On the Design of Ultra-fast Electro-Mechanical Actuators

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On the Design of Ultra-Fast Electro-Mechanical

Actuators

ARA BISSAL

Licentiate Thesis

Stockholm, Sweden 2013

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ISSN 1653-5146

ISBN 978-91-7501-713-6

SE-100 44 Stockholm SWEDEN Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framlägges till offentlig granskning för avläggande av teknologie licentiatexamen onsdagen den 8 maj 2013 klockan 10.00 i F3, Kungl Tekniska högskolan, Lindstedtsvägen 26, Stockholm.

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Abstract

The continuously increasing demand for connecting electric grids with remote renewable energy sources such as wind power and photovoltaic cells has rekindled interest in high voltage direct current (HVDC) multi-terminal networks. Although HVDC networks have numerous benefits, their adoption relies entirely on the availability of HVDC circuit breakers which, compared to traditional alternating current circuit breakers, have to operate in a time frame of milliseconds.

This thesis deals with the design of ultra-fast electro-mechanical actua-tors based on the so-called Thomson coil (TC) actuator. The simulation of a (TC) actuator constitutes a multi-physical problem where electromagnetic, thermal, and mechanical aspects must be considered. Moreover, it is complex since all those variables are co-dependent and have to be solved for simulta-neously. As a result, a multi-physics simulation model that can predict the behavior and performance of such actuators with a high degree of accuracy was developed.

Furthermore, other actuator concepts were also investigated and modeled in light of searching for a drive with a superior efficiency. The theory behind the force generation principles of two different types of ultra-fast electrome-chanical actuators, the TC and the double sided coil (DSC), were compared by the use of static, frequency, and comprehensive transient multi-physics finite element simulation models.

Although, simulation models serve as a powerful tool for modeling and designing such state of the art actuators, without validation, they are weak and prone to errors since they rely on approximations and simplifications that might not always hold. Therefore, a prototype was built in the laboratory and the model was validated experimentally.

Finally, it is important to note that the drives in this thesis are intended to actuate metallic contacts. As such, their behavior and performance upon mechanical loading was studied. Furthermore, some scaling techniques were applied to boost their performance and efficiency.

Keywords: Electro-mechanical drive, Circuit breakers, HVDC transmission, Eddy currents, Finite element, Electromagnetic, Thermal, Mechanical, Coils, Armature, Image motion analysis.

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Acknowledgements

Firstly, I would like to express my gratitude to Prof. Göran Engdahl for his guid-ance, innovative ideas, and numerous comments.

A lot of this thesis work was made possible due to the collaborative and fruitful working atmosphere at ABB AB Corporate research. Therefore, I would like to thank Dr. Mikael Dahlgren and Magnus Backman for employing me and giving me the opportunity to work with Dr. Thomas Eriksson and Dr. Ener Salinas. Their advice and guidance was very beneficial and they have helped me quite much especially when it came to the building of some of the prototypes.

I would also like to thank my friend Jesper Magnusson, with whom I always end up discussing interesting ideas.

Furthermore, I would like to thank Dr. Henrick Breder, and Dr. Lars Liljestrand at ABB for answering a lot of my questions.

Additionally, I would like to express my gratitude to Patricia for her sweetness and kind support and especially my friends, Samer Shisha, Andreas Krings, Shuang Zhao, and Antonios Antonopoulos whom I have visited their offices numerous times for help and guidance.

I would also like to give my special thanks to my sister Jessy for her nice jokes and kind heart.

As for my mother, there are simply no words to thank her enough for all the things I have put her through and for that reason I dedicate this thesis in her name. Thank you very much. I would have never made it so far without you. I hope I make you as proud as I am for having such a great mum like you.

Ara Bissal

Stockholm, May 2013

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Introduction

1.1 Background

After the adoption of the Kyoto Protocol in 1997, the industrialized countries were bound to reduce greenhouse gases. As a result, a lot of incentives for investing in green power emerged. Moreover, the development of more efficient wind turbines and solar panels have made these renewable technologies more attractive. Unfor-tunately, very often, many of these renewable energy based generation plants are located in remote areas. One main issue lies in integrating offshore wind power [16, 24, 25] and solar thermal generation built in desert terrains to the electric grid with minimum transmission losses.

The continuously increasing demand for connecting the grids with those re-mote renewable energy sources has rekindled interest in high voltage direct current (HVDC) multi-terminal networks. One of the major sources of large-scale renewable energy sources in Europe, is offshore wind power. Recently, ABB won a 1 billion dollar contract to connect large wind farms to the German grid. This project will avoid more than 3 million tons of carbon dioxide emissions and upon its completion in 2015, it will deliver clean renewable electric power via an HVDC link to more than 1.5 million households [1].

Why HVDC?

Although consumers receive electric power in the form of alternating voltage (AC), it is a must to resort to HVDC for long cable links since it is one of the viable solutions at hand with the existing technology. Cables can be seen as distributed capacitors that constantly need charging under ac voltage. At some critical length, this charge current will be equal to the maximum transmission capacity of the cable hindering power transmission.

The advantages of HVDC are not limited only to long cables. Some of the other advantages compared to AC systems are the following [42]

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• Lower losses

• Asynchronous interconnections • Lower environmental impact • Lower investment costs • Added controllability

Limitations of the HVDC network

Although HVDC networks have numerous benefits, their adoption relies entirely on the availability of HVDC circuit breakers [10, 14]. HVDC circuit breakers have much more stringent requirements than conventional AC circuit breakers rendering them a key technology for the new emerging multi-terminal HVDC network. Due to the absence of a natural current zero crossing as in AC systems and due to the low inductive nature of the network, these circuit breakers must be able to interrupt fault currents very quickly before the current increases in magnitude. As a result, it is critical to have ultra-fast opening times in direct current (DC) circuit breakers compared to AC breakers.

Currently, three types of (DC) circuit breakers exist: mechanical, power elec-tronic, and hybrid breakers all suitable only for low and medium voltage applica-tions. The state of the art in 2009 for mechanical breakers is an operation time of 60 ms and that of a power electronic breaker is 1 µs [18]. As for the world’s first state of the art hybrid HVDC breaker released by ABB in 2012, it has an operation time of 5 ms [11].

Limitation of DC breakers

DC breakers face a challenge when a short circuit occurs due to the low impedance in an HVDC network [11]. Large currents complicate the design of the breakers because these are harder to interrupt. Moreover, all other equipments have to be rated to withstand enormous fault currents [15]. Equipments rated for large fault currents become bulky and expensive and in some cases, an extra cooling system must be designed to dissipate the excess heat. These cooling systems require further monitoring and maintenance.

Mechanical breakers are cheap and have low on state losses but are relatively slow in operation time. On the other hand, although solid state breakers have a fast operation time that is in the order of micro seconds, they are expensive and suffer from high on state losses. Nowadays, if state of the art mechanical breakers are used in an HVDC system, the grid owners usually compensate for the extra operational time by inserting an extra inductance in the system to limit the rate of rise of fault currents so that they do not have to overrate all other equipments to excessively large fault currents. This however is not an optimum solution since it introduces losses and voltage fluctuations. The main aim of this PhD thesis work

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1.2. PURPOSE OF THE THESIS 3

is to develop promising designs of ultra-fast actuators that can be used to actuate contact systems to reduce the opening time of appropriate current switches from the range of hundreds of milliseconds down to the order of hundreds of microseconds.

