Development of a High Quality Resonant Coil for Low Frequency Wireless Power Transfer

DEVELOPMENT OF A HIGH QUALITY RESONANT COIL FOR LOW FREQUENCY WIRELESS POWER TRANSFER

Kevin Van Acker

Supervisor: Prof. dr. ir. Guillaume Crevecoeur Counsellor: Ir. Matthias Vandeputte

Master's dissertation submitted in order to obtain the academic degree of Master of Science in de industriële wetenschappen: elektromechanica

Department of Electrical Energy, Metals, Mechanical Constructions & Systems Chair: Prof. dr. ir. Luc Dupré Faculty of Engineering and Architecture Academic year 2017-2018

DEVELOPMENT OF A HIGH QUALITY RESONANT COIL FOR LOW FREQUENCY WIRELESS POWER TRANSFER

Kevin Van Acker

Supervisor: Prof. dr. ir. Guillaume Crevecoeur Counsellor: Ir. Matthias Vandeputte

Master's dissertation submitted in order to obtain the academic degree of Master of Science in de industriële wetenschappen: elektromechanica

Department of Electrical Energy, Metals, Mechanical Constructions & Systems Chair: Prof. dr. ir. Luc Dupré Faculty of Engineering and Architecture Academic year 2017-2018

Preface

Before you lies the dissertation “Development of a High Quality Resonant Coil for Low Frequency Wireless Power Transfer”, the basis of this dissertation is to find a way to easily develop an optimal resonant coil for low frequency wireless power transfer by only having some boundary conditions for the dimensions of this coil. The dissertation has been written to obtain the academic degree of Master of Science in de industriële wetenschappen: elektromechanica at the faculty of Engineering and Architecture at Ghent university.

I was engaged in researching and writing this dissertation from January to June 2018. The title and goal of this dissertation was formulated by my supervisor, Prof. dr. ir. Guillaume Crevecoeur, and my counsellor, ir. Matthias Vandeputte. Throughout the researching period the goal of this dissertation was refined by my counsellor and myself.

The research I’ve done was challenging and seemingly never-ending, but I kept being interested in the subject and the diversity of the things (like coding, math, designing, manufacturing, etc …) that I have done for this research really did help with that.

First, I would like to thank my counsellor ir. Matthias Vandeputte for his great counselling. He always came up with great ideas to research. When I had a problem he always made time to help me solve them, he basically taught me a new programming language (Matlab) from scratch in a few weeks. The list goes on but I have to keep this short so I will just say that I am very grateful to have had a counsellor like ir. Vandeputte. Secondly, I have to also thank my supervisor Prof. dr. ir. Guillaume Crevecoeur. He followed up my master’s dissertation very well, made time for my rehearsal presentations in his busy schedule and gave valuable feedback.

I would also thank my loving girlfriend to provide mental support throughout this sometimes stressful period and listening to my complaints and moans. My friends and family I would also like to thank for supporting me. My parents deserve a special thanks for making my studies possible and also to be so understanding of my absence these months, it was sometimes hard to accept that evenings and weekends were used to do work for this dissertation and school, especially because I was the first in my near family to pursue a university/college degree, but through all this time they still kept supporting me.

I hope you enjoy your reading

Kevin Van Acker

"De auteur geeft de toelating deze masterproef voor consultatie beschikbaar te stellen en delen van de masterproef te kopiëren voor persoonlijk gebruik. Elk ander gebruik valt onder de bepalingen van het auteursrecht, in het bijzonder met betrekking tot de verplichting de bron uitdrukkelijk te vermelden bij het aanhalen van resultaten uit deze masterproef."

"The author gives permission to make this master dissertation available for consultation and to copy parts of this master dissertation for personal use. In the case of any other use, the copyright terms have to be respected, in particular with regard to the obligation to state expressly the source when quoting results from this master dissertation."

Ghent, 24 June 2018

Abstract

In order to obtain the academic degree of Master of Science in de industriële wetenschappen: elektromechanica, this master's dissertation was submitted to the department of Electrical Energy, Metals, Mechanical Constructions & Systems in the academic year 2017-2018. The chair of this department is Prof. dr. ir. Luc Dupré. In this master’s dissertation a way to develop a high quality resonant coil for wireless power transfer is created. The principles of the strongly coupled magnetic resonance will be explained and investigated. Analytical and empirical formulas were used from literature to obtain characteristics of resonators that would be used by giving only the dimensions of the resonators. This was then all combined in a model that could predict the power transfer efficiency. Some optimization studies were conducted to investigate the optimal dimensions for power transfer with strongly coupled resonators. A way to find the optimal load resistance was found by using an optimization function with the model. An easy way to optimize the dimensions of the resonators was also found by using a Matlab code. The optimal position for a five-resonator power transfer system was also investigated and found, the optimal position for the resonators is not uniformly spacing these resonators over the desired transfer distance but rather have the second resonator be closer to the first resonator where the source is connected to and the second last resonator closer to the last resonator.

Extended abstract EN Development of a High Quality Resonant Coil for Low Frequency Wireless Power Transfer

Kevin Van Acker

Supervisor: Prof. dr. ir. Guillaume Crevecoeur Counsellor: Ir. Matthias Vandeputte

Abstract: In this dissertation wireless power The currents that flow in these resonators will be transfer via strongly coupled resonant coils will investigated and modelled. After that a model that be investigated. The basics of how energy is can predict these currents will be made with formulas transferred will be shown. The main focus is to found in literature. The formulas will be validated with create a method to develop a high quality some measurements. When the individual formulas resonant coil. This will be done by creating a seem to be correct for these measurements, predicting model in Matlab that will predict the measurements to validate the whole model can then power transfer efficiency of a certain resonator be performed. system in resonance.

II. PRINCIPLES OF SCMR I. INTRODUCTION Using the strongly coupled magnetic resonance In recent years wireless power transfer has wireless power transfer method, power is transferred gained in popularity for consumer devices, e.g. by using a magnetic field. The magnetic field is charging smartphones and other devices wirelessly. created by the current that runs through the coil of There are a few ways to transfer energy wirelessly. the first resonator, as stated by Ampères law. An The method that will be discussed here is strongly alternating magnetic field is induced by the coupled magnetic resonance. This method has a alternating current that runs through the first coil. A lower power transfer efficiency than other methods second resonator is placed near the first resonator for short (a few centimetres) and long distances (a so that the coils of the resonators are magnetically few dozens of metres to kilometres) but seems to be coupled with each other. The coil of the second the best solution for mid-range applications (range of resonator is now in a changing magnetic field a few centimetres to a couple of metres). because of the alternating current in the first Applications for strongly coupled magnetic resonator. Because of Faraday’s law, the changing resonance can be electric vehicles, robots and other magnetic flux of the magnetic field will generate a electronic devices that needs to be charged at a changing electric field in the coils in the form of an distance for optimal convenience. electromotive force. This electromotive force is then Strongly coupled magnetic resonance (SCMR) the source in the second resonator that will drive will be investigated in this dissertation. This method current through the components of the second of wireless power transfer uses at least two resonator. resonators. Resonators are electric circuits that consists out of a coil and a capacitor. To transfer power with these resonators a high frequency voltage source needs to be connected to one resonator (primary resonator). A load then needs to Figure 1: Circuit of a two resonator power transfer system. be connected to the other resonator (secondary resonator). In Figure 1 a representation of the two resonators can be seen. When the power of the source is

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derived by using circuit theory the following The expression for the efficiency that was previously expression for the input power of the system can be shown was for a two-resonator system that was stated (Huang, Zhang, & Zhang, 2014): directly fed by a voltage source. There is also 2 2 2 2 {푅1[(푅2 + 푅퐿) − 푋1 ] + (휔푀12) (푅2 + 푅퐿)}푉푠 another method that is used to achieve SCMR, this 푃푖푛 = 2 2 2 [푅1(푅2 + 푅퐿) − 푋1푋2 + (휔푀12) ] + [푅1푋2 + (푅2 + 푅퐿)푋1] method is with a dedicated driver and load coil. These driver and load coil are meant to be connected The same can be done to derive the output power of to the source (driver coil) and to the load resistor the system (the power that the load resistor 푅퐿 gets): (load coil). An illustration can be seen in Figure 3. 2 2 (휔푀12) 푉푠 푅퐿 푃표푢푡 = 2 2 2 [푅1(푅2 + 푅퐿) − 푋1푋2 + (휔푀12) ] + [푅1푋2 + (푅2 + 푅퐿)푋1]

The efficiency 휂 of the power transfer of the circuit in Figure 1, where the angular frequency is 휔, then becomes: Figure 3: Schematic of the setup of the coils for SCMR 2 power transfer with a dedicated driver coil and load coil. (휔푀12) 푅퐿 휂 = 2 2 2 (1) 푅1[(푅2 + 푅퐿) − 푋2 ] + (휔푀12) (푅2 + 푅퐿) The driver and load coil in this method are usually made out of 1 turn that is very strongly coupled with It can be seen that the efficiency reaches a maximum the closest resonator coil (for the driver coil this is the when there is resonance. Because in resonance 푋 2 primary resonator coil, for the load coil this is the becomes zero because 푋 = 휔퐿 − 1/휔퐶 = 0 when 2 2 2 secondary resonator coil). The currents in these coils 휔 equals the resonance angular frequency. were simulated by using Simulink from Matlab. In A factor that can indicate the efficiency of a resonator Figure 4 the currents that run in these coils can be is the quality factor or Q-factor. This factor indicates found along with their phase-change relative to the the energy loss relative to the stored energy of the phase of the voltage of the source. resonator. A resonator with a high Q-factor indicates a lower energy relative to the stored energy of the resonator. The Q-factor also describes how underdamped a resonator is, it is an indicator for the bandwidth of the resonator relative to the resonance frequency. The formula for the Q-factor can be written in the following ways: 1 퐿 휔 푄 = √ 표푟 푄 = 0 푅 퐶 ∆휔

In the next Figure the induced current in a resonator is shown relative to the normalised angular frequency of the resonator for different Q-factors.

Figure 4: Maximum currents (in Amperes) of driver, primary and secondary coil and voltage (in Volt) over the load resistor in the load coil. These maximum values with there respective phase are plotted on a polar system for different frequencies. The minimum frequency used is 130 kHz and the maximum frequency is 240 kHz. The Figure 2: The induced current for an arbitrary RLC circuit resonance frequency of the resonators is 185 kHz. The red increased drastically around resonance for higher Q- dots show the currents at resonance frequency. factors, while the bandwidth decreases.

It can be seen that the voltage over the load resistor two-resonator system this optimal load resistance

푈4 is maximum when there is resonance (red dot in could be derived (Wei, Wang, & Dai, 2014) from the the Figure). At resonance frequency the current in expression of the power transfer efficiency (1). The the driver coil 퐼1 becomes minimal. So it can be said optimal load resistance becomes: that the efficiency of the power transfer again 휔2푀2 푅 √ 12 2 푅퐿휂푚푎푥 = + 푅2 becomes maximal at resonance frequency. 푅1 For a SCMR power transfer system with more than two resonators the optimal load resistance was III. CREATING A PREDICTING MODEL found with an optimisation algorithm 푓푚푖푛푐표푛 that The goal of this dissertation is to create a way to was available in Matlab. optimize the resonators for power transfer. To do this An example was made for a SMCR power a model was created that could predict the efficiency transfer system with three resonators. This this of the power transfer for given dimensions of the example the outer resonators were set at a distance resonators. The dimensions that needed to be of 0.2 meters from each other, the middle resonator known were: the diameter of the coil, the length of was placed between these resonators. The optimal the coil, the number of windings of the coil, the load resistance was calculated for different positions conductor that was used for the windings, the of the middle resonator. In Figure 5 the efficiencies distance between the resonators, the frequency and can be found of the of the power transfer for different voltage of the source that is connected to the first positions of the middle resonators for different resonator and the load resistance that is connected optimal load resistors (a bigger version of this figure to the last resonator. With these dimensions of the can be found in Figure 58). It can be seen that the resonators the resistance, inductance and optimal position of the middle resonator is at 0.1 capacitance could be calculated with empirical and meters and the optimal load resistor is 0.55 Ω. analytical formulas found in literature. By having this values the following expression could be solved: 1 핍 = ℝ핀 + 푗휔핃핀 + ℂ핀 푗휔 With:

푅 0 퐿 푀 ℝ = [ 1 ] ; 핃 = [ 1 12] ; 0 푅2 + 푅퐿 푀12 퐿2

1/퐶 0 푉 퐼 ℂ = [ 1 ] ; 핍 = [ 1] ; 핀 = [ 1] 0 1/퐶2 0 퐼2

When the currents were calculated, the power of the Figure 5: Efficiency of power transfer for a three- voltage source could be calculated easily because resonator-system when moving the middle resonator (and the current in the first resonator was then known and choosing the optimal load resistor for that position). the voltage was also known. The power that the load IV. CASES resistor obtained could also be calculated easily because the resistance of the load resister was When the model was created and validated for a known along with the current that run through it. The range of different coils some cases were made to efficiency of the power transfer was then the division further investigate the optimal dimensions of the of the two powers. resonators (for cases with boundary conditions) and When looking back at the formula for the the optimal spacing of a five-resonator system. The efficiency of the power transfer (1) it can be seen that optimal values were found with the 푓푚푖푛푐표푛 the power transfer efficiency is also dependent on optimisation algorithm and the efficiency of the the load resistance. This means that for each power transfer was calculated with the model. The resonator system with a certain distance between cases were then also made in the lab and the the resonators there is an optimal resistance. For a efficiency of the power transfer was measured. The

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calculated efficiencies of the model and the V. CONCLUSION measured efficiencies of the testing setup were compared and the accuracy of the model was for all A model was made that can accurately predict the the cases better than 97 %. power transfer efficiency of a SCMR power transfer The effect on the efficiency of power transfer of system with n-resonators. The accuracy of these using more resonators was also investigated. It can predictions was around 97% when comparing then be seen in Figure 6 that the efficiency dramatically to real world measurements. It is important to note increases when more resonators are used to transfer that this accuracy was only validated for low power over a long distance. But at a certain number frequencies (<100 kHz). This was because of the of resonators this increase in efficiency start to slow limitations of the equipment that was used. Also, the down. equations used to get the power transfer efficiency were derived by using quasistatic magnetic field theory. When the frequency becomes higher, there will be a point where the error of using this approximated equations becomes too high to be neglected. Only the dimensions of the coils need to be known to use the obtained model and optimization code. This means that the optimal efficiency can be calculated for applications were only the dimensions are known that the resonators can be made in.

Figure 6: Graph that shows the effect of using multiple resonators to transfer power efficiency over a distance of 0.4 meters.

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Extended abstract NL Ontwikkeling van een Resonante Spoel met Lage Weerstand voor Draadloze Energieoverdracht bij Lage Frequentie

Kevin Van Acker

Promotor: Prof. dr. ir. Guillaume Crevecoeur Begeleider: Ir. Matthias Vandeputte

Abstract: In deze masterproef zal draadloze met een resonator (primaire resonator). Een energieoverdracht via sterk gekoppelde belasting moet dan worden verbonden met de resonante spoelen worden onderzocht. De basis andere resonator (secundaire resonator). van deze energieoverdracht zal worden getoond. De stromen die in deze resonatoren stromen, De meeste focus is hier gelegd op het maken van zullen worden onderzocht en gemodelleerd. Daarna een manier om de beste resonante spoel te zal een model dat deze stromingen kan voorspellen verkrijgen. Dit wordt gedaan via een worden gemaakt met formules uit de literatuur. De voorspellende model die gemaakt is in Matlab. formules worden gevalideerd met verschillende Dit model zal de efficiëntie van een gegeven metingen. Wanneer de afzonderlijke formules resonator systeem bepalen. correct lijken te zijn voor deze metingen, kunnen I. INTRODUCTIE vervolgens metingen worden uitgevoerd om het hele model te valideren. In de afgelopen jaren is draadloze II. BEGINSELEN VAN SCMR energieoverdracht in populariteit toegenomen voor consumentenapparaten, zoals het draadloos Met behulp van de sterk gekoppelde magnetische opladen van smartphones en andere elektronische resonatoren wordt vermogen overgedragen door apparaten. Er zijn een paar manieren om draadloos gebruik te maken van een magnetisch veld. Het energie over te dragen. De methode die hier wordt magnetisch veld wordt gecreëerd door de stroom die besproken, is sterk gekoppelde magnetische door de spoel van de eerste resonator loopt, zoals resonantie. Deze methode heeft een lagere vastgelegd door de wet van Ampères. Een wisselend efficiëntie voor de energieoverdracht dan andere magnetisch veld wordt geïnduceerd door de methoden voor korte (enkele centimeters) en lange wisselstroom die door de eerste spoel loopt. Een afstanden (enkele tientallen meters tot kilometers), tweede resonator wordt nabij de eerste resonator maar lijkt de beste oplossing te zijn voor geplaatst zodat de spoelen van de resonatoren toepassingen in het middel lange afstanden (bereik magnetisch met elkaar gekoppeld zijn. De spoel van van enkele centimeters tot een paar meters). de tweede resonator bevindt zich nu in een Toepassingen voor sterk gekoppelde magnetische veranderend magnetisch veld vanwege de resonantie kunnen elektrische voertuigen, robots en wisselstroom in de eerste resonator. Vanwege de andere elektronische apparaten zijn die op afstand wet van Faraday zal de veranderende magnetische moeten worden opgeladen. flux van het magnetisch veld een veranderend Sterk gekoppelde magnetische resonantie elektrisch veld genereren in de spoelen in de vorm (SCMR) zal in deze masterproef worden onderzocht. van een elektromotorische kracht. Deze Deze methode van draadloze energieoverdracht elektromotorische kracht is dan de bron in de tweede maakt gebruik van ten minste twee resonatoren. resonator die stroom door de componenten van de Resonatoren zijn elektrische circuits die uit een spoel tweede resonator zal sturen. en een condensator bestaan. Om vermogen over te dragen met deze resonatoren moet een hoogfrequente spanningsbron worden verbonden

In de volgende figuur wordt de geïnduceerde stroom in een resonator getoond ten opzichte van de genormaliseerde hoekfrequentie van de resonator voor verschillende Q-factoren. Figuur 1: Circuit van een twee-resonator systeem voor energieoverdracht.

In Figuur 1 is een weergave van de twee resonatoren te zien. Wanneer het vermogen van de bron wordt afgeleid met behulp van de circuittheorie, kan de volgende uitdrukking voor de ingangsvermogen van het systeem worden bekomen (Huang, Zhang, & Zhang, 2014): 2 2 2 2 {푅1[(푅2 + 푅퐿) − 푋1 ] + (휔푀12) (푅2 + 푅퐿)}푉푠 푃푖푛 = 2 2 2 [푅1(푅2 + 푅퐿) − 푋1푋2 + (휔푀12) ] + [푅1푋2 + (푅2 + 푅퐿)푋1]

Hetzelfde kan gedaan worden om het uitgangsvermogen van het systeem af te leiden (het vermogen dat de belasting weerstand 푅퐿 krijgt): (휔푀 )2푉2푅 Figuur 2: De geïnduceerde stroom voor een willekeurig 푃 = 12 푠 퐿 표푢푡 [푅 (푅 + 푅 ) − 푋 푋 + (휔푀 )2]2 + [푅 푋 + (푅 + 푅 )푋 ]2 RLC-circuit stijgt drastisch rond resonantie voor hogere Q- 1 2 퐿 1 2 12 1 2 2 퐿 1 factoren, terwijl de bandbreedte afneemt. The efficiency 휂 of the power transfer of the circuit in De uitdrukking voor de efficiëntie die eerder werd Figure 5, where the angular frequency is 휔, then getoond was voor een twee resonatorsysteem dat becomes: direct werd gevoed door een spanningsbron. Er is 2 (휔푀12) 푅퐿 ook een andere methode die wordt gebruikt om 휂 = 2 2 2 (1) 푅1[(푅2 + 푅퐿) − 푋2 ] + (휔푀12) (푅2 + 푅퐿) SCMR te bereiken, deze methode is met een aparte driver spoel en een load spoel. Deze driver en laad Het is duidelijk dat de efficiëntie een maximum spoel is bedoeld om te worden aangesloten op de bereikt wanneer er resonantie is. Want bij resonantie bron (driver spoel) en op de belasting weerstand wordt 푋2 nul omdat 푋2 = 휔퐿2 − 1/휔퐶2 = 0 wanneer (load spoel). Een illustratie is te zien in Figuur 3. 휔 aan de resonantie hoekfrequentie. Een factor die de efficiëntie van een resonator kan aangeven is de Q-factor. Deze factor geeft het energieverlies ten opzichte van de opgeslagen energie van de resonator aan. Een resonator met Figuur 3: Schema van de setup van de spoelen voor een hoge Q-factor geeft een lagere energie aan ten SCMR-energieoverdracht met een aparte driver spoel en opzichte van de opgeslagen energie van de load spoel. resonator. De Q-factor beschrijft ook hoe De driver en load spoel in deze methode zijn meestal ondergedempt een resonator is, het is een indicator gemaakt van 1 wikkeling die zeer sterk gekoppeld is voor de bandbreedte van de resonator ten opzichte aan de resonatorspoel die het dichtste bij is (voor de van de resonantiefrequentie. De formule voor de Q- driver spoel is dit de primaire resonatorspoel, voor factor kan op de volgende manieren worden de load-spoel is dit de secundaire resonatorspoel). geschreven: De stromen in deze spoelen werden gesimuleerd door Simulink van Matlab te gebruiken. In Figuur 4 1 퐿 휔 푄 = √ 표푟 푄 = 0 zijn de stromen die in deze spoelen lopen te vinden 푅 퐶 ∆휔 samen met hun faseverandering ten opzichte van de fase van de spanning van de bron.

die is verbonden met de laatste resonator. Met deze afmetingen van de resonatoren konden de weerstand, inductantie en capaciteit worden berekend met empirische en analytische formules die in de literatuur worden gevonden. Door deze waarden te hebben, zou de volgende uitdrukking kunnen worden opgelost: 1 핍 = ℝ핀 + 푗휔핃핀 + ℂ핀 푗휔 Met: 푅 0 퐿 푀 ℝ = [ 1 ] ; 핃 = [ 1 12] ; 0 푅2 + 푅퐿 푀12 퐿2

1/퐶 0 푉 퐼 ℂ = [ 1 ] ; 핍 = [ 1] ; 핀 = [ 1] 0 1/퐶2 0 퐼2 Wanneer de stromen werden berekend, kon het vermogen van de spanningsbron gemakkelijk worden berekend omdat de stroom in de eerste resonator toen bekend was samen met de spanning. Figuur 4: Maximale stromen (in Ampère) van de driver, Het vermogen dat de belasting weerstand kreeg kon primaire en secundaire spoel en spanning (in Volt) over de ook gemakkelijk worden berekend omdat de belasting weerstand in de load spoel. Deze maximale weerstand van de last weerstand bekend was samen waarden met hun respectieve fase zijn uitgezet op een polair assenstelsel voor verschillende frequenties. De met de stroom die er doorheen liep. De efficiëntie minimale gebruikte frequentie is 130.000 Hz en de van de energieoverdracht was toen de deling van de maximale frequentie is 240.000 Hz. De twee vermogens. resonantiefrequentie van de resonatoren is 185.000 Hz. Wanneer er wordt teruggekeken naar de formule De rode stippen tonen de stromen op resonantiefrequentie. voor de efficiëntie van de energieoverbrenging (1), kan worden gezien dat de efficiëntie van de Er kan worden gezien dat de spanning over de energieoverbrenging ook afhankelijk is van de belasting weerstand 푈4 maximaal is wanneer er belasting weerstand. Dit betekent dat voor elk resonantie is (rode stippen in de figuur). Bij resonatorsysteem met een bepaalde afstand tussen resonantiefrequentie wordt de stroom in de f driver de resonatoren er een optimale weerstand is. Voor spoel 퐼1 minimaal. Dus kan worden gezegd dat de een resonatorsysteem met twee resonatoren kan efficiëntie van de vermogensoverdracht opnieuw deze optimale belasting weerstand worden afgeleid maximaal wordt bij resonantiefrequentie. (Wei, Wang, & Dai, 2014) aan de hand van de uitdrukking van de efficiëntie van de III. EEN VOORSPELLENDE MODEL MAKEN vermogensoverdracht (1). De optimale belasting weerstand wordt: Het doel van deze masterproef is om een manier te creëren om de resonatoren voor efficiënte 2 2 휔 푀12푅2 energieoverdracht te optimaliseren. Hiervoor is een 푅 = √ + 푅 퐿휂푚푎푥 푅 2 model gemaakt dat de efficiëntie van de 1 krachtoverdracht voor bepaalde dimensies van de Voor een SCMR-energieoverbrenging met meer resonatoren kan voorspellen. De afmetingen die dan twee resonatoren werd de optimale belasting bekend moesten zijn waren: de diameter van de weerstand gevonden met een optimalisatie- spoel, de lengte van de spoel, het aantal windingen algoritme 푓푚푖푛푐표푛 die beschikbaar was in Matlab. van de spoel, de geleider die werd gebruikt voor de Een voorbeeld was gemaakt met een SMCR- windingen, de afstand tussen de resonatoren, de energieoverdrachtsysteem met drie resonatoren. De frequentie en spanning van de bron die is verbonden buitenste resonatoren werden op een afstand van met de eerste resonator en de belasting weerstand 0,2 meter van elkaar geplaatst, de middelste

resonator werd tussen deze resonatoren geplaatst. resonatoren werd ook onderzocht. In Figuur 6 is te De optimale belasting weerstand werd berekend zien dat de efficiëntie dramatisch toeneemt wanneer voor verschillende posities van de middelste meer resonatoren worden gebruikt om vermogen resonator. In Figuur 5 zijn de rendementen te vinden over een lange afstand over te dragen. Maar bij een voor de verschillende posities van de middelste bepaald aantal resonatoren begint deze toename in resonator met zijn verschillende optimale belasting efficiëntie af te nemen. weerstanden (een grotere versie van deze figuur is te vinden in Figure 58). Er kan gezien worden dat de optimale positie van de middelste resonator 0,1 meter is en dat de optimale belasting weerstand 0,55 Ω is.

