Electromagnetic modelling and testing of a Thomson coil ...1110818/FULLTEXT01.pdf · system are much ’closer’ electrically to the fault. This means fast-rising and high fault

IN DEGREE PROJECT ELECTRICAL ENGINEERING,SECOND CYCLE, 30 CREDITS

, STOCKHOLM SWEDEN 2017

Electromagnetic modelling and testing of a Thomson coil based actuator

BENCE HÁTSÁGI

KTH ROYAL INSTITUTE OF TECHNOLOGYSCHOOL OF ELECTRICAL ENGINEERING

Electromagnetic modelling and testing of a Thomson coilbased actuator

BENCE HATSAGI

Master of Science Degree Project in Electrical Energy Conversionat the School of Electrical Engineering

Royal Institute of TechnologyStockholm, Sweden, June 2017.

Supervisor: Staffan NorrgaExaminer: Oskar Wallmark

TRITA-EE 2017:065

Electromagnetic modelling and testing of a Thomson coil based actuatorBENCE HATSAGI

c© BENCE HATSAGI , 2017.

School of Electrical EngineeringDepartment of Electrical Energy ConversionKungliga Tekniska hogskolanSE–100 44 StockholmSweden

Abstract

The aim of the present thesis is to improve and optimize a Thomson coil based actuatorfor medium voltage vacuum interrupters. The Thomson coil based actuator’s concept isdiscussed. The thesis presents analytical as well as finite element models of the actuatoralong with a comparison of their results. Several experimental setups have been built forthis degree project and they are described in the thesis. Measurements from these setupsare also compared to simulation results. The thesis concludes by drawing conclusionsfrom the compared results and proposes possible directions for additional work in thenear future.

Keywords: Actuator design, FEM simulation, HVDC, hybrid circuit breaker, Thomsoncoil, vacuum interrupter

iii

iv

Sammanfattning

Malsattningen for denna uppsats ar att forbattra och optimera en aktuator for mel-lanspanningsvakuumbrytare baserad pa en Thomsonspole. Aktuatorkonceptet analyserasoch diskuteras. Uppsatsen presenterar analytiska modeller saval som numeriska modellerfor FEM av aktuatorn, samt jamforelser av resultaten fran simuleringar gjorda av dessa.Flera experimentuppstallningar har byggts under detta examensprojekt och beskrivs idenna uppsatsen. Matningar fran dessa uppstallningar jamfors ocksa med resultaten fransimuleringarna. Uppsatsen drar slutligen slutsatser utifran resultaten och foreslar mojligavagar for ytterligare arbete pa omradet inom en snar framtid.

Nyckelord: Aktuatordesign, FEM-simulering, HVDC, hybridbrytare, Thomsonspole,vakuumbrytare

v

vi

Acknowledgements

The present degree project has been carried out jointly at SCiBreak in Jarfalla and at theDepartment of Electrical Energy Conversion of the School of Electrical Engineering atthe Royal Institute of Technology, KTH in Stockholm.

I would like to thank Staffan Norrga, my supervisor and employer at SCiBreak forhis valuable feedback and for letting me participate in this interesting and challengingproject.

I would also like to thank Lennart Angquist and Simon Nee for their guidance andhelp in or outside the laboratory.

I would especially like to thank Antoine Baudoin for his help with both theoreticaland practical issues during the thesis. His patience and support are much appreciated.

Moreover, I would like to thank Oskar Wallmark, my examiner at KTH for pro-viding regular feedback and suggestions during the semester. Also, thanks go to MojganNikouie Harnefors for the discussions that helped me through the long office hours inwintertime.

Last but not least, I would like to express my gratitude to my family for their supportand love.

Bence HatsagiSolna, SwedenJune 2017

vii

viii

Contents

Abstract iii

Sammanfattning v

Acknowledgements vii

Contents ix

1 Introduction 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Thomson coil based actuator . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Analytical model 52.1 Simple analytical model . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Extended models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Two turns on the primary side . . . . . . . . . . . . . . . . . . . 112.2.2 Two coils on the secondary side . . . . . . . . . . . . . . . . . . 122.2.3 Generalized model . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 Finite element model 213.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 First model group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2.1 Model with single-turn coils . . . . . . . . . . . . . . . . . . . . 233.2.2 Model with multiple-turn coils . . . . . . . . . . . . . . . . . . . 24

3.3 Second model group . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.3.1 Model with aluminum plate . . . . . . . . . . . . . . . . . . . . 253.3.2 Model with copper plate . . . . . . . . . . . . . . . . . . . . . . 26

3.4 Third model group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.4.1 Model for designing a prototype . . . . . . . . . . . . . . . . . . 263.4.2 Model to check a built prototype . . . . . . . . . . . . . . . . . . 27

ix

Contents

4 Experimental setups 434.1 Smaller experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . 434.2 First prototype . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.3 Targeted prototype . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5 Comparison of results 515.1 Comparison of Matlab and COMSOL results . . . . . . . . . . . . . . . 515.2 Comparison of the small scale setup’s measurements and COMSOL . . . 515.3 Comparison of the first prototype’s measurements and COMSOL . . . . . 53

6 Conclusion 596.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

A Distance measurement 61

References 65

x

Chapter 1

Introduction

1.1 Motivation

The growth of electric power generation challenges already existing power transmissionsystems. In developed regions (e.g. Europe, North America) the electrical grid may facedifficulties in integrating additional power sources.A solution could be the reinforcement of the AC grids, either by increasing transmissionvoltage levels or by extending the grids with new lines. If this is not viable becauseof technical, economical or other types of consideration, converting already existing ACgrids to DC transmission systems may provide means to overcome the problems. In caseof new transmission lines, DC might be the only possible option (e.g. long-distance seacables). A high voltage DC (HVDC) system could have, among others, the followingadvantages compared to a high voltage AC system (as detailed in [1]):

• There are no reactive power flows in a HVDC system.

• The reactive components in the line’s circuit do not impose a limit on the powerthat can be transferred.

• Frequency stability is not an issue.

• The HVDC grid can interconnect asynchronous AC systems or reinforce weak ACgrids.

• Cables can be considerably longer which is useful for cross sea connections as wellas for grids that supply densely populated urban areas.

However, there are numerous challenges to overcome when building an HVDC grid(refer to [1]).

• Power flow control.

1

CHAPTER 1. INTRODUCTION

• The need for notably faster protection systems and circuit breakers than for an ACsystem.

• Compatibility of HVDC grids that are using different converter technologies (asof 2017, most of the HVDC links use line commutated converters, while voltagesource converters are more promising for building multiterminal HVDC grids inthe future)

• Presently, standardization is in its initial state.

As discussed in [1], HVDC transmission becomes economically more attractive than ACtransmission as distances and the amount of power to be transmitted increase. Moreover,with the emerging renewable technologies (e.g. wind, solar and tidal power generation)and with increasing trade of electricity come new power flow patterns that could be prob-lematic to accommodate by an AC grid. The larger scale renewable sources are oftenremotely located from the main power grid (examples are the wind farms installed inthe North Sea), thus imposing technical challenges that may be more manageable withHVDC solutions.

Until recently, the majority of HVDC grids have been point-to-point connections( [1]). Just a few multiterminal HVDC grids exist, see [2], [3] and [4] for example. Onemajor challenge in a multiterminal HVDC grid is the required performance of the circuitbreakers. Since such a grid has considerably lower line impedances than an AC grid, incase of a fault (either pole to ground or pole to pole) the converters feeding the HVDCsystem are much ’closer’ electrically to the fault. This means fast-rising and high faultcurrent levels. Moreover, a system voltage collapse can propagate much faster than inan AC system. Therefore, it is crucial that the circuit breaker is capable of breaking thecurrents in just a few milliseconds. This is at least one order of magnitude faster than thecurrently available AC breakers that utilise e.g. a spring mechanism to provide the neededforce.

Further discussion of challenges related to a HVDC grid is outside of the scope ofthis thesis, so the interested reader is referred to [1], [5], [6] and [7] for more informationon such grids, especially VSC-based HVDC.

1.2 Thomson coil based actuator

A solution for a fast enough circuit breaker is ABB’s breaker concept [8]. This attractedresearchers in the recent years [9], [10], [11], [12]. This concept combines the advan-tageous features of a breaker based purely on semiconductor devices (i.e. high speed,current breaking can take less than 1 ms) and a breaker with traditional metallic contacts(i.e. low on-state power losses). In short, the current path in normal operating conditionsleads through the metallic contacts. When the protection system detects the fault on the

2

1.2. THOMSON COIL BASED ACTUATOR

Figure 1.1: Illustration, taken from [10]

DC line, a smaller semiconductor switch instantly commutates the current to the mainbreaker’s path, where the stacked high voltage and high power semiconductor switchesare able to break the current. The breaker concepts that are detailed in the previously men-tioned papers usually contain either snubbers or a third, parallel branch which consists ofcomponents that are designed to dissipate the DC line’s magnetic energy (see Figure 1.1).These can be linear resistors or nonlinear arresters (a common material for the arresters isZnO).