1.2 Purpose of the thesis

This thesis is focuses on using the Thomson coil (TC) based principle to simu-late and develop ultra-fast actuators. The simulation of a TC actuator constitutes a multi-physical problem where electromagnetic, thermal, and mechanical aspects must be considered. Moreover, it is complex since all involved variables are co-dependent and have to be solved for simultaneously. As a result, one of the main goals of this thesis is to develop a multi-disciplinary simulation model that can pre-dict the behavior and performance of such actuators with a high degree of accuracy. Another goal is to build a prototype to validate the model experimentally.

A further objective is to design and build a flexible test bench, primarily to study the parameters influencing the generation of those high impulse forces and secondly, to identify and study critical materials and key components influencing the force transmission.

Finally, based on the acquired knowledge, different configurations are to be presented with suggestions of possible improvements.

1.3 Outline of the thesis

Chapter 1:

This chapter introduces TC based actuators and issues that are treated in this thesis.

Chapter 2:

This chapter provides a background of what has been done and the state of the art of TC based actuators. It also presents a brief introduction to the topic and a description of the actuators used in this thesis.

Chapter 3:

In this chapter, a multi-physics FEM model is derived and presented. The use of such a complicated model is justified by providing test cases that treat such actuators with and without involving thermal and mechanical aspects.

Chapter 4:

In this chapter, to add credibility, the developed multi-physics sim-ulation model is experimentally validated.

Chapter 5:

This chapter highlights the influence of the mechanical loading of the actuator performance. Scaling techniques are presented to improve performance and efficiency.

Chapter 6:

This chapter concludes the thesis, the results are summarized, and future work is suggested.

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1.4 Contributions

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

• A comprehensive description of the physics involved when generating ultra-fast mechanical impulses by TC like actuators.

• A description of the nature of the involved nonlinear transients.

• A development and experimental verification of a multi-physics finite element simulation model of TC actuators.

• A description of the behavior of such actuators when loaded mechanically. • Scaling techniques to improve actuator efficiency and performance.

1.5 Publications

The work presented in this thesis has resulted in the international conference papers listed below:

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

• A. Bissal, J. Magnusson, E. Salinas, G. Engdahl, A. Eriksson, "On the Design 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 scaling as-pects of Thomson based ultra-fast actuators," in 13th Internation Conference

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Chapter 2

Ultra fast actuators

This section presents the state of the art, explains the principle of generating an ultra fast impulsive force, and provides insight into the geometry and operating mechanisms of the TC and the double sided coil (DSC) actuator concepts.

2.1 State of the art

Traditionally, switching devices such as circuit breakers and disconnecters are com-posed of springs and gears. Recently, some of them have been replaced by perma-nent magnet based actuators due to their advantages of being maintenance free and highly reliable with applications in vacuum and gas circuit breakers [37, 38, 7, 8]. They exist in many forms such as the helix coil launcher, the reconnection gun, the coil gun etc... Recently, ultra-fast actuators are being studied more and more due to their diverse applications in robotics, automotive, steel punching, and switch-ing devices. New research is performed on repulsive actuators based on Thomson drives.

2.1.1 Applications

Due to the modern needs of power networks, there is a greater need for high-performance current limiting or interrupting devices. To achieve these ultra-fast switching requirements, Thomson drives are being used to develop new circuit breakers. For example, in [26], a Thomson drive is used in a hybrid DC circuit breaker consisting of a mechanical switch, two power IGCTs connected in paral-lel, diodes, and a metal-oxide varistor for dissipating the energy. In [40], a detailed description of a high speed single phase Mitsubishi circuit breaker is given. A Thom-son drive incorporating a closing coil, an opening coil, and a repulsion plate is used. Moreover, the vacuum interrupter, the moving contacts, and the bellows are shown. This circuit breaker has an opening time of 1 ms and a breaking time of around 20 ms. Yet another example is shown in [41], where a high voltage circuit breaker

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is described based on an electromagnetic repulsion drive and a permanent magnet spring. This drive incorporates two fixed coils and a movable coil in between. The current pulse is fed to the top and movable coils for an opening operation and to the bottom and movable coils for a closing operation. Additionally, in [9], a DC hybrid current limiting breaker is shown. This breaker consists of a Thomson drive and a fast switch. The contact in this circuit breaker opens in 200 µs and can attain a speed of 10 m s−1_{. Furthermore, in [39], a conducting aluminum ring is used to} bridge two current carrying contacts. To operate this repulsion drive, a current pulse is injected in the driving coil causing a repulsive force. The armature reaches speeds of 20 m s−1 _{and contact separation is achieved in as short as 100 µs. A drive} with a slightly larger opening time is shown in [17], where a high speed switch that is capable of opening within 1 ms was developed and integrated into a hybrid drive. Moreover, in [36], a repulsion drive is used to investigate the interruption capability of a DC current by injecting a high frequency counter current. Other works done with these drives are shown in [34, 5, 4], where the electromagnetic drive mecha-nism of a fast acting circuit breaker is examined, a high speed repulsion actuator is analyzed and compared with a a permanent magnet actuator, and the effect of two consecutive discharges through a Thomson drive are shown respectively. Lastly, in [22], a TC is implemented in an arc eliminator.

Thomson drives have not been limited to linear actuation. Rotational uses of Thomson drives have also been implement as for example in [12], where a rotational repulsion drive is shown that has an extremely short breaking time (within a few 100 µs). Although this drive mechanism can achieve velocities up to 50 m s−1_{, it} suffers from an efficiency of less than 5 %.

Thomson drives also play an important role in superconductive fault current limiters. Superconductors may be damaged due to excessive heating resulting from large currents. Therefore, it is vital to use a high speed drive. In [13], a study of a superconductive fault current limiter using a vacuum interrupter driven by an electromagnetic repulsion force is shown.

2.1.2 Modeling

Modeling of TCs is crucial to be able to design such complex drives for the new emerging switching devices. Traditionally, simple circuit based schemes are usually preferred to minimize computational effort. In [35] an equation based modeling of a Thomson drive is implemented. While, in [21], an adaptive equivalent circuit modeling method is shown. Furthermore, an analytical model based on the tableau method is shown in [40] and a reduced modeling of an eddy current-driven elec-tromechanical drive is explained in [20]. Finally, a general method for modeling fast-acting solenoid actuators has been done in [33] using an interpolation function and electric equivalent networks to account for the eddy currents.

Subsequently, researchers have focused on optimization studies to improve the efficiency of the TC using equivalent circuit methods as in [23].

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popu-2.2. GENERATION OF THE FAST IMPULSE 7

larity, not much effort has been done to merge all necessary physics in one simulation model that describes the behavior of a TC as done for other applications. One ex-ample of such a coupling can be seen in [19] for similar types of actuators where the field computations are coupled with circuits to model an electromagnetic de-vice. Moreover, in [31] for example, a dynamic simulation using the finite element method (FEM) for an electric screw driver was performed to calculate the induced torque. Additionally in [6], a coupled magneto-mechanical model is used to investi-gate the dynamics of electromagnetic valves. In [28], motion is incorporated in a 3D eddy current transient problem to model electromagnetic devices containing mov-ing conductmov-ing parts. Furthermore in [32], magneto-thermal-mechanical coupled simulation problems for low frequency electrical engineering devices ranging from small actuators to large synchronous generators are discussed. Finally in [29], an algorithm is presented that is comprised of a dynamical field-circuit coupled simula-tion to design linear electromagnetic actuators aimed at automated control systems. Therefore, one of the objectives of this thesis is to present a unified multi-physical validated model that accurately predicts the behavior of TC based actuators.

