UPTEC F13 020
Examensarbete 30 hp
Juni 2013
Simulation of a Self-bearing
Cone-shaped Lorentz-type Electrical
Machine
Jim Ögren
Abstract
Simulation of a Self-bearing Cone-shaped Lorentz-type
Electrical Machine
Jim Ögren
Teknisk- naturvetenskaplig fakultet
UTH-enheten
Besöksadress:
Ångströmlaboratoriet
Lägerhyddsvägen 1
Hus 4, Plan 0
Postadress:
Box 536
751 21 Uppsala
Telefon:
018 – 471 30 03
Self-bearing machines for kinetic energy storage have the advantage of integrating
the magnetic bearing in the stator/rotor configuration, which reduces the number
of mechanical components needed compared with using separated active magnetic
bearings. This master's thesis focus on building a MATLAB/Simulink simulation
model for a self-bearing cone-shaped Lorenz-type electrical machine. The concept
has already been verified analytically but no dynamic simulations have been made.
The system was modeled as a negative feedback system with PID controllers to
balance the rotor. Disturbances as signal noise, external forces and torques were
added to the system to estimate system robustness. Simulations showed stability
and promising dynamics, the next step would be to build a prototype.
Telefax:
018 – 471 30 00
Hemsida:
http://www.teknat.uu.se/student
Handledare: Johan Abrahamsson
Ämnesgranskare: Urban Lundin
Examinator: Tomas Nyberg
ISSN: 1401-5757, UPTEC F13 020
Sammanfattning
Ett problem med många förnyelsebara energikällor är att de är intermittenta. För att undkomma detta problem är energilagring av stort intresse.
Speciellt viktigt är lagring av elektrisk energi för elbilar. Kinetisk energilagring är ett lovande koncept där elektrisk energi lagras i en roterande massa,
ett svänghjul. Svänghjul kan till skillnad från batterier urladdas snabbt
och ofta, utan att livslängden påverkas. Denna egenskap gör svänghjul bra
som e↵ektbu↵er för elbilar och kan användas i kombination med batterier
där batterierna levererar den kontinuerliga energin och svänghjulet tar de
snabba ändringarna i energiflöde.
För att minimera friktionsförlusterna används aktiva magnetlager. Då
har rotorn ingen mekanisk kontakt med resten av motorstrukturen. Positionssensorer mäter rotorns position och ett styrsystem justerar strömmarna
till elektromagneterna vilket justerar kraften som verkar på rotorn. Genom
att integrera magnetlagret i stator/rotor-konfigurationen, dvs. göra maskinen självbärande, kan antalet komponenter dramatiskt reduceras.
I en permanentmagnetiserad synkronmaskin verkar krafter på rotorn
genom att de strömförande statorlindningarna upplever ett roterande magnetfält. I konventionella synkronmaskiner är bara krafter som ger vridmoment i rotationsriktningen önskade. Men genom att ändra rotorns och
statorlindningarnas geometri kan man få krafter i alla riktningar, vilket är
en förutsättning för en självbärande maskin.
Ett nytt koncept, patenterat av Electric Line Uppland AB och Johan
Abrahamsson, består av en koniskt formad rotor och stator. Statorlindningarna är skjuvade vilket gör att den resulterande kraften har komponenter i alla riktningar. Genom att införa strömobalanser i de olika lindningar kan resulterande krafter skapas utan att total ström och vridmoment
påverkas.
I detta examensarbete har en simuleringsmodell för en konisk, självlagrad
maskin byggts i MATLAB/Simulink. Rotorn har simulerats som en fritt
svävande, stel kropp. Positionssensorer ger information of rotorns position
och orientering och PID-regulatorer används för att styra rotorn. Utifrån
rotorns position och orientering alstrar PID-regulatorerna börvärden för
krafter och vridmoment som behövs för att balansera rotorn upprätt. Krafterna och vridmomenten översätts till styrströmmar genom att lösa ett underbestämt ekvationssystem. För att ställa in PID-regulatorerna användes
Ziegler-Nicholsmetodens inställningsregler. Den proportionella delen av regulatorerna valdes genom att manuellt välja styvhet för systemet.
När ett vridmoment appliceras på en roterande kropp uppstår gyroskopiska e↵ekter. Ett kompenseringsblock för gyroskope↵ekter implementerades i styrsystemet för att dämpa dessa oönskade e↵ekter och göra styrningen mer e↵ektiv.
För att simulera en dynamisk miljö, t.ex. i fallet att maskinen skulle
3
sitta i en elbil, adderades externa krafter och vridmoment. Magnituderna
på de externa krafterna baserades på uppmätt data från ett tidigare projekt. Vidare, samplat brus från kontrollsystemet till ett svänghjul med
aktiva magnetlager introducerades i simuleringsmodellen för att uppskatta
systemets störningskänslighet. Brus adderades till både strömsignalerna och
positionsignalerna och systemet uppvisade tillräcklig robusthet i båda fallen.
Osäkerheter i modellen består i bland annat idealiseringen av magnetfältet
som en perfekt fyrkantsvåg. Vidare, krafterna som beräknades baseras på
de analytiska värdena. I en verklig prototyp skulle geometrin inte bli perfekt och således skulle de resulterande krafterna avvika från de analytiskt
beräknade krafterna.
Simuleringsmodellen kan utvecklas vidare i många avseenden. Magnetfältet skulle kunna approximeras som en sinusvåg för en mer realistisk
uppskattning. Slutligen, nästa steg vore att bygga en prototyp för att testa
och demonstrera konceptets funktionalitet.
Index terms – Self-bearing machine, bearingless machine, cone-shaped flywheel, 6 degrees of freedom control.
4
Contents
Acronyms
6
1 Introduction
1.1 Background . . . . . . . . . . . . . .
1.1.1 Flywheels for Energy Storage
1.1.2 Active Magnetic Bearings . .
1.2 Self-bearing Electrical Machines . . .
1.3 Aim of Thesis . . . . . . . . . . . . .
1.4 Outline of Thesis . . . . . . . . . . .
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2 Theory
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2.1 Unconstrained Rigid Body Dynamics . . . . . . . . . . . . . . 10
2.1.1 Dynamics of Position . . . . . . . . . . . . . . . . . . . 11
2.1.2 Dynamics of Orientation . . . . . . . . . . . . . . . . . 11
2.1.3 Complete Dynamics . . . . . . . . . . . . . . . . . . . 12
2.1.4 Gyroscopic E↵ects . . . . . . . . . . . . . . . . . . . . 12
2.2 Lorentz Force . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Underdetermined System of Linear Equations . . . . . . . . . 14
2.3.1 Solving Systems of Linear Equations Using MATLAB 15
3 Self-bearing Cone-shaped Lorentz-type Electrical
3.1 Topology . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Stator Windings . . . . . . . . . . . . . . .
3.1.2 Complete System . . . . . . . . . . . . . . .
3.1.3 Position Sensors . . . . . . . . . . . . . . .
3.2 Operation Principles . . . . . . . . . . . . . . . . .
3.3 Force and Torque Coefficients for One Strand . . .
3.3.1 Total Force from One Strand . . . . . . . .
3.3.2 Total Torque from One Strand . . . . . . .
3.4 Geometrical Parameters and Physical Properties .
3.4.1 Dimensions . . . . . . . . . . . . . . . . . .
3.4.2 Limitations . . . . . . . . . . . . . . . . . .
4 Method and Model Development
4.1 Complete System . . . . . . . . . . . . . . .
4.2 Mechanical System . . . . . . . . . . . . . .
4.3 Electromagnetic System . . . . . . . . . . .
4.3.1 Generating the Control Currents . .
4.3.2 Generating Currents . . . . . . . . .
4.3.3 Stator System . . . . . . . . . . . . .
4.4 Control System . . . . . . . . . . . . . . . .
4.4.1 Compensation for Gyroscopic E↵ects
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Machine
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5 Results
5.1 Step Response Simulation . . . . . . . . . . . . . .
5.2 System Robustness . . . . . . . . . . . . . . . . . .
5.2.1 Dynamic Environment . . . . . . . . . . . .
5.2.2 Position Signal Noise . . . . . . . . . . . . .
5.2.3 Current Noise . . . . . . . . . . . . . . . . .
5.3 Optimizing Control Currents and Resistive Losses
5.4 Orientation Control and Gyroscopic E↵ects . . . .
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4.5
4.4.2 Tuning the PID Controllers
Non-ideal Conditions . . . . . . . .
4.5.1 Dynamic Environment . . .
4.5.2 Position Signal Noise . . . .
4.5.3 Current Signal Noise . . . .
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6 Conclusions
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7 Discussion
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7.1 Concept Feasibility . . . . . . . . . . . . . . . . . . . . . . . . 52
7.2 Sources of Error . . . . . . . . . . . . . . . . . . . . . . . . . 53
7.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
A Appendices
57
A.1 SolidWorks Model . . . . . . . . . . . . . . . . . . . . . . . . 57
A.2 Force and Torque Coefficients for all Strands . . . . . . . . . 58
Acronyms
AC
CM
DC
DOF
FES
PID controller
PM
alternating current.
center of mass.
direct current.
degrees of freedom.
flywheel energy storage.
proportional-integral-derivative controller.
permanent magnet.
6
1
Introduction
An issue with many renewable energy sources, such as wind, solar and wave
power, are the fact that the output power is intermittent and unpredictable.
Means of storing electrical energy with high efficiency is of great interest
in order to increase the fraction of renewable energy in the grid. Furthermore, electrical storage is essential for electrified applications that are not
connected to the grid, e.g. electrical vehicles. Flywheels are a promising
way to mechanically store energy for such applications.