A drawback of these breakers is the semiconductor-based main DC-breaker’s price.The utilised switches with sufficient ratings (components with rated voltage of a few kilo-volts and rated current breaking capacity of a few kiloamperes are stacked in order to yielda device that meets the requirements) are expensive. It would be economically beneficialto substitute the semiconductor breaker by a fast enough mechanical breaker. This thesisfocuses on the development of an actuator that offers a solution to the problems related toa HVDC breaker discussed earlier.

A Thomson coil’s operating principle in short is the next: discharging a capacitorthrough a flat coil causes a fast current surge with high amplitude in that coil. If a plateof conducting material is placed close enough to the coil, then the fast-changing magneticinduction (caused by the current surge) induces considerable eddy currents in the plate.The interaction of these eddy currents with the magnetic induction yields a repellent forcethat tries to move the coil and the plate away from one another.

If a shaft is attached to the plate, then the action of the plate leaving the Thomsoncoil can be used as well for the shaft to make or break a current path by establishingor terminating a contact. With the right selection of power electronics (e.g. a correctlytuned resonant circuit) and control, it is possible to reduce the current flowing throughthe breaker for the instant of breaking so that can be performed. Moreover, the breakercan accelerate the mechanical system so fast that the distance of the contacting surfaces

3

CHAPTER 1. INTRODUCTION

can be sufficiently high by a prescribed time period (in order to withstand the voltagesappearing over the breaker after a successful breaking). This way, considerable savingson the semiconductor devices can be achieved.

This makes the Thomson coil-based actuator a promising candidate for middle andhigh voltage breaker appliances. This concept has also been investigated by several re-searcher teams in recent years (e.g. [13], [14], [15]).

The thesis discusses the theoretical and modeling aspects of a Thomson coil-basedcircuit breaker. It details the actuator models implemented on different platforms (Matlab,COMSOL) as well as the models’ correspondence with each other. The report analyzesalso measurement data that were yielded by experimental setups.

The thesis’s structure can be divided the next way. Chapter 2 details both theoryand simulation of the analytical models. Chapter 3 discusses the finite element modeling:short description of theory and the implemented models. Chapter 4 describes the physicalmodels along with obtained measurements. Chapter 5 compares the results of the differentmodels, while Chapter 6 gives a short conclusion and sets out potential goals for futurework in this subject.

4

Chapter 2

Analytical model

The aim of this chapter is to present an analytical model of the Thomson coil. The forceproduced by the coil shall be discussed by using analytical formulae for mutual induc-tances. The first model is taken directly from a SCiBreak memo [16] and is detailed herein order to put the extensions in context. However, the interested reader can find moredetailed theoretical discussion in published papers, e.g. in [17].

2.1 Simple analytical model

First, the Thomson coil is modeled as only one circular closed conducting loop. The platethat should be repelled by the force is also modeled as one circular closed conducting loopin the first approach. The two coils are assumed to be coaxial. The storage that suppliesenergy to the Thomson coil is modelled as a capacitor charged to an initial voltage level.

Let the two coils have radii rp1 and rs1 and be displaced by the distance d accordingto Figure 2.1.

Let Lp1 and Ls1 denote the self inductance of the Thomson coil (primary side) andthe plate (secondary side), respectively. Their mutual inductance shall be denoted by M.Moreover, Rp1 and Rs1 denote the primary and the secondary coils’ electrical resistance,respectively.

All but one variables in this chapter are functions of time, so this dependency willnot be explicitly stated in the equations (i.e. the time function x(t) will be denoted by x).The variable that is a function of another parameter is the mutual inductance (which is afunction of distance). Also, this dependency is pointed out here and may not be denotedexplicitly in further equations.

At an arbitrary instant in time let ip1 and is1 denote the currents flowing in theprimary and the secondary coils. With concurrent reference directions the instantaneouspower (p) supplied to the magnetic field can be written as

p = up1ip1 + us1is1 =(Lp1

dip1

dt+M

dis1dt

)ip1 +

(Ls1

dis1dt

+Mdip1

dt

)is1 (2.1)

5

CHAPTER 2. ANALYTICAL MODEL

Figure 2.1: Coaxial circular loops

Here up1 and us1 denote the voltages over the primary and secondary coils, respectively.It can be considered that the closed loops are opened at a negligibly short part of theircircumferences and an external circuit connected to these coil ends provides the voltages.

The equation for the power can be written also as

p =d

dt

(1

2Lp1i

2p1 +

1

2Ls1i

2s1 +Mip1is1

)(2.2)

Therefore, the integration of this equation with respect to time should give the energystored in the magnetic fields of the coils. Its expression becomes

Wm =1

2Lp1i

2p1 +

1

2Ls1i

2s1 +Mip1is1 (2.3)

Now, if the two coils’ axial distance is changed by a virtual displacement ∆d, there will bea change in the coils’ mutual inductance (∆M ) and, consequently, in the stored magneticenergy. The latter change amounts ∆Mip1is1.

The change in the stored magnetic energy corresponds to the mechanical energyrequired to separate the coils by the distance ∆d. By denoting the force acting on thecoils by F, the next expression can be written:

F∆d = ∆Mip1is1 (2.4)

As the virtual displacement becomes infinitesimally small, the following equation can beestablished:

F + ip1is1∂M

∂d= 0 (2.5)

6

2.1. SIMPLE ANALYTICAL MODEL

So, the force acting on the coils can be computed if the coil currents and the mutual induc-tance’s derivative with respect to the distance are known. Both analytical approximations(see, e.g. [18]) and arbitrarily accurate computations [19] are available for calculatingthe mutual inductance of two conductors with the geometry concerned in this case. Themethod discussed below uses Maxwell’s formula, as detailed in [18] (although with dif-ferent notation of parameters). The mutual inductance of two, circular closed loops thatare coaxial, can be expressed as a function of distance between the coils as

M(d) = µ0√rp1rs1g(k) (2.6)

where

k =2√rp1rs1√

(rp1 + rs1)2 + d2(2.7)

and

g(k) =(2

k− k)K(k)− 2

kE(k) (2.8)

Here K(k) and E(k) are the complete elliptic integrals of the first and second kind, re-spectively. They are defined by

K(k) =

π2∫

0

dφ√1− k2 sin2 φ

(2.9)

and

E(k) =

π2∫

0

√1− k2 sin2 φdφ (2.10)

The mutual inductance’s derivative with respect to the distance between the coils can bewritten as

∂M

∂d= µ0

√rp1rs1

dg

dk

∂k

∂d(2.11)

where

∂k

∂d= − k3d

4rp1rs1(2.12)

and (with some rearrangements)

dg

dk=

2− k2

k2(1− k2)E(k)− 2

k2K(k) (2.13)

7


In the last expression, the following equations for the elliptical functions’ derivatives havebeen used:

dK(k)

dk=

E(k)

k(1− k2)− K(k)

k(2.14)

dE(k)

dk=E(k)−K(k)

k(2.15)

With the expressions derived above, it is possible to establish the system of differen-tial equations which describes this simple physical model’s electromechanical behavior.For this arrangement (one primary and one secondary coil) together with the externalcircuit, there are five state variables:

• Capacitor voltage (uc)

• Primary coil current (ip1)

• Secondary coil current (is1)

• Coil separation speed (v)

• Distance between the coils (d)

Figure 2.2: Thomson coil dynamic system model for two single-turn coils

Assuming a short-circuited secondary coil (i.e. the plate is not connected to an externalcircuit, the eddy currents’ paths are closed within the plate, see Figure 2.2) and negligible

8

2.1. SIMPLE ANALYTICAL MODEL

air drag, the five differential equations for the respective state variables are:

uc = Rp1ip1 + Lp1dip1

dt+d(Mis1)

dt= Rp1ip1 + Lp1

dip1

dt+∂M

∂dvis1 +M

dis1dt

(2.16)

0 = Rs1is1 + Ls1dis1dt

+d(Mip1)

dt= Rs1is1 + Ls1

dis1dt

+∂M

∂dvip1 +M

dip1

dt(2.17)

Cducdt

= −ip1 (2.18)

mdv

dt= F (2.19)

dd

dt= v (2.20)

where m is the total mass of object to be moved i.e. the plate and its (at this point fictive)attachments.In order to model the real device, a certain nonzero conductor size must be assigned tothe coils. This way, the resistances can be calculated, moreover the expressions of selfinductances can only be evaluated for conductors with nonzero cross section.