2.2 Generation of the fast impulse

Several techniques are used to generate forces where some of them are based on mechanical springs, magnets, electromagnets, hydraulics, and pneumatics. As dis-cussed in the introduction, in the case of low inductive systems, ultra-fast actuation is required. Therefore, TC based actuators capable of generating the required high impulsive forces are considered.

Mechanical springs are traditionally used as means of energy storage. The springs are maintained in a compressed state and released when needed. This principle is simple and effective but lacks ultra-fast speeds. Similarly, magnets can be used to generate attractive or repulsive forces. However, this force generation mechanism also falls short of generating large impulsive forces. One way to achieve these required forces within fractions of milliseconds can be realized by inspecting Biot and Savart’s law applied on two current carrying wires in proximity of each other.

A long conductive straight wire, carrying a current I1generates a magnetic field at a distance r from the wire given by:

B1=

µ0I1

2πr (2.1)

If in the presence of another conductive straight and parallel wire with a length L and carrying a current denoted by I2, a force will act on wire 2 that is given by:

F = Lµ0I1I2

2πr (2.2)

This force can be tuned by varying the magnitude of the currents and the length of the wires. Repelling or attractive forces can be generated depending

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(a) TC (b) DSC

Figure 2.1: Sketches of the drives

on current directions in both wires. From (2.1), it is evident that in order to generate substantial forces, currents in the order of tens of kilo Amperes are needed. Such high amplitude impulsive currents can be generated by discharging a series and parallel combination of capacitors. If the conductor cross section is not large enough, excessive temperatures can arise and destroy the coil. In this thesis, the ultra fast force impulse generation capabilities of a TC and a DSC based on the above principle are analyzed.

2.3 Description of the TC and DSC actuators

The Thomson coil (TC), originally discovered by Elihu Thomson [3], consists of a coil with a conducting ring on top. By applying a varying AC voltage source, the field of the coil induces currents in the ring and generates a repulsive force. Thomson proved this concept by levitating objects. This section presents a modification of the original TC to be suited for ultra-fast actuators and compares it with its counterpart, the DSC.

The TC used in this thesis consists of a spiral shaped flat multi-turn coil with an electrically conducting object in its proximity, while the DSC consists of two mirrored spiral coils that are connected in series. The main difference between the two configurations is the currents in the propagating armature. In the TC, the time derivative of the axial magnetic flux density results in azimuthal eddy currents in the armature. However, in the DSC the same current that flows in the primary coil also flows in the secondary coil but in opposite direction. In principle, the product of this azimuthal current with the radial magnetic flux density produces the required axial forces to repel the armature. Additionally, undesired compressive radial forces are also generated from the axial component of the flux density. These

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2.3. DESCRIPTION OF THE TC AND DSC ACTUATORS 9

forces are considerably lower than the axial forces and are manageable mechanically. It is important to define both actuators as identical as possible to be able to compare them. The base coils in this study are identical and equipped with con-ductors of rectangular cross section with a width of 2 mm and height of 4 mm. The cross section is dimensioned to satisfy the electro-magnetic, thermal, and mechani-cal constraints. The coil is made of copper and consists of 10 turns that are isolated with a 0.1 mm layer of enamel. However, these inter-turn gaps are neglected in the simulations. The inner and outer diameters are 50 mm and 90 mm respectively. For the DSC, the actuating armature consists of yet another series connected identical coil while for the TC, the armature consists of a copper ring with the same dimen-sions as that of the coil. The armature axis and the main axis of the coil coincide. The coil to coil electrical connection can be made by using brushes but they even-tually may wear out. Therefore, they have to be maintained more often. Another feasible solution is to use flexible leads of copper wire. These also in the long run will break due to fatigue. Figure 2.1 shows the geometry of both configurations.

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Chapter 3

Modeling

In this section, the theory behind the force generation principles of the two differ-ent types of ultra-fast electromechanical actuators are modeled starting with simple models and gradually building up into a comprehensive multi-physics finite element simulation model.

3.1 Static model (DC)

The aim of carrying out a static model is to perform a preliminary study of the behavior of both actuator concepts under DC excitation and analyze the flux and current distributions of both actuators. In this study, an energizing circuit is cou-pled to the terminals of both coils in the FEM models. This circuit is comprises a 500 V DC voltage source in series with a 5 mΩ resistor representing the cable leads. The equations used for the DC simulation in the FEM model are:

1

µ∇ × (∇ × A) = Je (3.1)

Je=

σeVcoil

2πr (3.2)

where, µ is the magnetic permeability, A is the magnetic vector potential, Jestands

for the external current density, σeis the electrical conductivity, Vcoilis the voltage

across the coil that is coupled to a SPICE circuit [27], and r is the radial position. Exciting the TC with a DC voltage results in a homogeneous current distribution in the coil without inducing any currents in the armature as shown in Figure 3.1. This current leads to the creation of a constant magnetic flux density enveloping the coil as can be seen in Figure 3.2. It has a maximum value exceeding 20 T and is mostly concentrated adjacent to the innermost turn of the coil. This field penetrates the armature totally and is not influenced by any electrically conductive bodies in its proximity.

Applying the same principle to a DSC results in a completely different magnetic flux density distribution. Although the currents in both coils are homogenous as

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r [mm] z[ m m ] 15 20 25 30 35 40 45 50 55 0 1 2 3 4 5 6×109 -10 -5 0 5 10 15

Figure 3.1: A homogenous current density in [A m−2_{] only in the coil of a TC} subjected to a DC voltage excitation.

r [mm] z[ m m ] 15 20 25 30 35 40 45 50 55 5 10 15 20 -10 -5 0 5 10 15

Figure 3.2: Magnetic flux density in [T] of a TC subjected to a DC voltage excita-tion. The field penetrates the armature and is highest close to the innermost coil turn.

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3.2. STATIONARY MODEL (AC) 13 r [mm] z[ m m ] 15 20 25 30 35 40 45 50 55 2 4 6 8 10 12 14 16 18 -10 -5 0 5 10 15

Figure 3.3: Magnetic flux density in [T] for a DSC subjected to a DC voltage excitation mostly confined in the air gap between the primary and secondary coil turns.

shown in Figure 3.4, the magnetic flux density is mostly confined in the air gap separating both coils and peaks at 18 T (see Figure 3.3). It is lower than that of a TC since in this case, the main current drawn from the power supply is smaller due to the added resistivity coming from the secondary coil. A smaller current results in a smaller field but leads to the generation of the desired repulsive electromagnetic forces.

In principle, to generate an axially directed electromagnetic force, the armature should have an azimuthal current and be subjected to a radial field. Although a DSC generates smaller magnetic fields compared to a TC, it has currents circulating in its secondary coil. Therefore, unlike a TC, a DSC is capable of generating a repulsive force when excited with a DC voltage source.

3.2 Stationary model (AC)

The aim of carrying out a stationary simulation model is to study the behavior of both actuator concepts under AC conditions and analyze their flux and current distributions when influenced by skin and proximity effects. In this study the actuator consists of a lumped impedance added in series with a 500 V AC voltage source to represent the cable leads under AC excitation. The impedance consists of a 5 mΩ resistor and a 1.5 µH inductor. This circuit is coupled to an FEM model as explained before. The equations used are as follows:

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r [mm] z[ m m ] 15 20 25 30 35 40 45 50 55 -4 -3 -2 -1 0 1 2 3 4×109 -10 -5 0 5 10 15

Figure 3.4: Homogenous current densities in [A m−2_{] in the primary and secondary} coil of a DSC subjected to a DC voltage excitation.

