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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

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Page 1: MASTER'S THESIS - diva-portal.org1031874/FULLTEXT02.pdfPhysics and Electrical Engineering at Lulea University of Technology, LTU. During my project work course I rst came into contact

MASTER'S THESIS

Optimization of Wireless Power

Oskar Rönnbäck2013

Master of Science in Engineering TechnologyEngineering Physics and Electrical Engineering

Luleå University of TechnologyDeptartment of Computer Science, Electrical and Space Engineering

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Optimization of Wireless Power

Oskar Ronnback

Lulea University of TechnologyDept. of Computer Science, Electrical and Space Engineering

December 5, 2013

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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 creat-

ing 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

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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

v

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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

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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

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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

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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.

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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

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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.

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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.

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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 ResistanceResistance 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

Rdc =ρl

A. (2.2)

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

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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.

δ =

√2ρ

ωµ. (2.3)

Where ω is the angular frequency and µ is the absolute mag-

netic permeability of the conductor. The resistance for highFigure 2.1: Skin depth

frequency currents therefore has a more complex expression [11]

Rac =ρ√

2δπrw

[Ber(q)Bei′(q)−Bei(q)Ber′(q)

Ber′(q)2 +Bei′(q)2

]Ω/m (2.4)

where

q =

√2rwδ

, (2.5)

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 proxim-

ity 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

values, developed by a manufacturer, is described in [12].Figure 2.2: Litz wire

2.2 InductanceInductance is a property of a conductor, the electromagnetic definition of inductance L

is the ratio of magnetic flux linkage λ to the current I

L =λ

I. (2.6)

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.

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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]

L = n2µ0 (2r − rw)

[(1− k2

2

)K (k)− E (k)

](2.7)

where

k =

√4r(r − rw)

(2r − rw)2, (2.8)

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 spi-

ral coils, differ from Eq. 2.7. A better approximation

of the inductance in pancake coils is given by [13]

L =r2n2

8r + 11w, (2.9)

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

coil2.2.3 Inductor quality factor

Quality factor is a measure on how ideal an inductor is and is defined by

Q =2πfL

R(2.10)

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

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10 The second chapter

[14]

M12 = M21 =λ12

I2

=λ21

I1

, (2.11)

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

be parallel and perfectly aligned, as M12 ∝ 1/d312. For two circular single turn wire coils,

with radius r1 and r2, the mutual inductance is given by

M12 =2µ

α

√r1r2

[(1− α2

2

)K(α)− E(α)

]; α = 2

√r1r2

(r1 + r2)2 + d212

(2.12)

K(α) =

∫ π/2

0

dφ√1− α2sin2(φ)

; E(α) =

∫ π/2

0

√1− α2sin2(φ)dφ (2.13)

For coils with multiple turns the mutual inductance becomes

Mtot =

N1∑i=1

N2∑j=1

Mij. (2.14)

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

k =M12√L1L2

. (2.15)

2.3 InductionInduction 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]

Vemf = −N dΨ

dt, (2.16)

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

Vemf = −N d

dt

∫S

B · dS, (2.17)

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

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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 ResonanceResonance 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]

I(ω) ∝(

Γ2

)2

(ω − Ω)2 +(

Γ2

)2 (2.18)

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.

Vs

R L

C

Figure 2.4: Series RLC-circuit

The impedance is given by

Z(jω) = R +1− ω2LC

jωC. (2.19)

At the frequency ω = 1/√LC the imaginary parts cancels and the system starts to

resonate.

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12 The second chapter

2.5 Wireless power using magnetic resonanceTo 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 TheoryCoupled 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

L

i

C

v

Figure 2.5: Parallel LC-circuit

v = Ldi

dt, (2.20)

i = −Cdvdt. (2.21)

These equations, 2.20 and 2.21, can be combined to a second order differential equation

d2v

dt2+ ω2v = 0, (2.22)

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2.6. Coupled Mode Theory 13

where the resonant frequency is ω = 1/√LC. Using coupled mode theory a complex

amplitude is defined as

a(t) =

√C

2

(v(t)− j

√L

Ci(t)

), (2.23)

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)

dt= −jωa(t). (2.24)

