modeling and verification of ultra-fast electro-mechanical actuators

Modeling and Verification of Ultra-Fast

Electro-Mechanical Actuators for HVDC Breakers

ARA BISSAL

PhD Thesis

Stockholm, Sweden 2015

TRITA-EE 2015:010ISSN 1653-5146ISBN 978-91-7595-480-6

Electromagnetic EngineeringSchool of Electrical Engineering, KTH

SE-100 44 StockholmSWEDEN

Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framläggestill offentlig granskning för avläggande av teknologie doktorexamen onsdagen den22 maj 2015 klockan 10.00 i F3, Kungl Tekniska högskolan, Lindstedtsvägen 26,Stockholm.

© Ara Bissal, May 2015

Tryck: Universitetsservice US AB

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Abstract

The continuously increasing demand for clean renewable energy has rekin-dled interest in multi-terminal high voltage direct current (HVDC) grids.Although such grids have several advantages and a great potential, theirmaterialization has been thwarted due to the absence of HVDC breakers. Incomparison with traditional alternating current (AC) breakers, they shouldoperate and interrupt fault currents in a time frame of a few milliseconds.The aim of this thesis is focused on the design of ultra-fast electro-mechanicalactuator systems suitable for such HVDC breakers.

Initially, holistic multi-physics and hybrid models with different levelsof complexity and computation time were developed to simulate the entireswitch. These models were validated by laboratory experiments. Followinga generalized analysis, in depth investigations involving simulations comple-mented with experiments were carried out on two of the sub-components ofthe switch: the ultra-fast actuator and the damper. The actuator efficiency,final speed, peak current, and maximum force were explored for differentdesign data.

The results show that models with different levels of complexity shouldbe used to model the entire switch based on the magnitude of the impulsiveforces. Deformations in the form of bending or elongation may deterioratethe efficiency of the actuator losing as much as 35 %. If that cannot beavoided, then the developed first order hybrid model should be used sinceit can simulate the behaviour of the mechanical switch with a very goodaccuracy. Otherwise, a model comprising of an electric circuit coupled to anelectromagnetic FEM model with a simple mechanics model, is sufficient.

It has been shown that using a housing made of magnetic material suchas Permedyn, can boost the efficiency of an actuator by as much as 80 %.In light of further optimizing the ultra-fast actuator, a robust optimizationalgorithm was developed and parallelized. In total, 20520 FEM models werecomputed successfully for a total simulation time of 7 weeks. One outputfrom this optimization was that a capacitance of 2 mF, a charging voltage of1100 V, and 40 turns yields the highest efficiency (15 %) if the desired velocityis between 10 m/s and 12 m/s.

The performed studies on the passive magnetic damper showed that theHalbach arrangement gives a damping force that is two and a half times largerthan oppositely oriented axially magnetized magnets. Furthermore, the 2Doptimization model showed that a copper thickness of 1.5 mm and an irontube that is 2 mm thick is the optimum damper configuration.

Keywords: Actuators, Armature, Capacitors, Circuit breakers, Coils, Damp-ing, Eddy currents, Elasticity, Electro-mechanical devices, Electromagneticforces, Finite element methods, HVDC transmission, Image motion analy-sis, Magnetic domains, Magnetic flux, Magnetic forces, Magnetic materials,Magnets, Thermal analysis.

Sammanfattning

En ständigt ökande efterfrågan på ren, förnybar energi har återuppväcktintresset för högspänd likström (HVDC) i elnät med fler än två ändstationer.

Trots att dessa nät har flera fördelar och en stor potential har förverk-ligandet förhindrats på grund av frånvaron av HVDC-brytare. I jämförelsemed en brytare av växelström (AC) måste en HVDC-brytare kunna bryta enfelström inom loppet av ett fåtal millisekunder.

Syftet med denna avhandling är inriktat på utformningen av ultrasnabbaelektromekaniska aktuatorsystem som lämpar sig för sådana brytare. Dessa ärbaserade på så kallade Thomsonspolar vilka består av en platt spiralformadlindning vilken är kopplad till en uppladdad kondensator som sedan laddasur genom spolen som därvid alstrar en kraftimpuls.

Initialt utvecklades multifysik- och hybridmodeller med olika komplexitetoch erforderlig beräkningstid för att simulera dessa aktuatorsystem.

Dessa validerades sedan med laboratorieexperiment. Därefter genomfördesen ingående modelleringsstudie av aktuatorsystemets ingående komponenter,den ultrasnabba aktuatorn och magnetiska dämparen. Aktuatorn verknings-grad, sluthastighet, toppström och maximala kraft utforskades bl a för olikadesigndata.

Resultaten visar att modeller av olika grad av komplexitet bör använ-das för att modellera hela aktuatorn beroende på storleken av den alstradekraftimpulsen. Deformationer i form av böjning eller töjning kan försämraverkningsgraden så mycket som 35 %.

Om dessa effekter inte kan undvikas, bör den fullständiga hybridmodellenanvändas eftersom den kan simulera aktuatorns beteende med en mycket godnoggrannhet. Annars är en modell bestående av en elektrisk krets med enenkel mekanikmodell kopplad till en elektromagnetisk FEM modell tillräcklig.

Utförda studier visar att användning av en aktuatorfixtur gjord av mag-netiskt material, exempel Permedyn, kan öka verkningsgraden hos aktuatornmed upp till 80 %.

Med syfte att ytterligare optimera den ultrasnabba aktuatorn utveckladesen robust optimeringsalgoritm för användning av parallella kärnor. Med dennagenomfördes under 7 veckors tid totalt 20520 FEM-beräkningar. Ett resultatfrån denna optimering var att en kapacitans på 2 mF, en laddningspänningav 1100 V och 40 lindningsvarv ger en högsta verkningsgrad (15%) om denönskade sluthastigheten ligger mellan 10 m/s och 12 m/s.

Genomförda studier av den magnetiska dämparen visar bl.a. att om de an-vända permanentmagneterna utförs i form av ett Halbacharrangemang medfördetta en dämpande kraft som är två och en halv gånger större än om motsattaxiellt orienterad magnetiserade magneter används.

Acknowledgements

Firstly, I would like to express my gratitude to my supervisor, Prof. Göran Engdahl,for his guidance, innovative ideas, and numerous comments throughout this PhDthesis. Special thanks goes to Prof. Rajeev Thottappillil for his continuous support,and to Dr. Nathaniel Taylor for being one of the reviewers of this thesis. I’d alsolike to thank Prof. Anders Eriksson, for being my pillar in mechanics.

A lot of this thesis work was made possible thanks to the collaborative andfruitful working atmosphere at ABB AB Corporate Research. Therefore, I wouldlike to thank Dr. Mikael Dahlgren and Magnus Backman for trusting in me andgiving me the opportunity to work with Dr. Thomas Eriksson, Dr. Ener Salinas,and Dr. Lars Liljestrand. Their advice and guidance was very beneficial especiallywhen it came to answering a lot of my technical questions. I was fortunate to getto meet people like Dr. David Schaeffer, Dr. Zichi Zhang, Stephan Halen, LarsJonsson, Ola Jeppsson, Dr. Elisabeth Lindell, and Dr. Roberto Alves at ABB ABCorporate Research. In my opinion, a nice working environment strongly dependson the people whom you work with.

During my PhD studies, I also had the privilege of working at ABB DE Corpo-rate Research in Ladenburg, Germany, for a mobility of six months. I learned a lotfrom this experience and had the chance to work with many competent professionalswith different backgrounds such as Dr. Octavian Craciun, Dr. Veronica Biagini,Dr. Christian Simonidis, Dr. Jörg Gebhadrt. Also, I would like to thank Dr.Markus Schneider, Wolfgang Waldi, Dr. Guenther Mechler, Dr. Gregor Stengel,Dr. Alexander Horch, and Dr. Ulf Ahrend. Last but not least, I would like toexpress my gratitude to Dr. Andreas Decker, Dr. Arne Wharburg, Dr. KimListmann, Dr. Jan Schlake, Andreas Schader, and Dietmar Post with whom I havehad very interesting discussions on the train from Darmstadt to Ladenburg. Ingeneral, the hospitality and kindness of all my newfound German colleagues knewno limits.

I will always cherish my time at KTH especially because of outstanding friendssuch as Dr. Samer Sisha, Dr. Andreas Krings, Dr. Shuang Zhao, Dr. Respi-cius Kiiza, David Fernando Ariza González, Cong-Toan Pham, and Dr. AntoniosAntonopoulos. A very special thanks goes to my friend and colleague JesperMagnusson, with whom I have worked the most with throughout my PhD. I wouldalso like to thank Peter Lönn not only for his help in IT matters at KTH, but also

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for his jolly character.I deeply appreciate having invaluable friends like Rami Bou Hadir, Hani Dakhil,

Joseph Noufaily, Anthony Saliba, and Elie Karaa who have travelled all the wayfrom Lebanon to Sweden to support me for my PhD defense.

I would also like to thank my sister for her sweetness, love, and support. Specialthanks goes to my mother for sacrificing her life to raise and support me every stepof the way. I hope one day I can make you as proud of me as I am of you.

Last but certainly not least, I would like to thank Silvia Lohfink for her extremekindness and never ending love. She was like a candle in the midst of darkness,illuminating the countless winter nights I spent writing this PhD thesis.

Ara Bissal

Stockholm, May 2015

Contents

Contents viii

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Why HVDC? . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Limitations of Multi-terminal HVDC Grids . . . . . . . . . . 21.1.3 HVDC Breakers . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Objectives of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Scientific Contributions . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 List of Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 The Mechanical Switch 92.1 Requirements of the Mechanical Switch . . . . . . . . . . . . . . . . 92.2 Switch Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.1 Electromagnetic Modeling . . . . . . . . . . . . . . . . . . . . 132.3.2 Thermal Modeling . . . . . . . . . . . . . . . . . . . . . . . . 172.3.3 Mechanical Modeling . . . . . . . . . . . . . . . . . . . . . . . 18

2.4 Model Validations by Experiments . . . . . . . . . . . . . . . . . . . 222.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.5.1 Models With Finer Segmentations . . . . . . . . . . . . . . . 32

3 The Ultra-Fast Actuator 353.1 State of the art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.1.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.1.2 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.3 Experimental Validation . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3.1 The Experimental Setups . . . . . . . . . . . . . . . . . . . . 383.3.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 41

3.4 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.4.1 The Electrical Circuit . . . . . . . . . . . . . . . . . . . . . . 44

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CONTENTS ix

3.4.2 The Magnetic Circuit . . . . . . . . . . . . . . . . . . . . . . 513.4.3 The Shape of the Actuator . . . . . . . . . . . . . . . . . . . 59

3.5 Brute Force Optimization . . . . . . . . . . . . . . . . . . . . . . . . 723.5.1 Setup of the optimization model . . . . . . . . . . . . . . . . 743.5.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 75

4 The Composite Magnetic Damper 974.1 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 984.4 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.4.1 Static modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 994.4.2 Transient Modeling . . . . . . . . . . . . . . . . . . . . . . . . 99

4.5 Model and Concept Verification . . . . . . . . . . . . . . . . . . . . . 1004.5.1 Small Scale Prototypes . . . . . . . . . . . . . . . . . . . . . . 1004.5.2 Large Scale Prototype . . . . . . . . . . . . . . . . . . . . . . 103

4.6 Optimal Damper Design . . . . . . . . . . . . . . . . . . . . . . . . . 105

5 Conclusions and Future Work 1115.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

A Math Operators 113

B Symbols and Acronyms 115

Bibliography 119

List of Figures 128

Chapter 1

Introduction

This section provides the background, motivation, and objectives of this researchstudy. Furthermore, the thesis outline, the scientific contributions, and list ofpublications are given.

1.1 Background

The world has seen a steady increase in population since the beginning of mankind.Their needs and expectations in life have been constantly increasing. The standardof living has increased dramatically especially with the advancement of mankind.Nowadays, their insatiable desire for electric energy is mostly met with non renew-able sources of power generation such as power plants based on fossil fuels. Thesepower plants are harmful to the environment and contribute to global warmingbecause they release large amounts of green house gases.

The Kyoto Protocol, a treaty with currently 192 parties, binds all of its par-ticipants to fight global warming by reducing the emission of green house gases.A post-Kyoto protocol agreement, will be discussed at the 21st United NationsClimate Change Conference, which will be held in Paris, France in 2015. For thefirst time in over twenty years, the objective of this conference is to establish bindingtargets for the reduction of green house gas emissions in all nations.

Consequently, countries have initiated a lot of initiatives and subsidies to en-courage clean, green, and renewable technologies. One such technology is offshorewind turbines cable of transforming wind into renewable energy. Another example isphotovoltaics, a method that transforms solar energy into electric energy by makinguse of semiconducting material exhibiting the photovoltaic effect. Due to theseincentives, a lot of research has been conducted towards increasing their efficiency.One factor that hinders adoption of renewable energy is the transportation of thisenergy over long distances to consumers.

Power generation and consumption are usually separated by large geographicdistances. Offshore wind power is one such example. Wind turbines are installed

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2 CHAPTER 1. INTRODUCTION

far out in the sea. The generated energy has to be transported over long distancesbefore reaching local grids. Another such example are solar panels installed inremote deserts such as the Mongolian desert. Such forsaken areas are attractivedue to the presence of large amounts of solar energy and vast areas of land.

One of the main challenges in harnessing the potential of renewable energy isto be able to transport it with minimum transmission losses [1–3]. Thus, a lot ofresearch has been conducted on high voltage direct current (HVDC), as an efficientsolution to transport power efficiently [4–13].

The European HVDC study group is putting a lot of effort to define “Technicalguidelines for first HVDC grids” [14]. A new coming trend in HVDC is the transitionfrom traditional point to point interconnections into multi-terminal and multi-vendor HVDC systems. Compared to point to point connections, a multi-terminalHVDC grid has fewer terminals, hence a lower cost and lower losses. Moreover, thefailure of a DC line at a terminal does not interrupt the power flow between otherterminals. Each terminal can operate with different current and power ratings in afully controllable manner.

All these initiatives have rekindled interest in multi-terminal high voltage directcurrent grids. Integrating renewable energy in local grids to meet the increasingdemands of electricity significantly decreases the release of green house gases. Onerecent project that is estimated to be commissioned in 2015 is BorWin1 [15]. Inthis project, HVDC cables connect the German grid to an offshore cluster of windfarms 130 km out to sea. The power rating of this wind farm is 400 MW. Onceoperational, 1.5 million households will receive clean renewable energy avoiding theemission of more than 150 thousand tons of carbon dioxide per annum.

1.1.1 Why HVDC?

Alternating current (AC) cables are limited in their transmission capability depend-ing on the length of the cable among other cable characteristics [16]. The longer thecable, the larger is the capacitance, that constantly needs charging when subjectedto an AC voltage. The charging current will be equal to the transmitted current atsome critical cable length. This hinders power transmission and can be avoided byusing a direct current (DC) system.

At some break-even distance, a DC system will be less costly than an AC system.For distances below 600 km it is cheaper to use an AC system. An HVDC systemwill be more expensive since AC-DC conversion and vice versa is required.

One very large advantage of HVDC is that it allows asynchronous intercon-nections [17]. Furthermore, it has significantly lower losses, a lower environmentalimpact, and is highly controllable.

1.1.2 Limitations of Multi-terminal HVDC Grids

Although HVDC technology has major advantages over traditional high voltagealternating current (HVAC), it has not been accepted due to the absence of an

1.1. BACKGROUND 3

HVDC breaker. The adoption of this technology relies entirely on the existenceof a HVDC breaker [18, 19]. A conventional HVAC breaker cannot be used foran HVDC breaker due to the absence of a current zero crossing. Unlike an ACsystem, a DC system has a constant current. Moreover, DC systems have a verysmall impedance, thus fault currents increase quickly in magnitude and attaindangerously high levels [20]. Large currents can destroy all components connectedin the network unless circuit breakers intervene promptly and interrupt these faultcurrents. If a slow breaker is used, then it should be dimensioned also for such largecurrents, complicating its design [21]. Moreover, it will be large and bulky, and insome cases would need even extra cooling. Thus, it is of crucial importance to havea fast breaker to avoid these complications.

1.1.3 HVDC Breakers

Recent research is devoted to understanding, designing, modeling, and testing ofHVDC breakers [22–37]. To interrupt a HVDC current, three different HVDCbreaker topologies could be used: a power electronic topology, a purely mechanicaltopology, or a combination of both, also known as a hybrid topology.

The power electronic or solid state breaker may consist of any active turn-offhigh power semiconductor such as an insulated gate bipolar transistors (IGBTs),in parallel with a metal oxide varistor (MOV). In a normally conducting mode,the IGBTs conduct the nominal current in the main path. IGBTs are placed inseries and parallel such that they can withstand the voltage and current ratingsrespectively. If bidirectional current conduction is a requirement, then the numberof components have to be doubled. Once a fault current emerges, the IGBTs areturned off and the current commutates to the MOV branch. The MOV starts toconduct and starts to absorb the energy in the circuit. The voltage drop across theMOV must be equal to or larger than the system voltage to ensure that the faultcurrent decays to zero. The state of the art for interrupting a fault current with apower electronic breaker is 1 µs, 5 kA, and 640 kV [37].

The main advantages of using a solid state breaker are the absence of mechanicalparts and extremely fast turn-off times. The major disadvantage is that it has veryhigh on-state losses. Thus, its operating cost is very high. Excessive losses overheatthe breaker, requiring a cooling system. This increased complexity increases thecost of such a topology even more.

The relatively high costs and high on-state losses of a solid state breaker mo-tivate the search for other methods with lower losses. One other method is apure mechanical breaker composed of three parallel branches. In the first branch,an ultra-fast mechanical switch is located. In a normally conducting mode, thecurrent passes through the metallic contacts of the mechanical switch. Dependingon the system voltage, the contacts might be inserted in a vacuum bottle or in anyother type of gas to increase the dielectric strength of the medium. The secondbranch consists of a resonance circuit. A capacitor in series with an inductor isconnected in parallel with the mechanical switch. At the instant a fault current


is detected, the mechanical switch is activated. As the contacts open, an arc isestablished. Once the contacts have separated with a minimum distance such thatthe mechanical switch can withstand the reverse recovery voltage, the capacitorbank generates a high frequency oscillating current that forces a current zero inthe mechanical switch. Thus, the current is interrupted and commutates to thethird and last parallel branch, the branch containing the MOV. Finally, the MOVabsorbs the magnetic energy stored in the circuit. The fault current starts to decayuntil it is completely interrupted.

With such a scheme, the losses in the main path are minimized. However, thestate of the art mechanical breakers in 2009 have an operational time of 60 ms.This is quite slow and will result in relatively large fault currents before they areinterrupted. One way of limiting the current rise is by installing an inductor inthe main circuit path such that when a fault occurs, it limits the di

dt. The major

disadvantage of this scheme is arcing. The arc will erode the contacts significantlydecreasing their lifetime. Hence, they should be maintained more frequently.

The advantages of both schemes are combined in a third topology, the hybridHVDC breaker. The hybrid breaker also consists of three parallel stages. In anormally conducting mode, the ultra-fast mechanical switch conducts the nominalcurrent. In case of a fault, the solid state breaker sitting in the second stageis triggered so that it starts conducting. Afterwards, the mechanical switch isactivated such that its contacts open to commutate the current to the second stage.Once the mechanical switch is fully open, the solid state breaker is turned off tointerrupt the current. The current then commutates to the third and final branch,the one containing the MOV. The MOV limits the current which starts to decayuntil it reaches zero. The world’s first hybrid HVDC breaker was released by ABBin the year of 2012. It has an operation time of 5 ms [20].

This scheme has very low on-state losses, is relatively fast, and has less arcingtime than a purely mechanical breaker. Another advantage is that the demandson the power electronic switch will be reduced since it does not have to conductthe nominal current continuously. One disadvantage is that it is still orders ofmagnitude slower than a pure solid state solution.

1.2 Objectives of the Thesis

Grid operators introduce a series inductance to limit fault currents in HVDC girds.This however is not an optimum solution since the inductor is costly, results inunnecessary losses, may introduce voltage fluctuations, and might also requirean additional cooling system. Moreover, arcing significantly reduces lifetime andreliability increasing maintenance costs. All these drawbacks can be mitigated bydecreasing the operational time of the electro-mechanical switch.

Increasing the speed of the electro-mechanical switch by designing high per-formance high efficiency actuators can improve the functionality, reliability, andlifetime of the breaker. These high performance switches can be installed in both

1.3. THESIS OUTLINE 5

the purely mechanical and the hybrid topologies.The objective of the thesis is to firstly develop multi-physics simulation models

with different levels of accuracy and computation time. Such models can help tounderstand the behaviour of these devices in which electromagnetics, mechanics,and thermal aspects must be considered. Secondly, another objective is to carryout a sensitivity analysis to identify the sensitivity of the actuator to all involvedparameters. This opens the door to study the influence of critical materials andkey components that can significantly boost the efficiency and performance of theactuator. Thirdly, another objective is to design a generic, robust, optimizationmodel to optimize these ultra fast actuators and boost their performance andefficiency. Finally, the last objective is to model, design, and build a passivemagnetic damper that can be used for decelerating the actuator in a controllablemanner.

1.3 Thesis Outline

The main focus of this thesis is the study and design of ultra-fast electro-mechanicalswitches. This thesis is presented using the general-to-specific pattern. Initially,simulations and experiments are done on a holistic perspective, that is, on the entiremechanical switch. They are then followed with simulations of and experiments onthe detailed components. The thesis is divide into the following five chapters:

Chapter 1: This chapter is the current chapter. It presents the background,motivation, and the objectives of the work.

Chapter 2: This chapter provides a holistic description of the mechanical switchand its sub-components; the actuator, the push/pull rod, the contacts, and thedamper. Finite element method based multi-physics models are derived with dif-ferent levels of complexity and are validated by experiments.

Chapter 3: In this chapter, one of the most important sub-components of themechanical switch is studied, the ultra-fast actuator. The state of the art isinvestigated. A sensitivity analysis is carried out using experimentally validatedmulti-physics models. Finally, an optimization scheme is presented.

Chapter 4: In this chapter, another switch sub-component is studied, the damper.A passive magnetic damper is designed, modeled, and experimentally validated.

Chapter 5: This chapter concludes the thesis with a summary of the mainfindings and touches on future work that can be valuable in this field.


1.4 Scientific Contributions

The work of the author has resulted in the following contributions to the state ofthe art:

• A comprehensive description of the required physics for modeling large im-pulsive forces in large structures within small time scales.

• A detailed description of an ultra-fast switch and necessary requirements toboost its efficiency.

• Experimentally validated multi-physics models with different levels of com-plexity and computation time.

• Sensitivity analysis of an ultra-fast actuator.

• A novel method to boost the efficiency of ultra-fast actuators.

• A robust generic brute force optimization algorithm to optimize ultra-fastactuators with variable objectives and constraints.

• Experimentally validated magnetic damper simulation models.

• The design of an optimal passive composite magnetic damper.

1.5 List of Publications

The work presented in this thesis has resulted in a mix of conference publications,journal publications, and patents. The first author is the main corresponding authorand has sole responsibility for the paper. All publications are listed in reversechronological order.

Conference Publications

• A. Bissal, E. Salinas, J. Magnusson, G. Engdahl, “Magnetic Flux Conductorsfor Ultra-Fast Actuators,” submitted for review to IEEE Energy ConversionCongress an Exposition (ECCE), Montreal, Canada, September 2015.

• A. Bissal, E. Salinas, J. Magnusson, G. Engdahl, “On the Design of a LinearComposite Magnetic Damper,” accepted for publication in IEEE Interna-tional Magnetics Conference (INTERMAG), Beijing, China, May 2015.

• A. Bissal, J. Magnusson, E. Salinas, G. Engdahl, “Electrical to MechanicalEnergy Conversion of Linear Ultra-Fast Electro-Mechanical Actuators Basedon Stroke Requirements,” in International Conference of Electrical Machines(ICEM), Berlin, Germany, September 2014.

1.5. LIST OF PUBLICATIONS 7

• C. Chen, A. Bissal, E. Salinas, “Numerical Modeling and ExperimentalTesting of Eddy-Current Dampers” in 14th International Conference on NewActuators, Bremen, Germany, 2014.

• A. Bissal, J. Magnusson, and G. Engdahl, “The influence of the VelocityTerm and Thermal Effects in Force Impulse Generator Simulations,” IEEEConference on Electromagnetic Field Computation CEFC 2012, Oita, Japan,November 2012.

• A. Bissal, J. Magnusson, E. Salinas, G. Engdahl, A. Eriksson, “On theDesign of Ultra-Fast Electromechanical Actuators: A Comprehensive Multi-Physical Simulation Model,” Electromagnetic Field Problems and Applications(ICEF), 2012 Sixth International Conference on, pp.1-4, June 2012.

• A. Bissal, J. Magnusson, G. Engdahl, E. Salinas, “Loadability and Scal-ing Aspects of Thomson Based Ultra-Fast Actuators,” in 13th InternationConference on New Actuators, Bremen, Germany, June 2012.

Journal Publications

• A. Bissal, J. Magnusson, M. Backman, E. Salinas, G. Engdahl, “Electricalto Mechanical Energy Conversion of Linear Ultra-Fast Electro-MechanicalActuators Based on Stroke Requirements,” accepted for publication in IEEETransactions on Industrial Applications, 2014.

• A. Bissal, A. Eriksson, J. Magnusson, G. Engdahl, “Hybrid MultiphysicsModeling of an Ultra-Fast Electro-Mechanical Actuator,” submitted for re-view to The international journal of Electric Power Systems Research, 2014.

• A. Bissal, J. Magnusson, G. Engdahl, “Multi-physics Modeling and experi-mental verification of Ultra-Fast Electro-Mechanical Actuators,” submittedfor review to The International Journal of Applied Electromagnetics andMechanics, 2014.

• A. Bissal, J. Magnusson, and G. Engdahl, “Comparison of Two Ultra-Fast Actuator Concepts,” IEEE Transactions on Magnetics, vol.48, no.11,pp.3315-3318, November 2012.

Other Publications

The author has also done minor contributions in the following publications listedbelow.

• J. Magnusson, J. Martinez-Velasco, A. Bissal, G. Engdahl, “Optimal De-sign of a Medium Voltage Hybrid Fault Current Limiter” Energy Conference(ENERGYCON), 2014 IEEE International.


• J. Magnusson, A. Bissal, G. Engdahl, R. Saers, Z. Zhang, L. Liljestrand,“On the Use of Metal Oxide Varistors as a Snubber Circuit in Solid-StateBreakers,” in Innovative Smart Grid Technologies Europe (ISGT EUROPE),2013 4th IEEE/PES.

