Simulation of a Self-bearing Cone-shaped Lorentz-type Electrical Machine

UPTEC F13 020

Examensarbete 30 hp Juni 2013

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 Telefax:

018 – 471 30 00 Hemsida:

http://www.teknat.uu.se/student

Abstract

Simulation of a Self-bearing Cone-shaped Lorentz-type Electrical Machine

Jim Ögren

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.

Ämnesgranskare: Urban Lundin Handledare: Johan Abrahamsson

Sammanfattning

Ett problem med m˚anga f¨ornyelsebara energik¨allor ¨ar att de ¨ar intermit- tenta. F¨or att undkomma detta problem ¨ar energilagring av stort intresse.

Speciellt viktigt ¨ar lagring av elektrisk energi f¨or elbilar. Kinetisk energila- gring ¨ar ett lovande koncept d¨ar elektrisk energi lagras i en roterande massa, ett sv¨anghjul. Sv¨anghjul kan till skillnad fr˚an batterier urladdas snabbt och ofta, utan att livsl¨angden p˚averkas. Denna egenskap g¨or sv¨anghjul bra som e↵ektbu↵er f¨or elbilar och kan anv¨andas i kombination med batterier d¨ar batterierna levererar den kontinuerliga energin och sv¨anghjulet tar de snabba ¨andringarna i energifl¨ode.

F¨or att minimera friktionsf¨orlusterna anv¨ands aktiva magnetlager. D˚a har rotorn ingen mekanisk kontakt med resten av motorstrukturen. Posi- tionssensorer m¨ater rotorns position och ett styrsystem justerar str¨ommarna till elektromagneterna vilket justerar kraften som verkar p˚a rotorn. Genom att integrera magnetlagret i stator/rotor-konfigurationen, dvs. g¨ora maski- nen sj¨alvb¨arande, kan antalet komponenter dramatiskt reduceras.

I en permanentmagnetiserad synkronmaskin verkar krafter p˚a rotorn genom att de str¨omf¨orande statorlindningarna upplever ett roterande mag- netf¨alt. I konventionella synkronmaskiner ¨ar bara krafter som ger vrid- moment i rotationsriktningen ¨onskade. Men genom att ¨andra rotorns och statorlindningarnas geometri kan man f˚a krafter i alla riktningar, vilket ¨ar en f¨oruts¨attning f¨or en sj¨alvb¨arande maskin.

Ett nytt koncept, patenterat av Electric Line Uppland AB och Johan Abrahamsson, best˚ar av en koniskt formad rotor och stator. Statorlind- ningarna ¨ar skjuvade vilket g¨or att den resulterande kraften har kompo- nenter i alla riktningar. Genom att inf¨ora str¨omobalanser i de olika lind- ningar kan resulterande krafter skapas utan att total str¨om och vridmoment p˚averkas.

I detta examensarbete har en simuleringsmodell f¨or en konisk, sj¨alvlagrad maskin byggts i MATLAB/Simulink. Rotorn har simulerats som en fritt sv¨avande, stel kropp. Positionssensorer ger information of rotorns position och orientering och PID-regulatorer anv¨ands f¨or att styra rotorn. Utifr˚an rotorns position och orientering alstrar PID-regulatorerna b¨orv¨arden f¨or krafter och vridmoment som beh¨ovs f¨or att balansera rotorn uppr¨att. Kraft- erna och vridmomenten ¨overs¨atts till styrstr¨ommar genom att l¨osa ett un- derbest¨amt ekvationssystem. F¨or att st¨alla in PID-regulatorerna anv¨andes Ziegler-Nicholsmetodens inst¨allningsregler. Den proportionella delen av reg- ulatorerna valdes genom att manuellt v¨alja styvhet f¨or systemet.

N¨ar ett vridmoment appliceras p˚a en roterande kropp uppst˚ar gyro- skopiska e↵ekter. Ett kompenseringsblock f¨or gyroskope↵ekter implementer- ades i styrsystemet f¨or att d¨ampa dessa o¨onskade e↵ekter och g¨ora styrnin- gen mer e↵ektiv.