1.1
1.1.1
Background
Flywheels for Energy Storage
The idea of storing kinetic energy in a rotating mass is old and has been
used in various mechanical applications over time. Modern flywheels have
taken this concept a few steps further. A dual-directional electrical motor/generator is connected to a rotor and in this way electrical energy can
be stored as mechanical energy. By having a rotor of a strong and light material, commonly a carbon composite, high rotational speeds can be achieved
and thus a lot of energy can be stored by a relative small rotational mass.
There are several advantages of using flywheel energy storage (FES).
High energy density, high power density and a constant storage capacity,
i.e. the capacity of a flywheel is independent on the number of discharge
cycles and the depth of discharge [1].
Flywheels play an important role between ultracapacitors, which have
high specific power but low specific energy, and batteries, which have high
specific energy but low specific power. Therefore, flywheels are well-suited
for applications with rather high energy needs and frequent discharge cycles,
e.g. electrical vehicles. They can be used either as main energy supply for
urban usage or as a power bu↵er in combination with batteries [2].
1.1.2
Active Magnetic Bearings
In order to make modern high-speed flywheels efficient, the mechanical losses
must be reduced to a minimum. Therefore, the rotor is enclosed in a vacuum
chamber to diminish air friction losses. Furthermore, the rotor is suspended
by magnetic bearings and the electrical energy is transferred to the rotor
via an electrical interface, i.e. the rotor is an electrical motor itself and thus
not in mechanical contact with anything. A flywheel is displayed in Fig. 1.
Earnshaw’s Theorem dictates that if only electrostatic interactions are
present, a charged particle cannot be kept in a stable equilibrium. As a
consequence the same holds for the magnetostatic case. This means that any
configuration consisting of only permanent magnets will always be unstable
in at least one direction [3]. Therefore, magnetic bearings are active and are
7
using electromagnets together with position sensors and power electronics.
The layout for an active magnetic bearing is shown in Fig. 2.
Figure 1: A modern flywheel with vacuum enclosure and magnetic bearings [4].
Figure 2: An active magnetic bearing consisting of two electromagnets that controls
the rotor position in one degree of freedom. A voltage V drives an electrical current
I in the electromagnet creating a magnetic field B. A net force can be created by
adjusting the currents I1 and I2 [2].
1.2
Self-bearing Electrical Machines
A self-bearing, also referred to as bearingless, Lorentz-type electrical machine has the magnetic bearing integrated in the motor/generator. In common synchronous alternating current (AC) electrical machines with permanent magnet rotors, the currents in the stator windings are subjected to
the Lorentz force which creates a rotational torque. In the normal case the
rotor is suspended by mechanical bearings and the only desired net force is
8
the one that gives torque in the direction of rotation, the sum of all other
forces is zero. However, by skewing the windings and/or altering the rotor
geometry, a net force in any direction can be realized. Thus, for a sufficient
number of control currents, the rotor can be self-bearing using the stator
windings as magnetic actuators. This will be explained further in Section 3.
The company Electric Line Uppland AB and inventor Johan Abrahamsson at the Division of Electricy at the Ångström Laboratory in Uppsala, have
a patent for a cone-shaped self-bearing Lorentz-type electrical machine [5].
The concept has been shown analytically but no dynamical simulations have
been made to determine if the concept is realizable.
1.3
Aim of Thesis
The aim of the thesis is to build a computer model in MATLAB/Simulink
for a self-bearing cone-shaped Lorentz-type electrical machine. Dynamic
simulations will be used to check concept feasibility and could in the future
lie as foundation for designing a prototype. The main question of this thesis
that will be attempted to answer: is it possible to control the rotor and make
it self-bearing? Other issues that will be addressed are:
• System robustness: simulations with non-ideal conditions.
• Di↵erent ways of determining the control currents
1.4
Outline of Thesis
In Section 2 some theory regarding rigid body dynamics and the Lorentz
force is presented. Section 3 presents the analytical description of the concept, shown by Johan Abrahamsson. The theory in Section 2 and the analytical description in Section 3 are then used for building the model, which
is explained step by step in Section 4. The results of the simulations are
presented in Section 5. Finally, conclusions and discussion are presented in
Section 6 and 7, respectively.
9
2
Theory
2.1
Unconstrained Rigid Body Dynamics
The dynamics of unconstrained rigid bodies must be understood in order
to simulate the behavior of the self-bearing machine. A general threedimensional object with a constant shape can be modeled as a rigid body.
Since rigid bodies occupy space they have both position and orientation,
and hence six degrees of freedom (DOF): translation in three directions and
rotation around three axes. It is convenient to use two coordinate systems
to describe the object’s motion over time – body space and world space.
In body space the coordinate system is fixed with it’s origin at center of
mass (CM) and in world space the coordinate system is fixed. The motion
over time of the rigid body can be described with the motion of the body
space coordinate system over time. The two coordinate systems are shown
in Fig. 3.
Figure 3: Body space and world space coordinate systems for a rigid body [6].
The motion of the rigid body can be calculated by rotating the body
space coordinates around CM and then translating the body. In general,
any point p0 in body space have, after time t, the location p(t) in world
space according to
p(t) = R(t)p0 + x(t),
(1)
where R(t) is the 3x3 rotation matrix and x(t) the position of CM over
time [6].
10
2.1.1
Dynamics of Position
The relationship of the position x(t) and the velocity v(t) of the object’s
CM is simply given by
d
v(t) = x(t).
(2)
dt
Furthermore, we know from Newton’s Second Law that the rate of change
of the linear momentum p(t) equals the sum of the external forces acting on
the body’s CM as
d
p(t) = F (t).
(3)
dt
2.1.2
Dynamics of Orientation
Angular velocity can be related to the rotation matrix in a similar way as
linear velocity can be related to position. Also, in a similar way as the
change of linear momentum is related to the sum of the external forces, the
angular momentum is related to the sum of external torques. However, when
working with orientation dynamics the equations get more complicated and
the concepts far less intuitive.
The rate of change of the rotation matrix can be shown, see [6], to relate
to the angular velocity !(t) as follows
d
R(t) = !(t)R(t).
dt
(4)
Any force acting on a rigid body can be divided into two parts: a force
causing the CM to accelerate and a torque causing the body to rotate. The
total external torque can be calculated as
X
⌧ (t) =
(r i (t) x(t)) ⇥ F i (t),
(5)
where r i (t)
x(t) is the distance between the point of action of the force
F i (t) and the center of mass x(t). The total external torque is then related
to the angular momentum according to
d
L(t) = ⌧ (t).
dt
(6)
Further, the angular momentum is dependent on the angular velocity
!(t),
L(t) = I!(t),
(7)
where I(t) is the inertia tensor. The inertia tensor of a rigid body can be
written as
0
1
Ixx Ixy Ixz
I = @ Iyx Iyy Iyz A .
(8)
Izx Izy Izz
11
The diagonal elements are calculated as
Z
Ixx = m
(y 2 + z 2 )dV ,
(9)
V
where m is the objects mass. The non-diagonal elements are calculated as
Z
Ixy = m
(xy)dV .
(10)
V
For objects with rotational symmetry, the axis of rotation is a principal
axis. If principal axes of the body are chosen for calculating the inertia
tensor all non-diagonal elements will be zero. Cylinders, cones etc. are
called symmetrical tops and if the z-axis is aligned with the axis of rotation
then Ix = Iy 6= Iz holds true and all non-diagonal elements are zero.
2.1.3
Complete Dynamics
If we let X(t) denote the state vector of the rigid body then the complete
dynamic equation of motion for a rigid body [6] can be written as
0
1 0
1
x(t)
v(t)
C B
C
d
d B
B R(t) C = B !(t)R(t) C .
X(t) =
(11)
dt
dt @ p(t) A @ F (t) A
L(t)
⌧ (t)
Any general three-dimensional object’s motion can be characterized merely
by the object’s mass and inertia tensor. This means that once Eq. 11 has
been implemented any object can be simulated by changing the inertia tensor and mass.
2.1.4
Gyroscopic E↵ects
Rotating objects sometimes behave quite unexpectedly and non-intuitively,
as can be demonstrated with a simple gyroscope. This must be taken into
account when attempting to balance a rotating object. In our application
the axis of rotation is aligned with the z-axis. A convenient way to define
the angles of displacement from the vertical position is using the right-hand
rule and let ✓x and ✓y be defined in the directions of the motion of the object
when a torque is applied in the positive x- and y-direction, respectively, see
Fig. 4.
For a rotating system, Eq. 6 can be written [7] as
✓ ◆
✓ ◆
dL
dL
⌧ =
=
+ ! ⇥ L.
(12)
dt F ixed
dt Body
12
(a) yz-plane projection.
(b) xz-plane projection.
Figure 4: Definition of the displacement angles from the vertical orientation.
If the object has rotation !m in the z-direction Eq. 12 yields
0
1 0
1 0
1
!˙ x
!x
0
⌧ = I · @ !˙ y A + @ !y A ⇥ @ 0 A .
!˙ z
!z
I z !m
(13)
Rewriting Eq. 13 on component form gives
⌧x = Ix ✓¨x + !z Iz ✓˙y
⌧y = Iy ✓¨y !z Iz ✓˙x
(14b)
⌧z = Iz ✓¨z
(14c)
(14a)
which are the gyroscopic equations as presented in [8]. These equations will
later be used for orientation control of the rotor.
2.2
Lorentz Force
The basic concept of the machine presented in this thesis is a synchronous
electrical motor with a permanent magnet (PM) rotor. The AC currents
in the stator windings create a rotating magnetic field which the magnetic
field of the rotor locks onto. This forces the rotor to rotate synchronously
with the stator magnetic field. This can be understood by using the Lorentz
force, which states that a moving charge in a magnetic field is subjected to
a force. According to Newton’s third law the rotor must then be subjected
to a force of the same magnitude but with opposite direction.
