝑥
Instituto Superior Técnico, Lisboa – Portugal 10
Fig. 27 - Transverse cut of the single-layer machine geometry
showing slot thickness definition. In this case, we have n=2
slots per phase for a total slot thickness of δc per phase.
In Table 10, the results are shown as a percentage increase
in electromagnetic torque due to slotting in relation to the
slotless design for the double and single-layer geometry.
Table 10 - Electromagnetic torque percentage increase of the
slotted machine in relation to its slotless counterpart for the
double-layer (DL) and single-layer (SL) geometries.
n
δc 1 2 4 8 12
1° DL 15,47
- - - - SL 73,26
2° DL 13,35 15,95
- - - SL 74,28 85,25
3° DL 10,69 12,30 20,35
- - SL 72,21 82,54 103,11
4° DL 7,85% 8,65 15,92 24,14
- SL 69,12 78,85 96,53 108,90
5° DL 4,79 4,63 11,32 18,84 22,03
SL 68,10 74,26 88,11 98,99 102,83
We can see that the single-layer geometry has a higher
increase in torque due to the slotting then the double-layer
geometry. In Table 11 the temperature difference as a
percentage between the slotted and slotless machine design is
shown for both geometries.
Table 11 – Percentage temperature difference between
slotted and slotless machine for double and single-layer
geometries.
δc Double-layer Single-layer
1° -0,86 % -1,64 %
2° -1,72 % -3,29 %
3° -2,58 % -4,93 %
4° -3,44 % -6,58 %
5° -4,29 % -8,22 %
VI. CONCLUSION
Throughout this paper analytical and FEA models for a
spherical induction machine were created in order to study its
characteristics and behaviour. We conclude that there is a
significant potential for improvement when using a single-layer
copper geometry instead of a double-layer one. The analytical
models show a maximum 29,74% increase in starting torque
and an increase in efficiency over double-layer geometry. 3D
FEA results show an 84,49% increase in torque for the single-
layer “Star” geometry. Slotting in the double-layer geometry
results in a maximum electromagnetic torque increase of
24,14% while in the single-layer geometry a 108,90% increase
is reached. The planar and spherical prototypes analysed
showed data consistent with theoretical study, validating the
models developed.
VI. REFERENCES
[1] Kahlen, Klemens, et al. "Torque control of a spherical
machine with variable pole pitch." IEEE Transactions on power
electronics 19.6 (2004): 1628-1634.
[2] Lee, Kok-Meng, Hungsun Son, and Jeffry Joni. "Concept
development and design of a spherical wheel motor (SWM)."
IEEE International Conference on Robotics and Automation.
Vol. 4. IEEE; 1999, 2005.
[3] Dehez, Bruno, et al. "Development of a spherical induction
motor with two degrees of freedom." IEEE Transactions on
Magnetics 42.8 (2006): 2077-2089.
[4] Kumagai, Masaaki, and Ralph L. Hollis. "Development and
control of a three DOF spherical induction motor." Robotics and
Automation (ICRA), 2013 IEEE International Conference on.
IEEE, 2013.
[5] J. F. P. Fernandes and P. J. C. Branco, "The Shell-Like
Spherical Induction Motor for Low-Speed Traction:
Electromagnetic Design, Analysis, and Experimental Tests,"
IEEE Transactions on Industrial Electronics, vol. 63, no. 7, pp.
4325-4335, July 2016.
[6] Lienhard, J. H. “A heat transfer textbook”, Cambridge
Massachusetts, Phlogiston Press, Fourth edition, 2011.
[7] Holfman, J. P., “Heat Transfer”, McGraw.Hill Series in
Mechanical Engineering, 10th Edition, McGraw-Hill, 1997.
[8] Nave, R. (2005). “Laminar Flow”, HyperPhysics, Georgia
State University. Retrieved 23 November 2010.
[9] Barelko, V. V., and E. A. Shtessel. "Heat transmission by
natural convection in cylindrical and spherical interlayers."
Journal of engineering physics 24.1 (1973): 1-6.
[10] Martynenko, Oleg G., and Pavel P. Khramtsov. Free-
convective heat transfer: with many photographs of flows and
heat exchange. Springer Science & Business Media, 2005.
[11] Hӧganӓs AB 3P Somaloy® brochure
https://www.hoganas.com/globalassets/media/sharepoint-
documents/BrochuresanddatasheetsAllDocuments/Somaloy_
Technology_for_Electric_Motors.pdf
𝛿𝑐2
𝛿𝑐4
𝛿𝑐4
Phase B