• J. Magnusson, A. Bissal, G. Engdahl, J.A. Martinez-Velasco, “Design As-pects of a Medium Voltage Hybrid DC Breaker ,” in Innovative Smart GridTechnologies Europe (ISGT EUROPE), 2013 4th IEEE/PES.

• S. Mousavi, A. Krings, G. Engdahl, A. Bissal, “Novel Method for Measure-ment of Anhysteritic Magnetization Curves,” Conference on the Computationof Electromagnetic Fields, Compumag 2013.

Licentiate Thesis Publication

• A. Bissal “On the Design of Ultra-Fast Electro-Mechanical Actuators,” Li-centiate Thesis at "The Royal Institute of Technology (KTH)", Stockholm,Sweden, May 2013.

Patents

• P.O. Karlström, E. Salinas, T.R. Eriksson, A. Bissal, “A High VoltageCurrent Interrupter and an Actuator System for a High Voltage CurrentInterrupter”, Patent Number: WO2014000790, June, 2012.

Some parts of the listed publications are included and elaborated more indetail in this thesis. Materials based on publications in conferences or journalsare copyrighted by their respective associations.

Chapter 2

The Mechanical Switch

This chapter provides a holistic description of the mechanical switch and its sub-components; the actuator, the push/pull rod, the contacts, and the damper. Theentire switch is modeled and validated by laboratory experiments.

2.1 Requirements of the Mechanical Switch

The mechanical switch of an HVDC breaker normally consists of an ultra fastactuator, a damper, bi-stables, a push/pull rod, contacts, and springs.

Due to the low impedance in an HVDC grid, fault currents can increase quicklyin magnitude, attaining large values in a matter of milliseconds. Therefore, acircuit breaker should be able to operate within this time scale such that it is ableto interrupt these fault currents promptly before they become large and harder tointerrupt.

An example of a typical HVDC breaker located in a 320 kV grid would be tointerrupt a fault current within 2 ms. Assuming a maximum over voltage of 10 %,this breaker should ideally be able to withstand 350 kV. If sulfur hexafluoride isused as an insulating medium with a dielectric breakdown strength that is roughlytwo and a half times that of air, i.e. 7.5 kV/mm, then the contacts should beseparated by 50 mm. If the contacts are subjected to a constant acceleration of25,000 m/s2, then they would attain a speed 50 m/s and travel 50 mm in 2 ms.Assuming a total moving mass of 3 kg, including the armature, the push/pull rod,and one end of the contact, then a constant force of 75 kN is required. In reality,since the contacts start from rest, then the actuator has to be able to generate apeak force of 150 kN after 1 ms. Such a combination involving large forces, shorttime scales, and a length scale set by the length of the push/pull rod, usually inthe order of several hundreds of millimeters, poses a mechanical challenge.

9

10 CHAPTER 2. THE MECHANICAL SWITCH

(a) TC

(b) DSC

Figure 2.1: Sketches of the drives

2.2 Switch Design

To be able to generate such impulsive forces, an ultra fast electromagnetic actuatoris required. Two topologies that are able to generate such forces are considered,the Thomson coil (TC), and the double sided coil (DSC) (see Fig. 2.1). The TC iscomposed of a flat spirally shaped coil consisting of several turns, with an electricallyconducting armature in its proximity. The armature has a flat profile and can besituated directly on top or at the bottom of the coil, as close as possible to minimizethe air gap. To generate such large currents, capacitors are connected in series andin parallel to increase the charging voltage and increase the capacitance respectively.Such a unit is charged using a DC power supply and is referred to as a capacitorbank. Upon the discharge of the capacitor bank through the coil, a large currentsurge is generated. This current generates a time varying magnetic field that willbe mostly confined in the air gap between the coil and the armature. A voltage,proportional to the time derivative of the axial component of the field generatedby the coil is induced in the armature. Thus, azimuthally directed eddy currentsappear directly in close proximity of the coil within the armature. The productof these currents with the radial component of the field generated by the coil andpenetrating the armature causes large repulsive forces.

The main difference between a DSC and a TC is the currents in the armature.The armature of a DSC consists of yet another coil that is connected in series withthe primary coil such that currents of the opposite direction flow. Currents in prox-imity to each other flowing in opposite directions cause a repulsive force similar tothe phenomenon explain above. However, the DSC has several advantages. It doesnot rely on a time varying field to induce currents in its armature. Furthermore, thegenerated force is less sensitive to the linearly expanding air gap, causing it to havea higher efficiency than the TC. One major disadvantage limiting its use is due to

2.2. SWITCH DESIGN 11

1

2a

2b

3

4

56a7

6b

Figure 2.2: A sketch of a DC breaker showing the current carrying contacts (2),the push/pull rod (4), the armature (5), the coils (6), and the bistables (7).

the required flexible copper cable connecting the primary coil with the armature.This connection is quite sensitive and can break easily severely diminishing thelifetime of the actuator. Yet another difficulty with the DSC is to wind a coil andattach it firmly to the push/pull rod ensuring it does not deform. As for the TC,an aluminum alloy can be manufactured out of one solid piece that generates theforces and can be easily incorporated in the insulating push/pull rod.

This actuator cannot be directly connected to the metallic contacts. One strictrequirement of a HVDC breaker is to actuate its contacts using an electricallyinsulating material to prevent the fault currents to flow elsewhere. An insulatingmaterial that electrically isolates the contacts from the armature and yet allowsthe transmission of forces between them is required. This component is the above-mentioned push/pull rod ((4) in Fig. 2.2) and is usually made of composite materialsor ceramics. The impulsive forces are generated in the armature and have to be firsttransmitted through the push/pull rod before they arrive to the desired location,i.e. the metal contacts. The main part of the body forces are generated withinthe first few millimeters inside the armature which is located directly on top ofthe coil. Depending on the system voltage level, the push/pull rod has to have acertain minimum length denoted by LPr. The usual length of such push/pull rodsis around 600 mm. The time it takes for a wave to propagate through the armatureto the contacts denoted by ttravel can be estimated using the velocity of waves (vp)


in a homogeneous isotropic medium with

ttravel =LPr

vp

, (2.1)

vp =

√

K + 4G/3

ρ, (2.2)

where K , G, and ρ are the bulk modulus, shear modulus, and density of themedium.

In this study, a push/pull rod manufactured of fiberglass reinforced epoxy(FR4) is investigated since its properties are significantly superior to the traditionalmaterial used for such applications. It is electrically insulating and mechanicallyvery strong with a Young’s modulus of 24 GPa and a yield tensile stress exceeding310 MPa. It has a shear modulus of 11 GPa and a density of 1850 kg/m3. Fur-thermore, it can be machined in any desired shape and size even if smoothnessin shape should be strived for. If the travel time in the aluminium armatureis neglected, then the time it takes for the wave to travel from one end of thepush/pull rod to the other is around 160 µs, which is in the time scale of the HVDCbreaker operation. This time may be even longer if a longer push/pull rod is usedor the push/pull rod is manufactured of a material with a lower Young’s modulus.Therefore, in the following sections, different models are examined in order to studythe influence of the push/pull rod mechanics on the force generation mechanismand the displacement of the armature.

2.3 Modeling

From this section on, the TC actuator is studied since it is more robust than theDSC. Modeling such an actuator is quite challenging since large deformations, largedisplacements, and high currents are involved. This constitutes a multi-physicstransient problem, whereby electromagnetics, mechanics, and thermal equationsmay be needed to be able to accurately predict the behaviour of the actuator.

The modeling of the actuator is divided into two parts, a circuit model and afinite element method (FEM) model that are implemented in the software ComsolMultiphysics (version 4.3b, Comsol AB, Stockholm, Sweden) as shown in Fig. 2.3.An electrical source consisting of series and parallel connected capacitors, a thyris-tor, and cable leads connecting to a Thomson coil (TC) is modeled as a lumpedcircuit. On the other hand, the TC actuator that comprises a primary stationarycoil and a mobile conductive armature, is modeled using a FEM model. These twomodels are then coupled together and solved simultaneously. To generate a largeimpulsive current, several parallel and series connected capacitors are charged to avoltage denoted VC. The capacitor bank can be modeled as an effective capacitancein series with its effective resistance denoted by RC. As for the thyristor and theleads connecting the capacitor bank to the actuator, their impedance is lumped

2.3. MODELING 13

Vc

Rstray

RTC

Lstray

LTC

Arm

atu

re

Rc

Figure 2.3: A SPICE circuit coupled to a FEM model for a Thomson coil. The coiland the armature are modeled using FEM, since the resistance and inductance ofthe coil and armature, RTC and LTC, are nonlinear and changing dynamically asthe armature moves away. The capacitor, diode, thyristor, and cables are modeledby lumped parameters.

together and designated by Rstray and Lstray, respectively. A free-wheeling diodeis furthermore placed in parallel with the capacitor bank to prevent the build upof a negative voltage, since an electrolytic capacitor must be charged with only onepolarity.

The finite element method is used to model the spiral coil and the armature,solving the electromagnetic and mechanical equations at every time step. In themodel, concentric rectangles representing the width and depth of each coil conduc-tor turn are drawn in a two dimensional axi-symmetric geometry, avoiding the needto use 3D simulations, and significantly reducing computation time.

2.3.1 Electromagnetic Modeling

The inputs to the electromagnetic models are the capacitance and the chargingvoltage of the capacitor bank. The voltage (Vcoil) across the spiral coil in the FEMmodel, serves as the connecting point to the circuit model. The total voltage at theterminals of the coil is divided into the different concentric turns such that,

Jen =σeVn

2πrn

, (2.3)

Vcoil =

n∑

i=1

|Vi| , (2.4)

In =

∫

Jn.ds , (2.5)

Icir = I1 = I2 = ... = In , (2.6)


J(A/m2)

Figure 2.4: The current density 100 µs after the discharge of the capacitor bankis distributed in only small portions of the geometry. Positive current densitiesappear in the top part of the coil conductors, while negative currents are inducedin a piece of the armature situated directly above the coil.

which ensures that the current (In) flowing through each and every turn (n) is thesame. Here, the externally applied current density is denoted by Jen and is dueto the voltage across each coil turn Vn. The electrical conductivity is denoted byσe, the current density by Jn, and the surface area of each conductor by s. Theaverage radius of each turn is denoted by rn and is given by rin+rout

2, the average

of the inner and outer radii of every turn.The current density induced in a moving conductive armature Ji,arm, is given

by

Ji,arm = σe(E + v × B) , (2.7)

where, v is the velocity of the moving object, E is the electric field and B themagnetic flux density. The currents in a primary coil are simpler since it isstationary. The induced current density is given by

Ji,coil = σeE . (2.8)

The total current densities, in both, the coil and the armature, are expressed

2.3. MODELING 15

by

Jcoil = Je + Ji,coil , (2.9)

Jarm = Ji,arm , (2.10)

respectively, where Je, is the externally applied current density. An example ofthe current densities in the coil and the armature of a TC can be seen in Fig. 2.4.Based on Maxwell’s equations,

∇ × H = J , (2.11)

∇ × E = −∂B

∂t, (2.12)

B = ∇ × A , (2.13)

the magnetic equation for the primary coil, TCp, is written as

σe

∂A

∂t+

1

µ∇ × (∇ × A) = Je , (2.14)

where A is the magnetic vector potential, H the magnetic field intensity, J thecurrent density, ∇ the nabla operator, and µ the magnetic permeability. Themagnetic equation for the mobile armature, is given by

σe

∂A

∂t+

1

µ∇ × (∇ × A) − σev × (∇ × A) = 0 . (2.15)

The magnetic flux density upon the discharge of the capacitor bank can be seenin Fig. 2.5. A current in the presence of a perpendicularly oriented magnetic fieldresults in a body force fem, also know as a Lorentz force (see Fig. 2.6).

fem = J × B . (2.16)

Similar to the current density distribution, the force density in the armatureis inhomogeneous (see Fig. 2.6). It is highest in the domains that have the largeproduct of the current density and the magnetic flux density. Thus, since the fieldcannot penetrate deep in the armature, the force density is largest close to thesurface of the armature located directly on top of the coil. At the center of thearmature, there is very little force. This creates a torque which might be destructiveto the armature. If the armature is not strong enough, it might bend and deformplastically.

To be able to resolve such small skin depths, second order triangular elementsare used to mesh the surface. The maximum allowed element size is limited tohalf a millimeter to catch these gradients. Although setting such small mesh ele-ments increases memory demand and computation time, it is necessary to increaseaccuracy and ensure stability and convergence.


B(T)

Figure 2.5: The magnetic flux density, in (T), 100 µs after the discharge of thecapacitor bank reaches 5 T and is highest in the air gap separating the coil and thearmature.

A variable time step is implemented such that the model uses very small timesteps only when needed. Nonlinear components, such as the diode model in thecircuit, require a very small time step down to 50 ns. Such a small time step shouldonly be used when the free-wheeling diode is activated. The diode is activatedwhen the voltage across the capacitor bank becomes zero. Although this smalltime step does not improve the accuracy in the computation of the magnetic field,it is necessary for the diode model to ensure convergence. Otherwise, time steps aslarge as 10 µs can be used without sacrificing accuracy.

The generated forces accelerate the armature, dynamically changing the sizeof the air gap. Thus the computed displacement is used to implement a movingmesh based on the Arbitrary Lagrangian-Euler method. If the movement is notaccounted for, then unrealistically large forces will be computed. Furthermore, thesystem resistance and inductance are highly dependent on the size of the air gap. Asthe air gap increases in size, inductance increases since the field from the primarycoil is not cancelled as effectively by the armature. A large inductance limits thedidt

. A slower current rise results in a larger skin depth. Thus the current diffusesmore into the material, decreasing resistance.

Once the mesh is distorted and violates a mesh criterion, the simulation ispaused and the geometry is re-meshed. Prior to re-meshing, the results are ex-tracted from the mesh, interpolated, and set in the newly generated mesh. Astop criterion can be the mesh quality or the maximum allowed elongation of the

2.3. MODELING 17

Fem(Pa)

Figure 2.6: The force density, in (Pa), 100 µs after the discharge of the capacitorbank. The coil is subjected to compressive forces while the armature is subjected toan axially directed body force that is mostly concentrated in the first few millimetersclosest to the coil.

element.

2.3.2 Thermal Modeling

Inrush currents with large current densities and small skin depths may lead to atemperature increase. The temperature rise in a stationary current-conducting coilcan be expressed by,

ρCp

∂T

∂t= ∇ · (k∇T ) + Q , (2.17)

where ρ is the density of the material, Cp is its heat capacity, T is the temperature,k is its thermal conductivity, and Q is the heat source density. The heat sourcedensity is equal to the resistive losses and can be calculated by

Q =J2

σe

. (2.18)

To simulate the temperature distribution in a moving object, a velocity termshould be added if the system of equations is solved using a stationary referenceframe. Thus the temperature equation in the armature of a TC or a DSC movingwith a velocity v is given by,

ρCp

(

∂T

∂t+ v · ∇T

)

= ∇ · k∇T + Q . (2.19)


If a large temperature rise exists, then the electrical conductivity in the domainsexposed to these high temperatures deteriorate. This deterioration is a function oftemperature and can be quantified by

σe = σe0[1 + α(T − T0)]−1 . (2.20)

If special attention is invested in the dimensioning of the coil conductor, andhigh current densities with small skin depths do not appear for a long time, then thethermal equations can be disregarded. To avoid high temperatures, a rectangularcoil is used whereby the depth of the coil can be dimensioned such that it can actas a heat reservoir. The larger the depth of the conductor, the larger the mass, andthe larger is the energy required to increase its temperature. However, the length ofthe conductor cannot be increased indefinitely as it will affect the generated force.Although increasing the depth of the conductor does decrease the net resistanceand limits the temperature rise, it also results in less current densities and longerpath for the magnetic field, increasing the reluctance. Therefore, there exists anoptimum conductor size. A more detailed study on the influence of temperature onthe actuator can be found in [38].

2.3.3 Mechanical Modeling

The developed electromagnetic model and the following mechanical models are co-simulated such that the generated electromagnetic force is used as an input to themechanical models. The mechanical behaviour of the actuator has a significantinfluence on the generation of the electromagnetic forces. Several FEM basedmechanical models are developed with different levels of complexity. Model 1 is themost detailed and accurate model. Although it is too computationally demanding,it has a very high accuracy and hence is used as a benchmark model. Other lesscomputationally demanding models are developed and investigated to determine ifsimpler models can be used without compromising accuracy.

Model 1: Full multi-physics model

The full mutli-physics model simulates the behaviour of the breaker when subjectedto an impulsive force. All components, i.e. the actuator comprising of the coil andthe armature, and the push/pull rod that is at one end attached to the armature andthe other to the copper contacts are meshed and simulated. Due to the existence ofmoderate strains and large displacements, the second Piola-Kirchhoff stress tensorS and the Green Lagrange strain tensor Em are implemented [39]. Although thesemight somewhat collide with the electromagnetic symbols described earlier, theyare used nonetheless. Cauchy’s first equation of motion can be expressed by

DivP + Fem = ρ0

D2U

Dt2, (2.21)

2.3. MODELING 19

where P is the first Piola-Kirchhoff stress tensor,

DivP = FS , (2.22)

F the deformation gradient, and D is the total derivative. The electromagneticforces described in the material reference frame,

Fem = Jfem , (2.23)

are computed by multiplying the calculated electromagnetic forces in the spatialcoordinates fem by

J = detF , (2.24)

the determinant of F. In Eq. 2.21, ρ0 is the density of the reference material andU is the displacement vector given by

U = x − X , (2.25)

where x and X are position vectors in the current and reference configurations,respectively. The deformation gradient and the Green Lagrange strain tensor canbe computed by

F = GradU + I , (2.26)

Em =1

2(GradT U + GradU + GradT UGradU) , (2.27)

where I is the identity matrix, and GradT is the gradient transpose operator withrespect to the material reference frame.

To avoid deformations of any kind, the armature is manufactured out of amaterial that is mechanically strong, yet electrically conductive at the same time.The material is classified as the aluminum alloy 7075 T651. Such a material isused since it has both a high Young’s modulus and a relatively high electricalconductivity compared to steel alloys. This material is represented by a linearelastic model

S = C : Em , (2.28)

where C is the fourth order stiffness tensor and ":" is the double contraction operator(see Appendix A for more details). These derived equations are also used for thepush/pull rod.

These equations are only valid as long as the material remains in the elasticregion. If the material deforms plastically, then they are no longer valid. Modelingplastic material is not of interest in this chapter since all components integrated ina HVDC breaker should have a high reliability. They should be able to successfullysurvive at least ten thousand operations.

Finally, this switch is modeled as an axi-symmetric continuum problem, us-ing Comsol Multiphysics. The developed model consists of 26,000 quadraticallyinterpolated elements.


Model 2: Hybrid multi-physics first order model

A hybrid model is defined as a model consisting of FEM based modeling andanalytical equations. The armature and the coil are modeled using FEM i.e. by(Eq. 2.21-Eq. 2.28) since it is very difficult to compute the electromagnetic forcesusing another method. The push/pull rod on the other hand is modeled using asimplified version of the Kelvin-Voigt model [40]. It is modeled as a first orderspring-mass model. This spring has a stiffness k and is bounded by a mass ateach side. The first mass, denoted m1, equals half the mass of the push/pull rod(m1 = m/2). On the other hand, the other mass connected at the lower terminalof the spring is equal to the sum of the masses of half of the push/pull rod andM , the mass of the attached contact system (m2 = m/2 + M). The sketch of themodel is shown in Fig. 2.7.

If the stresses in the material are assumed to be purely axial, then the stiffnessof the push/pull rod can be computed by the linear stress-strain relation

σ = S11 = Eǫ =F

A, (2.29)

where E is the Young’s modulus, ǫ is the strain in the material, F the force actingon the push/pull rod, and A is the push/pull rod’s cross sectional area. The forcegenerated by a elongating a spring by δx is given by,

Fspring = kδx , (2.30)

where k is the stiffness. The axial stiffness of a push/pull rod having a length L,can be represented by

k =EA

L, (2.31)

using the reference geometry.This model can be generalized into an n segment hybrid model as shown in

Fig. 2.7 (C). Dividing the push/pull rod into n segments yields n springs and n + 1masses. Each spring will have a stiffness of nk and each mass will be m

n+1, except

that the final mass, placed at the very end, will be equal to the mass of the contactsin addition to the mass of one of these partially distributed masses.

Model 3: Push/pull rod assumed infinitely stiff

Model 3 is a simpler model than model 2. In this model, the push/pull rod isassumed to be infinitely stiff. It can also handle the mechanical stresses. Thus, it ismodeled as a lumped element having only a mass. The masses of the push/pull rodand the copper contacts are lumped together and attached to the armature. Onthe other hand, since the armature is exposed to bending moments, it is modelledusing equations (2.21-2.28).

2.3. MODELING 21

m/(n+1) + M

nk

m/(n+1)

m/(n+1)

nk

k

m/2

m/2 + M

(B) (C)(A)

Figure 2.7: A sketch showing the modeling of the armature (top, in silver), thepush/pull rod (in green), and the contacts (in dark red), by a full multi-physicsmodel in (A), by a first order hybrid model in (B), and by a generalized n segmenthybrid model in (C). The push/pull rod’s total mass is represented by m and itsstiffness along elongation by k. The mass of the copper contact is represented byM .

Model 4: Armature and push/pull rod assumed infinitely stiff

Model 4 is the simplest and least computationally demanding model of them all.In this case, all components are modeled using simple mechanics. In other words,the stresses and strains are not computed and their influence on the generatedelectromagnetic forces is neglected. The developed electromagnetic equations areused to compute the repulsive body force. The axial component of this force, inthe z direction (nz) is then integrated to compute the velocity and the position of


Figure 2.8: A picture showing the slim and large mushroom shaped armatures.Although both are designed to withstand the mechanical stresses, one is flexibleand prone to bending while the other is more robust and stiffer.

the actuator as∫∫∫

Fem · nzrdrdθdz = Mtot

dv

dt, (2.32)

x =

∫

vdt , (2.33)

where Mtot is the lumped mass of the armature, the push/pull rod, and the metalliccontacts.

2.4 Model Validations by Experiments

Two experimental setups were built to study the validity of the developed models fordifferent cases. The first experiment was done with a slim armature. If exposed toan impulsive force that is large enough for the intended operating speed, it is proneto bending. Even though it will survive, it will deform elastically. The main reasonfor designing such an armature was to study the influence of finite deformationsand bending on the performance of the drive. In the second experiment the slimarmature was replaced by a larger, more robust, and bulky armature. This wasdesigned such that it can generate and deliver the electromagnetic forces to thedestination with minimum bending. The slim and the large armatures can be seenside by side in Fig. 2.8. Using two differently dimensioned armatures paves theway to validate the full multi-physics model, model 1, and the infinitely stiff model,model 4.

The experimental setup showing the actuator and the steel frame can be seenin Fig. 2.9. In this figure, the mushroom shaped armature is lying on top of aspirally shaped coil. Eight markers are attached on the bakelite frame to center thearmature after every operation. If the armature and the coil are not well aligned,then a larger force may be induced on only one side of the armature leading to a

2.4. MODEL VALIDATIONS BY EXPERIMENTS 23

Figure 2.9: A picture showing the experimental setup. The slim mushroomarmature is sitting on a flat spiral coil that is connected with large cables to acapacitor bank. It is mounted with a 3.6 kg steel mass.

rotational moment. The stem of the armature has threads such that it can carrya load. This weight represents the mass of the metal contacts. Two cylindricallyshaped steel masses, one weighing 3.6 kg and the other 1.5 kg, were attached firmlyto the slim and large armatures, respectively. Two large black cables connect theterminals of the coil to the capacitor bank. A capacitor bank, having a capacitanceof 33 mF and capable of being charged up to 500 V, was used to power the smallarmature. A capacitor bank, having a slightly larger electrical energy, was used topower the larger armature. This capacitor bank has a capacitance of 11 mF and iscapable of being charged up to 900 V,

The actuator is installed in a large and bulky steel frame that weighs around300 kg. This steel frame was clamped to a large two-ton steel table to avoid defor-mations and vibrations. To decelerate the armature smoothly without breaking thecomponents, a Newton mass damper was implemented. The top of the armaturecollides with the damper and transfers its kinetic energy coming to a halt. Thenitrogen-oil damper then slowly damps the energy.

The measured parameters were, the voltage of the capacitor bank, the deliveredcurrent pulse, the acceleration and the position of the armature. Once the thyristorwas triggered, a sudden voltage rise could be measured across its cathode andground. This was used to send a transistor-transistor logic (TTL) to trigger the


Time [ms]

Curr

ent

[kA

]

exp 100 V

sim 100 V

exp 200 V

sim 200 V

exp 300 V

sim 300 V

exp 400 V

sim 400 V

exp 500 V

sim 500 V

0 0.5 1 1.5 2 2.5 30

2

4

6

8

10

Figure 2.10: A picture showing the measured currents (solid lines) and the simulatedcurrents (dashed lines) of the slim armature for increasing charging voltages in stepsof 100 V.

high speed camera. The current pulse was measured using a Pearson probe and thearmature’s acceleration was measured using accelerometers. These accelerometershowever did not survive the high accelerations. After impact, the accelerometerswere broken. Thus, the plan to measure the acceleration was abandoned. Instead,a high speed camera filming at a rate of 100,000 fps was used. The armatures weremarked with high contrast markers and were filmed. Afterwards, the collectedimages were calibrated to track the motion of the armature. The velocity of thearmature was computed by taking the time derivative of the measured positions.

With the help of the camera, different parts of the mushroom armature couldbe tracked. To be able to measure bending, the tip of the armature, i.e. the pointof the armature having the largest radius was tracked. Another point that wasmarked and tracked was the top flat part of the armature. Tracking these twopoints enabled the bending of the armature to be computed.

The measured and simulated current pulses for the slim armature can be seen inFig. 2.10. Model 1 was used to simulate the actuator since bending was expected.At low voltages, the current pulse looks smooth and normal. It has only one peaksituated at 750 µs. However, when the charging voltage was ramped up, therebyincreasing the initial energy, oscillations were introduced in the current pulse. Thisdistortion increased with increasing charging voltages. Following the discharge ofthe capacitor bank charged with 500 V, two distinct peaks could be seen in thecurrent pulse, one at 550 µs, and one at 850 µs. The second peak is even larger thanthe first. This observation is referred to as the camel hump effect.

To investigate the camel hump behaviour introduced in the current pulse, thebending of the mushroom armature was simulated and measured (see Fig. 2.11).