F¨or att simulera en dynamisk milj¨o, t.ex. i fallet att maskinen skulle

sitta i en elbil, adderades externa krafter och vridmoment. Magnituderna p˚a de externa krafterna baserades p˚a uppm¨att data fr˚an ett tidigare pro- jekt. Vidare, samplat brus fr˚an kontrollsystemet till ett sv¨anghjul med aktiva magnetlager introducerades i simuleringsmodellen f¨or att uppskatta systemets st¨orningsk¨anslighet. Brus adderades till b˚ade str¨omsignalerna och positionsignalerna och systemet uppvisade tillr¨acklig robusthet i b˚ada fallen.

Os¨akerheter i modellen best˚ar i bland annat idealiseringen av magnetf¨altet som en perfekt fyrkantsv˚ag. Vidare, krafterna som ber¨aknades baseras p˚a de analytiska v¨ardena. I en verklig prototyp skulle geometrin inte bli per- fekt och s˚aledes skulle de resulterande krafterna avvika fr˚an de analytiskt ber¨aknade krafterna.

Simuleringsmodellen kan utvecklas vidare i m˚anga avseenden. Mag- netf¨altet skulle kunna approximeras som en sinusv˚ag f¨or en mer realistisk uppskattning. Slutligen, n¨asta steg vore att bygga en prototyp f¨or att testa och demonstrera konceptets funktionalitet.

Index terms – Self-bearing machine, bearingless machine, cone-shaped fly- wheel, 6 degrees of freedom control.

Contents

Acronyms 6

1 Introduction 7

1.1 Background . . . 7

1.1.1 Flywheels for Energy Storage . . . 7

1.1.2 Active Magnetic Bearings . . . 7

1.2 Self-bearing Electrical Machines . . . 8

1.3 Aim of Thesis . . . 9

1.4 Outline of Thesis . . . 9

2 Theory 10 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 Machine 16 3.1 Topology . . . 16

3.1.1 Stator Windings . . . 17

3.1.2 Complete System . . . 17

3.1.3 Position Sensors . . . 18

3.2 Operation Principles . . . 19

3.3 Force and Torque Coefficients for One Strand . . . 20

3.3.1 Total Force from One Strand . . . 22

3.3.2 Total Torque from One Strand . . . 22

3.4 Geometrical Parameters and Physical Properties . . . 23

3.4.1 Dimensions . . . 23

3.4.2 Limitations . . . 24

4 Method and Model Development 25 4.1 Complete System . . . 25

4.2 Mechanical System . . . 27

4.3 Electromagnetic System . . . 28

4.3.1 Generating the Control Currents . . . 31

4.3.2 Generating Currents . . . 32

4.3.3 Stator System . . . 32

4.4 Control System . . . 32

4.4.1 Compensation for Gyroscopic E↵ects . . . 34

4.4.2 Tuning the PID Controllers . . . 35

4.5 Non-ideal Conditions . . . 37

4.5.1 Dynamic Environment . . . 37

4.5.2 Position Signal Noise . . . 37

4.5.3 Current Signal Noise . . . 39

5 Results 41 5.1 Step Response Simulation . . . 41

5.2 System Robustness . . . 41

5.2.1 Dynamic Environment . . . 41

5.2.2 Position Signal Noise . . . 43

5.2.3 Current Noise . . . 43

5.3 Optimizing Control Currents and Resistive Losses . . . 47

5.4 Orientation Control and Gyroscopic E↵ects . . . 47

6 Conclusions 52 7 Discussion 52 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 alternating current.

CM center of mass.

DC direct current.

DOF degrees of freedom.

FES flywheel energy storage.

PID controller proportional-integral-derivative controller.

PM permanent magnet.

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. Further- more, 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 Background

1.1.1 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 mo- tor/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 ma- terial, 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

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 ma- chine has the magnetic bearing integrated in the motor/generator. In com- mon synchronous alternating current (AC) electrical machines with perma- nent 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

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 Abrahams- son at the Division of Electricy at the ˚Angstr¨om 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 con- cept, shown by Johan Abrahamsson. The theory in Section 2 and the ana- lytical 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.