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

decay and the excitation by Fs, where |s|2 is the input power. The model then becomes

da(t)

dt= − (jω + Γ) a(t) + Fs(t). (2.25)

2.6.2 Model of lossy 2-coil coupled system

Vs

RsC2

L2

R2

k23

R3

L3

C3

RL

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

da2(t)

dt= − (jω2 + Γ2) a2(t) + Fs(t) + jka3(t), (2.26)

da3(t)

dt= − (jω3 + Γ3 + ΓL) a3(t) + jka2(t), (2.27)

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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

Pi = 2Γi|ai|2 (2.29)

and in the load

PL = 2ΓL|aL|2. (2.30)

With Eq. 2.28 and 2.34 the efficiency can be stated as

η2−coil =PLPs

=1

1 + Γ3

ΓL

[1 + Γ2Γ3

K223

(1 + ΓL

Γ3

)2] , (2.31)

where K23 is the coupling rate.

2.6.4 WPT expanded to 3- and 4-coil systems

Vs

RsC2

L2

R2

k23

C3

L3

R3

k34

R4

L4

C4

RL

Figure 2.7: Circuit diagram of 3-coil system

Vs

RsC1

L1

R1

k12

R2

L2

C2

k23

C3

L3

R3

k34

R4

L4

C4

RL

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

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2.7. Reflected Load Theory 15

of coils

PS =m∑i=1

Pi + PL. (2.32)

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

ηm−coil =PLPs

=ΓL

Γm + ΓL +∑m−1

i=1 Γi

∣∣∣ Ai

Am

∣∣∣2 . (2.33)

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

η3−coil =K23K34ΓL

Γ2 [K234 + Γ3 (Γ4 + ΓL)]

2+K2

23

[Γ3 (Γ4 + ΓL)2]+K2

23K234 (Γ4 + ΓL)

. (2.34)

2.7 Reflected Load TheoryReflected 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]

Rref = k223ωL2QL3 (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

η2 =k2

23Q2Q3L

1 + k223Q2Q3L

· Q3L

QL

. (2.36)

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

ηi,i+1 =k2i,i+1QiQ(i+1)L

1 + k2i,i+1QiQ(i+1)L

. (2.37)

The PTE for the full system with m coils is achieved by

ηm =m−1∏i=1

ηi,i+1 ·Q3L

QL

. (2.38)

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16 The second chapter

2.8 Unified theoryIt 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

η3 =k2

23k234Q2Q3Q4L[

(1 + k223Q3Q4L)

2+ k2

23Q2Q3 (1 + k223Q3Q4L)

] · Q4L

QL

(2.39)

and The PTE of a 4-coil system, Fig. 2.8, by setting m = 4 in Eq. 2.38

η4 =(k2

12Q1Q2) (k223Q2Q3) (k2

34Q3Q4L)

[(1 + k212Q1Q2) (1 + k2

34Q3Q4L) + k223Q2Q3] [1 + k2

23Q2Q3 + k234Q3Q4L]

·Q4L

QL

(2.40)

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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 expres-

sions 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 simulationsThe 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 loadRL 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

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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

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3.1. Matlab simulations 19

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

90

100

PT

E [%

]

d23

[m]

(a) High and short plateau

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

90

100

PT

E [%

]

d23

[m]

(b) Low and wide plateau

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

PT

E [

%]

d23

[m]

(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.

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20 The third chapter

2-coil system

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

90

100

PT

E [%

]

d23

[m]

Rs = 0Ω

Rs = 1Ω

Rs = 10Ω

Figure 3.2: PTE for the magnet wire 2-coil system

with different source resistances

Design choices

Coil type Pancake

Wire type Cu, singel strand

Frequency 2MHz

Optimal parameters

r2 45mm

r3 45mm

rw2 1mm

rw3 1mm

n2 20

n3 1

Table 3.2: Parameters used for

the magnet wire 2-coil system

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

90

100

PT

E [%

]

d23

[m]