2.4. MODEL VALIDATIONS BY EXPERIMENTS 25

Time [ms]

Ben

din

g[m

m]

MeasurementSimulation

0 0.2 0.4 0.6 0.8

0

0.2

0.4

0.6

0.8

1

Figure 2.11: Comparison of the measured and simulated bending of the mushroomarmature upon the discharge of a capacitor bank charged with 500 V. The bendingcannot be measured for longer time scales since the picture loses focus with largedisplacements rendering the tracking unreliable.

The focus of the camera was shifted to the head of the mushroom. The first pointtracked was point A as indicated in Fig. 2.15. This point is situated at the wing ofthe armature. The second point that was marked and tracked was a point locatedat the top part of flat part of the armature. Both points were tracked with a highspeed camera simultaneously. Bending is defined as the change in vertical positionof one point with respect to another. If both points move vertically with the samedistance, then no bending exists.

The comparison of the simulated and observed bending is shown in Fig. 2.11. Itcan be seen that although there is a small discrepancy in absolute value, the trendshown by simulations and experiments matches very nicely. Some of the sourcesof error are due to filming with a small angle and loss of focus after 800 µs. Thistrend is also visible in the current pulse. This is yet another confirmation of howbending affects the current pulse. The second larger peak obtained at 850 µs is dueto the smaller air gap caused when the wing of the armature bends back towardsthe coil. After the armature starts bending away from the coil, the internal elasticforces start to increase. Once these forces become larger than the electromagneticforce, the armature bends backwards towards the coil. This causes a smaller airgap and a sudden decrease in inductance. Reducing the inductance causes a largeincrease in current thus creating a second current peak that is larger in magnitude.

The slim armature’s measured and simulated velocities, due to the dischargeof the capacitor bank that is initially charged with different charging voltages are


Vel

oci

ty[m

/s]

Time [ms]

exp 100 V

sim 100 V

exp 200 V

sim 200 V

exp 300 V

sim 300 V

exp 400 V

sim 400 V

exp 500 V

sim 500 V

0 2 4 6 8 100

2

4

6

8

10

12

Figure 2.12: A picture showing the measured velocities (solid lines) and thesimulated velocities (dashed lines) of the slim armature for increasing chargingvoltages in steps of 100 V.

shown in Fig. 2.12. The model predicts the behaviour of the armature nicely.The plots for the large armature’s simulated and measured current pulses are

shown in Fig. 2.13. Model 4 was used for the simulation since this armature wasdesigned such that deformations were minimized. Although this assumption fitsperfectly for small charging voltages, the deviations between the simulations andthe observed velocities increased with increasing charging voltages. The simulatedcurrent pulse following the discharge of a capacitor bank charged with 300 Valmost coincides with the measured current pulse. However, for larger voltages,the simulated current pulse always peaks prior to the measured current pulses.This is an indication of less inductance in the simulations. Since deformations arenot computed, then the total system inductance in the simulation model will besmaller. In reality, the armature bends slightly and moves away from the coil. Thusthe magnetic field from the primary coil is not counteracted as efficiently, therebyresulting in a larger inductance. Finally, although some bending does exist, it isvery small since all current pulses exhibit only one peak.

The large armature’s simulated and measured velocities are shown in Fig. 2.14.It was expected that the simulated armature velocities would be slightly larger thanthe measured values for large charging voltages. This is simply because the currentpeaks in the simulation are larger than those measured current peaks. This leadsto larger induced forces. To sum up, for this design with such a bulky armature,model 4 can be used with high confidence. Although a small discrepancy does exist,it is still very accurate.

2.5. RESULTS AND DISCUSSION 27

Time [ms]

Curr

ent

[kA

]

exp 300 V

sim 300 V

exp 400 V

sim 400 V

exp 500 V

sim 500 V

exp 600 V

sim 600 V

exp 700 V

sim 700 V

exp 800 V

sim 800 V

exp 900 V

sim 900 V

0 0.5 1 1.5 2 2.5 30

2

4

6

8

10

12

Figure 2.13: A picture showing the measured currents (solid lines) and the simulatedcurrents (dashed lines) of the large armature for increasing charging voltages insteps of 100 V.

2.5 Results and Discussion

After validating the developed models with experiments, the entire switch includingthe push/pull rod is included in the simulations. The push/pull rod is much moreelastic than the armature itself. As an initial start, model 1 is utilized to simulatethe switch. The velocity distribution, following the discharge of a capacitor bank,can be seen in Fig. 2.15. As expected, 150 µs after discharge, the velocity in thewings of the armature attain 38 m/s. However, this time scale was not enough forthe force to propagate to the contacts located at the bottom of the push/pull rod.This is a very large velocity gradient and is a big drawback for the operation ofthe switch. This result however was not a surprise and was expected initially fromEq. (2.1).

The displacement of the points A, B, and C computed by using model 1, canbe seen in Fig. 2.16. The bending of the armature is defined as the difference


Time [ms]

Vel

oci

ty[m

/s]

sim 300 V

exp 300 V

sim 400 V

exp 400 V

sim 500 V

exp 500 V

sim 600 V

exp 600 V

sim 700 V

exp 700 V

sim 800 V

exp 800 V

sim 900 V

exp 900 V

0 1 2 3 4 50

2

4

6

8

10

12

14

16

18

Figure 2.14: A picture showing the measured velocities (solid lines) and thesimulated velocities (dashed lines) of the large armature for increasing chargingvoltages in steps of 100 V. The simulation error increases with increasing impulsiveforces since even the large armature is elastic and will therefore bend eventually.

between points A and B, while the elongation of the push/pull rod is defined bythe difference between points B and C. Evidently, most of the bending occurs in thefirst 600 µs. The armature bends in the order of several millimeters. Although thisdoes deteriorate the performance of the actuator, the elongation due the elasticityof the push/pull rod is much more severe. 300 µs after discharge, point A, locatedon the wing of the armature, is displaced by as far as 6 mm. In the mean time, thecontacts are still stationary.

Following the simulations done using simulation model 1, it is important toinvestigate performance of each of the developed mechanical models. The currentpulses, computed from models 1 to 4, can be seen in Fig. 2.17. Evidently, simulationmodel 4 gives the largest current peak, followed by model 3, and finally by models1 and 2. Model 4 has the least inductance since the armature and the push/pullrod are assumed to be infinitely stiff. A lower inductance leads to a larger andearlier current peak. Model 3 gives a slightly smaller current pulse than model 4that peaks at a later time. In model 3, the effect of the armature’s bending andelongation can be seen since the push/pull rod is modeled as infinitely stiff. Thusthe decrease in the magnitude of the current pulse is not that severe as in the


v(m/s)

Figure 2.15: A 3D picture showing the velocity profile of the system in [m/s] after150 µs. (B) is a zoom in of the entire actuator shown in (A). The armature, whichis situated directly on top of the coil, is threaded into a push/pull rod and attachedfirmly. Point A is located at the outermost extremity of the mushroom armature.Point B is located at the top of the stem of the armature, i.e. just below therounded corner joining the head of the mushroom to its stem. Point C is locatedon the bottom of the push/pull rod, i.e. where the copper contacts are attached.

following cases. Models 1 and 2 give a current pulse that is significantly smallerthan the current pulse simulated by using model 4. In this case, taking into accountthe elastic behaviour of all components has a large impact on the current pulse.After subjecting the armature to an impulsive force, it is suddenly repelled beforethis force is felt by the contacts. Thus a premature air gap is created increasingsystem inductance. This sudden increase in inductance limits the di

dt. Consequently,

a smaller current peak is generated. To sum up, the higher the elasticity of the


Time [ms]

Poin

tp

osi

tions

[mm

]fr

om

Model

1

ptA

ptB

ptC

0 1 2 3 40

20

40

60

80

100

Figure 2.16: The displacements of three different points characterizing the dynamicmotion of the breaker. The bending of the head of the mushroom can be computedby subtracting the axial displacement of pt B from that of pt A. Similarly, theelongation of the push/pull rod and the stem of the mushroom armature can becomputed by subtracting the displacement of pt C from that of pt B.

incorporated components, the larger is the impact on the current pulse due tohigher inductances.

This not only negatively impacts the current pulse, but also the efficiency ofthe actuator. An infinitely stiff system will attain a velocity of 29 m/s while adeformable system will attain only 25 m/s keeping everything else constant. Thus,the efficiency of an infinitely stiff system is 35 % larger than the latter.

Evidently, the models that give lower currents will give smaller forces (seeFig. 2.18). In fact, the generated forces are inversely proportional to the square ofthe length of the air gap. Thus, the difference in force between all four models ismuch larger than their difference in the current pulses. Model 4 gives the largestpeak force, followed by model 3 and finally by models 1 and 2. Although models 1and 2 have identical force peaks, they seem to oscillate out of phase. An infinitelystiff system will result in the largest force impulse.

The displacements of the contacts versus time from all the developed models aregiven in Fig. 2.19. Models 1 and 2 have very similar displacements. However, theyoscillate out of phase. Models 3 and 4 travel a longer distance for a given amountof time. Therefore, it can be clearly concluded that model 2 is able to predict thebehaviours of the contacts with very good accuracy. The only noticeable differenceis that model 1 and 2 oscillate out of phase. If the current pulse, the force impulse,and the displacements are used as judging criteria, then it can be clearly concludedthat model 2 has similar results in comparison with model 1, the benchmark model.Model 3 is less accurate, and model 4 has the least accuracy. In this case, model


Time [ms]

Curr

ent

[kA

]

Model 1

Model 2

Model 3

Model 4

0 1 2 3 40

5

10

15

20

Figure 2.17: The arising current pulse in the four models following the discharge ofthe capacitor bank in the spirally shaped coil.

Time [ms]

Forc

e[k

N]

Model 1

Model 2

Model 3

Model 4

0 0.5 1 1.5 20

50

100

150

200

Figure 2.18: The arising force impulse in the four models following the dischargeof the capacitor bank in the spirally shaped coil.

4 overestimated the peak force by 100 %. Thus, in highly elastic systems, model2 could be used to simulate the behaviour of the switch without compromisingaccuracy. This does not mean that models 3 and 4 are useless. Models 3 and4 could be used as long as their underlying assumptions hold. Models 3 and 4demand even less computational power than model 2. Model 3 can be used if thepush/pull rod is very stiff. If the generated forces are not enough to elongate the


Time [ms]

ptC

Posi

tion

[mm

]

Model 1

Model 2

Model 3

Model 4

0 1 2 3 40

20

40

60

80

100

Figure 2.19: The displacement of the load with respect to each model.

push/pull rod, then this model is quite accurate and sufficient. Lastly, model 4could be used if all components are over-dimensioned such that deformations areminimized. Model 4 can provide valuable insight and is the least computationallydemanding model.

2.5.1 Models With Finer Segmentations

Since model 2 gave excellent results, further work was carried out to investigate thepotential of using models with finer segmentations. Thus three additional hybridmodels were developed denoted by models 5, 6, and 7. Similarly, as has been donein model 2, models 5, 6, and 7 represent the push/pull rod by 2, 3, and 10 springelements respectively (see Fig. 2.7). The contact displacement from all models canbe seen in Fig. 2.20. Surprisingly, although increasing the segmentation numberincreases complexity, models 5, 6, and 7 are not more accurate than model 2. Thus,increasing the number of spring elements does not necessarily give better results.Nonetheless, those additional models are still better than models 3 and 4.

A quick rough estimation for the number of necessary lumped elements canbe assessed by considering the wavelength of the force wave. Using Eq. 2.2 andthe length of the push/pull rod, the speed of sound in the push/pull rod canbe estimated to be around 3750 m/s. The force impulse peaks at around 400 µs.Assuming the force impulse is sinusoidal, then it will have a period of 4× 400 µs =1.6 ms. This results in a frequency of 625 Hz. Thus, the corresponding wavelengthis λ = 3750

625= 6 m. This wavelength is ten times larger than the length of the

push/pull rod. Thus, one spring-mass segment in the lumped element model shouldbe enough.

To sum up, if a stiff system is used, then Model 4, that couples the electro-


Time [ms]

ptC

Posi

tion

[mm

]

Model 1

Model 2

Model 3

Model 4

Model 5

Model 6

Model 7

0 1 2 3 40

20

40

60

80

100

Figure 2.20: The displacement of the load with respect to each model. Model 2:first order hybrid model. Model 5: second order hybrid model. Model 6: thirdorder hybrid model. Model 7: Tenth order hybrid model.

magnetic FEM model to a simple mechanics model can be used. However, if anelastic system is used, then a more complex model is required. The developed firstorder hybrid model has been shown to be able to simulate the behaviour of themechanical switch with a very good accuracy. Hybrid models with higher numberof segmentations have been shown to have a limited advantage.

Chapter 3

The Ultra-Fast Actuator

After having provided an overall description of the mechanical switch with all therequired components, this chapter focuses on the design of the actuator itself whenloaded with a lumped mass. Therefore, the push/pull rod and the contacts areconsidered to be infinitely stiff to study the key parameters influencing the efficiencyof the actuator. The main topics studied are the electrical circuit, i.e. the energizingsource, the magnetic circuit, and actuator shape optimization.

3.1 State of the art

Traditionally, most switching devices comprised of springs and gears. Permanentmagnet based switches have recently become attractive since they have fewer me-chanical components having a higher reliability. They have been used in vacuumand gas circuit breakers [41–44]. To meet the requirements of low loss HVDCbreakers, ultra-fast actuators are required.

Many forms of actuators exist for generating ultra fast impulsive forces some ofwhich are: the reconnection gun, the helix coil launcher, the coil gun, and the railgun etc ... The rail gun is very famous for attaining speeds up to mach 7 and ismostly used in the military for firing high speed projectiles that annihilate a targetupon impact just by the projectile’s kinetic energy.

Of all these ultra-fast actuators, the Thomson coil (TC) has gained attentionmostly for the use in HVDC breakers.

3.1.1 Applications

There are many examples where Thomson drives are used to generate impulsiveforces. In [45], a TC is used in a hybrid DC breaker consisting of a mechanicalswitch, two parallel connected gate-commutated thyristors (IGCTs), diodes, andan MOV to dissipate the energy and interrupt fault currents. Mitsubishi is anothercompany that has made use of the TC in its high speed single phase circuit breaker

35

36 CHAPTER 3. THE ULTRA-FAST ACTUATOR

[46]. A repulsion plate is positioned between a closing and an opening coil. Theoperational time of this circuit breaker is 1 ms. It is capable of interrupting a faultcurrent in 20 ms. Another example of an electromagnetic repulsion drive installedin a HVDC breaker can be found in [47]. In [48], a current limiting DC hybridbreaker is shown whose contacts can open in 200 µs attaining speeds of 10 m/s.In [49], injecting a current pulse in the primary coil generates a repulsive force thatallows the armature to attain speeds of 20 m/s. The contacts separate in only 100 µs.A slightly slower drive is shown in [50]. This high speed switch was used in a hybriddrive and could open in 1 ms. To study the interruption capability of injecting highfrequency counter currents, a repulsion drive was used to open the current carryingcontacts in [51]. In [52], the circuit breaker’s electromagnetic drive mechanism isexamined and in [53], a permanent magnet actuator is compared with a high speedrepulsion actuator. In [54], two consecutive discharges were implemented. Lastly,a TC has been used in an arc eliminator in [55].

Although the preceding examples were linear topologies, rotational uses ofThomson drives have also been implemented. One such example is in [56]. Ithas been shown that achieving a breaking time within a few 100 µs is possible.The armature attained velocities up to 50 m/s. This ultra fast speed came at theexpense of efficiency: this drive has an efficiency that is less than 5 %.

Other applications for Thomson drives are for superconductive fault currentlimiters in which temperature is a critical and crucial parameter [57]. In theseapplications, ultra-fast drives are needed to interrupt fault currents promptly.

3.1.2 Modeling

Due to the limited available computational power, traditionally, most TC modelswere crude requiring little computational effort. In [58], equation based modelingis used for a TC. In [59], an adaptive equivalent circuit modeling has been im-plemented. In [46], an analytical model is derived based on the tableau method.In [60], a reduced modeling technique of an eddy current drive is explained. In [61],an interpolation function with electrical equivalent networks that take into accounteddy currents has been implemented. In an attempt to improve the efficiency ofthe TC in [62], optimization studies were carried out.

Recently, an increasing trend to study the mutual influence of different physicshas begun. Multi-physics simulations started to increase in popularity. Examplescan be found in [63–68]. One aim of this thesis is to develop a multi-physicssimulation model to accurately simulate the behaviour of a TC in light of improvingits efficiency.

3.2 Modeling

This section presents the development procedure of a model that is accurate, robust,and simple such that it is capable of simulating ultra-fast actuators with a gooddegree of accuracy in a reasonable time scale.

3.2. MODELING 37

The capacitor bank is modeled as a SPICE circuit with C representing itscapacitance, and Rc denoting to its internal equivalent series resistance. The cablesconnecting the output terminals of the capacitor bank to the actuator consisting ofa TC or a DSC are also modeled by lumped parameters Rstray and Lstray denotingthe cable’s resistance and inductance.

The actuator is modeled using the FEM. A simple mechanics model is usedwhere deformations and wave propagations are neglected. The generated forcedensity is integrated to compute the acceleration of the armature, its velocity, andits position with respect to time. The position of the armature is used to move andstretch the mesh based on the ALE method as explained in the modeling section ofthe mechanical switch. Moreover, temperature effects are also neglected to reducethe computational demand of the model. These approximations are reasonable andare validated using experiments as presented in the following section.

The mesh element size should be chosen carefully since it may impact theaccuracy of the solution. In these studies, the mesh element size is kept constantfor all simulation cases. The maximum mesh element size in the armature and thecoil is chosen to be small enough to resolve the worst case scenario dominated bysmall skin depths due to proximity and skin effects.

The main factors influencing the mesh size are the capacitance, the chargingvoltage, the number of coil turns, the coil inner radius, and the electrical conductiv-ity of the armature. A small capacitance charged with a high voltage, a small overallsystem inductance, and a copper armature with a higher electrical conductivitycompared to aluminum leads to the fastest transients and small skin depths. Thesystem inductance is mostly dominated by the size of the coil. Minimizing the innerradius of the coil and its number of turns minimizes inductance. These combinationsresult in a fast transient whereby eddy currents are induced in the coil, preventingthe main current from flowing homogeneously and making use of the entire availableconductor cross section.

Although choosing the same mesh size for all possible combinations it is not op-timum and results in a larger computation burden for cases where slower transientsare present, this approach is safe and reliable since it will be able to fully resolvethe skin depth of all other combinations as well. Moreover, it does not lead to adrastic increase in computation time since the geometries of the armature and thecoil are small.

The air gap in these simulation models is divided into three layers. The inner-most layer of the air gap is allowed to stretch whereas the two other outermostlayers that are in contact with the coil and armature respectively are not allowedto stretch. The first layer of the air gap, in contact with the top of the coil, ismaintained fixed, while the other layer of the air gap, in contact withe the armature,is set to move with the same speed as that of the armature to ensure that themesh is not stretched and kept constant. This method increases computationalaccuracy. Finally, to further speed up the model, the air gap is meshed using aparametric square mesh such that when it is stretched, it cannot result in invertedmesh elements. This eliminates the need of re-meshing.


Figure 3.1: A schematic explaining the test setup and the different components ofthe prototypes.

A common problem with FEM models is convergence and stability. Since acomprehensive parametric study and optimization will be carried out, a robustmodel is needed to avoid crashing. A Matlab script was written and coupled withComsol Multiphysics. Furthermore, an error-resolving algorithm was implementedthat is able to detect failed cases, set tighter tolerances, and attempt to re-solvethem again. This was done in three repetitive stages whereby the tolerances weretightened even further until the model successfully solved. If the model failed afterattempting to solve it with the three contingency cases, then this particular set ofinput parameters was disregarded. This is only an example; more contingency casescan be developed to maximize robustness, but there was no need to complicate themodel further for this case. The developed model was able to successfully solve allinput cases as shown in the coming sections.

Finally, to reduce computation time, the code was parallelized to run on severalCPU cores at the same time. The gain in speed was significant and was almostdirectly proportional to the number of cores used.

3.3 Experimental Validation

To verify the developed model, many parameters were varied including the shapesand sizes of the coil and the overlying armature. The velocity of the armature ismeasured by tracking its motion using a high speed camera. The current pulse andcapacitor voltage are measured using a Pearson current probe and a voltage probe,respectively (see Fig. 3.1). The main aim of this section is to compare the resultsof the simulations with experiments based on several variations. The variables thathave been systematically varied are the capacitance, the charging voltage, the sizeof the coil, the coil’s number of turns, the armature’s shape and material, and themass to be actuated. As will be shown, all these variables have a significant impacton the behaviour of the actuator.

3.3.1 The Experimental Setups

Two flexible setups have been built for the purpose of validating this model. Thefirst consisted of a small scale prototype as shown in Fig. 3.2. Its energizing source

3.3. EXPERIMENTAL VALIDATION 39

Figure 3.2: A picture showing the high speed camera, a capacitor bank, a Pearsonprobe, and the small scale prototype.

Figure 3.3: A twenty turn spirally shaped flat coil for the large scale prototype.

consisted of a 33 mF capacitor capable of being charged up to 500 V. This wasconnected to a nine turn, spirally wound flat coil similar to the one shown inFig. 3.3. Two types of armatures were used to generate the impulsive forces, onemade of aluminum 6082 T651 as shown in Fig.3.4 and the other manufacturedout of oxygen-free high conductivity copper (OFHS). Initially, a teflon guide wasinstalled so that the armature can be guided with minimum friction. This howeverwas removed later since following the discharge of the capacitor bank, the armaturehas seen to follow a perfectly rectilinear trajectory.

A second real scale prototype shown in Fig. 3.5 was developed to study theinfluence of loading on the performance of these actuators. Moreover, all compo-nents were over-dimensioned to be able to operate this prototype with much highervelocities. Accelerating heavy armatures with speeds up to 20 m/s in a millisecondrequires a robust foundation according to Newton’s second law, F = ma, and thirdlaw, “Every action has an equal and opposite reaction”. A strong foundation isimportant to avoid shooting down the coil. If the coil is not immobilized firmly,then even a slight movement in the order of sub millimeters will cause a larger airgap, limiting the magnitude of the generated forces. Therefore, the coil is firstly


(A)

(B) (C)

Figure 3.4: Three armature variants, (A) showing a disk made of oxygen-free highconductive copper, (B) showing a disk made of Aluminum 6082 T651, and (C)showing a mushroom shaped armature made of Aluminum 7075 T651. Armatures(A) and (B) are used in the small scale prototype while armature (C) is used in thelarge scale prototype.

embedded in bakelite and covered with a strong epoxy. Moreover, the bakeliteis inserted in a steel housing that is resting on a massive sturdy steel structureclamped to a two ton table. Steel clamps are used to avoid any vibrations.

The energizing source of the large scale prototype consisted of an 11 mF capac-itor bank capable of being charged up to 900 V. The energy of this capacitor bankis slightly larger than that of the other capacitor bank. However, it was speciallydesigned with a small capacitance and higher charging voltages to generate largerand faster impulsive currents. A higher di

dtcauses larger accelerations. This bank

was connected to a twenty turn coil, significantly larger than the one in the smallscale setup to increase the surface area between the coil and the armature (seeFig. 3.3). Increasing this surface area will result in a wider force density distribution.

Several simulations have been done to optimize the shape of the armaturesuch that it can withstand these impulsive forces. An optimum armature is anarmature that has a high electrical conductivity and is capable of delivering thegenerated forces within the armature in the region located directly above the coil,to the mass usually attached at its stem, without any deformations. After severaltransient time-dependent mechanical simulations, it was found that an optimumshape capable of meeting these requirements consists of a mushroom. The fillet ofthe mushroom connecting its top to its stem and the width of the stem have to


Figure 3.5: The large scale prototype showing the coil firmly attached to the centerof a massive steel construction designed to avoid vibrations.

be designed such that the maximum stress is kept below 300 MPa and elongationsare minimized. Moreover, the armature was designed to be as light weight aspossible without jeopardizing its mechanical integrity. Although high electricalconductivities are essential, the armature is manufactured out of aluminum 7075T651, the strongest available aluminum alloy in the market which is mostly usedin aeronautics. In this case, its Young’s modulus and its ultimate yield strengthwere more important than the added electrical conductivity that could have beengained by using the aluminum alloy 6082. Moreover, copper cannot be used sinceit has a high density. A large mass will limit the acceleration and the maximumattained velocity. The ideal material properties of such armatures are high electricalconductivity, low density, high Young’s modulus and high ultimate tensile strength.

3.3.2 Results and Discussion

To verify the model’s validity under such a large parametric variation, two flexibletest setups were prepared. The velocity of the armature was decided to serve asthe assessment criterion.

The velocity of the copper and aluminum armatures following the dischargeof a 33 mF capacitor charged with 50 V through a nine turn coil is shown in


Time (ms)

Arm

atu

revel

oci

ties

(m/s)

Exp Al 50 V

Sim Al 50 VExp Cu 50 V

Sim Cu 50 V

0 2 4 6 8 100

0.5

1

1.5

2

2.5

Figure 3.6: Simulations compared with experimentally measured armaturevelocities made of copper and aluminum upon the discharge of a capacitor bankwith a capacitance of 33 mF charged up to 50 V.

Time (ms)

Arm

atu

revel

oci

ties

(m/s)

Exp Al 100 V

Sim Al 100 VExp Cu 100 V

Sim Cu 100 V

0 2 4 6 8 100

2

4

6

8

Figure 3.7: Simulations compared with experimentally measured armaturevelocities made of copper and aluminum upon the discharge of a capacitor bankwith a capacitance of 33 mF charged up to 100 V.

Fig. 3.6. Afterwards, the charging voltage was increased to 100 V and the processwas repeated as shown in Fig. 3.7. The measured and simulated armature velocitiesare in good agreement.

Although copper has a higher electrical conductivity than aluminum, the cop-


Time (ms)

Vel

oci

ty(m

/s)

300 V Sim300 V Exp

400 V Sim400 V Exp

500 V Sim500 V Exp

600 V Sim600 V Exp

700 V Sim700 V Exp

800 V Sim800 V Exp

900 V Sim900 V Exp

0 1 2 3 4 50

5

10

15

Figure 3.8: Simulations compared with experimentally measured armaturevelocities upon the discharge of an 11 mF capacitor bank charged from 300 V to900 V in steps of 100 V.

per armature attains a significantly lower velocity than an aluminum armature.Since copper has a higher electrical conductivity larger currents are induced in itcompared with aluminum. This results in a larger electromagnetic force. Thisgained force, however, is not enough to compensate for its weight. Copper hasa larger density and is thus significantly heavier than aluminum. Therefore, thecopper armature experiences a smaller acceleration and attains a smaller steadystate velocity.

