Master's Thesis

MASTER'S THESIS

Optimization of Wireless Power

Oskar Rönnbäck 2013

Master of Science in Engineering Technology Engineering Physics and Electrical Engineering

Luleå University of Technology Deptartment of Computer Science, Electrical and Space Engineering Optimization of Wireless Power

Oskar R¨onnb¨ack

Lule˚aUniversity of Technology Dept. of Computer Science, Electrical and Space Engineering

December 5, 2013

ABSTRACT

Today, the limit of wireless devices lays in the way they are powered. Imagine a device that doesn’t need a charger or even a battery, which instead gets the power wirelessly over the air. To make such a device possible the transfer distance of currently known systems have to be increased. That will be the aim of this thesis, to investigate how to increase the transfer distance of a wireless power system, WPS, purposed to charge low power electronic devices. In order for the system to be usable certain design limits are set to restricts the size of the coils, flat spiral coils with diameter < 90mm and wire diameter < 2mm, and thereby also narrowing the scope of the thesis. This thesis starts with a presentation of the theoretical framework behind wireless power, including techniques for modeling a complete system. The framework is then broken down to its basic components which generates expressions with geometrical and material properties as variables. These expressions are implemented in Matlab creating a simulator, which finds optimal values of geometrical and material properties that maximizes the transfer distance. The simulator is set up and ran for each system, 2, 3 and 4 coils, this because each system behaves differently and all have some desirable properties. The findings are implemented in Comsol which provides verification and illustrates the electromagnetic fields that are generated. The results from Comsol and Matlab are similar and shows that a 2-coil system can transfer power with 40% efficiency over a distance of ≈ 150mm. While 3- and 4-coil systems significantly improve the transfer distance and can transfer power with the same efficiency over a distance of ≈ 350mm. As a last step were WPS’s built using the findings from the simulations. The coils were made according to the optimal parameters and capacitors were added to tune them to the same resonance frequency. An E-class amplifier was designed and built to excite the transmitting coil in the real system. The measurements made are the power delivered to the amplifier and the power delivered to the load. From that the efficiency of the complete system can be calculated. The measurements made in this thesis don’t hold up to the simulations in the sense of transfer distance. The main reasons for that is that the amplifier is included in the measured PTE and not in the simulations and that the coils are not perfectly built or tuned.

iii

PREFACE

This thesis work were conducted as the last part of the Master Programme in Engineering Physics and Electrical Engineering at Lulea University of Technology, LTU. During my project work course I ﬁrst came into contact with wireless power and I thought is was a fascinating technology. Seeing the possibilities for wireless power it is clear that it will play a huge part in the future of electronics. I was not aware of any research in this area in Sweden, therefore I made the thesis work as a project on my own initiative which I carried out at LTU.

I would like to thank Kalevi Hyyppa for his understanding and guidance and my family for always supporting me.

Oskar Ronnback

CONTENTS

Chapter 1 – Introduction 1 1.1 Wireless power today ...... 1 1.2 Beneﬁts of wireless power systems ...... 2 1.2.1 Environmental ...... 2 1.2.2 Social ...... 2 1.3 Baseline ...... 3 1.4 Delimitations ...... 3 1.5 Goal ...... 4 1.6 Outline ...... 4 1.7 Frequently used variables and abbreviations ...... 4

Chapter 2 – Theory 7 2.1 Resistance ...... 7 2.1.1 Resistance in a wire ...... 7 2.1.2 Litz wire resistance ...... 8 2.2 Inductance ...... 8 2.2.1 Self inductance ...... 8 2.2.2 Inductance of pancake coil ...... 9 2.2.3 Inductor quality factor ...... 9 2.2.4 Mutual inductance ...... 9 2.2.5 Coupling coefficient ...... 10 2.3 Induction ...... 10 2.4 Resonance ...... 11 2.4.1 Electrical Resonance ...... 11 2.5 Wireless power using magnetic resonance ...... 12 2.6 Coupled Mode Theory ...... 12 2.6.1 Lossy model with source excitation ...... 13 2.6.2 Model of lossy 2-coil coupled system ...... 13 2.6.3 Wireless Power Transfer Efficiency ...... 14 2.6.4 WPT expanded to 3- and 4-coil systems ...... 14 2.7 Reflected Load Theory ...... 15 2.7.1 Expanded for m-coil systems ...... 15 2.8 Unified theory ...... 16 Chapter 3 – Simulations 17 3.1 Matlab simulations ...... 17 3.1.1 Finding the optimal PTE ...... 18 3.1.2 Simulation results ...... 19 2-coil system ...... 20 3-coil system ...... 21 4-coil system ...... 22 3.2 Comsol ...... 23 3.2.1 Simulation setup ...... 23 3.2.2 Simulation results ...... 24 2-coil system ...... 24 3-coil system ...... 26 4-coil system ...... 28

Chapter 4 – Electronic design 31 4.1 Tuning of the coils ...... 31 4.2 Source ...... 31 4.3 E-class ampliﬁer ...... 32 4.4 Simulation ...... 33 4.4.1 Component selection ...... 33 4.4.2 PSpice ...... 34

Chapter 5 – Real testing 37 5.1 Measurement setup ...... 37 5.2 Test procedure ...... 37 5.3 Measurements on 2-coil system ...... 38 5.4 Measurement on 3-coil system ...... 39 5.5 Measurements on 4-coil system ...... 40

Chapter 6 – Discussion 41 6.1 Simulations ...... 41 6.2 Real tests ...... 42 6.3 Conclusions ...... 42 6.4 Health issues ...... 43 6.5 Future work ...... 43

viii CHAPTER 1 Introduction

Wireless power is an old concept, Nikola Tesla experimented with it in the late 1800’s. He was considering it as an alternative to building the electric grid. History tells us that wireless power were never realized at a consumer level and the concept was almost forgotten. Induction stoves and transformers transfers power ”wirelessly” and have been around for some time but they all work over negligible distances. In 2007 scientists at MIT issued a press release describing how to transfer power wirelessly using magnetic resonance and presented results of transfer distances up to a couple of meters [1] . Since then interest in this technology have boomed and it is easy to see why. Wireless power could be used in a wide range of applications stretching from mobile devices cell phones, tablets, laptops, sensors, medical implants to electric cars, trains and buses. Estimations indicate that wireless power could be a billion dollar industry within the next 10 years.