The Lorentz force for a particle with charge q moving with velocity v in
an electric field E and magnetic field B is
F = q [E + (v ⇥ B)] .
13
(15)
In the case of a wire carrying a current in the presence of a magnetic field
but no external electric field (E = 0), the Lorentz force take the di↵erential
form as
dF = Idl ⇥ B,
(16)
where I is the current flowing through the wire and dl a small line segment.
Eq. 16 needs to be integrated over the entire volume V that the wire occupies
in order to get the total force, with the final expression written as
Z
F =
(Idl ⇥ B) dV .
(17)
V
2.3
Underdetermined System of Linear Equations
In order to determine the control currents an underdetermine system of
linear equations will have to be solved. A general m ⇥ n system of linear
equations can be written as
8
>
> a11 x1 + a12 x2 + . . . + a1n xn = b1
>
>
>
>
< a21 x1 +
a22
+ . . . + a2n xn = b2
(18)
.
.
..
.. .
..
>
..
..
>
.
.
=
.
>
>
>
>
:
am1 x1 + am2 x2 + . . . + amn = bm
A system of linear equation is called underdetermined if the number of
equations (constraints) are fewer than the unknowns, i.e. m < n. The
system can also be written on matrix form as Ax = b, where A is the
coefficient matrix, x the unknowns and b the right-hand side. Writing the
system in 18 on matrix form becomes
0
1
0
1
0
1
x
B 1 C
a11 a12 . . . . . . a1n
b1
B
C
B
C B x2 C B
C
CB
C B
C
B
B a21 a22 . . . . . . a2n C B . C B b2 C
CB . C B
C
B
(19)
B ..
..
.
.. C B . C = B .. C .
..
CB . C B . C
B .
. ..
.
.
@
A B .. C @
A
B
C
@
A
am1 am2 . . . . . . amn
bm
xn
Furthermore, a system is said to be consistent if at least one solution
exists. According to a linear algebra theorem of consistency for a m ⇥ n
system, the system is consistent if the matrix A have full row rank, i.e.
Rank(A) = m. A matrix have full row rank if all the rows are linearly
independent. This means that any two constraints of the problem cannot
be linearly dependent. If the system is underdetermined with full row rank
then the system must have infinitely many solutions and r = n rank(A)
parameters [9].
14
2.3.1
Solving Systems of Linear Equations Using MATLAB
MATLAB has several ways of solving systems of linear equations. A convenient and powerful method is writing the system on matrix form and
use the backslash operator x = A\b. The backslash operator is MATLAB
built-function that uses di↵erent algorithms depending of the structure of
the system. If the backslash operator is applied to a consistent, underdetermined system MATLAB will not return the parametrized general solution.
Instead, one single solution will be returned, optimized in such a way that
the solution vector will have as many zero elements as possible [10].
Another optimizing criterion would be to minimize the Euclidean norm
of the solution vector x. The Euclidean norm is the square root of the sum
of the elements to the power of two and written as
q
||x|| = x21 + x22 + . . . + x2n .
(20)
In MATLAB, this solution can be found using the Moore-Penrose pseudo
inverse, given by the command pinv(A). The vector on the right hand side
of the equation is then multiplied with the pseudo inverse to get the solution
with the smallest norm, x = pinv(A)*b, see [11].
15
3
Self-bearing Cone-shaped Lorentz-type Electrical Machine
In this section the topology and operational principles of a self-bearing coneshaped Lorentz-type electrical machine are explained. This type of machine
has been investigated analytically by Johan Abrahamsson in [5], [12] and
[13]. The concept will from here on be referred to as the electric machine
or simply the machine.
3.1
Topology
The electric machine described in this thesis is a cone-shaped synchronous
AC machine with a PM rotor. The stator is core-less with skewed windings
wound on a conical surface. This geometry of the windings results in a force
with non-zero components in all three directions {x̂, ŷ, ẑ}. In addition to
providing the machine with motoring torque, ⌧z , the forces from the stator
winding will also create the self-bearing e↵ect.
The machine consists of two units, each unit has one stator part placed
between two conical rotor parts. The middle rotor part is shared by the
two units, which gives a total of three conical rotor parts. The insides of
the rotor parts are covered with PM blocks creating an almost uniform and
constant magnetic field. Fig. 5 shows the machine with the two units axially
separated from each other and a cross section of the machine.
Figure 5: The electrical machine. Left: the two units are separated with the upper
unit’s permanent magnetic blocks removed making the skewed windings visible.
Right: A cross-section of the machine showing the rotor parts and the skewed
windings located between the rotor parts [13].
16
3.1.1
Stator Windings
The stator windings have three phases for each unit {AU , B U , C U } and
{AL , B L , C L } where U and L denote upper and lower unit. Each phase
is divided into two parallel branches, e.g. subphases {A1 , A2 }, and each
winding is wound in two turns connected in series. One turn consists of two
skewed strands, which gives a total of four strands per winding. A crosssection of the stator segment is shown in Fig. 6 and one single strand located
between two PM blocks is displayed in Fig. 7.
Figure 6: A cross section of a unit showing the stator windings, the PM blocks and
the direciton of the magnetic field. The magnetic field alters direction four times
per revolution. Each winding connects four strands. E.g. for branch A1 , first the
two (A1) in the left half are wound N turns, then connected in series to the two
(A1) in the right half of the stator slice, which are also wound in N turns [13].
3.1.2
Complete System
The complete system consists of two units attached to the same shaft but
with some distance apart along the rotational axis. This distance between
the two units makes torques in the x- and y-direction possible, see Fig. 7.
A larger distance between the two units would allow for larger torques.
On the other hand a smaller distance gives a more compact structure. In the
extreme case there would be only one rotor unit with one stator but with
two sets of windings, i.e. with opposite skew directions wound onto each
17
Figure 7: One strand placed between
two blocks of permanent magnets.
Each winding consist of four strands.
Illustration by Johan Abrahamsson.
Figure 8: The two rotor units connected to the same shaft. An axial distance between the two rotors makes torques in the x- and ydirection possible. Illustration by Johan Abrahamsson.
other, between mutual blocks of PMs. This would, if enough torque could
be generated, greatly reduce the need of material and make the machine
very compact.
The two stator parts would need to be mounted on a frame holding the
structure. Mechanical bearings with a diameter a few millimeters wider
than the axis diameter could be used as backup bearings and limit the rotor
movement in the radial direction. In the axial direction passive magnetic
bearings could be used to limit the vertical movement. This could support
the weight of the rotor and thus reduce the requirement of a vertical force
produced by the stator windings.
3.1.3
Position Sensors
The position and orientation of the rotor must be known in order for the
control system to be able to adjust forces and torques and make it selfbearing. For this purpose five position sensors are used. Two upper radial
position sensors give the upper x- and y-position of the upper part of the
axis. Same for the lower part of the axis. Finally, an axial position sensor
is placed below the axis to give the z-position. Suggested placements of the
position sensors are shown in Fig. 9.
Since all displacements are small, in the range of a few millimeters, the
18
Figure 9: Locations for the position sensors. Left: the upper and lower radial
position sensor pairs. Right: one radial position sensor pair.
small angle approximation sin(✓) ⇡ ✓ can be used. If hz is the distance
between the axial sensor and CM, which is situated at mid-distance between
the upper and the lower sensor, the position of the center of mass is given
by
✓
◆
xp,u xp,l yp,u yp,l
xCM =
,
, zp + hz ,
(21)
2
2
where xp,u , xp,l , yp,u , yp,l , zp are the position sensor outputs. Furthermore,
for the orientation the displacement angles, as defined in Section 2.1.4, are
calculated as
✓x =
✓y =
yp,u
yp,x
hp
xp,l
xp,u
hp
,
(22a)
(22b)
where hp is the axial distance between the upper and lower position sensor.
3.2
Operation Principles
From rotational symmetry it is obvious that if currents with the same magnitude, i.e. IA = IB = IC , flow through the stator windings then all forces
and torques in the x- and y-direction must cancel out. In the z-direction
however, for one unit all the strands contribute with force and torque in
the same direction which gives a net torque and net force. The idea is to
have the second unit, connected to the same axis as the first unit, but with
windings skewed in the opposite direction.
Changing the skewness to the opposite direction will change the direction
of the force in the z-direction while the motoring torque ⌧z will remain
unchanged. This means that a direct current (DC) imbalance between the
19
upper and lower units can be introduced to create a net axial force while
keeping motoring torque constant [12]. The currents for phase A can be
written as
U
IA
= IP (t)
IP
(23)
L
IA
= IP (t) + IP
where IP (t) is a periodic current, e.g. sinusoidal or trapezoidal, and shifted
by 120 electrical from phase B and 240 electrical from phase C. This will
result in a constant motoring torque determined by the magnitude of IP (t)
and a net axial force linearly dependent only on the imbalance current IP .
If IP = 0 the forces in the z-direction cancel out since the same currents
flow through the upper and lower unit thus creating axial forces of the same
magnitude but opposite direction.
Furthermore, each phase is divided into two parts as can be seen in
U , can be written as
Fig. 6. The current in phase A in the upper unit, IA
U
U
IA1
= IA
U
IA
U
U
IA2
= IA
+
U
IA
(24)
U is an imbalance current that will create a net force. Since the
where IA
U is added to I U the total current is
same current that is subtracted from IA1
A2
constant. Similar imbalance currents are introduced to the other two phases
U , I U , ...} are the control currents
B and C. These imbalance currents { IA
B
that make it possible to achieve net forces in all directions. This allows for a
self-bearing e↵ect while keeping total current and motoring torque constant.