Figuur 6: Graph that shows the effect of using multiple resonators to transfer power efficiency over a distance of 0.4 meters.

V. CONCLUSION Figuur 5: Efficiëntie van energieoverbrenging voor een drie resonator-systeem bij het verplaatsen van de middelste Er is een model gemaakt dat de efficiëntie van de resonator (en het kiezen van de optimale belasting weerstand voor die positie). energieoverdracht van een SCMR- energieoverdracht systeem met n-resonatoren nauwkeurig kan voorspellen. De nauwkeurigheid IV. TOEPASSINGSGEVALLEN van deze voorspellingen was ongeveer 97% bij vergelijking met metingen. Het is belangrijk op te Toen het model werd gemaakt en gevalideerd voor merken dat deze nauwkeurigheid alleen werd een reeks verschillende spoelen werden enkele gevalideerd voor lage frequenties (<100 kHz). Dit voorbeeld gevallen berekend. Zoals de optimale kwam door de beperkingen van de apparatuur die afmetingen van de resonatoren (voor gevallen met werd gebruikt. Ook werden de vergelijkingen, die randvoorwaarden) van een systeem bepalen. Een gebruikt werden om de efficiëntie van de ander geval dat werd onderzocht was de optimale energieoverdracht te verkrijgen, afgeleid door quasi afstand van een vijf resonatorsysteem. De optimale statische magnetisch veldentheorie te gebruiken. waarden werden gevonden met het 푓푚푖푛푐표푛- Wanneer de frequentie hoger wordt, zal er een punt optimalisatie-algoritme en de efficiëntie van de zijn waarop de fout bij het gebruik van deze energieoverdracht werd berekend met het model. De benaderde vergelijkingen te hoog wordt om te gevallen werden vervolgens ook in het labo gemaakt worden verwaarloosd. en de efficiëntie van de energieoverdracht werd Alleen de afmetingen van de spoelen moeten gemeten. De berekende efficiënties van het model bekend zijn om het model en de optimalisatiecode te en de gemeten efficiënties van de testopstelling gebruiken. Dit betekent dat de optimale efficiëntie werden vergeleken. De nauwkeurigheid van het kan worden berekend voor toepassingen waarbij model was voor alle gevallen beter dan 97%. alleen de dimensies bekend zijn waarvan de Het effect op de efficiëntie van resonatoren kunnen worden gemaakt. energieoverbrenging bij het gebruik van meerdere

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List of contents

Preface ...... i Abstract ...... ii Extended abstract EN ...... iii Extended abstract NL ...... ix List of contents ...... xiii List with abbreviations ...... xvii List with symbols ...... xvii List of Figures and tables ...... xviii List of tables ...... xxi

1 Introduction ...... 1 1.1 Magnetism ...... 2 1.1.1 Magnetic field B ...... 2 1.1.1.1 Ampère’s Law for magnetic field ...... 2 1.1.1.2 Calculating magnetic fields ...... 3 1.1.2 Magnetic field H ...... 7 1.1.3 Magnetic hysteresis ...... 8 1.1.4 Static and magnetoquasistatic fields ...... 9 1.1.5 Magnetic flux ...... 9 1.1.5.1 Gauss’s Law of magnetism ...... 10 1.1.5.2 Magnetic flux coupled with a closed current loop ...... 10 1.2 Passive components ...... 11 1.2.1 Resistance ...... 11 1.2.2 Frequency dependent resistance model for inductors ...... 12 1.2.2.1 Skin effect ...... 12 1.2.2.2 Proximity effect ...... 13 1.2.2.3 Litz wire ...... 14 1.2.2.4 Conclusion ...... 14 1.2.3 Inductance ...... 15 1.2.3.1 Inductance of a solenoid ...... 15 1.2.3.2 Conclusion ...... 16 1.2.4 Mutual induction ...... 16 1.2.5 Coupling factor ...... 17 1.2.6 Capacitance ...... 17

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1.3 Electromagnetic induction ...... 18 1.3.1 Faraday’s Law ...... 18 1.3.2 Induction phenomena ...... 18 1.3.3 Electromagnetic force of self-induction ...... 18 1.3.4 Electromagnetic force of mutual-induction ...... 19 1.3.5 Conclusion ...... 19 1.4 Resonator ...... 19 1.4.1 Resonance ...... 20 1.4.2 Quality factor ...... 20 1.4.3 Conclusion ...... 21 1.5 Wireless power transfer via strongly coupled magnetic resonance ...... 22 1.5.1 First experiment ...... 22 1.5.2 Second experiment ...... 24 1.5.2.1 Practical setup ...... 25 1.5.2.2 Simulation ...... 27 1.5.2.3 Reflected impedance ...... 31 1.5.2.4 Circuit theory of the experiment ...... 33 1.6 Frequency splitting phenomena ...... 38 1.6.1 Conclusion ...... 38

2 Developing a predicting model for resonators ...... 39 2.1 Formula of AC resistance of the conductor used for the windings ...... 40 2.2 Empirical formulas for inductance of a coil ...... 41 2.3 Mutual inductance calculation between circular filaments arbitrarily positioned in space ...... 42 2.4 Formula for Capacitance ...... 42 2.5 Validating the found expressions ...... 43 2.5.1 Comparing wire configuration and different capacitors ...... 43 2.5.1.1 Conclusion ...... 52 2.5.2 Validation of formulas for inductance and mutual inductance: ...... 53 2.5.2.1 Conclusion ...... 53 2.6 Determine dimensions of resonator for further measurements ...... 55 2.6.1 Boundary constraints ...... 55 2.6.2 Dimensions ...... 56 2.6.3 Conclusion ...... 58 2.7 Determining the optimal load resistance for a two-resonator-system ...... 59 2.7.1 Conclusion ...... 62 2.8 Optimal load resistance for a multiple resonator system ...... 63

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2.8.1 Conclusion ...... 63 2.9 Models made ...... 65 2.9.1 eff_dim function ...... 65 2.9.2 eff_autosize function ...... 66 2.9.3 Conclusion ...... 66

3 Making resonator prototypes ...... 67 3.1 Additive manufacturing ...... 68 3.2 Prototype features ...... 70 3.3 Used resonator prototypes and coils for measurements ...... 70 3.3.1 First resonator prototype ...... 70 3.3.2 Printed coils to compare the resistance for more wire configurations...... 72 3.3.3 Second resonator prototype ...... 73 3.3.4 Third resonator prototype ...... 74 3.3.5 Summary ...... 75

4 Finding the optimal dimensions of a resonator ...... 76 4.1 Optimization function fmincon ...... 76 4.1.1 Descriptions of the arguments of the fmincon function ...... 77 4.2 Implementation of the optimization code ...... 78 4.2.1 Optimization of resonator dimensions and load resistor ...... 78 4.2.2 Optimization of distance between resonators in a multi resonator power transfer system ...... 79 4.3 Principles of fmincon ...... 80 4.4 Conclusion ...... 80

5 Cases...... 81 5.1 Optimal dimensions for a resonator that can be built in the lab to transfer power over a distance of 0.1 meter with 2 identical resonators...... 81 5.1.1 Conclusion ...... 83 5.2 Predicting the power transfer of an existing testing setup ...... 84 5.2.1 Conclusion ...... 85 5.3 Get the optimal spacing of the 3 middle resonators of a 5-resonator system, with a distance of 0.4 meter between the first and the last resonator ...... 86 5.3.1 Conclusion ...... 88 5.4 Effect on efficiency of using multiple resonators to transfer power over a distance of 0.5 meters. 89 5.4.1 Conclusion ...... 91

Conclusion ...... 92 Future work ...... 94 List of references ...... 95 Appendices ...... 97

xvi

List with abbreviations

3D Three-Dimensional AC Alternating Current CAD Computer Aided Design DC Direct Current EMF ElectroMagnetic Force (휀) ESR Equivalent Series Resistance KVL Kirchhoff’s Voltage Law SCMR Strongly Coupled Magnetic Resonance

List with symbols

Symbol Description Unit

퐸푒푚푓 Electromotive force V µ = µr ∗ µ0 H/m µ0 Permeability of free space = 4π∗10-7 H/m µr Relative magnetic permeability - B Magnetic flux density T C Capacitance F d Diameter in meter M f Frequency of the AC current Hz F Force N H Magnetic field intensity A/m I Current A L Inductance H l Length M M Mutual induction H N Number of windings - Nstrands Number of strands - Q Electric charge C Q-factor Quality factor - R Resistance Ω S Surface m² V Voltage V v Velocity m/s X Reactance Ω Z Impedance Ω δ Skin depth m -12 -1 ε0 Vacuum permittivity ≈ 8.854×10 F⋅m εr Relative static permittivity - Φ Magnetic flux Wb Ψ Coupled magnetic flux Wb ω Angular frequency of the AC current - 휌 Resistivity Ω⋅m

xvii

List of Figures and tables

Figure 1: Representation of Ampère's Law. (Dupré, 2016) ...... 2 Figure 2: Magnetic induction B inside and outside a cylindrical straight wire were a current I flows through. (Dupré, 2016) ...... 4 Figure 3: Magnetic field of a current loop. (Dupré, 2016)...... 5 Figure 4: Magnetic induction of a solenoid. (Dupré, 2016) ...... 6 Figure 5: Representation of a cylindrical coil. (Dupré, 2016) ...... 7 Figure 6: The relationship between the magnetization M of a ferromagnetic material and the field strength H. (Magnetic Hysteresis, 2010) ...... 8 Figure 7: The magnetic flux through both surfaces are equal because of Gauss's Law. (Dupré, 2016) ...... 10 Figure 8: A simplified representation of a conductor. (Omegatron, 2007) ...... 11 Figure 9: Skin depth. (Biezl, 2008) ...... 12 Figure 10: The alternating current induces Eddy currents which oppose the current in the centre of the conductor. (Biezl, 2008) ...... 12 Figure 11: Proximity effect in 4 parallel conductors wherein current flows in the same direction. (Stepien, 2012) ...... 13 Figure 12: A representation of the cross section of a litz wire. This wire consists of 19 strands that are insulated from each other. (Zureks, 2009) ...... 14 Figure 13: Definition of inductance of a closed loop C. (Dupré, 2016) ...... 15 Figure 14: A representation of a parallel-plate capacitor. (Fabian, 2013) ...... 17 Figure 15: LC-circuit...... 19 Figure 16: RLC-circuit...... 20 Figure 17: The induced current for an arbitrary RLC circuit increased drastically around resonance for higher Q-factors, while the bandwidth decreases. (Vandeputte, 2016) ...... 21 Figure 18: Circuit of first experiment. 2 inductors are coupled, each inductor is part of a circuit with a capacitor and a resistor. One circuit is connected to a voltage source...... 22 Figure 19: Graph of the voltage over the resistor in the second circuit...... 23 Figure 20: Close up of the previous Figure in the resonant region...... 24 Figure 21: Schematic representation of second experiment. (Wei, Wang, & Dai, 2014) ...... 25 Figure 22: Setup of resonator experiment...... 25 Figure 23: Close up of primary coil and driver coil...... 26 Figure 24: Close up of the secondary coil and load coil...... 27 Figure 25: Circuit of resonator simulation...... 28 Figure 26: Simple circuit of 2 magnetically coupled coils...... 29 Figure 27: Maximum currents (in Amperes) of driver, primary and secondary coil and voltage (in Volt) over the load resistor in the load coil. These maximum values with there respective phase are plotted on a polar system for different frequencies. The minimum frequency used is 130 000 Hz and the maximum frequency is 240 000 Hz. The resonance frequency of the resonators is 185 000 Hz...... 30 Figure 28: Equivalent circuit of system with 2 coils that are coupled. (Wei, Wang, & Dai, 2014) ...... 31 Figure 29: Simplified circuit of the system of 2 magnetically coupled coils by replacing the secondary coil with an equivalent impedance in the primary coil. (Wei, Wang, & Dai, 2014)...... 32 Figure 30: Simplified circuit of the system of 2 magnetically coupled coils by replacing the primary coil with an equivalent impedance and voltage source in the secondary coil. (Wei, Wang, & Dai, 2014) ...... 32 Figure 31: The (black striped) graph of the analytical results of the behaviour of the current of the coils plotted on the (red lined) graph of the simulated relationship of the currents in the coils...... 34 Figure 32: A representation of the effect the mutual inductions of non-neighbouring coils have on the signals...... 36 Figure 33: Plot of maximum current in all coils with annotations...... 37 Figure 34: Efficiency of an arbitrary two-resonator WPT system plotted against the frequency. (Huang, Zhang, & Zhang, 2014) ...... 38

xviii

Figure 35: The normalized transfer power of the system of Figure 34 plotted against the frequency. (Huang, Zhang, & Zhang, 2014) ...... 38 Figure 36: Circuit of a two-resonator power transfer system with annotations...... 39 Figure 37: Cylindrical coil with rectangular winding cross-section. (Van den Bossche, 2008) ...... 41 Figure 38: Arbitrary position of a coil in space...... 42 Figure 39: plotting the resistance of the 2 different wire types for different frequencies...... 44 Figure 40: Comparing wire diameters and the effect of using multiple strands...... 45 Figure 41: AC resistance of a conductor for different frequencies...... 45 Figure 42: Proximity effect on the resistance of a conductor for different frequencies with a bigger range in frequency...... 46 Figure 43: AC resistance for regular and Litz wire...... 46 Figure 44: Resonators used for measurements of the resistance of the resonator for different resonance frequencies...... 47 Figure 45: Measuring setup for getting the resistance of the resonator for different frequencies...... 48 Figure 46: Graph of the resistance measurements of the resonator...... 48 Figure 47: Calculated and measured values of the resistance in the resonator for different frequencies. ... 49 Figure 48: A picture of all the coils that were made to validate the AC resistance formula...... 50 Figure 49: A graph of all the measurements and calculations of the resistances of the coils...... 51 Figure 50: Q-factors that were measured and calculated for the coils...... 54 Figure 51: Calculation of Q-factor for the same resonator, only the diameter of the strand is different. Note that when the diameter of the strands change, the length of the coil also changes...... 56 Figure 52: Calculation of Q-factor for the same resonator, only the length of the coil is different. Note that when the length changes, the number of windings of the coil also changes...... 57 Figure 53: Calculation of Q-factor for the same resonator, only the radius of the coil is different...... 58 Figure 54: Equivalent circuit model of 2 coils with a mutual inductance...... 59 Figure 55: Plot of the efficiency of the power transfer for different relative axial positions...... 60 Figure 56: Circuit of the simulation for 2 resonator coils...... 61 Figure 57: Comparison of the simulated efficiency of the power transfer to the measured efficiency...... 61 Figure 58: Efficiency of power transfer for a three-resonator-system when moving the middle resonator (and choosing the optimal load resistor for that position)...... 64 Figure 59: First sketches/ideas for a resonator design...... 67 Figure 60: First resonator that was made...... 67 Figure 61: A representation of the '.stl' file format. The left Figure is a 3D part that needs to be converted and the right Figure is the collection of triangles that the 3D part is converted to in a '.stl'-file. (Cardon, 2017) ...... 68 Figure 62: A representation of the moving print head for Fused Deposition Modelling...... 69 Figure 63: A picture of the 3D printer printing a resonator prototype...... 69 Figure 64: Printed version of the parts of the first resonator prototype...... 70 Figure 65: A picture of the wired first resonator prototype with litz wire...... 71 Figure 66: All the coils that were made to compare the resistance for more wire configurations...... 72 Figure 67: Parts of the coils. The right part is the base of the coil were the wire is wounded on. The left part is the lid of the coil which holds the wires in place...... 72 Figure 68: A picture of the second resonator prototype...... 73 Figure 69: A picture of the third and last resonator prototype...... 74 Figure 70: Picture of prototype 1...... 75 Figure 71: Picture of prototype 2...... 75 Figure 72: Picture of prototype 3...... 75 Figure 73: Picture of the measurement setup to validate the optimisation model in the first case...... 83 Figure 74: Testing setup of case 2...... 84 Figure 75: Setup to get power transfer over a distance of 0.4 m with 5 resonators...... 86

xix

Figure 76: Testing setup to validate the optimal positions of the resonator found by the optimisation code for five resonators...... 88 Figure 77: Graph that shows the effect of using multiple resonators to transfer power efficiency over a distance of 0.4 meters...... 90 Figure 78: Extended graph that shows the efficiency of using more than ten resonators...... 91

List of tables

Table 1: Parameters of the components used in the Simulink circuit...... 22 Table 2: Explanation of the variables used in the resonator determining program...... 55 Table 3: Overview of all the variables of the resonator chosen to be used in the variable spacing experiment...... 58 Table 4: Characteristics of the resonator used for finding the optimal load resistor...... 59 Table 5: Explanation of the arguments of the eff_dim function...... 65 Table 6: Advantages and disadvantages of all prototypes...... 75 Table 7: Fixed values and variables of the resonators for the first case...... 81 Table 8: Optimal dimensions for the resonators, calculated by the optimization function...... 82 Table 9: Dimensions and parameters of the resonators that were made in the lab...... 82 Table 10: Dimensions and values of testing setup of case 2...... 84 Table 11: Inputs of the distances and load resistance of the optimisation in case 3...... 87 Table 12: Optimal values to the distances of the resonators and optimal load resistor for this case...... 87 Table 13: Table of the power transfer efficiencies, that were calculated with the model and the measured power transfer efficiencies, for optimal spacing and uniform spacing...... 88 Table 14: Dimensions of resonators used for calculation of the efficiencies for different numbers of resonators ...... 89

xxi

1 Introduction

Wireless power transfer has been on the minds of mankind for quite some time (Brown, The history of power transmission by radio waves, 1984). It can be traced back to as early as 1889 where Nikola Tesla invented the Tesla coils, which could transfer power wireless. In the recent years the popularity of wireless power transfer (WPT) has gained tremendously. The gained attention for WPT in consumer application can mainly be thanked to the popularity of portable electronic devices like cell phones, laptops, tablets and smart watches. WPT could eliminate the traditional charging by wire of the batteries in these electronic devices. The advantages of wireless charging are obvious. The main advantage for consumers is convenience, charging an electronic device is a daily routine for many, so a lot of consumers see the advantage of having a more convenient way of charging their device cordless. There is also an increase in product life by charging wirelessly, by not using a connector port every time that a device needs to be charged, connector fatigue and failure can be avoided. Another advantage is the ability to create hermetically sealed devices as ports would be unnecessary. Electric Vehicles (EVs) are another great example of a promising application for WPT. EVs are now rapidly gaining market share of the transportation sector and wirelessly charging the batteries of these EVs can be done on many different occasions (from charging in a garage to charging while driving on the road). Even for greater distances WPT has been considered the solution for future renewable energy production (Glaser, 1968), where the idea of having massive sunlight collecting satellites (Brown, Status of the mirowave power transmission components for the solar power satellite (SPS)., 1981) around the earth that send energy back to earth by beaming energy back to earth via microwaves has been considered for a long time.

Wireless power transfer can be subdivided into three classes: electromagnetic radiation mode, electric field coupling mode and magnetic field coupling mode. In the first mode, electromagnetic radiation mode, the electric energy is first converted into electromagnetic energy like microwaves or laser beams like in the example of sunlight collecting satellites. The antennas that can be used are parabolic dishes, rectennas, lasers, photocells, etc…. This mode provides a good solution for long range power transfer. Electric field coupling mode (Leyh & Kennan, 2008) uses a high-voltage and high-frequency driver source to transfer power. It excites a resonant transmitter which generates an alternating electric field. This field is then coupled with a resonant receiver. The range of this mode is short. The magnetic field coupling mode can be classified as short to mid-range. There are two subcategories of this mode: electromagnetic induction (Covic & Boys, 2013) and strongly coupled magnetic resonance (SCMR) (Kurs, et al., 2007). Electromagnetic induction can transfer power with a high efficiency but the range is very short, just some centimetres. While the efficiency of SCMR is a little lower, it does perform better in mid-range transfer distances.

In this dissertation strongly coupled magnetic resonance will be investigated. SCMR can fill the void for mid- range distance applications and has a lot of potential as it is still under development. SCMR will be analysed by using circuit theory and simulations that are based on circuit theory. There will not be any conclusions on the viability of SCMR or a comparison between SCRM and other technologies in this dissertation. The goal of this dissertation is to obtain an easy way to optimize the dimensions of a high-quality resonance coil. This will be done by developing a model that predicts the power transfer between 2 or more magnetic resonators. The model will be using equations that predict power transfer between resonators that are derived of circuit theory. Analytical and imperial equations will be used to obtain an estimation of properties of the resonators. All these equations will be shown in the following chapters. So, the objective of this dissertation is first to construct and validate analytical and/or numerical models of the inductance and resistance of various coils that will be used for SCMR wireless power transfer. Secondly a comparative study or optimization needs to be conducted to determine the optimal coil shape and wire diameter. Other effects on the power transfer will also be studied.

This dissertation is structured in the following way: First in the basic principles will be shown about magnetism, electrical components, resonators and wireless power transfer using resonators. After that the formulas that were used to make the model that predicts the power transfer will be shown. These formulas were also validated with a lot of experiments and measurement, these measurements will also be shown. To make the measurements some resonator prototypes needed to be made so there will be a chapter about how these resonators were made. When the model is then made and validated, it will be used to determine the best resonators for some cases/examples with some boundary conditions for some dimensions and properties of the resonators and power transfer. Finally, some conclusions will be given about this dissertation and future work that can be done about this subject. 1.1 Magnetism

To study wireless power transfer by using strongly coupled magnetic resonance, magnetism needs to be studied. Here the basics of magnetism will be shown. Magnetism needs to be understood because this is the way how energy will be transferred wirelessly. Together with the Section about passive components and inductance the power transfer can be understood. 1.1.1 Magnetic field 퐵

1.1.1.1 Ampère’s Law for magnetic field

Ampère’s Law states that the integral of the magnetic field around a closed loop is related to the sum of the currents flowing through the enclosed surface of the loop.

푛

∮ 퐵푑푙 = µ0 ∑ 퐼푖 퐿 푖=1

Figure 1: Representation of Ampère's Law. (Dupré, 2016)

When looking at the example in Figure 1, the closed loop integral of the magnetic field gives:

∮ 퐵푑푙 = µ0 (퐼1 − 퐼2 + 퐼3) 퐿

1.1.1.2 Calculating magnetic fields

There are a few ways to calculate magnetic field, in this Section a couple will be discussed and used to get the magnetic fields for a straight wire, a current loop and a solenoid.

1) Biot-Savart’s Law

Biot-Savart’s law gives the contribution to the magnetic field 푑퐵⃑ of the current element 퐼푑푙⃗ in an arbitrary ⃗ point in space. 푑푙 is the vector along the path of the conductor and 1⃑ 푟 is the unit vector that shows the direction of the current element relative to the arbitrary point.

µ (퐼푑푙 × 1⃑ ) 푑퐵⃑ = 0 푟 4휋푟2

The magnetic field in point P due to the electric current 퐼 in a conductor then the line integral has to be calculated along the conductor:

µ (퐼푑푙 × 1⃑ ) ⃑ 0 푟 퐵(푃) = ∫ 2 푐표푛푑푢푐푡표푟 4휋푟

2) Ampère’s Law

As stated previously, the integral of the magnetic field round a closed loop is related to the sum of the currents flowing through the enclosed surface of the loop.

푛

∮ 퐵푑푙 = µ0 ∑ 퐼푖 퐿 푖=1

This equation can be used to get the magnetic field for symmetric problems and uniform charge distributions.

Magnetic field induced by a straight wire

To get the magnetic field by a straight conductor Ampère’s law is applied to a circular loop around the wire. The magnetic field 퐵 is symmetrically around the wire. Outside the wire, with radius 푟 and that conducts a current 퐼, the magnetic field is given by:

µ 퐼 퐵 = 0 2휋푟

Inside the wire, with a current density 퐽, the magnetic field is:

µ0퐼푟 퐵 = 2 2휋푟0

Figure 2: Magnetic induction B inside and outside a cylindrical straight wire were a current I flows through. (Dupré, 2016)

Magnetic field of a current loop

Figure 3: Magnetic field of a current loop. (Dupré, 2016)

The magnetic field of a current loop can be found using Biot-Savart’s law:

µ (퐼푑푙 × 1⃑ ) 푑퐵⃑ = 0 푟 4휋푟2

By symmetry there is only a 퐵푧 component. µ 퐼푑푙 푑퐵 = 0 4휋푟2

µ 퐼푑푙 푑퐵 = 0 푐표푠휃 푧 4휋푟2

When looking at Figure 3, it can be seen that:

푟2 = 푧2 + 푅2 푅 푅 푐표푠휃 = = 푟 √푧2 + 푅² Integrating around the loop gives:

µ 퐼푅² 퐵 = 0 푧 2(푧2 + 푅2)3/2

The magnetic field in the centre of the loop gives (z=0):

µ 퐼 퐵 = 0 푧 2푅

Magnetic field of a solenoid

Figure 4: Magnetic induction of a solenoid. (Dupré, 2016)

Consider a solenoid with n circular current loops per unit length. The magnetic field in point P needs to be calculated. When looking at a piece of the solenoid with a width of 푑푧 it can be seen as a circular current loop from the previous calculation. In this piece there are 푛푑푧 loops. The piece of magnetic field 푑퐵 of this piece of solenoid in point P is then:

µ 퐼푛푑푧푅² 푑퐵 = 0 2(푧2 + 푅2)3/2

By applying some trigonometry, the expression can be simplified to:

µ 푛퐼 푑퐵 = − 0 푠푖푛휙푑휙 2

The total magnetic field becomes then:

µ 푛퐼 퐵 = 0 [푐표푠훼 − 푐표푠훽] 2

For an infinitely long solenoid the axial field becomes uniform and is equal to:

퐵 = µ0푛퐼

This expression will be used later to get the inductance of a solenoid.

The magnetic field in a very short coil 푙 << 푅 will be practically uniform (except on the outer edges of the coil). The magnetic field in the coil can be approximated to:

µ 푁퐼 퐵 = 0 2푅

1.1.2 Magnetic field H

The magnetic induction 퐵 is induced by the magnetic field 퐻. In a magnetized material the magnetic field can differ by the magnetization 푀 of the material of the material.