If the actuator is unloaded and has no load to accelerate except for the massof its armature, then, a lightweight aluminum armature is better than a copperarmature. If on the other hand, the mass of armature is insignificant compared toits load, then naturally, a copper armature should be used. The thickness of thiscopper armature is not easy to determine but may be estimated roughly based onskin depth. Otherwise, an optimization model has to be done to maximize velocity.

An unloaded TC actuator is unrealistic unless it is intended to be used as aweapon. If integrated in a breaker, then it should accelerate the current-carryingmetallic contacts. Such contacts may weigh anywhere between half a kilogram andten kilograms. Therefore, a mushroom shaped armature was designed such that abulk mass can be attached to it.

The measured and simulated velocities of the mushroom armature followingthe discharge of a 11 mF capacitor bank through a twenty turn coil are shown inFig. 3.8. The charging voltage was increased from 300 V to 900 V in steps of 100 V.As can be seen, the measured and simulated velocities are almost overlapping forsmall voltages.


Cir C

w

Cd ...

Ad

Aw

Sym

me

try lin

e

N

Figure 3.9: A figure showing the defined actuator variables; more precisely, thevariables parameterizing the coil and the armature.

Although a small discrepancy can be seen with increasing voltages, it is verysmall. This discrepancy is due to the fact that a simplified mechanics model isused, neglecting deformations as explained int he previous chapter. To sum up,this model can predict the behaviour of the TC actuator with a good accuracy.

3.4 Sensitivity Analysis

The electrical circuit, the magnetic circuit, and the influence of the shape and sizeof the actuator have been studied in this section. Modeling the ultra fast actuatoris a complex problem since there are many decision variables involved where a smallchange in the input variables can lead to a drastic change in the output. Fig. 3.9shows all the studied variables of the actuator’s geometry. The coil inner radius isrepresented by Cir, the coil width by Cw, the coil depth by Cd, the number of turnsby N , the armature depth by Ad, and the armature width by Aw.

3.4.1 The Electrical Circuit

The main aim of this section is to highlight the main causes limiting the efficiencyof such drives, and to design an optimal energizing source.

One of the main causes that leads to low efficiencies for a TC and a DSC is dueto an increasing air gap. Regardless of the capacitance and the actuator topology,Fig. 3.10 and Fig. 3.11 demonstrate the importance of the air gap by comparingthe generated forces in a clamped armature with the forces generated in a mobilearmature in that of a TC and a DSC respectively. Although the clamped armatureexperiences the force, it does not move. In contrast, the mobile armature is freefrom constraint and hence, repels from the coil freely. As can be seen, a significantportion of the force is lost due to motion away from the field of the stationary coil.These lost forces are denominated as forgone forces.

3.4. SENSITIVITY ANALYSIS 45

Time (ms)

Forc

e(k

N)

Forgone force

Mobile TC

0 0.5 1 1.5 20

50

100

150

Figure 3.10: The force impulse of a mobile TC along with its forgone force dueto its stroke. The area shaded in blue is the force impulse of a mobile TC duringthe discharge of a 10 mF capacitor bank whereas the area under the curve shadedin red is the forgone force due to the increased air gap with contact separation.In essence, if the armature of the TC is clamped, then the total generated forceimpulse is given by the sum of the areas under both curves (i.e. the sum of theareas shaded in blue and in red).

Methodology

One way of increasing the efficiency of such drives is by minimizing their stroke. Toreduce the stroke, the contacts can be inserted in a medium that has a high dielectricstrength. Another method would be to design a switch that has several series-connected contacts. An actuator that is capable of separating all these contactsat once can create a larger arc voltage reducing the demands on stroke. However,these methodologies increase the complexity, cost, and reliability of the switch.

Instead of limiting the stroke, one approach would be to design an energizingsource that can deliver an impulsive current with a rise time that is significantlysmaller than the mechanical response of the system. The main factors controllingthe rise time of the current pulse are the capacitance and the system inductance.This circuit can be seen as an LRC circuit. Therefore, its resonance frequency isgiven by

fres =1

2π√

LC, (3.1)

where L and C are the inductance and the capacitance of the system. The in-ductance is dependent on the shape and size of the actuator and hence is fixed.Some of the factors that influence the system inductance are the coil’s width, its


Time (ms)

Forc

e(k

N)

Forgone force

Mobile DSC

0 1 2 3 4 50

10

20

30

40

50

Figure 3.11: The force impulse of a mobile DSC along with its forgone force dueto its stroke. The area shaded in blue is the force impulse of a mobile DSC duringthe discharge of a 100 mF capacitor bank whereas the area under the curve shadedin red is the forgone force due to the increased air gap with contact separation.In essence, if the armature of the DSC is clamped, then the total generated forceimpulse would be the total area under both curves (i.e. the sum of the areas shadedin blue and in red).

inner radius, its number of turns, and the armature’s conductivity, its shape andsize, and its proximity to the coil. The inductance is quite complex to calculatesince it is highly dependent on the length of the air gap. As the air gap increases,the coupling between the armature and the coil decreases. This results in a largersystem inductance. The inductance increases as a function of the length of the airgap. Its maximum can be calculated by simulating the coil without an armature.Although initially a typical system inductance is in the order of 2 µH, it may increaseup to 20 µH.

The objective of this study is to design an energizing source that maximizes theefficiencies of these actuators. Therefore, the shape of the actuators are maintainedfixed while the capacitor bank’s capacitance and charging voltage are systematicallyvaried while making sure that the electrical energy (EE) for all configurations is thesame and is equal to 2640 joules. The capacitance, and accordingly, the chargingvoltages are varied from 10 µF to 100 mF and from 230 V to 23 kV respectively (seeTable. 3.1).

Results and discussion

Table 3.1 shows the results of the simulations for the TC and the DSC for sixdifferent sets of parameters. The left of the table, designated by energizing source


Table 3.1: The efficiency of a clamped TC (TCc), a mobile TC (TCm), a clampedDSC (DSCc), and a mobile (DSCm) for different energizing source configurations.Red highlights the highest efficiency.

Energizing source η (%)EE (J) C V TCc TCm DSCc DSCm

2640 10 µF 22.978kV 3.5 2.6 3.9 3.22640 100 µF 7.266 kV 11.8 8.1 13.5 9.52640 1 mF 2.298 kV 33.1 14.5 40.1 19.12640 10 mF 726.64 V 53.7 14.7 75.7 23.22640 33 mF 400 V 46.4 10.2 84.5 19.92640 100 mF 229.78 V 26.3 5.2 87.7 13.8

shows the chosen capacitance and charging voltage combination. The electricenergy is also shown for all sets to emphasize that the electrical energy is maintainedconstant for all configurations. It can be computed by 1

2CV 2. The column located

in the right section of the table shows the efficiency (η = mv2

CV 2 ) for four differentactuator topologies. The mass of the armature and the load are lumped togetherand denoted by m. The mass of the load is assumed to be 1 kg in this case. Thefirst topology, designated by TCc, refers to the prospective efficiency of a clampedTC. The term clamped in this case refers to the armature, indicating that thearmature of the TC is fixed and thus cannot move. The prospective velocity is thehypothetical ideal velocity that this actuator would have attained if it were allowedto move. To calculate this velocity (v), the induced forces (Fem) in the armaturewere integrated with respected to time according to Newton’s law,

∫∫∫

Fem · nzrdrdθdz = mdv

dt, (3.2)

This prospective velocity is ideal and can only be attained if the generated forceswere not affected by the air gap or if the air gap was maintained constant. Thesecond topology (TCm), refers to a mobile TC. The armature of a mobile TC isfree to move and is prone to force deterioration due to an increasing air gap. Thisis a realistic case and is used to compare it with the clamped TC. Similarly, DSCc

and DSCm refer to a clamped and a mobile DSC respectively. The clamped studyis used as a benchmark to be able to asses all TC and DSC mobile configurations.

A current pulse with the highest resonance frequency results in the lowestactuator efficiencies for all cases. The first configuration in the table generatesa current pulse with the highest frequency. Proximity and skin effect significantlyreduce the effective conductor cross section area utilized by the current. Currentdensities of opposite signs concentrate mostly on the top of the coil’s conductorand on the bottom of the armature as shown in Fig. 3.12. This skin depth (δ) is


r (mm)

z(m

m)

J(A/m2)

20 30 40 50

-2

0

2×1010

-5

0

5

10

Figure 3.12: Current density in (A/m2) at 8 µs shown in the cross sections of thecoil and armature, just before the first current peak, for the first energizing sourceof a mobile TC shown in Table 3.1.

inversely proportional to the frequency (f) and can be expressed by

δ =

√

ρ

πfµ, (3.3)

where ρ and µ are the conductor’s resistivity and magnetic permeability respec-tively. Although this leads to a better coupling reducing inductance, it does so atthe expense of increased resistance. The effective resistance increases significantlyreducing the peak current and the resultant forces. High current densities areassociated with high temperatures. Hot spots may arise in regions with highcurrent densities that may deteriorate the electrical conductivity of the material.The electrical conductivity of the actuator in per unit, 88 µs after the discharge ofthe first energizing source shown in Table 3.1, is shown in Fig. 3.13. The electricalconductivity in the top part of the coil’s conductor becomes less than half its initialvalue. This drop in conductivity may increase the total electrical resistance if thisarea becomes substantial.

To alleviate the effect of temperature, a large conductor thickness should beused. Even though the skin depth dictates that only a very small portion of theconductor will be used, a large conductor should in any case be used to allow theheat to diffuse deeper into the coil. Active cooling cannot be used to decreasethe coil’s temperature since it is embedded in epoxy. Furthermore, the time scaleleading to such a rise in temperature is very small. Thus, using a bulky conductorallows the temperature to diffuse from hot regions to cooler regions. This limitsthe deteriorating effect of temperature on the electrical conductivity.

Yet another reason to avoid high temperatures is to extend the actuator’slifetime and increase the frequency of operations. If currents are discharged thoughthe coil without allowing enough time for it to cool down, then the coil will expand


r (mm)

z(m

m)

σe (pu)

20 30 40 50

0.5

0.6

0.7

0.8

0.9

-5

0

5

10

Figure 3.13: Electrical conductivity in per unit at 88 µs shown in the cross sectionsof the coil and armature for the first energizing source shown in Table 3.1.

r (mm)

z(m

m)

J(A/m2)

20 30 40 50

-5

0

5

10

15×108

-5

0

5

10

Figure 3.14: Current density in (A/m2) at 500 µs shown in the cross sections of thecoil and armature, just before the first current peak, for the last energizing sourceof a mobile TC shown in Table 3.1.

and crack the epoxy permanently deforming the actuator. This has been witnessedin the lab during an experiment.

On the other hand, the other extreme, i.e. the configuration with the lowestresonance frequency, is not optimal either. Although the current density for acapacitance of 100 mF is rather homogenous, as shown in Fig. 3.14, and makesuse of the entire available conductor area, this configuration is not optimal. Theefficiencies of a mobile TC and a mobile DSC are 5.2 % and 13.8 % respectively. Theslow rising current pulse causes the armature to repel prematurely before the currentincreases in magnitude substantially. This causes a large air gap and severely limitsthe induced induced forces in both actuators.

Although the efficiencies of the TC and the DSC deteriorate in response to acurrent pulse with a low resonant frequency, the TC is much more sensitive to an


Time (ms)

Vel

oci

ty(m

/s)

C: 10 µFC: 100 µFC: 1 mFC: 10 mFC: 33 mFC: 100 mF

0 2 4 60

5

10

15

20

25

30

Figure 3.15: The velocity profile of a mobile TC drive shown for the six differentenergizing source cases shown in Table 3.1.

increasing air gap than its counterpart. The efficiencies for a mobile TC and a DSCfor a capacitance of 100 mF are 5.2 % and 13.2 % respectively. This difference islargest for low frequencies and becomes smaller for higher frequencies. The TCrelies on a time varying axial component of a magnetic field and on its radialcomponent to generate repulsive forces. Both of these components are inverselyproportional to the length of the air gap. Therefore, the Lorentz force is inverselyproportional to square of the air gap for a TC. However, for a DSC, the Lorentzforce relies only on the radial component of the field from the coil and hence is onlyinversely proportional to the length of the air gap.

A current pulse with an intermediary resonance frequency leads to the maximumefficiency for both a mobile TC and a mobile DSC. A capacitance of 10 mF and acharging voltage of 727 V leads to efficiencies of 14.7 % and 23.0 % for a mobile TCand a DSC respectively. This clearly signifies that there exists one optimum. Theregion of interest where the optimum efficiency lies is between a capacitance of 1 mFand a capacitance of 33 mF for this particular shape and size of these actuators.Although the efficiencies of the mobile TC and mobile DSC are maximized for thiscase, their efficiencies are still far from their ideal velocities that they can achieve.This indicates that the increasing air gap still has a significant impact on efficiency.

For a clamped DSC, the optimal configuration is one with the lowest resonancefrequency. Since the DSC only relies on a constant radial component of the magneticfield to generate repulsive forces, its prospective velocity is highest when operatedwith a DC current.

The velocity versus time for a mobile TC and a mobile DSC are shown inFig. 3.15 and in Fig. 3.16 respectively.

In conclusion, a compromise between the resonance frequency and effective


Time (ms)

Vel

oci

ty(m

/s)

C: 10 µFC: 100 µFC: 1 mFC: 10 mFC: 33 mFC: 100 mF

0 1 2 3 4 50

10

20

30

40

Figure 3.16: The velocity profile of a mobile DSC drive shown for the six differentenergizing source cases shown in Table 3.1.

resistance exists. The factor limiting the efficiency of these actuators for smallfrequencies is the air gap. The other factor contributing to a poor efficiency forhigh frequencies is the increased resistance due to the skin and proximity effects.

3.4.2 The Magnetic Circuit

As discussed in the introduction, the TC consists of a spiral coil and a conductivearmature in its proximity. The coil is wound from a copper conductor having athin enamel coating. The enamel coating is usually 0.1 mm thick and is used forelectrically insulating the coil from its surrounding.

Newton’s third law states, “every force has an equal and opposite reaction force”.When the TC generates large impulsive forces in its armature, the coil experiencesthe same force, equally strong but opposite in direction. Thus, depending on themagnitude of these forces, a stable foundation is required to withstand them.

To ensure mechanical stability, the coil is inserted in a groove in a bakeliteframe and covered with epoxy. Epoxy is used to ensure that the coil is electricallyinsulated and isolated from the environment. Depending on the magnitude of thegenerated forces, the experimental setup has to be dimensioned accordingly. Thesmall-scale experimental setup shown in Fig. 3.2 has no solid foundation and is quiteflexible. This was designed for small forces. The large scale setup on the other handhas a solid foundation whereby the coil is further embedded in a steel housing (seeFig. 3.5). Different housings are commonly used in technical equipment with highinternal forces to add mechanical stability such as steel, stainless steel, and differentalloys of aluminum.

If the housing and foundation are not mechanically strong and rigid, they will


vibrate, shortening the lifetime of the actuator. Most of the force density is inducedin regions of the armature that are directly on top of the coil. There is very littleforce induced in the center of the armature. Therefore, the armature and the coilexperience shear forces where their outer extremities start to move. In the meantime, their centers are standing still. This induces a mechanical oscillation andleads to violent vibrations. Such oscillations initiate cracks in the epoxy. If theseoscillations are large in magnitude, the epoxy chips off exposing the coil.

Another drawback with a flexible foundation is a premature air gap creation.If the coil is placed on a flexible foundation, then the generated electromagneticforces will cause the coil to repel from the armature. If the coil is allowed to movein the opposite direction with respect to the armature, then the air gap betweenthe coil and the armature increases. The magnitude of the magnetic flux densityis inversely proportional to the distance from the coil. Furthermore, the Lorentzforces are even more sensitive to the air gap. In a TC actuator, the Lorentz forcesare inversely proportional to the square of the length of the air gap. Thus, creatinga premature air gap, reduces the generated forces and reduces the efficiency of suchactuators.

To increase robustness a housing constructed of steel or aluminum is often used.What is often overlooked is the influence of the material’s electrical conductivityon the performance of the actuator. Therefore, this section is aimed to studythe actuator’s sensitivity to housings having different material properties. Threedifferent housings are studied, one constructed out of steel, one out of a magneticmaterial called Permedyn MF1, and one out of stone.

Steel is a rigid and mechanically strong material. It is a popular choice forsuch actuators. It has an electrical conductivity of 1.67 × 106 S/m. Permedynis a composite soft magnetic material that exhibits a high maximum magneticpermeability and saturation magnetization. Its BH curve can be found in [69]. Ithas an electrical conductivity of 400 S/m. Although its electrical conductivity willlead to eddy current losses, the purpose of using such a material is to investigatewhether it can boost the performance of the actuator due to its magnetic properties.The last housing is made of stone such as granite. Stone is mechanically verystrong and has a negligible electrical conductivity and magnetic susceptibility. Thepurpose of using stone is so that it can serve as a benchmark material for comparingit with other housings made of steel, and MF1.

Modeling

To study the behaviour of these materials, the developed FEM model is used.Permedyn is modeled using its nonlinear BH curve. The BH curve of Permedynis ensured to have a relative magnetic permeability of 1, behaving like air, onceit saturates. Although this significantly adds complexity and drastically increasesthe computation time, it provides valuable insight. The magnetic equation for theprimary coil, the armature, and the housing are given by,


σe

∂A

∂t+

1

µ∇ × (∇ × A) = Je , (3.4)

σe

∂A

∂t+

1

µ∇ × (∇ × A) − σev × (∇ × A) = 0 , and (3.5)

σe

∂A

∂t+ ∇ × H = 0 , (3.6)

where A is the magnetic vector potential, µ is the magnetic permeability, ∇ thenabla operator, and Je, the externally applied current density due to the appliedvoltage. The magnetic field intensity is denoted by H. The electromagnetic forcedensity, is integrated to calculate the velocity by

∫∫∫

Fem · nz rdrdθdz = mdv

dt. (3.7)

The position of the armature is calculated by integrating the velocity. Based onthe calculated position, the armature is moved using a moving mesh based on thearbitrary lagrangian method. If the mesh is stretched to a point where it becomesdistorted, it is re-meshed.

Results

The currents in the coils of the three actuators with three different housings havingdifferent material properties are shown in Fig. 3.17. An actuator with a steelhousing results in the highest peak current, followed by one with a stone housingand last, by a housing made of MF1. The current peak of an actuator with an MF1housing is 9 kA whereas that with a steel housing is 12 kA.

The main difference between the three configurations is inductance. Accordingto Lenz’s law, the induced emf is

e = −ndφ

dt, (3.8)

where n stands for the number of turns, and dφdt

denotes to the derivative of themagnetic flux with respect to time. Any material in the proximity of a time varyingmagnetic field and having an electrical conductivity, will have have eddy currentsinduced in it. These eddy currents circulate in a manner to counteract the change inthe magnetic field. Thus for a TC with a steel housing, both the steel housing andthe armature induce currents of opposite direction to the main current circulating inthe coil. This leads to a better cancellation of the magnetic flux limiting inductance.Since a TC with a stone housing only induces currents in its armature, it will havea lower inductance since the magnetic field will not be cancelled as efficiently.Last, a TC with a magnetic housing will enhance the build up of the magneticfield. Although it does have a little electrical conductivity, its relative magnetic


Time (ms)

Coil

curr

ent

(kA

)

StoneSteelMF1

0 1 2 3 4 50

2

4

6

8

10

12

14

Figure 3.17: The current pulses through the coils of three identical Thomson coilactuators whereby their coils are embedded in three different housings, steel, stone,and MF1.

permeability compensates for the eddy current losses by boosting the magneticflux density. Thus a magnetic housing exhibits the largest inductance. This limitsthe buildup of the current pulse. As soon as the armature separates even furtherfrom the coil, the induced currents decrease in magnitude. Thus system inductanceincreases even more limiting current rise.

The eddy currents induced in the armatures of TCs with different housings areshown in Fig. 3.18. All currents are negative since as expected, they are opposingthe growth of the main current pulse. Although an armature with a steel housinghas the largest coil current, its armature has the least induced currents. Someof the magnetic field lines were cancelled by the conductive steel housing. Thus,less eddy currents were induced in its armature in comparison with a TC with astone or MF1 housing. It seems that a TC with a stone housing and a TC with anMF1 housing have almost identical current peaks in their armatures. However, itis evident that the time integral of the current pulse in the armature of a TC withan MF1 housing is larger.

The force impulses generated in the armatures of TCs with different housings areshown in Fig. 3.19. The highest peak force is in an actuator having an MF1 housingfollowed by a stone housing and last, by a steel housing. Since the actuators havingstone and MF1 housings have larger currents in their armature, naturally they willhave a larger force in comparison with a TC with a steel housing. Although a TCwith a stone housing and one with an MF1 housing had similar current peaks intheir armatures, the force impulse generated by a TC with a Permedyn housing islarger since a larger magnetic flux density is created.


Time (ms)

Arm

atu

recu

rren

t(k

A)

StoneSteelMF1

0 1 2 3 4 5-250

-200

-150

-100

-50

0

50

Figure 3.18: The current pulses through the armatures of three identical Thomsoncoil actuators whose coils are embedded in three different housings: steel, stone,and MF1.

Time (ms)

Forc

e(k

N)

StoneSteelMF1

0 0.5 1 1.5 20

50

100

150

200

Figure 3.19: The repulsive forces of three identical Thomson coil actuators wherebytheir coils are embedded in three different housings, steel, stone, and MF1.

Not only does a TC with an MF1 housing have the largest peak force, its timeintegral of the force is also the largest. As a result, the velocity of the armatureof a TC embedded in an MF1 housing is largest followed by a TC with a stonehousing, and last by a TC with a steel housing (see Fig. 3.20). Although a TC witha housing made of Permedyn does have the highest inductance, its acceleration


Time (ms)

Vel

oci

ty(m

/s)

StoneSteelMF1

0 1 2 30

10

20

30

40

Figure 3.20: The velocity of the armature of three identical Thomson coil actuatorswhereby their coils are embedded in three different housings, steel, stone, and MF1.

does not decrease and it has the highest steady state armature velocity.The mobile efficiencies of TCs with steel, stone, and MF1 housings, are 18 %,

25 %, and 32 % respectively. This shows that the housing of these ultra fastactuators plays a big role in determining the efficiency and performance of theseactuators.

To analyse how the magnetic housing responds to the current pulse, a transientmagnetic relative permeability (µr) is defined and is computed by

µr =B

µ0H, (3.9)

at every time step, where B is the magnetic flux density, H the magnetic fieldintensity, and µ0 the magnetic permeability of free space.

The relative magnetic permeability of MF1 prior to the discharge of the capac-itor bank is shown in Fig. 3.21. Initially, the permebaility of MF1 is largest and is405. The coil and the armature have a relative magnetic permeability near 1.

Three hundred microseconds following the discharge of the capacitor bank, therelative permeability of MF1 in regions closest to the coil starts to diminish (seeFig. 3.22). The magnetic housing starts to saturate. Saturated regions can be easilyidentified since they have a relative magnetic permeability of 1.

Saturation increases even further 800 µs after the discharge of the capacitorbank (see Fig. 3.23). As the current increases in magnitude, a larger field is created.Moreover, the armature accelerates and repels a few millimeters away from the coil.Since the armature is now located at a distance that is further away, it does nothave a chance to counteract the increase of the magnetic field as efficiently. Thus


Figure 3.21: The relative permeability (µr), prior to the discharge of the capacitorbank. The MF1 housing has an initial homogenous relative permeability of 405,while the relative magnetic permeability of the coil and the armature are 1.

Figure 3.22: The relative permeability (µr), 300 µs after the discharge of thecapacitor bank. Regions of the MF1 housing in proximity to the coil start tosaturate, whereby their relative permeability gradually decreases to 1.

this magnetic field penetrates more into the housing, saturating regions that arefurther away from the coil.

To determine whether Permedyn is always the best choice of material amongthe three, its performance was studied by varying the number of turns and thecoil’s inner radius. The efficiencies of two TC actuators, one having a coil with asteel housing and one having a coil with a Permedyn housing, were investigated. Atotal of 480 simulations were done for each case. It took three days to simulate aTC with a solid housing. However, due to the non linear BH curve of Permedyn,it took eight days to simulate the TC with a Permedyne housing. A very finemesh was used for the Permedyn and the maximum time step size was reduced toensure convergence. An identical energizing source was used for both actuators. Itconsisted of a 5 mF capacitor bank charged to 1500V.

The efficiency of a TC actuator with a stone housing versus the coil inner radiusand its number of turns is shown in Fig. 3.24. Its maximum efficiency is 25 %. The


Figure 3.23: The relative permeability (µr), 800 µs after the discharge of thecapacitor bank. As the magnetic field increases in magnitude, a larger portionof the MF1 housing is saturated.

PSfrag replacemen

Number of turnsCoil inner radius (mm)

Effi

cien

cy(%

)

Efficiency (%)

1020

3040

0

5

10

15

20

25

30

2040

6080

100120

0

10

20

30

Figure 3.24: The efficiency of a TC actuator versus the coil inner radius and numberof turns whereby the coil is embedded in a stone housing

efficiency of a TC actuator having a Permedyne housing versus the coil inner radiusand its number of turns is shown in Fig. 3.25. It has a peak efficiency of 33 %. Itcan also be noticed that the efficiency of a TC with a Permedyn housing is alwayshigher than that of a TC with a stone housing. This shows that a Permedyn housingcontributes positively to the efficiency of a TC no matter the considered number ofturns or coil inner radii.

To sum up, the larger the conductivity of a housing, the worse it is for aTC actuator. A conductive housing not only decreases the efficiency of a TC,it also leads to higher current peaks. High current peaks are not desirable since allcomponents in the main electrical circuit have to be dimensioned for currents with



Effi

cien

cy(%

)

Efficiency (%)

1020

3040

0

5

10

15

20

25

30

2040

6080

100120

0

10

20

30

Figure 3.25: The efficiency of a TC actuator versus the coil inner radius and numberof turns whereby the coil is embedded in a MF1 housing

such magnitudes. This increases the cost and the size of all components.To sum up, although Permedyn, a soft magnetic material with a low electrical

conductivity, increases inductance and eddy current losses, it boosts the efficiencyof the TC actuator in comparison to a traditional steel housing by 80 %.