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 three- dimensional 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].

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

v(t) = d

dtx(t). (2)

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

dtp(t) = F (t). (3)

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

dtR(t) = !(t)R(t). (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

⌧ (t) =X

(r_i(t) x(t))⇥ Fi(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

dtL(t) = ⌧ (t). (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

I = 0

@ I_xx I_xy I_xz I_yx I_yy I_yz I_zx I_zy I_zz

1

A . (8)

The diagonal elements are calculated as I_xx= m

Z

V

(y²+ z²)dV , (9)

where m is the objects mass. The non-diagonal elements are calculated as I_xy = m

Z

V

(xy)dV . (10)

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

d

dtX(t) = d dt

0 BB

@ x(t) R(t) p(t) L(t)

1 CC A =

0 BB

@

v(t)

!(t)R(t) F (t)

⌧ (t) 1 CC

A . (11)

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 ten- sor 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 dt

◆

F ixed

=

✓dL dt

◆

Body

+ !⇥ L. (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

⌧ = I· 0

@

˙!_x

˙!_y

˙!z

1 A +

0

@

!_x

!_y

!z

1 A ⇥

0

@ 0 0 Iz!m

1

A . (13)

Rewriting Eq. 13 on component form gives

⌧x= Ix✓¨x+ !zIz˙✓y (14a)

⌧_y = I_y✓¨_y !_zI_z˙✓x (14b)

⌧_z = I_z✓¨_z (14c)

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)] . (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

F = Z

V

(Idl⇥ B) dV . (17)

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>

>>

>>

><

>>

>>

>>

:

a₁₁x₁ + a₁₂x₂ + . . . + a_1nx_n = b₁ a₂₁x₁ + a₂₂ + . . . + a_2nx_n = b₂

... ... . .. ... = ...

a_m1x₁ + a_m2x₂ + . . . + a_mn = b_m

. (18)

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 BB BB BB

@

a₁₁ a₁₂ . . . a_1n a21 a22 . . . a2n

... ... . .. ... ... a_m1 am2 . . . amn

1 CC CC CC A

0 BB BB BB BB BB

@ x₁ x2

... ... xn

1 CC CC CC CC CC A

= 0 BB BB BB

@ b₁ b2

... b_m

1 CC CC CC A

. (19)

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].

2.3.1 Solving Systems of Linear Equations Using MATLAB MATLAB has several ways of solving systems of linear equations. A con- venient 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, underdeter- mined 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

||x|| = q

x²₁+ x²₂+ . . . + x²_n. (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].

3 Self-bearing Cone-shaped Lorentz-type Electri- cal Machine

In this section the topology and operational principles of a self-bearing cone- shaped 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 {ˆx, ˆy, ˆz}. 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].

3.1.1 Stator Windings

The stator windings have three phases for each unit {A^U, B^U, C^U} and {A^L, 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 cross- section 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

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 con- nected to the same shaft. An ax- ial distance between the two ro- tors makes torques in the x- and y- direction possible. Illustration by Jo- han 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 self- bearing. 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

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

x_CM =

✓x_p,u x_p,l

2 ,y_p,u y_p,l

2 , z_p+ h_z

◆

, (21)

where x_p,u, x_p,l, y_p,u, y_p,l, z_p are the position sensor outputs. Furthermore, for the orientation the displacement angles, as defined in Section 2.1.4, are calculated as

✓x= yp,u yp,x

h_p (22a)

✓y = x_p,u x_p,l

h_p , (22b)

where h_p 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 mag- nitude, i.e. I_A = I_B = I_C, 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

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

I_A^U = I_P(t) I_P

I_A^L= I_P(t) + I_P (23)

where I_P(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 I_P(t) and a net axial force linearly dependent only on the imbalance current I_P. If I_P = 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 Fig. 6. The current in phase A in the upper unit, I_A^U, can be written as

I_A1^U = I_A^U I_A^U

I_A2^U = I_A^U + I_A^U (24)

where I_A^U is an imbalance current that will create a net force. Since the same current that is subtracted from I_A1^U is added to I_A2^U the total current is constant. Similar imbalance currents are introduced to the other two phases B and C. These imbalance currents{ IA^U, I_B^U, ...} are the control currents 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 param- eterized as a function of height h as