Ampère’s law can be stated as:

푛 퐵 ∮ 푑푙 = ∑ 퐼푖 µ0 퐶 푖=1

Figure 5: Representation of a cylindrical coil. (Dupré, 2016)

When applying Ampère’s law to a cylindrical coil with a magnetic material like in Figure 5, it can be proved that the formula is transformed to: 퐵⃑ ∮ ( − 푀⃑⃑ ) 푑푙 = 푁퐼 퐶 µ0

With 푀⃑⃑ being the magnetization in the material that is present in the coil.

The magnetic field 퐻⃑⃑ can be stated as: 퐵⃑ 퐻⃑⃑ = − 푀⃑⃑ µ0

In many materials the relation between 푀 and 퐻 (for example ferromagnetic materials) can be stated as:

푀 = 휒푚퐻

Where the volume magnetic susceptibility is a property of a material that indicates the degree of magnetization in response to an applied magnetic field. 퐻⃑⃑ can also be called the magnetic field intensity with unit 퐴/푚. The most general form of Ampère’s law becomes:

푛

∮ 퐻⃑⃑ 푑푙 = ∑ 퐼푖 퐶 푖=1

As a side note Ampère’s law also can be written in function of the current density 퐽 in a surface 푆 that is enclosed by the loop 퐶.

∮ 퐻⃑⃑ 푑푙 = ∬ 퐽 푑푆 퐶 푆

1.1.3 Magnetic hysteresis

When a ferromagnet (for example iron) is located in a magnetic field, atomic dipoles in the ferromagnet will align themselves with the field. When the magnetic field is then removed the dipoles will remain the alignment to a certain degree. It can then be said that this ferromagnet has become magnetized. The ferromagnet can only be demagnetized by having it in a magnetic field in the opposite direction or by heating the material. A representation of the magnetization of ferromagnetic materials can be seen in Figure 6.

Figure 6: The relationship between the magnetization M of a ferromagnetic material and the field strength H. (Magnetic Hysteresis, 2010)

When the ferromagnetic material is demagnetized 퐻 and 푀 are equal to zero, this the state 푎 in Figure 6. When the magnetic field strength is then increased the material follows the curve 푎 − 푏 until it is magnetically saturated (state 푏). If the magnetic field strength is then reduced the magnetisation of the material follows the curve 푏 − 푐 − 푑. When the magnetic field equals zero (state 푐) there is still a degree of magnetisation. This amount is called remanence. To reduce the magnetisation of the material, it then has to be in a magnetic field with an opposite direction as the original field at which the material was magnetized. This is state 푑. But when the strength of this opposite field is then to high the material can then also become saturated in the opposite direction as initially (state 푐). The curve in Figure 6 shows all the states a ferromagnetic material can be in.

1.1.4 Static and magnetoquasistatic fields

Previous equations define static magnetic fields. These equations slightly change when examining slowly oscillating magnetic fields. These class of electromagnetic fields are called magnetoquasistatic fields.

For Static fields Ampère’s law can be stated as follows:

∮ 퐻⃑⃑ 푑푙 = ∬ 퐽 푑푆 퐶 푆

But for magnetoquasistatic fields Ampère’s law becomes:

푑 ∮ 퐻⃑⃑ 푑푙 = ∬ 퐽 푑푆 + ∫ 휀퐸푑푆 퐶 푆 푑푡 푆

This means that when a H-field is changing a E-field will be induced by it. And that E-field will then induce an additional H-field that will be opposite to the initial H-field. This correction is for many applications very small.

The correction can be considered small when the device that induces the magnetic field is small these devices are then considered “static”. A device is small when the device is smaller than the wavelength of light with a frequency that is the same as the frequency of the magnetic field of the device. So, for example a power line with a length of 1500 km is static device because its operating frequency is 60 Hz and the wavelength of light with 60 Hz is 5000 km which is bigger than the device.

1.1.5 Magnetic flux

The magnetic flux Φ through a surface is determined by integrating the normal component of the magnetic field 퐵푛 over the complete surface 푆.

Φ = ∫ 퐵푛푑푆 푆 The SI unit of the magnetic flux is the Weber. Magnetic flux can be visualized by a discrete number of field lines with a direction. The flux density (Tesla [T]) can be illustrated by drawing more field lines through a certain surface. This representation is purely illustrative as flux is a continuous quantity.

1.1.5.1 Gauss’s Law of magnetism

Gauss’s law for magnetism states that the magnetic flux through a closed surface (S) is zero. This is because there are no magnetic monopoles unlike in electrostatics. therefore, no “magnetic charge” can be build up in any point in space.

Φ = ∮ 퐵⃑ ∙ 푑푆 = 0 푆 A consequence of this law is that the magnetic flux through a surface with a certain edge curve 퐶 is only dependent on the edge curve 퐶 and not the surface. When looking at Figure 7, the flux through surface 퐴1 is equal to the flux through the surface 퐴2.

Figure 7: The magnetic flux through both surfaces are equal because of Gauss's Law. (Dupré, 2016)

1.1.5.2 Magnetic flux coupled with a closed current loop

Consider a current-carrying conductor with edge curve 퐶 and the surface that is enclosed by this edge curve is 푆. The magnetic flux that goes through this surface 푆 is only dependent on the edge curve 퐶. This magnetic flux is called the magnetic flux that is coupled with the conductor.

The coupled flux is then given by: Φ = 퐵 ∙ 푆 When looking at a solenoid that consists of 푁 windings the same magnetic flux Φ goes through all of these windings. This means that every winding is coupled with this flux Φ. The total coupled flux is then equal to:

Ψ = 푁Φ Φ is called the physical flux of the coil and Ψ is then called the coupled flux of the solenoid.

1.2 Passive components

To have SCMR power transfer a circuit needs to be built with some passive components. In this Section these passive components will be shown along with their properties. 1.2.1 Resistance

To have wireless power transfer by SCMR, coils are needed. These coils are made out of a conductor that is wound around a cylinder. A conductor always had an electrical resistance, this is the difficulty of the electric current has to pass through that conductor. The resistance is the ratio of the voltage (V) across an object to the current (I) through it. This is called Ohm’s law: 푉 푅 = 퐼

The DC resistance of a conductor can also be calculated with some properties of the conductor. The cross- section and the length of the conductor determines the resistance of it along with the material it is made of. Different materials have different electrical resistivity ρ. The electrical resistivity is dependent on the amount of delocalized electrons that are free to move through the material. For example, materials like metals have a high number of delocalized electrons. The formulation of the resistance of a conductor can be stated as:

푙 푅 = 휌 퐴

Figure 8: A simplified representation of a conductor. (Omegatron, 2007)

This formula is an approximation because it assumes the current density in the conductor is uniform. When Alternating current (AC) is used this the formula should be adjusted for effects like skin effect and proximity effect. These effects will be explained in the following Paragraph.

The SI unit of resistance is ohm (Ω).

1.2.2 Frequency dependent resistance model for inductors

1.2.2.1 Skin effect

When an alternating current is running through a conductor the current density becomes the largest near the surface of the conductor. The current flows mainly at the skin of the conductor hence the name “skin effect”. The effective resistance of the conductor will increase at higher frequencies because of this effect because the area of the conductor is virtually smaller. The area where the current mainly flows is described with the skin depth δ, this is the depth from the surface of the conductor as seen in Figure 9. Within a depth 4 times larger than the skin depth 98% of the current will flow.

Figure 9: Skin depth. (Biezl, 2008)

The cause of the skin effect is due to eddy currents. These are electrical currents that are induced within the conductor by the alternating magnetic field which is produced by the alternating current that runs through the conductor. Because of the change in current the magnetic field changes. The change of the magnetic field creates an electric field which is opposite to the change in current. This causes eddy currents to flow in the opposing direction as the direction of the current in the middle of the conductor flowing in the same direction in the outer layers of the conductor. This causes the current to flow mainly in the outer shell of the conductor. This altering of the distribution of the current causes an increase of AC resistance. In Figure 10 a simple representation can be seen of this effect.

Figure 10: The alternating current induces Eddy currents which oppose the current in the centre of the conductor. (Biezl, 2008)

1.2.2.2 Proximity effect

When alternating current is flowing within a conductor, it creates an alternating magnetic field. This alternating magnetic field induces eddy currents in nearby conductors. These eddy currents alter the distribution of current flowing within the conductors.

A simple representation can be seen in Figure 11. 4 parallel conductors are placed closely together. The currents in these conductors flows in the same direction. The current is concentrated to the outer corners of the conductors (red area) farthest away from the other conductors.

Figure 11: Proximity effect in 4 parallel conductors wherein current flows in the same direction. (Stepien, 2012)

Proximity effect causes current to be concentrated in areas of the conductor farthest away from nearby conductors carrying current in the same direction.

1.2.2.3 Litz wire

Litz wire is a special wire that consists of multiple strands that are insulated from each other. A representation of litz wire can be seen in Figure 12.

Figure 12: A representation of the cross section of a litz wire. This wire consists of 19 strands that are insulated from each other. (Zureks, 2009)

Litz wire is designed to reduce the AC resistance of the wire. Or in other words reduce effects like skin effect and proximity effect. Skin effect on the wire is reduced because litz wire consists of multiple strands of insulated wire and an individual strand has a radius that is often smaller than the skin depth (usually up to 1 MHz). Another reason that skin effect is greatly reduced is because the twisting pattern of the strands are in a special way. This pattern lets each strand is on the outside of the bundle for a certain distance and then on the inside of the bundle for a certain distance. Therefore, the strand will be located in weak EM (Electromagnetic) field and strong EM field for a certain distance. This way the current can be distributed on the inside of the bundle as well as the outside. The special twisting pattern also is made in such way that they have a reduces tendency to generate an opposing EM field in other strands.

Litz wire is very effective for frequencies below 500 kHz this because the skin depth is then larger than the radius of an individual strand. When the frequency gets higher than 1 MHz the benefits gets offset by the parasitic capacitance between the strands.

1.2.2.4 Conclusion

Because SCMR power transfer will use relatively high frequencies, the effects that are discussed in these Section are very important to consider when calculating the efficiency of the power transfer. These effects will be further discussed in Section 2.1 where a formula will be shown that can predict the effect these losses on the overall resistance.

1.2.3 Inductance

The inductance of a coil is also important to understand how the power transfer happens for SCMR. This inductance also needs to be known when the power transfer needs to be calculated. The explanation of the inductance can be found here.

Figure 13: Definition of inductance of a closed loop C. (Dupré, 2016)

Looking at Figure 13, the inductance 퐿 of an electrical conductor with the shape of a closed loop C can be defined as the ratio of the magnetic flux Φ through the area A (this is then the coupled flux Φ푐) to the current that flows in the closed loop C. this definition can be represented as: Φ 퐿 = 푐 퐼 1.2.3.1 Inductance of a solenoid

The magnetic field in an infinitely long solenoid is uniform. When 푛 is the number of windings per unit length of the infinite solenoid, the magnetic induction 퐵 of this solenoid that was found in the previous Section on magnetic fields was:

퐵 = µ0푛퐼

When 퐴 is the cross-section of the solenoid, then the magnetic flux Φ is:

Φ = 퐵퐴 = µ0푛퐼퐴

The number of windings in a zone with length 푙 of the solenoid is:

푁 = 푛푙

The coupled flux of the section is:

푁2퐼퐴 Ψ = 푁Φ = µ 0 푙

The inductance of the section is: Ψ 푁2퐴 퐿 = = µ 퐼 0 푙

This formula can also be used for a finite solenoid where the length of the solenoid is much longer than the diameter of the solenoid.

1.2.3.2 Conclusion

A coil (e.g. solenoid) can be used to make a magnetic field. This will be used to transfer power wirelessly. One coil will create a magnetic field by having an alternating current. Another coil will be used to ‘catch’ this magnetic field and convert it back to a current, this will be further discussed in the following section.

1.2.4 Mutual induction

The power transfer happens between two coils, the coupling of these two coils can be stated as mutual induction. Consider two coils that are placed in such way that the flux that goes through the windings of the first coil, also goes through the windings of the second coil. When a current 퐼1 flows through the first coil while there is no current in the second coil, a physical flux 휙1is created by 퐼1. The coupled flux of coil 1 is represented as 휓11. The coupled flux of coil 2 is 휓21. The coupled fluxes are defined as:

휓 = 푁 휙 = 퐿 퐼 { 11 1 1 1 1 휓21 = 푁2휙1 = 푀21퐼1

푀21 is the mutual induction of coil 1 on coil 2. From the formulas above can be found that:

푁 퐿 1 = 1 (1) 푁2 푀21

By having no current in the first coil and a current 퐼2 in the second coil. the coupled fluxes can be written as: 휓 = 푁 휙 = 푀 퐼 { 12 1 2 12 2 휓22 = 푁2휙2 = 퐿2퐼2

푀12 is the mutual induction of coil 2 on coil 1. From the formulas above can be found that:

푁 푀 1 = 12 (2) 푁2 퐿2 By using (1) and (2): 푀 퐿 12 = 1 퐿2 푀21 Or:

퐿1퐿2 = 푀12푀21

By now having current in coil 1 and 2 at the same time the total coupled flux of coil 1 휓1 and coil 2 휓2 can be written as:

휓1 = 휓11 + 휓12 = 퐿1퐼1 + 푀12퐼2 휓2 = 휓21 + 휓22 = 퐿2퐼2 + 푀21퐼1

These formulas were made by assuming that the fluxes that are created by both coils are in the same direction. When this is not the case, the coupled fluxes are defined as:

휓1 = 휓11 + 휓12 = 퐿1퐼1 − 푀12퐼2 휓2 = 휓21 + 휓22 = 퐿2퐼2 − 푀21퐼1

It can be shown that 푀21 = 푀12. This can be done by moving a coil from very close to infinity and comparing the work W that is needed for those movements. By equalling the mutual inductions to 푀 it can be seen that:

푀 = √퐿1퐿2

1.2.5 Coupling factor

Previously, it was assumed that the physical fluxes through both coils were the same. this will almost never happen, there will almost always be a leakage flux. When the coils are further away this leakage flux will be higher. Therefore, a coupling factor 푘 is added to the previous formula for mutual induction:

푀 = 푘√퐿1퐿2 Where:

−1 ≤ 푘 ≤ 1

When 푘 is 0, the coils are not magnetically coupled. When 푘 is 1, the coils are perfectly coupled and there is not any leakage flux.

1.2.6 Capacitance

The last passive component needed for making SCMR is a capacitor. This capacitor has a certain capacitance (퐶). This is the ability of a component or circuit to store energy in the form of electrical charge separation . Capacitance is the ratio of the electric charge to the potential difference (i.e., voltage) between 2 conductors.

푞 퐶 = 푉

A capacitor is a component made to have a specific capacitance. The capacitance of a capacitor is dependent on the geometry and the electric properties of the dielectric material. The electric property of the material that influences the capacitance is permittivity ε. Permittivity is the resistance the material gives to forming an electric field. A capacitor can be made in different ways, an example of a capacitor is a parallel-plate capacitor, see Figure 14.

Figure 14: A representation of a parallel-plate capacitor. (Fabian, 2013)

As stated before, a capacitor has 2 conductors. In Figure 14, this is represented by the top and bottom plate with an area of A. These plates are placed at a distance d from each other. Between the plates is a dielectric material with permittivity ε. The capacitance of this example can be written as: 퐴 퐶 = 휀 휀 푟 0 푑 The SI-unit for capacitance is Farad (F).

1.3 Electromagnetic induction

The way how the changing magnetic field (flux) induces an electric field can be seen in Faraday’s Law. The principles of electromagnetic induction and self-induction can be seen in this Section. 1.3.1 Faraday’s Law

An electric field is generated by a changing magnetic field. Faraday’s law states that the electromotive force

(퐸푒푚푓) (EMF) is given by the negative rate of change of the magnetic flux (휙푚):

푑Φ 퐸 = − 푚 푒푚푓 푑푡

The negative sign indicates that any current that is generated by the changing magnetic field opposes that change in magnetic field that induced it. Tis is called Lenz’s law. Note that 휙푚is the coupled flux of the current loop. When a coil is considered the expression should be changed to:

푑Φ 푑Ψ 퐸 = −푁 = − 푒푚푓 푑푡 푑푡 1.3.2 Induction phenomena

There are 2 kinds of induction phenomena. Induction phenomena caused by a movement of conductors in a magnetic field, this is called “induction phenomena of the first kind”.

퐸푒푚푓 = 퐵푙푣

“Induction phenomena of the second kind” is then induction when there is no movement. This induction is caused by a changing magnetic field, which is created by another (nearby) coil.

푑Ψ 퐸 = − 푒푚푓 푑푡

1.3.3 Electromagnetic force of self-induction

Consider a coil, with internal resistant 푅, connected to a source 퐸 and a switch. When closing the switch, the current will increase from 0 to a maximum value of: 퐸 퐼 = 푅

During this transition period the current will increase, which in turn increases the physical flux Φ. Because of Faraday’s Law an EMF is created in the coil because of induction:

푑Ψ 퐸 = − 푒푚푓 푑푡

This relation can be written in function of the current through the coil because Ψ = 퐿푖.

푑(퐿푖) 퐸 = − 푒푚푓 푑푡

1.3.4 Electromagnetic force of mutual-induction

When 2 coils are magnetically coupled the EMF can be stated for the first coil as:

푑(푀푖 ) 퐸 = − 2 푒푚푓,1 푑푡

1.3.5 Conclusion

In this Section and previous Sections, it can be seen that an electrical field can be induced in a coil that is located in a changing magnetic field of another coil. The changing magnetic field is induced by an alternating current in the ‘sender’-coil and because there is a certain mutual induction between the ‘sender’- and ‘receiver’-coil, an certain electrical field will be induced in the ‘receiver’-coil. Because of the electrical field in the second coil, a current can flow in this coil when there is a load connected to this coil (or it is shorted).

1.4 Resonator

Wireless power transfer between coils is explained in the previous Sections along with some passive components. For SCMR a passive component is needed to be added to these coils to make a resonator. this component is a capacitor. Together the inductor (퐿) and capacitor (퐶) form an ideal resonator. In Figure 15 the circuit can be seen of these components along with the induced voltage (휀) of the coil (Section 1.3).

Figure 15: LC-circuit.

In an inductor, energy is stored in the form of magnetic energy. This happens when a current is flowing through it. When an inductor is in a changing magnetic field, the coupled magnetic flux with the inductor changes as well. This changing coupled flux leads to an opposing voltage across the inductor, this is called the electromotive force.

In a capacitor, electrostatic energy is stored as potential energy when there is a difference in charge across a dielectric material. When the charge leaves the capacitor (in the form of current), the voltage over the capacitor will decrease.

Going back to a resonator circuit and starting with a situation where there is a voltage across the capacitor: The capacitor will generate a current that leaves it until the voltage over the capacitor equals zero. The inductor will oppose this generation of current. Therefore, the current will only build up over a certain time. Because of this magnetic energy is stored. When the capacity of the capacitor is ‘empty’ and the voltage is 0, a current is built in the inductor that will not just decrease on its own. But if the current stays, the charge distribution in the capacitor will build up again. The voltage in the capacitor will increase and oppose the current until the current is 0. Now the situation back to the starting situation.

Because there is no resistance in the circuit this cycle will continue forever. But in real world applications there will always be internal resistance which leads to damped oscillation. The internal resistance of the conductor and capacitor will have a dampening effect because a part of the energy will be transferred to heat because of heat dissipation in the resistance.

1.4.1 Resonance

Because an inductor and a capacitor have frequency dependent features, the circuit will also be frequency dependent. A system that is frequency dependent has a resonance frequency. This resonance frequency can be found when the amplitude of the impedance of the series circuit is at a minimum. So, for Figure 15 this means:

푍 = 푋퐿 + 푋푐 1 푍 = 푗휔퐿 + 푗휔퐶 1 − 휔2퐿퐶 푍 = 푗휔퐶

To have a minimum impedance the nominator must be zero:

0 = 1 − 휔2퐿퐶 1 휔0 = √퐿퐶

With 휔0 = 2휋푓0: 1 푓0 = 2휋√퐿퐶

With 푓0 as the frequency of the source, the total impedance Z will behave like an impedance of zero. In reality only the imaginary part of the impedance will be zero, the real part will be equal to the resistance of the conductor and other internal resistance of the components.

1.4.2 Quality factor

To get a better representation of a real-world resonator, the circuit is adjusted by adding a resistor to represent the equivalent series resistance (ESR) of the inductor, capacitor and conductor of the circuit as shown in Figure 16.

Figure 16: RLC-circuit.

The impedance of the circuit becomes: 1 푍 = 푅 + 푗 (휔퐿 − ) 휔퐶

When the driving frequency is f0, the impedance of L and C will cancel out, such that Z becomes:

푍푚푖푛 = 푅

The resistor is damping the effect of resonance, it will lower the current flowing through the circuit compared to having no resistance but is still higher than the current when there is the driving frequency is not f0. The bandwidth is defined as the frequency range where the amplitude of the current is higher than 50% of the peak amplitude (with f0). By lowering the peak amplitude, the bandwidth Δω will increase. A bigger bandwidth means easier tuning of the resonator. A way to show this relationship is by the quality factor Q: 휔 푄 = 0 ∆휔 Bandwidth in relation to the damping factor ζ:

푅 ∆휔 = 2휁 = 퐿 휔 푄 = 0 ∆휔 1 퐿 푄 = ∗ √퐿퐶 푅 1 퐿 푄 = √ 푅 퐶

So, a circuit with high quality factor means a narrow bandwidth and a high amplitude. In Figure 17 the effect of Q on the induced current is illustrated.

Figure 17: The induced current for an arbitrary RLC circuit increased drastically around resonance for higher Q-factors, while the bandwidth decreases. (Vandeputte, 2016)

1.4.3 Conclusion

Now a clear picture can be seen of how power can transfer wirelessly in strongly coupled magnetic resonance mode. It can be seen in the formula for the Q-factor that the resistance needs to be as low as possible to have a high-power transfer efficiency. The inductance and capacitance of the resonator don’t have a direct impact of the power transfer efficiency because they compensate each other in resonance. But the resonance frequency of the resonator and the magnetic coupling do affect the power transfer efficiency so the inductance and capacitance still are very important to calculate the efficiency.

1.5 Wireless power transfer via strongly coupled magnetic resonance

Some experiments were conducted to better understand the power transfer with strongly coupled magnetic resonance. The following paragraphs will explain the experiments that were conducted. The experiments show critical understanding and aspects of the power transfer via strongly coupled magnetic resonance. By showing these examples, the principles will be easier to understand. 1.5.1 First experiment

The first experiment was a simulation in Simulink from MathWorks. In this simulation a simple setup was simulated to see how the efficiency of transfer behaves as the frequency of the AC source approaches the resonance frequency of the circuit. The electric scheme of the simulation can be seen in Figure 18.

Figure 18: Circuit of first experiment. 2 inductors are coupled, each inductor is part of a circuit with a capacitor and a resistor. One circuit is connected to a voltage source.

The scheme consists of 2 circuits that are connected through the magnetic coupled inductors. Each circuit has an inductor, capacitor and resistor. One of the circuits is connected to a voltage source, the frequency of the voltage source can be changed with simple commands and will be variable to see how the voltage over Resistor 2 changes. All variables of the components are shown in Table 1.

Table 1: Parameters of the components used in the Simulink circuit. Component on diagram Specific parameter Value

Peak amplitude 10 V AC Voltage source Phase shift 0° frequency variable

Resistor 1 Resistance 1 Ω Capacitor 1 Capacitance 1 µF Resistor 2 Resistance 1 Ω Capacitor 2 Capacitance 1 µF Inductance L1 50 µH Mutual inductor Inductance L2 50 µH Coefficient of coupling 0.1

The frequency of the voltage source is controlled by a small program that was run in Matlab. The code of this program can be found in appendix A: code 1. As seen in the paragraph about the resonance frequency of a resonator the resonance frequency of this resonator is: 1 1 푓푟푒푠 = == = 22 508 퐻푧 2휋√퐿퐶 2휋√50 ∗ 10−6 ∗ 1 ∗ 10−6 In the experiment the frequency was variable over a range of 1000 to 40 000 Hz. This range was chosen to have a clear view of what happens to the circuit in resonance. For every frequency the voltage over resistor 2 was measured (in the simulation) and plotted in a graph that can be seen on the next page. On the horizontal axis the frequency can be found and on the vertical axis the maximum value of the voltage over resistor 2 can be found for that particular frequency.

Figure 19: Graph of the voltage over the resistor in the second circuit.

Figure 20: Close up of the previous Figure in the resonant region.

In Figure 20 can be seen that the maximum voltage over resistor 2 is at a frequency of approximately 22 600 Hz. This is at the theoretical resonance frequency of 22 508 Hz that was calculated earlier. This shows that the best transfer of power is at resonance of the circuit. This is because the Imaginary parts of the impedance of L and C cancel each other out such that the impedance is minimized and the induced current (and thus the voltage over de resistor) is maximized. No useless reactive power is generated in the imaginary impedance. (as this represents energy transfer from the capacitor to the inductor and vice-versa) 1.5.2 Second experiment

The second experiment was meant to show how the energy was transferred using resonators. There was also an objective to make a computer simulation in Simulink with the same components, prove that this gave the same results as measurements done on the real circuit and then make some other simulations to show how the energy transfer behaved.

The resonators just consist of an inductor, a capacitor and the internal resistance of those components and the wire. These components were placed in series in a closed loop. To put energy in these resonator, a driver coil was placed with a good magnetic coupling to the primary coil. To get energy out of the resonators, a load coil was placed with a good magnetic coupling to the secondary coil. A schematic representation can be found in Figure 21.

Figure 21: Schematic representation of second experiment. (Wei, Wang, & Dai, 2014)

1.5.2.1 Practical setup

To practically make this circuit, a rudimentary setup was made, this can be seen in Figure 22. The coils were wound around a PVC tube. The wire that was used to make the coils was litz wire (see Section 1.2.2.3 for more information on litz wire).

Figure 22: Setup of resonator experiment.

To have the best coupling between the driver coil (which was just 1 winding and a resistor in series) and the primary coil, the winding of the driver coil was placed on top and in the centre of the primary windings. This can be seen better in Figure 23.

Capacitor of primary circuit

1 loop driver coil 8 loop primary coil

Resistor of driver circuit

Figure 23: Close up of primary coil and driver coil.

