Design and Analysis of a 1-DOF magnetic bearing

MARCUS GRANSTRÖM

Master of Science Thesis Stockholm, Sweden 2011

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Design och Analys av ett magnetlager med 1-frihetsgrad

av

Marcus Granström

Examensarbete MMK 2011:51 MDA 280 KTH Industriell teknik och management

Maskinkonstruktion SE-100 44 STOCKHOLM

Marcus Granström

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Sammanfattning Ett nytt magnetlager med ett aktivt reglersystem i en frihetsgrad, 1-DOF, utvecklas och analyseras. Lagret är ett konventionellt radial permanent magnetlager, PMB, med en integrerad spole för axiell stabilitet med tillhörande reglersystem för en riktning. En förenklad 2,5D-modell härledds, och resultaten jämförs med FEM resultat, samt med resultatet från en 3D-analytisk metod. Resultaten visar på god överensstämmelse och avvikelser diskuteras. Steg svar för varierande belastning visar på att ett nytt jämviktsläge snabbt uppnås, där strömmen och lagerförlusterna går mot noll.

Examensarbete MMK 2011:51 MDA 280

Design och Analys av ett magnetlager med 1-frihetsgrad

Marcus Granström

Godkänt 2011-06-23

Examinator Jan Wikander

Handledare Bengt Eriksson

Uppdragsgivare Lembke Innovation

Kontaktperson Torbjörn Lembke

Fig. 1.1 Rotor koncept med två permanent magnetlager och integrerad spole i en av dem

Controller

ROTOR

Passivt magnetlager, med integrerad spole för axiell stabilitet

Passivt magnetlager

Axialläges sensor

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Master of Science Thesis MMK 2011:51 MDA 280

Design and Analysis of a 1-DOF magnetic bearing

Marcus Granström

Approved 2011-06-23

Examiner Jan Wikander

Supervisor Bengt Eriksson

Commissioner Lembke Innovation

Contact person Torbjörn Lembke

Abstract A novel magnetic bearing with one actively controlled degree of freedom, 1-DOF, is analyzed. The bearing is a conventional radial permanent magnetic bearing, PMB, with an integrated coil for axial stability. A simplified 2,5D-model is derived, and results are compared to FEM-results, as well as a 3D-analytical method. Results show good agreement, and the deviations are discussed. Step response to varying load shows that a new equilibrium position is quickly reached, where the current and bearing losses approaches zero.

Fig. 1.2 Rotor concept with two permanent magnetic bearings and integrated coil in one of them

Controller

Displacement sensor

,Passive magnetic bearing withintegrated coil for axial stability

ROTOR

Passive magnetic bearing

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Acknowledgement I would like to express a general gratitude to the people who have helped me in various ways with this research and development and also made a pleasant atmosphere. It’s been inspiring and pleasant to know Torbjörn Lembke who is the initiator of this thesis and the rest of the family Lembke. This work would never been possible without them. Thank you Torbjörn for everything you taught me about magnetic bearings, electric motors and actuators. You have been a great inspirer to move the limits. I will never forget how much help you were during the months I spent integrating the force equations and later finding all the errors and you gave me advice to use a variable substitution to spot were it went wrong. You are the best. Special thanks also to Bengt Eriksson more known as Benke at the Mechatronic department at KTH for supervising me with this research. A special gratitude to Doctor Matthias Lang at Hochschule Zittau, who sent me his code designed for force calculation between magnets. Lang developed this code for his Doctoral thesis and had another approach than me. I could use his code to verify my equations and found how accurate my equations were. I would also like to thank my boss and my colleagues at ABB Machines in Västerås for believing in me and that I would become finished with my Master of Science Thesis. ABB has not taken part of this work, but since I got a work there with what I love: To develop electric motors I have learnt a great deal from them that inspire me. Finally I would like to thank my wife for her endurance during these last years. Thanks for your appreciation and the children you gave birth to. Without your contribution I would have been forced to simplify the problems. I would also like to thank my parents, sister and relatives and closest friends for their support during all the years. I would not be what I am today without you all. You are always in my heart, thoughts and prayers. To the rest whom I have not mentioned, you are not forgotten and always in my heart. At last but not least all the people at the Mechatronic department at KTH for their support and taking part in this research and for letting me split my research in two parts when it expanded to more than the double amount. Thanks to my schoolmate Madeleine Ymerson who I had the opportunity to learn to know during my last course at KTH. Madeleine took over the second part of this research and further developed a control system using the stiffness and damping from my calculated models. She did this at the same time as my writings of this thesis.

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Contents

1 GENERAL INTRODUCTION ....................................................................................1

1.1 BACKGROUND .........................................................................................................1 1.2 REQUEST FOR NEW MAGNETIC BEARINGS ..................................................................1 1.3 ANALYTICAL METHOD .............................................................................................2 1.4 LIMITATIONS OF THE ANALYSIS ................................................................................2 1.5 RESEARCH MOTIVATION AND OBJECTIVES .................................................................2 1.6 CONTRIBUTIONS OF THE THESIS ................................................................................2 1.7 RESULT ...................................................................................................................3 1.8 OUTLINE OF THE THESIS ...........................................................................................3

2 PREVIOUS WORK ON MAGNETIC BEARINGS ...................................................5

2.1 HISTORICAL DEVELOPMENT OF MAGNETIC BEARINGS ................................................5 2.2 PERMANENT MAGNETIC BEARINGS, PMB ..................................................................7 2.3 ACTIVE MAGNETIC BEARINGS, AMB ...................................................................... 10 2.4 SUPERCONDUCTING BEARINGS, SCB ...................................................................... 11 2.5 ELECTRODYNAMIC BEARING, EDB ......................................................................... 11

3 NOVEL HYBRID MAGNETIC BEARING .............................................................. 15

3.1 BEARING CONFIGURATIONS .................................................................................... 15

4 BEARING ANALYSIS ............................................................................................... 17

4.1 MODEL DESCRIPTION ............................................................................................. 17 4.2 ANALYTICAL DERIVATION OF BEARING FORCES....................................................... 18

4.2.1 Force between two long parallel current carrying wires .................................... 18 4.2.2 Axial force between two concentric circular coils .............................................. 19 4.2.3 Modelling permanent magnets ........................................................................... 19 4.2.4 Lift force between a long magnet and a thin parallel wire ................................. 21 4.2.5 Lift force between two long parallel magnets ..................................................... 23 4.2.6 Lift force between a long magnet and a thick parallel wire ................................ 27 4.2.7 Axial forces between two concentric ring magnets ............................................. 30 4.2.8 Axial force between a circular magnet and a concentric coil ............................. 32

4.3 EQUATION VERIFICATION ....................................................................................... 34 4.3.1 Magnetic field calculation methods used for verification comparison ................ 34 4.3.2 Equation verification for two concentric ring magnets ...................................... 35 4.3.3 Equation verification for concentric ring magnet and coil ................................. 36 4.3.4 Comments on equation verification result .......................................................... 36

4.4 OPTIMISATION OF BEARING CONFIGURATIONS ......................................................... 37 4.4.1 Dimensioning the magnets ................................................................................. 37 4.4.2 Dimensioning the coil ........................................................................................ 38 4.4.3 Bearing configuration drawing 10100 ............................................................... 41

4.5 FEMM SIMULATION OF BEARING CONFIGURATIONS ................................................ 42 4.5.1 Bearing configuration 10 100-1 ......................................................................... 43 4.5.2 Bearing configuration 10 100-2 ......................................................................... 43 4.5.3 Bearing configuration 10 100-3 ......................................................................... 44 4.5.4 Bearing configuration 10 100-4 ......................................................................... 44 4.5.5 Bearing configuration 10 100-5 ......................................................................... 45 4.5.6 Bearing configuration 10 100-6 ......................................................................... 45

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4.5.7 Bearing configuration 10 100-7 ......................................................................... 46 4.5.8 Bearing configuration 10 100-8 ......................................................................... 46

5 CONTROLLER .......................................................................................................... 47

5.1 SIMPLE CONTROL SYSTEM GENERATED FROM THE BEARINGS EQUATION OF MOTION . 47 5.2 FURTHER DEVELOPMENT OF THE CONTROL SYSTEM WITH MATLAB AND SIMULINK .. 47

6 CONCLUSIONS AND SUGGESTIONS FOR FUTURE RESEARCH ................... 53

6.1 CONCLUSIONS ........................................................................................................ 53 6.2 SUGGESTIONS FOR FUTURE WORK ........................................................................... 53

REFERENCES ................................................................................................................... 55

APPENDIX A VARIABEL INTEGRATION ................................................................ A-1

EQUATION A1 ................................................................................................................ A-1 EQUATION A2 ................................................................................................................ A-2

APPENDIX B SCILAB PROGRAMS ............................................................................. B-1

FORCE FUNCTION MAGNET VERSUS MAGNET WITH CONSTANTS ....................................... B-1 FORCE FUNCTION SHEET VERSUS SHEET .......................................................................... B-2 FORCE FUNCTION MAGNET VERSUS MAGNET ................................................................... B-3

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Nomenclature

Magnetic field (induction, flux density) [Vsm-2] or [Tesla]

Current density [Am-2] Surface current density or Magnetic field (excitation) [Am-1] or [H] Magnetic permeability of vacuum [VsA-1m-1] Relative magnetic permeability [1] Magnetic permeability of material [VsA-1m-1]

, Current [A] , Number of coil turns

Diameter of coil part or thread [m] Bearing stiffness in axial direction due to coil [N/A] Bearing stiffness in axial direction due to magnets [N/m] Outer diameter [m] Inner diameter [m]

Thickness [m] Displacement in axial direction [m] Force [N] Force per unit length [Nm-1]

Axial force between two circular wires [N] Axial force between circular magnet and a concentric coil [N]

Axial force between two concentric ring magnets [N] Mass of rotor [kg] Proportional term of the controller

Integral term of the controller Derivative term of the controller

Indexes

wire magnet

coil rotor stator

magnet in rotor magnet in stator

Abbreviations 1-DOF One Degree Of Freedom PMB Permanent Magnetic Bearing AMB Active Magnetic Bearing SCB Superconducting Bearings EDB, ECB Electrodynamic Bearings or Eddy Current Bearings FEM / FEMM Finite Element Method / FEM-code used in this thesis [18]

, rB B

,J J

074 10

r

I iN n

ik

skDdtzFf

w wF

m cF

mr msFmPID

wmcrsmrms

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1 GENERAL INTRODUCTION The ambition to reach higher rotating speeds in smaller electric engines with a higher efficiency and energy density together with quicker frequency converters have led to the research and development of modern high-speed electric machines. When traditional bearings reach its limitation due to weak shaft, since higher speeds need much smaller bearings, engineers have to consider fluid film bearings with all needed accessories for pressurizing and cooling the oil in its circulating system. Next step in reaching higher speed or “oil free” demands due to the environment and other demands or problems with noise and vibrations or friction losses in bearings, engineers have to consider magnetic bearings.

1.1 Background Higher consideration for environmental issues, maintenance free operation and lower energy consumption are some reasons to why magnetic bearing in higher extends replace ball bearings and fluid film bearings. In a magnetic bearing the bearing forces constitute of magnetic forces keeping it floating in its stable force equilibrium providing a non-contacting, friction-free motion in rotary applications. In most cases they require a specialized fast feedback controller regulating its position. These bearings are called active magnetic bearings (AMB). Another kind of magnetic bearing on the market is the permanent magnetic bearing (PMB) existing of permanent magnets. Permanent magnetic bearing have the possibility to be constructed in ways that they are stable in all dimensions except for one dimension, depending on how they are configured. Two other magnetic bearings that exist are the superconducting bearing that demands high cooling and the electrodynamic bearing that demands movement or speed to induce a magnetic lifting force that decreases as the shaft reaches its equilibrium and no energy consumption will then be in the bearing. The electrodynamic bearing is the most promising in energy consumption aspects and also the latest one to be developed. This thesis intend to develop, analyze and optimise a new magnetic bearing based on a passive bearing in the radial direction and with a integrated active magnetic bearing acting in the axial direction.