(jωσe− ω2ǫ)A + 1

µ∇ × (∇ × A) = Je (3.3)

where, ω is the angular frequency, and ǫ is the electric permittivity.

Contrary to the behavior of a TC with DC excitation, when it’s subjected to an AC source, currents are induced in the armature as shown in Figure 3.5. Unlike before, the magnetic field is now confined in the air gap and peaks at slightly over 16 T as shown in Figure 3.6.

At low frequencies, the resistance of a TC and that of a DSC behave differently. Increasing frequency does not maximize the generated force (see Figure 3.7) of a TC due to a rapid increase of resistance in the system. Initially, for the TC, a steep resistance increase can be noticed since more and more currents start flowing in the armature with increasing frequencies. The magnitude of the induced currents depends on both the frequency and number of coil turns. Therefore, similarly to the operating behavior of a transformer, the resistance of the armature is now added to the system. This effect can be clearly seen in Figure 3.8 for frequencies up to 100 Hz where the TC’s resistance changes from 4.6 mΩ to 6.25 mΩ whereas that of the DSC’s resistance remains constant at a value of 9.2 mΩ. It is also important to note that at DC, the DSC’s resistance is twice as large as that of the TC clearly expressing the influence of the induced currents in the secondary coil.

Another factor influencing the generated force at low frequencies is the induc-tance. The inductance of the TC decreases drastically from 8.7 µH down to 5.6 µH

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3.2. STATIONARY MODEL (AC) 15 r [mm] z[ m m ] 15 20 25 30 35 40 45 50 55 -3 -2 -1 0 1 2 3 4 5×109 -10 -5 0 5 10 15

Figure 3.5: Current densities in [A m−2_{] in the primary coil and the armature of a} TC at peak force when subjected to an AC voltage excitation with a frequency of 120 Hz. r [mm] z[ m m ] 15 20 25 30 35 40 45 50 55 2 4 6 8 10 12 14 16 -10 -5 0 5 10 15

Figure 3.6: Magnetic flux density in [T] of a TC confined in the air gap when subjected to an AC voltage excitation with a frequency of 120 Hz.

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Frequency [Hz] F o rc e [k N ] TC DSC 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600

Figure 3.7: Existence of an optimal frequency for a TC. The force of a DSC decays with an increase in frequency.

when the frequency is increased from 0 to 100 Hz. This is due to the fact that the induced currents cancel out the change of magnetic field generated from the pri-mary coil decreasing inductance. As for the DSC operating in the same frequency range, its inductance remains constant at 3.8 µH since equal currents of opposite magnitudes circulate in both coils at all frequencies (see Figure 3.9).

At higher frequencies, the resistance and the inductance of both the DSC and the TC behave similarly and change drastically due to the combined influences of skin and proximity effects acting on both systems. Resistance increases with frequency due to decreasing skin depths as shown in Figure 3.10. The current density is no longer homogenous and is mostly concentrated on the facing surfaces of the coil and the armature. This increase in resistance leads to a smaller current and a smaller magnetic flux density peak. By comparing Figure 3.6 and Figure 3.11, it can be seen that the magnetic flux density peak of a TC drops from 16 T down to 7 T and is confined more in the air gap at higher frequencies. Due to decreasing skin depths, opposing current densities get closer to each other and hence, the TC and the DSC inductances decrease with frequency.

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3.2. STATIONARY MODEL (AC) 17 Frequency [Hz] R es is ta n ce [m Ω ] DSC TC DSC 0 100 200 300 400 500 600 700 800 900 1000 4 5 6 7 8 9 10 11 12 13

Figure 3.8: Increase in resistance of a TC and a DSC with an increase in frequency.

Skin effect arises due to oppositely oriented eddy currents induced in the pres-ence of a time varying alternating current. Currents tend to circulate more in the outer shell of a conductor [30]. Skin effect cannot be disregarded and has to be accounted for in most electric power devices since it can contribute significantly to resistive losses and heating. At 120 Hz, the skin effect is not an issue for the TC or the DSC since the skin depth of copper at this operating frequency is 6 mm which is larger than the coil conductor depth. The skin depth δ can be calculated as follows:

δ ≈ √ 1

πf µσe

(3.4) The proximity effect on the other hand refers to the constraint of current distribu-tions to smaller regions due to the influence of other currents in nearby conductors. The analytical treatment of the skin depth is rather complicated and will not be elaborated further in this thesis.

Another important aspect to study is the degree of influence of the coil to coil separation distance on resistance, inductance, and peak force. Increasing the air gap increases inductance due to reduced coupling (see Figure 3.13). The opposing

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Frequency [Hz] In d u ct a n ce [µ H ] TC DSC 0 100 200 300 400 500 600 700 800 900 1000 3 4 5 6 7 8 9

Figure 3.9: Decrease in inductance of a TC and a DSC with an increase in frequency.

currents are now further apart and do not cancel out to the same extent as for small separation distances. Resistance on the other hand decreases with increased separation due to the decreased influence of proximity effect (see Figure 3.14). This change in resistance and inductance affects the peak force. Figure 3.12 shows the maximum generated force with respect to frequency at different separation distances. The force decreases and peaks at lower frequencies with increased sep-aration distances. The decreases in force is due to lower magnetic flux densities and lower induced armature currents as the separation distance is increased. The resonance frequency of the system is inversely proportional to the square root of the inductance. Since inductance increases with increasing separation distances, the force will peak at lower frequencies as the coil to coil distance is increased.

In conclusion, the obtained results show that a time varying magnetic flux den-sity is needed to induce currents in the armature of a TC and generate a repul-sive force at the expense of increased resistance. However, there exists a critical frequency at which the generated force of a TC is maximized as can be seen in Figure 3.7. Increasing frequency further results in increased losses attributed with skin and proximity effects. As for the DSC, the generated force is maximized at DC since the required currents for generating a repulsive force do not depend on a

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3.2. STATIONARY MODEL (AC) 19 r [mm] z[ m m ] 15 20 25 30 35 40 45 50 55 -3 -2 -1 0 1 2 3×109 -10 -5 0 5 10 15

Figure 3.10: Current densities in [A m−2_{] of a TC subjected to an AC voltage} excitation with a frequency of 1000 Hz influenced by skin and proximity effects.

r [mm] z[ m m ] 15 20 25 30 35 40 45 50 55 1 2 3 4 5 6 7 -10 -5 0 5 10 15

Figure 3.11: Magnetic flux density in [T] of a TC confined mostly in the air gap when subjected to an AC voltage excitation with a frequency of 1000 Hz.

time varying magnetic flux density. Therefore, exciting a stationary DSC with an AC voltage source results in unnecessary losses.

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Frequency [Hz] F o rc e [k N ] 1.3 mm 2 mm 3 mm 4 mm 5 mm 0 100 200 300 400 500 600 700 800 900 1000 0 50 100 150 200 250 300 350 400

Figure 3.12: Overall decrease in generated force with larger coil to armature air gaps.