1.1 Wireless power today

Today, six years since that press release hardly any products have hit the market. At least not using magnetic resonance or who can transfer the power over a signiﬁcant distance. The scientists at MIT that was behind this technology started a company, WiTricity, to commercialize their discovery. Since then most of their work are kept secret and protected by hundreds of patents. Their primary targets are OEM’s that can embed their technology directly into their products. But no such products have been released yet. They have four own products, all using magnetic resonance [2]. Three of them are low power development kits aimed to showcase the technology for developers. The fourth one is a high power system for charging electrical cars. The products that are starting to pop up are charge pads/mats most of them uses the qi standard which is created by the Wireless Power Consortium [3], WPC. The WPC consists of over 140 members including industry leaders in mobile phones, batteries and consumer electronics. Their qi standard is made for low power wireless charging, <5W,

1 2 Introduction and specifies coil geometries, frequencies, communication, control and electric sources. The standard enables some design freedoms and is said to work with both direct induction and magnetic resonance. Most products today uses the first technique and the maximum transfer distance for a qi product today is 4cm. The WPC are working on a standard for medium power < 120W, but the specification for that is not made public yet. Their goal is to make worldwide standards for wireless power which is compatible for all devices, similar to Wi-Fi. Many of the large companies are doing their own research in this area e.g. Apple, Qual- comm, Duracell and Texas Instrument. But most is kept secret and the only available products are a few development kits and short distance charge pads/mats. There are research going on in a wide range of other applications also, from medical implants, consumer electronics to electrical cars and electric roads [4]. The medical implant research focuses on low power transfer using small coils. A 2-coil solution using direct induction, similar to a transformer, has been present for some time. Research for implementing a 4-coil system which is much more efficient and can work over longer distances have been made [5].

1.2 Beneﬁts of wireless power systems

1.2.1 Environmental

One can argue that wireless chargers are not environmental friendly because they have lower efficiency than regular chargers i.e. will consume more power while charging. Be that as it may, wireless chargers can be made with high efficiency similar to regular chargers. But the biggest benefit will come if a global medium-range-wireless standard is implemented. A standard that enables charing of all mobile devices, phone, tablet, laptop, sensors etc, using the same charger. Then all devices don’t need to have an own charger, which in turn will save a lot of resources and energy. Another benefit is in the battery area, sensors, remotes etc could run without batteries and phones, tablets etc could be fitted with smaller ones because they would be charged in many places. Decreasing the need for batteries will have a big effect on the environment because of the hazardous materials they are made of.

1.2.2 Social

In todays society a lot of people, especially young, carry their chargers with them most of the time. This is because the battery of todays phones don’t last the entire day. If universal medium-range-wireless systems are developed. There could be charge zones everywhere e.g in cars, coﬀee shops, class rooms etc. and thereby eliminate this need. Similar advantages to WiFi could be achieved and there would be a truly wireless society. 1.3. Baseline 3

1.3 Baseline

Published work in this area utilizes different technologies, coils sizes and load resistance making a baseline for the transfer distance a bit hard to set. The qi standard have reached a maximum of 4cm but it is not specified if direct induction or magnetic resonance is used for that case [3]. Other research have proven that adding 1-2 resonating coils /repeaters could significantly improve the transfer distance and power can be transfered up to two times the coil radius with reasonable efficiency [1], [6], [7], [8]. WiTricity [2] claims to be able to transfer the power up to a couple of feet but no values or coil setups are published. A common conception is that it is possible to transfer power a distance a couple of times longer than the coil diameter. A good baseline would therefore be a transfer distance twice the coil diameter, which is limited to 90mm in this thesis, thus 180mm.

1.4 Delimitations

A system that charges mobile devices has to be somewhat small in size, no one will use a bulky system, preferably the receiver is small enough to embed in the product. Mobile devices are primary flat and therefore the coils have to be flat as well, flat spiral coils were then a natural choice. The coil and wire diameters don’t have an obvious limit and were chosen to be 90mm and 2mm, which seemed reasonable. Litz wire were also simulated but is was only available with diameter of 0.78mm. The frequency of the AC current that drives the system is chosen to 2MHz and kept constant. This because the gate driver ,which drives the E-class amplifier in the real tests, doesn’t support frequencies higher than 2MHz and the amplifier has to be designed around a specific frequency. The load resistance is chosen to 10Ω which is a ballpark value of the resistance in mobile devices. Table 1.1 summarizes all the design limits.

Coil diameter < 90mm Coil type Pancake, ﬂat spiral Wire type Litz/magnet Magnet wire diameter < 2mm Litz wire diameter 0.78mm Load Resistance 10Ω Frequency 2MHz

Table 1.1: Design limits 4 Introduction

1.5 Goal

The goal with this thesis is to ﬁnd a way to optimize the transfer distance and by doing that design a system that can transfer power longer than twice the coil diameter.

1.6 Outline

The outline of this thesis will be as follows, starting in Ch. 2 with a presentation of the basic theories of resistance, inductance, induction, resonance as well as techniques for modeling WPS’s. In Ch. 3 this framework is implemented in Matlab [9], in an attempt to ﬁnd optimal parameters for increasing the transfer distance for WPS’s. The system, with optimal parameters, is then also simulated in Comsol [10], to verify the results and provide data on the magnetic and electric ﬁelds. Ch. 4 describes the electrical circuits used for the real tests described in Ch. 5 and the last chapter summarizes the work and discusses the results.

1.7 Frequently used variables and abbreviations

WPC: Wireless Power Consortium.

WPS: Wireless Power System.

PTE: Power Transfer Eﬃciency.

L1: The transmitting coil inductance in a 4-coil system.

L2: The transmitting coil inductance in 2- and 3-coil systems / resonator coil inductance in a 4-coil system.

L3: The receiving coil inductance in a 2-coil system / resonator coil inductance in 3- and 4-coil systems.

L4: The receiving coil inductance in 3- and 4-coil systems.

C1,2,3,4: The capacitance to make each coil resonate at the desired frequency. r1,2,3,4: Coil radius. rw1,w2,w3,w4: Wire radius. 1.7. Frequently used variables and abbreviations 5

n1,2,3,4: Number of turns in the coil. di,j: Distance between coil i and j, when the coils are parallel.

CHAPTER 2 Theory

This chapter describes the theoretical framework used in this thesis. Starting with basic electrical and electromagnetic deﬁnitions, moving on to wireless power and techniques to model wireless power systems.

2.1 Resistance

Resistance is deﬁned as the opposition to pass current through a conductor. Losses will always be present when a current moves through a conductor. The power dissipated by the resistor will mainly be in the form of heat and is given by

Pdiss = V · I. (2.1)

2.1.1 Resistance in a wire The resistance of a wire for DC or low frequencies is given by the resistivity of the material ρ, the cross section area A and the length of the wire l ρl R = . (2.2) dc A When the frequency increases, the current distribution in the wire changes. It goes from uniformly distributed to concentrated along the surface of the conductor.

7 8 The second chapter

The skin depth δ is a measure on how far this change has gone and is deﬁned as the distance from the outer surface to where the current density has fallen to 37 % of its value at the surface, Fig. 2.1. r 2ρ δ = . (2.3) ωµ Where ω is the angular frequency and µ is the absolute mag- Figure 2.1: Skin depth netic permeability of the conductor. The resistance for high frequency currents therefore has a more complex expression [11]

ρ Ber(q)Bei0(q) − Bei(q)Ber0(q) √ Rac = 0 2 0 2 Ω/m (2.4) 2δπrw Ber (q) + Bei (q) where √ 2r q = w , (2.5) δ rw is the wire radius and Ber and Bei is the real and imaginary part of the Bessel function.