1.2 Request for new magnetic bearings With increased demands and special limitations with the bearings used today like ceramic ball bearings, air and fluid film bearings, the relatively new magnetic bearing technology open up new possibilities and therefore becomes very popular in designing of high speed machines. They can be used for rotary applications and linear displacement like high speed levitated trains. Completely new designs are possible if magnetic bearings are considered early in the development stage. It is therefore important to know the benefits of the different types and their possibilities and advantages. In some cases new magnetic bearings give the possibility to integrate the motor into the load and thereby skip the rotor. This is not possible with ball bearings that have to be small due to high speed and therefore make the shaft weak and limits the performance of the application. A good example of design possibilities is the design of a new vacuum cleaner. If the motor/fan used magnetic bearing it could rotate with twice the speed and be four times smaller and maybe fit into the hose with the bag on the end of the hose. With magnetic bearing no vibrations are transmitted to the housing that often works as a resonance box. This would make a more silent and compact vacuum cleaner. The development is not there yet but in a near future. It needs a cheap and less complicated magnetic bearing that is a hybrid of

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different magnetic bearing technology that uses the advantage of different bearings. This could be the scenario to have a cheap mass produced bearing for advanced home products.

1.3 Analytical method The analytical method for this thesis after presenting the major different magnetic bearing categories is to present a novel hybrid bearing design and a bearing description with different configurations to be evaluated. To dimension and optimise the configuration can be difficult work with FEMM since the model dimensions have to be specified in the FEMM model and the model have to be updated between every computation. Therefore a lot of the work was laid on deriving a mathematical expression for the forces of interest in the bearing. With this mathematical expression the bearing configurations could be pre optimised. When introducing iron in the model the mathematical model is not valid and FEMM is used to optimise the configurations further or to make a final computation of the bearing forces as function of displacement and current. This method has been applied on the novel hybrid bearing concept to optimise the configurations. Equations of motion and control system will then be developed to optimise the current to be as low as possible for the system. The whole system covers the rotor with bearings as well as the controller with an amplifier and a displacement sensor.

1.4 Limitations of the analysis The study is limited to the axial degree of freedom, which is actively controlled. The radial and tilt degrees of motion are passively stabilized with permanent magnet, and are not analyzed in this report.

1.5 Research motivation and objectives Lembke [6] has previously shown that electrodynamic bearings give an outstanding opportunity to acquire cheep magnetic bearings without any losses, for shaft suspension, never previously believed possible that don’t need any electronic for the suspension. On the other hand the magnetic bearing called AMB that needs expensive electronic has also dropped in price making it possible to be further developed to reach the mass-market with its advantages but is still to complex as a system. To meet the higher demands on mobility and to reach the mass-market the system shall be optimised for battery operation, at which special consideration shall be taken to acquire low energy consumption and low weight. Through integrating active controlling in 1-dimension into as for the rest passively bearing one acquire a cheep, robust and stable magnetic bearing that obtain nearly equally low loss as with electrodynamic bearings but in contrast with them also work at low speeds down to zero.

1.6 Contributions of the thesis To find a passive magnetic bearing through FEM (Finite Element Method) simulation can be difficult, long and hard work. The ability to use powerful equations to analyse the geometry and optimise it before using FEM simulations could be a great advantage. A lack of good analytical methods and equations has been a disadvantage. In this thesis equations are derived to solve the force that arises between two parallel or cylindrical magnets and the force that arise if one of the magnets is replaced by a coil. These equations have been used to optimise the PMB (Permanent Magnetic Bearing). When introducing iron as pole shoes in the PMB, FEM has to be used for further optimisation.

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1.7 Result The result is a method of pre calculating and optimising bearing with round magnets and coils through simplified 2,5D-model which is derived, and results are compared to FEM-results, as well as “the” 3D-results. Results show good agreement, and deviations are discussed. The results can also be used to calculate forces between straight magnets if they are parallel orientated. It is shown that it is possible to design a controller with Step response to varying load and the bearing reaches a new equilibrium position quickly, where the current and bearing losses approaches zero. This was developed to meet the higher demands on mobility and optimization for battery operation with low energy consumption and low weight. Low weight was achieved through using few and compact bearing parts in a smart configuration. Low energy consumption was achieved when the electronic becomes smaller when only one-degree of freedom needs to be controlled instead of five and when the current can be feedback to get zero current.

1.8 Outline of the thesis Chapter 2 details the previous work on magnetic bearing. Four major categories are presented. Chapter 3 introduces a novel hybrid magnetic bearing design. Chapter 4 provides the deviation of mathematical model of the bearing together with analysis

of the novel hybrid bearing design. Chapter 5 presents a design of the controller that optimises the current in the bearing to

approach zero. Chapter 6 presents conclusions and provides specific suggestions for continuing work.

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2 PREVIOUS WORK ON MAGNETIC BEARINGS

Magnetic bearing have opened up for new design possibilities of mechanical machines and reduce maintenance and eliminate lubrication, which is needed for traditional ball and slide bearings. To make a new design it is essential to know the different types and their advantages and disadvantages and also to know the history. The developments of new magnetic bearings have taken many new steps forward and will keep on with that. There are four major categories of magnetic bearings of which are:

Permanent magnetic bearing (PMB) Active magnetic bearing (AMB) Superconductive bearings (SCB) Electrodynamic bearing (EDB)

All of the above mentioned bearing categories can be found in several configurations. There is a wider classification of them in Gerhard Schweitzer’s book on Active magnetic bearings [1], that have become a bible in this field. The reason that many engineers don’t know what a magnetic bearing is depends on that it is fairly new and expensive and often uses the latest electronics to get it to work. Therefore it has only been used in designs were no other bearing technology can be used to meet the most demanding requirements or environments like laboratory and space satellites. The magnetic bearing technology has not yet reached the consumer market. This will be changed as the magnetic bearings become smaller and smarter with fewer components, need less power, electronics and become cheaper. This would provide the consumer market with non-contact, oil-free, maintenance-free, friction-free motion in rotary applications were noise and vibration levels will be reduced because there is no contact between the rotor and the support and outer casing that often work as a resonance box. The limitation in many extreme applications is the ball and slide bearings. With magnetic bearings all the necessary equipment for lubrication is eliminated. A thin shaft diameter is required for ball bearing, especially in combination with high rotating speeds, but not required for all magnetic bearings. This makes it possible to make a redesign with a built-in motor integrated in the equipment instead of having the motor on the end of the application and thereby needs bearings between the motor and the equipment. The critical speed for bending increases and influence less if the shaft is shorter for a flywheel, turbine and fan to mention some applications. It can be trickier to cool the motor if it is integrated.

2.1 Historical development of magnetic bearings For several centuries there have been attempts to develop a magnetic bearing as it is known today. Long before ball bearing was invented researchers thought in terms of using magnetic forces to get something to levitate without external support and get a magnetic bearing. This occupied many brains. Attempts were made by balancing magnets above each other with the poles towards one another. Regardless of all attempts and designs the magnets rotated and stacked together. It was easier to get a magnetic bearing that worked in some degree of freedom but not all five at the same time. They had to have an external support that stabilised it in at least one degree of freedom. These bearings are called permanent magnetic bearings (PMB).

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When the mathematician by name Earnshaw[2] as early as 1842 was able to present a mathematic proof showing that there was no way to get a magnetic monopole to levitate independent in a equilibrium by only magnetic forces or electrostatic forces, there were many researchers that lost their enthusiasm. In Earnshaws theorem, it is also mentioned a “BUT”, if a material can be developed with a variable magnetism it could be possible to get something to levitate. Through variable magnetism, the magnetic field can be changed in direction and intensity. There is no permanent magnets were this is possible, these properties are locked. There is no other material with this characteristic. Later discovery of superconductors and diamagnets is the closest to a material and state one can get to variable magnetism in a passive system. As a parenthesis it can be mentioned that a diamagnetic is a material that can change poles and magnetic field and can levitate above a magnet regardless what side is down towards the magnet beneath. Some tried to prove that Earnshaws theory was wrong and many got patent for solutions that did not work. Some of these possible and impossible ones was shown by Geary[3]. New hope for magnetic bearing came when the discovery was made that one could get a variable magnetic field in a coil by changing the current, this is also called electro magnet. This gave new possibility for a magnetic bearing. The big breakthrough for magnetic bearing came when the computer was invented. With the electronic used in computer, a control system was built that controlled the electromagnet and got feedback from a sensor that registered were the rotor was. By this way a constant distance could be obtained to an object. A rotor with five axis of freedom needed at least five independent control systems. This kind of system was realized after the Second World War when a team of experts and researchers that had taken part in developing the atomic bomb and the V2 rocket, got together and invented a working system. This was the first active magnetic bearing (AMB) and was a very big and complicated system. A new era in developing new kinds of magnetic bearings was when superconductors were discovered and their ideal properties for making a magnetic bearing out of them. In 1987 two men named J. Georg Bednorz and K. Alexander Müller got the Nobel Prize in physics [4] for the discovery of superconductivity in a ceramic material. The superconducting properties appear for the material when it is cooled down to a low critical temperature and the resistance in the material reaches zero. A superconductor has the unique property that it produces an opposite directed and equally strong magnetic field as the one trying to penetrate it. This is called the Meissner-effect. The two persons mentioned above discovered superconductivity in an oxide material at a temperature 12°C higher than any other previously achieved. This was a big achievement since this new temperature was not unreachable. The new idea was that copper atoms in the new material could be used to transport electrons. The copper atoms collaborated stronger with the ambient crystals. A device based on one of these “high temperature superconductors” could be cooled down with floating nitrogen, which is more efficient, cheaper and easier to handle than floating helium. The same year this was presented and resulted in a Nobel Prize, Torbjörn Lembke commissioner of this Master of Science thesis, developed ideas on how similar results as superconductor could be achieved in room temperature. Lembke introduced the expression “magnetic mirror”. This is described in his report, “Design of a magnetic induction bearing” [5]. This further led to the developing of the electro rodyamicdynamic bearing (EDB), also referred to as eddy current bearings (ECB).

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2.2 Permanent magnetic bearings, PMB Permanent magnetic bearing is a passive kind of magnetic bearing, meaning that no electronic control system has to be used for it to work. Its weakness is that it can only stabilize up to four out of five degrees of freedom at the same time. Thus it has to be combined with another kind of bearing along the rotor shaft. It can be constructed either to take up radial or axial load. PMB are based on the principle of repulsion or attraction of two permanent magnets. Radial or axial bearings are established by a suitable arrangement. In Fig. 2.1 is shown how this is achieved by repelling and attracting forces. For the radial bearings in Fig. 2.1 and Fig. 2.2 the rotor has to be of a sufficient length, so the possible tilting angle is small.

Fig. 2.1 (A-H) Structural forms of PMB

Axial bearing

Rep

ellin

g R

epel

ling

Attr

actin

g A

ttrac

ting

Radial bearing

A) E)

B) F)

C) G)

D) H)

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One of the first commercial products with magnetic bearings offering big advantage and added value to the customer was the turbo molecular vacuum pump. Only magnetic bearing could achieve hydrocarbon free pumps. This was a great achievement at this time and would change the whole market. The first mass produced pump with magnetic bearing had a PMB that resembled the first figure in the first row and first column seen in Fig. 2.1. It was a vertical rotor with one ball bearing at the bottom on the exhaust side and a permanent magnetic bearing on the high vacuum side at the top with the motor between the bearings. By doubling the amount of magnets seen in Fig. 2.2, one achieved approximately 3 times the stiffness according to Lembke [6]. An optimization of PMB geometry was recently done by Lang & Fremerey[7], who showed that many thin magnets should be used instead of a few thick ones.

Fig. 2.2 Radial passive bearing with doubled amount of magnets

Passive bearings of these kinds are unstable in one direction according to the Earnshaw theorem. This can be seen in Fig. 2.3. If the inner magnet on the rotor is in the centre of the outer magnet the axial force is zero, but when it’s not kept in its centre it is shot out in that direction in which it was unaligned in. The axial force F increases as the rotor is displaced until it reaches it maximum and than decreases. In the case of the earlier mentioned pump the axial force is taken up by the ball bearing, were some force is good to keep contact in the ball bearing.