3.4.3 The Shape of the Actuator

To be able to compare the influence of different parameters on the actuator, fourcriteria are selected. The criteria taken into consideration to judge a good actuatordesign are: peak current, peak force, steady state velocity, and efficiency. Thepeak current is crucial to be able to dimension all the involved electronics in thecapacitor bank. The ratings of the capacitors, the diode, and the thyristor haveto be larger than the peak current. The peak force also serves as an importantmeasure for mechanical reasons. All components integrated to the actuator suchas copper contacts have to be able to withstand the arising stresses. These stressesare proportional to the peak force. The larger the stress, the more the fatigue,and the lower is the lifetime of the actuator or breaker. The end velocity is usedas a measure to determine whether the actuator can reach its target velocity, thatis, whether it is able to separate the current carrying contacts within the specifiedtime requirement. Finally, the efficiency is also studied since it is an importantparameter. One of the aims of this thesis is to provide new designs and boost thelow efficiency of a TC actuator.


Case study 1

The first study is aimed to vary the width (Cw) and the depth (Cd) of the coil’sconductor. The coil width is increased from 0.25 mm to 5 mm in steps of 0.25 mm,while its depth is increased from 1 mm to 25 mm in steps of 1 mm. The load isfixed to 1 kg and the coil’s inner radius (Cir) is fixed to 10 mm. This study wasconducted for four different numbers of coil turns (N) that was increased from 5to 26 in steps of 7. An aluminum disk was chosen with a fixed thickness (Ad) of15 mm. Although this thickness is substantially bigger than the skin depth and isnot necessary for the force generation, it is important for mechanical reasons. Itis over-dimensioned to minimize the stresses and avoid large deformations. In allcase studies, the armature’s radius (Aw) is variable and is a function of the coil’ssize. To be able to make use of all the generated flux efficiently, it was set to bealways 15 mm larger than the outer radius of the coil. Naturally, the outer radiusof the coil increases with increasing number of turns. Finally, the energizing sourceconsists of a capacitor bank whose capacitance is 5 mF, that is initially charged to1500 V. A total of 500 multiphysics FEM based time dependent simulations weredone. The approximate time to complete all simulations for this case study was 11days.

The peak current in a five turn coil versus the conductor’s width and depth isshown in Fig. 3.26. Such a small coil has very little inductance and resistance. Thiscauses a large in-rush current exceeding 60 kA to flow through the coil. It can beseen that the magnitude of the current peak is not as sensitive to the chosen widthof the conductor as it is to the conductor’s depth. The resistance of a conductor,

R = ρe

l

A, (3.10)

is proportional to the length of a conductor (l) and inversely proportional to its areaA. The electrical resistivity of the material is given by ρe. Increasing the conductorwidth does increase the area of the conductor but the coil becomes larger too. Themean radius of the coil increases, increasing resistance. The increased width due toincreased coil size also results in added inductance. On the other hand, increasingthe conductor’s depth increases the cross sectional area of the conductor withoutextending the coil. Therefore, the peak current increases with increasing Cd.

The current pulse in a larger 12 turn coil can be seen in Fig. 3.27. The same colorbar is used for all current peaks to be able to compare them. The peak current inthe larger coil is around 40 kA. Naturally, resistance and inductance increase withincreasing number of turns limiting the current rise. The current peaks of a 19 turnand a 25 turn coil are in the order of 25 kA and 10 kA respectively (see Fig. 3.28and Fig. 3.29), respectively, significantly smaller than that of a small 5 turn coil.

Although the 5 turn coil has the largest peak currents, it has the smallest peakforce. The peak forces for a 5 turn coil, a 12 turn coil, a 19 turn coil, and a 26turn coil can be seen in Fig. 3.30, Fig. 3.31, Fig. 3.32, and Fig. 3.33 respectively.The same color bar is again used for all figures to simplify the comparison. Large


Conductor width (mm)Conductor depth (mm)

Pea

kcu

rren

t(k

A)

Peak current (kA)

12

34

5 10

20

30

40

50

60

510

1520

250

20

40

60

Figure 3.26: The current peak in a coil consisting of 5 turns versus the conductor’swidth (Cw) and its depth (Cd) according to the parameters described in case study1.


Pea

kcu

rren

t(k

A)

Peak current (kA)

12

34

5 10

20

30

40

50

60

510

1520

250

20

40

60


currents are needed to generate large forces. However, alone they are not enough.The domains in the armatures exposed to a radial field should be maximized. Forthis specific energizing source, a 19 turn coil has the largest peak forces. Increasingthe number of turns to 26 leads to a lower force since resistance and inductance are



Pea

kcu

rren

t(k

A)

Peak current (kA)

12

34

5 10

20

30

40

50

60

510

1520

250

20

40

60



Pea

kcu

rren

t(k

A)

Peak current (kA)

12

34

5 10

20

30

40

50

60

510

1520

250

20

40

60


added. This shows that there is an optimum number of turns that maximizes thepeak force. Finally, it seems that the peak force is more sensitive to the Cw thanit is to Cd.

The final velocity of an armature for a 5 turn coil ranges from 5 m/s to 25 m/s



Pea

kfo

rce

(kN

)

Peak force (kN)

12

34

5 50

100

150

200

250

510

1520

250

100

200

Figure 3.30: The peak force in the armature of a coil consisting of 5 turns versus theconductor’s width (Cw) and its depth (Cd) according to the parameters describedin case study 1.


Pea

kfo

rce

(kN

)

Peak force (kN)

12

34

5 50

100

150

200

250

510

1520

250

100

200


(see Fig. 3.34). For larger coils with more turns, the steady state velocity is muchmore sensitive to Cw. For a 12 turn coil, the region with the highest peak velocitylies inside a contour bounded by a Cw between 2.5 mm and 4.5 mm and by a Cd

bounded between 2 mm and 5 mm (see Fig. 3.35). Increasing the size of the coil



Pea

kfo

rce

(kN

)

Peak force (kN)

12

34

5 50

100

150

200

250

510

1520

250

100

200



Pea

kfo

rce

(kN

)

Peak force (kN)

12

34

5 50

100

150

200

250

510

1520

250

100

200


even further, by increasing the number of turns to 19, shifts the location of theoptimum contour towards the origin. For a 19 turn coil, the region with the highestpeak velocity lies inside a contour bounded by a Cw between 2 mm and 2.5 mm, andby a Cd bounded between 2.5 mm and 7.5 mm (see Fig. 3.36). Last, for a 26 turn



Fin

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oci

ty(m

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5 5

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Figure 3.34: The steady state velocity of the armature versus the conductor’s width(Cw) and its depth (Cd) according to the parameters described in case study 1. Thecoil consists of 5 turns.

coil, the region with the highest peak velocity lies inside a contour bounded by Cw

between 1.5 mm and 2 mm and by Cd between 4 mm and 12.5 mm (see Fig. 3.37).In conclusion, to maximize the end velocity a large Cd and a small Cw should bechosen, with large coils having many turns. This indicates that there seems to existan optimum area defined by,

A = π(C2or − C2

ir) , (3.11)

= π((Cir + NCw)2 − C2ir) , (3.12)

= π(2NCirCw + N2C2w) , (3.13)

where Cir and Cor are defined as the coil’s inner and outer radii. Although increasingCd does increase the cross sectional area and hence decreases resistance, it alsoincreases inductance and allows the current density to distribute more in areasfurther away from the armature. There exists a region with an optimum Cd fora specified number of turns. This region is very hard to characterize by a simpleequation and is therefore presented using simulations based on FEM.

The efficiency of a 5 turn coil is very small and is in the order of 5 % (seeFig. 3.38). Increasing the number of turns significantly increases efficiency. Thepeak efficiencies of a 12 turn coil, a 19 turn coil, and a 26 turn coil, are around 20 %,25 %, and 30 % respectively (see Fig. 3.39, Fig. 3.40, and Fig. 3.41). It also showsthat the larger the (Cw), the larger is the efficiency. This unfortunately shows thatthe optimum steady state velocity and the optimum efficiency lie in different zones.If velocity is chosen as the most important requirement, then it would be best tochose an operating point at the rightmost edge on the contour having the highest



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velocity, i.e., choose the point with the greatest (Cw) among all points that havethe highest velocities.



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Figure 3.38: The actuator’s efficiency versus the conductor’s width (Cw) and itsdepth (Cd) according to the parameters described in case study 1. The coil consistsof 5 turns.

Case study 2

The second case study is aimed to vary the coil’s size and study the influence ofloading on the actuator. In this case, the coil’s size is controlled by specifying the



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inner radius of the coil (Cir) and its number of turns (N). The coil’s inner radiuswas increased from 5 mm to 120 mm in steps of 5 mm. The number of turns wasincreased from 1 to 40 in steps of 2. In this case, the width and depth of the coil’sconductor are chosen to be 2 mm and 4 mm respectively. The armature width, its



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depth, and the energizing source are kept the same as was specified in case study 1.Finally, this study was done in total for three different loads, 1 kg, 2 kg, and 3 kg.The same armature depth and energizing source were used as in case study 1. Thecapacitor bank has a capacitance of 5 mF and is initially charged to 1500V.

The peak force versus the coil’s inner radius and the number of turns for a 1 kgload is shown in Fig. 3.42. The peak force is more sensitive to the number of turnsthan to the coil’s inner radius. The maximum attained force is around 200 kN andis largest for a small coil inner radius. Increasing Cir decreases the peak force.

Increasing the load from 1 kg, to 2 kg, and to 3 kg, increases the peak force to225 kN and 260 kN (see Fig. 3.43 and Fig. 3.44). Adding a heavier load hindersthe armature to repel prematurely before the current peaks. This leads to a highercurrent peak and a higher magnetic field. The combination of inducing larger eddycurrents in the armature and enlarging the radial component of the magnetic fluxdensity results in a larger repulsive force.

Although heavy loads increase the actuator’s peak force, the gained force is notenough to compensate for the added mass. Thus their acceleration and their steadystate velocity decreases. The armature’s final velocity versus Cir and N for a 1 kgload is shown in Fig. 3.45. The peak velocity is achieved for a number of turnsbetween 10 and 20 and for a minimum coil inner radius. This maximum velocitydecreases from 37 m/s, to 27 m/s, and to 22 m/s for loads of 1 kg, 2 kg, and 3 kgrespectively (see Fig. 3.46 and Fig. 3.47).

As the load increases, the optimum operating point for maximizing velocityshifts rightwards, i.e. towards increasing number of turns. The optimum operating



Pea

kfo

rce

(kN

)

Peak force (kN)

1020

3040

0

50

100

150

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250

2040

6080

100120

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Figure 3.42: The peak force versus the coil’s inner radius and its number of turnsfor a load of 1 kg.


Pea

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region for a 1 kg load is between 10 and 20 turns. For a 2 kg load, the optimumoperating region is between 20 and 30 turns, and finally for a 3 kg, the optimumoperating region ranges from 20 to beyond 40 turns. For all loads, it would be bestto minimize the coil’s inner radius and make use of the armature’s entire surfacearea.

The efficiency of the TC versus Cir and N for a 1 kg load is shown in Fig. 3.48.



Pea

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Figure 3.45: The armature’s final velocity versus the coil’s inner radius and itsnumber of turns for a load of 1 kg.

An efficiency up to 26 % can be attained with a Cir between 20 mm and 60 mm anda total number of turns greater than 20. Similarly to the steady state velocity, theactuator’s efficiency also decreases with increasing loads. The peak efficiency of aTC when loaded with 2 kg is 23 %, and when loaded with 3 kg, it becomes 20 %.



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3.5 Brute Force Optimization

In this section a brute force optimization algorithm was implemented. To optimizethe TC actuator, its energizing source and the geometry of the actuator itself,comprising of the coil and the armature, should be optimized. After exploringthe simulation space of the previously presented studies, the variables having most

3.5. BRUTE FORCE OPTIMIZATION 73


Effi

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)

Efficiency (%)

1020

3040

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Figure 3.48: The actuator’s efficiency versus the coil’s inner radius and its numberof turns, for a load of 1 kg.


Effi

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)

Efficiency (%)

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Figure 3.49: The actuator’s efficiency versus the coil’s inner radius and its numberof turns, for a load of 2 kg.

impact on the behaviour of the TC are chosen. These are the capacitance of theenergizing source, its charging voltage, the coil’s shape and size, and the armature’sshape and size.



Effi

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)

Efficiency (%)

1020

3040

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Figure 3.50: The actuator’s efficiency versus the coil’s inner radius and its numberof turns for a load of 3 kg.

3.5.1 Setup of the optimization model

To get a control of all these parameters, three independent variables and one de-pendent variable were chosen. The three independent variables are the capacitanceof the energizing source, its charging voltage, and the number of coil turns. Thedependent variable was chosen to be the radius of the armature.

A brute force algorithm was implemented to explore the entire simulation space.Using such a method reveals what can be achieved based on all possible chosen setsof parameters. Thus, the objective function and the constraints can be definedafter the computation has ended. The results that meet the specified objectivefunction and lie within the required constraints can be easily filtered out of the entiresimulation space. Thus, using the results from this optimizing method, severalobjective functions and constraints could be easily studied. The results matchingthe requirements could be easily extracted out of the entire saved simulation space.

The first independent variable, the capacitance, was increased from 1 mF to30 mF in steps of 1 mF. The second variable, the charging voltage was increasedfrom 200 V to 2000V in steps of 100 V. Finally, the last independent variable, thecoil’s number of turns, was increased from 5 turns to 40 turns in steps of 1 turn.This amounts to 30 capacitance values, 19 voltage values, and 36 number-of-turnsvalues. Their product amounts to a total of 20,520 FEM simulations. A grid wasprepared to compute the entire simulation space. From each simulation point, thefollowing were saved: the time vector, the capacitor voltage at every time step, thecurrent pulse, the force impulse, the acceleration of the armature, its velocity, itsposition, the peak current, the peak force, the input electrical energy (1

2CV 2), and

the efficiency of the actuator, mv2

CV 2 , the ratio between the kinetic energy and the


electrical energy.The radius of the armature,

Aw = Cir + NCw + 15 mm (3.14)

was chosen to be 15 mm larger than the coil’s radius. The armature’s depthwas fixed to 15 mm. The mechanical forces are what define the thickness of thealuminum armature. If the armature is not chosen to be thick enough, it willbe prone to bending. Otherwise, to generate the electromagnetic forces, a 5 mmarmature would have been enough. The inner radius of the coil was specified to be5 mm since it leaded to the highest steady state velocity in the previously studiedcases. The required load to actuate is usually specified and is mostly determinedby the masses of the contacts, and the push/pull rod. A lumped mass of 1 kg waschosen as load for the actuator. Finally, a housing made of stone was selected toavoid eddy current losses in domains that are not required. Permedyn was notchosen since the non-linear BH curve of Permedyn requires a significantly longercomputation time. More importantly, the benefits of using Permedyn were notverified experimentally.

3.5.2 Results and Discussion

To perform all 20,520 simulations and process the results, a small cluster wasdesigned and used. The cluster consisted of an Intel Xeon processor E5-1620 with 4physical cores and 8 logical processors. The CPU had a clock frequency of 3.6 GHz.A total of 32 GB random access memory (RAM) was installed. For storage, threehard disks were used. To boost the cluster’s performance, a 500 GB solid statedrive was chosen as the primary hard disk. This was complemented with twoother normal hard drives, two terabyte each. These disks were mirrored to ensurecontinuous availability and redundancy in case of failure. This is also known asRaid 1. Finally, an AMD FirePro W700 graphics card with 4 GB GDDR5 memorywas installed.

The simulation was parallelized such that four simulations were carried outin parallel. The 20,520 simulations were distributed onto four CPUs. The totalsimulation time amounted to seven weeks. If this task was solved serially, the totalsimulation time would have been roughly four times longer amounting to almosthalf a year.

To be able to visualize the velocity at every single point, a 4D graph is needed.The first dimension of the graph would be the capacitance of the capacitor bankshown on the x axis. The second dimension would be the initial charging voltageshown on the y axis. The third dimension was specified to be the coil’s number ofturns, and is shown on the z axis. Finally, the velocity of each point, which is thefourth and last dimension, is represented by color.

Due to the presence of a large number of densely populated points, they have tobe plotted in either a very large graph, or in several graphs showing points lying indefined velocity intervals. Thus five figures are shown where the difference between


Capacitance (mF)Voltage (V)

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s


1020

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5001000

15002000

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Figure 3.51: A 4D plot showing all points having a velocity between 0 m/s and15 m/s. The first dimension is capacitance shown on the x axis, the seconddimension is the charging voltage shown on the y axis, the third dimension isthe number of turns shown on the z axis, and the fourth dimension is the velocityof the actuator, represented by color.

the minimum velocity and the maximum velocity is 15 m/s. All figures have thesame color bar for comparison purposes.

Fig. 3.51 shows all points having a velocity between 0 m/s and 15 m/s versusthe capacitance, C, the charging voltage V , and the number of coil turns, N . Mostof the points that lie in this velocity interval, have either a high capacitance and alow charging voltage, or a high charging voltage and a small capacitance. A thirdclass of points are concentrated close to the origin.

Fig. 3.52 shows all points having a velocity between 15 m/s and 30 m/s versus C,V , and N . In this figure, a clear Pareto front can be seen. This is a clear indicationthat a minimum electrical energy is required such that the actuator can attainthe desired speeds. The Pareto front, full of dark blue colored bubbles, representsall combinations that yield a velocity of 15 m/s. As expected, the Pareto front isnot linear since the electrical energy is proportional to the square of the chargingvoltage. The bubbles that are concentrated at the back of the Pareto frontier, i.e.those located furthest away from the origin, have a lighter color. These points havea higher electrical energy and thus have speeds near to 30 m/s.

Fig. 3.53 shows all points having a velocity between 30 m/s and 45 m/s versusC, V , and N . The Pareto front for this velocity range is located further away fromthe origin in comparison with the Pareto front of Fig. 3.52. To attain a higherspeed interval, a higher electrical energy is required.

Fig. 3.54 shows all points having a velocity between 45 m/s and 60 m/s versus



Num

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s


1020

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15002000

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Figure 3.52: A 4D plot showing all points having a a velocity between 15 m/sand 30 m/s. The first dimension is capacitance shown on the x axis, the seconddimension is the charging voltage shown on the y axis, the third dimension is thenumber of turns shown on the z axis, and the fourth dimension is the velocity ofthe actuator, represented by color.


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C, V , and N . The Pareto front has shifted more towards higher electrical energies.For these velocity ranges, it can be seen that there are no feasible points that havea capacitance of less than 10 mF. Any point that had a charging voltage of lessthan 1000 V is also infeasible

Fig. 3.55 shows all points having a velocity between 60 m/s and 75 m/s versusC, V , and N . These points have the highest electrical energies and are confinedmostly in the corner opposite to the origin.

After having inspected these figures, it can be seen that each velocity intervalhas thousands of points. In other words, there are thousands of feasible solutionshaving different values for C, V , and N . Which point among all these is theoptimum point and how can we visualize it?

To answer this question is not easy. Before proceeding, the objective functionshould be stated to decide which point should be chosen. There can be manyobjective functions and constraints. One objective function could be to maximizethe velocity of the armature for a given capacitance and charging voltage. In thiscase, since C and V are given, the only variable left is the number of turns. Thusthe number of points has to be chosen such that it maximizes the end velocity ofthe actuator.

In case such an actuator should be inserted in a limited space, it may havegeometrical constraints. One such example could be to design an actuator thatshould attain a certain speed without exceeding a maximum specified diameter.



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Thus the number of turns that can be used will be limited.In the following, different case studies have been designed to demonstrate the

power of using the developed brute force optimizer. The most common and realisticrequirements are given by case studies 1 and 2.

Case study 1

The objective function in case study 1 is to maximize the efficiency of the actuatorsuch that the actuator attains a speed between 10 m/s and 12 m/s (see Fig. 3.56).

In this case, a 5D graph is needed. The first dimension is represented bycapacitance and is shown on the x axis, the second dimension represents thecharging voltage and is shown on the y axis, the third dimension represents thenumber of turns and is shown on the z axis, the fourth dimension is the velocityof actuator represented by color, and finally, the fifth dimension is the actuator’sefficiency represented by the size of the bubble.

Fig. 3.56 shows the velocity and the efficiency of the points versus C, V , and N .The minimum and maximum efficiencies among all these feasible solutions rangebetween 0.8 % and 15 %. The smallest efficiency is represented by the bubble withthe smallest diameter and the largest efficiency is represented by the bubble withthe largest diameter. As can be seen, there is quite a big difference in efficiency.If an actuator with a velocity of 12 m/s is required that has the highest efficiency,then the optimum point is the largest red bubble shown in the Fig.Fig. 3.56. A



Num

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Figure 3.56: A 5D plot showing all combination sets leading to a velocity between10 m/s and 12 m/s. The first dimension is capacitance shown on the x axis, thesecond dimension is the charging voltage shown on the y axis, the third dimension isthe number of turns shown on the z axis, the fourth dimension is the velocity of theactuator represented by color, and the fifth dimension is the actuator’s efficiencyrepresented by the size of the bubble.

red color means that this point has a velocity of 12 m/s. The size of the bubbleis a measure of its efficiency. The largest bubble, regardless of its color, signifiesthat this point is the optimum point having the highest possible efficiency for aparticular velocity.

The alternative with the highest efficiency (15 %) has a capacitance of 2 mFand a charging voltage of 1100 V and 40 turns. It has an electrical energy of 1210 Jand an end velocity of 12 m/s.

To better illustrate this 5D figure, a 4D perspective of it is done showing onlytwo of the three x, y, z parameters at a time. In Fig. 3.57, an x-z view of Fig. 3.56can be seen showing the capacitance versus the number of turns. In Fig. 3.58, ay-z view of Fig. 3.56 can be seen showing the charging voltage versus the numberof turns. Finally, in Fig. 3.59, an x-y view of Fig. 3.56 can be seen showing thecapacitance versus the charging voltage.


Capacitance (mF)

Num

ber

of

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s

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4 7 11 15

Efficiency (%)

5 10 15 20 25 3010

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Figure 3.57: An x-z view: Capacitance versus number of turns perspective ofFig. 3.56. A 5D plot showing all combination sets leading to a velocity between10 m/s and 12 m/s. The first dimension is capacitance shown on the x axis, thesecond dimension, (not shown in this figure), is the charging voltage, the thirddimension is the number of turns shown on the z axis, the fourth dimension isthe velocity of the actuator represented by color, and the fifth dimension is theactuator’s efficiency represented by the size of the bubble.

Case study 2

The objective function in case study 2 is to minimize the required input electricalenergy such that the speed of the actuator attains a value between 10 m/s and12 m/s.

Surprisingly, the minimum and maximum electrical energy of all feasible so-lutions in Fig. 3.60 ranges between 875 J and 9600 J. The smallest energy isrepresented by the bubble with the smallest diameter and the largest energy isrepresented by the bubble with the largest diameter. Thus, the optimum point tochoose is a bubble having the smallest diameter. It is clear from this figure thatthe smaller the number of turns, the more is the required electrical energy. As thenumber of turns is increased, the required electrical energy decreases. Increasing


Voltage (V)

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Figure 3.58: A y-z view: Charging voltage versus number of turns perspective ofFig. 3.56. A 5D plot showing all combination sets leading to a velocity between10 m/s and 12 m/s. The first dimension, (not shown in this figure), is capacitance,the second dimension is the charging voltage shown on the y axis, the thirddimension is the number of turns shown on the z axis, the fourth dimension isthe velocity of the actuator represented by color, and the fifth dimension is theactuator’s efficiency represented by the size of the bubble.

the number of turns increases the surface area between the coil and the armature.Thus a force density can be induced in a larger area.

Ten alternatives gave the minimum electrical energy. All points had a capaci-tance of 7 mF and a charging voltage of 500 V. Their number of turns ranged from17 turns to 26 turns in steps of 1. All these alternatives have an end velocity of10 m/s. Their efficiencies range from 10 % to 12 %.

To better illustrate this 5D figure, a 4D perspective of it is done showing onlytwo of the three x, y, z parameters at a time. In Fig. 3.61, an x-z view of Fig. 3.60can be seen showing the capacitance versus the number of turns. In Fig. 3.62, ay-z view of Fig. 3.60 can be seen showing the charging voltage versus the numberof turns. Finally, in Fig. 3.63, an x-y view of Fig. 3.60 can be seen showing the


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age

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Figure 3.59: An x-y view: Capacitance versus charging voltage perspective ofFig. 3.56. A 5D plot showing all combination sets leading to a velocity between10 m/s and 12 m/s. The first dimension is capacitance shown on the x axis,the second dimension is the charging voltage shown on the y axis, the thirddimension, (not shown in this figure), is the number of turns, the fourth dimensionis the velocity of the actuator represented by color, and the fifth dimension is theactuator’s efficiency represented by the size of the bubble.

capacitance versus the charging voltage.

Case study 3

The objective function in case study 3 is to minimize the peak current such that thespeed of the actuator is between 10 m/s and 12 m/s. Components incorporated inthe capacitor bank such as the diode and the thyristor have to be able to withstandthe peak current.

Fig. 3.64 shows the velocity and the peak current of all feasible points versus C,V , and N . The minimum and maximum peak current among all these feasible solu-tions ranges between 3.4 kA and 80 kA. The smallest peak current is represented bythe bubble with the smallest diameter, and the largest peak current is represented



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510

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Figure 3.60: A 5D plot showing all combination sets leading to a velocity between10 m/s and 12 m/s. The first dimension is capacitance shown on the x axis, thesecond dimension is the charging voltage shown on the y axis, the third dimensionis the number of turns shown on the z axis, the fourth dimension is the velocity ofthe actuator represented by color, and the fifth dimension is the initial electricalenergy represented by the size of the bubble.

by the bubble with the largest diameter. To minimize the peak current, a smallbubble should be chosen. The larger the number of turns, the smaller is the bubblediameter. Increasing the number of turns increases resistance hence limiting thepeak current.

The alternative with the smallest peak current has a capacitance of 25 mF anda charging voltage of 300 V and 40 turns. This alternative has an electrical energyof 1125 J and an efficiency of 11 %. It yields a final velocity of 10 m/s.

To better illustrate this 5D figure, a 4D perspective of it is done showing onlytwo of the three x, y, z parameters at a time. In Fig. 3.65, an x-z view of Fig. 3.64can be seen showing the capacitance versus the number of turns. In Fig. 3.66, ay-z view of Fig. 3.64 can be seen showing the charging voltage versus the numberof turns. Finally, in Fig. 3.67, an x-y view of Fig. 3.64 can be seen showing thecapacitance versus the charging voltage.