Assuming the conducting material’s electrical resistivity (ρcond) and a circular con-ductor cross section with a radius rcond, the resistance of the two coils denoted Rp1 andRs1 can be computed by

Rp1 = ρcondlp1

Acond

= ρcond2πrp1

Acond

= ρcond2rp1

r2cond

(2.21)

and

Rs1 = ρcondls1Acond

= ρcond2πrs1Acond

= ρcond2rs1r2cond

(2.22)

If the current density is not uniform in the conductors, a further correctional term (Kp1skin

and Ks1skin) can be introduced in the above equations to take the skin effect into account.Also, if the circuit interconnecting the capacitor and the Thomson coil has to be modelled,a parasitic external resistance term can be added to the primary coil’s resistance. Theexpressions of the coil resistances become

Rp1 = Kp1skinρcond2rp1

r2cond

+Rp1ext (2.23)

Rs1 = Ks1skinρcond2rs1r2cond

(2.24)

The expression for the self inductance can be written as the sum of two parts, onecorresponding to the inductance of the conductor (hence it is called the inner inductance)and the other corresponding to the inductance of the loop (formed by the coil, so it is

9


called the outer inductance). As described for example in [20], the inner inductance Li ofa conductor of length lc is

Li =µ0

8πlc = µ0

rloop4

(2.25)

The second equation is true when the conductor is assumed to form a circular closed loopso that lc = 2πrloop. The outer inductance of the same loop (assuming a circular conductorcross section with radius rc) is approximately

Lo = µ0rloop ln(8rloop

rc− 2)

(2.26)

If the parasitic inductance of the coil interconnecting the capacitor and the Thomson coilis also taken into consideration, a corresponding term (Lp1ext) can be added to the primarycoil’s self inductance.

Plugging in the given geometrical parameter values to the expressions above, theequations for the self inductances become

Lp1 = µ0rp1

(1

4+ ln

( 8rp1

rcond

)− 2)

+ Lp1ext (2.27)

Ls1 = µ0rs1

(1

4+ ln

( 8rs1rcond

)− 2)

(2.28)

In order to solve the system of equations, the current derivatives must be computed.The first two equations can be rearranged so they yield these derivatives.

The rearranged equations rewritten in matrix form are[Lp1 M

M Ls1

][dip1dtdis1dt

]=

[uc −Rp1ip1 − ∂M

∂dvis1

−Rs1is1 − ∂M∂dvip1

](2.29)

This leads to the expression for the current derivatives:[dip1dtdis1dt

]=

[Lp1 M

M Ls1

]−1 [uc −Rp1ip1 − ∂M

∂dvis1

−Rs1is1 − ∂M∂dvip1

](2.30)

This way, on the left-hand sides of the system of equations stand the state vari-ables’ derivatives, while the right-hand side contains the state variables as well as knownparameters. By known initial conditions of the state variables, the system of differen-tial equations can be solved. In softwares like Matlab or Octave, the integration can beimplemented by an Euler backward scheme or by a type of Runge-Kutta method.

2.2 Extended models

The simple method detailed so far can be extended. A natural choice is to increase thenumber of turns either on the primary or on the secondary side, ultimately on both sides.

10

2.2. EXTENDED MODELS

2.2.1 Two turns on the primary side

First, in order to give a hint on how a more complex model can be derived, an additionalturn on the primary side is discussed. Since the primary turns are connected in series,the state variable of the first turn (i.e. ip1) is the same as the state variable of the secondturn (which could be denoted by ip2). Because of this, it is clear that the system of dif-ferential equations describing the model does not have any additional state variables. Itholds that the coupling between the second turn of the primary side and the secondarycoil should be taken into account as it changes both the electrical and the mechanical dy-namics. Therefore there should be new terms in the differential equations, but the numberof state variables remains the same as before, i.e. five. For the mechanical dynamics isformally the same as for the previous model, only the electrical circuit is displayed inFigure 2.3.

From here on, let the mutual inductance of the primary turns be denoted by Mpmn

where the mth and nth turns on the primary side are concerned. With linear magneticconditions (in this case air is assumed to surround the coils), it holds that Mpmn = Mpnm.The notation M11 is introduced to take over the role of M of the previous model, i.e. tonote the mutual inductance between the two sides’ first turns. Let it be assumed that thesecond turn on the primary side forms a circular closed loop (which is coaxial with thefirst turns) of radius rp2 with the same conductor size rcond. By denoting its resistanceand its self inductance and the mutual coupling between it and the secondary coil by Rp2,Lp2 and M21, respectively (whose way of computation is exactly the same as the methodspresented earlier), the system of differential equations applicable for this model can beestablished as

uc = (Rp1 +Rp2)ip1 + (Lp1 + Lp2 + 2Mp12)dip1

dt+(∂M11

∂d+∂M21

∂d

)vis1

+ (M11 +M21)dis1dt

(2.31)


+(∂M11

∂d+∂M21

∂d

)vip1 + (M11 +M21)

dip1

dt(2.32)

Cducdt

= −ip1 (2.33)

mdv

dt= F = F11 + F21 (2.34)

dd

dt= v (2.35)

Note that the force component F11 is due to the currents in the first turns of bothsides while component F21 corresponds to the interaction of currents in the second turnon the primary and the secondary coil.A matrix formulation for the state variable current derivatives can be arranged in a similar

11


Figure 2.3: Extended Thomson coil model with two-turn primary side

way as for the initial system detailed before.[dip1dtdis1dt

]=

[Lp1 + Lp2 + 2Mp12 M11 +M21

M11 +M21 Ls1

]−1uc −Rp1ip1 −

(∂M11

∂d+ ∂M21

∂d

)vis1

−Rs1is1 −(

∂M11

∂d+ ∂M21

∂d

)vip1

(2.36)

The same techniques can be used as in case of the first model (one turn on eachside) in order to solve the system of differential equations applicable for this model.

2.2.2 Two coils on the secondary side

Next, a model is discussed where the Thomson coil has only one turn but the plate isrepresented by two coils. Since these coils are not the turns of the same coil, their currentsare not forced to be the same as it was so for the previous model (where it was true thatip1 = ip2). Therefore, the number of state variables that are required to describe themodel’s electromagnetic behavior will be higher than in the first, simplest model. Oneextra coil on the secondary side means one more state variable, its current (is2). Themodel is depicted in Figure 2.4.

From here on, let the mutual inductance of the secondary coils be denoted by Msmn

where the mth and nth coils on the secondary side are concerned. Assuming linear mag-netic conditions as in the previous models, the equality Msmn = Msnm holds. Let thesecond coil on the secondary side be assumed as a circular closed loop with radius rs2.Also, it is assumed that the conductor forming this second coil has the radius rcond. With

12


methods already presented in this chapter, it is possible to calculate the second coil’s resis-tance, self inductance and the mutual inductance between it and the primary coil as wellas the mutual inductance between it and the first coil on the secondary side, all denotedby Rs2, Ls2, M12 and Ms12, respectively.

Taken the second coil on the secondary side as short-circuited, the system of differ-ential equations that describes the model can be written as

uc = Rp1ip1 + Lp1dip1

dt+∂M11

∂dvis1 +

∂M12

∂dvis2 +M11

dis1dt

+M12dis2dt

(2.37)


+∂M11

∂dvip1 +M11

dip1

dt+Ms12

dis2dt

(2.38)


+∂M12

∂dvip1 +M12

dip1

dt+Ms12

dis1dt

(2.39)

Cducdt

= −ip1 (2.40)

mdv

dt= F = F11 + F12 (2.41)

dd

dt= v (2.42)

Figure 2.4: Extended Thomson coil model with two-coil secondary side

The force component F11 is due to the currents in the first turns of both sides whilecomponent F12 corresponds to the interaction of currents in the second coil on the sec-ondary and the primary coil.It is worth to mention that some terms have been omitted in the second and the thirdequations that describe the current dynamics on the secondary side. A careful look into

13


the derivation of these equations would reveal that the omitted terms contain the factor∂Ms12

∂d. Since the plate is assumed to be one unit with fixed geometry, it is evident that the

coupling between the secondary coils is constant. So, there will be no impact on currentdynamics and no force contribution from the change of this coupling.The matrix formulation of the equations regarding the current derivatives can be obtainedin the similar way as before.

dip1dtdis1dtdis2dt

=

Lp1 M11 M12

M11 Ls1 Ms12

M12 Ms12 Ls2

−1 uc −Rp1ip1 − ∂M11

∂dvis1 − ∂M12

∂dvis2

−Rs1is1 − ∂M11

∂dvip1

−Rs2is2 − ∂M12

∂dvip1

(2.43)

Again, solving the system of equations is possible with the implemented algorithms inMatlab or Octave.