Fig. 2.3 Description of force as function of displacement

z

F

+ -

F [N]

z [mm]

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Another thing to take in consideration is the radial stiffness and force. The radial centering force is highest when the rotor and stator magnets are aligned in axial direction. As length of z in Fig. 2.3 increases from zero the concentric force in radial direction get more instable and at certain place the rotor and stator magnets will attract each other instead of repelling and it is not a contact free bearing any more. When a ball bearing is combined with a PMB one wants the ball bearing to have some pre axial load which can be achieved with a small displacement of the rotor magnets. In the same way as a spring has a spring constant two permanent magnets have a magnetic stiffness k [N/m] or resistance toward displacement from equilibrium. The radial stiffness is as follow:

(2.1)

For permanent magnetic bearing the correlation between axial and radial stiffness is given by the following inequality: (2.2) The equal sign is valid for ideal permanent magnets having a rectangular hysteresis loop with the relative permeability in the medium . The relative permeability r, is the ratio of the permeability of a specific medium to the permeability of free space given by the magnetic constant 0 [VsA-1m-1]

(2.3)

In the continued development of vacuum pumps the ball bearings were replaced with an active magnetic bearing giving axial stability and with the PMB to provide radial stability. Advantages:

No wear High rotation speed limit Silent bearing Unlimited life length Small temperature increase No required lubrication and sealing Compact and simple construction Not necessary with small rotor diameter at the bearing for high speed applications

Disadvantages:

Lower bearing stiffness At high temperature over 350 C the magnetic property decreases Needs to be combined with another bearing technology.

F Fk

2axial radialk k

1,0ru

74 10

0r

10

2.3 Active magnetic bearings, AMB The active magnetic bearing is an advanced machine element of mechatronic type for levitation. AMB:s are active in the sense that they use electromagnets to attract the levitating object. AMB uses mostly attractive forces compared to PMB that can use repelling forces as well. This is because the AMB:s most commonly use steel rotors that is attracted to the coil in the electromagnet irrespective direction of the current and the coils pole direction thereby. To keep the rotor on a fix distance there has to be an electromagnet on both sides of the rotor in all degree of freedoms. A contact free distance sensor measures the rotors position and an analogue or digital controller connected to the electromagnets stabilizes the rotor and compensate for the error in the displacement. A shaft that is supposed to be levitated with AMB have five degree of freedom, two radial (x- and y-direction) in each end of the rotor and one axial (z-direction). Each of these five has to use an electromagnet pair on opposite sides of the rotor because they can only use attracting forces. Thereby, a levitated rotor needs 10 electromagnets and 5 sensors for distance measuring. What started the development of the first AMB was knowledge of the cost to change batteries and the cost to transfer energy out in the space. All losses in ball bearings and wear, were not insignificant, and could be fatal. Using oil in the bearing for lubrication was not an option in vacuum applications because it vanishes. When it came to developing something for space it was allowed to cost. The AMB concept has now existed for over 30 years. The technology has improved and led to the developing of the other kinds of magnetic bearings. The AMB have become more efficient and the electronic have become smaller and less parts and need less cooling. All of this has made them cheaper to produce but they are still quite expensive and complicated because the fundamental design are very similar. There are many advantages as a consequence of the friction free and contact free bearing but also some disadvantages. Advantages:

No wear High rotation speed limit Silent bearing Unlimited life length Small temperature increase No required lubrication and sealing Can be used in a corrosive environment Not necessary with small rotor diameter at the bearing for high speed application Control algorithms and damping can be changed according to machine dynamics. Critical speeds can be passed by changing damping and stiffness coefficients, moving

the critical speed until passed. The centre of rotation can be changed by programming to avoid unbalance influence. Works in a wide temperature range from -150 to +450 C

Disadvantages:

Necessary with power supply to work and power backup. Necessary with backup bearing often consisting of a to big ball bearing that is not in

contact with the rotor unless the magnetic bearing fails, thereby needing smaller shaft

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tap making the shaft longer and weaker than necessary to obtain safety. The backup bearing has to be changed after every failure and usage because the high strain.

The AMB is very expensive due to high complexity and advanced electronic. The active magnetic bearings can today be found in a variety of applications like compressors, gas turbines, motors, flywheel, gyroscopes, fans and machine tool spindles.

2.4 Superconducting bearings, SCB A superconductor has the properties that if it is cooled down to its superconducting state before a permanent magnet is placed above the surface; the magnet will “see” a mirror image under the superconductor’s surface. The mirror image act like a real magnet and the force on the real magnet is repulsive, making the magnet levitate. The “mirror” follows if the magnet moves above the surface. The superconductor has another property that also makes it possible to achieve attracting forces between the superconductor and the magnet. The attracting forces are achieved if the magnet is at or above the superconductors surface when cooled down to its superconducting state, not all flux is expelled. This attraction makes the magnets frozen in radial and axial direction even if it turns upside down [6]. The magnitude of the force decreases the further away the magnet is from the surface when cooled down. This makes it possible to get the right centering force on a rotor when the bearing is cooled down, by lifting the rotor above its centreline.

Fig. 2.4 The superconductive levitation principle.

2.5 Electrodynamic bearing, EDB Electrodynamic bearing (EDB), also referred to as Eddy current bearings (ECB), is a class of bearings based on the electrodynamic repulsion principle. The operating principle is similar to that of superconducting bearings except for the fact that the resistivity is not zero. The eddy current bearings rest on the principle of Lenz law. Lenz law declares that if a conductor moves in a magnetic field a current is induced in the conductor that have a magnetic field that tries to resist the external field change. This is described in Lembkes report “Design and Analysis of a Novel Low Loss Homopolar Electrodynamic Bearing”[6]. It is also explained in the authors earlier report “Konceptförslag på svänghjul för energilagring” (Eng. translation: Concept on flywheel for energy storage)[8]. The author had the opportunity to work together with Lembke during the years 2000 until 2002 at the company MAGNETAL, with developing eddy current bearings. The following two figures Fig. 2.5, Fig. 2.6 and the one above, Fig. 2.4, are with permission copied from Lembkes publication

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[6]. In the following illustration Lembke introduced the expression “magnetic mirror”. When a magnet is moved over a conducting surface as seen in Fig. 2.5, due to Lenz law the magnet is subject for a repelling force whose vector is trying to slow down the speed and causes a lifting force from the surface. The magnetic mirror, moves in front of the moving magnet and repels.

Fig. 2.5 The electrodynamic levitation principle.

To make use of this principle in making a magnetic bearing for rotation it is possible to use a shaft of conducting material as copper or aluminium and put long magnets in the stator around the rotor [5]. This works well for “slow” rotation but for higher speed it acts as an induction oven and a lot of heat loss is induced. To make this into a more efficient bearing for high speed one can use round magnets that are polarized in axial direction and a copper cylinder that is rotating inside or outside the magnet depending on inner or outer rotor construction. As long as the copper cylinder rotates in the middle nothing happens since the rotating cylinder doesn’t feel any change in magnetic flux. When the rotating copper cylinder is moved from its centreline by external influence currents are induced in the copper cylinder so that on the one side of the cylinder attract the magnet and on the other side repel and thereby bringing the rotor to the centre as seen in Fig. 2.6. The current in the rotor is opposite directed to the one from the magnet, according to Lenz law. For this to work the relative surface speed between the rotor and magnet must be high. In the same way as in Fig. 2.5 were the “magnetic mirror” is not straight beneath the real magnet this results for the rotor in a force vector that is not only concentre Fig. 2.6a but as the speed or frequency increase the concentre force increase Fig. 2.6b.

Fig. 2.6a,b. Restoring eddy current circuits.

By using two magnets as seen in Fig. 2.6 instead of one the bearing forces get approximate three times stronger. The magnets have the poles towards each other with an iron pole shoe between the magnets, to direct as much as possible of the magnetic flux into the rotor.

13

The eddy current bearings can be divided in two categories at least. The above mentioned was patented 1998 by Torbjörn Lembke [9]. The second kind of eddy current bearing was invented by Richard F. Post [10]. In the journal “Science & Technology Review April 1996” [11] is described how Post changed the copper rotor to a winded coil. In the stator he used long magnets in a special arrangement called Halbach array to get a lot of magnetic flux into the rotor and high frequency in the induced coil when it rotated. The more magnets that were used in the Halbach array, the more poles the stator part of the bearing got leading to higher frequency in the alternating current. So far this is similar to the first kind of eddy current bearings if the coil is short-circuited. Instead Richard F. Post series connected the coil in such a way that the induced voltages eliminated each other. A prototype of this can be seen in Fig. 2.7.

Fig. 2.7 A prototype from Lausanne where Richard Post’s bearing was evaluated.

Since eddy current bearings need speed to get a lift force, they need initial bearing like sliding bearing or air film bearing [12]. An electric engine speeds up very fast, so it’s only in contact for a short time. Devices with eddy current bearings instead of active magnetic bearings operate more silently, since there is no amplifier noise being transmitted to the rotor and housing. Eddy current bearings enable higher speed than passive or active magnetic bearings. EDBs show the best result compared with other bearing technologies when it comes to reliability, maintenance and lubrication according to Impinna F. and his PhD thesis [13]. Both AMBs (active magnetic bearing) and EDBs have already been used successfully for industrial applications according to Impinna F. There are some disadvantages with EDB compared with AMB such as the lack of monitoring and tuning. EDBs are not stable under a certain rotating speed that have to be considered as a design parameter, therefore they need some kind of run-up bearing for the operation at low speed or at rest, could be an Air Bearing. The EDBs load capacity and achievable stiffness are lower than the one characterize AMBs but instead since it is a passive technology the advantages is lower realization cost, higher reliability, less complex, less maintenance if any and smaller size. Imprina F. presents Lembkes electrodynamic bearing and also referees to other that have done similar studies later. The literatures about the EDBs are poor compared with AMBs and are also missing some crucial aspects and therefore Lembkes and Imprina F. dissertations will fill those lacks.

14

The main issues are modelling, stability and design methodology. The design of an EDB is more delicate than for an AMB, because there is no way to correct a mistake and therefore important to follow a design procedure for EDB so that the static and dynamic for the whole system are well known. Advantages:

No wear High rotation speed limit Silent bearing Unlimited life length Small temperature increase No required lubrication and sealing Can be used in a corrosive environment Compact and simple construction but a redesign of the construction might be

important, to get all the benefits. Not necessary with a small rotor diameter at the bearing for high speed applications Quicker to compensate for shocks than active magnetic bearing at design speed. EDBs show the best result compared with other bearing technologies when it comes to

reliability, maintenance and lubrication. Disadvantages:

Low bearing stiffness Load capacity is speed dependent as with hydrodynamic bearings. Requires some kind of starting bearing due to a low lift force at low speed.

15

3 NOVEL HYBRID MAGNETIC BEARING The passive bearing in Fig. 2.2 can be improved for axial stability using a coil which can be integrated into the design as seen on the left bearing on the rotor in Fig. 3.1 where a novel hybrid magnetic bearing to the left is compared with a passive on the right. The coils current interact with the rotors magnetic field to produce axial forces. The current in the coil and thereby the force from the coil on the rotor are regulated by the controller connected to the displacement sensor. A levitating system of that type requires at least two bearings of which at least one is a novel hybrid magnetic bearing, an example can be seen in Fig. 3.1.

Fig. 3.1 Rotor concept with two permanent magnetic bearings and integrated coil in one of them Depending on system requirements different coil arrangements can be used. A selection of such magnetic bearing configurations will be studied. Common for these are that they are a hybrid of passive and active magnet bearings and combines the advantages of both types. In literature they are sometimes referred to as one degree of freedom bearings, or 1-DOF bearings. The first inventor to use this principle was K. Fremery [14] [15] and Karl Boden. According to the patent text Karl Boden used bearing according to Fig. 2.1D, with attracting forces, while the novel bearing configuration here presented use design according to Fig. 2.1A, with repelling forces. The advantage and disadvantage of the different novel bearing configurations will be presented together with the performance to help chose the bearing for the prototype.

3.1 Bearing configurations In Fig. 3.2 is shown a cross-section of five different configurations aligned on the rotor. The different configurations have equally rotor part but illustrate how the stator part of the bearing can be varied, size of stator magnets, amount of coil, amount of pole shoes if any and a coil shoulder in the last configuration. All of this gives different data to the bearing when the interaction between the magnetic fields make the magnetic field lines take different routes. What is optimal and how can optimality be calculated are some questions.

Controller

Displacement sensor

,Passive magnetic bearing withintegrated coil for axial stability

ROTOR

Passive magnetic bearing

16

Fig. 3.2 Bearing configurations for evaluation

The difference between the configurations in Fig. 3.2 is that for a given pair of rotor and stator magnets vary the amount of coil and the amount of iron on a given thickness and length of the magnetic bearing to get as good comparison between the different configurations as possible. Using iron will be on the coils expense. The iron in shape of washer or cylinder function as a pole shoe and conduct the magnetic field to be focused to the air gap between the rotor and the stator to get the best benefit and penetration of the magnetic field and repelling forces together with as good coil influence as possible. In bearing configuration 5 in Fig. 3.2 the coil is inside an iron cylinder with iron washers on the ends. Through this the magnetic field distance around the coil will only be equal to two air gaps (the distance it travels through the air). It travels around the coil through the iron and from one stator pole shoe to the rotor pole shoe and back again to the second stator pole shoe. This bearing configuration will be very good if only concerning to the bearings axial performance from the coil. The disadvantage of the iron in the stator is that the radial stiffness decrease due to the attractive forces between the rotor magnet and the stator pole shoes. The bearing consists of parts listed in Fig. 3.3 below.