= 8<

:

⇢(h) = h tan ↵ (t) = ✓(h) z(h) = h

(25)

where

✓(h) = csc ↵

✓

sin ¹ C h1

sin ¹ C h

◆

(26) and

C = h₁h₂sin ( sin ↵)

ph²₁+ h²₂ 2h₁h₂cos ( sin ↵). (27)

Figure 10: The winding for one phase. Illustration by Johan Abrahamsson.

The magnetic field in cylindrical coordinates can be written as

B = 8<

:

B⇢=±B cos ↵ B = 0

B_z =⌥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

dF = 8>

>>

>>

>>

<

>>

>>

>>

>:

dF_⇢= BIC tan ↵ ph² C² dh dF = BI

cos↵dh dFz = BIC

ph² C²dh

. (29)

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

resulting torque is

d⌧ = 8>

>>

>>

><

>>

>>

>>

:

d⌧_⇢= BI(h_c h) cos ↵ dh d⌧ = BIChctan ↵

ph² C² dh d⌧_z= BIh tan ↵

cos ↵ dh

(31)

where h_c 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 dF_zcomponent in Eq. 29:

F_z= Z h2

h1

dF_z = K_{F z}BI (32)

where the force coefficient, K_{F z}, is given by K_{F z}= BC ln

ph²₂ C² h₂ ph²₁ C² h₁

!

. (33)

The radial forces can be calculated in the same manner and are given by F_x= K_{F x}BI

K_{F x}= 1 cos ↵

Z h2

h1

✓

sin ✓ + C sin ↵ cos ✓ ph² C²

◆

dh (34)

F_y = K_{F y}BI K_{F y} = 1

cos ↵ Z _h2

h1

✓

cos ✓ C sin ↵ sin ✓ ph² C²

◆

dh. (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 = Z h2

h1

d⌧z = K⌧ zBI (36)

where

K_{⌧ z}= tan ↵

2 cos ↵(h²₂ h²₁). (37)

In the same way, ⌧_x and ⌧_y can be written as

⌧_x= K_{⌧ x}BI K_{⌧ x} = 1

cos ↵ Z h2

h1

✓

(h h_c) cos ✓ + Ch_csin ↵ sin ✓ ph² C²

◆

dh (38)

⌧_y = K_{⌧ y}BI K_{⌧ y} = 1

cos ↵ Z h2

h1

✓

(h h_c) sin ✓ Ch_csin ↵ cos ✓ ph² C²

◆

dh. (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 (h_c), 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 Value

↵ [deg] 20 [deg] 40 h₁ [m] 0.5 h2 [m] 1 hc [m] 0.1

d [m] 0.03

B [T] 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

I = 0

@ 1.14 0 0

0 1.14 0

0 0 1.57

1

A kg ⇤ m². (40)

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

A_stator= ⇡(r_out² r²_in) (41)

where

rin = h1tan ↵ d

r_out = h₁tan ↵. (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

A_strand= 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 collec- tion 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 contin- uous 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.

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-integral- derivative 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.

1MATLAB verison R2012b was used in this project.

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.

26

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

2The 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.

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 gen- erator block converts the reference forces and torques given from the PID controllers to control currents. Then currents for all the phases are gen- erated 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 {Fx^ref, Fy^ref, Fz^ref} and torques {⌧x^ref, ⌧y^ref} Outputs Actual forces {Fx, Fy, Fz} and torques {⌧x, ⌧y} generated by the stator windings.

During ideal conditions the actual values {Fx, F_y, F_z, ⌧_x, ⌧_y} would be the same as the reference values {Fx^ref, F_y^ref, F_z^ref, ⌧_x^ref, ⌧_y^ref} 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.

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.

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.