2.1.2 Litz wire resistance

Litz wires are designed for reducing the skin effect i.e reducing the HF resistance. It consists of multiple small strands, iso- lated from each other and braided in a specific pattern, Fig. 2.2. The multiple strands will give a larger surface area at high frequencies and the braiding pattern reduces the proximity effect between the strands. Calculations of the resistance gets very complex and no available method for it was found. An approximative method based on measurements and table Figure 2.2: Litz wire values, developed by a manufacturer, is described in [12]. 2.2 Inductance

Inductance is a property of a conductor, the electromagnetic deﬁnition of inductance L is the ratio of magnetic ﬂux linkage λ to the current I

λ L = . (2.6) I

2.2.1 Self inductance In electronics, inductors make use of the principle described by Eq. 2.6. A changing current ﬂows through the windings of an inductor, creating a changing magnetic ﬁeld. 2.2. Inductance 9

Each winding of the inductor captures this flux and produces an induced voltage, back EMF, which is why it is called self inductance. According to Faraday’s law Eq. 2.16 the induced voltage will oppose the change in flux which gives inductors the property of resisting changes in current. The value of the inductance L is purely defined by its material and geometrical properties. For a circular wire coil it is approximated by [11] k2 L = n2µ (2r − r ) 1 − K (k) − E (k) (2.7) 0 w 2 where s 4r(r − rw) k = 2 , (2.8) (2r − rw) n is the number if turns, r is the radius of the coil, rw is the wire radius, µ0 is the permeability and K and E are the complete elliptic integrals.

2.2.2 Inductance of pancake coil The inductance of pancake coils, single layer ﬂat spiral coils, diﬀer from Eq. 2.7. A better approximation of the inductance in pancake coils is given by [13]

r2n2 L = , (2.9) 8r + 11w where r is the radius to the half winding width in inches, w is the width of the windings in inches and n is the number of turns, Fig 2.3. The approximation is good for frequencies below 30MHz and is correct within 5% if w > 0.2r. Figure 2.3: Illustration of how r and w are defined for a pancake coil 2.2.3 Inductor quality factor Quality factor is a measure on how ideal an inductor is and is defined by 2πfL Q = (2.10) R where f is the frequency, L is the inductance and R is the wire resistance. An ideal inductor have no resistive loss resulting in an infinite quality factor. But all real inductors have at least some wire resistance.

2.2.4 Mutual inductance

Similar to self inductance, two inductors carrying currents I1 and I2 in close proximity interacts magnetically. Both inductors induces a voltage in each other, this is deﬁned by 10 The second chapter

[14] λ12 λ21 M12 = M21 = = , (2.11) I2 I1 where λij is the ﬂux linkage form i to j. The equality M12 = M21 can be proven, by energy concepts, for linear mediums surrounding the inductors e.g. air. The mutual inductance can be calculated from the geometrical and material properties, similar to the self inductance, but the main parameter is the distance between the inductors, must 3 be parallel and perfectly aligned, as M12 ∝ 1/d12. For two circular single turn wire coils, with radius r1 and r2, the mutual inductance is given by 2 r 2µ√ α r1r2 M12 = r1r2 1 − K(α) − E(α) ; α = 2 2 2 (2.12) α 2 (r1 + r2) + d12

Z π/2 dφ Z π/2 K(α) = ; E(α) = p1 − α2sin2(φ)dφ (2.13) p 2 2 0 1 − α sin (φ) 0 For coils with multiple turns the mutual inductance becomes

N N X1 X2 Mtot = Mij. (2.14) i=1 j=1

2.2.5 Coupling coefﬁcient From self and mutual inductance can a coupling coeﬃcient k be derived. It is a measure on how much two coils interact, 1 being totally interacted and 0 no interaction M k = √ 12 . (2.15) L1L2 2.3 Induction

Induction is the creation of voltage, electromotive force, when a conductor is placed inside a time varying magnetic field. If the conductor forms a closed circuit then this is expressed by Faraday’s law [14] dΨ V = −N , (2.16) emf dt where N is the number of turns in the circuit and Ψ is the flux through each turn. The negative sign indicates that the induced voltage acts in such a way to oppose the flux producing it. Eq. 2.16 can be written in terms of the magnetic field B d Z Vemf = −N B · dS, (2.17) dt S where S is the surface area of the closed circuit. A transformer works according to this concept; a time varying current is applied to the primary windings causing a time varying 2.4. Resonance 11 magnetic field, the secondary windings placed inside this magnetic field will have an induced voltage. When an electrical load is applied to the secondary windings, current will flow. In order to make this transfer as efficiently as possible, all of the magnetic field created has to flow through the secondary windings making the distances of power transfer negligible.

2.4 Resonance

Resonance occurs in many areas of physics and describes the tendency of a system to oscillate with larger amplitude at some speciﬁc frequencies. The response of a resonant system depends highly on the physical parameters of the system. The intensity of a lightly damped linear oscillating system can often be approximated with the formula [15]

Γ 2 I(ω) ∝ 2 (2.18) 2 Γ 2 (ω − Ω) + 2 where Ω is the resonance frequency and Γ is a parameter depending on the damping.

2.4.1 Electrical Resonance In electrical systems resonance occurs at a particular frequency where the imaginary parts of the systems impedance cancels out. An example of this is the series RLC circuit, Fig. 2.4.

R L

Vs C

Figure 2.4: Series RLC-circuit

The impedance is given by 1 − ω2LC Z(jω) = R + . (2.19) jωC √ At the frequency ω = 1/ LC the imaginary parts cancels and the system starts to resonate. 12 The second chapter

2.5 Wireless power using magnetic resonance

To transfer power with magnetic resonance capacitors are added to the transmitting and receiving circuits according to the circuit diagram in Fig. 2.6. These capacitors tunes the circuits to achieve resonance. When the oscillating source, Vs, excites the transmitter circuit energy is stored in the transmitter. The transmitter is coupled to the receiver by their mutual inductance, an analogy for the energy transfer can be made using two pendulums connected by a spring [8]. In the case of the pendulums, the spring is equivalent to the mutual inductance or coupling between the coils. The stiffness/coupling determines how much energy is transfered in each cylce, the rate of the energy transfer. The efficiency is not affected by the spring/coupling, it is only defined by the losses in the system, friction/winding resistance, etc. That is the major difference between direct induction and magnetic resonance. When you extract work from the receiver you add constrains to the system, the amount of power transfered to the receiver must be enough to drive the load else the magnitude of the oscillation will decrease. This gives the system a region where there is an equilibrium and beyond that region the system can not drive the load at maximum efficiency and the magnitude decays. A more in depth analysis is found in [1].