The second coil and load coil were made in a similar way on a smaller PVC tube. There was no actual reason to use a smaller PVC tube other than the fact that there was not any tube left with the same diameter.

The primary and secondary coil must have the same resonance frequency to have efficient resonant magnetic coupling. As the resonance frequency of a circuit is dependent on the inductance and the capacitance of the circuit, one of these parameters should be variable. For this reason, the secondary coil was made from 2 sets of windings that could move relative to each other. By moving the sets toward or away from each other the inductance changed, making tuning of the resonator possible. On Figure 24 you can see the secondary coil has a set of windings that is mounted on the PVC tube with a part of this set mounted on a piece of paper that can be moved over the tube.

Figure 24: Close up of the secondary coil and load coil.

The inductance of the different coils was measured with a RLC bridge (Rohde & Schwarz HM8118) to accurately measure what the induction and resistance was of the different coils. With that information capacitors were calculated to get corresponding resonance. 1.5.2.2 Simulation

The measurements were also used as parameter values for a computer simulation. The inductance of the driver and load coil, which were made of just 1 loop, were not measured because their impedance would be too small for the measuring tool. For this reason, the inductance of the 1 loop coils was derived from the formula of the inductance of a solenoid. This is not entirely accurate because the length of these coils is not very long so the formula for solenoid cannot be used. In the formula the obtained inductance of the primary coil will be used to predict the inductance of the driver coil by using the formula of the solenoid. As seen on the formulas below the area of the coils and the length of the coils is assumed to be equal. But the lengths of the driver/load coil and the primary/secondary coil were not actually. But these wrong assumptions are not that important because only a rough estimation was needed, the order of magnitude was enough. The following formula was used, with 훿 being the geometric form factor of the coils:

2 µ0. 푁푎 . 퐴 퐿푎 µ0. 퐴 퐿푎 = 훿 ↔ 2 = 훿푎 (1) 푙 푁푎 푙 µ . 푁2. 퐴 퐿 = 0 푏 훿 (2) 푏 푙 푏 2 (1) 푖푛 (2) 푁푏 → 퐿푏 ≈ 2 퐿푎 푁푎 The objective was to make a simulation in Simulink that will behave the same as the practical setup. It would be far easier to investigate the behaviour on the simulation. Therefore, the behaviour of the power transfer could be better investigated on the simulation. The circuit of the simulation can be seen in Figure 25.

Figure 25: Circuit of resonator simulation.

All the values of the components were given to the simulation with an initialisation program. Some values were slightly changed to get a better result for full resonance. The code can be found in Appendix A, code 2.

A formula was derived to get the mutual induction between the driver coil and the primary coil (represented as M12), primary and secondary coil (M23) and the mutual induction between the secondary coil and the load coil (M34). The way this formula was derived can be found on the next pages.

Figure 26: Simple circuit of 2 magnetically coupled coils.

When looking at a simple circuit with 2 coils that are magnetically coupled, see Figure 26, it can be stated that the current flowing in the second coil is induced by the current flowing in the first coil. The variable that makes this happen is the mutual induction. The relationship is stated by Kirchhoff’s voltage law (KVL): 1 (푗휔퐿2 + + 푅2) 퐼2 = −푗휔푀12퐼1 푗휔퐶2

There will be resonance in the resonator, this means that the inductance and capacitance will cancel each other out resulting in the following Equation.

푅2퐼2 = −푗휔푀12퐼1

Writing this Equation to solve to 푀12:

푅2퐼2 푅2퐼2 푀12 = = 푗 −푗휔퐼1 휔퐼1

This means that in order to get the mutual induction between 2 coils, the currents and the resistance in the second coil need to be obtained. So, the currents and the resistance of the coil was measured in the practical setup.

In the simulation M13 (mutual induction between driver and secondary coil), M14 (mutual induction between driver and load coil) and M24 (Mutual induction between primary and load coil) are put at 0. These are purposefully neglected because they do not have a great effect on the currents in comparison to coils that are next to each other (like M12, M23 and M34). The comparison between using these smaller mutual induction values in the system and not using these mutual inductions can be found further, first the calculations and results of the simulation with these parameters will be shown.

With these values the simulation is fully defined. For the experiment in the simulation the maximum amplitude of the current and voltages of the coils were generated and plotted along with the phase difference compared to the phase of the current in the driver coil. In Figure 27 the current phasor in the driver, primary and secondary coil and the voltage over the load resister in the load coil can be found for different frequencies of the AC source.

Figure 27: Maximum currents (in Amperes) of driver, primary and secondary coil and voltage (in Volt) over the load resistor in the load coil. These maximum values with there respective phase are plotted on a polar system for different frequencies. The minimum frequency used is 130 000 Hz and the maximum frequency is 240 000 Hz. The resonance frequency of the resonators is 185 000 Hz.

In Figure 27 can be seen that the current in the driver coil changes a little as the frequency changes. This is because there is an inductor in this coil. As the frequency changes, so does the reactance, leading to a change in the total impedance of the coil. Also, the effect known as the reflected impedance changes the current. This effect will be explained in the following Section.

1.5.2.3 Reflected impedance

The reflected impedance is the effect that the coupled coils have on the main coil. Take for example the simple situation of 2 circuits that both have an inductor, these inductors are coupled with each other. The equivalent circuit can be seen on Figure 28.

Figure 28: Equivalent circuit of system with 2 coils that are coupled. (Wei, Wang, & Dai, 2014)

When using circuit theory based on mutual inductance the following equations can be found for this circuit:

For the primary coil:

1 푉푠 = (푅2 + 푅푠)퐼2 + 푗 (휔퐿2퐼2 + 휔푀23퐼3 − 퐼2) 휔퐶2 For the second coil:

1 0 = 푅3퐼3 + 푗 (휔퐿3퐼3 + 휔푀23퐼2 − 퐼3) 휔퐶3

These equations can be shortened when the impedance 푍 = 푅 + 푗푋:

푉푠 = 푍2퐼2 + 푗휔푀23퐼3 0 = 푍3퐼3 + 푗휔푀23퐼2

When solving the second equation to 퐼3 the following expression can be found:

−푗휔푀23퐼2 퐼3 = 푍3

Now 퐼3 can be replaced in the equation of the first coil:

−푗휔푀23퐼2 푉푠 = 푍2퐼2 + 푗휔푀23퐼3 = 푍2퐼2 + 푗휔푀23 ( ) 푍3

By doing this the simple equation for 2 magnetically coupled coils is:

2 2 휔 푀23 푉푠 = 푍2퐼2 + 퐼2 푍3

It can be seen that the part of the equation that represents the coupling of the 2 coils now have become an impedance this is called reflected impedance, as the second coil acts as an added impedance in the first coil. The system in Figure 28 can now be simplified to the circuit in Figure 29.

Figure 29: Simplified circuit of the system of 2 magnetically coupled coils by replacing the secondary coil with an equivalent impedance in the primary coil. (Wei, Wang, & Dai, 2014)

The same can be done when the signals of the secondary coil is wanted. When rewriting the equation of the primary coil, the following relationship of 퐼2 can be found:

푉푠 푗휔푀23퐼3 푉푠 = 푍2퐼2 + 푗휔푀23퐼3 → 퐼2 = − 푍2 푍2

Replacing 퐼2 in the equation of the second coil, the following equivalent relationship can be found for the system:

2 2 푗휔푀23 휔 푀23 0 = 푍3퐼3 + 푉푠 + 퐼3 푍2 푍2 It can now be seen that the impedance of the primary coil is reflected on the secondary coil but also that the voltage source of the primary coil is also reflected in the secondary coil. The representation of this can be found in Figure 30.

Figure 30: Simplified circuit of the system of 2 magnetically coupled coils by replacing the primary coil with an equivalent impedance and voltage source in the secondary coil. (Wei, Wang, & Dai, 2014)

1.5.2.4 Circuit theory of the experiment

The experimental setup can also be described using circuit theory. The equations of all the coils can be written as the following: 1 핍 = ℝ핀 + 푗휔핃핀 + ℂ핀 푗휔

With:

푅1 0 0 0 퐿1 푀12 푀13 푀14 0 0 0 0 푉1 퐼1 0 푅 0 0 푀 퐿 푀 푀 0 1/퐶 0 0 0 퐼 ℝ = [ 2 ] ; 핃 = [ 12 2 23 24] ; ℂ = [ 2 ] ; 핍 = [ ] ; 핀 = [ 2] 0 0 푅3 0 푀13 푀23 퐿3 푀34 0 0 1/퐶3 0 0 퐼3 0 0 0 푅4 푀14 푀24 푀34 퐿4 0 0 0 0 0 퐼4

When writing these equations separately we get:

푉1 = 푅1퐼1 + 푗(휔퐿1퐼1 + 휔푀12퐼2 + 휔푀13퐼3 + 휔푀14퐼4) 1 0 = 푅2퐼2 + 푗 (휔퐿2퐼2 + 휔푀12퐼1 + 휔푀23퐼3 + 휔푀24퐼4 − 퐼2) 휔퐶2 1 0 = 푅3퐼3 + 푗 (휔퐿3퐼3 + 휔푀13퐼1 + 휔푀23퐼2 + 휔푀34퐼4 − 퐼3) 휔퐶3 0 = 푅4퐼4 + 푗(휔퐿4퐼4 + 휔푀14퐼1 + 휔푀24퐼2 + 휔푀34퐼3)

It was mentioned previously that the mutual inductions of non-neighbouring coils would be dropped from the equations because the equations will be simpler and the effect these mutual inductions have is negligible. By simplifying these relationships, the induction matrix will be reduced to a tridiagonal matrix:

퐿1 푀12 0 0 푀 퐿 푀 0 핃 = [ 12 2 23 ] 0 푀23 퐿3 푀34 0 0 푀34 퐿4

And the separate equations are shortened to:

푉1 = 푅1퐼1 + 푗(휔퐿1퐼1 + 휔푀12퐼2) 1 0 = 푅2퐼2 + 푗 (휔퐿2퐼2 + 휔푀12퐼1 + 휔푀23퐼3 − 퐼2) 휔퐶2 1 0 = 푅3퐼3 + 푗 (휔퐿3퐼3 + 휔푀23퐼2 + 휔푀34퐼4 − 퐼3) 휔퐶3 0 = 푅4퐼4 + 푗(휔퐿4퐼4 + 휔푀34퐼3)

These are the basic equations of the circuit and predict the behaviour of the current in the coils theoretically. A new graph was made to prove the simulation is based on these equations. In a script these equations were transformed into equations that solve these relationships to the currents, this script can be found in Appendix A code 3. The graph of the analytically found relationship of the currents were plotted on top of the graph of the simulated circuit:

Figure 31: The (black striped) graph of the analytical results of the behaviour of the current of the coils plotted on the (red lined) graph of the simulated relationship of the currents in the coils.

It can be seen on Figure 31 that the simulations is based on the theoretical equations that were found earlier because the behaviour of the currents matches perfectly.

Previously it was stated that the mutual inductions of non-neighbouring coils were neglected from the calculations because the effects of these mutual inductions were very small compared to the mutual inductions of neighbouring coils. To show this the mutual inductions of non-neighbouring coils were calculated using equations that calculate the mutual inductance between air-cored coils (Anele, et al., 2015), this equations will be shown further in section 2.3 . By using these additional inductions, the currents in all the coils were calculated again and compared to the currents that were calculated by neglecting these additional mutual inductions in Figure 32.

As seen in Figure 32 the effect of including these mutual inductions (of non-neighbouring coils) are rather small. The most noticeable effect these mutual inductions have is on the phase-change of the voltage 푈4. The phase of the voltage over the load resister is slightly different (a couple of degrees) if the mutual inductions are taking into account but the amplitude of the voltage still the same. So, the error on the currents and voltages in the coils by neglecting the mutual inductions of non-neighbouring coils is very small.

In Figure 33 the first found signals of the simulation are shown again like they were shown in Figure 27, but here some annotations were added. Red lines were added next to the graphs to show were the frequency is equal to (almost) zero and an arrow shows the direction of the curve if the frequency is building up. Red dots were added to show in which point resonance were obtained in the resonators.

When looking at the current in the driver coil (푖1) it can be seen that when the source is at resonance frequency the current is in phase with the voltage of the source. This is because of the fact that in resonance the inductance and capacitance cancel each other out and the impedance is reduced to just the ohmic resistance of the resonator (real part of the reactance). It can be seen that 푖1 is smaller when there is resonance as when the frequency of the source is very low, this is because of the effect of reflected impedance that was explained earlier. A way of seeing this is as if the other coupled coil is an impedance for the source, which causes the current to be smaller and when there is resonance the impedance of this other coil will also be reduced to the ohmic resistance of the coupled coil.

It can also be noticed that the phase of the currents in the other coils are always 90° shifted in comparison to the previous current when there is resonance. This phase change can be seen clearly when looking at the circuit theory of the relationship of the currents at the start of this Section (p. 46). When looking at the voltages of the second resonator for example the equation was:

1 0 = 푅2퐼2 + 푗 (휔퐿2퐼2 + 휔푀12퐼1 + 휔푀23퐼3 + 휔푀24퐼4 − 퐼2) 휔퐶2

When neglecting the mutual induction of non-neighbouring coils and when there is resonance, this equation is reduced to:

−푗휔푀12퐼1 = 푅2퐼2 + 푗휔푀23퐼3

It can be seen that the coupling with the first coil is the source for the second coil. And it can be seen that this “source voltage” is equal to −푗휔푀12퐼1 which is has a phase difference of 90° compared to 퐼1.

When looking at the currents if the frequency of the source is not equal to the resonance frequency it can be seen the phase change and amplitude of the currents are not that easy to explain. This is because the equations of the circuit cannot be reduced and so the relation between all the currents stay complicated.

Figure 32: A representation of the effect the mutual inductions of non-neighbouring coils have on the signals.

Figure 33: Plot of maximum current in all coils with annotations.

1.6 Frequency splitting phenomena

The frequency splitting phenomena have an influence on the transfer power of magnetic resonant coupling WPT, this is shown in the paper “Frequency Splitting Phenomena of Magnetic Resonant Coupling Wireless power transfer” (Huang, Zhang, & Zhang, 2014). It was seen previously that the efficiency of the wireless power transfer was optimal at resonance frequency. In Figure 34 the efficiency of an arbitrary two-resonator system is shown for different frequencies.

Figure 34: Efficiency of an arbitrary two-resonator WPT system plotted against the frequency. (Huang, Zhang, & Zhang, 2014)

But when plotting the power that is transferred against the frequency in Figure 35, it can be seen that the maximum power transfer is split into two peak values. Furthermore, the frequencies at which these peaks occur or not equal to the resonance frequency of the system, where the efficiency is optimal.

Figure 35: The normalized transfer power of the system of Figure 34 plotted against the frequency. (Huang, Zhang, & Zhang, 2014)

1.6.1 Conclusion

The frequency splitting phenomena is in respect to the frequency characteristic of transfer power and not to that of system efficiency. In this dissertation the optimal power transfer efficiency is desired so these phenomena will not be discussed later.

2 Developing a predicting model for resonators

In this Chapter a model will be made in Matlab that can predict the power transfer efficiency of the resonator system. The power transfer efficiency is dependent on het power that the AC voltage source gives to the first resonator and the power that the load resistor receives in the last resonator.

Because the voltage of the voltage source will be known the only other parameter of the system that needs to be calculated is the current to know the input power. The same goes for the output power of the load resistor, because the load resistor will also be self-chosen.

So, this means the currents in the resonators must be predicted. This will be done by using circuit theory just like in Section 1.5.2.4. In Section 1.5.2.4. the circuit theory was shown for two resonators with one resonator being strongly magnetically coupled with a driver coil and the other resonator strongly coupled to a load coil. To simplify this expression the expression will be given of the voltages the are in the resonators of a two- resonator system that are coupled with each other. The circuit can be seen in Figure 36.

Figure 36: Circuit of a two-resonator power transfer system with annotations.

The voltages in the resonator can be stated as: 1 핍 = ℝ핀 + 푗휔핃핀 + ℂ핀 푗휔

With:

푅 0 퐿 푀 1/퐶 0 푉 퐼 ℝ = [ 1 ] ; 핃 = [ 1 12] ; ℂ = [ 1 ] ; 핍 = [ 1] ; 핀 = [ 1] 0 푅2 + 푅퐿 푀12 퐿2 0 1/퐶2 0 퐼2

푅1 and 푅2 represent the AC resistance of the windings of resonator 1 and 2. 퐿1 and 퐿2 are the inductances of the resonators and 푀12 is the mutual inductance between the coils of the resonators. 퐶1 and 퐶2 are the capacitances of the resonators. 푅퐿 is the load resistor of the second resonator where the power needs to be delivered. 푉1 is the amplitude and 휔 the angular frequency of the source voltage. By rewriting this expression, the currents in the resonators can be stated as: 핍 핀 = 1 ℝ + 푗휔핃 + ℂ 푗휔

So, to know the currents in the resonators this equation can be used. The variables that need to be known of the resonators are: the resistances of the windings, the inductances of the coils, the mutual inductances between the coils, the capacitors that will be used in the resonator, the angular frequency of the source and the amplitude of the voltage of the source.

The angular frequency and the amplitude of the source do not need to be calculated by the model because this will self-chosen. Also, the load resistor will be self-chosen (note that there is an optimal resistor for each resonator system for a certain frequency, this will be shown later). The variables that need to be calculated are the resistances, the inductances, the mutual inductance and capacitances. These will be calculated with formulas found in literature, the formulas for these variables will be shown next. 2.1 Formula of AC resistance of the conductor used for the windings

The DC resistance of a conductor can be easily calculated if the length 푙 of the conductor, the cross-section area 푆 and the electrical resistivity of the material of the conductor 휌 is known: 휌 ∗ 푙 푅 = 푐표푝푝푒푟 퐷퐶 푆 But the AC resistance is in contrast much more difficult to calculate because of the effects of eddy current like skin and proximity effect, especially when the conductor is wounded around a coil. A formula was found that could calculate the AC resistance of transformer winding. This formula was derived by P.L. Dowell in “Effects of eddy currents in transformer windings” were the full derivation can be found.

The formula for the AC resistance (Dowell, 1996) is:

(푚2 − 1)푅푒(퐷) 푅 = 푅 (푅푒(푀) + ) 퐴퐶 퐷퐶 3 푀 = 훼ℎ푐표푡ℎ(훼ℎ) 훼ℎ 퐷 = 2훼ℎ푡푎푛ℎ ( ) 2 푗휔휇 휂 훼 = √ 0 휌 푎 휂 = 푁 푙 푏 With 푎 being the width of the square conductor used, ℎ is the height of the conductor, 푏 the width of the coil,

푁푙 the number of turns per layer, 휌 the resistivity, 휇0 the permeability and m the number of winding layers. Because this formula uses square conductors the width 푎 and the height ℎ of the conductor should be corrected because circular conductors are used in the experiments and calculations. This is done by the following expression, the correction is done by stating that the surface of the conductor must be the same for square and circular conductors:

퐴 = ℎ2 푑2 . 휋 퐴 = 푤 4 푑 . √휋 ℎ = 푤푖푛푑푖푛푔 2

2.2 Empirical formulas for inductance of a coil

a) Air coils with rectangular cross-section

The experimental equation of Welsby give the induction for air coils that are wound in a rectangular shape, see Figure 37. The accuracy should be around 2%. Solenoids (small 푐) and disc coils (small 푏) also give a good result. Inductance of coils with both small 푏 and 푐 compared to 푎 cannot be determent accurately with this equation (Murgatroyd, 1986).

푁2휋푎2 1 퐿 = µ0 푎 푐 푐 푏 1 + 0.9 + 0.32 + 0.84 푏 푎 푏

Figure 37: Cylindrical coil with rectangular winding cross-section. (Van den Bossche, 2008)

b) Finite-length air coils (solenoids)

The inductance of a finite-length solenoid with round wire can be calculated by using one of the Wheeler (Wheeler, 1942) formulas with good accuracy. The accuracy should be better than 1% if the length 푙 of the solenoid is larger than 0.8푎. 10휋µ 푁2푎2 퐿 = 0 9푎 + 10푙

For shorter solenoids where 푙 < 푎, the Nagaoka (Grover, 1946) formula can be applied.

2 2 퐿 = 휋µ0푁 푎 푙퐾

Where 퐾 is a constant depending on the ratio of the diameter (퐷) to the length of the coil (푙). Lundin’s formula (Knight, 2016) defines this constant very accurately (<0.0002%):

2푙 [ln (4퐷/푙)][1 + 0.383901(푙/퐷)2 + 0.017108(푙/퐷)4] 퐾 = [ + 0.093842(푙/퐷)2 + 0.002029(푙/퐷)4 휋퐷 [1 + 0.258952(푙/퐷)2] − 0.000801(푙/퐷)6]

2.3 Mutual inductance calculation between circular filaments arbitrarily positioned in space

The mutual inductance needs to be calculated for coils that are placed in a arbitrarily relative position. For this calculation a paper on mutual inductance named “Computation of the Mutual Inductance between Air- Cored Coils of Wireless Power Transformer” (Anele, et al., 2015) was consulted. In this paper the mathematical models for evaluating the mutual inductance between coils. The relative position can be with or without angular and lateral misalignment.

The formulas used are shown below. 푟1 and 푟2 are the radiuses of the 2 coupled coils, 푁1 and 푁2 are the number of windings of the respective coils. 휃 is the angular misalignment of a coil, 푑 is the lateral misalignment and c is the distance between the centers of the coils, see Figure 38. The mutual inductance is then given as: 푑 휋 [1 − 푐표푠휙] Ψ(푘) 2푁1푁2µ0 푟2 푀 = √푟1푟2 ∫ 푑휙 3 휋 0 푘√푉 Where 2 푟2 푐 2 4훼푉 푘 훼 = , 훽 = , 휉 = 훽 − 훼푐표푠휙푠푖푛휃, 푘 = 2 2 , Ψ(푘) = (1 − ) 퐾(푘) − 퐸(푘) 푟1 푟1 (1 + 훼푉) + 휉 2 푑 푑2 2 2 푉 = √1 − 푐표푠 휙푠푖푛 휃 − 2 푐표푠휙푐표푠휃 + 2 푟2 푟2 휋/2 1 휋/2 퐾(푘) = ∫ 푑휃 , 퐸(푘) = ∫ √1 − 푘²푠푖푛²휃 푑휃 0 √1 − 푘²푠푖푛²휃 0

Figure 38: Arbitrary position of a coil in space.

2.4 Formula for Capacitance

The capacitance in the resonator can be easily calculated. The angular frequency will already be given and the inductance of the coil will also be calculated, this means that if there must be resonance in the system the capacitor of the resonator should be equal to:

1 1 휔0 = ⇒ 퐶 = 2 √퐿퐶 휔0 퐿

2.5 Validating the found expressions

The expressions for the resistance, induction and mutual induction that were found in the previous Sections needed to be tested and validated.

2.5.1 Comparing wire configuration and different capacitors

The main goal of these measurements was to show the influence of skin effect and proximity effect on wires with different sizes. 2 resonators were used. The coil of the first resonator was made with litz wire. The litz wire was made with 60 strands, each strand was made with a diameter of 0.15 mm. The second resonator was made with regular wire. The diameter of this wire was calculated to get the same section as the litz wire. The diameter was found with the following calculations: 푑2 ∗ 휋 0.152 ∗ 휋 푆 = 푁 ∗ 퐿푖푡푧 = 60 ∗ = 1.06 푚푚2 퐿푖푡푧 푠푡푟푎푛푑푠 4 4 4 ∗ 푆 4 ∗ 1.06 푑 = √ 퐿푖푡푧 = √ = 1.16 푚푚 푒푞 휋 휋

The equivalent diameter is 1.16 mm. A wire diameter of 1.15 mm was used because this was the closest that was found to the theoretical value. Because the area is equal for the 2 wires, the DC resistance should be the same for these wires as the formula for the resistance is: 휌 ∗ 푙 푅 = 푐표푝푝푒푟 퐷퐶 푆 By using the same area of the section of the wire (S), the same length (l) and using the same kind of copper, resistance of the wire should be the same for DC and low frequency applications. But with high frequency (which will be used because resonance frequency is relatively high) influences like skin effect and proximity effect will have a greater effect on the resistance. Therefore, litz wire is used. By using litz wire these effects are suppressed.

To show the skin effect, a code was written in Matlab to compare the resistance of the 2 types of wire to each other. Formulas used in the simulation were the following:

To get the skin depth 훿 for a certain frequency the following formula (Vander Vorst, Rosen, & Kotsuka, 2006) was used: 2 ∗ 휌 훿 = √ 휔 ∗ µ

If the skin depth, this is the depth at which the current will flow in the conductor, is smaller than the radius of the conductor, then the useful area of the conductor is decreased. This area is defined by:

푑2 ∗ 휋 (푑 − 2 ∗ 훿)2 ∗ 휋 푆 = − 푢푠푒푓푢푙푙 4 4 If the skin depth is larger than the radius of the conductor the useful area of the conductor is equal to:

푑2 ∗ 휋 푆 = 푢푠푒푓푢푙푙 4 The total resistance of the conductor is then equal to:

휌 ∗ 푙 푅 = 푐표푝푝푒푟 푆푢푠푒푓푢푙푙

Plotting these formulas for a frequency range of [1000; 120 000] with a step of 1000 and values for length equal to the measured value of the length of the conductor that was used in the resonator that was made before, the following graph was made.

Figure 39: plotting the resistance of the 2 different wire types for different frequencies.

In Figure 39 can be seen that for small frequencies the resistance of both the litz wire (red line that represents 60 strands of wire with a diameter of 0.15 per strand) and the regular wire (blue line that represents 1 strand with a diameter of 1.15) is almost the same. The reason that there is a difference in resistance is due to the equivalent diameter of 60 strands of Ø 0.15 is 1.16… and not 1.15 to the resistance is a little less because the surface is a little bigger. As the frequency gets higher skin effect becomes higher. For a frequency of approximately 15 kHz the skin depth becomes less than the radius of the conductor making the useful area of the conductor smaller which in turn makes the resistance of the conductor bigger. The resistance almost doubles in value with a frequency of 120 kHz. In Figure 40 multiple wire diameters with 1 strand are plotted along with equivalent wire diameters with 5 strands to show how the resistance changes with bigger diameters.

