INDUCTIVE AND
WIRELESS ENERGY
TRANSFER
IN RESIDENTIAL APPLICATIONS
INDUCTIVE AND
WIRELESS ENERGY
TRANSFER
IN RESIDENTIAL APPLICATIONS
Proefschrift
ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus prof. ir. K.C.A.M. Luyben, voorzitter van het College voor Promoties,
in het openbaar te verdedigen op maandag15 oktober 2012 om 12.30 uur door Fredrik Frank Arie VAN DER PIJL
Elektrotechnisch Ingenieur geboren te Amstelveen.
Dit proefschrift is goedgekeurd door de promotor: Prof. dr. J.A. Ferreira
Copromotor: Dr. ir. P. Bauer
Samenstelling promotiecommissie: Rector Magnificus, voorzitter
Prof. dr. J.A. Ferreira, Technische Universiteit Delft, promotor Dr. ir. P. Bauer, Technische Universiteit Delft, copromotor Prof. dr. M.A.E. Andersen, University of Denmark, Denemarken Prof. dr. J.J. Smit, Technische Universiteit Delft
Prof. dr. J.R. Long, Technische Universiteit Delft Em. prof. dr. J.H. Blom, Technische Universiteit Eindhoven
Dr. ir. M. Castilla, Polytechnic University of Catalonia, Spanje
Dit onderzoek is gefinancierd als onderdeel van het Innovatiegericht Onderzoeksprogramma Elektromagnetische Vermogenstechniek (IOP-EMVT) van het Nederlandse ministerie van Economische Zaken.
Omslagontwerp door Astrid Koekoek en Maarten van der Pijl Gedrukt door Romein Grafisch
ACKNOWLEDGEMENTS
The research presented in this thesis was carried out at Delft University of Technology in the Electrical Power Processing group.
Writing this thesis took me a little too long. For some reason however, during the extra years my supervisor Pavol Bauer and I discussed most often and also most relaxed. Paul, I would like to thank you for your support and enthusiasm regarding inductive energy transfer. Sitting next to you in your big white automobile was my first electric driving experience.
I would also like to thank prof. Braham Ferreira for the opportunity he gave me to start this still fascinating PhD research. Braham, I will not forget your very useful remarks during my turns at the think-tank meetings of our group.
The first year of my research I was supervised by Henk Polinder. Henk, thank you for being critical to my rather unconventional (ok: wrong) ideas about inductive energy transfer back then.
Miguel Castilla from the Universitat Politècnica de Catalunya, I want to thank you for your essential contributions to my papers. I came back from the trips to Barcelona with new insights, inspiration and a healthier skin colour. Miguel, un fuerte abrazo, thank you for your time, I learned very much from you.
Rob Schoevaars, it was perfect to have you in our laboratory. Rob, you turned my technical drawings into a working practical electronic circuit. Your very first green garden hose implementation of the cable motivated its name.
Hans, my father-in-law, thank you for turning my initial “Shakespearean poetry” into normal sentences. Femke, daughter of my father-in-law, thank you for your
thesis revisions and for taking the time to understand the fundamentals of this thesis.
Mattia, Deok-je, Aleksandar, Balázs, you were perfect roommates. You living in Delft was the only bad thing about living in Amsterdam. Dongsheng, Milan, Marcelo, to name a few with whom I started. Well, our group grew a lot in the last few years. Thank you all for being my colleagues.
And yes, I had the pleasure of having a student working for me on his way to his electrical engineering degree. Martin Lievense, thank you for your effort and contribution.
Jeroen, we started college together and we finished our PhD at the same time. Ok, that is not exactly true. It took me two more years to really put de puntjes op de i. I hope that you will find your way in the non-electrical world.
Cara famiglia, as we always say; love you!
Also I would like to express my gratitude towards my thesis committee members, for taking the time to read my thesis and for making useful remarks accordingly.
This work was funded as part of the Innovatiegericht Onderzoeksprogramma
Elektromagnetische Vermogenstechniek (IOP-EMVT) program of the Dutch
Ministry of Economic Affairs. The program is almost at its end, but I think the initiative is worth being continued in some way. I would like to thank all participators in the program in the persons of its chairman Geert Wessel Boltje and my co-PhD student in Eindhoven, Christoph Sonntag.
Well, it is time to move on now. It is business time. Han, Annegreeth, Paul, Ronald, Henk, Dominique, Rick, Astrid, thank you for enabling this dream. Probably I forgot to mention many people. Please be angry with me and send me an e-mail at frank@optimos-apto.com.