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Figure 3.61: An x-z view: Capacitance versus number of turns perspective ofFig. 3.60. A 5D plot showing all combination sets leading to a velocity between10 m/s and 12 m/s. The first dimension is capacitance shown on the x axis, thesecond dimension, (not shown in this figure), is the charging voltage, the thirddimension is the number of turns shown on the z axis, the fourth dimension is thevelocity of the actuator represented by color, and the fifth dimension is the initialelectrical energy represented by the size of the bubble.

Case study 4

The objective function in case study 4 is to minimize the peak force such that thespeed of the actuator is between 10 m/s and 12 m/s. If the peak force is very large,it may require a very bulky armature to avoid bending. Moreover, larger peakforces can cause higher stresses reducing the lifetime of the actuator. Swapping alarge peak force with a larger force time integral can be better for reducing thestresses in all integrated mechanical parts.

Fig. 3.68 shows the velocity and the peak force of all feasible points versus C, V ,and N . The minimum and maximum peak forces among all these feasible solutionsranges between 20 kN and 101 kN. The smallest peak force is represented by thebubble with the smallest diameter and the largest peak force is represented by the


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35

40

Figure 3.62: A y-z view: Voltage versus number of turns perspective of Fig. 3.60.A 5D plot showing all combination sets leading to a velocity between 10 m/s and12 m/s. The first dimension, (not shown in this figure), is capacitance, the seconddimension is the charging voltage shown on the y axis, the third dimension is thenumber of turns shown on the z axis, the fourth dimension is the velocity of theactuator represented by color, and the fifth dimension is the initial electrical energyrepresented by the size of the bubble.

bubble with the largest diameter. If the smallest peak force is required to reducethe stresses in the material, then the smallest bubble should be chosen. Since theforce is proportional to the current pulse, then a smaller peak current will alsoresult in a smaller peak force. Hence, the larger the number of turns, the smalleris the peak current and the smaller is the peak force.

The alternative with the smallest peak force has a capacitance of 25 mF and acharging voltage of 300 V and 40 turns. This alternative has an electrical energy of1125 J and an efficiency of 11 %. It yields a final velocity of 10 m/s.

To better illustrate this 5D figure, a 4D perspective of it is done showing onlytwo of the three x, y, z parameters at a time. In Fig. 3.69, an x-z view of Fig. 3.68can be seen showing the capacitance versus the number of turns. In Fig. 3.70, a


Capacitance (mF)

Volt

age

(V)

Velocity (m/s)

2.6 4.9 7.2 9.6

Energy (kJ)

5 10 15 20 25 30 10

10.5

11

11.5

200

400

600

800

1000

1200

1400

1600

1800

2000

Figure 3.63: An x-y view: Capacitance versus number of turns perspective ofFig. 3.60. A 5D plot showing all combination sets leading to a velocity between10 m/s and 12 m/s. The first dimension is capacitance shown on the x axis, thesecond dimension is the charging voltage shown on the y axis, the third dimension,(not shown in this figure), is the number of turns, the fourth dimension is thevelocity of the actuator represented by color, and the fifth dimension is the initialelectrical energy represented by the size of the bubble.

y-z view of Fig. 3.68 can be seen showing the charging voltage versus the numberof turns. Finally, in Fig. 3.71, an x-y view of Fig. 3.68 can be seen showing thecapacitance versus the charging voltage.

Based on all the case studies presented, it is in general better to have a largernumber of turns. Having too few turns results in a very large current peak anda very inefficient system. On the other hand, choosing a very large number ofturns significantly increases resistance. There exists a delicate balance betweencapacitance, charging voltage, and number of turns that should be respected basedon the desired requirements. With the help of the developed optimization model,this delicate balance can be discovered. Hence, the desired optima can be extractedfrom the presented simulation space.



Num

ber

of

turn

s

Velocity (m/s)

23 44 61 80

Peak Current (kA)

510

1520

2530

0.8

10

10.2

10.4

10.6

10.8

11

11.2

11.4

11.6

11.8

200400

600800

10001200

14001600

18002000

5

10

15

20

25

30

35

40

Figure 3.64: A 5D plot showing all combination sets leading to a velocity between10 m/s and 12 m/s. The first dimension is capacitance shown on the x axis, thesecond dimension is the charging voltage shown on the y axis, the third dimensionis the number of turns shown on the z axis, the fourth dimension is the velocityof the actuator represented by color, and the fifth dimension is the peak currentrepresented by the size of the bubble.


Capacitance (mF)

Num

ber

of

turn

s

Velocity (m/s)

23 44 61 80

Peak Current (kA)

0.5

5 10 15 20 25 3010

10.5

11

11.5

5

10

15

20

25

30

35

40

Figure 3.65: An x-z view: Capacitance versus number of turns perspective ofFig. 3.64. A 5D plot showing all combination sets leading to a velocity between10 m/s and 12 m/s. The first dimension is capacitance shown on the x axis, thesecond dimension is the charging voltage, (not shown in this figure), the thirddimension is the number of turns shown on the z axis, the fourth dimension is thevelocity of the actuator represented by color, and the fifth dimension is the peakcurrent represented by the size of the bubble.


Voltage (V)

Num

ber

of

turn

s

Velocity (m/s)

23 44 61 80

Peak Current (kA)

0.5

10

10.5

11

11.5

200 400 600 800 1000 1200 1400 1600 1800 20005

10

15

20

25

30

35

40

Figure 3.66: A y-z view: Voltage versus number of turns perspective of Fig. 3.64.A 5D plot showing all combination sets leading to a velocity between 10 m/s and12 m/s. The first dimension is capacitance, (not shown in this figure), the seconddimension is the charging voltage shown on the y axis, the third dimension isthe number of turns shown on the z axis, the fourth dimension is the velocityof the actuator represented by color, and the fifth dimension is the peak currentrepresented by the size of the bubble.


Capacitance (mF)

Volt

age

(V)

Velocity (m/s)

23 44 61 80

Peak Current (kA)

5 10 15 20 25 30 10

10.5

11

11.5

200

400

600

800

1000

1200

1400

1600

1800

2000

Figure 3.67: An x-y view: Capacitance versus number of turns perspective ofFig. 3.64. A 5D plot showing all combination sets leading to a velocity between10 m/s and 12 m/s. The first dimension is capacitance shown on the x axis, thesecond dimension is the charging voltage shown on the y axis, the third dimension isthe number of turns, (not shown in this figure), the fourth dimension is the velocityof the actuator represented by color, and the fifth dimension is the peak currentrepresented by the size of the bubble.



Num

ber

of

turn

s

Velocity (m/s)

30 55 80 101

Peak Force (kN)

510

1520

2530

0.8

10

10.2

10.4

10.6

10.8

11

11.2

11.4

11.6

11.8

200400

600800

10001200

14001600

18002000

5

10

15

20

25

30

35

40

Figure 3.68: A 5D plot showing all combination sets leading to a velocity between10 m/s and 12 m/s. The first dimension is capacitance shown on the x axis, thesecond dimension is the charging voltage shown on the y axis, the third dimension isthe number of turns shown on the z axis, the fourth dimension is the velocity of theactuator represented by color, and the fifth dimension is the peak force representedby the size of the bubble.


Capacitance (mF)

Num

ber

of

turn

s

Velocity (m/s)

101

30 55 80 101

Peak Force (kN)

5 10 15 20 25 3010

10.5

11

11.5

5

10

15

20

25

30

35

40

Figure 3.69: An x-z view: Capacitance versus number of turns perspective ofFig. 3.64. A 5D plot showing all combination sets leading to a velocity between10 m/s and 12 m/s. The first dimension is capacitance shown on the x axis, thesecond dimension is the charging voltage, (not shown in this figure), the thirddimension is the number of turns shown on the z axis, the fourth dimension is thevelocity of the actuator represented by color, and the fifth dimension is the peakforce represented by the size of the bubble.


Voltage (V)

Num

ber

of

turn

s

Velocity (m/s)

30 55 80 101

Peak Force (kN)

0

10

10.5

11

11.5

200 400 600 800 1000 1200 1400 1600 1800 20005

10

15

20

25

30

35

40

Figure 3.70: A y-z view: Charging voltage versus number of turns perspective ofFig. 3.64. A 5D plot showing all combination sets leading to a velocity between10 m/s and 12 m/s. The first dimension is capacitance, (not shown in this figure),the second dimension is the charging voltage shown on the y axis, the thirddimension is the number of turns shown on the z axis, the fourth dimension isthe velocity of the actuator represented by color, and the fifth dimension is thepeak force represented by the size of the bubble.


Capacitance (mF)

Volt

age

(V)

Velocity (m/s)

30 55 80 101

Peak Force (kN)

5 10 15 20 25 30 10

10.5

11

11.5

200

400

600

800

1000

1200

1400

1600

1800

2000

Figure 3.71: An x-y view: Capacitance versus charging voltage perspective ofFig. 3.64. A 5D plot showing all combination sets leading to a velocity between10 m/s and 12 m/s. The first dimension is capacitance shown on the x axis, thesecond dimension is the charging voltage shown on the y axis, the third dimensionis the number of turns, (not shown in this figure), the fourth dimension is thevelocity of the actuator represented by color, and the fifth dimension is the peakforce represented by the size of the bubble.

Chapter 4

The Composite Magnetic Damper

After having presented the first subcomponent of the mechanical switch, i.e. theactuator, this chapter focuses on the design of second crucial subcomponent ofthe electromechanical switch, the damper. Although different damper designs canbe incorporated, a passive composite magnetic damper was chosen, designed, andexperimentally tested. The main purpose of a damper is to decelerate the actuatorwith a controllable force.

4.1 State of the Art

Eddy current dampers have been used in many applications. In [70], an eddycurrent damper has been used in a superconducting maglev system. In [71], theaccuracy in the modeling of eddy current dampers has been increased by using themethod of images. Other works on eddy current dampers can be found in [72–76].

4.2 Theory

If a relative motion between a permanent magnet and an electrical conductive tubeexists, as in the example of a set of magnets falling in a copper tube (see Fig. 4.1),then a current density Ji,

Ji = σv × B , (4.1)

is induced in the tube where σ is the electrical conductivity of the tube, v is therelative velocity, and B is the magnetic flux density in the tube. The cross productof the axial component of the velocity with the radial component of the magneticflux density produces currents in the azimuthal direction. The damping force,

Fem = Ji × B , (4.2)

= σv × B2 , (4.3)

97

98 CHAPTER 4. THE COMPOSITE MAGNETIC DAMPER

Symmetry line

Iron spacer

(A) (B)

Figure 4.1: A stack of magnets and iron spacers travelling inside a copper tube (seeFig. 4.1 A). The transparency of the copper tube is increased to show the magnetsfalling through it. Its axi-symmetric simulation model is shown in Fig. 4.1 B.

can be calculated by taking the cross product of the induced current density withthe magnetic flux density. The cross product of the current density oriented inthe azimuthal direction with the radial component of the magnetic flux densityproduces a Lorentz force in the axial direction that damps the motion. It can beseen that the damping force is proportional to the square of the radial componentof the magnetic flux density and is proportional to the velocity.

4.3 Methodology

The main concept behind the linear damper studied in this chapter consists ofmagnets travelling inside an electrically conductive tube. To minimize the geom-etry, size, and cost of the magnetic damper, it should be optimized. The maincomponents to optimize are the tube and the magnet topology. The objective ofthis optimization was to maximize the damping force while minimizing the numberof magnets and the size of the construction.

Initially, the aim was to build a small-scale experimental setup to study thedamping force and verify the developed simulation models for some magnet topolo-gies. Subsequently, the selected magnet design was then used to build a large scalesetup to test the damping behaviour when subjected to higher speeds. Accordingly,

4.4. MODELING 99

a large scale prototype involving higher speeds was built, modeled, and testedexperimentally.

After verifying the developed FEM models by experiments, numerical explo-ration was carried out. An in depth investigation was done to determine the opti-mum magnet and tube topology. The optimization process was carried out in parts.Initially, different magnet topologies were presented and optimized. Afterwards, thetube was optimized based on the chosen magnet arrangement.

4.4 Modeling

If the magnets are fully immersed in the copper tube and the ends of the coppertube are far away, then a two dimensional axi-symmetric static model can be used(see Fig. 4.1). If however the magnets penetrate the copper tube and the end effectscannot be neglected, then a transient simulation should be used.

4.4.1 Static modeling

The magnetic field equation in the air, is expressed by

1

µ∇ × B = 0 . (4.4)

Ampere’s law for the magnets simplifies to,

1

µ∇ × (B − Br) = 0 , (4.5)

where Br is the remanent magnetic flux density for the magnets. The remanentmagnetic flux density for radially and axially magnetized magnets are given by:

Br = Brr , and (4.6)

Br = Brz , (4.7)

respectively.The magnetic field equation in the copper tube can be expressed by,

1

µ∇ × (∇ × A) − σev × (∇ × A) = 0 . (4.8)

4.4.2 Transient Modeling

The magnetic field equation in the air, is expressed by

σe

∂A

∂t+

1

µ∇ × B = 0 . (4.9)


In the transient model, Ampere’s law for the magnets simplifies to,

σe

∂A

∂t+

1

µ∇ × (B − Br) = 0 . (4.10)

As for the magnetic field equation in the copper tube, it can be expressed by,

σe

∂A

∂t+

1

µ∇ × (∇ × A) − σev × (∇ × A) = 0 . (4.11)

The damping force,Fem = J × B , (4.12)

is integrated to calculate the velocity,

∫∫∫

Fem · nz rdrdθdz = mdv

dt, (4.13)

where m is the total mass of the moving part. Finally, the relative position of themagnets with respect to the tube versus time is calculated by,

x =

∫

vdt . (4.14)

This is needed for the moving mesh. The advantage of using a transient simulationis that end effects can be taken into account. Once the mesh quality factor wasviolated, the simulation was paused to re-mesh the entire geometry. Subsequently,the results were interpolated into the new mesh and the simulated was resumed.

The ferromagnetic spacers were modeled with a constant relative permeabilityof 500. However, if there was a risk for saturation, as in the case of using very thincomposite tubes, then the nonlinear BH curve of the material was used.

4.5 Model and Concept Verification

To verify the model and the concept of the magnetic damper, small scale and largescale prototypes have been built and tested. Initially, small scale setups were builtfor two reasons. The first reason is to gain some experience with handling magnetsand develop an assembly scheme. Large magnets can be very dangerous if nothandled with care. Another reason was to identify the best configuration and builda large scale prototype using the best configuration.

4.5.1 Small Scale Prototypes

To select an optimum magnet array, five different topologies were initially proposed.Different combinations of axially magnetized magnets with plastic and ferromag-netic spacers were studied as shown in Fig. 4.2. Configuration A consists of twoaxially magnetized magnets stacked on top of each other. This configuration is

4.5. MODEL AND CONCEPT VERIFICATION 101

(A) (B) (C) (D) (E)

Plastic spacer Ferromagneticspacer

Magnet Symmetry line

Figure 4.2: A diagram showing the initial investigated magnet topologies.

quite easy to assemble since the magnets attract each other. Plastic spacers arescrewed from both sides to fix the magnets in place. Configuration B is similar toconfiguration A; the only difference is that it has an extra plastic spacer separatingthe two magnets. In configuration C, the magnets are placed such that they areopposing each other. A plastic spacer is used to separate them to simplify theassembly process and avoid the risk of breaking the magnets due to the repulsiveforces. The two magnets are fixed from both ends by screwing two plastic spacers.Configuration D comprises of an alternating magnet and ferromagnetic spacers.The magnets are positioned such that they attract each other. This configurationwas quite simple to assemble. Finally, configuration E is similar to D except for theorientation of the magnets. The magnetization directions of the magnets opposeeach other.

Although configuration C was difficult to assemble, the assembly process ofconfiguration E was quite easy due to the ferromagnetic spacers. Initially, the firstferromagnetic spacer is placed. Secondly, the first magnet polarized in the positivez direction is inserted. Thus it is directly attracted to the spacer. When a secondspacer is introduced, it is also attracted to the magnet such that the magnet isnow sandwiched between two spacers. When a second magnet magnetized in thenegative z direction is introduced, it initially repels the other sandwiched magnet.If left on a guide, it will be suspended at a distance from the first magnet. However,by pushing the second magnet gently downwards, at some point, the repulsive forcewill immediately transition into an attractive force and the magnet will be attractedto the ferromagnetic spacer. This simplifies the stacking process.

The distance at which the repulsive force transitions into an attractive forcedepends on the thicknesses of the magnet and the ferromagnetic spacers. Themagnets and the ferromagnetic spacers are annulus shaped. The inner and outerdiameter of the magnets and the ferromagnetic spacers are 20 mm and 50 mmrespectively. They have a thickness of 10 mm.

Each of the configurations described earlier was mounted on a plastic shaft andplaced inside a copper tube (see Fig. 4.3 and Fig. 4.4). Afterwards, they weredropped such that they could fall freely inside the copper tube. A stop watch wasused to measure the travel time of the magnets inside the copper tube.


Figure 4.3: A picture showing the small scale setup whereby three differentarrangements of magnets are mounted on three shafts and inserted in copper tubes.

Figure 4.4: A picture exposing the shaft and the magnets.

Results and Analysis

The simulated and the measured times for the magnets to travel through the coppertube are shown in Table 4.1. Each test was repeated ten times to calculate the meanmeasured time. The measured and simulated times in some cases are identical andin some cases, they do not match perfectly. A small deviation exists. This ismainly because the magnets were not centered perfectly before dropping them.

4.5. MODEL AND CONCEPT VERIFICATION 103

Table 4.1: The simulated and measured times for configurations A to E.

Configuration Simulated time (s) Measured time (s)A 3.7 3.4B 4.4 3.8C 5.9 5.9D 1.9 2E 4.5 5

Furthermore, due to the absence of a guide, the magnets in some cases tilted andcollided with the inside of the copper tube. This results in an inhomogeneous airgap generating a nonlinear force. Furthermore, it introduces friction.

Although D and E are composed of the exact same components, their traveltimes differ significantly. The time it takes for configuration E to fall throughthe copper tube is significantly more than that of configuration D. This shows theimportance of choosing magnet arrangements with the optimal magnetization di-rections. Despite of having identical masses, configuration E generates significantlylarger damping forces.

Configuration C is a bit harder to compare with D and E since it is composed ofplastic spacers. It is significantly lighter. In this case the added damping force dueto the use of ferromagnetic spacers was not enough to compensate for the addedmass. If the only mass to damp is the weight of the magnets themselves, thenperhaps configuration C can be considered. However, if the weight of the load todamp is significantly larger than the weight of the magnets, then configuration Eis better since it has the largest damping force.

One of the main conclusions from this experimental study is that a guide isneeded. Furthermore, the magnets should be centered carefully. A guide should beinstalled such that the air gap is kept constant from all sides. The trajectory of themagnets’ travel path should be rectilinear and should be only in the z-direction. Ifthe magnets are tilted, air gaps with different sizes are created at different times asthe magnets travel through the electrically conductive tube. This causes nonlinearforces and is not possible to simulate using a two dimensional axi-symmetric model.

Although the simulation model compared to the experimental results does notmatch perfectly for some of the cases, this experimental setup was very useful todetermine which configuration generates the largest damping forces. This smallscale study served as a good pre-study before building the large scale prototype.

4.5.2 Large Scale Prototype

Out of the five tested configurations, a large scale prototype was built based onconfiguration E (see Fig. 4.5). Four magnets and five ferromagnetic spacers wereused. These were mounted on a nonmetallic and nonmagnetic guide and fixedfrom both sides using steel bolts. The magnets were inserted inside the copper


Figure 4.5: A figure showing the actuator and the damper. A stack of four magnetsand five iron spacers based on topology E are assembled and inserted in a coppertube.

tube such that end effects can be neglected. To ensure that the magnets followa rectilinear path, two guides are installed at the top and bottom. These guideswere constructed from teflon, a material with very little friction. A 300 mm longcopper tube was used and fixed to the frame. The thickness of the copper tubewas chosen to be significantly larger than the skin depth. At the very bottom,a material made of Polyoxymethylene (POM) was fixed to the shaft. POM is athermoplastic having a high stiffness and low friction. Its main purpose is to actas the collision interface to transfer the kinetic energy from the actuator to thedamper without leading to excessive stresses. The choice of material suited to beplaced at the location of the collision is quite important and requires an in depthmechanical study. Further research will be dedicated to classify such materials anddevelop appropriate simulation models.

The actuator consists of an armature and a coil. A mushroom shaped armatureconstructed from aluminum 7075 T651 initially rests on a flat spirally shaped coil.A mass of 1.5 kg is screwed onto the bottom of the armature. The discharge ofthe capacitor bank initiates the acceleration process. After operating the actuator,it attains a steady state speed and collides with the shaft of the damper thusaccelerating the magnets. A damping force is generated proportional to the speedof the magnets.

Results and Analysis

To verify the simulation model, the measured and simulated damper velocities werecompared and shown in Fig. 4.6. A high speed camera filming at a rate of 10,000frames per second was used to film the shaft on which the magnets were mounted.The shaft was marked with a marker. After filming, the captured pictures werecalibrated and tracked to plot the velocity of the damper. In this figure, the velocityof the armature was also shown. It could be seen that in the first millisecond, the

4.6. OPTIMAL DAMPER DESIGN 105

Time (ms)

Vel

oci

ty(m

/s)

Exp: ArmExp: DamperSim: Damper

0 10 20 30

0

1

2

3

4

Figure 4.6: Measured armature and damper velocities compared with the simulateddamper velocity.

TC actuator accelerates to attain a steady state speed of 4 m/s. After 15 ms, thetop part of the armature collides with the damper’s shaft. This accelerates thedamper to around the same speed of the armature and decelerates the armature.The damper attains an initial speed of around 4 m/s.

This initial speed was set in the simulation to model the deceleration of thedamper. It can be seen that a good agreement was obtained between the observedand simulated velocities. The acceleration phase due to the TC actuator was notsimulated. In principle, the entire velocity profile could be simulated. Initially,the developed axi-symmetric multi-physics simulation model for the actuator couldbe used to simulate the acceleration phase until the armature reaches steady statespeed. Afterwards, Newton’s equations could be used to model the slight drop inspeed due to gravity until it impacts the damper’s shaft. The collision could besimulated with a nonlinear solid mechanics model coupled to another axi-symmetricelectromagnetic model for the damper. Simulating the collision is not trivial andrequires a substantial amount of work.

4.6 Optimal Damper Design

After validating the developed simulation models with experiments, numerical ex-ploration was carried out to identify the optimum magnet arrangement. Subse-quently, the tube geometry and composition were optimized based on the selectedmagnet configuration.

To determine the optimum magnet array, several topologies were studied andlisted as shown in Fig. 4.7. Configurations A to E are composed of axially mag-


(A) (B) (C) (D)

(E)

Plastic spacer Ferromagneticspacer

Magnet Symmetry line

(F) (H)(G)

Figure 4.7: A diagram showing all investigated magnet topologies. Topologies A toE are comprised of axially magnetized magnets, topologies F and G are comprisedof radially magnetized magnets, and finally topology H, also known as a Halbacharrangement, comprises of a mixture of radially and axially magnetized magnets

netized magnets. Since the main conclusion from the theory section was that thedamping force is proportional to the square of the radial component of the magneticfield, radially magnetized magnets were also chosen to be studied. ConfigurationsF and G are composed of purely radially magnetized magnets. Configuration Fconsists of five vertically stacked magnets where each two consecutive magnetsare magnetized in opposite directions. In configuration G, all five radially mag-netized permanent magnets are magnetized in the same direction. Finally, thelast configuration denoted by H consists of a combination or radially and axiallymagnetized magnets. This configuration is also known as the Halbach arrangement.An example of an eddy current damper utilizing the Halbach arrangement wasstudied in [77]. Other examples of the Halbach arrangement can be found in [78–86].

The optimum configuration is identified by the one that is capable of deliveringthe largest damping force for a constant velocity of 5 m/s. Table 4.2 shows thesimulated damping force for different magnet arrangements travelling through acopper tube with a constant velocity of 5 m/s. Configuration A has a dampingforce of 251 N while configuration B has a damping force of 207 N. ConfigurationB in this case has a slight larger reluctance since the magnets are separated. Alarger reluctance leads to a smaller magnetic field and smaller currents induced inthe tube.

Configuration C has a damping of 268 N, larger than the damping forces gen-erated by configurations A and B. The fringe fields for configurations A and B aresmall in comparison with C. A smaller fringing field means only a small portion of


Table 4.2: The damping force for magnet configurations A to E for a given velocityof 5 m/s.

A B C D E F G HDamping force (N) 251 207 268 305 417 381 112 988

the magnetic field penetrates the copper tube leading to small induced currents.The smaller the induced currents, the smaller are the generated damping forces.Configuration C increases the fringe fields such that the magnetic field is pushedoutwards towards the copper tube and inwards towards the center. Thus, a largerportion of the tube is exposed to the magnetic field.

Configuration D has a damping force of 305 N, slightly larger than that ofconfiguration C. Replacing the plastic spacers with ferromagnetic spacers decreasesmagnetic reluctance and increases the magnitude of the magnetic flux densityexposed to the copper tube. This causes larger damping forces since the generatedforces are proportional to the square of the radial component of the magnetic fluxdensity.

Evidently, configuration E has a much larger damping force than configurationD since its magnets are also magnetized in opposite directions and the magneticreluctance of the circuit is reduced due to the use of ferromagnetic spacers.

Although the aim is to maximize the radial component of the magnetic fluxdensity to maximize the damping force, radially magnetized magnets functionvery poorly. Configurations F and G are composed of purely radially magnetisedmagnets. Their damping forces are 381 N and 112 N respectively. Contrary toexpectations, they have lower damping forces than configuration E. A purely radialarrangement results in a smaller radial field in the copper tube and has less fringingfields compared to configuration E. Configuration G results in high current inregions of the copper tube only facing the first and last magnet. This is werethe reluctance is smallest. At its center, this configuration has infinite reluctance.

Finally, a combination of radially magnetized and axially magnetized magnetsproduces the largest damping forces. Configuration H produces a damping force of988 N. This is more than twice the damping force of the second best configuration(E). The Halbach arrangement is very efficient in creating a one sided magneticfield. Configuration E pushes the field in both directions, outwards, and inwards.In contrast, the Halbach arrangement pushes all the field outwards. A larger portionof the field penetrates the copper tube. Moreover, the magnitude of the magneticfield in the copper tube is significantly larger than that of configuration E. Sincethe Halbach arrangement pushes fields only in one direction, the reluctance of itsmagnetic circuit is half that of configuration E. Thus, it is the optimum combinationleading to the largest one-sided fringing fields.