Fig. 3.3 Bearing description with used variables

rdrD

sdsD

64 and5

N S N S

N S N S

Config 1 Config 2 Config 3 Config 4 Config 5

Coil, N•I Iron pole Magnet

Iron coil shoulder

ROTOR

17

4 BEARING ANALYSIS A fairly simple model with equations will be derived in this section. The equations will be used to optimise the bearing. Model and equations are derived in section 4.1 - 4.2. The equations are verified with FEM and a 3D-analytical method in section 4.3 and the deviations are discussed. The bearing is optimised into eight different configurations in section 4.4. All the eight bearing configurations are FEM calculated in section 4.5 to derive the bearing stiffness coefficients and . With the bearing stiffness coefficients the systems stability can be calculated and the controller can be developed as in section 5. From the systems current limitation or maximum current to the coil the bearings operating range can be calculated and the resulting maximum forces that can be produced when the rotor is in its maximum allowed displacement in axial direction.

4.1 Model description

Fig. 4.1 Model description of the bearing to be analyzed

This model will be analyzed in several steps. The interaction between the permanent magnets will be separated from the interaction between the magnet and the coil. This simplification is true as long as the iron pole shoes are not saturated. This model will be curved into 2,5D model, which will be compared with a “true”3D model by Lang [17]. Later in the FEM analysis also partly saturated iron will be analyzed in two dimensions.

Fig. 4.2 Example of the broken down bearing studying the forces between two magnets in Fig. 4.1

How the forces are calculated will be derived in the following sections. To the Fig. 4.2 can be mentioned that the force Fr4 and Fr5 (resulting force on cross section side of magnet) will be negative in sign since it is pulling downward. The upward force Fr3+Fr6 will be greater than

sk ik

N

N

N

N

2 6 5 4 8 1 3 7

tms

Fr3

Dmr

F32

F31

F38

F37 Fr4

F41

F42

F47

F48 4 3

2 1

dms

dmr

Dms

z

tmr

3 4 5 6 3 4

The resulting force The force vector in axial direction onF = =Fr + Fr + Fr + Fr = =2 (Fr +Fr )

on the inner magnet the inner magnets respective sheetres

Fr3 Fr4 Fr5 Fr6

rd

smtrDsdsD

smt

rmt

N

N

N

N

N

N

N

N ct

rmt

18

downward Fr4+Fr5 and the resulting force will be upward. If the inner magnet is beneath the outer magnet all the resulting forces would change direction and sign.

4.2 Analytical derivation of bearing forces The novel magnetic bearing with the aligned stator and rotor magnets function thru repelling forces between the axially magnetised cylindrical magnets. The repelling forces cause the rotor to align concentric. In axial direction the bearing is unstable unless it had not been a coil that can produce a fluctuating magnetic field in both strength and direction connected to a control system regulating the current I in the coil as function of the displacement of the rotor in axial direction. To be able to optimise the bearing one would need the forces as function of all dimensions and the current. The forces to be derived are between two magnets and the force between a magnet and a coil and thereafter all forces can be summarized in a large equation. Well known and a start for this is the forces between two wires as seen in section 4.2.1 with equation (4.1). Further that equation is integrated two times to get the force equation between two magnets, equation (4.24), and three times to get the solution for the force between a magnet and a coil, equation (4.25). This procedure is also applied to get the force between circular magnets and a concentric coil that interact.

4.2.1 Force between two long parallel current carrying wires The force between two infinitely long parallel wires is well known and has until recently formed the basis of the international definition of the ampere: two long parallel wires separated by a distance of one meter influence on each other with a force of 2x10-7 Newton per meter of length when the current in each is one ampere. It is assumed that the diameters of the wires are negligible compared to their separation. Se Fig. 4.3 below.

Fig. 4.3 Direction of force between two leaders The force per unit length is

. (4.1)

For a sufficiently long pair of conductors each of length , neglecting end effects, the force between the wires, , can be expressed as

. (4.2)

resF

0

2a b

w wI IdFf

dlNm

lw wF

0

2a b

w wI IF l N

F F aI

bIF

F aI

bIl

19

4.2.2 Axial force between two concentric circular coils

Fig. 4.4 Direction of force between two circular leaders

(4.3)[16]

where

. (4.4)

4.2.3 Modelling permanent magnets According to AMPÈRE´S CIRCUITAL LAW a permanent magnet can be modelled as an air wound solenoid carrying a surface density, , and the surface current . A simplified but less accurate model is to exchange the surface current with a coil with only one turn, and the equivalent line current, , see Fig. 4.5 below.

02 2

sin2

a bres

I IF lz g

N

22tan tan

D dl

z zg D d

I

I

bI

aI

bI

aI

g

zresFresF

resF

resFD

d

20

Fig. 4.5 Model of permanent magnets surface current To calculate the current travelling around the magnet or the surface current density which is the source for the magnetic field , Ampère´s circuital law is used. To illustrate the derivation of the surface current , the illustration in Fig. 4.6 are used.

Fig. 4.6 Cross section of a magnet inside an iron yoke Ampère´s circuital law for the closed line integral around contour (closed curve) C states

(4.5)

IB

I

0 0 mCdl I t

mt crB

Iron yokeMagnet

I

Cylinder magnet

Ring magnet

N S

S

N N

S

S

N

Equivalent surface current sheets

Long straight magnet Simplified model using line conductors

21

Integrating left side of equ. (4.5), results in

(4.6) Resulting in the surface density

(4.7)

The total current on the side of the magnet and the magnetic permeability of vacuum,

results in the current as

. (4.8)[5]

(4.9) The surface density

(4.10)

4.2.4 Lift force between a long magnet and a thin parallel wire It was shown in section 4.2.3, that a magnet could be modelled using current sheets. The force between one such current sheets and a parallel wire will now be derived using equ. (4.1) and (4.9).

Fig. 4.7 Cross section of a straight magnet with sheet current and a leader With the definitions of the forces seen in Fig. 4.7 above is negative and is positive. Each sheet of the magnet is represented as many wires with current , thicknes and force contribution .

0r m mB t t

0

rB

I7 1 1

0 4 10u VsA m I

70 4 10

r mrm

B tBI t

mI t

74 10rB

1f 2fdI mdz

df

Cross section

N

I 1f2f

mtrB

gw

z

22

(4.11)

(4.12)

As previously mentioned permanent magnets can be modelled as current sheets. A current sheet can be said to be built up of an infinite number of wires, each carrying a current

(4.13) Thus equ.(4.11) can be integrated using equ.(4.12), (4.13) and (4.1) resulting in a force 1f , between the wire and the closest magnetic surface current sheet as follows.

(4.14)

In the same way as above the force can be obtained from the wire and the left side of the magnet with the new distance between them, .

1 1f df

11

sinw wfdf dI

I

mdI dz

2 2201

22 2

sinsin

2

m ctm

m cmtm

m c

z z zI g z z zf dz

g z z z

022 12

m cm c

mmm c

m

x z z zI z z z dz dx dz dxg z z z dz

2 20 02 2

0

1 ln2 2 2

rI I Bx dx g xg x

2220

0 2

ln4

mr

mr

tr

m c t

I Bg z z z

22

22

2ln4

2

mrc

r

mrc

tg z zI B

tg z z

Nm

2fg w

23

(4.15)

The resulting force on the magnet from the wire is

(4.16)

The forces and always have opposite sign and since is closer to the wire.

4.2.5 Lift force between two long parallel magnets It was shown in section 4.2.4, that a magnet could be modelled using current sheets. The force between two such current sheets will now be derived using the result from equ. (4.14)

Fig. 4.8 Cross section of two straight magnets with sheet currents

Force between a current sheet and a wire according to equ. (4.14)

(4.17)

As previously mentioned permanent magnets can be modelled as current sheets. A current sheet can be said to be built up of an infinite number of wires, each carrying a current

Resulting in a force . Thus equ.(4.17) can be integrated to get the total force between two current sheets. Thus

22

2 22

2ln4

2

mrc

r

mrc

tg w z zI Bf

tg w z z

Nm

1 2f f f Nm

1f 2f 1 2f f 1f

22

22

2ln4

2

mrs

s w rs w

mrs

tg z zdF I Bf

dl tg z z

Nm

sdI dz

s sdf

Cross section

NN

1f2f

mrtrB

gmrw

zmst

rB

msw

24

Integrating to get the total force per meter between two current sheets results in

(4.18)

Solving Part 1 of equ. (4.18):

2 12

222

22ln

4 22

mr

mrs

mrrs s

mrs

s

tu ztg z z

tBdf dI u ztg z z dI dz

221

222

ln .4

srs

s

g u zB dzg u z

2221

222

2

ln4

ms

ms

t

srs s s

t s

g u zBf dzg u z

2 22 22 2

1 2

2 2

ln ln4

ms ms

ms ms

tt

rs s s s s s

t t

Bf g u z dz g u z dz

221 1ln

1 1

s s s

ss

Substitutiong u z dz m u z

dm dz dmdz

2 2 2 2ln 1 lnSolved in

g m dm g m dmAppendix A

2 2 1ln 2 2 tan mm m g m gg

Part 1 Part 2

25

(4.19) Solving Part 2 of equ. (4.18): Part 2 is solved in the same way as Part 1 above with the exception that the variable m is different and this gives equation(4.20).

(4.20)

Substitution of equation (4.19) and (4.20) into equation (4.18) yields

(4.21)

1

12 2 1 2ln 2 2 tan

taken from above

2

s

mr

mrs

m u ztu zmm m g m g

gtm z z

22 1 2ln 2 2 tan

2 2 2

mrs

mr mr mrs s s

t z zt t tz z z z g z z gg

22 2 2 12 2ln ln 2 2 tan

2mr

s s s stmg u z dz m m g m g m u z z z

g

22 1 2ln 2 2 tan

2 2 2

mrs

mr mr mrs s s

t z zt t tz z z z g z z gg

2

2

2

2

22 1

4

22

2ln 2 2 tan2 2 2

ln 22 2

mss

mss

mss

mss

tz

tz

rs s tz

tz

mrs

mr mr mrs s s

Bf

mr mrs s

t z zt t tz z z z g z z gg

t tz z z z g 1 22 tan2

mrs

mrs

t z zt z z gg

26

. (4.22)

This can be rewritten with the substitutions as below

(4.23)

The total force per meter of magnet length including the force contributions from all four current sheets are

[N/m] (4.24)

22 1

22

2 2ln 2 2 tan2 2 2 2 2 2

ln 2 22 2 2 2 2 2

4

mr ms

mr ms mr ms mr ms

mr ms mr ms mr ms

rs s

t tzt t t t t tz z g z gg

t t t t t tz z g z

Bf

1

22 1

2 2tan

2 2ln 2 2 tan2 2 2 2 2 2

ln2 2 2 2

mr ms

mr ms

mr ms mr ms mr ms

mr ms mr ms

t tzg

g

t tzt t t t t tz z g z gg

t t t tz z2

2 1 2 22 2 tan2 2

mr ms

mr ms

t tzt tg z gg

2 2 1 11 1 1

2 2 1 21 2 2 2

2

33

4

ln 2 2 tan

ln 2 2 tan2 2

2 2 4

2 2

2 2

mr ms

mr ms r

mr ms

mr ms

tt t g t gg

Substitutiont t tt z t t g t g

gt t Bt z

t t tt z

t tt z

2 2 1 33 3

2 2 1 44 4 4

ln 2 2 tan

ln 2 2 tan

tt g t gg

tt t g t gg

Nm

1

2

1 2 3 43

4

2 2 1

2 2

2 2( , ) ( , ) ( , ) ( , ) ( , )

42 2

2 2

( , ) ln 2 2 tan

mr ms

mr ms

rmr mss s

mr ms

Substitutiont tt z

t tt zBt tf z g a t g a t g a t g a t gt z

t tt z

ta t g t t g t gg

,m mf z g

( , ) , , , ,m m s s s s ms s s mr s s ms mrf z g f z g f z g w f z g w f z g w w