Stationary simulations are insufficient to predict the behavior of these actuators since in reality, the position of the armature is changing dynamically. Therefore, the analysis provided above serves as a good understanding of the influence of some of the involved parameters and motivates the need for a time dependent simulation model that incorporates all influencing variables.

3.3 Transient model

The influence of the armature position was elaborated in the previous section with a frequency based simulation model. Although it served as a good example for demonstrating the effect of one of the dynamic variables, there are many more that have to been taken into account for. In reality a capacitor bank capable of sup-plying high currents is used as the energizing source. Therefore, a time dependent simulation model is developed to capture the transients upon its discharge in the TC and DSC coils.

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3.3. TRANSIENT MODEL 21 Frequency [Hz] In d u ct a n ce [µ H ] mm 1.3 mm 2 mm 3 mm 4 mm 5 mm 0 100 200 300 400 500 600 700 800 900 1000 3 4 5 6 7 8 9

Figure 3.13: Inductance increase of a TC with larger coil to armature air gaps.

either to a TC or a DSC. The electrical source is modeled as a SPICE circuit, while the TC and DSC are modeled as FEM models. Subsequently, the FEM and circuit models are coupled together and simulated for both actuator concepts as shown in Figure 3.16 and Figure 3.15.

The electrical source consists of an electrolytic capacitor bank charged up to 500 V, a diode, a thyristor, and the connecting cables to the coil. The diode’s main purpose is to prevent a negative voltage build up across the capacitor bank. After the capacitor bank is fully charged, the thyristor is triggered causing it to discharge a current through the coil configurations shown in Figure 3.15 and Figure 3.16. The resistance of the capacitor bank is assumed to be frequency independent and hence is represented by the use of a constant resistance denoted Rc. The thyristor and connecting cables are modeled as two lumped static stray parameters denoted

by Rstray and Lstray. RTC and LTC represent the resistance and the inductance

of the TC respectively while for the DSC, RCp, LCp, RCs, and LCs represent the impedances of its primary and secondary coils respectively. These parameters are variable and calculated dynamically with the FEM model.

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Frequency [Hz] R es is ta n ce [m Ω ] 1.3 mm 2 mm 3 mm 4 mm 5 mm 0 100 200 300 400 500 600 700 800 900 1000 4 5 6 7 8 9 10 11

Figure 3.14: Resistance decrease of a TC with larger coil to armature air gaps.

Vc Rstray RCp Lstray LCp Rc RCs LCs

Figure 3.15: SPICE circuit for a DSC

armature near one of its flat sides. Following the discharge of the capacitor bank, high current densities are created in the coil cross section increasing its temperature and resistance. Concurrently, in a matter of a millisecond, flux densities in the

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3.3. TRANSIENT MODEL 23

V

c

R

stray

R

TC

L

stray

L

TC

A

rm

a

tu

re

R

c

Figure 3.16: SPICE circuit for a TC

order of 5 T can build up in the coil and the conducting plane in proximity of the coil. The conductive armature then is subjected to a magnetic kick. The derivative of the axial component of the flux density induces eddy currents in the armature. As a result, the azimuthal eddy currents and the radial component of the flux density create an impulsive force that repels the armature. However, the force distribution is not homogeneous enough to cause a smooth vertically oriented linear displacement. Therefore, stresses are induced straining the material and causing deformations which in turn affect the system behavior. These deformations are relatively large so the nonlinear effect of the change in geometry needs to be taken into account. For that, the Green-Lagrange strain tensor and the second Piola-Kirchhoff stress tensor are used.

A moving mesh based on the Arbitrary Lagrangian-Euler method (ALE) is used since the induced forces are highly dependent on the proximity of the armature. The mesh is progressively stretched until a specified mesh quality factor is violated. Subsequently, it is re-meshed and this cycle is repeated.

3.3.1 Methodology

The movable armatures consisting of a copper ring for a TC and a secondary coil for a DSC are denoted by TCs and DSCs respectively. The induced current densities of such mobile armatures with a relative velocity v generated by the Lorentz force is given by:

JiTCs = JiDSCs= σe(E + v × B) (3.5)

, where JiTCs and JiDSCs are the current densities induced in TCs and DSCs

re-spectively, σe is the electrical conductivity of the material, E is the electric field and v represents the velocity of the moving armature. As for the stationary base coils of the TC and the DSC given by TCp and DSCp respectively, their induced

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current densities reduce to:

JiTCp= JiDSCp= σeE (3.6)

For a DSC, the current densities in both coils are composed of the sum of the induced and the externally applied currents since the innermost and outermost turn of DSCpand DSCsrespectively are connected to the terminals of a capacitor bank. Although DSCpand DSCshave different induced current formulations, they should have identical currents since they are series connected, and moreover, they should have identical current densities since according to the center of masses reference frame, both are moving relative to each other.

JDSC= JDSCp= JDSCs = Je+ Ji (3.7)

, where JDSCpis the current density in DSCp, JDSCsis the current density in DSCs,

Je is the externally applied current density, and Ji represents the induced current densities in each coil respectively.

Contrary to the DSC, the currents of a TC circulating in the coil and armature are different. The only currents circulating in the armature of a TC are due to the induced eddy currents since no external currents are applied. The TC currents are expressed by:

JTCp= Je+ JiTCp (3.8)

JTCs= JiTCs (3.9)

Based on Maxwell’s equations and by expressing the magnetic flux density in terms of the magnetic vector potential one has:

∇ × H = J (3.10)

∇ × E = −∂B

∂t (3.11)

B_{= ∇ × A} (3.12)

where, H is the magnetic field intensity, J is the current density, B is the magnetic flux density, and A is the magnetic vector potential. The magnetic equations for both TCp and DSCp are given by:

σe

∂A ∂t +

1

µ∇ × (∇ × A) = Je (3.13)

As for the mobile armatures, the magnetic equations of TCs and DSCsare respec-tively given by,

σe ∂A ∂t + 1 µ∇ × (∇ × A) − σev × (∇ × A) = 0 (3.14) σe ∂A ∂t + 1 µ∇ × (∇ × A) − σev × (∇ × A) = Je (3.15)

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The DSC has the intrinsic property of a very low initial inductance due to the oppositely chosen current directions and the good coupling between the primary and secondary coils. The mutual inductance will reduce the contribution of the self inductance of both coils resulting in a low inductive system in the order of some micro Henrys. This low resistive and low inductive system leads to high inrush currents causing a very steep current rise that is hard to estimate unless modeled with FEM. However, as the coils separate, the current tends to diffuse into the coil for three reasons. First, the current derivative decreases as the current pulse approaches its peak, secondly, the inductance of the system starts to increase limiting the current build up, and thirdly, the conductivity of the material, mainly in locations with high current densities, decreases due to temperature rise. This decrease in conductivity affects the current distribution as the current will deviate to take a less resistive path. A similar effect is also seen in a TC although it has a larger initial inductance.

