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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

[email protected]

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)


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.


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


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


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


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)


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


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


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


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