27

4.2.6 Lift force between a long magnet and a thick parallel wire Fig. 4.9 Cross section of a straight magnet with sheet current and a thick leader or many leaders. Lift force between two long parallel magnets is derived in equ. (4.24) and consists of four different forces from current sheets. If the stator magnet is exchanged for a thick coil, the coil can be regarded as a thick current sheet, each consisting of an infinite number of thin current sheets of thickness on the distance from the respective magnet sheet. Thus the total

force between the coil and the magnet is made up of only two force

contributions

, (4.25) where

(4.26)

and

. (4.27)

The force [N/m2], can be explained as follows: The coil can be regarded as a thick current sheet, built up of an infinite number of thin current sheets of the infinitesimal thickness , each to which the equation (4.23) is applicable. Now gap in the integrals (4.23), (4.26) and (4.27) is an integration variable between the current sheet, s, on the rotor magnet and the sheet of integration on the stator coil. The force is the force per unit length of the sheets, and per unit width of the coil. It differs from the force in equation (4.23) in that the current density [A/m2] is used instead of the sheet current density [A/m]. Here is the effective current density in a cross section of the coil and is defined by

, (4.28)

so that

cd

( , )m cf z g Nm

1 2,m c s c s cf z g f f

1, ( , )

cg w

s c s sg

f z g p z d

2, ( , )mr c

mr

g w w

s c s sg wf z g p z d

,s sp z

d g

s ( , )s sp z

s sf

effj

effj

effc c

n Ijt w

Cross- section

N

n I

N

1s cf2s cf

mrtz

cd

cw

mrw

2 1

ctg

28

(4.29)

Here it is recalled that , n=1..4, is a substitution and is derived in equation (4.23). Now the stator magnet with thickness is replaced with the coil with thickness which gives

(4.30)

With the equation

(4.31)

The total force on the magnet from the coil, using equation (4.26),(4.27) and (4.31) is

(4.32)

Using the same substitution as above, each integral representing the force on the sheet of the magnet from the coil, can be solved as follows:

(4.33)

1 2 3 4 2, ( , ) ( , ) ( , ) ( , ) .4eff r

s s

j B Np z a t a t a t a tm

nt

mst ct

1

2

3

4

2 2 1

,2 2

,2 2

,2 2

2 2

( , ) ln 2 2 tan .

mr c

mr c

mr c

mr c

t tt z

t tt z

t tt z

t tt z and

ta t g t t g t gg

1 2 3 4, ( , ) ( , ) ( , ) ( , )4

rs s

c c

n I Bp z a t a t a t a tt w

( , ) ( , )c

mr c

mr

g wg w w

m c s s s sg wg

f p z d p z d

max

min

max

min

min max 1,2

1 2 3 4

, , ,

( , ) ( , ) ( , ) ( , )4

ns c s sn

r

c c

f z p z d

n I B a t a t a t a t dt w

max max max max

min min min min

1 2 3 4( , ) ( , ) ( , ) ( , )4

r

c c

n I B a t d a t d a t d a t dt w

1 min max 2 min max 3 min max 4 min max( , , ) ( , , ) ( , , ) ( , , )4

r

c c

n I B A t A t A t A tt w

29

The solution of will now be derived explicit

(4.34)

The solution of and is derived explicit in Appendix A, Variable Integration, and the solution is

(4.35)

(4.36)

With equ. (4.35) and (4.36) inserted in equ. (4.34) gives equ. (4.37) with the boundary

.

(4.37)

( , )A t

2 2 1 2 2

1

( , ) ( , ) ln 2 2 tan ( , ) ln

( , ) tan

SubstitutiontA t a t d t t t d h t t

te t

( , ) 2 2 ( , ) ( , ) 2 2 ( , )t h t d t d e t d t h t d t e t d

( , ) 2 2 ( , )d dt H t t E t

( , )H t ( , )E t

2 2 2 2 1( , ) ( , ) ln ln 2 2 tanH u h u d u d u uu

2 1 2 1

1tan tan

( , ) ( , ) tan2

t t tt tE t e t d d

min max,

max

max

min

min

max

min

2 2 1

min max

2 1 2 1

ln 2 2 tan 2

, ,

tan tan

t t t tt

A tt t t

t

22 1 maxmax max max

22 1 minmin min min

max min

2 1 2 1 maxmax max

max

2 1min

ln 2 2 tan

ln 2 2 tan

2 2

tan tan

tan

t t tt

t t tt

t t

t t tt

2 1 minmin

min

tant t tt

30

4.2.7 Axial forces between two concentric ring magnets Turning the straight magnets of infinite length into rings, as seen in Fig. 4.10, with specific diameters results in the lengths equal the circular perimeters. The total force from the outer magnet on the inner rotor magnet only act in axial direction through the radial force component cancel out each other acting to centering the magnet in radial direction if not concentred.

Fig. 4.10 Cross section of two concentric ring magnets

The force on the rotor magnet, ,in Fig. 4.10, is two times the force on the right cross section of the inner rotor magnet, which is the sum of forces on sheet and from the four sheets in the cross section sheets , , and results to the following equation

(4.38)

The total force between the magnets, , on the rotor magnet taken the magnet sheet lengths into action results in the following equation

(4.39)

( , )mr msf z g

3S 4S

1S 2S 7S 8S

3 1 3 2

3 7 3 8

4 1 4 2

4 7 4 8

, ,

, ,( , ) 2

, ,

, ,

s s ms s s

s s mr s s ms mrmr ms

s s mr ms s s mr

s s mr mr s s ms mr mr

f z g w f z g

f z g D f z w g Df z g

f z w g w f z w g

f z g w d f z w g w d

Nm

( , )mr msF z g

3 1 3 2

3 7 3 8

4 1 4 2

, ,2 4 2 4

, ,2 4 2 4

( , ) 2, ,

2 4 2

ms mr ms mr ms mr mr mss s s s

mr ms mr ms mr ms mr mss s s s

mr msms mr mr ms ms mr

s s s s

D D D D d D D df z f z

D d D d D D D Df z f zF z g

D d d D d df z f z

4 7 4 8

4

, ,2 4 2 4

mr ms

mr ms mr ms mr ms mr mss s s s

d d

d d d d d D d Df z f z

N

mrd mrw

mrD g

mswmsd

msD

mst

z

s1 s3 s4 s5

mrt

s2 s7 s8

N

S

N

S

s6

31

Where the individual sheet force equation with resp. variable follow as

(4.40)

where

[A/m] (4.41)

1

2

1 23

4

2 2 1

2 2

2 2( , ) ( , ) ( , )

42 2

2 2

( , ) ln 2 2 tan

mr ms

mr ms

rmr mss s eff eff eff

mr ms

eff eff effeff

Substitutiont tt z

t tt z

Bt tf z g a t g a t gt z

t tt z

ta t g t t g t gg

3 4( , ) ( , )eff effa t g a t g

70 4 10r rB B

32

4.2.8 Axial force between a circular magnet and a concentric coil The magnet and coil in chapter 4.2.6 is turned into a circular magnet and a concentric coil as seen in Fig. 4.11 in the same way as was done with permanent magnets in the previous chapter.

Fig. 4.11 Cross section of a circular magnet and a concentric coil The total force on the magnet from the coil taken into action if the length if stretched out.

[N] (4.42)

The force on the magnet from the coil is the sum of the forces on respective sheet of the magnet from the coil as follow

[N/m] (4.43)

(4.44)

The force between a magnetic sheet and the coil is according to equ.(4.33)

(4.45)

And is according to equ. (4.37)

, ,2

c mm c m c

d DF z g f z g

m cf

1 2 3 4,m c s c s c s c s cf z g f f f f

, , , ,,

, , , ,s c c s c mr mr c

m cs c mr mr mr mr c s c mr mr c

f z g g w f z g w g w wf z g

f z g w d g w d w f z g D g D w

max

min

1 min max 2 min maxmin max

3 min max 4 min max

( , , ) ( , , ), , ,

( , , ) ( , , )4r

s c s sc c

A t A tn I Bf z f z dA t A tt w

min max( , , )A t

mrd mrw

mrD g

cwcd

cD

mstz

mrtNS

s1 s2 s3

n I

s4

33

(4.46)

With the sub equations as follow.

IF above.

22 1 maxmax max max

22 1 minmin min min

min max max min

2 1 2 1 maxmax max

max

ln 2 2 tan

ln 2 2 tan

, , 2 2

tan tan

t t tt

t t tt

A t t t

t t tt

2 1 2 1 minmin min

min

tan tant t tt

10 tan2

tt

1

2

3

4

,2 2

,2 2

2 2

.2 2

mr c

mr c

mr c

mr c

t tt z

t tt z

t tt z and

t tt z

34

4.3 Equation verification To verify the equations derived in chapter 4.2.7 and 4.2.8, two other ways of calculating the forces analytically will be used. The first of them is also a mathematic model entirely based on Matthias Lang’s Doctoral thesis [17]. The second method is through simulation by a Finite Element Method program called FEMM[18], made by David Meeker. The equations derived in chapter 4.2.8 and 4.2.7, will be referred to as Marcus G., the author, when compared with the mathematic method referred to as Matthias L. and the third referred to as the FEMM method. A short introduction to these methods will now follow.

4.3.1 Magnetic field calculation methods used for verification comparison To verify the equations derived in chapter 4.2.8 and 4.2.7 another way of calculating the forces analytically will be used, entirely based on Matthias Lang’s Doctoral thesis [17]. Matthias L. made a mathematic program using SCILAB[19], similar to MATLAB[20] where he used the equations in his thesis, to divide the magnets surface into small segments and calculated the force between these small segments on different magnets. He used the equation for elliptical integral to get the distance between the segments, and the forces were summarised. Matthias Lang’s code will not be presented here since it is presented in his thesis. It can also be mentioned that to be able to use Matthias Lang’s code for magnets versus magnets, to compare it with the equations for magnet versus coils forces in chapter 4.2.8, Matthias Lang´s code had to be modified. The coil with turns and current was divided into a large number of thin cylinders and the current in the coil was divided equally on these thin cylinders. Each of these thin cylinders with the current on radius behaved as a magnetic sheet with a sheet current equal to .

(4.47)

The force between each of these sheets and the magnet can be calculated and summarised. The number of cylinders, , that the coil was divided into was increased until a value with a desired amount of significant numbers in the solution was achieved. The additional Finite Element Method Magnetic program was introduced that is called FEMM[18], made by David Meeker. FEMM is suitable for solving low frequency electromagnetic problems on two-dimensional planar and axisymmetric domains. In this thesis the bearings axisymmetric domains are studied. FEMM is not a 3D finite element program and can therefore not calculate the radial stiffness when the magnets and coils are not concentric which is not studied in this thesis. Involving a coil and a magnet FEMM can calculate the magnetic forces in at least two ways, by using Lorentz Forces or with Weighted Stress Force. Using FEMM to calculate the forces between two magnets the method of using Weighted Stress Tensor can be used on each magnet and then calculate the average of them. Choosing right mesh size is important for the result. Smaller mesh size at the boundaries of the magnets, using up to 500 000 Nodes and 1 000 000 Elements to get good results in the simulations.

n CIm

SI r

SI

CS

I nIm

m

35

4.3.2 Equation verification for two concentric ring magnets

Fig. 4.12 FEMM simulation of forces between circular magnets

Table 4.1 Comparison between results and analysis of arrangement in Fig. 4.12

Fig. 4.13 Three methods compared for magnet versus magnet force calculation

Displacement Matthias L Marcus Gz Force Force rotor stator average

[mm] [N] [N] [N] [N] [N]0,0 0,0 0,0 -0,009 0,0013 -0,00510,0005 8,2741 8,2729 8,2652 -8,2863 8,27570,001 16,9614 16,9597 16,9423 -16,9832 16,96280,0015 25,9281 25,9258 25,9029 -25,9326 25,91780,002 34,068 34,0617 34,0408 -34,179 34,10990,0025 40,2038 40,1866 40,1717 -40,2036 40,18770,003 43,99 43,9533 43,947 -43,9968 43,97190,0035 45,5095 45,4445 45,4716 -45,4913 45,48150,004 44,8552 44,7539 44,8226 -44,8545 44,83860,0045 42,0292 41,8844 41,9755 -42,0197 41,99760,005 37,0521 36,8584 37,02 -37,0541 37,0371

FEMM simulationMathematic modelForce between two magnets, B=1,2 ur=1,0

Weighted stress tensor for

Magnet against magnet

-10

0

10

20

30

40

50

0 0,002 0,004 0,006 0,008 0,01

z[mm]

Forc

e[N

]

Matthias LMarcus GFEMM

N N

Dmr = 24 mmdmr = 10 mmtmr = 7 mm

Dms = 32 mmdms = 26 mmtms = 4 mm

Magnetic propertyBr = 1,2 T

r = 1,0

Dimension

z[m]

36

4.3.3 Equation verification for concentric ring magnet and coil

Fig. 4.14 FEMM simulation of forces between circular magnet and a concentric coil

Table 4.2 Comparison between results and analysis of arrangement in Fig. 4.14

Fig. 4.15 Three methods compared for magnet versus coil force calculation

4.3.4 Comments on equation verification result The difference between Matthias Lang’s 3D model and our curved 2,5D (referred to as Marcus G), model is primarily due to our estimated length in the air gap, the length of the curved model. The results show that the deviation is negligible within the bearings operating range and the deviations occur at larger lateral movements.