3.3
Force and Torque Coefficients for One Strand
The geometry of a strand must be known in order to calculate the force and
torque from one strand of a winding. Let ↵ denote the angle between the
z-axis and the surface of the cone, i.e. half the aperture. Furthermore, let
denote the angular displacement of the lower and upper ends of one strand,
see Fig. 10.
As have been shown in [12], the line segment of a strand can be parameterized as a function of height h as
8
< ⇢(h) = h tan ↵
(t) = ✓(h)
=
(25)
:
z(h) = h
where
✓
✓(h) = csc ↵ sin
and
1
C
h1
sin
1
C
h
◆
h1 h2 sin ( sin ↵)
C=p 2
.
h1 + h22 2h1 h2 cos ( sin ↵)
20
(26)
(27)
Figure 10: The winding for one phase. Illustration by Johan Abrahamsson.
The magnetic field in cylindrical coordinates can be written as
8
< B⇢ = ±B cos ↵
B =0
B=
:
Bz = ⌥B sin ↵
(28)
where B is the magnitude, which can be approximately be described as a
square wave or a sinusoidal periodical waveform in the azimuthal direction.
The Lorentz force on one conductor segment of height dh can be calculated
by inserting the expressions in Eq.s 25 and 28 in Eq. 16 and defining a
positive current in the direction of increasing h. Furthermore, according
to Newton’s Third Law, the force acting on the rotor is in the opposite
direction and can be written, see [12], as
8
>
BIC tan ↵
>
>
dF⇢ = p
dh
>
>
2
2
>
h
C
>
<
BI
dF =
.
(29)
dF =
dh
>
cos↵
>
>
>
>
BIC
>
>
dh
: dFz = p
2
h
C2
The resulting torque from the force above can be calculated by using
d⌧ = r ⇥ dF
(30)
where r is distance from point of action, i.e. the position of the dl segment,
to the CM of the rotor. Using the expression for the force in Eq. 29 the
21
resulting torque is
8
BI(hc h)
>
>
>
d⌧⇢ =
dh
>
>
cos ↵
>
<
BIChc tan ↵
d⌧ =
d⌧ = p
dh
>
>
h2 C 2
>
>
>
>
: d⌧ = BIh tan ↵ dh
z
cos ↵
(31)
where hc is the location of CM of the rotor, which from rotational symmetry
is somewhere on the z-axis.
3.3.1
Total Force from One Strand
The axial force can be calculated by integrating the dFz component in Eq. 29:
Z h2
Fz =
dFz = KF z BI
(32)
h1
where the force coefficient, KF z , is given by
p
h2 C 2
KF z = BC ln p 22
h1 C 2
h2
h1
!
.
(33)
The radial forces can be calculated in the same manner and are given by
Fx = KF x BI
KF x
1
=
cos ↵
Fy = KF y BI
KF y
1
=
cos ↵
Z
Z
h2
h1
h2
h1
✓
C sin ↵ cos ✓
sin ✓ + p
h2 C 2
✓
C sin ↵ sin ✓
p
h2 C 2
cos ✓
◆
dh
◆
dh.
(34)
(35)
The integrals in Eq.s 34 and 35 cannot be solved analytically and need
to be evaluated numerically.
3.3.2
Total Torque from One Strand
The motoring torque ⌧z can be calculated by integrating the z-component
of Eq. 31 from h1 to h2 , this gives
Z h2
⌧z =
d⌧z = K⌧ z BI
(36)
h1
where
K⌧ z =
tan ↵ 2
(h
2 cos ↵ 2
22
h21 ).
(37)
In the same way, ⌧x and ⌧y can be written as
⌧x = K⌧ x BI
K⌧ x =
1
cos ↵
⌧y = K⌧ y BI
K⌧ y
1
=
cos ↵
Z
Z
h2
h1
h2
h1
✓
✓
(h
(h
◆
dh
◆
dh.
hc ) cos ✓ +
Chc sin ↵ sin ✓
p
h2 C 2
hc ) sin ✓
Chc sin ↵ cos ✓
p
h2 C 2
(38)
(39)
Again, the integrals in Eq.s 38 and 39 have no closed-form expression
but can be solved numerically.
3.4
Geometrical Parameters and Physical Properties
The geometrical dimensions and physical properties used in the simulations
are presented in this section.
3.4.1
Dimensions
The values for half cone aperture (↵), skewness angle ( ), height of one rotor
unit (h1 and h2 ), position of CM (hc ), radial airgap (d) and magnetic flux
density (B) are listed in Table 1. These are the same values as used in [12].
Table 1: Geometrical Parameters.
Parameter
↵ [deg]
[deg]
h1 [m]
h2 [m]
hc [m]
d [m]
B [T]
Value
20
40
0.5
1
0.1
0.03
0.4
For the dynamic simulation the inertia tensor and mass of the rotor
must be known. A very simple CAD model were created in SolidWorks
to determine mass and inertia tensor for a cone, see Appendix A.1. The
SolidWorks model based on the parameters in Table 1 resulted in m = 17 kg
and inertia tensor
0
1
1.14
0
0
1.14
0 A kg ⇤ m2 .
I=@ 0
(40)
0
0
1.57
23
Adding a second unit and a rotational axis would change the inertia
tensor. For a general simulation the value in Eq. 40 is sufficient since the
rotor would still be a symmetric top and qualitatively the inertia tensor
would not change much.
3.4.2
Limitations
The main limitation in what forces and torques that can be realized by the
stator is set by the thermal limits of the stator windings. The horizontal
cross-sectional area of the stator part, as shown in Fig. 6, at the lower part
of the unit can be calculated as
2
Astator = ⇡(rout
where
rin = h1 tan ↵
2
rin
)
d
rout = h1 tan ↵.
(41)
(42)
In Fig. 6 there are 24 positions for the windings. Each position will
contain a collection of N strands since each winding contain N turns. The
cross-sectional area for a collections of strands in one position is given by
Astrand = F F · Astator /24
(43)
where FF is the fill factor. Inserting values from Table 1 into Eq. 41, 42
and 43 and using a fill factor of 25% gives an area of 328 mm2 per collection of strands. The maximum current density for a copper wire, under
the assumption that there is a cooling system provided, is 35 A mm 2 [12].
A maximum continuous current density of 4 A mm 2 should be within the
thermal limits with sufficient safety margin. This gives a maximum continuous current of ⇡ 1300 A per cross-sectional area of a collection of strands.
Thus, there should be sufficient margin to the thermal limits for the currents
needed in order to create the balancing forces and torques.
24
4
Method and Model Development
This section presents the MATLAB/Simulink1 model and how it was built.
The overall system is a negative feedback system using proportional-integralderivative controllers (PID controllers) to balance the rotor. Information
about the position and orientation of the rotor is given by position sensors.
Outputs from the PID controllers are reference values for forces and torques
required for balancing the rotor and creating the self-bearing e↵ect. The
reference values for forces and torques are converted to control currents in
the first part of the electromagnetic system block. The created currents are
inputs to the stator system which calculates the resulting forces and torques
acting on the rotor. Finally, these forces and torques are inputs to the
mechanical system, which simulates the motion of the rotor. An overview
of the model is shown in Fig. 11.
Figure 11: An overview of the simulation model.
4.1
Complete System
In the complete system reference values for both position and orientation
can be set. Step values can also be added to the reference points. To make
simulations more realistic noise signals are added to the force and torque
signals. This represents shaking the rotor to simulate the forces and torques
the rotor would be subjected to if placed in a moving environment, e.g. an
electrical vehicle. The complete system is shown in Fig. 12.
1
MATLAB verison R2012b was used in this project.
25
26
Figure 12: The complete model. A negative feedback system with PID controllers calculating reference torques and forces needed for
balancing the rotor. A value corresponding to the gravitational force is added to the vertical force reference in order to have the PID
controller outputs regulating around zero. Reference forces and torques are converted to control currents, and the generated currents are
converted to force and torques produced by the stator windings in the electromagnetic system. Finally the mechanical system simulates
the motion of the rotor.
4.2
Mechanical System
The mechanical system simulates the motion of the rotor. The inputs and
outputs are:
Inputs External forces {Fx , Fy , Fz } and torques {⌧x , ⌧y , ⌧z }
Outputs Position of CM {x, y, z}, orientation {✓x , ✓y } and rotational
speed !z .
The motoring torque ⌧z is a separate input to the mechanical system block
since it is not a part of the forces and torques balancing the rotor. The
mechanical system block is shown in Fig. 13.
Figure 13: The mechanical simulation block. Inputs are forces and torques and
outputs are position, orientation and rotational speed of the rotor.
The motion of the rotor is calculated by solving the dynamic equations
summarized in Eq. 11 in the rigid dynamic block2 . The mass and inertia
tensor are also inputs, in addition to external forces and torques, to the rigid
body dynamics block. This makes it easy to change the properties of the
object simulated for. Initial conditions such as angular momentum can be
set manually by changing the parameters of the rigid body dynamics block.
The output from the rigid body dynamics block is the solution vector X
which contains the position of CM x(t) and the rotation matrix R(t). This
is not the kind of information one would have at hand in a real application.
Therefore, the output X are converted to outputs that would be given by
position sensors in the position sensor system with the sensors placed as
suggested in Fig. 9. These outputs are then converted to the coordinates of
CM according to Eq. 21 in the position block and to displacement angles
according to Eq. 22 in the orientation block. The mechanical system is
shown in Fig. 14 and the position sensor system in Fig. 15.