SUMMARY
This thesis considers contactless electromagnetic energy transfer, applied to residential and office environments. Motivated by an analysis in Chapter 1, the objective is to investigate and design a (partially) contactless residential energy distribution network system for multiple electrical devices with a highly improved end-user flexibility. A special focus is system efficiency.
Power electronics engineers mainly use the quasi-static inductive approach, while telecommunication engineers are mainly interested in the radiative properties of antennas. Chapter 2 introduces an example where both inductive and radiative phenomena are present, in order to get a clear understanding of the possibilities and limitations of contactless energy transfer, with regard to the systems’ power efficiency. The example considers the energetic coupling of two dipoles. As a result the concepts of induction and radiation are brought together.
However, due to the point-source nature of the dipole model there is not enough information to conclude the general energy transfer characteristics. Therefore, the dipoles are given an actual size in Chapter 3 in order to complete the picture. The chapter explores the theoretical efficiency limit of a generalized contactless system as a function of relative air gap, assuming in the first place that such a limit can meaningfully be quantified. First, a general air coiled transformer is designed and (relatively) optimized from a theoretical physical base. Second, attempts are made to improve the design even more by respectively opting alternative coil shapes and by making use of magnetic core material.
It is shown that an efficiency limit as a function of relative contactless transfer distance exists, under the assumption of steady-state sinusoidal excitation
current waveforms. Up to this point in the thesis the latter result is taken for granted, so that the proposed contactless system in Chapter 4 does not have an air gap at all, in order to allow for a relatively optimal efficiency. Here a probable misinterpretation of the term “contactless” becomes apparent in that the proposed E-Snake system is electrically-contactless, although primary and secondary electrically isolated system sides do make physical contact.
The E-Snake contactless system consists of a cable where clamps can be attached to or removed from arbitrarily during operation. The system is designed to be functionally equivalent to the current plug-and-socket type of energy distribution in house and office environments. The name stems from the appearance of the main primary cable. Multiple load systems (i.e. clamps) might be attached and released from the E-Snake. High efficiency and flexibility make the system an interesting and serious alternative to the current power distribution architecture in-house or in-office.
The objective of Chapter 5 is to design a control for the power flow through the E-Snake system with the help of input and output power electronic converters. The following three complicating control issues made the control design to differ from what is usually encountered by the power electronics engineer:
Several intrinsic master controllers (i.e., instead of the more common master-slave(s) setup)
‘Blind’ input and output converters Large (effective) power operating range
A control solution is found in the modulation method of quantum conversion combined with sliding mode control theory. Quantum control has first been adopted for the simple reason that the E-Snake system should work as functionally required (i.e., 230 V - 50 Hz output waveform for each clamp). In the end it did work, but it was soon realized that the power efficiency was non-optimal.
The main contribution of Chapter 6 is a new sliding surface for the input converter, which has the property that it adapts its input power to the (unknown) load power demand, and without the need for a communication link.
The idea behind the new control is that the input converter monitors the energy content in the circuit’s resonant tank. The latter tank provides indirect information about the relative energy demand by all clamps together. Unaware of the absolute power demand, the input converter only requires the instantaneous commands ‘more power’ or ‘less power’. The fact that such strategy works is partially due to the quantum conversion modulation, for which the energy direction within each quantum pulse is always unambiguous.
At this point in the thesis, it is remarked that with the new sliding surface the fundamental problem of reduced efficiency in case of a larger air gap still remains. Chapter 7 therefore introduces a control strategy that intends to improve the latter wireless efficiency. In Chapter 3 it was proven that an increase in air gap between two system parts inevitably leads to a decrease in system efficiency, under the (silent) assumption of harmonic excitation electrical waveforms. In Chapter 7 the latter assumption is left behind, resulting in a different and new way of establishing inductive energy transfer.
SAMENVATTING
Dit proefschrift beschouwt contactloze energieoverdracht, toegepast in een huiselijke omgeving. Naar aanleiding van een analyse in Hoofdstuk 1 is het doel om een inductief huiselijk energieoverdrachtssysteem voor meerdere belastingen te onderzoeken en te ontwerpen met een sterk verbeterde flexibiliteit, gezien vanuit de eindgebruiker. Binnen dit ontwerpproces is er speciale aandacht voor het systeemrendement.