Figure 40: Comparing wire diameters and the effect of using multiple strands.

All these calculations were done only considering skin effect. But there is another effect that has influence on the resistance of a conductor in high frequency, proximity effect. Proximity effect is caused by the magnetic field of nearby conductors, because of this the current is concentrated to an area in de conductor that is farthest away from the interfering conductor. The effect has more influence on the resistance as the frequency is higher.

To show this the formula of Dowell was used as it can be seen in Section 2.1. This formula shows all the effects of eddy current so proximity and skin effect will be seen.

Plotting the AC resistance of an arbitrary conductor that has some windings around a coil for different frequency gives the following graph in Figure 41.

Figure 41: AC resistance of a conductor for different frequencies.

On Figure 41 it can be seen that the resistance of a conductor increases if the frequency increases. This starts from approximately 7kHz. At higher frequencies the increase in resistance is not as dramatic. To show this a bigger range was chosen in Figure 42.

Figure 42: Proximity effect on the resistance of a conductor for different frequencies with a bigger range in frequency.

In Figure 43 the AC resistance of litz wire (red) and regular one-stranded wire (bleu) are plotted. It can be seen that the effect of proximity effect is very high because the resistance of the regular one-stranded wire now quadruples for a frequency of 120 kHz. The resistance of litz wire only increases a little for higher frequencies.

Figure 43: AC resistance for regular and Litz wire.

To validate these calculations there had to be made some measurements. For this reason, 2 resonators were made. One was made with regular wire with a diameter of 1.15 mm and the other was made with Litz wire with 60 strands of diameter 0.15 mm wire. A picture of the resonators can be found in Figure 44.

Figure 44: Resonators used for measurements of the resistance of the resonator for different resonance frequencies.

The measurements were done with a bigger transmitter coil that was connected to an AC voltage source. For getting measurements with different resonance frequencies the capacitor was swapped out for different values of capacitance. The frequency where resonance took place in the resonator was measured, along with the current that was flowing through the transmitter coil and the current that was flowing through the resonator.

The resistance of the resonator was found using the following expressions:

First the system expressions were noted.

푉1 = 푅1퐼1 + 푗휔퐿1퐼1 + 푗휔푀12퐼2 { 1 0 = 푅2퐼2 + 푗휔퐿2퐼2 + 퐼2 + 푗휔푀12퐼1 푗휔퐶2

When looking at the second expression it can be noted that the terms for the capacitance and inductance cancel each other out when resonance takes place. The second expression becomes then.

0 = 푅2퐼2 + 푗휔푀12퐼1

Solving this Equation to 푅2 the following can be found.

−푗휔푀12퐼1 푅2 = 퐼2 This means the resistance can be found by only knowing the frequency, the currents in both coils and the mutual inductance. The mutual inductance was measured by doing some test with different frequencies. By measuring the resistance of the resonator for a certain frequency with a special measuring device and measuring the current, a mutual induction of 2.107 µ퐻 was found.

In Figure 45 the measuring setup can be seen.

Figure 45: Measuring setup for getting the resistance of the resonator for different frequencies.

With these measurements the following graph was plotted.

Figure 46: Graph of the resistance measurements of the resonator.

The blue dots in Figure 46 show to measuring points of the resistance of the resonator with litz wire. It can be seen that the resistance stays quasi the same for all frequencies. The red dots represent the measuring

points of the resistance of the resonator with regular wire with diameter 1.15 mm which has the surface as the litz wire. The resistance of the regular wire increases by increasing the frequency, this is due to the skin and proximity effect in the conductor. These effects do not increase the resistance in litz wire with a measurable value. The solid lines are trend lines of the measuring points.

By now drawing the graphs of the measuring values of the resistance of the resonator with the calculated values a comparison can be done. This graph can be seen in Figure 47.

Figure 47: Calculated and measured values of the resistance in the resonator for different frequencies.

It can be seen in Figure 47 that the calculated value of the regular wire (purple) is a very good representation for low frequencies. For higher frequencies the calculation starts to overstate the actual resistance. For litz wire the calculated resistance understates the actual measured resistance. It can be noted that the resistance of the capacitor is not included in the calculation (but this resistance is measured). This could explain why the calculated resistance of litz wire is lower than the measured resistance. The coil of the resonator with the regular wire was also not tightly wound at all, this can also explain why the resistance does not become as high as it is calculated. Further measurements were obviously needed to get rid of these issues of bad winding and unknown resistance of the capacitor.

Six new coils were made that were better wound. All the coils were different from each other. There were three coils that were wound with a conductor that consist of 1 strand with a diameter of 1.15 mm. The other three coils were made with a conductor that consisted of 2 strands with a strand diameter of 0.8 mm. The cross-section area of these 2 types of conductor were as good as the same. So, by comparing these two types of coils the effect of the number of strands (푛푠푡푟푎푛푑푠) will be seen.

The coils were also made with the same number of windings. But the layers of windings (푚) was different, there were coils with 1 layer of 20 windings, then there were coils with 2 layers of 10 windings and there were coils with 4 layers of 5 windings. If the resistances of these coils are then compared the difference in proximity effect is for these coils will be seen.

A picture of all the coils made can be seen in Figure 48.

Figure 48: A picture of all the coils that were made to validate the AC resistance formula.

To measure the resistances of the coil the same method was used as in the precious measurements. But here the resistances for the capacitors, that were place in series with the coils, were also measured for different frequencies. These resistances were than subtracted from the measurements to just have a comparison of the resistance of the coils.

On the next page a graph can be seen of the measured resistances of the coils and the calculated resistances of these windings by the expression of Dowell.

Figure 49: A graph of all the measurements and calculations of the resistances of the coils.

In Figure 49 it can be seen that the expression that calculates the AC resistance of the coils from Dowell is a very good representation of the measured values. Though, it should be noted that the radius of the wires was multiplied with a factor of 0.87 to get a better fit of the curves. It is safe to say that the formula for the AC resistance is validated for coils with a low number of windings and with low frequencies (< 100 kHz).

By comparing the resistances of the different coil, it can be seen that litz wire (conductor with 60 strands) has the lowest resistance for higher frequencies as it is designed to have (this was a coil with 1 layer of 20 windings). The second lowest resistance is of the coil with two wire strands and 1 layer of 20 windings (green curve with green dots). By looking at Figure 49 it can be said that more layers of windings increase the resistance of the coil drastically for high frequencies. This is due to the proximity effect, the windings are closer together so the EM field will be higher, creating more eddy currents. It can also be said that more numbers of strands also give lower resistance for higher frequencies. This is mainly due to skin effect because the radius of the strands is smaller than the skin depth for higher frequencies.

2.5.1.1 Conclusion

With these measurements the expression for the AC resistance is validated, it can be seen in Figure 49 that the calculated values for the resistance of the coils are very close to the measured values for the resistance. It must be noted that factor was added to the expression to get a better fit of these values. This factor was 0.87 and was multiplied to the radius of the wires, this was because the expression was derived for square wires, this radius was already corrected by making the useful area the same (see Section 2.1). It can also be seen that the for coils with four layers of windings (푚 = 4) the accuracy of the formula starts to drop. This accuracy drop is still very small but for more layers of windings this can maybe be worse. In this dissertation there will be a maximum of two layers of windings so no further investigation was done for more windings per layer as adding more layers of windings would only decrease the efficiency of the power transfer (as seen in Figure 49 the resistance becomes higher with by adding more layers of windings.

2.5.2 Validation of formulas for inductance and mutual inductance:

Other formulas for the inductance and the mutual inductance of the coils were also validated with these experiments. All the inductances were measured with a RLC bridge (HM8118 from HALEG) and then compared to the formulas of the inductance. The conclusion of these measurements was that these formulas are also very close to the measured values. There was always an accuracy of more than 95% of the inductances. The mutual inductance was also measured between some coils. These were done by measuring the currents and the resistance in the coils and using this formula:

푅2퐼2 푅2퐼2 푀12 = = 푗 −푗휔퐼1 휔퐼1 The explanation of how this formula was derived can be found in Section 1.5.2.2 .The accuracy of the expressions for mutual inductance was a bit lower but this is due to the difficulty of trying to measure the mutual induction of two coils.

Next the Q-factors for the resonators were measured and calculated, see Figure 50, using the formulas for the resistances and the inductances and then used in the next formula:

휔퐿 푄 = 푅

2.5.2.1 Conclusion

Again, the measurements and the calculations are very close to each other. And again can be said that litz wire has the best Q-factor. The formula for the inductance is also validated as the measurements (individually and by using Q-factor calculations) are close to the calculated values.

Figure 50: Q-factors that were measured and calculated for the coils.

2.6 Determine dimensions of resonator for further measurements

In the previous Section formulas were made to determine the variables of the resonator depending on its dimensions. By using these formulas, the best possible resonator can be found for a given situation. For the next experiment resonators were needed to investigate the spacing between the resonator and the optimal load resistor for the last resonator. For the experiment a platform was made with a long slot in the middle. Resonators could be mounted in this slot so they could move axially from each other. In this section it will be shown how the dimensions of the resonator for that application was determined. This is also a guide to develop other resonators for other applications.

A program was written in Matlab to get comparisons for different dimensions. In this program, all the dimensions of the resonator could be changed these dimensions are:

Table 2: Explanation of the variables used in the resonator determining program. Dimension of the Variable name in program Unit resonator

Frequency frequency Hz

Radius of the coil Resonator_A.geometry.radius m

Radius of the strand Resonator_A.geometry.radiuswire m

Number of strands Resonator_A.geometry.n_strands -

Length of the coil Resonator_A.geometry.length m

Number of turn per layer Resonator_A.geometry.n -

Number of layers Resonator_A.geometry.m -

The program and functions of the program can be found in Appendix A codes 4 to 7. 2.6.1 Boundary constraints

For all applications there will be some boundary constraints. For this experiment a platform was already available with a slot. This slot was 600 mm long and there needed to be space for a minimum of 5 resonators. Also, spacing between the resonator had to be sufficient. It was determined that the resonators should have a maximum length of 30 mm. The second condition was the diameter of the coil. because the frame of the coil had to be 3D printed a maximum was set of the diameter of the coil. This is because the print time of the resonator would be reasonable. The maximum diameter was set to 60 mm. The maximum number of strands of the wire was 2. This was because the coils were wound by hand. Using more than 2 strands would be more difficult to make and would also mean that there would be less wire turns (not to be confused with strands) around the coil. The area of the wire needed to be big enough to make sure that enough current could flow through the coil. 5 Amperes was set as a maximum rating for the coil. This means that the area of the wire needed to be larger than 0.75 mm². The last conditions of the resonator were the amount of actual turns. This needed to be larger than 20. This was to get a large enough inductance of the coil. Also, the formulas would be more accurate with more turns.

With these boundary conditions the dimensions of the coil could be determined.

2.6.2 Dimensions

The frequency of the voltage source was set at 60 kHz because the formulas for the Q-factor were validated with a good accuracy for this frequency. By looking at Figure 50 in the previous Section it can be seen that the number of strands has a positive impact of the Q-factor. So, 2 strands were picked because that was the maximum amount that could be used. When more strands could be picked it is recommended to have more, when looking at Figure 50 it can be seen that litz wire (which has the most strands in the graph) has the highest Q factor so the resonator would be more efficient.

Next the Q-factor was plotted in function of the diameter of the strands using the formulas found in previous sections for resistance and induction of coil. Arbitrary starting values for the dimensions of a resonator were chosen, these dimensions can be seen in the legend of the graph that was created. The Q-factor was plotted in Figure 51.

Figure 51: Calculation of Q-factor for the same resonator, only the diameter of the strand is different. Note that when the diameter of the strands change, the length of the coil also changes.

Choosing a smaller diameter gives a better Q-factor with the maximum at a strand diameter of around 0.0006 m. When the diameter of the wire is smaller, more windings can be placed on the same coil length. In this calculation the length if the coil remained the same, but the amount of winding was dependent on how many windings could actually be placed on that length. So, the smallest diameter should be chosen. But there is another boundary condition that constrains the diameter, the area that is needed to get 5 A safely through the coil. this means that the smallest diameter that can be chosen is 0.8 mm.

The Q-factor was also plotted in function of the length of the coil, see Figure 52.

Figure 52: Calculation of Q-factor for the same resonator, only the length of the coil is different. Note that when the length changes, the number of windings of the coil also changes.

On Figure 52 can be seen that the Q-factor gets higher as the coil is longer (and more windings are on the coil). When the coil gets longer, more windings can be wound around the coil which increases the inductance of the coil. This because the inductance is quadratically dependent on the number of turns.

푁2 퐿~ 푙 The Q-factor of the coil does not increase quadratically of the coil. This is because the length increases of the conductor that is wound around the coil when more windings are wound around the conductor. And when the length increases, so does the resistance of the conductor (also, proximity effect gets higher). Because of the correlation in Figure 52, the length is set as the maximum length 0.030 m. The usable length is 0.028 m because there were also borders on the coils. By knowing the length of the coil and the diameter of the strands, the number of turn per layer can be calculated. 16 turns could be made in one layer, and practically 14 (when calculating in the isolation of individual strands and winding imperfections). This is not enough, so there was decided to have two layers.

The last parameter that was calculated was the diameter of the coil. In Figure 53 the Q-factor is plotted in function of the radius of the coil.

Figure 53: Calculation of Q-factor for the same resonator, only the radius of the coil is different.

Because the Q-factor is directly proportional to the inductance, and the inductance value is quadratically proportional to the radius of the coil, the Q-factor will increase. But the Q-factor is inversely proportional to the resistance and this resistance increases because the length of the wire increases. Therefore, the curve is not strongly rising as the inductance suggested.

The maximum radius of the coil implied by the boundary constraints was also chosen here because it can be seen in Figure 53 that a higher radius gets a higher Q-factor. The maximum diameter was 0.06m (or radius 0.03), so this was chosen as the diameter. Now, all the variables of the resonator are chosen, an overview can be found in Table 3.

Table 3: Overview of all the variables of the resonator chosen to be used in the variable spacing experiment. Variable Value Frequency 60 kHz Diameter coil 0.06 m Diameter strand 0.0008 m Number of strands 2 Length 0.03 m Number of turns 28 Number of layers 2

2.6.3 Conclusion

This section shows how the dimensions are the resonator can be optimized. This is a long way of optimizing the dimensions, when a model is made the optimization will be fully automatical and more accurate because then the optimization can be done for all the dimensions at once.

2.7 Determining the optimal load resistance for a two-resonator-system

The transfer efficiency is dependent on the load resistance, this will be shown in this section.

Figure 54: Equivalent circuit model of 2 coils with a mutual inductance.

When applying the circuit theory on the circuit shown in Figure 54, the input power and the output power can be obtained (Huang, Zhang, & Zhang, 2014):

2 2 2 2 {푅1[(푅2 + 푅퐿) − 푋1 ] + (휔푀12) (푅2 + 푅퐿)}푉1 푃푖푛 = 2 2 2 [푅1(푅2 + 푅퐿) − 푋1푋2 + (휔푀12) ] + [푅1푋2 + (푅2 + 푅퐿)푋1]

2 2 (휔푀12) 푉1 푅퐿 푃표푢푡 = 2 2 2 [푅1(푅2 + 푅퐿) − 푋1푋2 + (휔푀12) ] + [푅1푋2 + (푅2 + 푅퐿)푋1]

The efficiency can be found by dividing the output power and the input power:

2 (휔푀12) 푅퐿 휂 = 2 2 2 푅1[(푅2 + 푅퐿) − 푋2 ] + (휔푀12) (푅2 + 푅퐿)

It can be seen that the efficiency is dependent on the load resistance 푅퐿. To show this an experiment was conducted. For this experiment resonators were made with certain dimensions using the method of the previous section to get the optimal dimensions and properties. In Table 4 the dimensions and properties can be seen of these resonators.

Table 4: Characteristics of the resonator used for finding the optimal load resistor. Resonator Variable Value Resonance frequency 46,5 kHz Diameter coil 0.06 m Diameter strand 0.0008 m Number of strands 2 Length 0.03 m Number of turns per layer 7 Number of layers 2 C 0,24 µF Calculated L 48,9 µH Calculated R 0,2886 Ω

Then three random load resistors were chosen: 0,75 Ω; 0,39 Ω and 0,2 Ω. These were all separately applied as the load of the second coil. In the experiment the input and output power of the circuit was measured for different relative positions of the resonators. The resonators were axially moved to get a plot of the optimal distance for the applied load resistance. By changing the axial position of the resonators, the mutual induction was changed. This can be seen in Section 2.3 where the mutual inductance was shown for coils that were arbitrarily positioned in space.

The results of this measurements can be seen in Figure 55.

Figure 55: Plot of the efficiency of the power transfer for different relative axial positions.

A simulation was done by using all the formulas that were found to calculate the resistance, self- and mutualinductance of the resonators. The circuit of the simulation can be seen in Figure 55.

Figure 56: Circuit of the simulation for 2 resonator coils.

The code that was used to get the results can be found in Appendix A (code 8). The comparison of the calculated efficiency and the measured efficiency can be seen in Figure 56.

Figure 57: Comparison of the simulated efficiency of the power transfer to the measured efficiency.

Figure 57 shows how the simulation predicts a slightly better efficiency then what was measured. In the simulation the mutual inductance, the internal resistance of the coils and the capacitors and the inductance of the coils are calculated using formulas that were found in previous sections. This along with the fact that

in the real world it is much harder to get perfect resonance in the resonators explains why these graphs are not perfectly aligned. But it still shows that the simulation is a very close prediction.

In Figure 57 it can be seen that for different distances of the coil there is an optimal resistance. This is clear when only looking at the simulated graphs (the striped ones). It can be seen that, for a distance of around 0.08 m, the load resistance of 0.39 Ohm (green) suddenly has a better efficiency than the load resistor 0.75 Ohm. This is because the optimal load resistance for a distance of 0.08m is closer to 0.39 Ohm than to 0.75 Ohm.

The phenomenon that there is an optimal load resistance for each distance is known. In “A Critical Review of Wireless Power Transfer via Strongly Coupled Magnetic Resonances” (Wei, Wang, & Dai, 2014) a formula was derived with circuit theory to get the optimal load resistance for a certain mutual induction between two resonators, because the mutual induction is dependent on the distance this formula also indicates what the optimal load resistance is for a certain distance between two resonators. The formula for the optimal load resistance is:

휔2푀2 푅 √ 12 2 푅퐿휂푚푎푥 = + 푅2 푅1

This formula can be only be used for a two-resonator system that are in a resonance state. The circuit can be seen in Figure 54 at the beginning of this Section.

2.7.1 Conclusion

It can be seen for every value of the mutual induction there is an optimal load resistance. And because the mutual induction changes when the distance between coils change it can be said that for every position of the coils there is an optimal load resistance.

2.8 Optimal load resistance for a multiple resonator system

It was shown in the previous Section how the load resistance affects the power transfer of a two-resonator system and how to calculate the optimal load resistor. In this Section it will be shown how the load resistor affects the power transfer efficiency when using more than two resonators.

A simulation was done with three resonators. The first resonator (driver resonator), where the source was connected to, was placed in the origin. The last resonator, where the load resistor was connected to, was placed at a distance of 0.2 meters from the first resonator. Then there was a middle resonator which had a variable distance from the first resonator. The goal of this simulation was to show that there is an optimal load resistance for each position of the middle resonator. Another goal of this simulation was to show at what position of the middle resonator the efficiency of the power transfer would be optimal.

The way how this simulation was done is as follows: First the position of the first resonator was set at 0 m (origin) and the position of the last resonator was set at 0.2 m. Second the position of the middle resonator was set at 0.04m, so somewhere between the first and the last resonator. Then an algorithm was used that calculated what the optimal load resistance was for that position of the middle coil. This algorithm was 푓푚푖푛푐표푛 this function will be explained further in this dissertation but for now it can be said that this function tries to optimize the efficiency of the power transfer between the first and last resonator by changing the load resistor and then returning which load resistor gave the optimal power transfer. When the optimal load resistor was found, the efficiency of the power transfer was plotted (for the three-resonator system with that particular load resistor) against the position of the middle resonator. So, the position of the middle resonator was than variable again to see how the efficiency changes when changing the position of the middle resonator. The plotted graph was saved and the process started over to get the optimal load resistor for the system when the middle resonator was at another position. With this new optimal load resistor the system was then tested again by plotting the power transfer efficiency to the position of the middle resonator. This was done 25 times.

In Figure 58, nine of these graphs or shown. Each coloured graph show the efficiency of the system for a certain load resistance for different positions of the middle resonator. The black graph in Figure 58 shows the envelope of all the plotted graph, it shows all the maximum efficiencies of the different resistors that were calculated by the black dots. In Figure 58 can be seen that the maximum efficiency in power transfer is when the middle resonator is perfectly positioned in the middle of the first resonator (position = 0.0) and the last resonator (position = 0.2). This efficiency is then 7.5%, this is with a load resistance of around 0.55 Ω. When the middle resonator is then moved to another position, for example 0.16 meters, the efficiency drops dramatically to 0.2%. But there can be a better power transfer efficiency for that position when another load resistor is used. This is when a load resistor with a resistance of 16.6 Ω is used, then the efficiency becomes approximately 2% (ten times more efficient). So, it can be seen that different load resistors can be the most efficient resistor for a certain distance.

It can also be seen that the load resistance does not have to be super accurate to get a good power transfer. When looking at the efficiency when the middle resonator is at a position of 0.1 m, it can be seen that the efficiency by using a load resistance of 0.55 Ω is 7.5 %. When a load resistance of 0.75 is used for the same position, the efficiency only drops to 7.3 %. The same can be said for the accuracy of the position of the middle resonator. When the middle resonator is at for example 0.105 m instead of 0.10 m, the efficiency only drops to 7.4 %. 2.8.1 Conclusion

It can be concluded that for a multiple resonator system there is still an optimal load resistor for each arrangement of the resonators (when distances between resonators change). The way of finding this optimal load resistance will be done with the model that will be obtained in Section 2.9 and using the optimization function (in Section 4) on the models. Examples will be shown in Section 5.

Figure 58: Efficiency of power transfer for a three-resonator-system when moving the middle resonator (and choosing the optimal load resistor for that position).

2.9 Models made

In previous sections expressions were found to obtain the resistance, inductance, mutual inductance and capacitance of resonators. These expressions were then combined in a model. Some functions were made that could return the power transfer efficiency of magnetically coupled resonators in resonance. 2.9.1 푒푓푓_푑푖푚 function

The first function that was made was the 푒푓푓_푑푖푚 function, this function can be found in Appendix A code 9. This function returns the power transfer efficiency between two identical resonators, where one resonator is connected to a voltage source and the other to a load resistor. The arguments of the function were: eff_dim(frequency,Vin, distance,R_load,r_coil,l_coil,r_strand, n_strands,LayersOfWindings) In Table 5, the explanation of the arguments can be seen.

Table 5: Explanation of the arguments of the eff_dim function. Input: Input name Frequency of the source (resonance Frequency frequency) Amplitude of the voltage source Vin Distance between the first and second coil distance Resistance of load resistor R_load Radius of the coil r_coil Length of the coil l_coil Radius of the wire strands r_strand Number of wire strands n_strands Number of identical winding layers LayersOfWindings

The function automatically calculates the number of windings that can be wound around the coil. This is calculated by using the length of the coil that is given and dividing it by the outer diameter of the conductor that is used for these windings. Because this conductor can be made out of multiple strands, a function 푂푢푡푒푟푑푖푎푚푒푡푒푟 is made that can return the outer diameter of a conductor with a certain strand diameter and a certain number of strands. This is an estimation that is done by using some values for the outer diameters of certain litz wires in a catalogue from a manufacturer (Elumeg GmbH, 2011).

After the number of windings per layer are calculated the resistance, inductance and capacitance of the resonators are calculated along with the mutual inductance of the two resonators. This is done with the expressions that can be found in Section 2.1 through 2.4 . The load resistance is added to the resistance of the second resonator because it is connected in series to the second resonator.

Then the currents are calculated of the two resonators using circuit theory:

1 −1 핀 = 핍 × [ℝ + 푗휔핃 + ℂ] 푗휔

With:

1 0 푅 0 퐿 푀 퐶 푉 퐼 ℝ = [ 1 ] ; 핃 = [ 1 12] ; ℂ = 1 ; 핍 = [ 1] ; 핀 = [ 1] 0 푅 + 푅 푀 퐿 1 0 퐼 2 퐿표푎푑 12 2 0 2 [ 퐶2]

After the currents in the resonators are calculated, the power of the source 푃1 and the power of the load resistor 푃2 can be calculated: 푅푒(퐼 ) × 푉 푃 = 1 푖푛 1 2 푎푏푠(퐼 )² × 푅 푃 = 2 퐿표푎푑 2 2 The efficiency of the power transfer 휂 is then calculated and returned: 푃 휂 = 2 푃1 2.9.2 푒푓푓_푎푢푡표푠푖푧푒 function

After some test were done with the 푒푓푓_푑푖푚 function some improvements were made to make it compatible for some new tests were more than 2 identical resonators would be used. The same inputs were required as the 푒푓푓_푑푖푚 function but one input extra was needed, the number of resonators was required. The arguments of the new 푒푓푓_푎푢푡표푠푖푧푒 function were: eff_autosize(frequency,Vin, distance,R_load,r_coil,l_coil,r_strand, n_strands,LayersOfWindings, Windingsperlayer, NumberOfResonators)

This function can be found in Appendix A code 11. The explanation of the other arguments can be found in Table 5. The windings per layer of the resonators are calculated in the same way as the 푒푓푓_푑푖푚 function. The resistances, inductances and capacitances of the resonators are made in a for-loop. The number of times the for-loop is run through is equal to the number of resonators that are given as an input. The mutual inductances are also made in another for-loop that is in the first for-loop. By doing that all the mutual inductances between all the resonators can be calculated. With these for-loops the following matrices can be filled in: 1 0 … 0 퐶 푅 0 … 0 퐿 푀 … 푀 1 푉 퐼 1 1 12 14 1 1 1 0 푅2 … 0 푀12 퐿2 … 푀2푛 0 … 0 0 퐼2 ℝ = [ ] ; 핃 = [ ] ; ℂ = 퐶 ; 핍 = [ ] ; 핀 = [ ] ⋮ ⋮ ⋱ ⋮ ⋮ ⋱ 2 ⋮ ⋮ 0 0 푅 + 푅 푀 푀 퐿 ⋮ ⋮ ⋱ 0 퐼 푛 퐿표푎푑 1푛 2푛 푛 1 4 0 0 [ 퐶푛]

The currents, powers and efficiency can then be calculated in the same way as in Section 2.9.1 .