It may seem backwards to convert the information in X to signals given
by fictitious position sensors and then calculate the position and orientation
2
The Rigid Body Dynamics block was implemented in C by Magnus Hedlund at the
Division of Electricity at Uppsala University. All other blocks are implemented by the
author of this report.
27
Figure 14: The mechanical system block. Forces, torques, mass and inertia tensor
are inputs to the rigid body dynamics block, which calculates the motion of the
rotor. Gravity is added as an external force. Ouput position and orientation are
given by position sensor system block.
again. The advantage of this approach is that noise can be added to the
sensor output signals in order to get a more realistic scenario of the known
information about the position of the rotor.
4.3
Electromagnetic System
The electromagnetic system block consists of two parts. The current generator block converts the reference forces and torques given from the PID
controllers to control currents. Then currents for all the phases are generated and entered into the stator system block, where the total force and
torque is calculated. Inputs and outputs of the electromagnetic system block
are:
Inputs Reference forces {Fxref , Fyref , Fzref } and torques {⌧xref , ⌧yref }
Outputs Actual forces {Fx , Fy , Fz } and torques {⌧x , ⌧y } generated
by the stator windings.
During ideal conditions the actual values {Fx , Fy , Fz , ⌧x , ⌧y } would be
the same as the reference values {Fxref , Fyref , Fzref , ⌧xref , ⌧yref } but due to
imperfections, noise, uncertainties etc. this is not the case in reality. By
building the simulation model this way, sensitivity analyses can be conducted
by adding noise to current signals. The electromagnetic system block is
shown in Fig. 16 and 17.
28
Figure 15: The position sensor system block. The coordinates of CM and rotation
matrix R are converted to signals that would be given by position sensors. Noise
is added to the sensor output signals. The sensor outputs are then converted to
information about the position and orientation.
Figure 16: The electromagnetic simulation block. Inputs are force, torque references
and the rotational speed. Outputs are the resulting forces and torques created by
the stator windings.
29
30
Figure 17: The electromagnetic system. The control currents generator converts the reference forces and torques to control currents. The
control currents are added to the motoring currents creating the currents for all windings. Noise is also added to the current signals. In
the stator system block the total forces and torques generated by the stator windings are calculated.
4.3.1
Generating the Control Currents
Generating the control current IP needed for a given reference force in the
z-direction is straightforward since all strands, due to symmetry, have the
same vertical force coefficient. Using the expression in Eq. 32 and the fact
that the two units have 24 strands each and the forces from the two units
have opposite directions, then for N number of turns per phase the total
force can be written as
Fztot = 24N KFUz BI U + 24N KFLz BI L
(44)
where
KFUz =
KFLz .
(45)
The currents can be written as
I U = IP (t)
IP
(46)
I L = IP (t) +
IP .
(47)
This results in a total force only dependent on the displacement current
IP and can be written as
Fztot = 48N KF z B IP
(48)
where KF z is given by Eq. 33. For a given reference force Fz the needed
current imbalance between the upper and lower unit, IP , can easily be
calculated by solving Eq. 48.
The x and y force and torque coefficients for each phase can be calculated
by using Eq. 34, 35, 38 and 39 for one strand and then multiply with a
rotation matrix to get the other strands. The angular distance between two
neighboring strands are 15 and the strands are ordered according to Fig. 6.
Then the coefficients for each direction and phase can be summed giving
coefficients for total force and torques in all directions. The same thing is
done for the other two phases. This is repeated for the second unit but with
opposite direction of the skewness and a di↵erent distance to CM, this is
explained further in Appendix A.2.
Going from reference forces and torques is a matter of solving the system
of linear equations written as
8
>
Fx
>
>
>
>
>
< Fy
>
>
⌧x
>
>
>
>
: ⌧
y
U + KU
U
U
U
L
L
L
L
L
L
= KFUx,A IA
F x,B IB + KF x,C IC + KF x,A IA + KF x,B IB + KF x,C IC
U + KU
U
U
U
L
L
L
L
L
L
= KFUy,A IA
F y,B IB + KF y,C IC + KF y,A IA + KF y,B IB + KF y,C IC
U + KU
U
U
U
L
L
L
L
L
L
= K⌧Ux,A IA
⌧ x,B IB + K⌧ x,C IC + K⌧ x,A IA + K⌧ x,B IB + K⌧ x,C IC
U + KU
U
U
U
L
L
L
L
L
L
= K⌧Uy,A IA
⌧ y,B IB + K⌧ y,C IC + K⌧ y,A IA + K⌧ y,B IB + K⌧ y,C IC
31
, (49)
where {KFUx,A , KFUx,B , ...} are the coefficients for the phases for the upper
U , I U , ...} are the control currents. The system in
and lower units and { IA
B
Eq. 49 can be written on matrix form as
0
KFUx,A KFUx,B KFUx,C
B
B U
B KF y,A KFUy,B
B
B U
B K⌧ x,A K⌧Ux,B
@
K⌧Uy,A K⌧Uy,B
KFLx,A KFLx,B KFLx,C
KFUy,C
KFLy,A KFLy,B
KFLy,C
K⌧Ux,C
K⌧Lx,A
K⌧Lx,B
K⌧Lx,C
K⌧Uy,C
K⌧Ly,A
K⌧Ly,B
K⌧Ly,C
0
1B
B
B
CB
CB
CB
CB
CB
CB
AB
B
B
@
U
IA
1
C 0
U C
IB
C
C B
C B
ICU C B
C=B
C B
L
B
IA C
C @
C
L C
IB
A
L
IC
Fx
1
C
C
Fy C
C.
C
⌧x C
A
⌧y
Since the number of equations are fewer than the unknows in Eq. 50
the system is underdetermined. Furthermore, all coefficients have unique,
non-zero values and thus it is safe to assume that the coefficient matrix will
have full row rank, i.e. all rows are linearly independent, and hence the
system have infinitely many solutions.
4.3.2
Generating Currents
The signal source blocks in Fig. 17 generate sine-wave currents IA , IB and IC
that are put in to the current generator block together with all displacement
U , I L , I U , ...}. This block creates the currents for all
currents { IP , IA
A
B
phases for the upper and lower unit, see Fig. 18. Generating currents fast
and with high precision is difficult in reality. Here, the current source is
assumed to be ideal.
4.3.3
Stator System
The currents from the current generator block enters the stator system block
where the resulting forces and torques are calculated. In this block, force
and torque coefficients for individual strands are used. The reason for this is
that each individual strand in each subphase carry the same current but are
not subjected to the same magnetic field. In an idealized case the magnetic
field could be approximated to a square waved signal and all strands of each
subphase would experience a magnetic field of the same magnitude.
4.4
Control System
The control system of the machine is divided into two separate PID controller
systems, see Fig. 19. One control system for position, i.e. keeping CM in a
given position by adjusting the forces and one control system for orientation,
32
(50)
Figure 18: The current generator block.
Inputs are the control currents
U
L
U
IP , IA
, IA
, IB
, etc., the motoring currents IA , IB , IC . Outputs are the six
subphase currents for the upper and the lower unit.
i.e. keeping the rotor vertical (displacement angles ✓x = ✓y = 0) by adjusting
x and y torques. Since the system is a negative feedback system, the inputs
to the the position control system are the error signals, i.e. the reference
signals minus the position signals given by the position sensors. The output
values are saturated in order to limit the reference force outputs, see Fig. 20.
The orientation control system also has the error signals as inputs and
saturates the output signals. Also, the displacement angles and the rotational velocity are inputs, which are needed in order to compensate for the
gyroscopic e↵ects, see Fig. 21. The gyroscopic e↵ect compensator block will
be explained in Section 4.4.1.
A vehicle in urban traffic is subjected to moderate accelerations during
normal operation, see Section 4.5.1. Therefore the saturation values are set
so that the PID controllers give reference forces of maximum 2mg N and
torques of maximum mg N m (force of 2mg N and 0.5 m lever arm), this is
related to the sti↵ness of the system see Section 4.4.2. It is important to
33
Figure 19: The control system with two separated systems for position and orientation control.
Figure 20: The position control system consist of a standard PID controller.
have a limit of the PID controller output since the output can reach very
high values when the derivative is taken on a signal containing noise.
4.4.1
Compensation for Gyroscopic E↵ects
As was shown in Section 2.1.4, gyroscopic e↵ects arises when an object
rotating about the z-axis rotates in the in the ✓x - or ✓y -direction. When
an orientation displacement in e.g. ✓x is detected by the position sensors
the PID controller will give a reference torque ⌧x . But a torque ⌧x applied
to the rotor will, due to gyroscopic e↵ects, also create a rotational motion
in the ✓y -direction since the rotor is freely levitating. This will of course
be compensated for by the control system when the position sensors have
registered the displacement. Then a reference torque ⌧y will be put out from
the PID controller. Again, this torque ⌧y will also create a rotation in ✓x
and so on. The result is a damped wobbling motion before the object will
be oriented vertically or worse, if the control system have too low damping,
34
Figure 21: The orientation control system with the gyroscopic e↵ect compensator
block.
gyroscopic e↵ects could lead to system instability.
The gyroscopic e↵ects could be accounted for by the control system
in order to minimize the impact of these e↵ects and make the process of
controlling the rotor more efficiently. By implementing Eq.s 14a and 14b in
the gyroscopic e↵ect compensator block the torque references from the PID
controller can be modified. A term dependent of the velocity ✓˙y is added
to the ⌧x reference torque and a term dependent of the velocity ✓˙x to ⌧y to
create torques in order to compensate for the rotational motion that will
arise due to the gyroscopic e↵ects. The gyroscopic compensation block is
shown in Fig. 22. Inputs are the PID controller output torque references,
the displacements ✓x and ✓y , rotational velocity !z and Iz .