Vermogenselektronisch ingenieurs benaderen een probleem vaak via de quasi-statische inductieve kant, terwijl ingenieurs in de telecommunicatie vrijwel alleen geïnteresseerd zijn in de elektromagnetische stralingseigenschappen van antennes. Hoofdstuk 2 schetst een situatie waarbij zowel inductieve als stralingsverschijnselen tegelijkertijd aanwezig zijn, om zo een goed beeld te kunnen vormen van de mogelijkheden en de beperkingen van contactloze energieoverdracht op het gebied van het vermogensrendement. De situatie beschouwt de energetische koppeling tussen twee dipolen. Het resultaat van deze oefening is dat er een brug geslagen wordt tussen beide werelden, inductief en radiatief.
Echter, vanwege het puntbronkarakter van het dipoolmodel is er niet voldoende informatie om een conclusie te kunnen trekken over de algemene eigenschappen van de energieoverdracht. Daarom krijgen de dipolen een reële afmeting in Hoofdstuk 3 om het model compleet te maken. Het hoofdstuk verkent de theoretische rendementslimiet voor een gegeneraliseerd contactloos system als functie van de relatieve luchtspleetlengte, aangenomen dat deze limiet in de eerste plaats op een zinvolle manier kan worden gekwantificeerd. Deze limiet blijkt te bestaan, onder de aanname van een harmonische bronvoeding. Dit resultaat wordt op dit punt in het proefschrift onderkend, zodat
logischerwijs het ontwerpvoorstel in Hoofdstuk 4 helemaal geen luchtspleet heeft, om zo tot een optimaal systeemrendement te kunnen komen.
Het E-Snake contactloze systeem bestaat uit een speciale kabel waarlangs op
elk punt stekkerklemmen kunnen worden aangekoppeld en losgekoppeld tijdens het in bedrijf zijn van het systeem.Het systeem is zo ontworpen dat het functioneel equivalent is aan het huidige huiselijke system van stekkers en stopcontacten. De naam van het systeem volgt uit het uiterlijk van de kabel als slang. Het uitzonderlijk hoge rendement en grote flexibiliteit maken dat het systeem een interessant en serieus alternatief is voor het huidige energiedistributiesysteem in huiselijke omgevingen.
Het doel van Hoofdstuk 5 is om een elektronische regeling te ontwerpen voor de vermogensoverdracht door het E-Snake systeem met behulp van vermogensomzetters in de slang en in de stekkerklemmen. De volgende drie complicaties maken dat het regelontwerp anders is dan wat vermogenselektronisch ingenieurs normaal tegenkomen:
Meerdere intrinsieke meesterregelaars (in tegenstelling tot de meer gebruikelijke meester-slaaf configuratie)
‘Blinde’ ingangs- en uitgangsomzetters Groot effectief vermogensbereik
Een regeloplossing werd gevonden in de basismodulatiemethode quantum conversion, gecombineerd met sliding mode control theory. De keuze viel op quantum conversion, simpelweg omdat het E-Snake system functioneel moest werken. Uiteindelijk werkte de regeling, maar snel werd duidelijk dat het behaalde systeemrendement niet optimaal was.
De hoofdbijdrage van Hoofdstuk 6 is een nieuwe sliding control surface voor de ingangsomzetter, die de eigenschap heeft dat het ingangsvermogen zich automatisch aanpast aan het gevraagde, maar onbekende, uitgangsvermogen. Hiervoor is geen aparte communicatieverbinding nodig tussen de slang en de stekkerklemmen.
Het idee achter de nieuwe vermogensregeling is dat de ingangsomzetter continu de energieinhoud in de resonante tank bekijkt. Deze tank levert
indirecte informatie over de relatieve energievraag van alle stekkerklemmen samen. Zich niet bewust van het absoluut gevraagde vermogen, heeft de ingangsomzetter genoeg aan de signalen ´minder energie´ of ´meer energie´. De reden dat deze strategie werkt is gedeeltelijk te danken aan de quantum conversion, waarbij de richting van de energiepulsen altijd eenduidig is.
Op dit moment in het proefschrift wordt opgemerkt dat het fundamentele probleem van een gereduceerd rendement in het geval van grotere luchtspleet nog steeds niet is opgelost. Daarom introduceert Hoofdstuk 7 een regelstrategie die tot doel heeft om het ´draadloze´ rendement te vergroten. In Hoofdstuk 3 werd geconcludeerd dat het rendement onherroepelijk afnam bij toenemende luchtspleet tussen de twee systeemdelen, onder de aanname van een harmonische bronvoeding. In Hoofdstuk 7 wordt deze aanname omzeild, wat resulteert in een andere en nieuwe manier voor het inductief overdragen van energie.