Displacement Matthias L Marcus G Lorentz force Weightedz Force Force Neg. (JxB) stress force

[mm] [N] [N] [N] [N]0,0 0,0 0,0 0,0 -0,01720,0005 -0,4709 -0,4667 -0,4709 -0,46970,001 -0,918 -0,9108 -0,9181 -0,93870,0015 -1,3223 -1,3143 -1,3224 -1,35070,002 -1,6729 -1,6665 -1,6729 -1,70080,0025 -1,9649 -1,963 -1,9649 -1,99150,003 -2,1971 -2,2021 -2,1971 -2,22660,0035 -2,3697 -2,384 -2,3696 -2,39980,004 -2,4835 -2,5094 -2,4835 -2,52410,0045 -2,5401 -2,5795 -2,5401 -2,56350,005 -2,5414 -2,5961 -2,5413 -2,5732

FEMM simulationMathematical modelForce between magnet and coil, N=40, I=5A, B=1,2 ur=1,0

Magnet against coil, N=40, I=5A

-3

-2,5

-2

-1,5

-1

-0,5

00 0,002 0,004 0,006 0,008 0,01

z[mm]

F[N

]

Matthias LMarcus GLorentz forceWeighted

N I

Dmr = 24 mmdmr = 10 mmtmr = 7 mmDc = 32 mmdc = 26 mmtc = 6 mm

Nc = 40 turnsØ = 1 mmI = 5 A

Magnetic propertyBr = 1,2 T

r = 1,0

Dimension

z[m]

37

4.4 Optimisation of bearing configurations The method for optimisation is to build up an equation for the force between the rotor and stator part of the bearing using the developed equations in section 4.2.7 and 4.2.8. This was done in Scilab, a similar program to Matlab. The Scilab code can be found in Appendix B for the bearing using different input for the dimensions of the different configurations. The two different constants that can be calculated for the bearing from the code, that tells the performance of the bearing is the bearing stiffness sk and ik . The equations are true as long as the magnets are not saturated or iron is used to lead the magnetic field. In the novel bearing iron is used to penetrate the magnetic field to the air gap to try to increase the performance. To analyse the performance increase with the added iron, FEM have to be used which is done in section 4.5 for the different configurations that are chosen in this section. As seen in Fig. 3.2 different bearing configurations are to be evaluated. But the first step will be to choose two sets of ring magnets, two for the rotor and two for the stator and stick to them in the different configurations. The other parts of the magnetic bearings configurations for dimensioning and evaluation is the amount of coil in the stator and the amount of pole shoes in the rotor and stator. The thicker pole shoe on the rotor the greater distance it will be from the centre of the coil to the centre of the rotor magnets, at the same distance as the rotor pole shoe expands.

4.4.1 Dimensioning the magnets One design criteria of the bearing are that the rotor and stator part of the bearing is equally thick which is logical. Since the coil is in the stator on “behalf” of magnetic material it is possible to have thicker magnets in the rotor than in the stator. Two sets of magnets who meet this design criterion were chosen. Since magnets with special customised dimensions can have very long time of delivery the magnets that best meet the criteria on the lab shelf are presented below. The coil is easier to vary and can be hand wind. Dimensions for Magnet 1 for the rotor and Magnet 2 for the stator can be seen in Fig. 4.16 together with the axial force as function of displacement between the mid planes. The air gap between the rotor and stator part is 1mm on each side.

Fig. 4.16 Force between one rotor magnet and one stator magnet as function of displacement

One magnet against another

-50-40

-30-20-10

010

2030

4050

-0,01 0 0,01 0,02 0,03 0,04 0,05

Displacement, z [m]

Forc

e [N

]

Dmr = 24 mmdmr = 10 mmtmr = 7 mmBr = 1,2 T

Dms = 32 mmdms = 26 mmtms = 4 mmBr = 1,2 T

Magnet 2

Magnet 1

38

Fig. 4.17 Force on the rotor from stator magnets in bearing configuration 10 100-8 from Fig. 4.21. The bearing stiffness due to magnet forces between the magnets is the slope coefficient

[N/m] (4.48)

4.4.2 Dimensioning the coil The second part of dimensioning the magnetic bearing is the coil. The nomenclature seen in Table 4.3 together with the current is used in the following formula derivations for the coil.

Table 4.3 Nomenclature for the coil

The fill factor for a hand wounded coil with litz wire usually is around 51% and is derived in the following equation.

(4.49)

The total length of the thread in the coil can be derived from the following equation

(4.50)

Force on the rotor from the stator magnets

-50

-40

-30

-20

-10

0

10

20

30

40

50

-0,0015 -0,001 -0,0005 0 0,0005 0,001 0,0015

Displacement, z [m]

Force [N]

sk

sFkz

I

Thickness Inner diam. Outer diam. Coil thread diam. Coil turns[m] [m] [m] [m]t c d c D c N

Coil

2 2

c

Coil turns=NCoil turns Thread area 2Fill factor % = Thread area=

Tot coil area 4 4D

Tot coil area=2

c c c

c c

ND d t

d t

2 2c c

2 22c

2

t DCoil volume=

4DCoil volume Fill factor 2Coil thread length m = Fill factor=

Thread area 4 2

Thread area=4

c

c

c c c c c

d

d NND d t D d

39

The resistance in the coil can be derived from the following equation

(4.51)

Four coils were chosen to suit the chosen bearing configurations who meet the total volume of its disposal and with fill factor equal to 51%. Data for the coil configurations can be seen in Table 4.4 and are used in the bearing configurations presented in Fig. 4.21.

Table 4.4 Data for the different Coil configurations

Force between one concentred rotor magnet inside the different coils depending on displacement, z is as.

Fig. 4.18 Force between one rotor magnet and the stator coil as the current I=7A

2 2c

2 2c-8 -8

22

DCoil thread length= 2 DCoil thread length 2Coil Resistance =1,67 10 1,67 10

Thread areaThread area=

4

c

cc c

c c

d Nd ND d

D d

Force between "One" rotor magnet and "The" coil configurations

-10

-8

-6

-4

-2

0

2

4

6

8

10

-0,008 -0,003 0,002 0,007

Displacement, z [m]

Forc

e [N

]

Coil 10 104-1

Coil 10 104-2

Coil 10 104-3

Coil 10 104-4

Thickness Inner diam. Outer diam. Coil thread diam. Coil turnst c d c D c N

Part no. [m] [m] [m] [m]Coil 10 104-1 0,00775 0,026 0,032 0,00055 42Coil 10 104-2 0,006 0,026 0,032 0,00055 39Coil 10 104-3 0,006 0,026 0,031 0,0006 32Coil 10 104-4 0,00775 0,026 0,032 0,00035 123

Data Coil configurations

40

Fig. 4.19 The total force on the rotor magnets from the coil as the current I=7A

The rotor is constraint from moving in axial direction by mechanical hinder so that it will not pop out but rest on a touch down surface. The maximum allowed axial travel for the rotor before touching mechanical surface is . The praxis for magnetic bearing is that the allowed travel in axial direction is half the one in radial direction, the air gap distance. Praxis is also that one want to achieve the variables sk and ik to be linear. This will simplify the controller. In following Fig. 4.20 is Fig. 4.19 zoomed in with the constraint for the maximum allowed axial travel for the rotor. Inside the constraint the different curves can be approximated with a straight line or a constant force F.

Fig. 4.20 Magnetic force on the rotor from the coil with axial travel 0,25mm and I=7A

The bearing stiffness due to magnet forces between the magnets and the coil can be calculated for each coil configuration according to

[N/A] (4.52)

Fig. 4.20 shows that is linear within the operating range that is a constant in this case. The force is proportional to the current.

Force between "Both" rotor magnets and "The" coil configurations

-20

-15

-10

-5

0

5

-0,008 -0,006 -0,004 -0,002 0 0,002 0,004 0,006 0,008

Displacement, z [m]

Forc

e [N

]

Coil 10 104-1

Coil 10 104-2

Coil 10 104-3

Coil 10 104-4

0, 25mm

Force between "Both" rotor magnets and "The" coil configurations

-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0-0,00025 -0,00015 -0,00005 0,00005 0,00015 0,00025

Displacement, z [m]

Forc

e [N

]

Coil 10 1004-1

Coil 10 1004-2Coil 10 1004-3Coil 10 1004-4

ik

iFkI

ik

41

4.4.3 Bearing configuration drawing 10100 The final eight bearing configurations for further evaluation are presented here. The values for

and in the Fig. 4.21 below comes from FEM analysis in next chapter. The eight bearing configurations have a number according to the format 10 100 - [1..8].

Fig. 4.21 Bearing configurations and assembly drawing 10100

ik sk

Drawing Nr:SUB: 1 2 3 4 5 6 7 8

Pos no. Part no. Description D d t N Ø1 2 2 2 2 2 2 2 2 10 101-1 Magnet 24 10 72 2 2 2 2 2 2 2 2 10 101-2 Magnet 32 26 43 1 1 1 1 1 10 102-1 Fe rotor pole shoe 24 10 1,754 2 2 10 102-3 Fe stator pole shoe 32 26 0,8755 1 1 10 103-1 Al rotor shims 24 10 1,756 2 2 10 103-2 Al stator shims 32 26 0,8757 1 10 104-1 Coil 31 26 6 32 0,558 1 1 1 10 104-2 Coil 32 26 6 39 0,559 1 1 10 104-3 Coil 31 26 7,75 42 0,610 1 10 104-4 Coil 32 26 7,75 123 0,3511 1 10 105-1 Coil shoulder 32 31 6

AMB t. [mm] 15,75 15,75 14 15,75 15,75 15,75 15,75 15,75ki [N/A]= 0,952 1,073 0,924 0,987 1,132 1,517 2,192 3,14236

ks [N/m]= 37716 34069 42205 37601 33719 48283 78922 34069ki/ks [m/A]= 2,5E-05 3,1E-05 2,2E-05 2,6E-05 3,4E-05 3,1E-05 2,8E-05 9,2E-05

Bearing Configuration10 100-

Quantity

Individual part properties Dimensions [mm]

32

42

4.5 FEMM simulation of bearing configurations In this section the eight bearing configurations from the previous Fig. 4.21, will be analyzed through FEMM (Finite Element Method Magnetic) to find out the bearings rotor stiffness due to the stator magnets, , and due to the coil, . These coefficients can be used in the control system in chapter 5 to analyse and optimise the control system. A short briefing on how the bearings coefficients, and are calculated from the FEMM results in the following sections to this chapter will follow. The FEMM models are presented in a FEMM analysis with the magnetic flux lines and concentration. To the right of these figures is the model presenting what materials are used, the magnets pole direction, the number of turns in the coil and the diameter of litz wire that the coil is made of. A table follows were the force results are presented from the FEMM analysis depending on displacement and if the current in the coil is turned OFF or ON . In the table at the different displacement are calculated according to

(4.53) The bearing stiffness, , due to magnet versus coil forces are calculated according to

[N/A] (4.54)

The bearing stiffness, , due to magnet versus magnet forces are calculated according to

[N/m] (4.55)

In some of the configurations aluminium (Al) cylinders are used and in the next configuration iron (Fe) is used. This it due to that aluminium can substitute air so that the two results can be compared to realize what effect it has by applying iron in the model on the stiffness coefficients. The dimensions of the different parts in the different bearing configurations can be seen in the previous Fig. 4.21. Another interesting equation is the minimum current for the amplifier to master/overcome the rotor from the touch down bearing at start up which is when the rotor is displaced 0,25mm. Further than 0,25mm the rotor can’t be axially displaced due to mechanical constraints in the final bearing design, beyond this thesis. Minimum current for the amplifier