2.2.3 Generalized model

The models discussed above can give a hint on how a more general representation ofthe actuator is derived and what can be ’expected’ from the matrix formulation of thecurrent derivatives. It can be concluded that adding more turns to the primary coil can berepresented by more terms added to the already existing elements in the matrix of self andmutual inductances. On the other hand, adding more coils to the secondary side resultsin newer state variables (i.e. coil currents), thus there will be newer rows and columnsintroduced to the inductance matrix.

Let the model of the device have P turns on the primary side and S coils on thesecondary side (naturally, P and S are positive integers). Let the jth turn on the primaryside have a radius rpj with the conductor radius rcond. The turn has the current ipj . Theturn’s resistance Rpj and its self inductance Lpj can be computed. All mutual inductancesbetween the turns of the primary side (Mpjk, where 1 ≤ j, k ≤ P , j 6= k and (j, k) ∈ N+)can be determined the same way as described earlier.

Furthermore, let the lth coil on the secondary side have a radius rsl with the con-ductor radius rcond. The coil has the current isl. The coil’s resistance Rsl and its selfinductance Lsl can be calculated. All mutual inductances between the coils of the sec-ondary side (Mslm, where 1 ≤ l,m ≤ S, l 6= m and (l,m) ∈ N+) can be determined aswell.

Finally, the coupling between the turns of two sides can be computed. Using theavailable geometrical data, the mutual inductances between the jth turn of the Thomsoncoil and the lth coil of the plate (Mjl, where 1 ≤ j ≤ P ,1 ≤ l ≤ S and (j, l) ∈ N+) canbe yielded.

With all these parameters and variables, the system of differential equations can be

14


written as

uc =( P∑

j=1

Rpj

)ip1 +

( P∑j=1

Lpj + 2P∑

j=1

P∑k=j+1

Mpjk

)dip1

dt

+S∑l=1

( P∑j=1

∂Mjl

∂d

)visl +

S∑l=1

( P∑j=1

Mjl

)disldt

(2.44)

0 =Rs1is1 + Ls1dis1dt

+( P∑

j=1

∂Mj1

∂d

)vip1

+( P∑

j=1

Mj1

)dip1

dt+

S∑l=2

(Ms1l

disldt

) (2.45)

0 =Rs2is2 + Ls2dis2dt

+( P∑

j=1

∂Mj2

∂d

)vip1

+( P∑

j=1

Mj2

)dip1

dt+

S∑l=1,l 6=2

(Ms2l

disldt

) (2.46)

...

0 =RsSisS + LsSdisSdt

+( P∑

j=1

∂MjS

∂d

)vip1

+( P∑

j=1

MjS

)dip1

dt+

S−1∑l=1

(MsSl

disldt

) (2.47)

Cducdt

= −ip1 (2.48)

mdv

dt= F =

P∑j=1

S∑l=1

Fjl (2.49)

dd

dt= v (2.50)

15


The matrix formulation of the coil current derivatives can be given asdip1dtdis1dtdis2dt...

disSdt

=

∑Pj=1 Lpj + 2

∑Pj=1

∑Pk=j+1Mpjk

∑Pj=1Mj1

∑Pj=1 Mj2 · · ·

∑Pj=1MjS∑P

j=1Mj1 Ls1 Ms12 · · · Ms1S∑Pj=1Mj2 Ms21 Ls2 · · · Ms2S

...∑Pj=1MjS MsS1 MsS2 · · · LsS

−1

·

uc −(∑P

j=1Rpj

)ip1 −

∑Sl=1

(∑Pj=1

∂Mjl

∂d

)visl

−Rs1is1 −(∑P

j=1∂Mj1

∂d

)vip1

−Rs2is2 −(∑P

j=1∂Mj2

∂d

)vip1

...

−RsSisS −(∑P

j=1∂MjS

∂d

)vip1

(2.51)

2.3 Simulation results

Some simulation results are presented here. These will be compared later to results fromfinite element simulations discussed in subsequent chapters.

First, results from the model with only one primary turn and one secondary coil aregiven. The capacitor was assigned the size of 10 mF and was charged to the initial voltageof 300 V. The additional attached mass equaled to 1 kg. The coils had the same radius,5 cm, and the conductors’ radius was 1 mm. Copper has been chosen as the material ofthe conductors, so the electrical resistivity was also known. It can be mentioned that themass of the ’plate’ i.e. the mass of the secondary coil was approximately 8.65 g, so it wasneglected when the mechanical dynamics was determined. The initial distance betweenthe coils was set to 2.1 mm. The parasitic inductance of the conductor connecting theprimary side to the capacitor was assumed to be 1 µH. The conductor’s resistance wasneglected.

The simulation was run in Matlab, using the Euler backward scheme. The timespan of the simulation was 1 ms, with a step of 1 µs. The plotted figures show the ca-pacitor voltage, the velocity of the secondary coil and its displacement with respect to theinitial state, respectively. It is interesting to calculate the energy efficiency of this mod-elled device, given as the ratio of the secondary side’s motional energy at the end of thesimulation and the circuit’s electrical energy in the initial state. Plugging in the resultsfrom Matlab to the equation

η =mv2

Cu2c

(2.52)

gives an energy efficiency of roughly 0.08%. It is often mentioned in the literature (see,

16

2.3. SIMULATION RESULTS

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t [ms]

-50

0

50

100

150

200

250

300

uC

[V

]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t [ms]

0

0.2

0.4

0.6

0.8

1

1.2

v [

m/s

]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t [ms]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

d [

mm

]

Figure 2.5: Capacitor voltage, plate velocity and separation distance, single-turn sides

e.g. [9] or [14]) that the Thomson coil-based actuators tend to be inefficient from anenergetic point of view.

Then, results from the model with several primary turns and secondary coils arepresented. The radii are enumerated in Table 2.1.

These were chosen to represent an ’arbitrary’ set of radii, in order to verify themodel derived above by comparing the results to the output of the finite element methodsolution. The rest of the parameters, namely initial separation, initial capacitor voltage,capacitor size, parasitic elements and added mass were the same as before.

The simulation’s time span was again 1 ms, with the step size of 1 µs. The plottedfigures show the capacitor voltage, the velocity of the secondary side and its displacementwith respect to the initial state, in this order.

17


Primary siderp1 5 cmrp2 4 cmrp3 3 cmrp4 1.5 cm

Secondary siders1 5 cmrs2 4.6 cmrs3 4.2 cmrs4 3.8 cmrs5 3.4 cmrs6 3 cmrs7 2.6 cmrs8 2.2 cmrs9 1.8 cmrs10 1.4 cm

Table 2.1: Radii

18

2.3. SIMULATION RESULTS

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t [ms]

-50

0

50

100

150

200

250

300

uC

[V

]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t [ms]

0

0.5

1

1.5

2

2.5

v [

m/s

]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t [ms]

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

d [

mm

]

Figure 2.6: Capacitor voltage, plate velocity and separation distance, several turns andcoils on both sides

19


20

Chapter 3

Finite element model

3.1 Background

With the computational capacity available even in a personal computer or in a laptop as of2017, it is often possible to run finite element-based simulations that are fine enough yetthey do not require a supercomputer. The main advantage of using finite element methods(FEMs) to simulate various physical processes is that a real device can be approximatedby careful model design with sufficient accuracy, so investing time and money to buildexperimental setups could be limited. If building such a setup is indeed the goal, then theFEM simulations can be used to predict outcomes, to analyze the mode of operation andto investigate different scenarios by changing appropriate model inputs.

It is outside of the scope of this thesis to discuss the applied mathematics behind theimplemented FEMs. The interested reader could find more on this subject e.g. in [21] orin [22]. It suffices to mention that simulations require relevant initial values for variablesand for physical properties to be set and pertinent boundary conditions to be prescribedso the simulation corresponds to physical phenomena intended to be investigated.

The FEM models built during this thesis project can be divided into three majorgroups. These groups share the same ’external circuit’, which lets the stored electric en-ergy to the Thomson coil. On the other hand, the coil and the plate have been modelledseparately in different groups. One group has been developed in parallel with the Thom-son coil-based actuator’s analytical models (implemented in Matlab) with the purpose ofvalidating the results from the FEM-based software and to cross check the correctnessof the analytical models. One group has been developed in order to model a small scaleexperimental setup already available. Since measurements could be made with the setup,the FEM models were used to ’reinforce’ the credibility of the finite element approach.More details on the small scale experimental setup can be found in subsequent chapters.The third group comprises the models that have been used to design prototypes of theactuator and to check measurements of one these prototypes.

The FEM-based software COMSOL has been used for modelling. The advantage

21

CHAPTER 3. FINITE ELEMENT MODEL

of COMSOL compared to some other FEM-based programs is that couplings among dif-ferent physical mechanisms can be modelled. This simulation of multiphysics not onlyallows to establish a model that approximates reality better than a model employing onlya single type of physics (e.g. in case of the Thomson coil-based actuator, modelling theelectromagnetic dynamics without taking the mechanics into account would lead to erro-neous results). It can also warn about risky scenarios that otherwise might be encounteredafter building an actual prototype only (e.g. the coupling of electromagnetics and mechan-ics yields the prediction that the plate is not strong enough to withstand the mechanicalstresses imposed on it by the evoked forces).