Despite the fact that configuration H leads to the largest damping forces, itsassembly is difficult and cumbersome. Since the magnets repel each other, glueshould be used to adhere them together. This also requires a lot of time and patience


a b

Figure 4.8: A magnet array based on configuration E is inserted in a composite tubecomposed of copper and iron as its innermost and outermost layers respectively.

for finalizing the assembly. Thus, due to the ease of assembly and relatively largedamping force, configuration E was chosen. In addition, configuration E is lesscostly since it only makes use of two magnets compared with five magnets utilizedby the Halbach arrangement.

Based on configuration E, four magnets were stacked on top of each otherand sandwiched by five ferromagnetic spacers. After choosing the desired magnetconfiguration, the next step was to optimize the shape and material of the tube.Thus, a composite tube composed of copper and iron was proposed as shown inFig. 4.8. The innermost layer is composed of a copper tube with a width “a”. Copperwas chosen as the innermost layer since it has a high electrical conductivity. Theoutermost layer is composed of iron with a width “b”. It has a relative magneticpermeability of 5000. Iron was chosen since it is magnetic and can close theflux path. Closing the flux path significantly reduces reluctance increasing themagnitude of the magnetic flux density that the copper tube is exposed to.

The damping force generated by a combination of magnets travelling at a speedof 5 m/s versus “a” and “b” is shown in Fig. 4.9. The thickness of the copper tube,“a”, is varied from 0.25 mm to 10 mm in steps of 0.25 mm while “b”, is varied from0.1 mm to 10 mm in steps of 0.1 mm. A grid was formed to calculate the dampingforce for each set of combinations. A total of 4000 FEM simulations were computed.

Increasing the thickness of the copper to 10 mm and coating it with an irontube that has a thickness of 0.25 mm yielded a damping force of 800 N. A similarmagnitude for the damping force was attained when “a” was limited to 0.25 mmand “b” was increased to 10 mm. Using a 0.25 mm thick copper is not enough tomaximize the damping force. The thickness of the copper can be determined byapproximating the skin effect. The copper’s thickness should be larger than the


a (mm)b (mm)

Forc

e(N

)

Force (N)

05

10

500

1000

1500

2000

0

5

100

500

1000

1500

2000

Figure 4.9: A 3D plot of the damping force as a function of the thickness of thecopper tube “a” and the thickness of the iron tube “b”. The 3D plot is also projectedon the “ab” plane to show the contour plot of the damping force.

skin depth. Using a small copper thickness increases electrical resistance. Thus,smaller eddy currents are generated leading to a smaller damping force.

On the other hand, increasing the thickness of copper to 10 mm is not a goodchoice either. Enlarging “a” increases the reluctance of the magnetic circuit sincethe iron tube’s inner diameter will be equal to the copper tube’s outer diameter.Placing the iron tube this far away has no effect. This can be noticed from Fig. 4.9.Using an iron tube with a thickness of 0.1 mm or a thickness of 10 mm with a coppertube that has a thickness of 10 mm makes no difference at all.

Using a very small thickness for an iron tube leads to saturation. This can benoticed only when the copper thickness is not that large. The region whereby theiron tube is saturated is close to the origin and is shown in dark blue. For smallcopper and iron thicknesses, the achieved damping force can be as little as 200 N.

The largest damping force is obtained at a copper thickness of 1.5 mm andan iron thickness of 10 mm. This however is not the most efficient point. Theoptimum point whereby a relatively large damping force is obtained for an affordablegeometry is at the knee point. The coordinates of the knee point are a copperthickness of 1.5 mm and an iron tube that is 2 mm thick.

Apparently, small changes in the magnetization direction lead to large changesin the damping force. The geometry, size, and material properties of the tube areas important as the used magnet topology. A composite magnetic damper basedon configuration E is the optimum choice for large damping force but with easyassembly. If ease of assembly is not an issue, then the Halbach topology is withoutquestion the best.

Chapter 5

Conclusions and Future Work

This chapter aims at summarizing the topics discussed throughout the thesis work.Some future work that can complement this thesis are also discussed.

5.1 Conclusions

HVDC breakers are one of the key components in the development of multi-terminalHVDC grids. If realized, clean renewable energy can be transported over vastdistances and made readily available for consumers living in big cities. The studydone in this thesis has targeted two of the weaknesses of an HVDC breaker; thelack of high efficiency ultra-fast actuators, and the lack of accurate multi-physicssimulation models.

To model an ultra-fast mechanical switch with high accuracy, a multi-physicssimulation model is required. Depending on the force impulse and the stiffness ofthe incorporated materials, models with different levels of complexity can be used.The underlying assumptions have to be carefully chosen to select a model thathas the minimum required complexity an yet maintains a high accuracy. Elasticcomponents significantly deteriorate the performance of the drive. The stiffness ofall materials in the switch should be maximized to maximize efficiency. If largedeformations are present, then the developed first order hybrid model can be usedto accurately simulate the switch within a reasonable time without compromisingaccuracy.

To increase the efficiency of the actuator a sensitivity analysis has been carriedout. The actuator is very sensitive to the energizing source, the shape and size ofthe coil and its armature, and, finally, to the coil’s housing.

The actuators of these switches need a housing to firmly keep in place theprimary coil. Using an electrically conductive housing deteriorates the efficiency ofthe actuator and yields high peak currents. It has been shown that a housing madeof a magnetic material has a potential to boost the efficiency of the actuator.

To further increase efficiency, the validated model was parallelized. The devel-

111

112 CHAPTER 5. CONCLUSIONS AND FUTURE WORK

oped algorithm was proven to be powerful and robust. It can be used to optimizeand design ultra-fast actuators to meet all desired objectives and constraints with ahigh confidence. If several or different objective functions or constraints are decidedto be studied, it can be done so very easily. The already existing solutions can beused to identify other optima satisfying other objective functions and criteria.

So far, all studies have been aimed to improve the acceleration process. Itis also important to decelerate the armature in a controllable manner. Thus anovel passive magnetic damper has been developed. It has been shown that theHalbach arrangement used together with a composite tube produces the largestdamping forces. Due to the cumbersome assembly of the Halbach arrangement,axially magnetized magnets oriented in opposite directions (configuration E) canalso be used, since this topology yields the second largest damping force after theHalbach arrangement and is easy to assemble.

One limitation of this work is that all developed models are computationallydemanding and require a lot of simulation time. Although a parallel computingalgorithm was implemented to reduce the simulation time of the FEM based models,a lumped element model could reduce the solving time even more. However, sucha model with reduced computation time, may be less accurate.

In conclusion, the performed work presented in this thesis brings the world astep closer towards ensuring its readiness to one of the global future trends; thetransformation of the traditional grid into a fully controllable smart grid.

5.2 Future Work

Future work will be dedicated to develop new actuator topologies with the aim ofdiscovering new methods that significantly boost the efficiency of ultra-fast actua-tors. More focus will also be dedicated to developing other optimization schemesto reduce the number of required computations. Furthermore, an investigation willbe done into the development of lumped element multi-physics models.

In some operation modes, there might be a need to operate these switchessuccessively. In other words, an open-close-open operation might be a requirement.Using a passive magnetic damper is slow and hence cannot be used in this case.Therefore, several active damping systems will be investigated, modeled, and tested.

Appendix A

Math Operators

Let us assume that A and B are two tensors. Then the double contraction operatoracting on A and B is given by:

A : B = tr(ATB) , (A.1)

where tr is the trace operator. The trace of a tensor is the sum of its diagonalelements, for example,

tr(A) = a11 + a22 + a33 + ...ann =

n∑

i=1

aii , (A.2)

where n denotes to the size of A and ann denotes to the entry of the n’th row andn’th column of A.

The transpose of A, is defined as:

[AT]ij = [A]ji . (A.3)

113

Appendix B

Symbols and Acronyms

Matrices are represented by bold symbols and all variables are italicized.Roman Letters

δx elongation [m]A magnetic vector potential [Wb/m]B magnetic flux density [T]C stiffness tensor [−]E electric field [V/m]Em Green Lagrange strain tensor [−]Fem force density in a material frame [N/m3]fem force density in a spatial frame [N/m3]H magnetic field intensity [A/m]I identity matrix [−]J current density [A/m2]Ji induced current density [A/m2]Ji,arm induced current density in the armature [A/m2]P Piola-Kirchhoff stress tensor [N/m2]s surface area [m2]U displacement vector [m]v velocity [m/s]X position vector in reference configuration [m]x position vector in spatial reference frame [m]F deformation gradient [−]Br remanent magnetic flux density [T]Jen externally applied current density [A/m2]Vn voltage across each turn [V]det determinant [−]Div divergence with respect to material reference frame [−]EE electrical energy [J]Grad gradient with respect to a material reference frame [−]

115

116 APPENDIX B. SYMBOLS AND ACRONYMS

TCp Thomson coil primary [−]A Area [m2]Ad armature depth [m]Aw armature width [m]C capacitance [mF]Cd coil depth [m]Cw coil width [m]Cp heat capacity [J/K]Cir coil inner radius [m]Cor coil outer radius [m]D total derivative [−]E Young’s modulus [N/m2]e induced emf [V]F Force [N]G bulk modulus [Pa]Icir current in circuit [A]In current in each turn [A]J determinant of the deformation gradient [−]Jn current density in each turn [A/m2]K shear modulus [Pa]k material stiffness [N/m]L length [m]l length [m]LPr push/pull rod length [m]Lstray stray inductance [H]LTC Thomson coil inductance [H]M mass of the metallic contacts [kg]m mass [kg]N number of turns [−]n turn number [−]Q heat source density [W/m2]R resistance [Ω]RC capacitor bank equivalent series resistance [Ω]Rstray stray resistance [Ω]RTC Thomson coil resistance [Ω]rn average turn radius [m]T Temperature [K]T0 reference temperature [K]ttravel travel time [s]V charging voltage [V]VC capacitor bank voltage [V]Vcoil voltage across coil [V]vp wave velocity [m/s]

117

Greek Letters

δ skin depth [m]ǫ strain [−]η efficiency [%]λ wavelength [m]σe0 reference electrical conductivity [S/m]µ magnetic permeability [H/m]µ0 magnetic permeability of free space [H/m]µr relative magnetic permeability [−]φ magnetic flux [Wb]ρ density [kg/m3]ρ0 density of reference material [kg/m3]ρe electrical resistivity [Ωm]σe electrical conductivity [S/m]

118 APPENDIX B. SYMBOLS AND ACRONYMS

Acronyms

AC Alternating CurrentCPU Central Processing UnitDC Direct CurrentDSC Double Sided CoilDSCc Clamped Double Sided CoilDSCm Mobile Double Sided CoilFEM Finite Element MethodFR4 Fiberglass Reinforced EpoxyHVAC High Voltage Alternating CurrentHVDC High Voltage Direct CurrentIGBT Insulated Gate Bipolar TransistorIGCT Integrated Gate Commutated ThyristorMOV Metal Oxide VaristorOFHS Oxygen-Free High Conductivity CopperPOM PolyoxymethyleneRAM Random Access MemoryTC Thomson CoilTCc Clamped Thomson CoilTCm Mobile Thomson CoilTTL transistor-transistor logic

Bibliography

[1] N. Kirby, L. Xu, M. Luckett, and W. Siepmann, “HVDC transmission for largeoffshore wind farms,” Power Engineering Journal, vol. 16, no. 3, pp. 135 –141,Jun. 2002.

[2] W. Lu and B.-T. Ooi, “Optimal acquisition and aggregation of offshore windpower by multiterminal voltage-source HVDC,” IEEE Transactions on PowerDelivery, vol. 18, no. 1, pp. 201 – 206, Jan. 2003.

[3] C. Meyer, M. Hoing, A. Peterson, and R. De Doncker, “Control anddesign of DC grids for offshore wind farms,” IEEE Transactions on IndustryApplications, vol. 43, no. 6, pp. 1475 –1482, Dec. 2007.

[4] M. Faruque, Y. Zhang, and V. Dinavahi, “Detailed modeling of CIGREHVDC benchmark system using PSCAD/EMTDC and PSB/SIMULINK,”IEEE Transactions on Power Delivery, vol. 21, no. 1, pp. 378–387, Jan. 2006.

[5] B. Andersen and L. Xu, “Hybrid HVDC system for power transmission toisland networks,” IEEE Transactions on Power Delivery, vol. 19, no. 4, pp.1884–1890, Oct. 2004.

[6] N. Hingorani, “High-voltage DC transmission: a power electronics workhorse,”IEEE Spectrum, vol. 33, no. 4, pp. 63–72, Apr. 1996.

[7] P. Bresesti, W. Kling, R. Hendriks, and R. Vailati, “HVDC connection ofoffshore wind farms to the transmission system,” IEEE Transactions on EnergyConversion, vol. 22, no. 1, pp. 37–43, Mar. 2007.

[8] L. Weimers, “HVDC light: A new technology for a better environment,” IEEEPower Engineering Review, vol. 18, no. 8, pp. 19–20, Aug. 1998.

[9] M. Bahrman and B. Johnson, “The ABCs of HVDC transmissiontechnologies,” IEEE Power and Energy Magazine, vol. 5, no. 2, pp. 32–44,Mar. 2007.

[10] H. Jiang and A. Ekstrom, “Multiterminal HVDC systems in urban areas oflarge cities,” IEEE Transactions on Power Delivery, vol. 13, no. 4, pp. 1278–1284, Oct. 1998.

119

120 BIBLIOGRAPHY

[11] N. Flourentzou, V. Agelidis, and G. Demetriades, “VSC-based HVDC powertransmission systems: An overview,” IEEE Transactions on Power Electronics,vol. 24, no. 3, pp. 592–602, Mar. 2009.

[12] S. Allebrod, R. Hamerski, and R. Marquardt, “New transformerless, scalablemodular multilevel converters for HVDC-transmission,” in IEEE PowerElectronics Specialists Conference, 2008. PESC 2008, Jun. 2008, pp. 174–179.

[13] S. Bozhko, G. Asher, R. Li, J. Clare, and L. Yao, “Large offshore DFIG-based wind farm with line-commutated HVDC connection to the main grid:Engineering studies,” IEEE Transactions on Energy Conversion, vol. 23, no. 1,pp. 119–127, Mar. 2008.

[14] V. Akhmatov, M. Callavik, C. Franck, S. Rye, T. Ahndorf, M. Bucher,H. Muller, F. Schettler, and R. Wiget, “Technical guidelines andprestandardization work for first HVDC grids,” IEEE Transactions on PowerDelivery, vol. 29, no. 1, pp. 327–335, Feb. 2014.

[15] “ABB wins $1 billion order for offshore wind power connection,” http://www.abb.com/cawp/seitp202/73b24f512a7591cac12578e0001a63c8.aspx.

[16] J. Song-Manguelle, T. Maja Harfman, S. Chi, S. Gunturi, and R. Datta,“Power transfer capability of HVAC cables for subsea transmission anddistribution systems,” in 2013 Record of Conference Papers IndustryApplications Society 60th Annual IEEE Petroleum and Chemical IndustryTechnical Conference (PCIC), Sep. 2013, pp. 1–9.

[17] H. Wang and M. Redfern, “The advantages and disadvantages of usingHVDC to interconnect AC networks,” in 45th International Universities PowerEngineering Conference (UPEC), 2010, Sep. 2010, pp. 1 –5.

[18] C. Franck, “HVDC circuit breakers: A review identifying future researchneeds,” IEEE Transactions on Power Delivery, vol. 26, no. 2, pp. 998 –1007,Apr. 2011.

[19] K. Kanngiesser, H. Ring, and T. Wess, “Simulator study on line fault clearingby DC circuit breakers in a meshed MTDC system,” in , InternationalConference on AC and DC Power Transmission, 1991, Sep. 1991, pp. 102–107.

[20] J. Häfner and B. Jacobson, “Proactive hybrid HVDC breakers - a keyinnovation for reliable HVDC grids,” Bologna, Italy, Sep. 2011.

[21] R. Kapoor, A. Shukla, and G. Demetriades, “State of art of power electronicsin circuit breaker technology,” in 2012 IEEE Energy Conversion Congress andExposition (ECCE), Sep. 2012, pp. 615 –622.

BIBLIOGRAPHY 121

[22] A. Hassanpoor, J. Hafner, and B. Jacobson, “Technical assessment of loadcommutation switch in hybrid HVDC breaker,” IEEE Transactions on PowerElectronics, vol. PP, no. 99, pp. 1–1, 2014.

[23] S. Tokuyama, K. Hirasawa, Y. Yoshioka, and Y. Kato, “Simulations oninterruption of circuit breaker in HVDC system with two parallel transmissionlines,” IEEE Transactions on Power Delivery, vol. 2, no. 3, pp. 772–778, Jul.1987.

[24] K. Arimatsu, Y. Yoshioka, S. Tokuyama, Y. Kato, and K. Hirata,“Development and interrupting tests on 250 KV 8 KA HVDC circuit breaker,”IEEE Transactions on Power Apparatus and Systems, vol. PAS-104, no. 9, pp.2452–2459, Sep. 1985.

[25] W. Premerlani, “Forced commutation performance of vaccum switches forHVDC breaker application,” IEEE Transactions on Power Apparatus andSystems, vol. PAS-101, no. 8, pp. 2721–2727, Aug. 1982.

[26] J. Anderson and J. Carroll, “Applicability of a vacuum interrupter as the basicswitch element in HVDC breakers,” IEEE Transactions on Power Apparatusand Systems, vol. PAS-97, no. 5, pp. 1893–1900, Sep. 1978.

[27] A. Mokhberdoran, A. Carvalho, H. Leite, and N. Silva, “A review on HVDCcircuit breakers,” in Renewable Power Generation Conference (RPG 2014),3rd, Sep. 2014, pp. 1–6.

[28] S. Negari and D. Xu, “A new solid-state HVDC circuit breaker topology foroffshore wind farms,” in 2014 IEEE 5th International Symposium on PowerElectronics for Distributed Generation Systems (PEDG), Jun. 2014, pp. 1–5.

[29] K. Sano and M. Takasaki, “A surgeless solid-state DC circuit breakerfor voltage-source-converter-based HVDC systems,” IEEE Transactions onIndustry Applications, vol. 50, no. 4, pp. 2690–2699, Jul. 2014.

[30] B. Pauli, G. Mauthe, E. Ruoss, G. Ecklin, J. Porter, and J. Vithayathil,“Development of a high current HVDC circuit breaker with fast fault clearingcapability,” IEEE Transactions on Power Delivery, vol. 3, no. 4, pp. 2072–2080, Oct. 1988.

[31] G. Hofmann, G. LaBarbera, N. Reed, L. Shillong, W. Long, and D. Melvold,“Field test of HVDC circuit breaker: Load break and fault clearing on thepacific intertie,” IEEE Transactions on Power Apparatus and Systems, vol. 95,no. 3, pp. 829–838, May 1976.

[32] R. Sander and T. Leibfried, “Considerations on energy absorption of HVDCcircuit breakers,” in Power Engineering Conference (UPEC), 2014 49thInternational Universities, Sep. 2014, pp. 1–6.

122 BIBLIOGRAPHY

[33] M. Sakai, Y. Kato, S. Tokuyama, H. Sugawara, and K. Arimatsu,“Development and field application of metallic return protecting breaker forHVDC transmission,” IEEE Transactions on Power Apparatus and Systems,vol. PAS-100, no. 12, pp. 4860–4868, Dec. 1981.

[34] G. Ge, M. Liao, X. Duan, and J. Zou, “HVDC hybrid circuit breaker based onSF6 interrupter and vacuum interrupter in series,” in 2013 2nd InternationalConference on Electric Power Equipment - Switching Technology (ICEPE-ST),Oct. 2013, pp. 1–4.

[35] T. Senda, T. Tamagawa, K. Higuchi, T. Horiuchi, and S. Yanabu,“Development of HVDC circuit breaker based on hybrid interruption scheme,”IEEE Transactions on Power Apparatus and Systems, vol. PAS-103, no. 3, pp.545–552, Mar. 1984.

[36] H. Darwish, M. Izzularab, and N. Elkalashy, “Real-time testing of hvdc circuitbreakers part 1: bench test development,” in 2004 International Conferenceon Electrical, Electronic and Computer Engineering, 2004. ICEEC ’04, Sep.2004, pp. 765–769.

[37] E. Koldby and M. Hyttinen, “Challenges on the road to an offshore HVDCgrid,” in Nordic Wind Power Conference, Sep. 2009.

[38] A. Bissal, On the Design of Ultra-Fast Electro-Mechanical Actuators.Stockholm, Sweden: Licentiate Thesis, Royal Institute of Technology, 2013,no. 978-91-7501-713-6.

[39] G. A. Holzapfel, Nonlinear Solid Mechanics: A Continuum Approach forEngineering. John Wiley & Sons Ltd, Feb. 2000.

[40] O. P. Agrawal and A. A. Shabana, “Automated visco-elastic analysis of largescale inertia-variant spatial vehicles,” Computers and Structures, vol. 22, no. 2,pp. 165–178, 1986.

[41] F. Shuhua, L. Heyun, Y. Chenfeng, L. Xiping, and G. Jian, “Magnetic fieldanalysis and control strategy of permanent magnet actuator for low voltagevacuum circuit breaker,” in 7th International Conference on Power Electronicsand Drive Systems, 2007 (PEDS 07), Nov. 2007, pp. 540 –543.

[42] F. Shuhua, L. Heyun, L. Xiping, and G. Jian, “Comparison evaluation forpermanent magnet arrangements of AC permanent magnet contactor,” inInternational Conference on Electrical Machines and Systems, 2007 (ICEMS),Oct. 2007, pp. 939 –942.

[43] S. Fang, H. Lin, and S. Ho, “Transient co-simulation of low voltage circuitbreaker with permanent magnet actuator,” IEEE Transactions on Magnetics,vol. 45, no. 3, pp. 1242 –1245, Mar. 2009.

BIBLIOGRAPHY 123

[44] S. Fang, H. Lin, S. Ho, X. Wang, P. Jin, and H. Liu, “Characteristics analysisand simulation of permanent magnet actuator with a new control method forair circuit breaker,” IEEE Transactions on Magnetics, vol. 45, no. 10, pp. 4566–4569, Oct. 2009.

[45] J.-M. Meyer and A. Rufer, “A DC hybrid circuit breaker with ultra-fastcontact opening and integrated gate-commutated thyristors (IGCTs),” IEEETransactions on Power Delivery, vol. 21, no. 2, pp. 646 – 651, Apr. 2006.

[46] T. Takeuchi, K. Koyama, and M. Tsukima, “Electromagnetic analysis coupledwith motion for high-speed circuit breakers of eddy current repulsion using thetableau approach,” Electrical Engineering in Japan, vol. 152, no. 4, pp. 8–16,2005.

[47] M. Tsukima, T. Takeuchi, K. Koyama, and H. Yoshiyasu, “Development ofa high-speed electromagnetic repulsion mechanism for high-voltage vacuumcircuit breakers,” Electrical Engineering in Japan, vol. 163, no. 1, pp. 34–40,2008.

[48] Y. Feng and X.-f. Zhang, “Analysis of current-limiting process for DC hybridcurrent-limiting breaker based on repulsion switch and fast fuse,” in 2011 IEEEPower Engineering and Automation Conference (PEAM), vol. 2, Sep. 2011, pp.176 –180.

[49] M. Steurer, K. Frohlich, W. Holaus, and K. Kaltenegger, “A novel hybridcurrent-limiting circuit breaker for medium voltage: principle and test results,”IEEE Transactions on Power Delivery, vol. 18, no. 2, pp. 460 – 467, Apr. 2003.

[50] Y. Kishida, K. Koyama, H. Sasao, N. Maruyama, and H. Yamamoto,“Development of the high speed switch and its application,” in The 1998 IEEEIndustry Applications Conference, 1998. Thirty-Third IAS Annual Meeting,vol. 3, Oct., pp. 2321–2328 vol.3.

[51] Z. Shi, S. Jia, M. Ma, X. Song, H. Yang, C. Liu, and L. Wang, “Investigation onDC interruption based on artificial current zero of vacuum switch,” in 2010 24thInternational Symposium on Discharges and Electrical Insulation in Vacuum(ISDEIV), Sep. 2010, pp. 158–161.

[52] R. Rajotte and M. G. Drouet, “Experimental analysis of a fast actingcircuit breaker mechanism-electrical aspects,” IEEE Transactions on PowerApparatus and Systems, vol. 94, no. 1, pp. 89–96, Jan.

[53] D. Enyuan, W. Yongxing, C. Jiyuan, and Z. Jiyan, “The analysis of high-speedrepulsion actuator and performance comparisons with permanent magneticactuator in vacuum circuit breaker,” in 23rd International Symposium onDischarges and Electrical Insulation in Vacuum, 2008. ISDEIV 2008, vol. 1,Sep. 2008, pp. 189 –191.

124 BIBLIOGRAPHY

[54] E. Dong, P. Tian, Y. Wang, and W. Liu, “The design and experimental analysisof high-speed switch in 1.14kV level based on novel repulsion actuator,”in 2011 4th International Conference on Electric Utility Deregulation andRestructuring and Power Technologies (DRPT), Jul. 2011, pp. 767 –770.

[55] W. Li and C. S. Koh, “Design of the thomson-coil actuator of an arc eliminatorusing adaptive equivalent circuit method,” in 2010 International Conferenceon Electrical Machines and Systems (ICEMS), Oct. 2010, pp. 1721 –1724.

[56] W. Holaus and K. Frohlich, “Ultra-fast switches- a new element for mediumvoltage fault current limiting switchgear,” in IEEE Power Engineering SocietyWinter Meeting, 2002, vol. 1, 2002, pp. 299 – 304 vol.1.

[57] T. Hori, A. Otani, K. Kaiho, I. Yamaguchi, M. Morita, and S. Yanabu, “Studyof superconducting fault current limiter using vacuum interrupter driven byelectromagnetic repulsion force for commutating switch,” IEEE Transactionson Applied Superconductivity, vol. 16, no. 4, pp. 1999–2004, Dec.

[58] D. Sadedin, “A study of the magnetic induction-repulsion accelerator,” inPulsed Power Conference, 1991. Digest of Technical Papers. Eighth IEEEInternational, Jun., pp. 68–72.

[59] W. Li, Y. W. Jeong, and C. S. Koh, “An adaptive equivalent circuitmodeling method for the eddy current-driven electromechanical system,” IEEETransactions on Magnetics, vol. 46, no. 6, pp. 1859 –1862, Jun. 2010.

[60] S.-M. Lee, S.-H. Lee, H.-S. Choi, and I.-H. Park, “Reduced modeling ofeddy current-driven electromechanical system using conductor segmentationand circuit parameters extracted by FEA,” IEEE Transactions on Magnetics,vol. 41, no. 5, pp. 1448 – 1451, May 2005.