30

4.3.1 Generating the Control Currents

Generating the control current I_P 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

F_z^tot = 24N K_{F z}^U BI^U+ 24N K_{F z}^L BI^L (44) where

K_{F z}^U = K_{F z}^L . (45)

The currents can be written as

I^U = I_P(t) I_P (46)

I^L= I_P(t) + I_P. (47)

This results in a total force only dependent on the displacement current I_P and can be written as

F_z^tot= 48N KF zB IP (48)

where K_{F z} is given by Eq. 33. For a given reference force F_z 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>

>>

>>

><

>>

>>

>>

:

F_x = K_{F x,A}^U I_A^U+ K_{F x,B}^U I_B^U+ K_{F x,C}^U I_C^U+ K_{F x,A}^L I_A^L+ K_{F x,B}^L I_B^L+ K_{F x,C}^L I_C^L Fy = K_{F y,A}^U I_A^U + K_{F y,B}^U I_B^U+ K_{F y,C}^U I_C^U+ K_{F y,A}^L I_A^L+ K_{F y,B}^L I_B^L+ K_{F y,C}^L I_C^L

⌧_x = K_{⌧ x,A}^U I_A^U+ K_{⌧ x,B}^U I_B^U+ K_{⌧ x,C}^U I_C^U+ K_{⌧ x,A}^L I_A^L+ K_{⌧ x,B}^L I_B^L+ K_{⌧ x,C}^L I_C^L

⌧y = K_{⌧ y,A}^U I_A^U+ K_{⌧ y,B}^U I_B^U + K_{⌧ y,C}^U I_C^U+ K_{⌧ y,A}^L I_A^L+ K_{⌧ y,B}^L I_B^L+ K_{⌧ y,C}^L I_C^L

, (49)

where {K_{F x,A}^U , K_{F x,B}^U , ...} are the coefficients for the phases for the upper and lower units and{ IA^U, I_B^U, ...} are the control currents. The system in Eq. 49 can be written on matrix form as

0 BB BB BB

@

K_{F x,A}^U K_{F x,B}^U K_{F x,C}^U K_{F x,A}^L K_{F x,B}^L K_{F x,C}^L K_{F y,A}^U K_{F y,B}^U K_{F y,C}^U K_{F y,A}^L K_{F y,B}^L K_{F y,C}^L K_{⌧ x,A}^U K_{⌧ x,B}^U K_{⌧ x,C}^U K_{⌧ x,A}^L K_{⌧ x,B}^L K_{⌧ x,C}^L K_{⌧ y,A}^U K_{⌧ y,B}^U K_{⌧ y,C}^U K_{⌧ y,A}^L K_{⌧ y,B}^L K_{⌧ y,C}^L

1 CC CC CC A

0 BB BB BB BB BB BB

@ I_A^U I_B^U I_C^U I_A^L I_B^L I_C^L

1 CC CC CC CC CC CC A

= 0 BB BB BB

@ F_x Fy

⌧_x

⌧_y 1 CC CC CC A

. (50)

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 I_A, I_Band I_C that are put in to the current generator block together with all displacement currents { IP, I_A^U, I_A^L, I_B^U, ...}. This block creates the currents for all 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,

Figure 18: The current generator block. Inputs are the control currents IP, I_A^U, I_A^L, I_B^U, 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 rota- tional 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

Figure 19: The control system with two separated systems for position and orien- tation 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,

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, K_P, 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 K_D = K_PT₀/3 [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

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 K_P K_I K_D

PID 0.6K₀ 2K_P/T₀ K_PT₀/8

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 sys- tem. The proportional part of the controller determines the sti↵ness of the system. The higher the value of K_P, 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, K_D, 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 K_D value is too high.

Finally, the integrating part of the controller handles stationary errors. Too high K_I can cause problems due to induced time lag [15].

In the case of a Lorentz-type self-bearing machine, sti↵ness could be re- lated 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

output at maximum displacement, i.e. maximum value of the error signal entering the controller. This force should be chosen larger than the maxi- mum external force to have sufficient sti↵ness. In that case the proportional part of the force and torque controller can be calculated as

K_{P,f orce}= aFmax

d_max (51)

K_P,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 coeffi- cients, 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 nor- mal 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 in- formation 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. Ex- ternal 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 ex- ternal 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