Rs = 0Ω

Rs = 1Ω

Rs = 10Ω

Figure 3.3: PTE for the litz wire 2-coil system with

different source resistances

Design choices

Coil type Pancake

Wire type Cu, Litz

Frequency 2MHz

rw2 0.39mm

rw3 0.39mm

Optimal parameters

r2 45mm

r3 45mm

n2 20

n3 1

Table 3.3: Parameters used for

the litz wire 2-coil system

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3.1. Matlab simulations 21

3-coil system

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

PT

E [%

]

d23

[m]

Rs = 0Ω

Rs = 1Ω

Rs = 10Ω

Figure 3.4: PTE for the magnet wire 3-coil system

with different source resistances

Design choices

Coil type Pancake

Wire type Cu, singel strand

Frequency 2MHz

Optimal parameters

d34 31mm

r2, r3, r4 45mm

rw2, rw3 1mm

rw4 0.4mm

n2, n3, n4 20

Table 3.4: Parameters used for

the magnet wire 3-coil system

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

PT

E [%

]

d23

[m]

Rs = 0Ω

Rs = 1Ω

Rs = 10Ω

Figure 3.5: PTE for the litz wire 3-coil system with

different source resistances

Design choices

Coil type Pancake

Wire type Cu, Litz

Frequency 2MHz

rw2, rw3 0.39mm

Optimal parameters

d34 31mm

r2, r3 45mm

r4 33mm

n2, n3 20

n4 14

Table 3.5: Parameters used for

the litz wire 3-coil system

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22 The third chapter

4-coil system

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

PT

E [%

]

d23

[m]

Rs = 0Ω

Rs = 1Ω

Rs = 10Ω

Figure 3.6: PTE for the magnet wire 4-coil system

with different source resistances

Design choices

Coil type Pancake

Wire type Cu, singel strand

Frequency 2MHz

Optimal parameters

d12 1mm

d34 31mm

r1, r2, r3, r4 45mm

rw1, rw2, rw3 1mm

rw4 0.4mm

n1, n2, n3, n4 20

Table 3.6: Parameters used for

the magnet wire 4-coil system

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

PT

E [%

]

d23

[m]

Rs = 0Ω

Rs = 1Ω

Rs = 10Ω

Figure 3.7: PTE for the litz wire 4-coil system with

different source resistances

Design choices

Coil type Pancake

Wire type Cu, Litz

Frequency 2MHz

rw1, rw2, rw2, rw2 0.39mm

Optimal parameters

d12 1mm

d34 31mm

r1, r2, r3 45mm

r4 33mm

n1, n2, n3 20

n4 14

Table 3.7: Parameters used for

the litz wire 4-coil system

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3.2. Comsol 23

3.2 ComsolTo 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 sim-

ulate, 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.

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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

Vs

RsC2

L2

R2

k23

R3

L3

C3

RL

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.

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3.2. Comsol 25

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

90

100

PT

E [

%]

d23

[m]

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

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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

Vs

RsC2

L2

R2

k23

C3

L3

R3

k34

R4

L4

C4

RL

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.

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3.2. Comsol 27

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

90

100

PT

E [

%]

d23

[m]

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

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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

Vs

RsC1

L1

R1

k12

R2

L2

C2

k23

C3

L3

R3

k34

R4

L4

C4

RL

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

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3.2. Comsol 29

transmitter is the top coil.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

90

100

PT

E [

%]

d23

[m]

Figure 3.18: PTE of 4-coil system simulated in Comsol

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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

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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 coilsA coil is primary inductive but the there are always wire resistance and parasitic capac-

itance present. The impedance of the coil will look like

Z(s) = Ls+Rw +1

Cps(4.1)

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

ω =1√LC

. (4.2)

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 SourceThe 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

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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 amplifierThe 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.

T1 Cs

L C

RL

Lc

+Vcc

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, VT1 is

suppose to behave as

VT1 = 0, (4.3)

dVT1

dt= 0. (4.4)

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4.4. Simulation 33

To fulfill these conditions the following expressions have been derived [18]. The power

delivered to the load is given by

RL =(Vcc − Vo)2

P0.576801

(1.001245− 0.451759

QL

− 0.402444

Q2L

), (4.5)

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

Cs =1

2πfRL

(π2

4+ 1)π2

(0.99866 +

0.91424

QL

− 1.03175

Q2L

)+

0.6

(2πf)2 Lc, (4.6)

where Lc is the choke inductor and the load network

C =1

2πfRL

(1

QL − 0.104823

)(1.00121 +

1.01468

QL − 1.7879

)− 0.2

(2πf)2 Lc(4.7)

and

L =QLRL

2πf. (4.8)

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

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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.