Due to the dynamics of the current density and large current density gradients, it is important to model the temperature distributions and their effect on system performance. The temperature equation for the stationary coils can be written as:

ρCp

∂T

∂t = ∇ · (k∇T ) + Q (3.16)

, where ρ is the density of the material, Cpis the heat capacity, T is the temperature,

kis the thermal conductivity, and Q is the heat source density and is considered to be originating from the power loss density. It is given by:

Q= J 2

σe

(3.17) To simulate the temperature in the armatures, the thermal equations of the stationary coils need to be slightly modified to account for the movements. A velocity term needs to be added since the system of equations are solved in a stationary reference frame. Therefore, the thermal equations for TCs and DSCs are given by:

ρCp(

∂T

∂t + v · ∇T ) = ∇ · (k∇T ) + Q (3.18)

Sufficiently large temperature changes are often associated with conductivity changes. It is important to calculate the change in conductivity as a function of temperature to reduce losses and design the conductor accordingly. Hence the change in conductivity is modeled by:

σe = σe0[1 + α(T − T0)]−1 (3.19) During the capacitor bank discharge, the high current densities result in a con-centration of the magnetic flux in between the stationary and moving objects re-sulting in a Lorentz force denoted by F as shown in (3.20).

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One way to calculate the velocity of the moving armature is to integrate the force density in accordance with Newton’s law as shown in (3.21) under the assumption that the armature is infinitely stiff, in other words, it is not prone to bending. The sum of the of the masses of the armature and that of the load to be actuated then are lumped together and given by m.

ZZZ

Frdrdθdz = mdv

dt (3.21)

This technique reduces computational effort since the mechanical stresses are not calculated. However, if a thorough simulation needs to be done to design a more complex armature, then the following equation is used where the stresses and strains of materials are computed:

ρ∂

2_u

∂t2 − ∇ · σm= fem (3.22)

, where u is the displacement vector, and σmis the mechanical stress tensor. In this case, the following assumptions are made:

• Thermal expansions of materials are neglected since the coefficient of linear thermal expansion for metals is in the order of 1 × 10−5 ◦_C−1_.

• The system is considered to be adiabatic since within these extremely short time scales, there is no opportunity for significant heat exchange with the surrounding medium. As a result, the normal component of the temperature gradient on the boundaries is set to zero.

• The temperature rise is assumed to be only in the coil and armatures. It is not modeled in the cable leads. Therefore, the temperature conduction from the coils to the cable leads are neglected.

• The thermal conductivity and heat capacity are assumed to be invariant due to an operating temperature range of 20◦_{C to 200}◦_C.

• Materials are assumed to be isotropic.

• The spiral coil is model as concentric rings and assumed to be rigid and incompressible. Therefore the displacement vector on the boundary of the coil is set to zero.

• Heat conduction between the coil turns is disregarded due to the concentric rings assumption.

• Mechanical damping, hardening, and plasticity are neglected since the actua-tor should not operate in this region. Otherwise, it will decrease in efficiency as the bending of the armature increases and eventually break downs com-pletely.

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3.3.2 Simulation results for three test cases

This section presents results of a chosen set of test cases to primarily explain the electromagnetic, mechanical, and thermal features of such an actuator, and sec-ondly, to justify the importance of multi-physics simulations and the importance of using the appropriate differential equations when needed.

The first two cases denoted by TC and DSC are used to show the electromagnetic features and serve as a base comparison between a TC and a DSC. The energizing source consists of a 10 mF capacitor bank charged up to 500 V. The second two cases denoted by "with SM", and "without SM", are used to demonstrate the bending of the armature of a TC and compare both outcomes using either equation (3.22) or (3.21) respectively. A copper disc is used as the actuating armature and it is centrally loaded with a mass of 1 kg. Furthermore, its thickness is reduced to 2 mm and the capacitance is increased to 33 mF to ease bending and study the consequential implications. The final set of test cases denoted by "with T", and "without T" are used to demonstrate the outcome of the simulations of a TC with and without using equations (3.18), (3.16), (3.17), and (3.19). The capacitance is decreased to 500 µF to get a faster current pulse. Accordingly, the charging voltage is increased to 10 kV to compensate for the decrease of electric energy by decreasing capacitance. To demonstrate a distinct difference, the coil conductor cross section is also reduced to a width of 0.5 mm and a depth of 2 mm.

Electromagnetic features

Figure 3.17 shows the current density distribution after 20 µs where the effect of inrush currents and proximity effect are witnessed. The current distributes in the conductor in a way that counteracts the generation of the magnetic field due to the injected current. Similarly as was shown in the frequency simulations, due to proximity and skin effects, the current densities are highest and mostly concentrated at the top of the coil conductor and the bottom layer of the conductive armature for a TC and at the top and bottom of the primary and secondary coils respectively for a DSC. The currents concentrate in vicinity of each other to reduce the total inductance resulting in a large amplitude of radial magnetic flux density confined in the air gap (see Figure 3.18). Similarly, the current also concentrates between consecutive conductor turns rendering the middle and bottom of the primary coil conductor almost entirely unused.

The current and magnetic flux density distributions evolve with time and are mostly dependent on the time derivative of the current pulse and armature proxim-ity. As the armature advances by some millimeters, the mutual inductance decreases thereby increasing the total inductance of the system and limiting the rate of rise of the current pulse. Furthermore, the proximity effect is less dominant and the current density tends to diffuse more into the material due to a lower current deriva-tive as shown in Figure 3.19. Similarly, although the magnetic flux density is still mostly confined in the air gap, it penetrates deeper into the conductive armature

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r [mm] z[ m m ] 20 25 30 35 40 45 50 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2×109 -5 0 5 10 15

Figure 3.17: Current densities concentrated on the top and bottom of the primary and secondary coils respectively of a DSC shown in [A m−2_{] 20 µs after the discharge} of a capacitor bank. r [mm] z[ m m ] 20 25 30 35 40 45 50 0.2 0.4 0.6 0.8 1 1.2 1.4 -5 0 5 10 15

Figure 3.18: The magnetic flux density in [T] after 20 µs barely penetrates the armature and is mostly confined in the air gap.

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3.3. TRANSIENT MODEL 29 r [mm] z[ m m ] 20 25 30 35 40 45 50 -2 -1.5 -1 -0.5 0 0.5 1 1.5×109 -5 0 5 10 15

Figure 3.19: Upon the discharge of the capacitor bank in a DSC, the current pulse peaks at 200 µs. The current density shown in [A m−2_{], tends to homogenize.}

r [mm] z[ m m ] 20 25 30 35 40 45 50 0.5 1 1.5 2 2.5 3 3.5 4 4.5 -5 0 5 10 15

Figure 3.20: At 200 µs, the magnetic flux density shown in [T] penetrates deeper into the material due to a zero flux time derivative.

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Time [ms] C u rr en t [k A ] DSC TC 0 0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 12

Figure 3.21: The peak current of a primary coil of a TC is larger than that of a DSC.

The current pulses of both actuators are shown in Figure 3.21. The DSC peaks at 9.5 kA and the TC peaks at 11.5 kA. Although the current flowing in the primary coil of a TC is larger than that of a DSC, the induced currents in its armature are lower than that of a DSC. This is because the current flowing in the armature of a DSC is the same as that flowing in it’s primary coil, and as for a TC, the armature currents are induced based on the time derivative of the current pulse and armature proximity.

Currents in the presence of a magnetic field result in electromagnetic forces. The product of the azimuthal current density with the radial magnetic flux density produces the forces required to repel the armature. Similarly, the coil conductors will also be subjected to an undesirable radially directed compressive force. How-ever, these forces are considerably lower than the axial forces and are manageable mechanically. Although the primary coil current of a TC is larger, its generated force impulse peaks at 33 kN and that of a DSC peaks at 36 kN as shown in Fig-ure 3.22. This shows that a DSC has a superior efficiency than that of a TC since it attains a larger steady state velocity with the same input energy as shown in Figure 3.23.