2.6 Coupled Mode Theory

Coupled mode theory, CMT, is an analytic tool for systems involving interacting oscil- lations and leads to solutions for oscillating and propagating waves [16]. Therefore it is used to model wireless power system that uses magnetic resonance. Considering an ideal LC circuit, Fig. 2.5, two coupled ﬁrst order diﬀerential equations can be stated i v

L C

Figure 2.5: Parallel LC-circuit di v = L , (2.20) dt dv i = −C . (2.21) dt These equations, 2.20 and 2.21, can be combined to a second order diﬀerential equation d2v + ω2v = 0, (2.22) dt2 2.6. Coupled Mode Theory 13

√ where the resonant frequency is ω = 1/ LC. Using coupled mode theory a complex amplitude is defined as r r ! C L a(t) = v(t) − j i(t) , (2.23) 2 C where the energy stored in the circuit is then given by |a|2. Eq. 2.22 can then be stated as one first order differential equation

da(t) = −jωa(t). (2.24) dt 2.6.1 Lossy model with source excitation To be able to consider a real system, the model Eq. 2.24 has to be expanded to account for losses and source excitation. The losses are represented by Γ which is the rate of 2 decay and the excitation by Fs, where |s| is the input power. The model then becomes da(t) = − (jω + Γ) a(t) + F (t). (2.25) dt s 2.6.2 Model of lossy 2-coil coupled system

R C2 C3 s k23

L2 L3 Vs RL R2 R3

Figure 2.6: Circuit diagram of 2-coil system

A resonant circuit is added as a load to the system described in Eq. 2.25, Fig 2.6, the transmitting circuit is denoted by the index 2 and receiving by 3. The two circuits are coupling to each other by the term k and the load resistor is taken into account by ΓL. The complete system can now be described by

da (t) 2 = − (jω + Γ ) a (t) + F (t) + jka (t), (2.26) dt 2 2 2 s 3 da (t) 3 = − (jω + Γ + Γ ) a (t) + jka (t), (2.27) dt 3 3 L 3 2 14 The second chapter

2.6.3 Wireless Power Transfer Efficiency In order to calculate the efficiency of the power transfer, the theory of energy conservation is applied. If the radiated power in the near field is neglected the following statement can be made [7]

PS = P2 + P3 + PL, (2.28) where the average power in each circuit, coil and capacitor, is

2 Pi = 2Γi|ai| (2.29) and in the load 2 PL = 2ΓL|aL| . (2.30) With Eq. 2.28 and 2.34 the eﬃciency can be stated as P 1 η = L = , (2.31) 2−coil 2 Ps Γ3 Γ2Γ3 ΓL 1 + 1 + 2 1 + ΓL K23 Γ3 where K23 is the coupling rate.

2.6.4 WPT expanded to 3- and 4-coil systems

R C2 C4 s k23 k34

L2 L3 L4 Vs C3 RL R2 R3 R4

Figure 2.7: Circuit diagram of 3-coil system

R C1 C4 s k12 k23 k34

L1 L2 L3 L4 Vs C2C3 RL R1 R2 R3 R4

Figure 2.8: Circuit diagram of 4-coil system

The efficiency can be significantly improved at larger distances by using more than two coils [1], [6], [7],[8]. The law of energy conservation can be stated for an arbitrary number 2.7. Reflected Load Theory 15 of coils m X PS = Pi + PL. (2.32) i=1 Form Eq. 2.32 an expression of the efficiency for a m-coil system can be derived in a similar way as for the 2-coil system P Γ η = L = L . (2.33) m−coil 2 Ps Pm−1 Ai Γm + ΓL + Γi i=1 Am In the 3-coil system an extra load circuit is added for impedance matching, Fig 2.7. From Eq. 2.33 the 3-coil efficiency given by

K23K34ΓL η3−coil = . (2.34) 2 2 2 2 2 2 Γ2 [K34 + Γ3 (Γ4 + ΓL)] + K23 Γ3 (Γ4 + ΓL) + K23K34 (Γ4 + ΓL) 2.7 Reﬂected Load Theory

Reflected load theory, RLT, is widely used by electrical engineers to analyze transformers but can also be applied to WPS’s. The theory is based on that the current in the primary coil is dependent on the load in the secondary coil. The load that is reflected to the primary coil is not the same value as the load present in the secondary coil. It can be shown that the√ highest PTE√ is found when both coils is tuned to the same resonance frequency 1/ L2c2 = 1/ L3c3. At resonance the reflected load will appear as function of the mutual induction between the coils [7]

2 Rref = k23ωL2QL3 (2.35) where k23 is the coupling coeﬃcient Eq. 2.15, QL3 = Q3QL/(Q3 + QL) and QL = RL/ωL3. The power applied to the primary coil will then be divided between R2 and Rref . The power transfered to the second coil will be divided between the load and the coil resistance. From this the PTE, from source to load, can be derived 2 k23Q2Q3L Q3L η2 = 2 · . (2.36) 1 + k23Q2Q3L QL

2.7.1 Expanded for m-coil systems

Assuming that the coupling between non-neighboring coils is negligible, the partial ηi,i+1 can be stated 2 ki,i+1QiQ(i+1)L ηi,i+1 = 2 . (2.37) 1 + ki,i+1QiQ(i+1)L The PTE for the full system with m coils is achieved by m−1 Y Q3L η = η · . (2.38) m i,i+1 Q i=1 L 16 The second chapter

2.8 Uniﬁed theory

It can be shown that both CMT and RLT will result in the same steady state equations for the PTE [7], [17]. The rate of decay in CMT can be expressed as Γi = ω/2Qi and the coupling rate as Kij = ωkij/2. Substituting this in Eq. 2.33 will lead to the same expression as for the RLT Eq. 2.38. The PTE of a 3-coil system, Fig. 2.7, can be expressed by setting m = 3 in Eq. 2.38

2 2 k23k34Q2Q3Q4L Q4L η3 = h i · (2.39) 2 2 2 2 QL (1 + k23Q3Q4L) + k23Q2Q3 (1 + k23Q3Q4L) and The PTE of a 4-coil system, Fig. 2.8, by setting m = 4 in Eq. 2.38

2 2 2 (k12Q1Q2)(k23Q2Q3)(k34Q3Q4L) Q4L η4 = 2 2 2 2 2 · (2.40) [(1 + k12Q1Q2) (1 + k34Q3Q4L) + k23Q2Q3] [1 + k23Q2Q3 + k34Q3Q4L] QL CHAPTER 3 Simulations

Simulations of the PTE are made both in Matlab [9] and in Comsol Multiphysics [10]. Matlab is a numerical computing environment, ideal for implementing analytical expressions and combining them into a simulator. Comsol is a finite element analysis, solver and simulation software. It can combine multiple physics into the same solution, making it very useful for this kind of problem. The framework presented in Ch. 2 is implemented in Matlab creating a simulator, which is designed to find optimal geometrical parameters for increasing the transfer distance. The results are then implemented in Comsol to verify the model and to get a picture of the electromagnetic fields.