4.4.2
Tuning the PID Controllers
Tuning PID controllers is often an iterative process and since there are many
unknown and uncertain factors involved in real-life applications it is hard to
have a robust theoretical method. The Ziegler-Nichols method is a heuristic
tuning method for PID controllers [14]. First all gains are set to zero. Then
the proportional gain, KP , is gradually increased till a sustained oscillatory
output with constant amplitude occurs. The value of the proportional gain,
K0 , and the oscillation period, T0 , are noted and the the PID gains are set
according to Table 2. To reduce overshoot a controller with higher damping
could be desired. In that case it is recommended that KD = KP T0 /3 [14].
The Ziegler-Nichols method will not work in its original form for this
system. Since the rotor is a freely levitating body no external forces except
gravity act on the body. Therefore, a position step response simulation with
only the use of proportional control will always result in an oscillatory stable
35
Figure 22: Gyroscopic compensation block. The torque references from the PID
controller are modified to compensate for gyroscopic e↵ects.
Table 2: Ziegler-Nichols Method.
Regulator
PID
KP
0.6K0
KI
2KP /T0
KD
KP T0 /8
output since there are no inherent damping in the system. For torques the
same e↵ect can be obtain by using the gyroscopic e↵ect compensator.
Instead it is useful to reason in terms of sti↵ness and damping of the system. The proportional part of the controller determines the sti↵ness of the
system. The higher the value of KP , the higher the force will be created for
a given position displacement. Very low sti↵ness requires exact knowledge
of the system and can be hard to realize in practice due to uncertainties.
Very high sti↵ness can lead to saturation and/or instability issues. Best is to
have an intermediate sti↵ness. In the case of conventional active magnetic
bearings this is related to the negative sti↵ness of the bearing [15].
Furthermore, the damping, i.e. the derivative part, KD , must be in
proportion to the sti↵ness. Higher sti↵ness requires higher damping for
satisfactory results. However, taking the derivative of a signal amplifies
the noise, which can lead to system instability if the KD value is too high.
Finally, the integrating part of the controller handles stationary errors. Too
high KI can cause problems due to induced time lag [15].
In the case of a Lorentz-type self-bearing machine, sti↵ness could be related to the maximum external force and torque that the rotor is subjected
to. One way of doing this is to determine what force the controller should
36
output at maximum displacement, i.e. maximum value of the error signal
entering the controller. This force should be chosen larger than the maximum external force to have sufficient sti↵ness. In that case the proportional
part of the force and torque controller can be calculated as
KP,f orce = a
Fmax
dmax
(51)
KP,torque = b
⌧max
,
✓max
(52)
where Fmax is the maximum external force and a, b are sti↵ness factors.
Appropriate values of the sti↵ness factors might be around 1...4. To ensure
reasonable ratios between the proportional, integral and derivative coefficients, the relationships in Table 2 could still be used once KP has been
chosen.
4.5
Non-ideal Conditions
Signal noise and external forces and torques are added in order to create a
realistic simulation and determine system robustness.
4.5.1
Dynamic Environment
A vehicle in urban traffic is subjected to moderate accelerations during normal operation. Experiments have shown that for a city bus the horizontal
accelerations rarely exceeds 0.5g and vertical accelerations are only slightly
higher, see [16]. Further, these accelerations vary slowly, with about 1.5 Hz.
In MATLAB white noise source blocks were used to create external forces
in all directions and torques in the x- and y-direction. The force variations
for a rotor mass 17 kg for a 0.5g acceleration becomes ±83 N. Unfortunately
there were no measurements of the angular accelerations and thus no information of the external torques. For the sake of simulation, a 0.5 m level
arm is assumed and the maximum forces as above, the torque then becomes
±41 N m. Fundamental frequency of variation is set to 2 Hz and a fractional
variation with 20 Hz is added on top to compensate for uncertainties. External force and torque signals in one direction are shown in Fig.s 23 and 24,
respectively. Similar signals are created for the other directions. These external force and torque signals were added to the force and torque signals
entering the mechanical block, see Fig. 12.
4.5.2
Position Signal Noise
Whenever signals are created in real applications noise always occurs. This
a↵ects the system and create uncertainties regarding e.g. the position of
the rotor. The signals from the position sensor will contain noise but a
37
100
80
60
External Force [N]
40
20
0
−20
−40
−60
−80
−100
0
2
4
6
8
10
12
14
16
18
20
Time [s]
Figure 23: External force in one direction for simulating a dynamic environment.
control system with sufficient robustness should be able to handle those
uncertainties without system instability.
A control system for a high-speed flywheel with active magnetic bearings has been designed and built at the Ångström Laboratory at Uppsala
University, see [17]. Signal data from this system has been extracted to get
a sense of noise levels for position and current signals.
A voltage signal from one of the position sensors was sampled at a fixed
rotor position. The noise was extracted by subtracting the signal average
value and then voltage values were translated to position values. The modified, sampled signal is shown in Fig. 25.
A Fourier transform of the sampled noise showed two frequency components of 25 and 500 Hz. Maximum noise signal amplitude was 3 µm. From
this information noise signals were created in MATLAB using the white
noise signal blocks and then added to the position sensor outputs as shown
in Fig. 15. By excluding higher frequencies in the creation of noise signals
this is an indirect way of filtering the signals. In a real application filtering
the signals would be crucial.
38
60
External Torque [Nm]
40
20
0
−20
−40
−60
0
2
4
6
8
10
12
14
16
18
20
Time [s]
Figure 24: External torque in one direction for simulating a dynamic environment.
4.5.3
Current Signal Noise
The current signals are created by power electronics in the control system.
These signals will contain noise and this will impact the system in the sense
that the resulting forces and torques might di↵er from the reference forces
and torques put out by the PID controllers.
Current signal noise was sampled from the control system in the lab.
Improvements of the current measurements have been done to minimize
noise, see [18]. Maximum amplitude of the noise was found to be 30 mA.
The extracted noise signal is shown in Fig. 26. A Fourier transform of
the signal showed two lowest frequency components of 200 Hz and 2035 Hz.
Noise signals was created in MATLAB using the white noise blocks with
frequency 200 Hz (higher frequencies are considered to be filtered) and amplitude 30 mA. This noise was added to the currents entering the stator
system as shown in Fig. 17.
39
−6
2.5
x 10
2
1.5
Position [m]
1
0.5
0
−0.5
−1
−1.5
−2
−2.5
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Time [s]
Figure 25: Sampled noise from a position sensor signal.
0.025
0.02
0.015
Current [A]
0.01
0.005
0
−0.005
−0.01
−0.015
−0.02
−0.025
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Time [s]
Figure 26: Sampled noise from a current signal.
40
0.18
0.2
5
Results
The system was simulated for a square wave-shaped magnetic field, rotational speed of 1000 rpm and no external torque in the z-direction. The
PID controllers coefficients were determined according to Section 4.4.2 for
sti↵ness constants a = b = 4, maximum position displacement dmax = 2 mm,
maximum displacement angle ✓max = 0.0067 rad (Eq. 22 and hp = 0.6 m),
damping constants KD with lower overshoot, lever arm r = 0.5 m and maximum external force magnitude Fmax = 83 N (0.5mg N), see Tables 3 and 4.
Table 3: Position PID controller coefficients.
KP
160,000
KI
5,000,000
KD
3,000
Table 4: Orientation PID controller coefficients.
KP
25,000
5.1
KI
1,000,000
KD
400
Step Response Simulation
Step response analyses were performed to determine system characteristics.
A position step response is shown in Fig. 27 and an orientation step response in Fig. 28. The system step response simulations showed a system
characteristics with reasonable rise-time and moderate over-shoot.
It should be mentioned that a step response have little real-life relevance
for an application like this. The reference value for the PID controllers will
always be to center the rotor, i.e. reference values will be constant during
operation. Since the rotor cannot suddenly change position in space, a step
scenario is not relevant for operational performance.
5.2
System Robustness
Dynamic simulations with non-ideal conditions were performed in order to
check system robustness. In all cases noise signals are added to simulate
external forces, sensor signals with noise etc., in order to see if the control system can manage to keep the rotor positioned within the mechanical
constraints: maximum displacements dmax = 2 mm and ✓max = 0.0067 rad.
5.2.1
Dynamic Environment
In Fig. 29 the position is plotted together with the external force. As can be
seen the rotor displacement never exceeds the mechanical constraints and
41
−3
x 10
3
2.5
Position [m]
2
1.5
1
0.5
x−position
Reference value
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Time [s]
Figure 27: Position step response simulation.
−3
9
x 10
8
Displacement angle [rad]
7
6
5
4
3
2
1
Thetax
Reference value
0
0
0.05
0.1
0.15
0.2
0.25
0.3
Time [s]
Figure 28: Orientation step response simulation.
42
0.35
0.4
−4
x 10
100
4
80
3
60
2
40
1
20
0
0
−1
−20
−2
−40
−3
−60
−4
−5
0
x−position
External Force
2
4
6
8
10
12
14
16
18
External Force [N]
Position [m]
5
−80
−100
20
Time [s]
Figure 29: The x-component of the position plotted together with the external
force in the x-direction.
when a quasi-stationary external force is applied the system can manage the
stationary error and quickly center the rotor. In Fig. 30 the orientation is
plotted together with external torque.
5.2.2
Position Signal Noise
Fig. 31 shows actual position and the position signal with noise given to the
PID controller. Fig. 32 shows actual orientation and the orientation that is
calculated on position signal with noise and given to the PID controller. The
simulation shows that the rotor position was kept well within the mechanical
constraints and no system instabilities occurred.