All models exploited the fact that both the Thomson coil and the plate that is to berepelled (and the shaft attached to the plate in models used as tools to design the actuator’sprototype) had axisymmetrical geometry. This way, it was not necessary to model thedevice in 3D, but was sufficient to use axisymmetrical 2D models. This reduced therequired time to run a single simulation by a factor of around 7 or 8 (for finite elementmeshes with same resolution).

The electromagnetics have been of the same type in all models. So, the equationsgoverning a time-dependent study of the magnetodynamics as well as the constraints canbe given collectively. These equations, where bold fonts designate vectors, are

∇×H = J (3.1)

B = ∇×A (3.2)

J = σE + σv ×B + Je (3.3)

E = −∂A∂t

(3.4)

where B, H, A and E denote the magnetic flux density, the magnetic field, the mag-netic vector potential and the electric field, respectively. Moreover, J and Je stand forthe current density and external current density in this order, while σ denotes the electri-cal conductivity (in matrix form if the material is anisotropic) and t denotes time. Thedependent variable is A. It can be noticed that COMSOL uses the low-frequency ap-proximation, i.e. Ampere’s law (which is the first equation above) does not contain thedisplacement current component.

Ampere’s law is applicable in every part of the model except for the Thomsoncoil. The coil constitutes the link between the FEM model and the external circuit (alsoimplemented in COMSOL), the latter forcing current through the former. The constraintof magnetic isolation prescribes A’s tangential component to be zero on the boundary ofthe whole modelled space.

The force that repels the plate is yielded by the integration of Maxwell’s stresstensor over the exterior surfaces of the chosen domains. The integrals are evaluated overthe plate (or over its model that consists of several circular conducting loops) and over the

22

3.2. FIRST MODEL GROUP

shaft, if the latter is included in the model. The equation of force calculation is

F =

∫dΩ

2πrnTdS (3.5)

where F denotes force, Ω denotes the surface of concern, n denotes the normal vector andT stands for the torque’s amplitude.

The mechanical dynamics also work the same way for all models, so the equationfor the time-dependent mechanics can be presented here as well. It is

ρ∂2u

∂t2= ∇ · S + Fv (3.6)

where Fv is the applied volume force, S is the second Piola-Kirchhoff stress, ρ denotesthe material’s density and u is the displacement vector. For mechanics, the dependentvariable is u.

As for the initial variable values and constraints, both the displacement vector andits derivative with respect to time have been set to zero in the beginning of the simula-tion. Depending on the models’ major groups, there have been different additional termsincluded. The group that was meant to compare results from COMSOL to those fromMatlab modelled the plus mass added to the plate but did not incorporate gravity, as thisforce was not included in Matlab neither, hence a closer similarity of models could havebeen achieved. The group which aimed to verify FEM results with the small experimentalsetup did not use additional mass on the plate, but included the effect of gravity. The thirdgroup modelled not only gravity and external mass applied to the shaft, but simulated theeffect of the spring mechanism that exerts a certain force on the shaft in order to keep thecircuit breaker reliably close (more details on the actuator’s prototype is presented laterin the chapter).

More in-depth description of what is solved and how it is solved can be found inthe software’s relevant manuals (see, e.g., [23] and [24]).

Next, the models built and their yielded results are described. The comparison ofthese results with those gotten from either the analytical model or the experimental setupis pivotal from the thesis’s standpoint, so it will be detailed in its own, separate chapter.

3.2 First model group

3.2.1 Model with single-turn coils

The first model presented is of the group that was used to check the COMSOL simulationresults in comparison with the analytical model. The same figure depicts the Thomsoncoil, the plate and the external circuit as Figure 2.2. The only difference is that the coiland the plate had been simulated with a FEM (Figure 3.1), while the analytical model

23


used a set of differential equations to compute the forces based on mutual inductances.The blue edges are limiting the space where the actuation is modelled. They impose aconstraint of

n×A = 0 (3.7)

that is, the dependent variable’s tagential component is set to zero on these edges. Duringmodelling, the edges were located far enough from the coil so that they did not influencethe magnetic flux density. They have been ’moved in’ only for illustration.

Figure 3.1: One primary and one secondary coil (first group)

With the same parasitic elements (Rp1ext is neglected, Lp1ext is assumed as 1 µH), sameexact geometries (one primary and one secondary turn with a radius of 5 cm, circular con-ductor with a radius of 1 mm, 2.1 mm initial separation distance) and material (conductorsmade of copper) and identical capacitor size (10 mF) as well as initial capacitor voltage(300 V), the results plotted are the capacitor’s voltage over the simulated time span, thesecond coil’s velocity and its distance with respect to the initial state (Figure 3.2).It can be noted that all COMSOL models simulated at least 2 ms (the reason for this isgiven later). In order to compare relevant results with the analytical model’s output, theacquired data over 1 ms had simply been removed from the plots.

3.2.2 Model with multiple-turn coils

The next model presented corresponds to the Matlab model detailed at the end of Chapter2. The difference of this model and the one discussed just above lies in the geometry ofthe coil model and the plate model (Figure 3.3). Now, the primary side has four turns andthe plate is modelled as ten coaxial circular loops. Their radii are given in Table 3.1.Figure 3.4 displays the capacitor’s voltage, the plate’s velocity and its distance with re-spect to the initial state.

24

3.3. SECOND MODEL GROUP

Primary siderp1 5 cmrp2 4 cmrp3 3 cmrp4 1.5 cm

Secondary siders1 5 cmrs2 4.6 cmrs3 4.2 cmrs4 3.8 cmrs5 3.4 cmrs6 3 cmrs7 2.6 cmrs8 2.2 cmrs9 1.8 cmrs10 1.4 cm

Table 3.1: Radii

3.3 Second model group

The models detailed next were built based on the experimental setup. The coil conductorswere assigned the same size and shape as the real conductors. The capacitor size was 2.2mF and the capacitor had been charged to the initial voltage of 200 V. Most importantly,the plate in the FEM simulations was modelled as a plate, and not as a group of differentcoils. Contrary to the models previously discussed, the parasitic elements of the wiresconnecting the coil to the capacitor had also been taken into account.

3.3.1 Model with aluminum plate

The aluminum plate was modeled as a 10 mm thick cylinder with the inner radius of12.5 mm and the outer radius of 50 mm. The coil was modelled with 12 turns, each ofrectangular cross section with 2 mm width and and 5 mm height. The innermost turn wasplaced as far from the axis of symmetry (in the z-direction) as the plate’s inner radius, andthe consecutive turns followed by leaving a gap of 0.5 mm between two adjacent turns.The stray inductance of the external wires has been estimated as 4 µH and the parasiticresistance as 30 mΩ.The simulated time interval was 3 ms. Figure 3.6 displays the current through the coil andthe plate’s distance with respect to starting state.

25


3.3.2 Model with copper plate

The copper plate’s model was nearly identical to that of the aluminum plate. The onlydifference was that the plate was 8 mm thick (instead of 10 mm). All other parametervalues were the same as presented for the aluminum plate’s model.The simulated time interval was again 2 ms. Figure 3.7 shows the current through the coiland the plate’s distance with respect to starting state, in this order.

3.4 Third model group

3.4.1 Model for designing a prototype

The idea for the ’base case’ for the third group of models has been taken from [13]: theidea was to attach the plate to a shaft that traverses vertically and to model the rest ofthe physical setup by simulating its effect as an additional mass and force (latter from aspring mechanism). The models’ finite element part (i.e. the external circuit excluded)therefore contained only a coil and a moving unit of shaft and plate. The method to meetthe targeted requirements was to first set the external circuit’s relevant parameters (mainlythe capacitor size and its initial voltage) that contributed to the length of separation undera certain time. The method then continued with changing the FEM model’s parameters(such as introducing additional fillets to reinforce the shaft-plate joint or increasing theplate’ thickness) in order to limit the mechanical stresses in the moving part.

As pointed out earlier, SCiBreak is constructing a prototype circuit breaker that isable to create a gap of 6 mm in 2 ms between the contacts (this is the reason why FEMmodels modelled at least 2 ms). The aim of modelling the device in COMSOL was to setthe parameter values in order to yield the outlined separation of the contacts within therequired time frame.

The arrangement of the turns was rather free, as the position of the coil for a givennumber of turns as long as it was with its whole breadth located under the plate, did nothave considerable impact on the shaft’s covered distance.