[61] M. Piron, P. Sangha, G. Reid, T. Miller, and D. Ionel, “Rapid computer-aided design method for fast-acting solenoid actuators,” IEEE Transactionson Industry Applications, vol. 35, no. 5, pp. 991–999, Oct.

[62] W. Li, Z. Y. Ren, Y. W. Jeong, and C. S. Koh, “Optimal shape design ofa thomson-coil actuator utilizing generalized topology optimization based onequivalent circuit method,” IEEE Transactions on Magnetics, vol. 47, no. 5,pp. 1246 –1249, May 2011.

[63] P. Kuo-Peng, N. Sadowski, N. Batistela, and P. Bastos, “Coupled fieldand circuit analysis considering the electromagnetic device motion,” IEEETransactions on Magnetics, vol. 36, no. 4, pp. 1458 –1461, Jul. 2000.

[64] T. Ota, Y. Mitsutake, Y. Hasegawa, K. Hirata, and T. Tanaka, “Dynamicanalysis of electromagnetic impact drive mechanism using eddy current,” IEEETransactions on Magnetics, vol. 43, no. 4, pp. 1421 –1424, Apr. 2007.

BIBLIOGRAPHY 125

[65] M. Ertl and M. Kaltenbacher, “Investigation of the dynamics of electromag-netic valves by a coupled magneto-mechanical algorithm including contactmechanics,” COMPEL: The International Journal for Computation andMathematics in Electrical and Electronic Engineering, vol. 30, no. 2, pp. 603–621, Mar. 2011.

[66] L. Nowak, “Movement simulation in 3D eddy current transient problem,”COMPEL: The International Journal for Computation and Mathematics inElectrical and Electronic Engineering, vol. 20, no. 1, pp. 293–302, Mar. 2001.

[67] M. Pantelyat, M. Shulzhenko, Y. Matyukhin, P. Gontarowskiy, I. Dolezel,and B. Ulrych, “Numerical simulation of electrical engineering devices:Magneto-thermo-mechanical coupling,” COMPEL: The International Journalfor Computation and Mathematics in Electrical and Electronic Engineering,vol. 30, no. 4, pp. 1189–1204, Jul. 2011.

[68] L. Nowak and K. Radziuk, “Dynamics of an electromagnetic linear actuatoroperating in error actuated control system,” COMPEL: The InternationalJournal for Computation and Mathematics in Electrical and ElectronicEngineering, vol. 26, no. 4, pp. 941–951, Aug. 2007.

[69] “Technical description, permedyn MF1,” Magnetic Components AB, Sweden,Tech. Rep., 2009.

[70] J. Zheng, Z. Deng, Y. Zhang, W. Wang, S. Wang, and J. Wang, “Performanceimprovement of high temperature superconducting maglev system by eddycurrent damper,” IEEE Transactions on Applied Superconductivity, vol. 19,no. 3, pp. 2148–2151, Jun. 2009.

[71] K. Pluk, T. van Beek, J. Jansen, and E. Lomonova, “Modeling andmeasurements on a finite rectangular conducting plate in an eddy currentdamper,” IEEE Transactions on Industrial Electronics, vol. 61, no. 8, pp.4061–4072, Aug. 2014.

[72] H. Zhang, B. Kou, Y. Jin, L. Zhang, H. Zhang, and L. Li, “Modeling andanalysis of a novel planar eddy current damper,” Journal of Applied Physics,vol. 115, no. 17, pp. 17E709–17E709–3, May 2014.

[73] W. Guan, M. Jin, J. Chen, J. Ruan, Z. Du, Y. Zhang, Y. Li, K. Dai, Y. Fan,H. Zhang, and Y. Wang, “Finite-element modeling of eddy current and forcedistribution for induction dampers,” IEEE Transactions on Plasma Science,vol. 41, no. 5, pp. 1061–1065, May 2013.

[74] N. Amati, A. Tonoli, A. Canova, F. Cavalli, and M. Padovani, “Dynamicbehavior of torsional eddy-current dampers: Sensitivity of the designparameters,” IEEE Transactions on Magnetics, vol. 43, no. 7, pp. 3266–3277,Jul. 2007.

126 BIBLIOGRAPHY

[75] M. Mehrtash and M. Khamesee, “Modeling and analysis of eddy-currentdamping effect in horizontal motions for a high-precision magnetic navigationplatform,” IEEE Transactions on Magnetics, vol. 49, no. 8, pp. 4801–4810,Aug. 2013.

[76] A. Gosline and V. Hayward, “Eddy current brakes for haptic interfaces: De-sign, identification, and control,” IEEE/ASME Transactions on Mechatronics,vol. 13, no. 6, pp. 669–677, Dec. 2008.

[77] Y. Le, J. Fang, and J. Sun, “Design of a Halbach array permanent magnetdamping system for high speed compressor with large thrust load,” IEEETransactions on Magnetics, vol. 51, no. 1, pp. 1–9, Jan. 2015.

[78] Z. Zhu and D. Howe, “Halbach permanent magnet machines and applications:a review,” Electric Power Applications, IEE Proceedings -, vol. 148, no. 4, pp.299–308, Jul. 2001.

[79] C. Xia, L. Guo, and H. Wang, “Modeling and analyzing of magnetic field ofsegmented Halbach array permanent magnet machine considering gap betweensegments,” IEEE Transactions on Magnetics, vol. 50, no. 12, pp. 1–9, Dec.2014.

[80] L. Zhang, B. Kou, F. Xing, and H. Zhang, “Analysis and comparison of twotwo-dimensional Halbach permanent magnet arrays for magnetically levitatedplanar motor,” Journal of Applied Physics, vol. 115, no. 17, pp. 17E704–17E704–3, May 2014.

[81] T. Shi, Z. Qiao, C. Xia, H. Li, and Z. Song, “Modeling, analyzing, andparameter design of the magnetic field of a segmented Halbach cylinder,” IEEETransactions on Magnetics, vol. 48, no. 5, pp. 1890–1898, May 2012.

[82] M. Markovic and Y. Perriard, “Optimization design of a segmented Halbachpermanent-magnet motor using an analytical model,” IEEE Transactions onMagnetics, vol. 45, no. 7, pp. 2955–2960, Jul. 2009.

[83] S.-M. Jang, S.-H. Lee, H. W. Cho, and S. K. Cho, “Design and analysis ofhelical motion permanent magnet motor with cylindrical Halbach array,” IEEETransactions on Magnetics, vol. 39, no. 5, pp. 3007–3009, Sep. 2003.

[84] R. Björk, C. Bahl, A. Smith, and N. Pryds, “Optimization and improvementof Halbach cylinder design,” Journal of Applied Physics, vol. 104, no. 1, pp.013 910–013910–9, Jul. 2008.

[85] R. Björk, “The ideal dimensions of a Halbach cylinder of finite length,” Journalof Applied Physics, vol. 109, no. 1, pp. 013 915–013915–6, Jan. 2011.

BIBLIOGRAPHY 127

[86] R. Praveen, M. Ravichandran, V. Achari, V. Raj, G. Madhu, and G. Bindu,“A novel slotless Halbach-array permanent-magnet brushless DC motor forspacecraft applications,” IEEE Transactions on Industrial Electronics, vol. 59,no. 9, pp. 3553–3560, Sep. 2012.

List of Figures

2.1 Sketches of the drives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 A sketch of a DC breaker showing the current carrying contacts (2), thepush/pull rod (4), the armature (5), the coils (6), and the bistables (7). 11

2.3 A SPICE circuit coupled to a FEM model for a Thomson coil. Thecoil and the armature are modeled using FEM, since the resistanceand inductance of the coil and armature, RTC and LTC, are nonlinearand changing dynamically as the armature moves away. The capacitor,diode, thyristor, and cables are modeled by lumped parameters. . . . . . 13

2.4 The current density 100 µs after the discharge of the capacitor bankis distributed in only small portions of the geometry. Positive currentdensities appear in the top part of the coil conductors, while negativecurrents are induced in a piece of the armature situated directly abovethe coil. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.5 The magnetic flux density, in (T), 100 µs after the discharge of thecapacitor bank reaches 5 T and is highest in the air gap separating thecoil and the armature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.6 The force density, in (Pa), 100 µs after the discharge of the capacitorbank. The coil is subjected to compressive forces while the armature issubjected to an axially directed body force that is mostly concentratedin the first few millimeters closest to the coil. . . . . . . . . . . . . . . 17

2.7 A sketch showing the modeling of the armature (top, in silver), thepush/pull rod (in green), and the contacts (in dark red), by a full multi-physics model in (A), by a first order hybrid model in (B), and by ageneralized n segment hybrid model in (C). The push/pull rod’s totalmass is represented by m and its stiffness along elongation by k. Themass of the copper contact is represented by M . . . . . . . . . . . . . . 21

2.8 A picture showing the slim and large mushroom shaped armatures.Although both are designed to withstand the mechanical stresses, oneis flexible and prone to bending while the other is more robust and stiffer. 22

2.9 A picture showing the experimental setup. The slim mushroom arma-ture is sitting on a flat spiral coil that is connected with large cables toa capacitor bank. It is mounted with a 3.6 kg steel mass. . . . . . . . . . 23

128

List of Figures 129

2.10 A picture showing the measured currents (solid lines) and the simulatedcurrents (dashed lines) of the slim armature for increasing chargingvoltages in steps of 100 V. . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.11 Comparison of the measured and simulated bending of the mushroomarmature upon the discharge of a capacitor bank charged with 500 V.The bending cannot be measured for longer time scales since the pictureloses focus with large displacements rendering the tracking unreliable. . 25

2.12 A picture showing the measured velocities (solid lines) and the simulatedvelocities (dashed lines) of the slim armature for increasing chargingvoltages in steps of 100 V. . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.13 A picture showing the measured currents (solid lines) and the simulatedcurrents (dashed lines) of the large armature for increasing chargingvoltages in steps of 100 V. . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.14 A picture showing the measured velocities (solid lines) and the simulatedvelocities (dashed lines) of the large armature for increasing chargingvoltages in steps of 100 V. The simulation error increases with increasingimpulsive forces since even the large armature is elastic and will thereforebend eventually. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.15 A 3D picture showing the velocity profile of the system in [m/s] after150 µs. (B) is a zoom in of the entire actuator shown in (A). Thearmature, which is situated directly on top of the coil, is threaded into apush/pull rod and attached firmly. Point A is located at the outermostextremity of the mushroom armature. Point B is located at the top ofthe stem of the armature, i.e. just below the rounded corner joining thehead of the mushroom to its stem. Point C is located on the bottom ofthe push/pull rod, i.e. where the copper contacts are attached. . . . . . 29

2.16 The displacements of three different points characterizing the dynamicmotion of the breaker. The bending of the head of the mushroom canbe computed by subtracting the axial displacement of pt B from that ofpt A. Similarly, the elongation of the push/pull rod and the stem of themushroom armature can be computed by subtracting the displacementof pt C from that of pt B. . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.17 The arising current pulse in the four models following the discharge ofthe capacitor bank in the spirally shaped coil. . . . . . . . . . . . . . . . 31

2.18 The arising force impulse in the four models following the discharge ofthe capacitor bank in the spirally shaped coil. . . . . . . . . . . . . . . . 31

2.19 The displacement of the load with respect to each model. . . . . . . . . 32

2.20 The displacement of the load with respect to each model. Model 2: firstorder hybrid model. Model 5: second order hybrid model. Model 6:third order hybrid model. Model 7: Tenth order hybrid model. . . . . . 33

3.1 A schematic explaining the test setup and the different components ofthe prototypes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

130 List of Figures

3.2 A picture showing the high speed camera, a capacitor bank, a Pearsonprobe, and the small scale prototype. . . . . . . . . . . . . . . . . . . . . 39

3.3 A twenty turn spirally shaped flat coil for the large scale prototype. . . 393.4 Three armature variants, (A) showing a disk made of oxygen-free high

conductive copper, (B) showing a disk made of Aluminum 6082 T651,and (C) showing a mushroom shaped armature made of Aluminum 7075T651. Armatures (A) and (B) are used in the small scale prototype whilearmature (C) is used in the large scale prototype. . . . . . . . . . . . . . 40

3.5 The large scale prototype showing the coil firmly attached to the centerof a massive steel construction designed to avoid vibrations. . . . . . . . 41

3.6 Simulations compared with experimentally measured armature velocitiesmade of copper and aluminum upon the discharge of a capacitor bankwith a capacitance of 33 mF charged up to 50 V. . . . . . . . . . . . . . 42

3.7 Simulations compared with experimentally measured armature velocitiesmade of copper and aluminum upon the discharge of a capacitor bankwith a capacitance of 33 mF charged up to 100 V. . . . . . . . . . . . . . 42

3.8 Simulations compared with experimentally measured armature velocitiesupon the discharge of an 11 mF capacitor bank charged from 300 V to900 V in steps of 100 V. . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.9 A figure showing the defined actuator variables; more precisely, thevariables parameterizing the coil and the armature. . . . . . . . . . . . . 44

3.10 The force impulse of a mobile TC along with its forgone force due toits stroke. The area shaded in blue is the force impulse of a mobile TCduring the discharge of a 10 mF capacitor bank whereas the area underthe curve shaded in red is the forgone force due to the increased airgap with contact separation. In essence, if the armature of the TC isclamped, then the total generated force impulse is given by the sum ofthe areas under both curves (i.e. the sum of the areas shaded in blueand in red). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.11 The force impulse of a mobile DSC along with its forgone force due toits stroke. The area shaded in blue is the force impulse of a mobile DSCduring the discharge of a 100 mF capacitor bank whereas the area underthe curve shaded in red is the forgone force due to the increased airgap with contact separation. In essence, if the armature of the DSC isclamped, then the total generated force impulse would be the total areaunder both curves (i.e. the sum of the areas shaded in blue and in red). 46

3.12 Current density in (A/m2) at 8 µs shown in the cross sections of the coiland armature, just before the first current peak, for the first energizingsource of a mobile TC shown in Table 3.1. . . . . . . . . . . . . . . . . 48

3.13 Electrical conductivity in per unit at 88 µs shown in the cross sections ofthe coil and armature for the first energizing source shown in Table 3.1. 49

3.14 Current density in (A/m2) at 500 µs shown in the cross sections ofthe coil and armature, just before the first current peak, for the lastenergizing source of a mobile TC shown in Table 3.1. . . . . . . . . . . 49

List of Figures 131

3.15 The velocity profile of a mobile TC drive shown for the six differentenergizing source cases shown in Table 3.1. . . . . . . . . . . . . . . . . 50

3.16 The velocity profile of a mobile DSC drive shown for the six differentenergizing source cases shown in Table 3.1. . . . . . . . . . . . . . . . . 51

3.17 The current pulses through the coils of three identical Thomson coilactuators whereby their coils are embedded in three different housings,steel, stone, and MF1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.18 The current pulses through the armatures of three identical Thomsoncoil actuators whose coils are embedded in three different housings: steel,stone, and MF1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.19 The repulsive forces of three identical Thomson coil actuators wherebytheir coils are embedded in three different housings, steel, stone, and MF1. 55

3.20 The velocity of the armature of three identical Thomson coil actuatorswhereby their coils are embedded in three different housings, steel, stone,and MF1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.21 The relative permeability (µr), prior to the discharge of the capacitorbank. The MF1 housing has an initial homogenous relative permeabilityof 405, while the relative magnetic permeability of the coil and thearmature are 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.22 The relative permeability (µr), 300 µs after the discharge of the capacitorbank. Regions of the MF1 housing in proximity to the coil start tosaturate, whereby their relative permeability gradually decreases to 1. . 57

3.23 The relative permeability (µr), 800 µs after the discharge of the capacitorbank. As the magnetic field increases in magnitude, a larger portion ofthe MF1 housing is saturated. . . . . . . . . . . . . . . . . . . . . . . . . 58

3.24 The efficiency of a TC actuator versus the coil inner radius and numberof turns whereby the coil is embedded in a stone housing . . . . . . . . 58

3.25 The efficiency of a TC actuator versus the coil inner radius and numberof turns whereby the coil is embedded in a MF1 housing . . . . . . . . . 59

3.26 The current peak in a coil consisting of 5 turns versus the conductor’swidth (Cw) and its depth (Cd) according to the parameters describedin case study 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61




3.30 The peak force in the armature of a coil consisting of 5 turns versus theconductor’s width (Cw) and its depth (Cd) according to the parametersdescribed in case study 1. . . . . . . . . . . . . . . . . . . . . . . . . . . 63

132 List of Figures




3.34 The steady state velocity of the armature versus the conductor’s width(Cw) and its depth (Cd) according to the parameters described in casestudy 1. The coil consists of 5 turns. . . . . . . . . . . . . . . . . . . . . 65

3.35 The steady state velocity of the armature versus the conductor’s width(Cw) and its depth (Cd) according to the parameters described in casestudy 1. The coil consists of 12 turns. . . . . . . . . . . . . . . . . . . . 66



3.38 The actuator’s efficiency versus the conductor’s width (Cw) and its depth(Cd) according to the parameters described in case study 1. The coilconsists of 5 turns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.39 The actuator’s efficiency versus the conductor’s width (Cw) and its depth(Cd) according to the parameters described in case study 1. The coilconsists of 12 turns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68



3.42 The peak force versus the coil’s inner radius and its number of turns fora load of 1 kg. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70



3.45 The armature’s final velocity versus the coil’s inner radius and its num-ber of turns for a load of 1 kg. . . . . . . . . . . . . . . . . . . . . . . . . 71


List of Figures 133


3.48 The actuator’s efficiency versus the coil’s inner radius and its number ofturns, for a load of 1 kg. . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.49 The actuator’s efficiency versus the coil’s inner radius and its number ofturns, for a load of 2 kg. . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.50 The actuator’s efficiency versus the coil’s inner radius and its number ofturns for a load of 3 kg. . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.51 A 4D plot showing all points having a velocity between 0 m/s and15 m/s. The first dimension is capacitance shown on the x axis, thesecond dimension is the charging voltage shown on the y axis, the thirddimension is the number of turns shown on the z axis, and the fourthdimension is the velocity of the actuator, represented by color. . . . . . 76

3.52 A 4D plot showing all points having a a velocity between 15 m/s and30 m/s. The first dimension is capacitance shown on the x axis, thesecond dimension is the charging voltage shown on the y axis, the thirddimension is the number of turns shown on the z axis, and the fourthdimension is the velocity of the actuator, represented by color. . . . . . 77




3.56 A 5D plot showing all combination sets leading to a velocity between10 m/s and 12 m/s. The first dimension is capacitance shown on the xaxis, the second dimension is the charging voltage shown on the y axis,the third dimension is the number of turns shown on the z axis, thefourth dimension is the velocity of the actuator represented by color,and the fifth dimension is the actuator’s efficiency represented by thesize of the bubble. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

134 List of Figures

3.57 An x-z view: Capacitance versus number of turns perspective of Fig. 3.56.A 5D plot showing all combination sets leading to a velocity between10 m/s and 12 m/s. The first dimension is capacitance shown on the xaxis, the second dimension, (not shown in this figure), is the chargingvoltage, the third dimension is the number of turns shown on the z axis,the fourth dimension is the velocity of the actuator represented by color,and the fifth dimension is the actuator’s efficiency represented by thesize of the bubble. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.58 A y-z view: Charging voltage versus number of turns perspective ofFig. 3.56. A 5D plot showing all combination sets leading to a velocitybetween 10 m/s and 12 m/s. The first dimension, (not shown in thisfigure), is capacitance, the second dimension is the charging voltageshown on the y axis, the third dimension is the number of turns shownon the z axis, the fourth dimension is the velocity of the actuatorrepresented by color, and the fifth dimension is the actuator’s efficiencyrepresented by the size of the bubble. . . . . . . . . . . . . . . . . . . . 82

3.59 An x-y view: Capacitance versus charging voltage perspective of Fig. 3.56.A 5D plot showing all combination sets leading to a velocity between10 m/s and 12 m/s. The first dimension is capacitance shown on the xaxis, the second dimension is the charging voltage shown on the y axis,the third dimension, (not shown in this figure), is the number of turns,the fourth dimension is the velocity of the actuator represented by color,and the fifth dimension is the actuator’s efficiency represented by thesize of the bubble. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.60 A 5D plot showing all combination sets leading to a velocity between10 m/s and 12 m/s. The first dimension is capacitance shown on the xaxis, the second dimension is the charging voltage shown on the y axis,the third dimension is the number of turns shown on the z axis, thefourth dimension is the velocity of the actuator represented by color,and the fifth dimension is the initial electrical energy represented by thesize of the bubble. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

3.61 An x-z view: Capacitance versus number of turns perspective of Fig. 3.60.A 5D plot showing all combination sets leading to a velocity between10 m/s and 12 m/s. The first dimension is capacitance shown on the xaxis, the second dimension, (not shown in this figure), is the chargingvoltage, the third dimension is the number of turns shown on the z axis,the fourth dimension is the velocity of the actuator represented by color,and the fifth dimension is the initial electrical energy represented by thesize of the bubble. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

List of Figures 135

3.62 A y-z view: Voltage versus number of turns perspective of Fig. 3.60.A 5D plot showing all combination sets leading to a velocity between10 m/s and 12 m/s. The first dimension, (not shown in this figure), iscapacitance, the second dimension is the charging voltage shown on they axis, the third dimension is the number of turns shown on the z axis,the fourth dimension is the velocity of the actuator represented by color,and the fifth dimension is the initial electrical energy represented by thesize of the bubble. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

3.63 An x-y view: Capacitance versus number of turns perspective of Fig. 3.60.A 5D plot showing all combination sets leading to a velocity between10 m/s and 12 m/s. The first dimension is capacitance shown on the xaxis, the second dimension is the charging voltage shown on the y axis,the third dimension, (not shown in this figure), is the number of turns,the fourth dimension is the velocity of the actuator represented by color,and the fifth dimension is the initial electrical energy represented by thesize of the bubble. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

3.64 A 5D plot showing all combination sets leading to a velocity between10 m/s and 12 m/s. The first dimension is capacitance shown on the xaxis, the second dimension is the charging voltage shown on the y axis,the third dimension is the number of turns shown on the z axis, thefourth dimension is the velocity of the actuator represented by color,and the fifth dimension is the peak current represented by the size ofthe bubble. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3.65 An x-z view: Capacitance versus number of turns perspective of Fig. 3.64.A 5D plot showing all combination sets leading to a velocity between10 m/s and 12 m/s. The first dimension is capacitance shown on the xaxis, the second dimension is the charging voltage, (not shown in thisfigure), the third dimension is the number of turns shown on the z axis,the fourth dimension is the velocity of the actuator represented by color,and the fifth dimension is the peak current represented by the size ofthe bubble. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3.66 A y-z view: Voltage versus number of turns perspective of Fig. 3.64.A 5D plot showing all combination sets leading to a velocity between10 m/s and 12 m/s. The first dimension is capacitance, (not shown inthis figure), the second dimension is the charging voltage shown on they axis, the third dimension is the number of turns shown on the z axis,the fourth dimension is the velocity of the actuator represented by color,and the fifth dimension is the peak current represented by the size ofthe bubble. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

136 List of Figures

3.67 An x-y view: Capacitance versus number of turns perspective of Fig. 3.64.A 5D plot showing all combination sets leading to a velocity between10 m/s and 12 m/s. The first dimension is capacitance shown on the xaxis, the second dimension is the charging voltage shown on the y axis,the third dimension is the number of turns, (not shown in this figure),the fourth dimension is the velocity of the actuator represented by color,and the fifth dimension is the peak current represented by the size ofthe bubble. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.68 A 5D plot showing all combination sets leading to a velocity between10 m/s and 12 m/s. The first dimension is capacitance shown on the xaxis, the second dimension is the charging voltage shown on the y axis,the third dimension is the number of turns shown on the z axis, thefourth dimension is the velocity of the actuator represented by color,and the fifth dimension is the peak force represented by the size of thebubble. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

3.69 An x-z view: Capacitance versus number of turns perspective of Fig. 3.64.A 5D plot showing all combination sets leading to a velocity between10 m/s and 12 m/s. The first dimension is capacitance shown on the xaxis, the second dimension is the charging voltage, (not shown in thisfigure), the third dimension is the number of turns shown on the z axis,the fourth dimension is the velocity of the actuator represented by color,and the fifth dimension is the peak force represented by the size of thebubble. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.70 A y-z view: Charging voltage versus number of turns perspective ofFig. 3.64. A 5D plot showing all combination sets leading to a velocitybetween 10 m/s and 12 m/s. The first dimension is capacitance, (notshown in this figure), the second dimension is the charging voltage shownon the y axis, the third dimension is the number of turns shown on thez axis, the fourth dimension is the velocity of the actuator representedby color, and the fifth dimension is the peak force represented by thesize of the bubble. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

3.71 An x-y view: Capacitance versus charging voltage perspective of Fig. 3.64.A 5D plot showing all combination sets leading to a velocity between10 m/s and 12 m/s. The first dimension is capacitance shown on the xaxis, the second dimension is the charging voltage shown on the y axis,the third dimension is the number of turns, (not shown in this figure),the fourth dimension is the velocity of the actuator represented by color,and the fifth dimension is the peak force represented by the size of thebubble. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.1 A stack of magnets and iron spacers travelling inside a copper tube (seeFig. 4.1 A). The transparency of the copper tube is increased to showthe magnets falling through it. Its axi-symmetric simulation model isshown in Fig. 4.1 B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

List of Figures 137

4.2 A diagram showing the initial investigated magnet topologies. . . . . . . 1014.3 A picture showing the small scale setup whereby three different arrange-

ments of magnets are mounted on three shafts and inserted in coppertubes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.4 A picture exposing the shaft and the magnets. . . . . . . . . . . . . . . 1024.5 A figure showing the actuator and the damper. A stack of four magnets

and five iron spacers based on topology E are assembled and inserted ina copper tube. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.6 Measured armature and damper velocities compared with the simulateddamper velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.7 A diagram showing all investigated magnet topologies. Topologies A toE are comprised of axially magnetized magnets, topologies F and G arecomprised of radially magnetized magnets, and finally topology H, alsoknown as a Halbach arrangement, comprises of a mixture of radially andaxially magnetized magnets . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.8 A magnet array based on configuration E is inserted in a compositetube composed of copper and iron as its innermost and outermost layersrespectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

4.9 A 3D plot of the damping force as a function of the thickness of thecopper tube “a” and the thickness of the iron tube “b”. The 3D plot isalso projected on the “ab” plane to show the contour plot of the dampingforce. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

modeling and verification of ultra-fast electro-mechanical actuators

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