Design och Analys av ettmagnetlager med 1-frihetsgrad

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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 ^ExaminerJan 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 with integrated 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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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

Repell ing Repell ing Attracting Attracting

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 F

k

axial 2 radial

k k

r 1,0 u 4 10 7

0 r

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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

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

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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.

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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.

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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

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.

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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 with integrated coil for axial stability

ROTOR

Passive magnetic bearing

(30)

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

dr

Dr

ds

Ds 6

4 and 5 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

(31)

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

ks k_i

N N

N

2

6 5 4

8 7 3 1

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 on

F = =Fr + Fr + Fr + Fr = =2 (Fr +Fr )

on the inner magnet the inner magnets respective sheet

res

Fr3

Fr4

Fr5

Fr6

dr

tsm

Dr

ds

Ds

tsm

trm

N

tc

trm

(32)

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)

Fres

0

2

a b

w w

I I f dF

dl

N m

l Fw w

0

2

a b

w w

F I I l N

F

a F I

Ib F

F Ia

Ib

l

(33)

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.

0

2 2 sin

2

a b

res

F I I l

z g N

2

tan tan 2 l D d

z z

g D d

I I

Ib

Ia

Ib

Ia

g

res z F Fres

Fres

D d

(34)

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) I

B I

0 0 m

dl I t

tm

c

Br

Iron yoke Magnet

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

(35)

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 .

0

r m m

B t t

0

Br

I

7 1 1

0 4 10

u VsA m I

7

0 4 10

r m

B B t

I t

I tm

4 10 7

Br

f1 f₂

dI dz_m

df

Cross section N

I f₂ f¹

tm

Br

w g

z

(36)

(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 f , ₁ 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 1

f df

1

1 w w sin

df f dI

I

dI dzm

2 2 2

1 0

2 2 2

sin sin 2

m c

tm

m c

m tm

m c

z z z

I g z z z

f dz

g z z z

0

2 2 1

2

m c

m m c m

m

x z z z

I z z z

dz dx

g z z z dz

2 2

0 0

2 2

0

1 ln

2 2 2

I x I Br

dx g x

g x

2 2 2

0

0 2

4 ln

mr

t r

m c t

I B g z z z

2 2

ln 2 4

2

mr c

r

mr c

g t z z

I B

g t z z

N m

f2

g w

(37)

(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

2 2

2

ln 2 4

2

mr c

r

mr c

g w t z z

f I B

g w t z z

N m

1 2

f f f N

m

f1 f₂ f₁ f₂ f₁

2 2

ln 2 4

2

mr s

s w r

s w

mr s

g t z z

dF I B

dl f t

g z z

N m

dI dzs

dfs s

Cross section N

N

f1

f2

tmr

Br

g wmr

z

tms

Br

wms

(38)

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

(4.18)

Solving Part 1 of equ. (4.18):

2 1

2

2 2 2

2 ln 2

4 2

2

mr

mr s

mr s s r

mr s

s

u t z

g t z z

t

df B dI u z

g t z z dI dz

2 2 1 2 2

2

ln .

4

r s

s s

g u z

B dz

g u z

2 2

2 1

2 2 2 2

4 ln

ms

t r s

s s s

t s

g u z

f B dz

g u z

2 2

1 2

2 2

ln ln

4

ms ms

t t r

s s s s s s

t t

f B g u z dz g u z dz

2 2

1 1

ln

1 1

s s s

s s

Substitution

g u z dz m u z

dm dz dm

dz

2 2 2 2

ln 1 ln Solved in

g m dm g m dm

Appendix A

2 2 1

ln 2 2 tan m

m m g m g

g

Part 1 Part 2

Design och Analys av ettmagnetlager med 1-frihetsgrad

Design and Analysis of a 1-DOF magnetic bearing

Design och Analys av ett magnetlager med 1-frihetsgrad

av

Marcus Granström

Contents

1 GENERAL INTRODUCTION

1.1 Background

1.2 Request for new magnetic bearings

1.3 Analytical method

1.4 Limitations of the analysis

1.5 Research motivation and objectives

1.6 Contributions of the thesis

1.7 Result

1.8 Outline of the thesis

2 PREVIOUS WORK ON MAGNETIC BEARINGS

2.1 Historical development of magnetic bearings

2.2 Permanent magnetic bearings, PMB

Axial bearing

Repell ing Repell ing Attracting Attracting

Radial bearing

2.3 Active magnetic bearings, AMB

2.4 Superconducting bearings, SCB

2.5 Electrodynamic bearing, EDB

3 NOVEL HYBRID MAGNETIC BEARING

3.1 Bearing configurations

4 BEARING ANALYSIS

4.1 Model description

4.2 Analytical derivation of bearing forces

c