Large current densities also lead to a temperature rise. Unlike a DSC, a TC has imbalanced current densities in its primary coil and armature. Since the currents in the primary coil of a TC are larger than those of a DSC, a higher temperature rise is witnessed as shown in Figure 3.24. On the other hand, the temperature rise of the armature of a TC is significantly lower than that of a DSC as seen in Figure 3.25. As a result, not only does a DSC have lower requirements than a TC in terms of powering source component ratings, it is also superior in performance

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3.3. TRANSIENT MODEL 31 Time [ms] F o rc e [k N ] DSC TC 0 0.5 1 1.5 -10 0 10 20 30 40

Figure 3.22: The DSC generates a larger peak force than a TC.

Time [ms] V el o ci ty [m / s] DSC TC 0 0.5 1 1.5 2 2.5 3 0 10 20 30 40 50 60 70

Figure 3.23: The steady state velocity of a DSC is larger than that of a TC.

and has a higher efficiency. Mechanical features

In these set of test cases, the electromagnetic forces generated by discharging a 4125 J capacitor bank are enough to bend a centrally loaded armature and to study its influence on the actuator while it is being repelled away.

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Time [ms] C o il te m p er a tu re [ ◦C ] DSC TC 0 0.5 1 1.5 2 2.5 3 20 22 24 26 28

Figure 3.24: The temperature rise in the primary coil of a TC exceeds that of a DSC. Time [ms] A rm a tu re te m p er a tu re [ ◦C ] DSC TC 0 0.5 1 1.5 2 2.5 3 20 21 22 23 24 25

Figure 3.25: The temperature rise in the armature of a TC is lower than that of a DSC due the lower primary coil to armature coupling.

densities are generated in only a portion of the armature that is situated directly on top of the primary coil. The central part of the disc, where it is axially loaded with 1 kg, experiences a maximum stress of 211 MN m−2 _{after 100 µs as shown in} Figure 3.26. These extreme accelerations result in high stresses that deform the armature even further before repelling it away. Figure 3.27 shows a severe bending of the armature with respect to its central axis as it is moving away. The maximum

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Figure 3.26: Von Mises stress of a TC armature in [N m−2_{] after 100 µs.}

Figure 3.27: Von Mises stress in [N m−2_{] after 320 µs severely deforming the TC} armature as it moves away.

stress that it is subjected to is in the order of 15 GN m−2_{clearly demonstrating that} this armature will deform plastically. The used model however does not take into consideration plastic deformation since in reality, the system should be designed to operate with stress levels that are significantly lower than the yield stresses and attain several thousands of operations during the course of its lifetime. To simulate the actuator, a moving mesh based on the Arbitrary Lagrangian-Euler method (ALE) is used. The mesh is progressively stretched until a 10 % mesh expansion or compression factor is violated. Subsequently, it is re-meshed and this cycle is repeated. For this specific example, 222 re-meshings have been done to simulate the actuator for 3 ms.

When the armature is subjected to such extreme deformations, the electromag-netic system is affected. As the armature bends away, proximity effect decreases and inductance increases considerably. Moreover, smaller currents are induced in

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Time [ms] C u rr en t [k A ] TC with SM TC without SM 0 0.5 1 1.5 2 2.5 3 0 5 10 15 20

Figure 3.28: The current pulse in the primary coil of a TC is distorted due to the rapid increase of inductance associated with the extreme bending of the armature.

the conductive armature since the effective air gap is increased thereby decreasing its efficiency. This bending effect can be clearly seen in the current pulse of the primary coil shown in Figure 3.28. The current pulse with SM begins to deviate compared to a simulation without SM at 100 µs. The slope of the current pulse decreases and then suddenly starts to increase once more at 320 µs after bending peak as shown in Figure 3.29. Once the strain energy becomes larger than the in-duced forces, the armature’s bending decreases as it heads backwards towards the coil behaving just like a spring until once again, at 500 µs, the induced forces exceed the elastic forces bending it once more time away from the coil. During this time period, the current pulse experiences a steeper slope after 320 µs since the armature tends to shrink the air gap as it relaxes. This phenomenon causes another peak current at 500 µs that is even larger than the peak current of a TC without SM. Finally, after 1 ms, the electromagnetic forces are almost zero allowing the armature to bend negatively and attain a negative peak of around −5 mm. Afterwards, it starts oscillating since mechanical damping is not modelled.

As explained before, the force generated by a TC is proportional to its induced currents that in turn are directly proportional to the time derivative of the primary current pulse. This coupling effect can be clearly seen in Figure 3.30. The force peaks at 30 kN just after 100 µs since the slope of the current pulse decreases here-inafter. However, as soon as as the slope of the current pulse increases once more at 320 µs, the force increases in magnitude and peaks at 500 µs exactly where the second current peak occurs.

Due to the bending, the generated force magnitude is smaller than when com-pared with an infinitely stiff armature. Consequently, the steady state velocity of

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3.3. TRANSIENT MODEL 35 Time [ms] A rm a tu re b en d in g [m m ] 0 0.5 1 1.5 2 2.5 3 -5 0 5 10

Figure 3.29: The bending of the armature taken as the difference in displacement between the center of the armature and a point located at its outer diameter.

Time [ms] F o rc e [k N ] TC with SM TC without SM 0 0.5 1 1.5 2 2.5 3 -20 0 20 40 60 80

Figure 3.30: Clear deterioration of the generated force due to the influence of the bending of the armature.

the armature is reduced by 5 m s−1_{severely impacting its performance and efficiency} (see Figure 3.31).

As for the temperature rise of the primary coil of a TC, it is larger when the armature deforms due to the second attained current peak (see Figure 3.32). On the other hand, the armature of a deformable armature results in a lower temperature rise since the induced currents due to bending are lower when compared with an

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Time [ms] V el o ci ty [m / s] TC without TC with SM TC without SM 0 0.5 1 1.5 2 2.5 3 0 5 10 15 20 25

Figure 3.31: The average velocity of a TC is significantly lower if finite deformations such as bending are taken into consideration.

Time [ms] C o il te m p er a tu re [ ◦C ] TC with SM TC without SM 0 0.5 1 1.5 2 2.5 3 20 25 30 35 40 45 50

Figure 3.32: The primary coil temperature of a TC is larger when deformations are taken into consideration.

infinity stiff system as can be seen in Figure 3.33.

In conclusion, it is important to design against bending to maximize the num-ber of operations, increase efficiency, and decrease the primary coil current peak. Bending can lead to destructive stresses deforming the armature plastically if not accounted for. One way of reducing stresses is to embed a conductive armature in a stiff material or simply increase its thickness enough such that it can withstand

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3.3. TRANSIENT MODEL 37 Time [ms] A rm a tu re te m p er a tu re [ ◦C ] TC with SM TC without SM 0 0.5 1 1.5 2 2.5 3 20 21 22 23 24 25 26 27

Figure 3.33: The currents induced in the armature decrease with increase in bending resulting in lower temperatures when compared with an infinitely stiff TC.

the generated forces. Another way to decrease mechanical stresses is to generate a force impulse with a smaller peak and prevail it for a longer time period. This however serves as a good topic for another study.

Thermal features

The final set of test cases denoted by "with T" and "without T" simulate the TC actuator with and without temperature dependence. The capacitance is decreased to 500 µF to get a fast electrical force impulse, but the charging voltage on the other hand is increased to 10 kV to compensate for the corresponding decrease in energy and to demonstrate the effect of temperature increase on conductivity and overall performance. Moreover, a uniform load of 5 kg is used to get a higher current peak by limiting the increase of inductance caused by the repulsion of the armature.