The optimization of the finite element model was a ’crude’ optimization. Eventhough there are papers on shape optimization of a Thomson coil-based actuator (e.g. [15]and [25]), the mathematical complexity involved had ruled this approach impractical.Hence, the different parameters (e.g. capacitor size, turn arrangement, shaft radius etc.)had been changed in discrete steps in ranges that were determined based on practicalconsiderations.

In order to model the physical device’s parts that were not modelled with FEMdirectly with sufficient margins, the added mass on top of the shaft has been taken as 3,4 and 5 kg. As expected, the largest added mass required the highest voltage level for agiven capacitor size to be moved at least 6 mm in 2 ms. Moreover, the spring that keeps

26

3.4. THIRD MODEL GROUP

the contacts steadily closed during normal conditions (by exerting 2 kN on the shaft)was taken into account by adding a nonphysical spring force which is 2000 N at zeroseparation and linearly reduces to 0 in 7 mm. More on the operating principle of thespring as well as illustrations can be found e.g. in [26]. The capacitor size was set to 10mF, 30 mF and to 50 mF. The restriction for initial capacitor voltage was not to exceed400 V, as above this voltage level additional care should have been taken to ensure theappropriate electrical insulation of the prototype setup’s circuit. The coil’s conductorswere assigned the same shape as for those of the models to validate FEM results withmeasurements (rectangular shape, width and height of 2 and 5 mm, 0.8 mm gap betweentwo adjacent turns).

The outer radius of the plate was decided to be 75 mm (as increasing it to 100 mmwith a step of 5 mm has not yielded any apparent advantages), so after determining theinner radius of the coil, the number of turns could be computed. The inner radius hasbeen chosen so that there is sufficient space for the shaft (whose radius was the output ofanother, mechanical engineering thesis on the actuator) and for the joints of the plate andshaft. Also, a larger inner coil radius means easier manufacturing as the wire does nothave to be bent that much. The number of turns has been computed as 15.

As for the capacitor size, 5 kg of added mass required 400 V of initial capacitorvoltage, when the capacitor size had 10 mF. On the other hand, a capacitor of 50 mF hasnot been completely discharged in 2 ms (i. e. the time frame of the simulation) which isnot optimal since it means that the energy stored in the capacitor was not fully utilized.So, since a 30 mF capacitor allowed to use voltages limited to 260 V, it was decided thatthis size can be used in the prototype.

It has also been concluded that the plate should be at least 8 mm thick, since witha thickness of 3-4 mm the predicted mechanical stresses were beyond acceptable limitsboth for aluminum and for copper plates (around 1.8-2 GPa).

The material of the plate has been chosen based on the FEM models, too. It waspredicted that a device with aluminum plate is capable of creating a slightly larger sepa-ration distance in 2 ms than that with a copper plate.

The cross section of the model is displayed in Figure 3.8. It can be noticed thatthe joints have been modelled as the extension of the plate, i. e. the detailed mechanicalstructure (e.g. bolts) has been simplified.

The capacitor voltage and the current through the device are plotted on Figure 3.9.The shaft’s velocity as well as its covered distance of separation are plotted on Figure3.10.

3.4.2 Model to check a built prototype

This model does not belong to the second group because it has been yielded by transform-ing the previously described model of the third group. The aim of this transformation was

27


to fit the model to the built prototype (more details about the prototype can be found inChapter 4). Figure 3.11 shows the modelled coil, plate and shaft.

Geometrical parameter values (shaft radius, plate thickness, plate’s inner and outerradii, number of coil turns, coil conductor dimensions etc.) and materials used in thesimulations have been set according to the prototype. As for the electrical parameters, thecapacitor size was known (30 mF) and the parasitic resistance and inductance of the wiresconnecting the coil to the capacitor were assumed to have the same values as in case ofthe previous model.

The capacitor’s initial voltage have been set to 150, 200 and 250 V for differentsimulations. The separation distance between the coil and the plate as well as the coilcurrent are plotted in Figures 3.12 - 3.14 for these cases. It is interesting to see the forcesacting on the coil’s turns, so Figures 3.15 - 3.17 display the radial and axial forces forthe three cases (averaged over the different turns of the coil). The axial forces have beennegative by default, but they are plotted as positive. The simulated time span was 4 ms.

28


Figure 3.2: Capacitor voltage, plate velocity, separation distance, single-turn coils (firstgroup)

29


Figure 3.3: Four primary turns and ten secondary coils (first group)

30


Figure 3.4: Capacitor voltage, plate velocity, separation distance, multiple-turn coils (firstgroup)

31


Figure 3.5: Experimental setup’s model with aluminum plate (second group)

Figure 3.6: Current through the coil with aluminum plate, plate velocity, aluminum plate(second group)

32


Figure 3.7: Current through the coil with copper plate, plate velocity, copper plate (secondgroup)

Figure 3.8: Large scale setup’s model with aluminum plate (third group)

33


Figure 3.9: Capacitor voltage, current through the coil (third group)

34


Figure 3.10: Shaft velocity, covered distance (third group)

Figure 3.11: Built prototype’s model (third group)

35


Figure 3.12: Current through the coil and separation distance for built prototype’s model,150 V (third group)

36



37



38


Figure 3.15: Axial and radial forces acting on the coil for built prototype’s model, 150 V(third group)

39



40



41


42

Chapter 4

Experimental setups

4.1 Smaller experimental setup

The small scale experimental setup has been used to measure relevant parameters (i.e.current through the coil, plate’s separation distance) and to compare these to the outputsof the FEM simulations. Since the experimental setup’s details were known (e.g. thick-ness of the plate, number of turns, capacitor size and initial capacitor voltage etc.), thesimulations’ parameters have been set accordingly. Therefore, the results could be inter-preted as the indicators of how reliable the COMSOL models are (in this specific case).It should be noted that while the current through the coil has been measured directly, thedistance measurement employed an ultrasound device, and the position of the plate hasbeen extracted from recorded raw data by means of polynomial curve fitting, in this case,with linear curve fitting. So, the measured distance is also an approximation to someextent.

The setup with the aluminum plate is displayed in Figure 4.1.The measured distance for the device with the aluminum plate as well as the dis-

charged current have been plotted (Figure 4.2).The same parameters are plotted for the device with a copper plate as well ( Figure

4.3).

4.2 First prototype

Because of the numerous parts required for the prototype had been ordered from variousmanufacturers, the deliveries did not arrive at the same time. Since it took several weeksfor several parts to arrive, it was decided that an initial prototype shall be constructed.The missing parts (e.g. the plates on which the coil, the spring mechanism etc. were tobe mounted) had been built using other materials, and there were extra pieces ordered forthe different components, so these allowed more than one prototype to be constructed at

43

CHAPTER 4. EXPERIMENTAL SETUPS

Figure 4.1: Small scale experimental setup

the same time. The assembled prototype is depicted in Figure 4.4 (at this phase withoutthe current and distance measuring devices, which would be added on top of the topmostwooden plate, thus they could not have been shown anyway).

Several tests have been run with this prototype. The current through the coil and theseparation distance between the coil and the plate have been measured for different initialcapacitor voltages, and a few measurements are plotted (Figures 4.5 - 4.7). It can be seenthat the current measurements contain strange, square wavelike ripples, but those are dueto the instrument that recorded the measurements. When only current measurements wereconducted (i.e. without distance measurement), the measured currents were ripple-free.The plotted distances have been yielded by smoothing the raw measured distance signalsby a third-order Savitzky-Golay filter, available in Matlab (more on distance measurementcan be found in Appendix A).

4.3 Targeted prototype

As of the submission of this thesis, although the components have arrived for this pro-totype, it would take one or two workdays more to assemble the setup and to run tests.Certain parameter values (material and thickness of the plate) that have been determinedby COMSOL served as base to order components. Other components have been designedeither by a student doing a degree project at SCiBreak in the field of mechanical engi-neering or by SCiBreak. Figure 4.8 shows the spring housing and the plate and Figure 4.9displays the coil (cast in epoxy).

44

4.3. TARGETED PROTOTYPE

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time [ms]

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Dis

tance [m

m]

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time [ms]

500

1000

1500

2000

Cu

rre

nt

[A]

Figure 4.2: Approximated distance and measured current, small scale setup with alu-minum plate

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time [ms]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Dis

tan

ce

[m

m]

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time [ms]

0

500

1000

1500

2000

Cu

rre

nt

[A]

Figure 4.3: Approximated distance and measured current, small scale setup with copperplate

45


Figure 4.4: First prototype

46


0 0.5 1 1.5 2 2.5 3 3.5 4

t [ms]

-1

0

1

2

3

4

5

6

d [

mm

]

-1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5

t [ms]

0

500

1000

1500

2000

2500

3000

3500

i [A

]

Figure 4.5: Approximated distance and measured current, built prototype, 150 V

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

t [ms]

-2

0

2

4

6

8

10

12

d [

mm

]

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4

t [ms]

0

1000

2000

3000

4000

5000

i [A

]


47


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

t [ms]

-2

0

2

4

6

8

10

12

14

16

d [m

m]

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

t [ms]

0

1000

2000

3000

4000

5000

6000

i [A

]


48


Figure 4.8: Targeted prototype’s spring housing

49


Figure 4.9: Targeted prototype’s coil

50

Chapter 5

Comparison of results

This chapter analyzes the results of simulations and measurements from the precedingchapters and compares them.