CONTENTS
1
INTRODUCTION
1
1.1 CONTACTLESS ENERGY TRANSFER ... 1
1.2 APPLICATION CLASSIFICATION FRAMEWORK ... 5
1.3 RESIDENTIAL CONTACTLESS ENERGY TRANSFER ... 9
1.3.1 Conventional residential contactless products ... 12
1.3.2 Classification of proposed residential application... 13
1.4 PROBLEM DEFINITION ... 15
1.5 DESIGN OBJECTIVES ... 16
1.5.1 Efficiency and relative contactless distance ... 16
1.5.2 System simplicity ... 16 1.5.3 Relative optimization ... 17 1.6 THESIS LAYOUT ... 17 1.7 CONTRIBUTIONS SUMMARY ... 19
2
ELECTROMAGNETIC THEORY
21
2.1 INTRODUCTION ... 222.2 ENERGETIC COUPLING BETWEEN DIPOLES ... 27
2.3 DISCUSSION ... 33
2.3.1 Radiative design approach ... 33
2.3.2 Inductive-coupled design approach ... 34
3
INDUCTIVE-COUPLED TRANSFER
39
3.1 INTRODUCTION ... 40
3.1.1 Inductive-coupled contactless system description ... 41
3.2 AIR COIL TRANSFORMER DESIGN ... 44
3.2.1 Air coil transformer volume and geometry ... 45
3.2.2 Inductance calculation ... 46
3.2.3 Parametric energy transfer optimization ... 49
3.2.4 Frequency considerations ... 54
3.3 TRANSFORMER IMPROVEMENT DISCUSSION ... 54
3.4 CONCLUSIONS ... 57
4
E-SNAKE
61
4.1 INTRODUCTION ... 624.2 E-SNAKE DESIGN ... 64
4.2.1 Design objectives ... 64
4.2.2 Equivalent circuit of inductive coupling ... 64
4.2.3 Snake cable design ... 64
4.2.4 Clamp design ... 66
4.3 CLAMP LOSS OPTIMIZATION PROCEDURE ... 69
4.4 E-SNAKE INDUCTANCES ... 73
4.4.1 Inductance versus resistance ... 76
4.5 CONCLUSION ... 77
5
E-SNAKE CONTROL DESIGN
79
5.1 CONTROL PROBLEM ... 805.2.1 LC-type series resonant converter topology ... 82
5.2.2 LLC-type series resonant converter topology ... 85
5.3 DESIGN OF THE PROPOSED CONTROL ... 87
5.3.1 Introduction ... 88
5.3.2 Averaged model of the E-Snake system ... 90
5.3.3 Sliding-mode control design ... 91
5.4 COMPUTER SIMULATED CONTROLLER VERIFICATION ... 96
5.4.1 Input and output controller schemes ... 96
5.4.2 Modification of the output quantum modulation... 99
5.5 CONCLUSION ... 101
6
IMPROVED E-SNAKE CONTROL DESIGN
103
6.1 PROPOSED INPUT SLIDING MODE CONTROL ... 1046.2 DESIGN OF THE PROPOSED CONTROL ... 109
6.2.1 Averaged model of the contactless system ... 109
6.2.2 Stability condition ... 110 6.2.3 Existence condition ... 111 6.2.4 Reaching condition ... 114 6.2.5 Design example ... 115 6.3 EXPERIMENTAL RESULTS ... 116 6.4 CONCLUSION ... 120
7
TAILORED WIRELESS CONTROL
123
7.1 INTRODUCTION ... 1247.2 WIRELESS INDUCTIVE-COUPLED SYSTEM DESCRIPTION ... 124
7.3 CONVERTER TOPOLOGY AND PROBLEM DESCRIPTION ... 127
7.5 PROPOSED CONTROL STRATEGY ... 132 7.5.1 Input converter control design ... 132 7.5.2 Output converter control strategy ... 136 7.6 EXPERIMENTAL RESULTS ... 138 7.6.1 Coupling factor and transferred power improvement ... 138 7.6.2 Harmonic distortion improvement ... 140 7.6.3 Efficiency improvement ... 140 7.7 CONCLUSION ... 141
Chapter 1
INTRODUCTION
This thesis considers efficient contactless electromagnetic energy transfer, applied to residential and office environments. ‘Contactless’ is sometimes replaced by ‘wireless’ or ‘non-contact’, but all words stand for the same: transferring electromagnetic power through a certain volume other than by using electrical conductors.