5.2.3
Current Noise
Noise was added to the current signals. The actual force and the force
reference in one direction are shown in Fig. 33. The position signal for the
same simulation is shown in Fig. 34. The simulation shows that although
the reference force and actual force di↵er the rotor was still kept within the
mechanical constraints.
43
−3
x 10
60
2
40
1
20
0
0
−1
−20
−2
−40
External Torque [Nm]
Displacement Angle [rad]
3
Thetax
External Torque
−3
0
2
4
6
8
10
12
14
16
18
−60
20
Time [s]
Figure 30: The ✓x displacement of orientation plotted together with the external
torque in the x-direction.
44
−5
Position [m]
1
x 10
0.5
0
−0.5
Position sensor signal
−1
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Time [s]
−6
Position [m]
5
x 10
0
Actual Position
−5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Time [s]
Figure 31: Position control in one direction with noise added to position sensor
signals. Upper: the position sensor output signal. Lower: the actual position of
the rotor.
Displacement Angle [rad]
−5
1
x 10
0.5
0
−0.5
Orientation sensor signal
−1
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Time [s]
−5
Orientation [rad]
1
x 10
0.5
0
−0.5
Actual Orientation
−1
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Time [s]
Figure 32: Orientation control in one direction with noise added to position sensor
signals. Upper: the orientation output signal. Lower: the actual orientation of the
rotor.
45
3
2
Force [N]
1
0
−1
−2
Fx
Fx−reference
−3
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Time [s]
Figure 33: The reference force and the actual force from the stator system when
noise is added to the current signals.
−5
3
x 10
2
Position [m]
1
0
−1
−2
Rotor Position
−3
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Time [s]
Figure 34: The rotor position for centering the rotor with noise added to the current
signals.
46
1050
1000
Normalized Resistive Losses [W/ohm]
950
900
850
800
750
700
650
600
550
Backslash operator
Pseudo inverse
500
0
1
2
3
4
5
6
7
8
9
10
Time [s]
Figure 35: The resistive losses using two di↵erent solving methods for determining
the control currents. Losses are lower for the pseudo inverse-method (blue) than
the backslas operator (red).
5.3
Optimizing Control Currents and Resistive Losses
The resistive losses in a conducting wire is proportional to the resistance
multiplied with the current squared, i.e. / RI 2 . Since a2 + a2 < (a + b)2 +
(a b)2 = a2 +a2 +2b2 higher resistive losses will occur when control currents
are added/subtracted to the motoring currents.
The simulation run above with dynamic environment was performed with
two di↵erent optimization methods for determining the control currents.
In the first case Eq. 49 was solved using the MATLAB built-in function:
the backslash operator x = A\b. In the second case the control current
solution vector norm was minimized by solving Eq. 49 with the pseudo
inverse-method x = pinv(A)*b in MATLAB. In both runs the total resistive
loss, normalized with resistance, were calculated by summing the currents
squared, the result is presented in Fig. 35. As expected, the resistive losses
are lower using the pseudo inverse-method.
5.4
Orientation Control and Gyroscopic E↵ects
Orientation step response simulations were performed with and without the
gyroscopic e↵ect compensator. A step response was performed in ✓y which
47
Displacement Angle [rad]
−3
1.5
x 10
1
0.5
0
−0.5
ThetaX
−1
Theta Reference
X
−1.5
0
0.1
0.2
0.3
0.4
0.5
0.6
Time [s]
Displacement Angle [rad]
−3
4
x 10
2
0
−2
−4
ThetaY
−6
ThetaY Reference
0
0.1
0.2
0.3
0.4
0.5
0.6
Time [s]
Figure 36: Orientation step response without gyroscopic e↵ect compensator.
due to the gyroscopic e↵ect also result in a motion in ✓x , see Fig. 36 and 37.
Due to sufficiently high sti↵ness of the system the gyroscopic e↵ect has a
small impact on the step response. Although, using the the gyroscopic e↵ect
compensator still decreases the motion in the unwanted direction ✓x with
two orders of magnitude. Fig. 38 and 39 show 2D plots of the rotor shaft’s
upper position for the same simulation.
Simulations were performed for a slower system in order to more clearly
demonstrate the gyroscopic e↵ect. By decreasing the rotational speed to
100 rpm and setting the PID controller coefficients to KP = 6, KI = 0.002
and KD = 8 the system was made slower and the sti↵ness greatly reduced.
The orientation step response simulations with and without the usage of
the gyroscopic e↵ect compensator are shown in Fig. 40 and 41. The lower
sti↵ness leads to a bigger gyroscopic impact and the system step response
time becomes significantly decreased when using the gyroscopic compensator
block. Also, the motion in the ✓x -direction was reduced.
48
Displacement Angle [rad]
−5
x 10
6
4
2
0
−2
ThetaX
−4
ThetaX Reference
−6
0
0.1
0.2
0.3
0.4
0.5
0.6
Time [s]
Displacement Angle [rad]
−3
x 10
4
2
0
−2
ThetaY
−4
Theta Reference
−6
Y
0
0.1
0.2
0.3
0.4
0.5
0.6
Time [s]
Figure 37: Orientation step response with gyroscopic e↵ect compensator.
−3
2
x 10
1.5
1
Y [m]
0.5
0
−0.5
−1
−1.5
Shaft Upper Position
Start
End
−2
−2
−1.5
−1
−0.5
0
X [m]
0.5
1
1.5
2
−3
x 10
Figure 38: Position of the rotor shaft’s upper position for a orientation step response
without the gyroscopic compensator. Same simulation as in Fig. 36.
49
−3
x 10
2
1.5
1
Y [m]
0.5
0
−0.5
−1
−1.5
Shaft Upper Position
Start
End
−2
−2
−1.5
−1
−0.5
0
0.5
1
1.5
X [m]
2
−3
x 10
Figure 39: Position of the rotor shaft’s upper position for a orientation step response
with the gyroscopic compensator. Same simulation as in Fig. 37.
Displacement Angle [rad]
−3
1
x 10
0
−1
−2
ThetaX
−3
ThetaX Reference
−4
0
5
10
15
20
25
30
35
Time [s]
Displacement Angle [rad]
−3
2
x 10
0
−2
−4
ThetaY
−6
ThetaY Reference
−8
0
5
10
15
20
25
30
35
Time [s]
Figure 40: Orientation step response without gyroscopic e↵ect compensator for a
system with low sti↵ness.
50
Displacement Angle [rad]
−6
5
x 10
0
−5
−10
ThetaX
−15
ThetaX Reference
−20
0
5
10
15
20
25
30
35
Time [s]
Displacement Angle [rad]
−3
2
x 10
0
−2
−4
ThetaY
−6
Theta Reference
Y
−8
0
5
10
15
20
25
30
35
Time [s]
Figure 41: Orientation step response with gyroscopic e↵ect compensator for a system with low sti↵ness.
51
6
Conclusions
A functioning simulation model of a self-bearing cone-shaped Lorentz-type
machine was built. A negative feedback system with PID controllers was
used to successfully control the system. The PID coefficients were determined by selecting sti↵ness for the position and orientation controller respectively and then use the Ziegler-Nichols method for the mutual relationships between the derivative and integral gains. These coefficients resulted
in a stable system.
A dynamic environment was simulated by adding external forces and
torques with magnitudes based on measured accelerations form urban traffic.
Noise on position signals and current signals, based on measured data, were
introduce to disturb the system. The system showed sufficient robustness
with all disturbances added and the rotor position and orientation were kept
within the fictitious mechanical constraints at all times.
Two methods were used for solving the underdetermined system of linear equations in order to calculate the control currents. Using the pseudo
inverse-method reduced the resistive losses compared to the MATLABfunction the backslash operator.
A gyroscopic compensator was added to the orientation controller in
order to reduce the impact of gyroscopic e↵ects. The perpendicular motion
for unidirectional orientation step responses was damped with two orders
of magnitude by the gyroscopic compensator. The gyroscopic compensator
was especially efficient for an orientation control system with low sti↵ness.
7
Discussion
The central part of this project was to build a complete simulation model.
Many parts of the model could have been treated more carefully and investigated in greater detail. But due to the time restrictions of a master’s thesis,
focus was put on having a complete and functioning dynamic model.
The control system for this simulation was mainly a tool for proving concept. Other methods, and possibly more suitable than the Ziegler-Nichols
method, for designing the PID controllers could have been used and a more
theoretical approach could have been taken but this was considered outside
the scope of this thesis.
7.1
Concept Feasibility
The most burning question regarding this patented concept is of course
whether it would be realizable or not? There is nothing in the simulations
of this report that would imply that it would not be possible to build a prototype of this machine. However, in the simulation model many approximations and idealizations have been made. In a practical application there are
52
many other limitations in terms of construction, geometrical imperfections
and signal qualities. Mechanical sti↵ness of the shaft, i.e. the connection
of the two units, and the ability of the power electronics to generate the
control currents are examples of other limitations.
The system in this report acquired sufficient robustness due to the sti↵ness of the PID controllers. In reality, signals containing high frequency
noise can cause problems when calculating the derivative. Therefore filters
are needed. However, filters introduce undesired time lags in the system.
It is a trade-o↵ between sufficient filtering and speed of the system. This
limits how high the damping can be set, which also limits the sti↵ness of
the system since high sti↵ness requires high damping.
7.2
Sources of Error
The magnetic field was approximated as square wave-shaped. This would
obviously not be the case in reality. A sine wave approximation of the
magnetic field is probably a better approximation. In that case each strand
of a winding experiences a magnetic field of di↵erent magnitude. This means
that the resulting forces would also di↵er for each strand. However, due to
rotational symmetry this should not induce imbalances but might result in
smaller forces from some of the windings. This should be managed by the
PID controllers.