[A] (4.56)

sk ik

sk ik

0I 5I A

F

( 5, ) ( 0, )F z F I z F I z

ik

5i

F z F zk z

I

sk

1

1

0, 0,n ns n

n n

F I z F I zk z

z z

1

min, 0,00025iamplifier

s

kIk

43

4.5.1 Bearing configuration 10 100-1 Description: Al pole shoe in rotor

Fig. 4.22 FEMM simulation of bearing 10 100-1

Table 4.5 Property of bearing 10 100-1

4.5.2 Bearing configuration 10 100-2 Description: Fe pole shoe in rotor instead of Al in bearing configuration 10 100-1.

Fig. 4.23 FEMM simulation of bearing 10 100-2

Table 4.6 Property of bearing 10 100-2

z F (I=0) F (I=5A) F ki ks ki/ks[m] N [N] [N] [N/A] [N/m] [m/A]0 0,00242 4,76413 4,76171 0,95234

0,0001 3,77402 8,53374 4,75972 0,95194 37716,03 2,52E-050,0002 7,50403 12,25860 4,75458 0,95092 37300,050,0003 11,19875 15,94460 4,74585 0,94917 36947,25

z F (I=0) F (I=5A) F ki ks ki/ks[m] N [N] [N] [N/A] [N/m] [m/A]0 -0,00370 5,35975 5,36345 1,07269

0,0001 3,40322 8,76453 5,36132 1,07226 34069,13 3,15E-050,0002 6,74314 12,09845 5,35532 1,07106 33399,200,0003 10,05405 15,40005 5,34600 1,06920 33109,15

44

4.5.3 Bearing configuration 10 100-3 Description: Without pole shoes, making the bearing shorter.

Fig. 4.24 FEMM simulation of bearing 10 100-3

Table 4.7 Property of bearing 10 100-3

4.5.4 Bearing configuration 10 100-4 Description: With Al pole shoes in both rotor and stator

Fig. 4.25 FEMM simulation of bearing 10 100-4

Table 4.8 Property of bearing 10 100-4

z F (I=0) F (I=5A) F ki ks ki/ks[m] N [N] [N] [N/A] [N/m] [m/A]0 0,00033 4,62133 4,62100 0,92420

0,0001 4,22087 8,83911 4,61825 0,92365 42205,39 2,19E-050,0002 8,41440 13,02575 4,61136 0,92227 41935,300,0003 12,56345 17,16305 4,59960 0,91992 41490,55

z F (I=0) F (I=5A) F ki ks ki/ks[m] N [N] [N] [N/A] [N/m] [m/A]0 -0,00687 4,92859 4,93546 0,98709

0,0001 3,75328 8,68650 4,93323 0,98665 37601,45 2,62E-050,0002 7,51187 12,43860 4,92674 0,98535 37585,900,0003 11,19080 16,10695 4,91615 0,98323 36789,35

45

4.5.5 Bearing configuration 10 100-5 Description: With Al stator pole shoe and Fe rotor pole shoe instead of Al.

Fig. 4.26 FEMM simulation of bearing 10 100-5

Table 4.9 Property of bearing 10 100-5

4.5.6 Bearing configuration 10 100-6 Description: With Fe stator pole shoe and Fe rotor pole shoe instead of Al.

Fig. 4.27 FEMM simulation of bearing 10 100-6

Table 4.10 Property of bearing 10 100-6

z F (I=0) F (I=5A) F ki ks ki/ks[m] N [N] [N] [N/A] [N/m] [m/A]0 -0,00475 5,65499 5,65974 1,13195

0,0001 3,36713 9,02419 5,65707 1,13141 33718,72 3,36E-050,0002 6,75012 12,39950 5,64938 1,12988 33829,950,0003 10,05675 15,69400 5,63725 1,12745 33066,30

z F (I=0) F (I=5A) F ki ks ki/ks[m] N [N] [N] [N/A] [N/m] [m/A]0 -0,00722 7,57648 7,58370 1,51674

0,0001 4,82105 12,40005 7,57900 1,51580 48282,71 3,14E-050,0002 9,66282 17,23550 7,57268 1,51454 48417,700,0003 14,42775 21,99630 7,56855 1,51371 47649,30

46

4.5.7 Bearing configuration 10 100-7 Description: With Fe pole shoe and Fe coil shoulder

Fig. 4.28 FEMM simulation of bearing 10 100-7

Table 4.11 Property of bearing 10 100-7

4.5.8 Bearing configuration 10 100-8 Description: This bearing is exact as 10 100-2 with the difference in the coil. The coil is thinner and has more turns, 123 turns instead of 42 and a thinner litz wire with 0,35mm. This configuration was an extra configuration to meet the need of using 24V (instead of 12V), in the lab for the test rig according to Madeleine Ymerson’s theses [21] where this bearing is implemented further in a control system than what is done in next chapter. The reason to why this configuration or configuration 10 100-2 would have been chosen for the prototype to be made and for Madeleine’s theses is because it is no iron in the stator that can influence negative in radial stiffness and it seen easier to test and develop the control system if it is not two many uncertain variables that can influence negative. Configuration 10 100-6 and 10 100-7 could be analysed in a 3D-FEM program to see that the radial stiffness is sufficient.

Table 4.12 Property of bearing 10 100-8

z F (I=0) F (I=5A) F ki ks ki/ks[m] N [N] [N] [N/A] [N/m] [m/A]0 0,01291 10,97185 10,95894 2,19179

0,0001 7,90507 18,82425 10,91919 2,18384 78921,59 2,77E-050,0002 15,78805 26,60780 10,81975 2,16395 78829,850,0003 23,57270 34,22840 10,65570 2,13114 77846,50

ki ks ki/ks[N/A] [N/m] [m/A]3,14236 34069 9,22352E-05

47

5 CONTROLLER An error in the magnetic bearings shaft displacement from the displacement sensor can be regulated with a digital control system.

5.1 Simple control system generated from the bearings equation of motion According to Newton’s second law where is the mass of the rotor and is the acceleration in axial direction. The total force also equals the force components from the magnetic repulsion and attraction forces which depend on the stiffness and displacement

and also on the force component from the coil depending on and current according to equation(4.57). (4.57) With equation (4.57) turned into a current control system with a PID regulator can be seen in Fig. 5.1.

Fig. 5.1 Magnetic bearing turned into a control system with a PID controller

5.2 Further development of the control system with Matlab and Simulink The control system in Fig. 5.1 are further developed according to the laboratory equipment and displacement sensor into four other control systems shown in this section, going from a simple to a very complex control system with its unique advantages. The sensor is of the differential kind with an internal gain of 8000 as seen in Fig. 5.2. In the development and testing of the different control system a [N/A], [N/m] and [kg] are used. The different gains in the PID controller are optimised to get a quick response and not to high peaks.

F m z m z

skz ik I

tot s iF m z k z k I

1.85ik 51 10sk1m

Controller

Sensor

Electromagnet

Passiv magnets

48

In the following control system in Fig. 5.2 the coefficients for the bearing and the lab sensor are applied. Step 1 is used to represent an axial displacement at start-up and step 2 symbolises an external axial force on the rotor. The external force could be the case if there is a propeller on the rotor such as for a hobby model aircraft or any other axial fan.

Fig. 5.2 Control system with laboratory equipment

Fig. 5.3 Current and displacement from control system in Fig. 5.2 Step 1 at t=0,1s is a force resulting in a permanent displacement. Step 2 at t=0,3s causes an external force on the rotor which the controller has to adjust.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-20

-15

-10

-5

0

5

10

15

time [s]

Cur

rent

[A]

Current

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

1

x 10-4 Displacement

time [s]

Dis

plac

emen

t [m

]

P=5I=150D=1/63

vaF zi

Step2t=1 =>10

Step1t=0.5 =>4

-8000

Prob2

8000

Prob1

5/1.85

Power-amplifier

PID

PID- controller

1*10^5

Ks

1.85

Ki

1s

Integr.2

1s

Integr.1

2.5

Gain2

1/5

Gain1

Displacement

Current

1

1/m

Vz=40V/mm, z=0 +/- 0.25 mm

0.1

0.3

Sensor

49

To reduce the current peak during step response to position changes (servo cases) half the position signal is passed by the PID controller, (Gain3).

Fig. 5.4 Control system with decreased effect from startup displacement

Fig. 5.5 Current and displacement from control system in Fig. 5.4

P=2.5I=150D=1/59

vaF zi

Step2t=0.3 =>10

Step1t=0.1 =>4

-8000

Prob2

8000

Prob1

5/1.85

Power-amplifier

PID

PID- controller

1*10^5

Ks

1.85

Ki

1s

Integr.2

1s

Integr.1

2.5

Gain3

2.5

Gain2

1/5

Gain1

Displacement

Current

1

1/m

Vz=40V/mm, z=0 +/- 0.25 mm

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-20

-15

-10

-5

0

5

10Current

time [s]

Cur

rent

[A]

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8x 10-4

time [s]

Dis

plac

emen

t [m

]

Displacement

Reduced peak

50

To reduce current consumption a current feedback loop is introduced.

Fig. 5.6 Control system with current feedback loop

Fig. 5.7 Current and displacement from control system in Fig. 5.6

P=2.5I=0D=1/59

vaF zi

Step2t=0.3 =>10

Step1t=0.1 =>4

-8000

Prob2

8000

Prob1

5/1.85

Power-amplifier

PID

PD- controller

1*10^5

Ks

1.85

Ki

1s

Integr.2

1s

Integr.1

500

I=500

2.5*5

Gain3

2.5

Gain2

1/5

Gain1

Displacement

1s

Current-Integrator

Current

1

1/m

Vz=40V/mm, z=0 +/- 0.25 mm

0 0.1 0.2 0.3 0.4 0.5-12

-10

-8

-6

-4

-2

0

2

4x 10

-5 Displacement

time [s]

Dis

plac

emen

t [m

]

New Equilibriumposition

5 10

0 0.1 0.2 0.3 0.4 0.5-15

-10

-5

0

5

10Current

time [s]

Cur

rent

[A]

Position step0Current I

Load step0Current I

51

Simulated bearing start up. Rotor almost touches touchdown bearing.

Fig. 5.8 Control system behaving like the hybrid bearing from start up and load compensation

Fig. 5.9 Current and displacement from control system in Fig. 5.8 Initial condition At t=0 the position z=0 and current i=0. After t0=0 the reference position is set to -0,25mm which corresponds to -10V according to switch 3. This is done in order to simulate a movement of the rotor to the starting position in an emergency bearing. Switch 1 and 2 guaranties that the integrator works as an ordinary PID regulator. That is the integrator tries to maintain the position. The rotor almost touch the touchdown bearing during the time before t=0,2s.

0 0.1 0.2 0.3 0.4 0.5-20

-10

0

10

20

30

40

50Current

time [s]

Cur

rent

[A]

0 0.1 0.2 0.3 0.4 0.5-2.5

-2

-1.5

-1

-0.5

0

0.5

1x 10

-4 Displacement

time [s]

Dis

plac

emen

t [m

]

3 0.4t s

2 0.3t s

1 0.2t s

0 0t s1&2

0

Switchspecial PID

regulating I

30

SwitchV

310

SwitchV

44

SwitchN

P=2.5I=0D=1/59

vaF xi

Step2t=0.4 =>10

Step1t< 0.3:-10t>=0.3: 0

Step1t< 0.2: 1t>=0.2: 0

Step1t< 0.2: 0t>=0.2: 1

Product1

Product

-8000

Prob2

8000

Prob1

5/1.85

Power-amplifier

PID

PD- controller

1*10^5

Ks

1.85

Ki

1s

Integr.2

1s

Integr.1

1s

I

2.5*5

Gain3

2.5

Gain2

1/5

Gain1

Displacement

500

Current IntegratorI=500

Current

1

1/m

Vz=40V/mm, z=0 +/- 0.25 mm

Switch 2

Switch 3

Switch 4

Switch 1

z

52

Start up from emergency or touchdown bearing or turning on electricity At t1=0,2s a simulated start up from the emergency bearing is activated. In this case this is not done by changing the reference position to zero, but by connecting the integrator to the current signal using switch 1 and 2. Turning off the effective component from the simulated initial state at touch down. At t2=0,3s the reference position is changed to zero by using switch 2. This has no effect on the position error but acts only as a disturbance. Simulated axial load At t3=0,4s load step simulates an extra weight on the shaft with a permanent position error caused by the current integrator which wants to find a new equilibrium position where the current is zero through moving the rotor in opposite direction from its earlier equilibrium so that the external force is taken up by the magnetic repelling forces and the current saturate at zero.