5.1 Comparison of Matlab and COMSOL results

The first figure (Figure 5.1) displays the capacitor voltage (uC), separation of primaryand secondary coils (d) and the secondary coil’s velocity (v) yielded by COMSOL andMatlab. The Thomson coil has been modeled as a single-turn coil on the primary side andthe plate has been modeled as a single-turn coil on the secondary side. Data regardingthe largest deviation is given in Table 5.1. The last column shows the error relative to thelargest simulated value during the simulated interval.

Figure 5.2 displays again the same parameters (uC, d and v), in this case for themodel with a four-turn primary side and ten single-turn secondary side coils. The largestdeviation of the Matlab and COMSOL models are analyzed in Table 5.2.

It can be seen that the models give quite similar results (the relative errors at thelargest discrepancies tend to be less than 20%). This can be exploited when running alarge number of simulations as the implemented Matlab code completes one simulationin a few seconds while COMSOL takes around 1-2 minutes for completing a simulation(in this specific case). On the other hand, Matlab cannot include multiphysics.

5.2 Comparison of the small scale setup’s measurementsand COMSOL

Figure 5.3 shows the measured and simulated coil currents as well as the measured andsimulated separation distances (between coil and plate) for the small scale experimentalsetup with the aluminum plate.

The discrepancy between the instants and magnitudes of the coil current peaks are

51

CHAPTER 5. COMPARISON OF RESULTS

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t [ms]

-50

0

50

100

150

200

250

300

uC

[V

]

COMSOL

Matlab

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t [ms]

0

0.2

0.4

0.6

0.8

1

d [m

m]

COMSOL

Matlab

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t [ms]

0

0.2

0.4

0.6

0.8

1

1.2

1.4

v [

ms]

COMSOL

Matlab

Figure 5.1: Capacitor voltage, separation distance and plate velocity, single-turn coils

given in Table 5.3 and the error of distances at the end of the analyzed time span is givenin Table 5.4 (error relative to the measurement).

Figure 5.4 displays the same exact parameters as above, but the experimental setupused a copper plate in this case.

Tables 5.5 and 5.6 describes the same discrepancies between measurement and sim-ulation as in the previous case.

It can be observed that the simulated coil currents were close to the measured onesin both cases. The discrepancies between measured and simulated separation distances arenot negligible (20-30%), but the simulations gave the correct prediction that the aluminumplate would cover approximately twice the distance as the copper plate in 2 ms.

One reason why simulations gave larger separation distances than measurementscan be that the wooden shaft of the experimental setup (see Chapter 4) was not perfectly

52

5.3. COMPARISON OF THE FIRST PROTOTYPE’S MEASUREMENTS ANDCOMSOL

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t [ms]

-50

0

50

100

150

200

250

300

uC

[V

]

COMSOL

Matlab

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t [ms]

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

d [

mm

]

COMSOL

Matlab

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t [ms]

0

0.5

1

1.5

2

2.5

v [

m/s

]

COMSOL

Matlab

Figure 5.2: Capacitor voltage, separation distance and plate velocity, multiple-turn coils

aligned vertically. Thus, some friction between plate and shaft was likely to prevent theformer to travel as far as it could in an ideal case. On the other hand, air drag is not likelyto make a difference between simulation and measurement as the surface of the plate issmall and its velocity is low. A FEM simulation confirmed that air drag can be neglected.

5.3 Comparison of the first prototype’s measurements andCOMSOL

Three sets of comparisons are made in this section, based on the setups with three differentinitial capacitor voltage levels. The measured coil currents and separation distances arecompared to those yielded by simulations.

Figure 5.5, Figure 5.6 and Figure 5.7 display the mentioned currents and distances

53


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

t [ms]

0

500

1000

1500

2000

2500

i [A

]

COMSOL

Measured

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

t [ms]

0

0.5

1

1.5

2

2.5

d [m

m]

COMSOL

Measured

Figure 5.3: Coil currents and separation distances, aluminum plate

for the setups with the initial capacitor voltages of 150 V, 200 V and 250 V, respectively.In these figures, it is apparent that the discrepancies between simulation and mea-

surement are large. As for the currents, the mismatch could be explained by the assump-tion of incorrect parasitic resistance and inductance values in the circuit. On the otherhand, increasing the stray inductance delays the current peak, which comes later than themeasured current peak already in the presented comparisons. An increase of the parasiticresistance (from 30 mΩ up to 80 mΩ) did not make considerable differences.

There can be several factors that rendered the simulated separation distances highlyinaccurate with regard to the measurements. Simulations do not take into account thefrictional forces between the shaft of the model and the structural plates (holding the coil,the spring mechanism etc.). These forces are slowing down the shaft and thus the plate.Another reason can be that the shaft of the first prototype was made of aluminum and itwas hollow (i.e. like a cylinder’s outer part). The axial forces that move the plate awayfrom the Thomson coil might have elongated the shaft due to its elasticity.

The distance during test could also have been measured incorrectly. Since the struc-tural plates have been built of wood (instead of PVC, the material of the ordered structuralplates), it could have happened that the forces pushed the plate (on which the coil is in-stalled) in the other direction. This way, the plate could have been bent instead of theThomson coil being repelled. Another source of error is the way in which the distance ismeasured. An additional PVC plate had been mounted on the shaft so that a sensor couldmeasure the distance based on this plate. Since it might not have been rigid enough, the

54


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

t [ms]

0

500

1000

1500

2000

2500

i [A

]

COMSOL

Measured

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

t [ms]

0

0.2

0.4

0.6

0.8

1

d [

mm

]

COMSOL

Measured

Figure 5.4: Coil currents and separation distances, copper plate

initial acceleration could have bent it in a way that its outer rims did not move in the firstmillisecond of the experiment. The distance sensor had been measuring at these outerrims.

The last, possible reason could explain why there is a delay of the action of moving,when measurements are compared to simulations. It might be that the sensor did not ’see’the plate moving.

55


0 0.5 1 1.5 2 2.5 3 3.5

t [ms]

0

1000

2000

3000

4000

5000

6000

i [A

]

COMSOL

Measured

0 0.5 1 1.5 2 2.5 3 3.5 4

t [ms]

-5

0

5

10

15

20

25

d [m

m]

COMSOL

Measured

Figure 5.5: Coil currents and separation distances, 150 V of initial voltage

Instant [ms] Absolute value Relative value [%]uC 0.179 23.2089 V 7.7363d 1 0.1796 mm 18.4733v 0.198 0.3874 m/s 30.9638

Table 5.1: Largest error, single-turn coils

Instant [ms] Absolute value Relative value [%]uC 0.213 30.9471 V 10.3157d 1 0.2246 mm 12.7835v 0.724 0.4108 m/s 18.6463

Table 5.2: Largest error, multiple turns

Instant [ms] Absolute value [A] Relative error [%]COMSOL 0.148 2312.2 A 1.9893

Measurement 0.083 2267.1 A 0

Table 5.3: Current peaks, aluminum plate

56


Absolute value [mm] Relative error [%]COMSOL 2.1987 33.9609

Measurement 1.6413 0

Table 5.4: Separation distance discrepancy, aluminum plate

Instant [ms] Absolute value [A] Relative error [%]COMSOL 0.146 2360.7 A -1.5473

Measurement 0.078 2397.8 A 0

Table 5.5: Current peaks, copper plate

Absolute value [mm] Relative error [%]COMSOL 0.9341 20.2188

Measurement 0.777 0

Table 5.6: Separation distance discrepancy, copper plate

0 0.5 1 1.5 2 2.5 3 3.5 4

t [ms]

0

1000

2000

3000

4000

5000

6000

7000

8000

i [A

]

COMSOL

Measured

0 0.5 1 1.5 2 2.5 3 3.5 4

t [ms]

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20

30

40

50

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m]

COMSOL

Measured


57


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t [ms]

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i [A

]

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Measured

0 0.5 1 1.5 2 2.5 3 3.5 4

t [ms]

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80

d [

mm

]

COMSOL

Measured


58

Chapter 6

Conclusion

6.1 Conclusions

As detailed in Chapters 2 and 3, both analytical and finite element methods have been usedto model a Thomson coil based actuator and the its feeding electrical circuit. While it isevident that a finite element method that allows for multiphysics to be used can modela complex device with higher accuracy than an analytical model with its unavoidableapproximations, the latter is considerably (around two orders of magnitude) faster than theformer (in the investigated cases). This can be exploited when the approximation of theanalytical model is sufficient and there is a need for running large number of simulations.