3.1 Matlab simulations

The simulator is built around the PTE as a function of the transfer distance, Eq. 2.40,

2.39 and 2.36. But the PTE is also dependent on the frequency f, the load RL and for each coil the quality factor Qi and the coupling coefficient kij. The task for the simulations is then to find an optimal set of these parameters, which gives as long transfer distance as possible and on the same time keeps the efficiency at reasonable levels. These parameters can be broken down further, Eq. 2.10 shows that the quality factor is dependent on f, the coil inductance Li and the coil resistance Ri. The coupling coefficient kij given by Eq. 2.15 depends on coil inductances Li and Lj and the distance between the coils dij. The coil inductance and resistance, Eq. 2.9 and 2.4, can be broken down to its geometrical and material properties, where the parameters are coil type, coil radius, wire radius, wire type, frequency and material. Design limits was set in order to keep the system small, use materials that are reasonable priced and available and narrow the scope of the simulations. To keep the system small and have the possibility to embed it in a mobile device, the coil radius was limited to 45mm. All mobile devices today are flat in some sense therefore the flat coil type, pancake

17 18 The third chapter coil, is a natural choice. The thickness of the coil is then only dependent on the wire radius, which in turn is limited to 1mm for magnet wires and for litz wires there are only one radius, 0.39mm, that is considered. All wires are made of copper, other wires like silver plated or super conductive could make a huge diﬀerence but their unavailability and price excludes them as an option in this thesis. Due to limitations in the gate drivers and the design of the ampliﬁer, for the real tests, the frequency is set at a constant 2MHz. Most mobile devices charges at 5V with 0.5-1A, making 10Ω a good choice for the load resistance. All the limits are summarizes in Table 3.1.

Coil radius <45mm Coil type Pancake Wire type Litz/magnet Magnet wire diameter <2mm Litz wire diameter 0.78mm Wire material Cu Load Resistance 10Ω Frequency 2MHz

Table 3.1: Design limits

Collecting all parameters and using the limits in Table 3.1 gives the following set of parameters for optimization, for each coil: radius, wire radius and number of turns. Systems with three or four coils will have additional parameters in form of distances between coils. These parameters are input to the simulator and it calculates all possible combinations to ﬁnd the optimal set. The simulations are made with an ideal source and with added source resistance. The added resistance will compensate for non ideal source properties.

3.1.1 Finding the optimal PTE To setup the simulator to find an optimal PTE, a definition on what an optimal PTE is has to be made first. This is not as straight forward as you might think, by varying all the parameters very different PTE curves can be achieved, Figures 3.1a - 3.1c shows examples of how theses curves can look. It becomes clear that a compromise between a curve that has a short but high plateau, curves with a maximum shifted from the origin and curves with a long and low plateau have to be made. If the PTE is integrated over the transfer distance you will get a measure of the total efficiency over the distance. That is a good starting point, but because curves with high and short plateaus have much higher values on short distances there will not be a good compromise just by looking at the biggest total efficiency e.g. Fig 3.1a, high and short, has a bigger total efficiency than Fig 3.1b, low and wide, but 3.1. Matlab simulations 19

100 100 60

90 90 50 80 80

70 70 40

60 60

50 50 30 PTE [%] PTE [%] PTE [%]

40 40

20 30 30

20 20 10

10 10

0 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 d [m] d [m] d [m] 23 23 23 (a) High and short plateau (b) Low and wide plateau (c) Shifted maximum

Figure 3.1: Examples of PTE curves on the same time has a shorter transfer distance at 40% eﬃciency. To make it better the integrating interval is narrowed, the best compromise is found with an interval of 0.2-1m. With the deﬁnition of the optimal PTE, the simulator calculates this value for all possible combinations of the parameters. The combination with the highest value will be the optimal set.

3.1.2 Simulation results There are two cases for all systems, coils made of magnet- and litz-wire. The Matlab simulator calculates all possible combinations of the parameters to ﬁnd an optimal set. Figures 3.2 - 3.7 shows the optimal PTE for each case and system, for three diﬀerent source resistances. 20 The third chapter

2-coil system

100 R = 0Ω s R = 1Ω 90 s Design choices R = 10Ω s Coil type Pancake 80 Wire type Cu, singel strand 70 Frequency 2MHz 60 Optimal parameters 50 r 45mm PTE [%] 2 40 r3 45mm 30 rw2 1mm

20 rw3 1mm n 20 10 2 n3 1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 d [m] 23

Figure 3.2: PTE for the magnet wire 2-coil system Table 3.2: Parameters used for with diﬀerent source resistances the magnet wire 2-coil system

100 R = 0Ω s R = 1Ω 90 s Design choices R = 10Ω s Coil type Pancake 80 Wire type Cu, Litz 70 Frequency 2MHz 60 rw2 0.39mm 50 r 0.39mm

PTE [%] w3 40 Optimal parameters

30 r2 45mm

20 r3 45mm n 20 10 2 n3 1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 d [m] 23

Figure 3.3: PTE for the litz wire 2-coil system with Table 3.3: Parameters used for diﬀerent source resistances the litz wire 2-coil system 3.1. Matlab simulations 21

3-coil system

80 R = 0Ω s R = 1Ω s 70 R = 10Ω s Design choices 60 Coil type Pancake

50 Wire type Cu, singel strand Frequency 2MHz 40

PTE [%] Optimal parameters

30 d34 31mm r2, r3, r4 45mm 20 rw2, rw3 1mm 10 rw4 0.4mm n2, n3, n4 20 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 d [m] 23

Figure 3.4: PTE for the magnet wire 3-coil system Table 3.4: Parameters used for with diﬀerent source resistances the magnet wire 3-coil system

80 R = 0Ω s R = 1Ω s Design choices 70 R = 10Ω s Coil type Pancake 60 Wire type Cu, Litz

50 Frequency 2MHz rw2, rw3 0.39mm 40

PTE [%] Optimal parameters

30 d34 31mm r2, r3 45mm 20 r4 33mm 10 n2, n3 20 n4 14 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 d [m] 23

Figure 3.5: PTE for the litz wire 3-coil system with Table 3.5: Parameters used for diﬀerent source resistances the litz wire 3-coil system 22 The third chapter

4-coil system

80 R = 0Ω s R = 1Ω s Design choices 70 R = 10Ω s Coil type Pancake 60 Wire type Cu, singel strand Frequency 2MHz 50 Optimal parameters 40 d 1mm PTE [%] 12

30 d34 31mm r1, r2, r3, r4 45mm 20 rw1, rw2, rw3 1mm

10 rw4 0.4mm n1, n2, n3, n4 20 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 d [m] 23

Figure 3.6: PTE for the magnet wire 4-coil system Table 3.6: Parameters used for with diﬀerent source resistances the magnet wire 4-coil system

80 Design choices R = 0Ω s R = 1Ω s Coil type Pancake 70 R = 10Ω s Wire type Cu, Litz 60 Frequency 2MHz

50 rw1, rw2, rw2, rw2 0.39mm Optimal parameters 40

PTE [%] d12 1mm

30 d34 31mm r1, r2, r3 45mm 20 r4 33mm 10 n1, n2, n3 20 n4 14 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 d [m] 23

Figure 3.7: PTE for the litz wire 4-coil system with Table 3.7: Parameters used for diﬀerent source resistances the litz wire 4-coil system 3.2. Comsol 23

3.2 Comsol

To make simulations in Comsol you have to have a CAD model of the system, it can be in 2D or 3D. Comsol provides and internal CAD software, it also enables geometrical properties as parameters that can be swept. 2D models are simple and quite fast to simulate, which works well for some systems. 3D models require more boundary conditions and larger mesh areas, which makes them more complex therefore also computationally heavy. As a middle ground, there is 2D-axisymetrical. It will revolve a 2D solution around a symmetry axis creating a 3D model. This will let you have the beneﬁts of both models, but the system have to have a symmetry axis.