Due to the proximity effect, large current densities appear in the top and bottom of the primary coil and armature of a TC respectively as explained before. These large current densities heat up the material at the facing surfaces and decrease their conductivity. Hot spots arise significantly deteriorating the conductivity of regions with high current density concentrations as shown in Figure 3.34. Temperature differences up to 220◦_{C appear causing nonlinear coil conductivity distributions} with differences as large as 50 % as can be seen in Figure 3.35. This change in conductivity affects the current distribution and forces it to peak at greater depths in the coil conductors with undisturbed conductivities.

The temperature difference initially is highest at the coil to armature interfaces and decreases progressively in magnitude with time as it diffuses into the mate-rial. Although temperature variations decrease, the coil temperature continues to

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r [mm] z[ m m ] 23 24 25 26 27 28 29 30 31 32 40 60 80 100 120 140 160 180 200 220 -1 0 1 2 3 4 5 6 7 8

Figure 3.34: Large temperature differences up to 220◦_{C, 40 µs after the discharge} of the capacitor bank.

increase until it attains 870◦_{C (see Figure 3.36). Unlike the coil temperature, the} temperature of the armature increases only up to 90◦_{C primarily because the} cur-rent densities in the armature are much smaller than those in the primary coil and secondly, since the armature cross section is larger than that of the coil conductors (see Figure 3.37).

In this case, the armature thickness was chosen intentionally to be larger than that of the coil to demonstrate the importance of dimensioning current carrying conductors to limit temperature increase. Electronics and processors are usually mounted with heat sinks. Air flows between the ducts of the heat sink and cools down the equipment by convection. However, in these short time scales, convection is not a solution since high temperatures can arise in some microseconds. Instead, these temperatures can be avoided by using a larger heat reservoir. This can be achieved by increasing the cross-section of the current carrying conductors. How-ever, the geometrical configuration of the wire cross-section has a great influence on the induced forces. In essence, the aim is to have as high current densities as possible adjacent to the armature without sacrificing number of turns. This can

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3.3. TRANSIENT MODEL 39 r [mm] z[ m m ] 23 24 25 26 27 28 29 30 31 32 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 -1 0 1 2 3 4 5 6 7 8

Figure 3.35: Large electrical conductivity variations plotted in per unit 40 µs after the discharge of the capacitor bank.

be solved by using rectangular shaped copper conductors where the width of the conductor can be adjusted to dimension the force impulse and underlying number of turns, while its depth can be increased depending on the required temperature limitation. Increasing the depth of the wire ensures a heat flow from the top to the bottom of the conductor. In this way, high concentrations of current densities due to the proximity effect will not serve as a bottle neck for the system.

Another significant advantage with limiting the temperature rise is to increase the frequency of operation. If successive operations are required, then the tempera-ture increase per operation has to be small enough such that the total temperatempera-ture rise after all operations is small enough to avoid destroying the epoxy or any other materials in the proximity of the conductors.

This temperature dependence increases resistance and leads to a smaller current pulse in the primary coil as shown in Figure 3.38. Hence, a lower current pulse generates a substantially smaller force impulse as can be seen in Figure 3.39. As a result, the performance of the system deteriorates significantly where the steady state velocity drops from 10 m s−1 _{down to 6 m s}−1 _{as shown in Figure 3.40.}

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Time [ms] C o il te m p er a tu re [ ◦C ] TC with T TC without T 0 0.1 0.2 0.3 0.4 0.5 0 200 400 600 800

Figure 3.36: Temperature up to 850◦_{C are attained in only 250 µs.}

Time [ms] A rm a tu re te m p er a tu re [ ◦C ] TC with T TC without T 0 0.1 0.2 0.3 0.4 0.5 20 40 60 80 100

Figure 3.37: The temperature of the armature increases up to 90◦_{C when} temper-ature is taken into consideration.

In conclusion, a temperature increase deteriorates performance. If possible, high temperatures should be avoided all together since they can reduce the lifetime of actuators, and increase their cost and complexity especially if a cooling system is chosen to be incorporated.

In this section, the effect of thermal and mechanical calculations were shown. Although it is of course better to avoid adding complexity when not needed, but if simplifications are introduced based on unrealistic assumptions, then the

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sim-3.3. TRANSIENT MODEL 41 Time [ms] C u rr en t [k A ] TC with T TC without T 0 0.1 0.2 0.3 0.4 0.5 0 5 10 15 20 25 30 35

Figure 3.38: Larger peak current for a TC with no temperature computations.

Time [ms] F o rc e [k N ] TC with T TC without T 0 0.1 0.2 0.3 0.4 0.5 0 100 200 300 400 500

Figure 3.39: Generated force of a TC decreases when temperature is taken into consideration.

ulations are not trustworthy at all. For example, if no deformations or bending are involved (as in the case of a ring type armature loaded homogeneously), then simulating such an actuator with (3.22) will only add complexity and increase com-putational effort without any beneficial gain. On the other hand, in case of a disc shaped thin armature that is centrally loaded, then deformations have to be taken into consideration as was described earlier. Additionally, temperature calculations can also be avoided if a slow electromagnetic force impulse is used and the

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conduc-Time [ms] V el o ci ty [m / s] TC with T TC without T 0 0.1 0.2 0.3 0.4 0.5 0 2 4 6 8 10

Figure 3.40: Performance of a TC suffers due to high temperature rises.

tors are dimensioned to withstand the temperature rise. In conclusion, a simulation model should aim for a minimum complexity that is sufficient enough to simulate the behavior of an ultra-fast actuator within the required tolerances.

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Chapter 4

Experimental verification

Although, simulation models serve as a powerful tool for modeling and designing the described ultra-fast actuators, without validation, they are weak and prone to errors since they rely on approximations and simplifications that might not always hold. Therefore, this chapter aims at adding credibility to the multi-physics simulation models by experimental validation.

There are several involved parameters that are easily accessible in the simulation context. However experimentally, it is relatively hard to get controllability on all of them since some of them are not even observable. For example, it is difficult to access different parts of the actuator such as the coil. The coil is embedded in epoxy and is initially in direct contact with the armature. Thus, it is hard to attach a temperature sensor to the coil and measure its temperature. An infra red camera cannot be used either since the actual hot spots are not visible. Thus low charging voltages are used and discharged in a relatively large conductor with a cross section of (2×4 mm2_{) to isolate the thermal effects. In these set of experiments, the effect of} bending and mechanical deformations are isolated as well by using a thick enough ring type armature subjected to low stresses and without any mechanical loading. Since the efficiency and performance of these actuators are of utmost importance, only the velocity of the armature will be validated.

4.1 The experimental setup

The coil consists of 10 turns and is embedded in a bakelite housing with a filling of epoxy to hold it in place. The bakelite is screwed to a steel plate and the whole structure is clamped down by 4 steel bars to a massive, rigid steel table to avoid vibrations as shown in Figure 4.1. A Pearson probe is used to measure the current pulse with a sampling rate of 2 MS/s. The voltage of the capacitor bank is also measured and used to trigger a high speed camera. To verify the accuracy of the simulation tool, the conductivity of the armature and the charging level of the capacitor bank are varied. One of the used armatures is made of oxygen-free high

On the Design of Ultra-fast Electro-Mechanical Actuators