53

6 CONCLUSIONS AND SUGGESTIONS FOR FUTURE RESEARCH

6.1 Conclusions A lot of work on this thesis was done analytical and to find programs that could solve different problems or verify the solutions. Two examples was to find a program that could help with the integrals not knowing how to continue and later finding a FEM program that could be used to verify the equations and learning how to use it and firstly verifying that it was used in the right way. After spending several weeks seeking the error in equations to why the results didn’t match with FEM. The last error was on the first page of the equations and everything had to be recalculated with new equations. How the integrals was solved was in a funny approach after both failing in using Matlab [20] and an internet page for integrals [22] without any luck. Both of these used Fourier transformation without showing how it was solved. This was not sufficient scientific. Finally I found an incredible program on my first mobile with Symbian OS platform called Formulae 1 (F1) version 1.0 [23] that could solve my integrals step by step where I got stuck. Another request from Lembke, the commissioner, was to try to use freeware and Excel as far as possible. Therefore the models and simulations was not only implemented in Matlab/Simulink [20] but also in the freeware Scilab/Scicos [19] The equations in this report were implemented in Scilab and also in Excels Visual Basic module. It was easy to rewrite the code as a function for Excel, with good results. For FEM simulations the freeware called FEMM [18] was used. FEMM is a program used for electromagnetic problems on two-dimensional planar and axisymmetric domains. One night I put my alarm clock on for every 20 min and woke up and wrote down the results and started next FEM simulation. After that I wrote a script in the scripting language Lua [24], so I could get many FEM simulations from FEMM while sleeping. By this thesis I have developed mathematic functions to pre optimise an axisymmetric model with magnets and coil that interact, in this case a magnetic bearing. A Lua script can be made that tries to optimise the FEMM bearing model further with a Newton-Raphson method to optimise the bearing to get the best performance. 3D FEM simulations still have to be made to find out the radial stability coefficients.

6.2 Suggestions for future work Some of the suggestions for future schedule: Optimisation of the circular perimeter in the force equations for the curved 2,5D model. Demonstration on a test rig. Analyse configuration 10 100-6 and 10 100-7 with a 3D-FEM program to see that the

radial stiffness is sufficient. Looking at a configuration without magnets on the rotor where the rotor get magnetised

from stator magnets, skipping rotating magnets but leading to less radial stiffness.

54

55

REFERENCES [1] Schweitzer G., Bleuler H., Traxler A., “Active magnetic Bearing”, vdf Hochschulverlag AG an der ETH Zürich, 1994, ISBN 3 7281 2132 0 [2] Earnshaw, S., “On the Nature of the Molecular Forces which Regulate the

Constitution of the Luminferous Ether”, Trans. Camb. Phil. Soc., 7, pp 97-112 (1842) [3] P. G. Geary, “Magnetic and electric suspensions”, Brittish Sientific Instrument Association, 1964 [4] http://www.nobel.se [Marc 2007] [5] Lembke, T., “Design of Magnetic Induction Bearing”, M.Sc. thesis, Chalmers

University of Technology, Göteborg 1990 [6] Lembke, T., ”Design and Analysis of a Novel Low Loss Homopolar Electrodynamic

Bearing”, Dr. thesis, Electrical Machines and Power Electronics, Royal Institute of Tech, Stockholm 2005,

http://www.magnetal.se/Dokument/PhDThesis.pdf [Mars 2011] [7] Lang, M. and Fremerey, J.K., “Optimization of Permanent-Magnet Bearings”, 6th

International Symposium on Magnetic Suspension Technology, Turin, October 2001. [8] Marcus Granström, Konceptförslag på svänghjul för energilagring, (eng. title

translation ”Flywheel concept proposal for energy storage”), Department of Material Engineering, Uppsala University 2002, ISRN UTH-INGUTB-EX—M-2001/08-SE

[9] U.S. Patent 6,118,199 “Magnetic bearings” [10] U.S. Patent 5,783,885 “Self-Adjusting Magnetic Bearing Systems”, Richard F. Post,

1998 [11] Science & Technology Review April 1996: http://www.llnl.gov/str/pdfs/04_96.2.pdf [Marc 2007] [12] U.S. Patent 6,469,411199 “Compliant foil fluid film bearing with eddy current

damper” [13] Impinna F., “Electrodynamic Bearings Modeling and Design”, PhD thesis, Politecnico

Di Torino, March 2010, http://www.cspp.polito.it/content/download/459/2571/version/3/file/Impinna_10+-

+Electrodynamic+Bearings.pdf [Mars 2011] [14] German Patent DE2444099 “Contact free magnetic bearing” [15] German Patent DE3808331 “Magnetic support with permanent magnets for absorption

of radial bearing stresses”

56

[16] Paul Lorrain and Dale Corson, Electromagnetic fields and waves, New York 1972,

ISBN 0-7167-0331-9 [17] Lang, M. ”Berechnung und Optimierumg von passiven permanent magnetischen

Lagern für rotierende Maschinen”, VDI-Verlag, Düsseldorf, 2004, ISBN 3-18-335721-6

[18] Meeker, D. ”FEMM - Finite Element Method Magnetics, http://femm.foster-miller.net [April 2007] [19] SCILAB, version 4.1, http://www.scilab.org [April 2007] [20] MATLAB, http://www.mathworks.com [April 2007] [21] Madeleine Ymerson, Implementering av ett magnetlager (eng. title Implementation of

a magnetic bearing), Master of Science Thesis MMK 2007:10 MDA 293, at KTH, Stockholm 2007

[22] Online integrator, www.integrals.com [April 2007] [23] Formulae 1 (F1) version 1.0, www.poliplus.com [Sept 2008] [24] Script language Lua, www.lua.org [Sept 2008]

A-1

Appendix A Variabel Integration Equation A1

1

2

12 2 2

21

tan

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )2

1( ) tan ( )1

tan2

ux dxx

f x g x dx F x g x F x g x dx

xIntegration by Part f x x F x f x dx

u d u ug x g xx dx x x uu

x

u x ux

2 12 2

2 22 2

2 1 2 12 2

2 2 2 2

22 1 2 1 2

2 22 2

tan

2 22

tan tan1

2 2 2 2

1tan tan

2 2 2

uxx u xxdx dx

x ux u

u ux xu u u ux xdx x dx

x u x u

u uux x u x u u dxu x u dxx x x ux u

1

2 1 2 1 2 13

2 1 2 1

2

tantan tan tan

2 2 2 2

tan tan

2

xu u xux x u x uu x ux x uu

u xx u x ux u

A-2

Equation A2

2 2

2 2

2 22 2 2 2

22 2 2 2

2 2 2 2

ln

( ) ( ) ( ) ( ) ( ) ( )

( ) 1 ( ) ( )

( ) ln

1 2( )

2ln ln 2

l

g m dm

f m y m dm F m y m F m y m dm

f m F m f m dm mIntegration by Part y m g m

d my m g mg m dm g m

m mm g m m dm m g m dmg m g m

m2 2

2 2 2 22 2 2 2

1

2 2 2 2 2 22 2

2 2 1

n 2 1 ln 2 1 2

tan1ln 2 2 ln 2 2

ln 2 2 tan

g gg m dm m g m dm dmm g m g

mg

m g m m g dm m g m m gm g g

mm g m m gg

B-1

Appendix B SCILAB PROGRAMS Force Function Magnet versus Magnet with constants function Fz_mr_ms_tot=Fz_mr_ms() // Filename: Fz_mr_ms_constants.sci dr=0.01; // Rotor inner diameter Dr=0.024; // Rotor Outer diameter t_mr=0.007; // Rotor magnet thickness //Kr=954929; // //zr=0.0027; // Nr=1; // ds=0.026; // Ds=0.040; // t_ms=0.007; // Ks=954929; // //zs=0.00; // Ns=1; // Br=1.2; Bs=1.2; z=0.0012; Fz_mr_ms_tot=2*(-fz_s_s(z,(Ds-Dr)/2,t_mr,t_ms,Nr*Br,Ns*Bs)*%pi*(Ds+Dr)/4 ... +fz_s_s(z,(ds-Dr)/2,t_mr,t_ms,Nr*Br,Ns*Bs)*%pi*(Dr+ds)/4 ... -fz_s_s(z,(Dr+ds)/2,t_mr,t_ms,Nr*Br,Ns*Bs)*%pi*(Dr+ds)/4 ... +fz_s_s(z,(Dr+Ds)/2,t_mr,t_ms,Nr*Br,Ns*Bs)*%pi*(Dr+Ds)/4 ... +fz_s_s(z,(Ds-dr)/2,t_mr,t_ms,Nr*Br,Ns*Bs)*%pi*(dr+Ds)/4 ... -fz_s_s(z,(ds-dr)/2,t_mr,t_ms,Nr*Br,Ns*Bs)*%pi*(dr+ds)/4 ... +fz_s_s(z,(dr+ds)/2,t_mr,t_ms,Nr*Br,Ns*Bs)*%pi*(dr+ds)/4 ... -fz_s_s(z,(dr+Ds)/2,t_mr,t_ms,Nr*Br,Ns*Bs)*%pi*(dr+Ds)/4); endfunction

B-2

Force Function Sheet versus Sheet function f=fz_s_s(z,g_eff,t_mr,t_ms,Br,Bs) // Filename: fz_s_s.sci u0=4*%pi*0.0000001; t1=t_mr/2+z-t_ms/2; t2=t_mr/2+z+t_ms/2; t3=-t_mr/2+z-t_ms/2; t4=-t_mr/2+z+t_ms/2; f=Bs*Br/(4*%pi*u0)*(a_s(t1,g_eff)-a_s(t2,g_eff)-a_s(t3,g_eff)+a_s(t4,g_eff)); endfunction function a=a_s(t,g_eff) // Filename: a_s.sci a=-t*log(t^2+g_eff^2)+2*t-2*g_eff*atan(t/g_eff); endfunction / //#include <math.h> //int n; n=0; for z=0:0.0001:0.002, F=MG_Fz_magnetlager2x2_konstanter(z), n=n+1; end

B-3

Force Function Magnet versus Magnet function Fz_mr_ms_tot = Fz_mr_ms(dr,Dr,t_mr,Br,Nr,mr_displacement,ds,Ds,t_ms,Bs,Ns,ms_displacement) // Filename: Fz_mr_ms.sci //Example [Fz_magnet_tot1]=Fz_magnet(dr,Dr,tr,Kr,mag_avst_rotor+z,-1,ds,Ds,ts,Ks,mag_avst_stator,-1); z=mr_displacement-ms_displacement; Fz_mr_ms_tot=2*(-fz_s_s(z,(Ds-Dr)/2,t_mr,t_ms,Br*Nr,Bs*Ns)*%pi*(Ds+Dr)/4 ... +fz_s_s(z,(ds-Dr)/2,t_mr,t_ms,Br*Nr,Bs*Ns)*%pi*(Dr+ds)/4 ... -fz_s_s(z,(Dr+ds)/2,t_mr,t_ms,Br*Nr,Bs*Ns)*%pi*(Dr+ds)/4 ... +fz_s_s(z,(Dr+Ds)/2,t_mr,t_ms,Br*Nr,Bs*Ns)*%pi*(Dr+Ds)/4 ... +fz_s_s(z,(Ds-dr)/2,t_mr,t_ms,Br*Nr,Bs*Ns)*%pi*(dr+Ds)/4 ... -fz_s_s(z,(ds-dr)/2,t_mr,t_ms,Br*Nr,Bs*Ns)*%pi*(dr+ds)/4 ... +fz_s_s(z,(dr+ds)/2,t_mr,t_ms,Br*Nr,Bs*Ns)*%pi*(dr+ds)/4 ... -fz_s_s(z,(dr+Ds)/2,t_mr,t_ms,Br*Nr,Bs*Ns)*%pi*(dr+Ds)/4); endfunction

AUTHOR

Marcus Granström (member) graduated from the Uppsala University in 2002 were he received his Bachelor of Science in Mechanical Engineering. He continued study at KTH in Stockholm and completed his Master of Science 2011, and at 2006 while writing this thesis he joined ABB as an order constructer at the Machine division in Västerås, SWEDEN. His research interests are maglev systems, design of electric machines and winding configurations and their applications and not to forget computer aided design, calculation and programming.

“Today, as I reflect upon my winding path in new mechanical element design, I’m amused by the things that helped me along the way: dreaming big, accepting life’s twists, seeking innovation, maintaining a level of fun, and, finally, recognizing a lightning strike when I saw one.” – Marcus Granström,