3.2.1 Simulation setup The model of the power transfer system has symmetry along the separation axis and can therefore be modeled as 2D-axisymetrical. The system is drawn up in 2D, the coils are made of rectangles with rounded corners. When revolved they make discs with a hole in the center. This will emulate a pancake coil made of wire, wound in a spiral pattern. The Comsol simulations are set up to use Magnetic Fields for the coils and the energy transfer. The discs are set to multi-turn coil domains, which makes them behave as they are made of wire. The coils are set to be made of copper and the surroundings of air. The mesh is set up using boundary layers for the coil, this is because the coils will suffer from skin effect and therefore most of the current will be on the surface and it is good with small mesh to not lose accuracy. The surroundings will have a fine triangular mesh. The rest of the system, source, load and resonance capacitors are simulated as Electrical circuits. The components are added and connected by their node numbers. The optimal parameters for the systems, found in Matlab, are implemented. The resonator capacitors have to be tuned separately for each coil, to achieve resonance everywhere. The distance between the coils are then being swept and the efficiency is plotted as power received/power sent, the data is then exported and plotted in Matlab. The magnetic and electric field plots are made by revolving the 2D model and then looking at the fields in a plane. 24 The third chapter

3.2.2 Simulation results 2-coil system Simulations are made with the optimal parameters for the 2-coil system, Table 3.2. Figure 3.8 shows an overview of the 2-coil setup. The disc in the top right corner is the transmitter and the one in the bottom left corner is the receiver. The electrical circuits used for the simulation is shown in Fig. 3.9, where L2 and L3 are the coils from Fig. 3.8.

Figure 3.8: Representative 3D picture of the 2-coil system

R C2 C3 s k23

L2 L3 Vs RL R2 R3

Figure 3.9: The electrical circuits simulated in Comsol, where L2,3 are the coils in Fig. 3.8

Figure 3.10 shows the PTE as a function of the distance between the transmitter and receiver. Figures 3.11a and 3.11b shows the electric and magnetic ﬁeld when d23 is 80mm. 3.2. Comsol 25

100

50 PTE [%]

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 d [m] 23

Figure 3.10: PTE for the 2-coil system simulated in Comsol

(a) Electric ﬁled (b) Magnetic ﬁled

Figure 3.11: Comsol simulations of the 2-coil system with a transfer distance, d23, of 80mm 26 The third chapter

3-coil system Simulations are made with the optimal parameters for the 3-coil system, Table 3.4. Figure 3.12 shows an overview of the coils in the setup. The disc in the top right corner is the transmitter, the middle disc is the resonator and the one in the bottom left corner is the receiver. The electrical circuits used for the simulation is shown in Fig. 3.13, where L2, L3 and L4 are the coils from Fig. 3.12.

Figure 3.12: Representative 3D picture of the 3-coil system

R C2 C4 s k23 k34

L2 L3 L4 Vs C3 RL R2 R3 R4

Figure 3.13: The electrical circuits simulated in Comsol, where L2,3,4 are the coils in Fig. 3.12

Figure 3.14 shows the PTE as a function of the distance between the transmitter and the resonating coil. Figures 3.15a and 3.15b shows the electric and magnetic ﬁeld when d23 is 80mm, the transmitter is the top coil. 3.2. Comsol 27

100

50 PTE [%]

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 d [m] 23

Figure 3.14: PTE of 3-coil system simulated in Comsol

(a) Electric ﬁeld (b) Magnetic ﬁeld

Figure 3.15: Comsol simulations of the 3-coil system with a transfer distance, d23, of 80mm 28 The third chapter

4-coil system Simulations are made with the optimal parameters for the 4-coil system, Table 3.6. Figure 3.16 shows an overview of the coils in the setup. The disc in the top right corner is the transmitter, the disc next to it and the one in the middle are resonators and the one in the bottom left corner is the receiver. The electrical circuits used for the simulation is shown in Figure 3.17, where L1, L2, L3 and L4 are the coils from Fig. 3.16.

Figure 3.16: Representative 3D picture of the 4-coil system

R C1 C4 s k12 k23 k34

L1 L2 L3 L4 Vs C2C3 RL R1 R2 R3 R4

Figure 3.17: The electrical circuits simulated in Comsol, where L1,2,3,4 are the coils in Fig. 3.16

Figure 3.18 shows the PTE as a function of the distance between both resonating coils.

Figures 3.19a and 3.19b shows the electric and magnetic ﬁeld when d23 is 80mm, the 3.2. Comsol 29 transmitter is the top coil.

100

50 PTE [%]

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 d [m] 23

Figure 3.18: PTE of 4-coil system simulated in Comsol 30 The third chapter

(a) Electric ﬁled (b) Magnetic ﬁled

Figure 3.19: Comsol simulations of the 4-coil system with a transfer distance, d23, of 80mm CHAPTER 4 Electronic design

This chapter describes the design of the electrical circuits used for the real tests and states the components that are used.

4.1 Tuning of the coils

A coil is primary inductive but the there are always wire resistance and parasitic capacitance present. The impedance of the coil will look like 1 Z(s) = Ls + Rw + (4.1) Cps At a certain frequency the inductance and capacitance will cancel each other and the coil is deﬁned only by the wire resistance. That is the resonance frequency and all coils in a WPS must operate at the same resonance frequency to have eﬃcient power transfer. The resonance frequency is given by 1 ω = √ . (4.2) LC To be able to tune this frequency a capacitor is added either in parallel or in series with the inductance, Eq. 4.2 is true for both cases. The added capacitor is usually much larger than the parasitic capacitance and therefore the parasitic capacitance does not need to be included in Eq. 4.2.

4.2 Source

The most important properties of the source is low output resistance and a good sine output. This is because the resistance will be directly added to the wire resistance of the primary coil, signiﬁcantly decreasing the quality factor, hence the eﬃciency. It also has

31 32 The fourth chapter to be able to handle high frequencies and have high output power. There are a variety of circuits that might be suitable for this: a power oscillator, high-frequency amplifier or a half/full bridge. A power oscillator has the advantage that it will run at resonance frequency without any tuning. But reliable and stable oscillators is hard to make. A bridge circuit has an external clock that sets the operating frequency which adds one more degree of tuning but the circuit is very stable. The downside is that it will have issues at frequencies above 1MHz. The RF amplifier also requires an external clock signal to set the frequency but it can be made very effective and made to work with very high frequencies. The E-class amp is therefore the choice in this thesis.