2.9.3 Conclusion

Two models were made. First, 푒푓푓_푑푖푚 was made to calculate the efficiency of a two-resonator. Then this model was improved to work for any number of resonates in a system, the new model name is 푒푓푓_푎푢푡표푠푖푧푒. These models are actually function that return the efficiency of a SCMR resonator system with given inputs for the dimensions of the resonators and other variables like voltage of source, frequency and load resistor.

3 Making resonator prototypes

In this section the prototypes that were made for the resonators will be discussed. This Section will show how the coils of the resonators were physically made and what the ideas behind the prototypes were. At the end of the section a comparison between the three most interesting prototypes will be made.

Figure 59: First sketches/ideas for a resonator design.

Some resonator and coils were made for testing and measurements. The coils were made with copper wires (sometimes litz wire) that were wound around a plastic tube. For first experiment, the coils were made with litz wire that was wound around a PVC tube that was found in the lab. In Figure 59, a picture can be seen of these coils.

Figure 60: First resonator that was made.

This was a very rudimentary way of making resonators. Because formulas and predictions of the simulation model needed to be validated, a lot of resonators needed to be made and this way of making resonators was not optimal. Other prototypes for resonators needed to be made by another way.

Different resonators with different dimensions were needed to validate the model and its formulas. This means the frames of the resonators needed to be manufactured. Three-dimensional (3D) printing (or additive manufacturing) was chosen as the manufacturing process. This was because additive manufacturing is very flexible, complicated designs are not a problem for it. 3D-printing was also the cheapest option because designs can be adjusted and printed in a very short time. A 3D printer was also available so the cost for printing a part was only the cost of the print material and the electricity that the machine used. It was important to know how a manufacturing process works when designing a part. Therefore, 3D-printing will be discussed in the following Section. 3.1 Additive manufacturing

3D printing or additive manufacturing is an enhanced (automated) method for fabrication of a physical model or prototype from a three-dimensional (3D) CAD file. A short explanation of the method of this manufacturing method will be shown. The 3D-printer that was used was a Reprap Prusa I3. This was a cheap DIY kit that was owned and assembled by the student. The specific manufacturing process this printer used was the extrusion based process, so only this process will be discussed. Note that there are many other processes available for 3D-printing but this is a very popular and cheap process.

To make a part with additive manufacturing a 3D CAD file is needed. For this dissertation the CAD file was made using the software packet NX 11 owned by Siemens PLM Software. When a part is drawn a CAD program it needs to be converted to the ‘.stl’ file format. STL is a file format for CAD files that has become an industry standard for additive manufacturing. When a CAD file is converted to an STL-format it converts the 3D model into a model that consists of a series triangles that describe the shape of the closed model. The end points of the triangles are then saved in this file system along with the vector perpendicular to the triangle that points away from the model.

Figure 61: A representation of the '.stl' file format. The left Figure is a 3D part that needs to be converted and the right Figure is the collection of triangles that the 3D part is converted to in a '.stl'-file. (Cardon, 2017)

This STL-file can then be opened in a slicer program which converts the 3D model into layers that are stacked on top of each other. Each individual layer is then just a 2D figure of the cross section of the model. These sliced layers are then converted to the machine code of the used machine. This machine code commands the machine to do XY-movements for each individual layer, it generates a tool path. When the layer is finished the machine can then begin with the following layer by first moving the model one step in the Z-direction to start printing on top of the previous level.

The way the Reprap Prusa I3 3D printer prints is by using the extrusion based process. This is commonly called Fused Deposition Modelling (FDM), this was a manufacturing technology developed and patented by Stratasys (Chua, Leong, & Lim, 2003). After the expiration of the patent large open source development communities (e.g. Reprap, …) enabled low cost 3D-printers to hit the market. The process of FDM is very simple and can be seen in Figure 62, a filament material (in most cases a polymer) is pushed through a liquefier chamber by a pinch roller feed system. In this chamber the filament material heats up and melts so it can barely flow through the nozzle at the end of the chamber. The melted material comes out of the nozzle

and touches the colder platform (or scaffold when there is already material printed) where it solidifies and sticks. The nozzle is moved in both X, Y and Z directions by stepper motors. The nozzle follows the fool path of the generated machine code. There is also a stepper motor that feeds the filament material by the pinch roller feed system, this motor is also controlled by the generated machine code and is dependent on the speed of the nozzle.

Figure 62: A representation of the moving print head for Fused Deposition Modelling.

Figure 63: A picture of the 3D printer printing a resonator prototype.

3.2 Prototype features

The resonator had to have some features which will be explained in this paragraph. First the resonance frequency had to be adjustable. This needed to be adjustable for tests with different frequencies but also for precise tuning of the resonators (tune its resonance frequency) so multiple resonators could be used with the same resonant frequency to get optimal energy transfer. There were only 2 parameters that can change the resonance frequency, this was the capacitance and the self-induction of the resonator. There was decided that both ways should be used to change the resonance frequency. The resonator consists of a coil and a capacitor. The capacitor was not soldered on the coil but a connector was soldered to the coil in this way capacitors can be easily swapped out to get very large changes in resonance frequency. To get more precise tunability of the frequency the self-induction of the resonator had to be variable. There were multiple ways to do this. The diameter of the coil could be changed, the amount of windings could be changed, … . But these types of changes are very labour and time intensive to do. There needed to be an easier way to change the inductance of the coil. Making the coil out of 2 parts was found to be the easiest way to do this. By changing the coupling of the parts of the coil the inductance would change. This can be done in different ways. The coils could be moved radially or axially away from each other. Different prototypes were made with different ways of moving the coils were made.

There were also other coils made that were used to make measurements on the resistance of different winding layouts at different frequencies. The frames for these coils were also made with 3D printing.

In the following Section the printed resonator frames and coils will be shown as well as pictures of the wounded coils. 3.3 Used resonator prototypes and coils for measurements

3.3.1 First resonator prototype

The first self-build resonator prototype was made from two coils that could move in a rotary motion relative to each other. Printed parts of the prototype can be found in Figure 64.

Figure 64: Printed version of the parts of the first resonator prototype.

As seen on the Figure 64 the halves of the coils are made with a same diameter for easy calculation of the self-induction of the coil when there in optimal aligning. The coils will be connected by a plastic bolt and nut. Steel parts should not be used in this prototype to not get interference of the magnetic field with steel parts.

On Figure 64 it can be seen that the printed parts have bars above the wires. These bars were support for the windings of the coil. Because for this prototype test were done for single layer helical coil, there was only place needed for just 1 layer of windings. By having this support beam, the wires were locked more into place.

When the wire was fully wrapped around the body, connectors were soldered to the ends of the wire. These connectors were used to easily swap capacitors out to get different resonance frequencies of the resonator. A picture of the wired resonator prototype can be found in Figure 65.

Figure 65: A picture of the wired first resonator prototype with litz wire.

This resonator was used to get the first measurements to get the resistance for different wire configurations. Two resonators were made of this kind, one with regular (1 stranded) wire and one with litz wire. These can be seen back in Figure 44.

There were several issues with this prototype that were addressed in further prototypes. First there were to many bars that were above the wires. This resulted in a more difficult process of winding the wires on the coils. Another issue was the difficulty of predicting the inductance of these resonators. This was because the windings of the first half and the windings of the second half were not directly next to each other but were displaced a few millimetres from each other. The displacement of the halves was because the sides of the coils were used to connect the two halves together with the bolts. This meant that another way of varying the inductance of the resonator was required. A smaller issue was the fact that there was an overhang of one side of the coil when printing, this meant that the print quality of the overhang was not that great, but this a more aesthetic issue. The last issue of this version was stability. When the rotary half of the resonator was not in the line with the fixed half, the resonator would tip over. This also needed to be addressed by changing the way of obtaining an adjustable inductance.

3.3.2 Printed coils to compare the resistance for more wire configurations.

Because predicting the resistance in the model was not very easy due to eddy currents, multiple coils were made to get a lot of measurements for different wire configurations. The coil frames were a simplified design of a half of the first resonator prototype. All the coils that were made can be seen in Figure 66.

Figure 66: All the coils that were made to compare the resistance for more wire configurations.

The frame of the coils consisted of 2 parts, the base part and a lit that fitted on that base part. These parts can be seen in Figure 67.

Figure 67: Parts of the coils. The right part is the base of the coil were the wire is wounded on. The left part is the lid of the coil which holds the wires in place.

The reason to make the lid separate was because of printing problems when printing the whole coil in one part. When printing it in one part it would mean that there would be an overhang at one side of the coil which resulted in a bad quality of that overhang. There was also a cut in the sides of the coils. This cut was made to get the beginning and end of the wire through. By doing that the beginning and end of the wire was stuck in place and could not move, which made winding the coils easier.

3.3.3 Second resonator prototype

A second resonator frame was made to explore another way of varying the inductance of the coil. There were still two coil halves that move relative to each other but here the coils moved radially from each other. By making the resonator this way, the inductance of the coil could be estimated more accurately because the sides of the coil could be made significantly thinner. Also, the stability problem of the first prototype was solved by making a stand on the resonator. A picture of the second resonator prototype can be seen in Figure 68.

Holder for the coil halves

Figure 68: A picture of the second resonator prototype.

In Figure 68 can be seen that the resonator was made out of two coil halves that could slide horizontally out of each other. Both these halves were mounted on a small beam-shaped holder that could easily slide in a slot as seen in Figure 68. This resonator version was used for most of the measurements that were done.

3.3.4 Third resonator prototype

Side of the coil frame that could slide in a rotary way

Cut where the end of the wire goes through

Figure 69: A picture of the third and last resonator prototype.

A third resonator prototype was made. This version used a whole other way to get an adjustable inductance of the coil. Other version used two halves that could move relative to each other to achieve a variable inductance. This prototype achieved a variable inductance by wounding up more wire. This was done by making a side of resonator frame loose so it could rotate. Because the side coil rotates, the cut where the end of the wire was in moved also along the edge of the coil. This meant that more or less wire could be wound around the coil. Which resulted in a change in inductance. This version was only used for the first case in Section 5.1 .

This way of changing the inductance was promising because only one coil needed to be made (along with a small side that could rotate in the coil). This prototype was not perfect because the side did not rotate easily so it was hard to adjust the induction but by adding some adjustments in the design this should be an easier way to create an adjustable resonator coil for lab usage.

3.3.5 Summary

Three resonator prototypes were made and used, all of the prototypes had a different way to tune the inductance (so that the resonance frequency could be tuned). All prototypes had their advantages and disadvantages:

Table 6: Advantages and disadvantages of all prototypes.

Prototype 1 Prototype 2 Prototype 3

Figure 70: Picture of Figure 71: Picture of Figure 72: Picture of prototype 1. prototype 2. prototype 3. Advantages: Very easy to tune Easy to tune Could be made fast

Good predictability of good predictability of induction induction

Disadvantages: Time to make is high Time to make is high Very hard to tune

bad predictability of induction

It is important to note that a lot improvement can be made to these prototypes. For example, if the borders of the first prototype can be made thinner the predictability of the induction of that resonator will be better. Another example is for prototype 3, if the rotating end part of the resonator (where the end of the wire goes through would be connected to the coil in some kind of grove, sliding would be way easier making tunability of the resonator easier.

4 Finding the optimal dimensions of a resonator

In Section 2.6 an early prototype was made by optimizing the dimensions of the resonator. The dimensions of the prototype were optimized by plotting the Q-factor of the resonator for each dimension that needed to be optimized, starting for some starting values for these dimensions. The Q-factor was calculated because the Q-factor of a resonator is an indicator of the efficiency of the power transfer.

This method was a time consuming and inaccurate way of finding the optimal dimension. It was time consuming because a graph needed to be made of the Q-factor in function of the to-be-optimized-dimension, then the value of the dimension corresponding with the maximum Q-factor must be extracted from that graph and be used in the calculations to find the next dimension that needed to be optimized. This had to be done for all the dimensions and properties of the resonator. The accuracy of this calculation was also not very high. This was because the Q-factor of the resonator needed to be calculated for all the values between the lower and upper bound of the dimension. So, to keep the calculation time low the step of the calculation needed to be high. Also, when one optimal dimension was found of the resonator, for example the length of the resonator, that value was used to find the next optimal dimension, for example the radius of the wires. Now when that second optimal dimension was found, the radius of the wire in this example, it should be used to find the first dimension again (the length) because the optimal value of the first found dimension of the resonator (the length) can now be different because the radius of the wire is now different in the calculation. This was not done in Section 2.6 because it would be too time consuming for just some early measurement and it was also not done because most optimal dimensions that were found were the maximum values that that dimension could be so it was safe to assume that it would still be the maximum value when some dimensions were slightly changed.

However, in this section a more accurate and automated (so faster) way of optimizing the dimensions will be shown. Because a function was already made that can find the efficiency of the power transfer between resonators with given distances from each other, dimensions of the resonators and properties of the resonators the function 푓푚푖푛푐표푛 could be used in Matlab. 4.1 Optimization function 푓푚푖푛푐표푛

푓푚푖푛푐표푛 is a function in Matlab that can find a minimum of a constrained nonlinear multivariable function. This function finds the minimum of a function by changing the desired variables of a problem specified by:

푐(푥) ≤ 0

푐푒푞(푥) = 0 min 푓(푥) 푠푢푐ℎ 푡ℎ푎푡 퐴 ∙ 푥 ≤ 푏 푥 퐴푒푞 ∙ 푥 = 푏푒푞 { 푙푏 ≤ 푥 ≤ 푢푏 The syntax of this function is:

푥 = 푓푚푖푛푐표푛(푓푢푛, 푥0, 퐴, 푏, 퐴푒푞, 푏푒푞, 푙푏, 푢푝)

A short description of these variables will be given in the following paragraph.

4.1.1 Descriptions of the arguments of the fmincon function

These are simplified descriptions of the arguments of the function, for a more in-depth description of this function and its arguments Matlab sources should be consulted (MathWorks, 2018).

푥 = 푓푚푖푛푐표푛(푓푢푛, 푥0, 퐴, 푏, 퐴푒푞, 푏푒푞, 푙푏, 푢푝, 푛표푛푙푐표푛, 표푝푡푖표푛푠)

푓푢푛: The first argument of the 푓푚푖푛푐표푛 function is 푓푢푛. This is the function that needs to be optimized. The arguments (of the ‘푓푢푛’ function) that must be used to find the minimum of the function need to be specified as 푥. 푥 can be a scalar or an array depending if there is one argument that must be optimized or multiple arguments.

푥0: The second argument must be filled in with the starting values of the to-be-optimized function.

퐴 푎푛푑 푏: 퐴 should be a matrix and b should be a vector when consists of multiple variables (when 푥 is an array of all the to-be-optimized arguments of 푓푢푛). These are optional arguments of 푓푚푖푛푐표푛 that can be filled in when a relationship between certain variables of 푓푢푛 can be stated in the form of “퐴 ∙ 푥 ≤ 푏”.

퐴푒푞 푎푛푑 푏푒푞: These arguments are similar to the 퐴 and 푏 arguments. But with the difference that 퐴푒푞 and 푏푒푞 should be used if an equality can be stated in the form of “퐴푒푞 ∙ 푥 = 푏푒푞”.

푙푏 푎푛푑 푢푏: These arguments represent respectively the lower bound and upper bound of the variables 푥.

푛표푛푙푐표푛: 푛표푛푙푐표푛 represent nonlinear constraints that needs to be fulfilled by 푥. Here a function can be used to calculate some other relationships 푥 needs to fulfill. The function than has to return variables 푐(푥) and 푐푒푞(푥). 푐(푥) is the array of nonlinear inequality constraints at x (푐(푥) ≤ 0) and 푐푒푞(푥) is the array of equality constraints (푐푒푞(푥) = 0) that 푓푚푖푛푐표푛 attempts to satisfy.

표푝푡푖표푛푠: With the 표푝푡푖표푛푠 argument of 푓푚푖푛푐표푛 some preferences can be specified. There are many options that can be changed from the default to another preference but some example for these preferences are: the optimization algorithm that will be used, the step tolerance on 푥, display iterations, … .

Some examples for these arguments can be seen on the next pages.

4.2 Implementation of the optimization code

In this Section codes can be used to optimize power transfer between 2 or more resonators will be briefly discussed. This can also be seen as some examples for the implementation of the 푓푚푖푛푐표푛 function. 4.2.1 Optimization of resonator dimensions and load resistor

This code should find the optimal dimensions of a resonator. A case with arbitrary distance between the resonator and voltage source with a certain frequency (f) and voltage (Vin) is imagined.

All the dimensions of the resonator need to be found by the optimization function 푓푚푖푛푐표푛. The dimensions are: the radius of the coil (r_coil), the length of the coil (l_coil), the radius of the wire strands (r_strand), the number of wire strands (n_strands) and the number of winding layers. Note that the number of windings per layer is calculated by dividing the length of the coil by the width of the equivalent wire of all the wire strands. The optimal load-resistor (R_load) (that is connected to the second resonator) also needs to be calculated by the 푓푚푖푛푐표푛 function. For all these variables a starting value is chosen and a lower and upper bound is specified. These values are just arbitrary.

With all these variables specified in the code, the only thing that makes up the code is the implementation of the 푓푚푖푛푐표푛 function:

Code example:

%%f,Vin,distance,R_load,r_coil,l_coil,r_strand,n_strands, LayersOfWindings objective = @(x) -eff_dim(f,Vin,distance,x(1) ,x(2) ,x(3) ,x(4) ,x(5) , x(6) ); x0 = [R_load,r_coil,l_coil,r_strand,n_strands, LayersOfWindings] ; A = []; b = []; Aeq = []; beq = []; lb = [R_loadm,r_coilm,l_coilm,r_strandm,n_strandsm, LayersOfWindingsm]; ub = [R_loadM,r_coilM,l_coilM,r_strandM,n_strandsM, LayersOfWindingsM];

OptimalValues = fmincon(objective,x0,A,b,Aeq,beq,lb,ub)

It can be seen above that the first line represents the objective of the 푓푚푖푛푐표푛 function, which is the efficiency of the power transfer. The function 푒푓푓_푑푖푚 returns the power transfer efficiency of a two-resonator system. A minus sign is placed in front of the function because the value that the function returns needs to be negative. This is because the function 푓푚푖푛푐표푛 can only find the minimum of a given function, not the maximum, so by making the function negative the minimum value of the function will actually the optimal efficiency. It can also be seen that the arguments where the dimensions are (like r_coil, l_coil) are replaced by an array element of 푥. This is because the 푓푚푖푛푐표푛 function will only change 푥- values.

The line under the objective shows the declaration of the starting values for 푥, represented as 푥0. There are no equality or inequality relations that need to be met so the matrices A, b, Aeq and beq are empty. 푙푏 should be an array of all the lower bounds of the variables that were specified. 푢푏 should than be an array of all the upper bounds. The last line shows than that the optimal values well be returned by the 푓푚푖푛푐표푛 function with the arguments that were specified above it.

4.2.2 Optimization of distance between resonators in a multi resonator power transfer system

Imagine five resonators that are spaced out over a certain distance. If the distance between the first resonator and the last resonator is set, what would be the optimal positions of the middle 3 resonators? this is the question that can be solved with the next example.

For this example, the function 푒푓푓_푎푢푡표푠푖푧푒 can be used because this function can return the efficiency of a given number of identical resonators with a given position of each resonator (note that the resonators have the same axis). All the dimensions of the resonators are set as well as the voltage source and the number of resonators (5). The only variables left are the positions of the resonators and the load resistor.

The arguments of the function 푒푓푓_푎푢푡표푠푖푧푒 can be seen here: eff_autosize(f,Vin,Distance,R_load,r_coil,l_coil,r_strand, n_strands, LayersOfWindings, NumberOfResonators)

These arguments are in order: Frequency of the voltage source in the first resonator, Amplitude of the voltage source in the first resonator, array of the positions of the resonators, load resistance of the last resonator, radius of the coil, length of the coil, radius of a wire strand, number of wire strands used in one full wire, number of winding layers and number of resonators used.

To have the positions of the resonators and the load resistance variable these arguments need to be changes to 푥(1: 5) for the five positions of the resonators and 푥(6) for the load resistance. This can be seen in the first line of the code example below. The starting positions of the resonators are written in the array 퐷푖푠푡푎푛푐푒 and the starting position of the load resistance is set in 푅_푙표푎푑. The lower bounds are written in 퐷푖푠푡푎푛푐푒푚 and 푅_푙표푎푑푚. The upper bounds are written in 퐷푖푠푡푎푛푐푒푀 and 푅_푙표푎푑푀.

Code example: objective = @(x) -eff_autosize(f ,Vin, x(1:5) ,x(6) ,r_coil,l_coil,r_strand, n_strands, LayersOfWindings, 5); x0 = [Distance, R_load] ; A = [1, -1, 0, 0, 0, 0; 0, 1, -1, 0, 0, 0; 0, 0, 1, -1, 0, 0; 0, 0, 0, 1, -1, 0; 0, 0, 0, 0, 0, 0; 0, 0, 0, 0, 0, 0;]; b = [0; 0; 0; 0; 0; 0]; Aeq = []; beq = []; lb = [Distancem, R_loadm]; ub = [DistanceM,R_loadM];

RL = fmincon(objective,x0,A,b,Aeq,beq,lb,ub,nonlcon)

Here an inequality constraint can be stated. The order of the resonator needs to be set. The first resonator needs to be at position ‘0 m’, this can be done by making the upper and lower bound of the first position 0 m. But the second, third and fourth resonator need to be variable but at the same time in position 2-3-4. The position of resonator 2 needs to be smaller than the position of resonator 3. The position of resonator 3 than also needs to be smaller than resonator 4. Or to be broader:

푃표푠푖푡푖표푛푅푒푠표푛푎푡표푟1 < 푃표푠푖푡푖표푛푅푒푠표푛푎푡표푟2 < 푃표푠푖푡푖표푛푅푒푠표푛푎푡표푟3 < 푃표푠푖푡푖표푛푅푒푠표푛푎푡표푟4 < 푃표푠푖푡푖표푛푅푒푠표푛푎푡표푟5

Or when using the form of 퐴 ∙ 푥 ≤ 푏:

1 −1 0 0 0 0 푃표푠푖푡푖표푛 0 푅푒푠표푛푎푡표푟1 0 1 −1 0 0 0 푃표푠푖푡푖표푛 0 푅푒푠표푛푎푡표푟2 0 0 1 −1 0 0 푃표푠푖푡푖표푛 0 ∙ 푅푒푠표푛푎푡표푟3 ≤ 0 0 0 1 −1 0 푃표푠푖푡푖표푛푅푒푠표푛푎푡표푟4 0 0 0 0 0 0 0 푃표푠푖푡푖표푛푅푒푠표푛푎푡표푟5 0 [0 0 0 0 0 0] [ 퐿표푎푑 푟푒푠푖푠푡푎푛푐푒 ] [0] This can also be seen in the code example above. 4.3 Principles of 푓푚푖푛푐표푛

The 푓푚푖푛푐표푛 function will try to find the minimum of a given function by changing the designated arguments of that function. It can do this by many algorithms. These algorithms will not be explained here but the way 푓푚푖푛푐표푛 finds the minimum of the function is by deriving the function and second derivative of it. The goal of the 푓푚푖푛푐표푛 is to make the derivative of the function zero. By knowing what the derivative is, it can see what action needs to be made to make this derivative more equal to zero (by making the variable that the function was derived to bigger or smaller.

When the function that needs to be optimized has multiple local minimums the optimization function can give a wrong minimum. Because in a local minimum the derivate is also zero as in the overall minimum. So, it is important to at least try multiple starting values to make sure that the resulted minimum that 푓푚푖푛푐표푛 finds is not a local minimum but an absolute minimum. 4.4 Conclusion

By using the optimization function on the created model all the dimensions and other important parameters of the resonator could be optimized. This is a quick and easy way to test some ideas and maybe use this for prototypes for applications.

5 Cases

In this Section some dimension or parameters will be calculated to get the optimal power transfer between 2 or more resonators. In the cases some boundary conditions will be given to make sure that the found dimensions can also be built in the lab. The resonators that were found with these calculations were also built in the lab and the power transfer efficiency was measured and compared to the calculated power transfer efficiency.

5.1 Optimal dimensions for a resonator that can be built in the lab to transfer power over a distance of 0.1 meter with 2 identical resonators

This first case has a simple goal: get the optimal power transfer efficiency by determining the dimensions of the resonators that will be used. The frequency and the voltage of the source will be set to a fixed value as well as the distance between the resonators. All the fixed values can be found in Table 7 as well as the boundary conditions for the dimensions of the resonators. There is also a starting value because the optimization function (that can be found in Appendix A code 12) needs somewhere to start. The boundary conditions were chosen in function of what was possible to make (and easy to make) in the lab. Note that the number of windings per layer cannot be changed as this is in function of the length of the coil and the width of the wires and is therefore calculated when these parameters are given.

Table 7: Fixed values and variables of the resonators for the first case. case 1

Fixed values:

Source voltage 1 V

Frequency source 75 000 Hz Distance between 0.1 m resonators Variables: Starting value Minimum Maximum Radius coil 0.03 m 0.01 m 0.03075 m Length coil 0.03 m 0.001 m 0.032 m Layers of windings 1 1 1 Windings per Was calculated by dividing the length of the coil with the equivalent diameter layer of the wire. radius strand 0.0008 m 0.000075 m 0.001 m number of strands 50 1 60 Load resistor 0.44 Ω 0.01 Ω 10 Ω

With these inputs the optimization function found the optimal dimensions for the resonators. These values for the optimal dimensions can be found in Table 8.