The magnetic field approximation has another issue. If the four PM
blocks were cut vertically the direction of the magnetic field would not
change at the same time over a given strand since the strands are skewed.
When a strand is in between two PM blocks one part of the strand would be
subjected to a magnetic field in the opposite direction compared to the other
part. In terms of motoring torque this is not a problem since the change of
the direction of the magnetic field coincide with the change of the direction
of the current. But the control current might not be zero at that time and
this could lead to forces in undesired directions.
This e↵ect could be limited, maybe eliminated completely, by building
the PM blocks with shapes that follows the skewness of a strand. This would
make the transition when the magnetic field changes direction over a strand
much smoother. PM blocks with a skewed shape could be constructed by
mounting small PM magnets onto a conical surface. Also, using a smaller
skewness angle would further reduce this problem.
Another idealization in the simulation model lies in the force and torque
coefficients. The analytical values for the coefficients were used to determine
the control currents. The same coefficients were then used for modeling the
stator system. Due to geometrical imperfections and uncertainties in exact
position of the windings the actual resulting forces in a real application
would di↵er from the analytical values. This idealization is amplified by
the approximations of the magnetic field. This could create big di↵erences
53
between the reference forces from the PID controllers and the actual forces.
However, if building a prototype the analytical values should be compared
and adjusted to measurements of the force and torque coefficients
7.3
Future Work
This simulation model could be worked on and extended in several ways.
The motoring and control currents could be simulated more realistic with
constraints in dynamics and a current loop feedback system simulating the
power electronics. Filters should be added to the PID controllers and allowing for simulations with noise containing higher frequencies. This would
become especially important for the derivative part of the controller since the
ability to filter signals is vital for having a high damping and high sti↵ness.
More measurements on the forces and torques a vehicle in urban traffic
are subjected to should be conducted for a more accurate simulation of a
dynamic environment. The external torques used in these simulations were
rough approximations related to the maximum force and might have little
relevance.
The next obvious step would be to build a small prototype to confirm and
demonstrate the basic operational principles of the concept. This simulation
model could be used when designing the prototype for simulating various
geometries and dimensions.
54
References
[1] B. Bolund, H. Bernho↵, and M. Leijon, “Flywheel energy and power
storage systems,” Renewable and Sustainable Energy Reviews, vol. 11,
no. 2, pp. 235 – 258, 2007.
[2] J. Abrahamsson and H. Bernho↵, “Magnetic bearings in kinetic energy
storage systems for vehicular applications,” J. Electrical Systems, vol. 7,
no. 2, pp. 225–236, 2011.
[3] R. Bassani, “Earnshaw (1805-1888) and Passive Magnetic Levitation,”
Meccanica Milano, vol. 41, no. 4, pp. 375–389, 2006.
[4] Wikipedia,
“Flywheel
energy
storage,”
2013.
Url:
http://en.wikipedia.org/wiki/Flywheel energy storage [Online; accessed 26-February-2013].
[5] Johan Abrahamsson and Electric Line Uppland AB, “A conical magnetic bearing,” 10 2012. International Patent no. WO/2012/134367,
Application no. PCT/SE2011/050388 01/04/2011, IPC: F16C 32/04
(2006.01), F16C 39/06 (2006.01), H02K 7/09 (2006.01), Url:
http://patentscope.wipo.int/search/en/.
[6] David
Bara↵,
Pixar
Animation
Studios,
“Physically
Based Modelling – Rigid Body Simulation,” 2001.
Url:
http://graphics.cs.cmu.edu/courses/15-869-F08/lec/14/notesg.pdf
[Online; accessed 15-March-2013].
[7] S. T. Thornton and J. B. Marion, Classical Dynamics of Particles and
Systems. United States: Thomsom Brooks/Cole, fifth ed., 2004.
[8] A. Chiba, T. Fukao, O. Ichikawa, M. Oshima, M. Takemoto and D. G.
Dorell, Magnetic Bearings and Bearingless Drives. Oxford: Newnes,
2005.
[9] Howard Anton and Chris Rorres, Elementary Linear Algebra. Wiley
International, ninth ed., 2005.
[10] MATLAB Documentation, “Systems of Linear Equations.” Url:
http://www.mathworks.se/help/matlab/math/systems-of-linearequations.html [Online; accessed 10-April-2013].
[11] MATLAB Documentation, “pinv.” Url: http://www.mathworks.se/
help/matlab/ref/pinv.html [Online; accessed 10-April-2013].
[12] J. Abrahamsson, “A cone-shaped lorentz-type self-bearing machine
with skewed windings.” Unpublished manuscript, May 2011.
55
[13] J. Abrahamsson, “Kinetic Energy Storage and Magnetic Bearings for
vehicular applications,” May 2011. Licentiate Thesis, Uppsala University.
[14] Torkel Glad and Lennart Ljung, Reglerteknik: grundläggande teori.
Lund: Studentlitteratur, 2006.
[15] G. Schweitzer and E. H. Maslen, ed., Magnetic Bearings – Theory,
Design and Application to Rotary Machinery. Berlin: Springer, 2009.
[16] Tomas Löwall, “Measuring Accelarations in Vehicles using the AccBox System – Results and Discussion.” Project Report, Divison
of Electricity, Uppsala University, 20-06-2012. Url: http://uu.divaportal.org/smash/record.jsf?searchId=1&pid=diva2:621342 [Online;
accessed 15-May-2013].
[17] J. Abrahamsson, M. Hedlund, T. Kamf and H. Bernho↵, “High-speed
kinetic energy bu↵er: Optimization of composite shell and magnetic
bearings,” IEEE Transactions on Industrial Electronics, vol. 60, June
2013.
[18] Fredrik Evestedt and Anders Karlsson, “Strömkontroll för aktiva magnetlager med noggrann och störningsreducerad strömmätnning.” Bachelor Thesis at the Divison of Electricity, Uppsala University, 2013. Url:
Diva adress när publicerad.
56
A
A.1
Appendices
SolidWorks Model
A simple SolidWorks model was made to retrieve information about mass
and inertia tensor. The model is shown in Figure 42.
Figure 42: A SolidWorks model of a cone with mass and inertia tensor calculated
by the software.
57
Table 5: Force and torque coefficients for one strand.
KF,x
-0.2738
A.2
KF,y
0.4515
KF,z
-0.1647
K⌧,x
-0.0788
K⌧,y
-0.0136
K⌧,z
0.1452
Force and Torque Coefficients for all Strands
How the force and torque coefficients were calculated will be explained in
more detail here. The coefficients for one single strand was calculated by
using the equations in Section 3.3.1 and 3.3.2. For the upper unit h1 =
0.5 m, h2 = 1.0 m, hc = 0.6 m, half the cone aperture ↵ = 20 and skewness
angle = 40 was used. The resulting force coefficients in [N A 1 T 1 ] and
torque coefficients in [N m A 1 T 1 ] are presented in Table 5.
To calculate the coefficients for the other strands the x and y-values can
be multiplied with the rotation matrix:
✓
◆
cos ✓
sin ✓
R=
(53)
sin ✓ cos ✓
The z-component of both force and torque remain the same around the
stator due to rotational symmetry. In Figure 6, consider the most left A1
strand to have the coefficients as in Table 5. The angle between two adjacent strands is ⇡/12 (15 ) and walking counterclockwise around the stator
slice in Figure 6 gives angles of [5, 6, 11] · ⇡/12 for the other three strands
of winding A1. Furthermore, the four strands of winding A2 have angles
[12, 17, 18, 23] · ⇡/12. Then the total force and torque coefficients for phase
A can be calculated.
Total force in x-direction from phase A with N number of turns is given
by
A1
A2
FxA = BN KF,x
IA1 + KF,x
IA2 .
(54)
Due to symmetry it is obvious that
A1
KF,x
=
A = 2K A1 . Since I
Let KF,x
A1 = IA
F,x
tion 56 can be written as
A2
KF,x
.
IA and IA2 = IA +
A
FxA = BN KF,x
IA .
(55)
IA Equa(56)
The same is done for the force coefficient in the y-direction and the two
torques. The same method gives the total coefficients for the other two
phases B and C, see Table 6.
The same procedure was repeated for the lower unit but with opposite
skewness direction, i.e.
= 40 . Also, the lower unit was placed 10 cm
below the upper unit, i.e. hc = 0.7 m. The coefficients for one strand is
shown in Table 7 and the total force and torque coefficients in 8
58
Table 6: Total force and torque coefficients for upper unit.
Phase
A
B
C
KF,x
-2.1692
1.9106
0.25862
KF,y
-0.95374
-1.4017
2.3554
KF,z
-1.3177
-1.3177
-1.3177
K⌧,x
0.014524
0.30333
-0.31786
K⌧,y
-0.35864
0.1919
0.16674
K⌧,z
1.1620
1.1620
1.1620
Table 7: Force and torque coefficients for one strand with opposite skewness direction.
KF,x
0.4999
KF,y
0.1699
KF,z
0.1647
K⌧,x
-0.0346
K⌧,y
0.0111
K⌧,z
0.1452
Table 8: Total force and torque coefficients for lower unit.
Phase
A
B
C
KF,x
-0.46293
-1.7811
2.244
KF,y
23239
-1.5629
-0.76106
KF,z
1.3177
1.3177
1.3177
59
K⌧,x
-0.069721
0.16262
-0.092894
K⌧,y
-0.14752
0.013379
0.13414
K⌧,z
1.1620
1.1620
1.1620