4.3 E-class ampliﬁer

The E-class is a switch mode ampliﬁer designed for very high power eﬃciency. It is non linear in the sense that the output amplitude does not corresponds to the input amplitude. In order to adjust the output amplitude the supply voltage can be adjusted. It is primary made for high frequency applications and is commonly used to create carrier waves for radio transmissions. Fig. 4.1 shows an ideal E-class circuit.

+Vcc

Lc L C

T1 Cs RL

Figure 4.1: E-class ampliﬁer

It consists of a RF choke Lc, a switch T1, a shunt capacitor (which includes the transistor capacitance) Cs, a load network L-C and a load RL. The switch is operated at the desired output frequency, for maximum efficiency a 50% duty cycle is used. The load network gives the required phase shift to prevent high current and high voltage at the switch ,transistor, and acts as a open circuit for the first harmonic, passing a sine wave to the load. There are some conditions that should be fulfilled in order to make it as efficient as possible, when the switch goes from open to closed the voltage over the switch, VT 1 is suppose to behave as

VT 1 = 0, (4.3) dV T 1 = 0. (4.4) dt 4.4. Simulation 33

To fulﬁll these conditions the following expressions have been derived [18]. The power delivered to the load is given by 2 (Vcc − Vo) 0.451759 0.402444 RL = 0.576801 1.001245 − − 2 , (4.5) P QL QL where Vcc is the supply voltage, Vo the transistor saturation voltage (0 for FET transis- tors), RL is the load, and QL is the loaded quality factor. The shunt capacitor is then given by 1 0.91424 1.03175 0.6 C = 0.99866 + − + , (4.6) s π2 π Q Q2 2 2πfRL 4 + 1 2 L L (2πf) Lc where Lc is the choke inductor and the load network 1 1 1.01468 0.2 C = 1.00121 + − 2 (4.7) 2πfRL QL − 0.104823 QL − 1.7879 (2πf) Lc and Q R L = L L . (4.8) 2πf

The design choices left to do is to specify the supply voltage Vcc, the output power P and the quality factor QL.

4.4 Simulation

4.4.1 Component selection

Setting Vcc to 5V , RL to 1Ω, Lc to 100µL and QL to 10 and using Eq. 4.5-4.8 gives the component values displayed in Table 4.1. The switch is replaced by an MOSFET transistor, IRFB5615PBF, which has a low Rdson and small gate charge in order to be eﬃcient and easy to drive. A gate driver is used to drive the transistor, to make it as close to an ideal switch as possible. The capacitors are polyester and for the resonators a variable capacitor is used for easier tuning. All the components used are shown in Table 4.1.

Transistor IRFB5615PBF Gate driver lm5104 C 6.6nF Cs 11.5nF L 1.1uH Load resistance 1.37Ω

Table 4.1: Design limits 34 The fourth chapter

4.4.2 PSpice With all components selected the E-class amplifier is implemented in PSpice [19]. In PSpice the transient behavior of the circuit is studied. What is important is that the voltage and current behaves similar to Eq. 4.3 and 4.4 and that the output is a nice sine wave. Sokal [18] describes in his report a way to fine tune the component values to get the best transistor current and voltage curves as possible. After the tuning the component values differ slightly from Table 4.1 and because the transistor capacitance is parallel with the shunt capacitor is it removed from the shunt value. The load RL is replaced by a series RLC circuit, representing the transmitting coil and capacitance. The resistor is the coil resistance and at the resonance frequency the inductor and capacitor will cancel out leaving only the resistance. The 100MΩ resistor is only added to make simulations possible and will not affect the circuit. Figure 4.2 shows the schematic of the complete and tuned circuit. The transistor current and voltage along with the output voltage are shown in Fig. 4.3.

50F F

000uH 000

000 000 00 200 2u0

00u 99 FuFn 0u0900u 0un00uun 000 u0Fuuu9u

50u0u00 50u n 200 0n0 50u0u0 F0FFFF0FFFF 00u0n 000g7g 00u0u0 00u0u00n 0u9 0Fu0u00n 0uuu0 F7u0u0uF77u F70u0uF77u 0 0 0 0 0

Figure 4.2: PSpice schematic of the E-class ampliﬁer 4.4. Simulation 35

Transistor voltage [V] 5 Transistor current [A] 4

1.066 1.067 1.068 1.069 1.07 1.071 1.072 1.073 1.074 1.075 −4 x 10

200

100

0 Voltage [V] −100

−200 1.06 1.062 1.064 1.066 1.068 1.07 1.072 1.074 1.076 1.078 1.08 −4 Time [s] x 10

Figure 4.3: PSpice simulation of the voltage over the transistor, the current through the transistor and the output voltage

CHAPTER 5 Real testing

This chapter contains the real tests of the WPS. The results from the simulations in Ch. 3 are tested to see how they hold up in reality.

5.1 Measurement setup

The E-class power ampliﬁer designed in Ch. 4 is built and serves as the source for all measurements. The supply voltage is taken from a voltage cube with a current limitation of 2A and the clock signal used to trigger the gate driver comes form a 5V square wave generator. The coils are made according to the optimal parameters found in the simulations, Ch. 3, and capacitors are added and tuned to make all coils resonate at the desired 2MHz. The transmitting circuit is connected to the E-class ampliﬁer and the receiving to a 10Ω resistor.

5.2 Test procedure

The transmitter creates a large magnetic field which makes it hard to make accurate measurements on the transmitter. The oscilloscope probe and ground clamp creates a wire loop in which there are induced voltage. Therefore the power delivered from the voltage cube will be measured instead. On the receiver it is possible to use a probe because the voltage over the load resistance is much bigger than the induced voltage in the probe. The PTE for the real test is then (power received by load)/(power delivered from voltage cube). The PTE will then differ from the simulations because of the losses in the amplifier. The measurements are related to the distance between the coils as in the simulations and are made every other centimeter to enable good plotting.

37 38 The fifth chapter

5.3 Measurements on 2-coil system

The E-class ampliﬁer is connected to the transmitter circuit which is tuned to resonate at 2MHz and a load resistor, 10Ω, is connected to the receiver circuit. Fig. 5.1 show the measurement setup and Fig. 5.2 shows the measured PTE.

Figure 5.1: Measurement setup of the 2-coil system

PTE [%] 8

0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 d [m] 23

Figure 5.2: PTE measurements of the whole 2-coil system 5.4. Measurement on 3-coil system 39

5.4 Measurement on 3-coil system

The E-class ampliﬁer is connected to the transmitter circuit which is tuned to resonate at 2MHz. A load resistor, 10Ω, is connected to the receiver circuit and a resonator circuit is added between the transmitter and receiver at a distance of 31mm from the receiver. Fig 5.3 show the measurement setup and Fig. 5.4 shows the measured PTE.

Figure 5.3: Measurement setup of the 3-coil system

20 PTE [%]

0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 d [m] 23

Figure 5.4: PTE measurements of the whole 3-coil system 40 The fifth chapter

5.5 Measurements on 4-coil system

The E-class ampliﬁer is connected to the transmitter circuit which is tuned to resonate at 2MHz. A load resistor, 10Ω, is connected to the receiver circuit and two resonator coils are added between the transmitter and receiver. One resonator is placed 1mm from the transmitter and the other 31mm from the receiver. Fig 5.5 show the measurement setup and Fig. 5.6 shows the measured PTE.