Table 8: Optimal dimensions for the resonators, calculated by the optimization function. Variables: optimal value Radius coil 0.03075 m Length coil 0.032 m Layers of windings 1 Windings per layer 17 radius strand 0.00009 m number of strands 60 Load resistor 0.1858 Ω

It can be seen that the optimal dimensions of the coil frame are the maximum values. This was found earlier too and this was because the bigger the coil is, the higher the inductance is of the coil (also the resistance of the coil gets higher because the wire needs to be longer). When the inductance gets higher, the Q-factor also gets higher which means a better power transfer. It can also be seen that the maximum number of strands were chosen which is due to effect like skin- and proximity-effect and the fact that a higher number of strands reduces these effects. The load resister is here a certain value, the explanation of why this resistor needs to be a certain value can be found in Section 2.7. The number of winding layers is here 1 because this was a boundary condition. The other 2 variables were the radius of the strands and the number of windings per layer. A compromise was found by the optimisation function for these to last dimensions. This is because the windings per layer increases the inductance of the resonator, this should be high but that would mean that the radius of the strands should be small (because the number of windings depend on the width of the wire) and when the radius of the strands are smaller, the resistance of the coil increases as well. So, it can be seen that there needs to be a compromise because the Q factor equals to: 휔퐿 푄 = 푅 The efficiency of this power transfer was also calculated. The efficiency is here 52.96 %. With these dimensions two identical resonators were made. Some dimensions were slightly changed due to availability of wires and small resistors. The dimensions of the adjusted resonators were:

Table 9: Dimensions and parameters of the resonators that were made in the lab. Variables: Actual value Radius coil 0.03075 m Length coil 0.032 m Layers of windings 1 Windings per layer 0.03 m radius strand 0.000075 m number of strands 60 Load resistor 0.44 Ω

The made resonators were then placed with a placing of 0.1 m out of each other and a AC voltage source with the 1 V and a frequency of around 75000 Hz was connected to the first resonator. Also, the load resistor

was connected to the second resonator. The input power was measured as well as the power through the load resistor.

Figure 73: Picture of the measurement setup to validate the optimisation model in the first case.

The power that the source put into the first resonator was 54.0 mW, the measured power through the load resistor in the second resonator was 26.7 mW. The efficiency of the power transfer is then the quotient of those two powers, 0.494 or 49.4 %.

Compared this efficiency to the efficiency that was found using the optimisation code (52.96 %) is can be seen that these efficiencies are very close. But this calculated efficiency was calculated with the optimal dimensions of the resonators (and the optimal resistor) which is not the same that was used in the measuring setup, so this explains the slight difference in efficiency. When using the same dimensions as the measuring setup in the model the actual predicted efficiency was calculated and this was then equal to 50.65 %. 5.1.1 Conclusion

The output resonator model is validated with a real-world measurement with a 97.5 % accuracy. So, it can be concluded that the resonator model is very good at predicting the power transfer efficiency for short distances, solenoid shaped coils, multi stranded wires and low range frequencies. Also, the optimisation function is also validated because it was a very close prediction. There was also no better efficiency found by slightly changing the resonators dimensions so it can be concluded that this optimisation function is correct.

5.2 Predicting the power transfer of an existing testing setup

Because the first case gave such good results of the predictions of the model, another resonator was used to see if the model can also predict the power transfer if the resonator is made with more winding layers and with regular wire (less strands). Another reason for this testing setup was to see if the model can give a good prediction of the power transfer using specifically these resonators because in the next case more of these resonators will be used.

Like in the first case, two resonators were used to transfer power over a distance of 10 centimetres. The resonators that were used were made with a two-stranded wire. The diameter of the strands were 0.8 millimetres. The radius and the length of the coil was kept the same as in case 1. The maximum number of windings per layer was wounded on the coil and there were 2 layers of windings. The dimensions of the resonators and other values of the testing setup can also be found in Table 10. A picture of this testing setup can be found in Figure 74.

Table 10: Dimensions and values of testing setup of case 2. Variables: Actual value Source voltage 3.3 V Frequency source 46 500 Hz Distance between resonators 0.1 m Radius coil 0.03075 m Length coil 0.029 m Layers of windings 2 Windings per layer 16 radius strand 0.0004 m number of strands 2 Load resistor 0.6 Ω

Figure 74: Testing setup of case 2.

The efficiency of the power transfer was measured and an efficiency of 16.6% was found.

The efficiency was also calculated with the model. The efficiency that the model predicted was 17.3%. This is a bit higher than the measured 16.6% but still very close. The model is also in this case more than 95% accurate. This means that for the next case, where the spacing of multiple resonators will be investigated, this resonator can be used without worrying that the accuracy of the power transfer of the model will influence the calculation.

5.2.1 Conclusion

Note that the efficiency of the power transfer of this setup (16.6%) is lower than the efficiency of the power transfer of the first case (49.4%). This not only because Litz wire (that is used in the first case) has less AC- resistance, but also because the resonators of the second case had lower resonance frequency than the resonators of the first case. The inductance of the resonators of the second case were actually higher (because more windings were used) but the increase in inductance was not enough to negate the effect of the increased AC resistance and lowered frequency on the Q-factor of the resonators.

5.3 Get the optimal spacing of the 3 middle resonators of a 5-resonator system, with a distance of 0.4 meter between the first and the last resonator

Figure 75: Setup to get power transfer over a distance of 0.4 m with 5 resonators.

In this third case the spacing of a multiple-resonator-power-transfer-system was calculated with the optimisation program that was made. In this case 5 resonators were used. The dimensions of the resonators were the same as in the second case. This was because these resonators could be made relatively fast and the predictions of the model for this particular resonator was already validated. The dimensions of these resonators can be found in Table 10 in the previous Section.

More than two resonators were used, this meant that the existing model that predicted the power transfer efficiency was not compatible. The model was modified to make the amount of resonators variable, in Section 2.9.2 the new 푒푓푓_푎푢푡표푠푖푧푒 model is explained.

To get the optimal spacing of the resonator the optimisation function fmincon was used. The function that fmincon used was the new model that was made. The inputs of the model were the dimensions of the resonator, the frequency and voltage of the source, the load resistor and the relative position of the resonators. The dimensions of the resonators, frequency and voltage of the source were fixed values that can be found in Table 10 in the previous section. The position of the first resonators (where the source was connected to) and the position of the last resonator (where the load resistor is connected to) were also set at 0 and 0.4 m. The position of the other three resonators and the value of the load resistor should be optimised so an upper and lower limit should be set for these variables. In Table 11 the starting value and limits can be seen on these values.

The optimization code can be found in Appendix A code 13.

Table 11: Inputs of the distances and load resistance of the optimisation in case 3.

Variables: Starting value Minimum Maximum Distance 1st coil 0 m 0 m 0 m Distance 2nd coil 0.06 m 0.05 m 0.35 m Distance 3rd coil 0.07 m 0.05 m 0.35 m Distance 4th coil 0.08 m 0.05 m 0.35 m Distance 5th coil 0.4 m 0.4 m 0.4 m Load resistance 0.6 Ω 0.01 Ω 50 Ω

In Table 10 can be seen that the starting positions of the three middle coils and the load resistance are arbitrary values. The order of the coils were also set in the fmincon function (the distance of the 2nd coil should be smaller than the 3rd coil, …).

The function was run and the result of the function can be found in Table 12.

Table 12: Optimal values to the distances of the resonators and optimal load resistor for this case. Variables: optimal value Distance 1st coil 0 m Distance 2nd coil 0.0893 m Distance 3rd coil 0.1999 m Distance 4th coil 0.3106 m Distance 5th coil 0.4 m Load resistance 0.706 Ω

It can be seen that the efficiency of the power transfer is 1.07 %. But more interesting is the optimal spacing of the resonators. It was assumed that the optimal spacing of the five resonators would be respectively 0; 0.1; 0.2; 0.3; 0.4. But this result shows that the optimal spacing is in fact: 0; 0.09; 0.2; 0.31; 0.4. The coils closest to the driver and load coils should be spaced closer to those coils than the middle coil.

Also the time that the computer needed to find these optimized values was shown in Matlab. This was approximately 20 minutes. This is nothing compared to the time that was needed to make this testing setup. Five identical resonators were made with the dimensions in Table 10. One resonator was made in 4 hours. Then the board were the resonators could be mounted on was also made. So, to make and measure this testing setup approximately 3 days were needed. A picture of this setup can be seen in Figure 76.

Figure 76: Testing setup to validate the optimal positions of the resonator found by the optimisation code for five resonators.

The power transfer efficiency was measured with optimal spacing as well as uniform spacing (spacing the resonators with the same distance between them). In Table 13 the results of these measurements can be seen as well as the results that the model gave for these spacing.

Table 13: Table of the power transfer efficiencies, that were calculated with the model and the measured power transfer efficiencies, for optimal spacing and uniform spacing. Optimal Uniform Spacing: (0 – 0.09 – 0.2 – 0.31 – 0.4) (0 – 0.1 – 0.2 – 0.3 – 0.4)

Predicted power transfer 1.07 % 0.93 % efficiency of the model: Measured power transfer 1.04 % 0.93 % efficiency:

5.3.1 Conclusion

In Table 13 can be seen that the model once again makes a very accurate prediction of the power transfer efficiency. This time it does it for multiple resonators. It can also be seen that indeed the optimal spacing for this system is actually not a uniform spacing but a modified spacing were the outer coils are closer to the driver and load coil then the middle coil.

5.4 Effect on efficiency of using multiple resonators to transfer power over a distance of 0.5 meters.

In this Section the effect on the efficiency of the power transfer of using multiple resonators will be discussed. The resonators that were used in Section 5.1 and 5.2 will be used for the calculation of the efficiency in this Section. The dimensions of these resonators can be found in the next table.

Table 14: Dimensions of resonators used for calculation of the efficiencies for different numbers of resonators Low strand count High strand count Variables: resonator resonator Source voltage 2 V 2 V Frequency source 46 500 Hz 75 000 Hz Distance between first and last resonator 0.4 m 0.4 Radius coil 0.03075 m 0.03075 m Length coil 0.029 m 0.032 m Layers of windings 2 1 Windings per layer 16 0.03 m radius strand 0.0004 m 0.000075 m number of strands 2 60 Load resistor Optimal resistor will be calculated for each test

The first resonator is a resonator with a small number of strands. This resonator was also used in Section 5.2 and Section 5.3. The second resonator has a high number of strands, this resonator was used in Section 5.1 . When comparing the efficiencies that were measured and calculated in 5.1 (50 %) and 5.2 (17 %) it can be seen that the resonator with high strand count performed better when the power transfer distance was 0.2 meters.

In this Section the power transfer efficiencies will be calculated using the model (and the optimization function to get the optimal load resistance). The transfer distance is set at 0.4 meters but the number of resonators used will be changed. The range of the number of resonators is from just two resonators (just a sender and receiver) to ten resonators. A maximum of ten resonators was chosen because the coils of the resonators have a thickness of 0.03 meters and the frame of the coils are a little wider so this is the maximum number of resonators that can be placed over a distance of 0.4 meters. The resonators will be spaced out equally, this means that the distance between two neighbouring resonators will be equal for all resonators. In Figure 77 a graph can be seen of the efficiency of the power transfer for a different number of resonators, using a low strand count resonator and a high strand count resonator.

It can be seen in Figure 77 that a higher number of resonators can dramatically increase the efficiency of the power transfer. The resonators with a high strand count obviously have a higher efficiency than with a low strand count, also the frequency of the power transfer is also higher for the high strand count resonator which also helps increase the efficiency of the power transfer (Q-factor becomes higher). It can also be seen in Figure 77 that the efficiency increase of adding a resonator to the system lowers after a certain amount of resonators. It can be seen for the High strand count resonators that the efficiency increase from eight resonators to nine resonators is bigger than the efficiency increase from nine to ten resonators. To show this effect some more measuring points were calculated for eleven, twelve and thirteen resonators. That many

resonators are obviously not physically possible for that transfer distance but it can be imagined that a more pancake-shaped coil can be made with the same inductance as this coil. In Figure 78 the extended graph can be seen that shows the efficiency of the power transfer system for more resonators.

Figure 77: Graph that shows the effect of using multiple resonators to transfer power efficiency over a distance of 0.4 meters.

The graph in Figure 78 looks like an S-shaped graph. It can be seen that adding more resonators in this system will not yield as much gain in efficiency than it did for lower number of resonators.

Figure 78: Extended graph that shows the efficiency of using more than ten resonators.

5.4.1 Conclusion

It can be concluded that using more resonators spaced out over a distance drastically increases the efficiency of the power transfer. After a number of resonators, the increase in efficiency by adding another resonator starts to slow down. This can be seen especially when a resonator with a high Q-factor is used, the example is shown in Figure 77 where the red dots represent a higher Q-factor resonator and the blue dots represent a lower Q-factor resonator. It can be seen in the graph that the increase in efficiency starts to slow down after using 9 resonators while the lower Q-factor resonator seems to keep increasing at the same rate. Obviously the total efficiency is way higher when using a higher quality factor resonator as was concluded in Section 1.4.2 .

Conclusion

The following conclusion can be made out of this dissertation:

The basics of strongly coupled magnetic resonance power transfer were discussed in this dissertation. Magnetism was discussed as well as the circuit theory of the resonators. Also some terms and effects were briefly discussed that linked to SCMR like reflected impedance, frequency splitting, … . Therefore, a full explanation of the strongly coupled magnetic resonance power transfer was discussed in this dissertation in Section 1.

A predicting model was made that could calculate the power transfer efficiency of a multiple resonator power transfer system. The predicting model uses analytical and empirical formulas that were used from literature. In this dissertation these formulas were shown and references to the literature were they were obtained were made. The formulas were validated by making measurements with different types of coils, various sizes and wire types were used to validate these formulas. The (Matlab) codes for this model can be found in the Appendix. The accuracy of this model was tested for a number of cases in the last section of this dissertation. The model showed an accuracy of over 97% for the power transfer efficiency when compared to the measured power transfer efficiency. Note that the model was tested for low frequency power transfer (< 100 kHz) also the number of windings of the coils of the resonators were also relatively low.

The predicting model was combined with an optimization algorithm. The optimization code can also be found in the Appendix. With these optimization code the dimensions of the resonator coils can be optimized along with the load resistor. Also, the spacing of the resonators (when more than two resonators are used to transfer power) can be optimized with this code as was done in Section 5.4.

When the optimal spacing for the resonators for a five-resonator power transfer system was calculated, it was found that the spacing of these resonators were not equal. The spacing of the first and second resonator is a bit smaller than the spacing between the second and third resonator. The same can be said for the spacing between the second last resonator.

It was shown that for every distance between two or more resonators, there was an optimal load resistance. For two resonators a formula was found in the literature that can calculate the optimal load resistance for a certain mutual induction between two coils, so when calculating the mutual induction for a certain distance this could be used to determine the optimal load resistance for a certain distance. An easier way of doing this is with the model that was made. In the model the distance between the resonators along with the basic dimensions of the resonators can be put in and with the help of the optimization function the optimal load resistance can be found.

The effect of using more resonators was also investigated. It was shown that the efficiency can drastically increase when using more than 2 resonators over a distance of 0.4 meters. For an arbitrary resonator it was calculated that the power transfer efficiency was less than 0.05 % when using two resonators over a distance of 0.4 meters, when four resonators were used over the same distance the efficiency increased to 2 %, when eight resonators were used over the same distance the efficiency increased to 33 %. The increases in efficiency did lower when more resonators were added (see Figure 78).

By looking at all the previous conclusions it can be seen that a model was created and validated that can accurately predict the SCMR power transfer efficiency between two or more resonators. This was the first objective (that was stated in Section 1). The second objective was to do a comparative study on the power transfer. This was done in Section 2.5.1 where the dimensions of a first prototype resonator were investigated, graphs were shown how the Q-factor, so power transfer efficiency, variated by changing the

dimensions of the resonator. But, in Section 4 (optimization) further investigation was done on the way how to optimize these dimensions. An optimization code was made that could change not only the dimensions of the coil but also change the load resistance, distances between every resonator used, frequency and wire configurations. In Section 5 some experiments were done by using this function.

The model van be used for many applications because of the versatility of the model. The inputs of the model are very simple, just basic dimensions of the resonators need to be put in along with the properties of the voltage source, distances between resonators and the load resistance of the last resonator. In addition to simple inputs, al the inputs can be optimized. So, this means that this model can be used for further investigation of SCMR wireless power transfer to get fast results on different testing setups. The model can also be used to investigate transferring power over very long distances by using multiple resonators that are positioned after each other without actually making all these resonators. But, this can also be used as a first investigation to create a certain product or application. For example, say a wireless charging application needs to be built for a certain product. This would mean that the load resistance is known, the maximum dimensions of the resonator is also known because it needs to be built into the product. By now filling these fixed values in and setting the minimum and maximum bounds for the other values an optimal power transfer system can be calculated. It is clear that the model and optimization code can be used for a lot of different applications.

Finally, a last conclusion can be made about the measurements that were done in the context of the efficiency. When looking at Figure 50 it can be seen that the Q-factors of the measured coils increase for higher frequencies. In this dissertation no higher frequencies were used even though the Q-factor (so efficiency) would increase because of it. This was because the frequency range was set to the low frequency range for this dissertation as stated in the introduction. Because of this low frequency range, the efficiencies in the measurements and cases that were done were relatively low. Higher efficiencies can be achieved when measurements and test would be done with higher frequencies, the methods in this dissertation that were used to conduct and optimize these tests can be used for these new tests as well.

Future work

The model that was created has some limitations and some areas were not investigated in this dissertation. Here are some things that can be investigated and limitations of the model that can be fixed:

Firstly, the power transfer efficiencies that were become in testing setups were not very high. The best efficiency that was measured was around 50 %. Some test can be done to show that efficiencies can be way higher with SCMR power transfer. For example, when some tests were done with the model efficiencies of over 95% were reached over a distance of 0.2 meters by just increasing the diameter of the coil to 0.2 meters. These and other calculations were not shown in this dissertation because there was no testing setup made for this so it couldn’t be validated and the efficiencies that are shown in this dissertation were all validated, so it was chosen not to include this.

Another sign of limitation was shown in Figure 49 in Section 2.5.1. There was a slight indication that the calculations for the resistance of coils were getting less accurate when more layers of windings were used. More could can be made with more layers of windings to see if the accuracy is actually lower when there are more layers and if the accuracy is lower maybe some modifications to the expression or a new expression could be derived to find the AC resistance of the conductor of an arbitrary coil.

The capacitor that should be used in the resonators can also be further investigated. This is because an actual capacitor is used to achieve resonance at lower frequencies (compared to using no physical capacitor and using the parasitic self-capacitance of coil windings for high frequency applications). The capacitor has an internal resistance so this will lower the quality factor, so power transfer efficiency, of the resonator.

The model that was made in Matlab could be made more efficient, code could be written more efficiently to reduce calculation time. Especially when multiple resonators are used, this calculation time for optimization can be up to hours (when using more than 10 resonators in the system). Maybe the model can also be made in an application with a more user-friendly interface compared to changing the code in Matlab.

Other applications for SCMR wireless power transfer can be investigated further with this model.

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Appendices

Programs in Matlab List of contents

Code 1. Showing the behaviour of the signals near resonance ...... - 3 - Code 2. Using a code to automatically fill in a Simulink circuit ...... - 4 - Code 3. Transformation of equations to determine relationship of currents of coils in function of components in the circuit ...... - 5 - Code 4. Resonator determining program ...... - 8 - Code 5. Function that predicts the resistance of a coil ...... - 10 - Code 6. Function that predicts the inductance of a short solenoid coil ...... - 11 - Code 7. Function that predicts the inductance of a rectangular coil ...... - 12 - Code 8. Code that was used to find the optimal position for a certain load resistance ...... - 13 - Code 9. 푒푓푓_푑푖푚 function ...... - 15 - Code 10. Outerdiameter function used in code 9 ...... - 16 - Code 11. 푒푓푓_푎푢푡표푠푖푧푒 function ...... - 17 - Code 12. Optimization code to optimize the resonators in a two-resonator system ...... - 19 - Code 13. Optimization code to optimize the positions of the resonators in a five-resonator system - 20 -

- 2 -

Code 1. Showing the behaviour of the signals near resonance clear %memory clearing

C = zeros(1, 101000) %create array of all zeros with 101000 elements f = 1000

for i= 1:1:10 %for loop, starts with i=1, ends with i=20 with steps of 1 set_param('Voorbeeld_simulink/AC', 'frequency', 'f') %places value f in AC voltage source a = sim('Voorbeeld_simulink') b = simout.signals.values; %Gets values out of the simulation into the workplace B = max(b) %Gets maximum value from the measured values C(1,f) = B %Places maximum value in array f = f+10000 %Sets frequency for measurement in next loop end

plot(C) %Makes a graph of the array

- 3 -

Code 2. Using a code to automatically fill in a Simulink circuit clear close all f = 185000; omega = 2*pi*f; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %inputs N1 = 1; R1 = 50; N2 = 8; L2 = 11.38*10^(-6); C2 = 67.7*10^(-9); L2 = 1/(omega^2*C2); %Corrected to get desired resonance Rl2 = 47.2*10^(-3); Rc2 = 310*10^(-3); R2 = (Rl2 + Rc2); N3 = 14; L3 = 14.7*10^(-6); C3 = 46.54*10^(-9); L3 = 1/(omega^2*C3); % Corrected to get desired resonance Rl3 = 99*10^(-3); Rc3 = 423.3*10^(-3); R3 = (Rl3 + Rc3); R4 = 10^9; N4 = 1; Rb = 10000; L1 = (N1^2/N2^2)*L2; %estimation, (not really that important) L4 = (N4^2/N3^2)*L3; %estimation, (not really that important)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %measurements V1 = 2.48; I1 = 0.0564*5; I2 = 286*5*10^(-3); I3 = 48.2*5*10^(-3); U4 = 0.301; Vin = 16.9; %calculations I1_check = (Vin-V1)/R1; %dicht bij gemeten I1 V1_check = Vin-I1*50; %dicht bij gemeten V1 M12 = V1/(omega*I2) R2_check = 2*pi*f*M12*I1/I2; %dicht bij gemeten R2 R2_checkb = V1*I1/I2^2; %alternatieve uitdrukking voor R2 M12 = 1/omega*sqrt(V1*R2/I1) %alternatieve uitdrukking M13 = 0; M23 = (R3*I3)/(2*pi*f*I2); M24 = 0; M34 = (U4)/(2*pi*f*I3); M14 = 0; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

- 4 -

Code 3. Transformation of equations to determine relationship of currents of coils in function of components in the circuit clear close all

%% acquiring equations of currents without M13 M14 M24 % solve this part separately to get these equations % these resulting equations are used below (the very long ones) syms Vin R1 R2 R3 R4 X1 X2 X3 X4 M12 M23 M34 I1 I2 I3 I4 t omega eq1 = Vin == R1*I1 + j*(X1*I1 + omega*M12*I2) eq2 = 0 == R2*I2 + j*(X2*I2 + omega*M12*I1 + omega*M23*I3) eq3 = 0 == R3*I3 + j*(X3*I3 + omega*M23*I2 + omega*M34*I4) eq4 = 0 == R4*I4 + j*(X4*I4 + omega*M34*I3)

[sol1, sol2, sol3, sol4] = solve([eq1,eq2,eq3,eq4],[I1,I2,I3,I4]) simplify([sol1; sol2; sol3; sol4])

%% Setting constants f = 185000; omega = 2*pi*f; N1 = 1; R1 = 50; N2 = 8; C2 = 67.7*10^(-9); L2 = 1/(omega^2*C2);%corrected to get resonance Rl2 = 47.2*10^(-3); Rc2 = 310*10^(-3); R2 = (Rl2 + Rc2); N3 = 14; C3 = 46.54*10^(-9); L3 = 1/(omega^2*C3);%corrected to get resonance Rl3 = 99*10^(-3); Rc3 = 423.3*10^(-3); R3 = (Rl3 + Rc3); R4 = 10^9; N4 = 1; L1 = (N1^2/N2^2)*L2; %schatting, maar is niet belangrijk L4 = (N4^2/N3^2)*L3; %schatting, maar is niet belangrijk Vin = 16.9;

V1t = 2.48; I1t = 0.0564*5; I2t = 286*5*10^(-3); I3t = 48.2*5*10^(-3); U4t = 0.301;

M12 = 1/omega*sqrt(V1t*R2/I1t); %alternatieve uitdrukking M13 = 0;%1.048*10^(-8); M23 = (R3*I3t)/(2*pi*f*I2t); M24 = 0;%0.958*10^(-7); M34 = (U4t)/(2*pi*f*I3t); M14 = 0;%8.5588*10^(-10);

- 5 -

%% LOOP, with the equations that were found in the part above, to get the relationships of the currents fmin = 5000; fmin2 = 180000; fmax2 = 190000; fmax = 365000; fstep = 2500; fstep2 = 250; frange = [fmin:fstep:fmin2-fstep,fmin2:fstep2:fmax2,fmax2+fstep:fstep:fmax]

I1m = zeros(1, length(frange)) ; %create array of all zeros with n elements I2m = zeros(1, length(frange)); I3m = zeros(1, length(frange)); I4m = zeros(1, length(frange)); PHI_1 = zeros(1, length(frange)); PHI_12 = zeros(1, length(frange)); PHI_23 = zeros(1, length(frange)); PHI_34 = zeros(1, length(frange)); for k= 1:length(frange) %for loop, starts with k=0, ends with k=fmax with steps

f = frange(k) %Sets frequency for measurement omega = 2*pi*f;

X1 = omega*L1; X2 = omega*L2-(1/(omega*C2)); X3 = omega*L3-(1/(omega*C3)); X4 = omega*L4;