0 0 0

0

0 0

000

00

99

F0FFFF0FFFF

000u0Fuuu9u

000000uH

200

FuFn

200

00u0n

50FF

000

0u0900u

2u0

0un00uun00u

n

0u9

0uuu0

50u

00u0u0

0Fu0u00nF7u0u0uF77uF70u0uF77u

50u0u00

00u0u00n

50u0u00n0

000g7g

Figure 4.2: PSpice schematic of the E-class amplifier

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4.4. Simulation 35

1.066 1.067 1.068 1.069 1.07 1.071 1.072 1.073 1.074 1.075

x 10−4

0

1

2

3

4

5

Transistor voltage [V]

Transistor current [A]

1.06 1.062 1.064 1.066 1.068 1.07 1.072 1.074 1.076 1.078 1.08

x 10−4

−200

−100

0

100

200

Voltage [V

]

Time [s]

Figure 4.3: PSpice simulation of the voltage over the transistor, the current through the tran-

sistor and the output voltage

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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 setupThe 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 procedureThe 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

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38 The fifth chapter

5.3 Measurements on 2-coil systemThe 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

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

2

4

6

8

10

12

14

16

18

PT

E [%

]

d23

[m]

Figure 5.2: PTE measurements of the whole 2-coil system

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5.4. Measurement on 3-coil system 39

5.4 Measurement on 3-coil systemThe 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

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

5

10

15

20

25

30

35

40

PT

E [%

]

d23

[m]

Figure 5.4: PTE measurements of the whole 3-coil system

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40 The fifth chapter

5.5 Measurements on 4-coil systemThe 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 0.05 0.1 0.15 0.2 0.25 0.3 0.350

0.05

0.1

0.15

0.2

0.25

PT

E [%

]

d23

[m]

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

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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 SimulationsChoosing 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

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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 testsThe measurements will always differ from the simulations because the PTE of the mea-

surements 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 ConclusionsThe 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

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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 simu-

lation 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 issuesA 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 1C 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 elec-

tromagnetic fields. But with current knowledge this technology can be regarded as safe

to use in consumer products.

6.5 Future workThe 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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REFERENCES

[1] A. Karalis, J. Joannopoulos, and M. Soljacic, “Efficient wireless non-radiative mid-

range energy transfer,” 2007.

[2] http://www.witricity.com.

[3] http://www.wirelesspowerconsortium.com.

[4] http://olev.kaist.ac.kr/en/.

[5] A. K. RamRakhyani, S. Mirabbasi, and M. Chiao, “Design and optimization of

resonance-based efficient wireless power delivery systems for biomedical implants,”

2011.

[6] B. Wang, K. H. Teo, T. Nishino, W. Yerazunis, J. Barnwell and J. Zhang, “Experi-

ments on wireless power transfer with metamaterials,” 2011.

[7] M. Kiani and M. Ghovanloo, “The circuit theory behind coupled mode magnetic

resonance based wireless power transmission,” 2012.

[8] A. P. Sample, D. A. Meyer and J. R. Smith, “Analysis, experimental results, and

range adaptation of magnetically coupled resonators for wireless power transfer,”

2011.

[9] MATLAB, version 8.1.0 (R2013a). Natick, Massachusetts: The MathWorks Inc.,

2013.

[10] COMSOL Multiphysics, version 4.3b. COMSOL AB, 2012.

[11] S. Ramo, J. R. Whinnery and T. Van Duzer, Field and Waves in Communication

Electronics. John Wiley & sons, 3rd ed., 1993.

[12] http://www.litzwire.com/nepdfs/Litz_Design_PDFs.pdf.

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

45

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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.

[17] E. Bou, E. Alarcon and J. Gutierrez, “A comparison of analytical models for resonant

inductive coupling wireless power transfer,” 2012.

[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.