This chapter provides the background for the thesis organization in Fig. 1.12. This background is essential to point out the main message of the thesis:
A specific contactless design effort is required to reduce the efficiency penalty as encountered in conventional contactless energy transfer systems.
The message states that stepwise building upon conventional knowledge does not likely lead to contactless energy transfer systems that compete in efficiency with conventional non-contactless systems. Instead, it is proposed that technology specifically tailored to contactless energy transfer is required.
1.1 Contactless energy transfer
During my PhD-research the movie “The Prestige” came out (2007), in which one of the movie characters was Nikola Tesla (1856-1943), a famous Serbian physician and inventor. The story suggested the invention of a science fiction
2 Contactless energy transfer
Fig. 1.1. Nikola Tesla’s Colorado Springs experimental site for wireless energy transfer.
human-transportation device by Tesla, but Tesla himself was introduced in the movie while doing a non-fictive experiment: isolated light-bulbs on the ground of his laboratory garden were wirelessly made to lighten-up (i.e., turned on) due to a distant electromagnetic energy source. It may be a surprise to know that Tesla actually performed the experiment successfully at his Colorado Springs laboratory test-site, even with a wireless energy transmitting distance of a couple of miles, see Fig. 1.1 [1].
Despite the (contactless) work of Tesla, nowadays it is standard to transfer energy by means of conductive transfer. Tesla suffered from a funding-stop and a fire that destroyed most of his laboratory and his paper work [1], but also without those circumstances there are clear reasons for the popularity of energy transfer by electrical (current) conduction:
Small loss due to high electrical conductivity Low-cost copper guiding material (used to be) Good tracking of energy flow (i.e., through wires)
Cable networks distribute energy all the way from the power plants up to the homes, industrial areas and office buildings. In-house, a network of cables, ending in wall electric outlets, supplies energy to electrical devices by means of plugs and device cables.
light bulb transmitting tower
Fig. 1.2. Relative contactless energy transfer distance.
There are other ways to transfer energy and well-known devices that use such other forms are transformers and capacitors. Magnetic field is the energy transfer carrier in transformers, while electric field is the corresponding carrier for capacitors. Electromagnetic waves, as applied in wireless information transfer, are another energy carrier.
Bundling the previous alternative ways to transfer energy, contactless in the context of this thesis is defined as the bundle of techniques to transfer electromagnetic energy along a path without using electrical conductors, while transmitter and receiver are both electrical devices. Intuitively, contactless energy transfer is associated by many with a ‘beam of energy’ that is transferred across certain distance. In this light, it is clear that concerning the subject of contactless there are many dreams, but few breakthroughs. In this thesis effort is put into separating dreams from reality by examining the latter collection of contactless techniques. Specifically, this thesis is about opportunities for applying and integrating contactless techniques into the energy distribution network in residential environments.
To have a preliminary general idea what contactless energy transfer is mostly about, there appears to be a single governing fundamental parameter: relative contactless energy transfer distance drel.
uncontrolled area: no energy guidance no deliberate materials d secondary electrical circuit primary electrical circuit Vsource r1 r2 Vload
4 Contactless energy transfer
Fig. 1.3. Concept of an application classification framework.
1 2
max , rel d d r r (1.1)Fig. 1.2 visualizes this measure, which is defined as the length of the uncontrolled transfer path d divided by the largest of the two radii r1 andr2 of the
energy source device and the load device respectively.
An uncontrolled transfer path is defined by the absence of design freedom. No materials can be deliberately put in the uncontrolled area in Fig. 1.2. This implies that the placement of magnetic or electric field guides is not allowed. Uncontrolled also means that the area is freely accessible to people. For example, contactless energy transfer systems in industrial environments are designed to act in a more or less controlled area which is not accessible to ‘unwanted electrical devices’ or people. Therefore, to appreciate certain contactless system design achievement it is sufficient to know the system’s efficiency and (uncontrolled) relative contactless energy transfer distance. The remainder of this chapter shows if and where it is meaningful to apply contactless techniques in the residential energy distribution network.
System 1 Intended functionality Obtained functionality Application System 2 Obtained functionality System n Obtained functionality Which system matches the intended functionality best?
Technique pool
For this purpose a general application classification framework is proposed in Fig. 1.3. In the framework an application is translated into an intended technical functionality.