I1 = (Vin*(M23^2*R4*omega^2 + M34^2*R2*omega^2 + M23^2*X4*omega^2*1i + M34^2*X2*omega^2*1i + R2*R3*R4 + R2*R3*X4*1i + R2*R4*X3*1i + R3*R4*X2*1i - R2*X3*X4 - R3*X2*X4 - R4*X2*X3 - X2*X3*X4*1i))/(M12^2*M34^2*omega^4 + M12^2*R3*R4*omega^2 + M23^2*R1*R4*omega^2 + M34^2*R1*R2*omega^2 + M12^2*R3*X4*omega^2*1i + M12^2*R4*X3*omega^2*1i + M23^2*R1*X4*omega^2*1i + M23^2*R4*X1*omega^2*1i + M34^2*R1*X2*omega^2*1i + M34^2*R2*X1*omega^2*1i - M12^2*X3*X4*omega^2 - M23^2*X1*X4*omega^2 - M34^2*X1*X2*omega^2 + R1*R2*R3*R4 + R1*R2*R3*X4*1i + R1*R2*R4*X3*1i + R1*R3*R4*X2*1i + R2*R3*R4*X1*1i - R1*R2*X3*X4 - R1*R3*X2*X4 - R1*R4*X2*X3 - R2*R3*X1*X4 - R2*R4*X1*X3 - R3*R4*X1*X2 - R1*X2*X3*X4*1i - R2*X1*X3*X4*1i - R3*X1*X2*X4*1i - R4*X1*X2*X3*1i + X1*X2*X3*X4)

I2 = -(M12*Vin*omega*(M34^2*omega^2 + R3*R4 + R3*X4*1i + R4*X3*1i - X3*X4)*1i)/(M12^2*M34^2*omega^4 + M12^2*R3*R4*omega^2 + M23^2*R1*R4*omega^2 + M34^2*R1*R2*omega^2 + M12^2*R3*X4*omega^2*1i + M12^2*R4*X3*omega^2*1i + M23^2*R1*X4*omega^2*1i + M23^2*R4*X1*omega^2*1i + M34^2*R1*X2*omega^2*1i + M34^2*R2*X1*omega^2*1i - M12^2*X3*X4*omega^2 - M23^2*X1*X4*omega^2 - M34^2*X1*X2*omega^2 + R1*R2*R3*R4 + R1*R2*R3*X4*1i + R1*R2*R4*X3*1i + R1*R3*R4*X2*1i + R2*R3*R4*X1*1i - R1*R2*X3*X4 - R1*R3*X2*X4 - R1*R4*X2*X3 - R2*R3*X1*X4 - R2*R4*X1*X3 - R3*R4*X1*X2 - R1*X2*X3*X4*1i - R2*X1*X3*X4*1i - R3*X1*X2*X4*1i - R4*X1*X2*X3*1i + X1*X2*X3*X4)

I3 = -(M12*M23*Vin*omega^2*(R4 + X4*1i))/(M12^2*M34^2*omega^4 + M12^2*R3*R4*omega^2 + M23^2*R1*R4*omega^2 + M34^2*R1*R2*omega^2 + M12^2*R3*X4*omega^2*1i + M12^2*R4*X3*omega^2*1i + M23^2*R1*X4*omega^2*1i + M23^2*R4*X1*omega^2*1i + M34^2*R1*X2*omega^2*1i + M34^2*R2*X1*omega^2*1i - M12^2*X3*X4*omega^2 - M23^2*X1*X4*omega^2 - M34^2*X1*X2*omega^2 + R1*R2*R3*R4 + R1*R2*R3*X4*1i + R1*R2*R4*X3*1i + R1*R3*R4*X2*1i +

- 6 -

R2*R3*R4*X1*1i - R1*R2*X3*X4 - R1*R3*X2*X4 - R1*R4*X2*X3 - R2*R3*X1*X4 - R2*R4*X1*X3 - R3*R4*X1*X2 - R1*X2*X3*X4*1i - R2*X1*X3*X4*1i - R3*X1*X2*X4*1i - R4*X1*X2*X3*1i + X1*X2*X3*X4)

I4 = (M12*M23*M34*Vin*omega^3*1i)/(M12^2*M34^2*omega^4 + M12^2*R3*R4*omega^2 + M23^2*R1*R4*omega^2 + M34^2*R1*R2*omega^2 + M12^2*R3*X4*omega^2*1i + M12^2*R4*X3*omega^2*1i + M23^2*R1*X4*omega^2*1i + M23^2*R4*X1*omega^2*1i + M34^2*R1*X2*omega^2*1i + M34^2*R2*X1*omega^2*1i - M12^2*X3*X4*omega^2 - M23^2*X1*X4*omega^2 - M34^2*X1*X2*omega^2 + R1*R2*R3*R4 + R1*R2*R3*X4*1i + R1*R2*R4*X3*1i + R1*R3*R4*X2*1i + R2*R3*R4*X1*1i - R1*R2*X3*X4 - R1*R3*X2*X4 - R1*R4*X2*X3 - R2*R3*X1*X4 - R2*R4*X1*X3 - R3*R4*X1*X2 - R1*X2*X3*X4*1i - R2*X1*X3*X4*1i - R3*X1*X2*X4*1i - R4*X1*X2*X3*1i + X1*X2*X3*X4)

I1m(k) = abs(I1); I2m(k) = abs(I2); I3m(k) = abs(I3); U4m(k) = abs(I4)*10^9;

PHI_1(k) = angle(I1); PHI_12(k) = angle(I2) - angle(I1); PHI_13(k) = angle(I3) - angle(I1); PHI_14(k) = angle(I4) - angle(I1); end

%% signalen tov I1 openfig('Signalen tov I1.fig') %hold on subplot(2,2,1) polar(PHI_1,I1m, 'red') title('I1') subplot(2,2,2) polar(PHI_12,I2m, 'red') title('I2') subplot(2,2,3) polar(PHI_13,I3m, 'red') title('I3') subplot(2,2,4) g=polar(PHI_14,U4m) set(g, 'LineWidth',2, 'color', 'red') hold on h= polar(PHI_14,U4m) set(h, 'LineWidth',1.2, 'linestyle','-.', 'color', 'black') title('U4')

- 7 -

Code 4. Resonator determining program %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% Resonator determining program %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Shows Q-factor and Resistance for a range of dimensions %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Resonator: geometry: radius (mean radius) % length (lenght of coil) % radiuswire (radius of strands) % n_strands (number of strands in wire) % n (number of turns per layer) % m (number of layers) % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Resonator initialisation clear close all i_range = 0.005:0.001:0.030; R_range = zeros(1,length(i_range)); Q_range = zeros(1,length(i_range)); n_range = zeros(1,length(i_range)); for i=1:length(i_range)

frequency = 60000; Resonator_A.geometry.radius = 0.03; Resonator_A.geometry.radiuswire = 0.0008/2; Resonator_A.geometry.n_strands = 2; Resonator_A.geometry.length = i_range(i); Resonator_A.geometry.n = Resonator_A.geometry.length/(Resonator_A.geometry.radiuswire*2*1.16)/Resona tor_A.geometry.n_strands; Resonator_A.geometry.m = 2;

n_range(i) = Resonator_A.geometry.n ; R_range(i) = Resistance(Resonator_A, frequency);

if Resonator_A.geometry.m == 1 L = Inductance_bc(Resonator_A); else L = Inductance_rectangularcrosssection(Resonator_A); end

Q_range(i) = (2*pi*frequency*L)/R_range(i); end

%% specific calculations % R_A = Resistance(Resonator_A, frequency) % % if Resonator_A.geometry.m == 1 % L_A = Inductance_bc(Resonator_A) % else % L_A = Inductance_rectangularcrosssection(Resonator_A) % end

%% graphs

- 8 -

leg = ['d_{strands} = ', num2str(2*Resonator_A.geometry.radiuswire),', ','N_{strands} = ', num2str(Resonator_A.geometry.n_strands),', ', 'd = ', num2str(Resonator_A.geometry.radius),', ', 'l = ', num2str(Resonator_A.geometry.length),', ', 'N_{Windings per layer} = ', num2str(Resonator_A.geometry.n),', ', 'N_{Layers} = ', num2str(Resonator_A.geometry.m)] ; Wat = ['Resistance'] tov = ['Frequency'] Extra = [''] Fig1Naam = ['_',Wat,'_','tov','_',tov,'_',Extra,'.fig'] fig1 = figure plot(i_range,R_range) title(Wat) xlabel('Frequency [kHz]') ylabel('Resistance [\Omega]') legend(leg)

Wat = ['Q-factors for different coil lenghts '] tov = ['d'] Extra = [''] Fig2Naam = ['_',Wat,'_','tov','_',tov,'_',Extra,'.fig'] fig2 = figure plot(i_range,Q_range) title(Wat) xlabel('l_{coil} [m]') ylabel('Q factor') legend(leg) fig3 = figure plot(i_range,n_range) title(Wat) xlabel('d_{strand} [m]') ylabel('n (per layer)') legend(leg)

- 9 -

Code 5. Function that predicts the resistance of a coil function [Resistance] = Resistance(Resonator1,frequency) %This function returns the Resistance of a resonator radius1 = Resonator1.geometry.radius; radiuswire = Resonator1.geometry.radiuswire; n1 = Resonator1.geometry.n; % # windings n1_strands = Resonator1.geometry.n_strands; % # strands m = Resonator1.geometry.m; % # layers %f = frequency; Rho = 1.75*1.e-8; % Electrical resistivity of copper at 20°C Mu = 4*pi*1.e-7; % Permeability (relative permeability of copper (=1) * permeabilty of free space) omega = frequency*2*pi; N = m*n1; % ammount of windings l = N*2*pi*radius1; % Total lenght of wire used for coil S = radiuswire^2*pi; % Surface of 1 strand

%% dw = radiuswire.*2; % diameter draad in m a = dw/2*sqrt(pi)*0.85; % width square conductor, recalculation for circular conductor h = a; % height = witdh for square conductor b = n1*2*radiuswire*1.2; % width of coil

nu = n1*a/b; alpha = sqrt(1i*omega*Mu*nu/Rho); M =alpha.*h.*coth(alpha*h); %D = 2*alpha.*h.*tanh(alpha*h/2); %origineel D = 2*alpha.*h.*tanh(alpha*h/2)*0.225; %aanpassing voor m=2

factor = (real(M)+(m^2-1)*real(D)/3);

%% solution

Resistance = (Rho*l)/(n1_strands*S)*factor; end

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Code 6. Function that predicts the inductance of a short solenoid coil function [Inductance] = Inductance_bc(Resonator1) %This function returns the Inductance of a resonator r = Resonator1.geometry.radius; l = Resonator1.geometry.length; n = Resonator1.geometry.n; % # windings

% % omega = frequency*2*pi; % Rho = 1.75*1.e-8; % Electrical resistivity of copper at 20°C Mu = 4*pi*1.e-7; % Permeability (relative permeability of copper (=1) * permeabilty of free space)

% volgens bron B D= 2*r; K= 2/pi*(l/D)*((log(4*D/l)- 0.5)*(1+0.383901*(l/D)^2+0.017108*(l/D)^4))/(1+0.258952*(l/D)^2); %p35 bron D if l > r Lb = (10*pi*Mu*n^2*r^2)/(9*r+10*l); else Lb = pi*Mu*r^2*l*n^2*K; end

Inductance = Lb; end

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Code 7. Function that predicts the inductance of a rectangular coil function [Inductance] = Inductance_rectangularcrosssection(Resonator1) %This function returns the inductance of a rectangular resonator r = Resonator1.geometry.radius; r_strands = Resonator1.geometry.radiuswire; n_strands = Resonator1.geometry.n_strands; l = Resonator1.geometry.length; n = Resonator1.geometry.n; % # windings m = Resonator1.geometry.m; N = m*n; %omega = frequency*2*pi; %Rho = 1.75*1.e-8; % Electrical resistivity of copper at 20°C Mu = 4*pi*1.e-7; % Permeability (relative permeability of copper (=1) * permeabilty of free space)

if n_strands == 1 d = 2*r_strands; else d_strands = 2*r_strands; A_t = (d_strands^2*pi)/4*n_strands; d = sqrt((4*A_t)/pi); d = d*1.28; % packing factor + sleeve end

% the rectangular resonator has a crosssection of the windings with dimensions: a = r; % distance from center of gravity of upper section coil to centerline resonator b = l; % width of coil section c = m*d; % height of coil section

% volgens bron 'passive components' in maps literatuur start p7 L = (Mu*N.^2*pi*a.^2)/(b*(1+0.9*(a./b)+0.32*(c./a)+0.84*(c/b)));

Inductance = L; end

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Code 8. Code that was used to find the optimal position for a certain load resistance clear close all lmin = 0.01; lstep = 0.01; lmax = 0.230; lrange = [lmin:lstep:lmax];

P1 = zeros(1, length(lrange)); P2 = zeros(1, length(lrange)); n = zeros(1, length(lrange)); for j= 1:length(lrange) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %inputs

Vin = 1.68/2; f = 46500; %Hz omega = 2*pi*f;

% Primary Coil Resonator_B.geometry.radius = 0.0308; Resonator_B.geometry.radiuswire = 0.0008/2; Resonator_B.geometry.n_strands = 2; Resonator_B.geometry.length = 0.029; Resonator_B.geometry.n = 14; Resonator_B.geometry.m = 2; Resonator_B.position.x = 0; Resonator_B.position.y = 0; Resonator_B.position.A = 0;

% Secondary coil Resonator_C.geometry.radius = 0.0308; Resonator_C.geometry.radiuswire = 0.0008/2; Resonator_C.geometry.n_strands = 2; Resonator_C.geometry.length = 0.029; Resonator_C.geometry.n = 14; Resonator_C.geometry.m = 2; Resonator_C.position.x = 0; Resonator_C.position.y = lrange(j); Resonator_C.position.A = 0;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Calculations

C2 = 0.12*10^(-6); par1= {[C2,C2]}; [C_eq,R_eq] = getNewCap(f,par1(1)); % Values of resistance and capacitance of capacitors Rcap = R_eq;

R1c = Resistance(Resonator_B, f); R2c = Resistance(Resonator_C, f);

R1 = R_eq;

- 13 -

R2 = R_eq; %Weerstand Condensator; Rb = 0.75; %Belastingsweerstand

L1 = Inductance(Resonator_B); % gemeten = 50.18 µH L2 = Inductance(Resonator_C);

C1 = 1/(L1*omega^2); C2 = 1/(L2*omega^2);

M12 = MutualInductance(Resonator_B,Resonator_C); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% a = sim('PraktischeOpstelling');

[faseverschil,frequentie] = function_faseverschil(I1gemeten.time,I1gemeten.signals.values,Ul1gemeten.ti me,Ul1gemeten.signals.values); faseverschil1 = faseverschil; I1 = I1gemeten.signals.values; I1 = I1(floor(length(I1)/2):end); %laatste helft b I1max = max(I1); U1 = Ul1gemeten.signals.values; U1 = U1(floor(length(U1)/2):end); %laatste helft b U1max = max(U1); P1(j) = -0.5*I1max*U1max*cos(faseverschil1);

[faseverschil,frequentie] = function_faseverschil(I2gemeten.time,I2gemeten.signals.values,U4gemeten.tim e,U4gemeten.signals.values); faseverschil2 = faseverschil; I2 = I2gemeten.signals.values; I2 = I2(floor(length(I2)/2):end); %laatste helft b I2max = max(I2); U2 = U4gemeten.signals.values; U2 = U2(floor(length(U2)/2):end); %laatste helft b U2max = max(U2); P2(j) = 0.5*I2max*U2max*cos(faseverschil2);

n(j) = P2(j)/P1(j); indicatie = j/length(lrange) end fig= openfig('berekening12.fig') plot(lrange, n, '--') legend('R_L = 0.2 Ohm', 'R_L = 0.39 Ohm', 'R_L = 0.75 Ohm', 'R_L = 0.2 Ohm berekend', 'R_L = 0.39 Ohm berekend', 'R_L = 0.75 Ohm berekend' ) title('Efficiency of resonators for variable distance') xlabel('Y distance (m)') ylabel('Efficiency') %%

- 14 -

Code 9. 푒푓푓_푑푖푚 function function [eff] = eff_dim(frequency,Vin, distance,R_load,r_coil,l_coil,r_strand, n_strands,LayersOfWindings, Imax) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % This function returns the efficienty of a two resonator system %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Inputs V1 = Vin; f = frequency; %Hz omega = 2*pi*f; d_outer = Outerdiameter(n_strands, 2*r_strand); Windingsperlayer = (l_coil/(d_outer*1.1)); disp(num2str([R_load,r_coil,l_coil,r_strand,round(n_strands), round(LayersOfWindings), round(Windingsperlayer)]))

% Resonator coil dimensions Resonator_A.geometry.radius = r_coil; % radius coil Resonator_A.geometry.length = l_coil; % length of coil

Resonator_A.geometry.radiuswire = r_strand; % radius wire strand Resonator_A.geometry.n_strands = n_strands; % # strands Resonator_A.geometry.n = Windingsperlayer; % # windings per layer Resonator_A.geometry.m = LayersOfWindings; % # layers

Resonator_A.position.x = 0; % radial position Resonator_A.position.y = 0; % axial position Resonator_A.position.A = 0; % angle

Resonator_B = Resonator_A; % resonators coils are the same Resonator_B.position.y = distance; % changing position of Resonator B

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Calculations of properties resonators C2 = 0.12*10^(-6); par1= {[C2,C2]}; [C_eq,R_eq] = getNewCap(f,par1(1)); % Values of resistance and capacitance of capacitors x = 1.09; %correction factor resistance %R1c =(0.297*(50-46.5)+0.327*(46.5-45))/(50-45)*0.95; % measured value R1l = Resistance(Resonator_A, f)*x; % resistance of coil R2l = Resistance(Resonator_B, f)*x;

R1c = R_eq; % Resistance of capacitor R2c = R_eq; RL = R_load; % Load resistance

R1 = R1l+R1c; R2 = R2l+R2c+RL;

L1 = Inductance(Resonator_A); L2 = Inductance(Resonator_B);

C1 = 1/(L1*omega^2); % making sure there is resonance

- 15 -

C2 = 1/(L2*omega^2);

M12 = MutualInductance(Resonator_A,Resonator_B);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Calculation currents syms I1 I2 I3 VV = [V1;0]; RR = [R1,0 ; 0,R2]; LL = [L1,M12 ; M12,L2]; CC = [1/C1,0;0,1/C2];

%VV = RR*II + 1i*omega*LL*II + (1/1i*omega)*CC*II %oplossen naar II %II = inv(RR + 1i*omega*LL + (1/(1i*omega)*CC))*(VV) II = (RR + 1i*omega*LL + (1/(1i*omega)*CC))\(VV) ;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Calculation efficiency P1 = real(II(1)'*V1)/2; P2 = abs(II(2))^2*RL/2; n = P2/P1;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %solution eff = n; end

Code 10. Outerdiameter function used in code 9 function [Outerdiameter] = Outerdiameter(n_strands, d_strand) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % This function returns the guessed outer diameter of Litz wire %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

A_t = (d_strand^2*pi)/4*n_strands; d_eq = sqrt((4*A_t)/pi); if (n_strands == 1) d = 1.1*d_strand; else coef = 1.1 + ((-0.12/58)*(n_strands-2)+0.32); d = coef*d_eq; end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %solution Outerdiameter = d; end

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Code 11. 푒푓푓_푎푢푡표푠푖푧푒 function function [eff] = eff_autosize(frequency,Vin, distance, R_load, r_coil, l_coil, r_strand, n_strands,LayersOfWindings, NumberOfResonators) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % This function returns the efficienty of a n-resonator system %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Inputs f = frequency; %Hz omega = 2*pi*f; d_outer = Outerdiameter(n_strands, 2*r_strand); Windingsperlayer = (l_coil/d_outer); disp(num2str([R_load,r_coil,l_coil,r_strand,n_strands, LayersOfWindings, Windingsperlayer, NumberOfResonators, distance ])) for j = 1:NumberOfResonators % Resonator{j} = Resonator_A; % All resonators are the same (except position of resonator) Resonator_A{j}.geometry.radius = r_coil; % radius coil Resonator_A{j}.geometry.length = l_coil; % lenght of coil

Resonator_A{j}.geometry.radiuswire = r_strand; % radius wire strand Resonator_A{j}.geometry.n_strands = n_strands; % # strands Resonator_A{j}.geometry.n = Windingsperlayer; % # windings per layer Resonator_A{j}.geometry.m = LayersOfWindings; % # layers

Resonator_A{j}.position.x = 0; % radial position Resonator_A{j}.position.y = 0; % axial position Resonator_A{j}.position.A = 0; % angle Resonator_A{j}.position.y = distance(j); % changing position of Resonator(j) end

VV = zeros(NumberOfResonators, 1); RR = zeros(NumberOfResonators, NumberOfResonators); LL = zeros(NumberOfResonators, NumberOfResonators); CC = zeros(NumberOfResonators, NumberOfResonators);

VV(1,1) = Vin; for j = 1:NumberOfResonators

% filling resistance matrix C2 = 0.12*10^(-6); par1= {[C2,C2]}; [C_eq,R_eq] = getNewCap(f,par1(1)); % resistance of capacitors R1l = Resistance(Resonator_A{1}, f)*1.09; % resistance of coil RR(j,j) = R1l + R_eq; % total resistance RR(NumberOfResonators,NumberOfResonators) = R1l + R_eq + R_load; % last resistance has the load resistance

% filling inductance matrix for k = 1:NumberOfResonators if j==k

- 17 -

LL(j,j) = Inductance(Resonator_A{j}); else LL(j,k) = MutualInductance(Resonator_A{j},Resonator_A{k}); end end

% filling capacitance matrix c = (1/(LL(j,j)*omega^2)); CC(j,j) = 1/c; end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Calculation currents syms I1 I2 I3

%VV = RR*II + 1i*omega*LL*II + (1/1i*omega)*CC*II %oplossen naar II %II = inv(RR + 1i*omega*LL + (1/(1i*omega)*CC))*(VV) II = (RR + 1i*omega*LL + (1/(1i*omega)*CC))\(VV);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Calculation efficiency P1 = real(II(1)'*Vin)/2; P2 = abs(II(NumberOfResonators))^2*R_load/2; n = P2/P1;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %solution eff = n; end

- 18 -

Code 12. Optimization code to optimize the resonators in a two- resonator system clear close all tic %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %inputs Rho = 1.68*10^(-8); % [Ohm.m] for copper

Vin = 1.9/2; %[V] x = 0.05; % max percentage of voltage loss over wires P = 50; % [W] power needed for load f = 100000; %[Hz] omega = 2*pi*f; distance = 0.1; %[m]

R_load = 0.44; %[Ohm] R_loadm = 0.01; R_loadM = 10;

% Coil r_coil = 0.0308; % radius coil r_coilm = 0.01; r_coilM = 0.0309; l_coil = 0.029; % lenght of coil l_coilm = 0.001; l_coilM = 0.029;

LayersOfWindings = 1; % # layers LayersOfWindingsm = 1; LayersOfWindingsM = 1; r_strand = 0.00015/2; % radius of 1 strand r_strandm = 0.00015/2; r_strandM = 0.002/2; n_strands = 2; n_strandsm = 1; n_strandsM = 60;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% toc %f,Vin,distance,R_load,r_coil,l_coil,r_strand,n_strands, LayersOfWindings objective = @(x) -eff_dim(f,Vin,distance,x(1) ,x(2) ,x(3) ,x(4) ,x(5) , x(6) ,I_max); x0 = [R_load,r_coil,l_coil,r_strand,n_strands, LayersOfWindings] ; %objective(x0) A = []; b = []; Aeq = []; beq = []; lb = [R_loadm,r_coilm,l_coilm,r_strandm,n_strandsm, LayersOfWindingsm]; ub = [R_loadM,r_coilM,l_coilM,r_strandM,n_strandsM, LayersOfWindingsM];

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%nonlcon = @CrossSectionGood; %nonlcon = @(x) CrossSectionGood(x,f,Vin, distance, I_max);

RL = fmincon(objective,x0,A,b,Aeq,beq,lb,ub) % optimal variabelen L = ['R_load',' r_coil',' l_coil',' r_strand',' n_strands',' LayersOfWindings', 'WindingsPerLayer']; disp(L) eff_dim(f,Vin,distance,RL(1),RL(2),RL(3),RL(4),RL(5), RL(6)) toc

Code 13. Optimization code to optimize the positions of the resonators in a five-resonator system

clear close all tic %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% inputs Rho = 1.68*10^(-8); % [Ohm.m] for copper

Vin = 1.9/2; % [V] x = 0.05; % max percentage of voltage loss over wires P = 50; % [W] power needed for load f = 46500; % [Hz] omega = 2*pi*f;

R_load = 0.6; %[Ohm] R_loadm = 0.01; R_loadM = 50;

% Coil r_coil = 0.0308; % radius coil l_coil = 0.029; % lenght of coil LayersOfWindings = 2; % # layers WindingPerLayer = 14; % # windings per layer r_strand = 0.0008/2; % radius of 1 strand n_strands = 2; distance = 0.1; %[m]

Distance(1) = 0; Distancem(1) = 0;%volgor DistanceM(1) = 0;%wille

Distance(2) = 0.1; Distancem(2) = 0.05; DistanceM(2) = 0.4;

Distance(3) = 0.2; Distancem(3) = 0.05; DistanceM(3) = 0.4;

Distance(4) = 0.3;

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Distancem(4) = 0.05; DistanceM(4) = 0.4;

Distance(5) = 0.4; Distancem(5) = 0.4; DistanceM(5) = 0.4; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% toc objective = @(x) -eff_autosize(f ,Vin, x(1:5) ,x(6) ,r_coil,l_coil,r_strand, n_strands, LayersOfWindings, 14, 5); x0 = [Distance, R_load] ; objective(x0) A = [1, -1, 0, 0, 0, 0; 0, 1, -1, 0, 0, 0; 0, 0, 1, -1, 0, 0; 0, 0, 0, 1, - 1, 0; 0, 0, 0, 0, 0, 0; 0, 0, 0, 0, 0, 0;]; b = [0; 0; 0; 0; 0; 0]; Aeq = []; beq = []; lb = [Distancem, R_loadm]; ub = [DistanceM,R_loadM]; nonlcon = []; %@(x) ranking(x(1:5));

RL = fmincon(objective,x0,A,b,Aeq,beq,lb,ub,nonlcon) % optimal variabelen L = ['d1',' d2',' d3',' d4',' d5',' Rload']; disp(L) % eff_dim(f,Vin,distance,RL(1),RL(2),RL(3),RL(4),RL(5), RL(6)) n = eff_autosize(f,Vin, RL(1:5) ,RL(6),r_coil,l_coil,r_strand, n_strands, LayersOfWindings, 14, 5) toc

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