Chapter 4 discusses two different experimental setups that have been used for mea-surement purposes. The measured parameters are also detailed. Moreover, another proto-type (under construction as of the submission of this thesis) has also been mentioned.

As analyzed in Chapter 5, the analytical approach and FEM give reasonably closeresults for the discussed Thomson coil models. The validity of the finite element modelsare also supported by measurements of the small scale experimental setups. Buildinga first prototype of the actuator let even more comparisons be possible. Although thediscrepancies of the measured and simulated parameter values warn that a finite elementmodel cannot (and should not) be relied upon unconditionally, the process of building andtesting the prototype let valuable practical knowledge and skills to be acquired. These willcertainly be useful for building more advanced prototypes.

6.2 Future work

The most evident future work related to the present thesis is the construction and testingof a more elaborate prototype. These tests will allow for more comparisons to be made,thus hopefully enabling the FEM models’ fine tuning.

On the longer run, there are plenty of work and research to be carried out related to

59

CHAPTER 6. CONCLUSION

fast hybrid circuit breakers. This thesis analyzed the actuator, but that is only one piecein a circuit breaker. For instance, the rest of the system has been modeled as extra massadded to the actuator. In reality, these parts have to be carefully designed as well in orderto the whole device could give the desired performance.

60

Appendix A

Distance measurement

The distance measurement device employes an ultrasound source and receiver. The soundsignal has to travel from the source to the receiver by getting reflected from the objectwhose distance is to be measured. There is a phase shift between the transmitted andthe corresponding received signals. As the object moves, this phase shift changes. Theprinciple is that as the source’s frequency is known, by assuming a certain speed of sound,the distance required for full period (360) of change in the phase shift can be yielded.

In the discussed experimental setups, the transmitter used a square wave sound sig-nal of roughly 41 kHz. Denoting this by fs and the wavelength by λ, with the assumptionof csound = 340m

sas the sound’s speed in air, it can be written that

λ =csoundfs

= 8.293mm (A.1)

That is, the object causes 360 change in the phase shift while it travels approximately4.15 mm (since the sound wave has to travel forth and back).

While SCiBreak already had one functioning distance measurement device, it wasfixed to the first experimental setup. (It can even be spotted on Figure 4.1.) It has beendecided that another, similar device should be built. The original circuit was modified,and the resulted schematics is plotted on Figure A.1. Although there are several simpli-fications in Figure A.1 compared to the original schematics, it can be seen that the inputsignal is generated by a tunable oscillator. Then, the received signal is amplified andby comparing both the received and transmitted signals, the phase shift of those can bedetermined.

A circuit had been built during the degree project in order to implement the schemat-ics of Figure A.1 and it is plotted in Fig A.2

The instrument recording and routing these sound signals could at most allocatetwenty samples per period for each signal. This means that a theoretical precision of

360

20· 4.15mm

360= 0.21mm (A.2)

61

APPENDIX A. DISTANCE MEASUREMENT

Figure A.1: Schematics of distance measurement device’s electrical circuit

can be achieved for measuring travelled distances.It can also be noted that due this instrument and the square wave signals from the

distance measurement device, the current measurements contained unreal ripples (as seenin e.g. Figure 4.5)

62

Figure A.2: The distance measurement device’s implemented circuit

63

APPENDIX A. DISTANCE MEASUREMENT

64

References

[1] Cigre Working Group B4.52, “HVDC Grid Feasibility Study,” tech. rep., 2013.

[2] G. Tang, Z. He, H. Pang, X. Huang, X.-P. Zhang, “Basic Topology and Key Devicesof the Five-Terminal DC Grid,” CSEE Journal of Power and Energy Systems, vol. 1,June 2015.

[3] H. Rao, “Architecture of Nan’ao Multi-terminal VSC-HVDC System and Its Multi-functional Control,” CSEE Journal of Power and Energy Systems, vol. 1, March2015.

[4] W. Wang, G. Wang, M. Andersson, “Developement in UHVDC Multi-Terminal andVSC Grid,” in International High Voltage Direct Current Conference, October 2016.

[5] C. M. Frank, “HVDC Circuit Breakers: A Review Identifying Future ResearchNeeds,” IEEE Transactions on Power Delivery, vol. 26, April 2011.

[6] N. Flourentzou, V.G. Agelidis, G. D. Demetriades, “VSC-Based HVDC PowerTransmission Systems: An Overview,” IEEE Transactions on Power Electronics,vol. 24, March 2009.

[7] M. P. Bahrman, B. K. Johnson, “The ABCs of HVDC Transmission Technologies,”IEEE power and energy magazine, March/April 2007.

[8] J. Hafner, B. Jacobson, “Proactive Hybrid HVDC Breakers - A key innovation forreliable HVDC grids,” in The electric power system of the future - Integrating su-pergrids and microgrids International Symposium, September 2011.

[9] A. Bissal, J. Magnusson, G. Engdahl, “Electric to Mechanical Energy Conversionof Linear Ultrafast Electromechanical Actuators Based on Stroke Requirements,”IEEE Transactions on Industrial Applications, vol. 51, July/August 2015.

[10] J. Magnusson, A. Bissal, G. Engdahl, J. A. Martinez-Velasco, “Design Aspects ofa Medium Voltage Hybrid DC Breaker,” in 5th IEEE PES Innovative Smart GridTechnologies Europe, October 2014.

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References

[11] C. Peng, I. Husain, A. Huang, B. Lequesne, R. Briggs, “A Fast Mechanical Switchfor Medium Voltage Hybrid DC and AC Circuit Breakers,” in IEEE Energy Conver-sion Congress and Exposition, October 2015.

[12] C. Peng, X. Song, A. Q. Huang, I. Husain, “A Medium Voltage Hybrid DC CircuitBreaker, Part II: Ultra-fast Mechanical Switch,” IEEE Transactions on IndustrialApplications, vol. 5, no. 1, 2015.

[13] C. Peng, I. Husain, A. Huang, B. Lequesne, R. Briggs, “Drive Circuits for Ultra-fast and Reliable Actuation of Thomson Coil Actuators used in Hybrid AC and DCCircuit Breakers,” in IEEE Applied Power Electronics Conference and Exposition,2016.

[14] D. S. Vilchis-Rodriguez, R. Shuttleworth, M. Barnes, “Finite element analysis andefficiency improvement of the Thomson coil actuator,” in 8th IET International Con-ference on Power Electronics, Machines and Drives, 2016.

[15] W. Li, Z. Y. Ren, Y. W. Jeong, C. S. Koh, “Optimal Shape Design of a Thomson-coilActuator Utilising Generalized Topology Optimization Based on Equivalent CircuitMethod,” IEEE Transaction on Magnetics, vol. 47, no. 5, 2011.

[16] L. Angquist, “Simulation of Thomson coil,” 2016.

[17] B. Roodenburg, B. H. Evenblij, “Design of a fast linear drive for (hybrid) circuitbreakers - Development and validation of a multi domain simulation environment,”Elsevier, Mechatronics, vol. 18, April 2008.

[18] E. B. Rosa, L. Cohen, “The mutual inductance of two circular coaxial coils of rect-angular section.,” Bulletin of the Bureau of Standards, vol. 2, no. 3, 1906.

[19] T. H. Fawzi, P. E. Burke, “The accurate computation of self and mutual inductancesof circular coils,” IEEE Transactions on Power Apparatus and Systems, vol. PAS-97,March/April 1978.

[20] Erik Hallen, Elektricitetslara. Almqvist & Wiksell, 1953.

[21] O.W. Eshbach, Eshbach’s Handbook of Engineering Fundamentals. John Wiley &Sons’ Inc., 5th ed., 2009.

[22] G. Strang, G. Fix, An Analysis of the Finite Element Method. Wellesley-CambridgePress, 2nd ed., 2008.

[23] COMSOL AC/DC Module User’s Guide, Version 5.2a, 2016.

[24] COMSOL Structural Mechanics Module User’s Guide, Version 5.2a, 2016.

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[25] M. T. Pham, Z. Ren, W. Li, C. S. Koh, “Optimal design of a Thomson-coil Actu-ator Utilizing a Mixed-Integer-Discrete-Continuous Variables Global OptimizationAlgorithm,” IEEE Transaction on Magnetics, vol. 47, October 2011.

[26] M. Al-Dweikat, Z. Zhang, T. Cheng, S. Gao, W. Liu, “”Research on OperatingMechanism for Ultra-fast 40.5 kV Vacuum Switches,” IEEE Transactions on PowerDelivery, vol. 30, December 2015.

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