Figure 5.5: Measurement setup of the 4-coil system

0.25

0.2

0.15 PTE [%]

0.1

0.05

0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 d [m] 23

Figure 5.6: PTE measurements of the whole 4-coil system CHAPTER 6 Discussion

This chapter summarizes the work done in this thesis and discusses issues and future work in this area. The key parts are as follows

• The WPS models for 2, 3 and 4 coils are extended with the theory of the individual components, implemented as Matlab models with actual material and geometrical properties as input parameters.

• An optimization technique is deﬁned and a simulator is created that ﬁnds the optimal set of input parameters in Matlab which results in the best PTE curve.

• The simulator have found optimal setups for 2, 3 and 4 coil systems and the validity of theses systems are backed up by FEM simulations of the electromagnetic ﬁelds made in Comsol Multiphysics

6.1 Simulations

Choosing design limits that makes the system reasonable sized and using these in the Matlab simulator for the different models gives a set of optimized parameters for each of the three cases. A Comsol model, for each of the cases are made and the PTE and electromagnetic fields are looked at. Comparing the PTE’s from Matlab and Comsol shows fairly good resemblance for all cases. Some deviations were to be expected due to the fact that the system models in Matlab and some component expressions were approximations. But the similarity of the curves and transfer distances gives the Matlab simulations validity. From the electromagnetic simulations it is evident that adding one or two resonating coils really helps to couple the magnetic and electric fields to the receiver and transmitter (4-coil system). It is due to that the resonator coils works as impedance matching systems for the source and load resistances. The electromagnetic field plots also show that both fields rapidly goes to zero in the free space.

41 42 The sixth chapter

The simulations with litz wire are very limited due to the lack of analytical formulas or approximations. The only available way to model it with geometrical properties is by tabled values from the manufacturers, therefore there are problems to sweep the wire radius. But the simulations using 0.39mm litz wire gives PTE curves that only have <20% shorter transfer distance than models using magnet wire with variable radius (0- 1mm). That indicates that litz wire probably is beneficial and a proper model would be of interest. The extended models of the WPS’s, implemented in Matlab, could be useful in a couple of ways, an existing magnetic resonance system could be simulated just by having the geometrical and material properties. The simulator can then be customized to suit different needs e.g. an existing 3- or 4-coil system could be made better without changing the coils only by finding optimal distance for the resonator coils. Or as presented in this thesis, an optimal system can be created from the ground just by setting the design limits.

6.2 Real tests

The measurements will always differ from the simulations because the PTE of the measurements is over the complete system including the amplifier, while the PTE of the simulations is only between the transmitter and receiver coils. From the figures 5.2, 5.4 and 5.6 it is evident that the measurements don’t hold up to the simulations. What can be seen in the 2- and 3-coil cases is that even though the transfer distance doesn’t hold up with the simulations the shape of the curves is similar. The transfer distance of the 3 coil case is, as in the simulations, significantly longer than with only two coils. This proves that adding at least one resonator coil will improve the transfer distance. The 4-coil case is definitely the worst with only 0.2 % efficiency at the most. But that is not a surprise, each coil adds degrees of freedoms and uncertainties to the system. All coils have to be carefully tuned because of the mutual inductance, the resonance frequency of one coil depends on the other coils especially the closest one. To manually tune four coil therefore becomes quite a difficult task, which might be a reason for the poor results. Another reason is that the coils are not perfectly flat which makes it hard to align the resonator coil precisely 1mm from the transmitter, thus the impedance matching might be off.

6.3 Conclusions

The Matlab simulator gives a way to optimize the PTE for a system given the design limits. Systems suitable for portable devices are designed which in the simulations can transfer power up to four times the coil diameter. In that sense the goal of the thesis is 43 met but the real system can not conﬁrm the results from the simulations. Looking at both the simulations and the real tests it is obvious that adding a resonator coil close to the receiver signiﬁcantly improves the transfer distance. The Matlab simulation also show that adding a second resonator coil, close to the transmitter, decreases the dependence on the source resistance enabling similar curves as with no resistance. This could not be simulated using my Comsol model because the PTE doesn’t include the source resistance.

6.4 Health issues

A question that comes to mind when hearing about wireless power is: Is it safe? No study regarding safety and health was made in this thesis. However, IEEE [20] and ICNIPR [21] have set up guidelines for exposure to RF electromagnetic fields. They both conclude that there is no established evidence that RF electromagnetic fields causes cancer, but there is evidence that they can cause heating in body tissue and stimulate muscle and nerve tissue. Both IEEE and ICNIPR concludes that even the most sensitive tissue is not adversely effected when the whole body average SAR level is less than 4 [W/kg], which corresponds to a maximum rise of 1◦C in body temperature. But they both recommend using a safety factor and therefore sets the limit at 0.08 [W/kg] for the general public. WiTricity have made a study [22] investigating the safety of their WPT system. They simulate SAR levels in a human body, for both their high and low power (3kW/5W) systems. The study concludes that it is safe and that the SAR levels are below the limit set for the general public, for both cases. The thing you can’t predict or simulate are the longterm effects of exposure RF electromagnetic fields. But with current knowledge this technology can be regarded as safe to use in consumer products.

6.5 Future work

The Matlab simulator could be improved by adding a model that supports litz wires with variable diameter, because of the beneﬁcial frequency properties. Then complete litz as well as mixed systems could be evaluated. To conﬁrm the Matlab model it would be a good idea to make better real tests with more professionally made coils, exact tuning as well as doing the measurements with a network analyzer.

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[13] H. A. Wheeler, “Simple inductance formulas for radio coils,” pp. 1398–1400, 1928.

45 46

[14] M. N. O. Sadiku, Elements of Electromagnetics. New York: Oxford University Press, 4th ed., 2007.

[15] A. E. Siegman, Lasers. University Science Books, 1986.

[16] H. A. Haus and W. Huang, “Coupled-mode theory,” 1991.

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[18] N. Sokal, “Class-e rf power ampliﬁers,” 2001.

[19] OrCAD Capture, version 16.6, PSpice Plugin. Cadence Design Systems, Inc., 2012.

[20] IEEE Std. C95.1-2005, “IEEE Standard for Safety Levels with Respect to Human Exposure to Radio Frequency Electromagnetic Fields, 3 kHz to 300 GHz,”

[21] ICNIRP Guidelines, International Commission on Non-Ionizing Radiation Protec- tion, Health Physics, 74, no. 4, “Guidelines for limiting exposure to time-varying electric, magnetic and electromagnetic ﬁelds (up to 300 ghz),” pp. 494–522, 1998.

[22] M. Keller, Highly Resonant Wireless Power Transfer: Safe, Eﬃcient, and over Dis- tance. WiTricity Corp., 2013.