In most cases a starting point is to have a clear application in mind. An engineer then selects one or more design tools (that are available in a technique pool) and implements a technical system that should resemble the intended functionality of the application. For example, System 1 might represent a modern system based on conventional energy transfer. It is then valuable to know beforehand if a next-generation candidate, System 2, which for example uses a combination of contactless and conventional energy transfer, is likely to achieve an improved functionality. The other way around, it is also possible to start redefining the application. Consequently, the corresponding intended functionality is moderated. System 1 then might suddenly perform poorly and chances for a next-generation System 2 increase. It will turn out that this last approach greatly influences the problem definition of this research in Section 1.6.
1.2 Application classification framework
The aim of this section is to develop the concept of a general classification framework to decide beforehand whether contactless transfer is useful to apply in certain arbitrary situation. The framework is introduced and illustrated with the example application of making a phone call. Although the example significantly differs from applying contactless energy transfer in a residential environment, it however demonstrates the fact that energy transfer across large wireless (relative) distances by itself is not a problem at all, contrarily to efficient energy transfer. With the huge modern success of wireless information transfer it is expected to be simply a matter of interpolation to have wireless energy transfer. This unfortunately is not the case, because the intended functionality from Fig. 1.3 for both information and energy transfer applications is opposite, which fact will be elaborated in the making a phone call example.
Initiating the classification framework build, consider that the primary objective of making a phone call is information transfer, but note that information cannot be transferred without energy. The example has been chosen because it perfectly answers the question why nowadays information transfer is mostly
6 Application classification framework
wireless and the mobile phone’s operational energy however is (still) supplied by batteries.
Translating the call-application in intended functionality leads to: Make a device that connects your phone to another phone Make a device that records your voice
Make a device that transfers the recording to a second person
For all three functions some type of energy transfer is involved. According to Fig. 1.4 there are six conceptual situations in which energy is transferred from a source to a load. Brief examples for each of the concepts are given in the following:
Concept 1: source = battery, load = electrical portable device
Concept 2: source = battery that is being charged somewhere, load = electric device that works on batteries
Concept 3: source = wall electric outlet, load = electric device with an electric cord (and plug)
Concept 4: source = rail track, load = train
Concept 5: no direct example available, but think of a large load in combination with a small source
Concept 6: source = transmitting antenna, load = receiving antenna Assume further that the state-of-art system is a cord-phone call system, which makes use of the concepts 3 and 4. Energy is generated at power plants and transported towards all houses: concept 4. The cord of the cord-phone is plugged into a wall outlet: concept 3. Furthermore, the cord-phone is plugged into the telephone network outlet (again concept 3), and the network itself is made available to all houses (again concept 4).
Then there is an idea for a (next-generation) mobile-phone call system, where the new intended functionality is to communicate more freely. For such system the inclusion of concept 6 seems reasonable in addition to the concepts 3 and 4. At this point Fig. 1.5 presents the proposed concept of an application area to determine the preliminary chances for the new system. Spanning the end-points of the classifier axes given an area that represents the theoretically ideal implementation of an application with full intended functionality. Each system, a
Fig. 1.4. Energy transfer concepts.
real-world implementation of the application, is evaluated to give a score on each classifier axis. Each classifier denotes a technical function that characterizes the application. Naturally the list of classifiers can be extended at will. Connecting the scores obtained with an actual system gives a score area. Repeating this process for each system results in a visual indication of their relative performances. Note that the classification area concept combines in principle incomparable functionalities like for example convenience and safety. The basic configuration consists of normalized axis lengths, but if it is felt that for example safety weighs more than convenience, the relative axis lengths are simply adjusted.
The systems cord-phone call and mobile-phone call have been classified and comparing the areas suggests that a mobile-phone call outperforms the cord phone call. Note that the scores are given on an intuitive basis: it is perfectly
1 2 3 4 5 6 load load source load load source source source load source non-material energy carrier
Energy transfer scheme
source
load
Description
source is always near the load bring the source to the load bring the load to the source
make the source available everywhere
make the load available everywhere
wireless energy transfer
8 Application classification framework
Fig. 1.5. Concept of an application area with 5 classifier axes. Projected are the application scores of the two systems phone call with cord and mobile phone call. A comparison of their respective areas gives an indication of their relative performance.
possible to start with such intuitive prognosis and then to gradually modify the scores during development when more insight is obtained. Stated differently, time information is built into the classifiers. For example, the system architecture classifier in the phone call example considers the difference between available and required phone network architecture. In the early days a dedicated architecture for mobile phones was largely absent and the existing satellite network then was only for military purposes. Therefore, the mobile phones score for the architecture classifier was very low. Indeed, it was not until the architecture had been standardized and largely implemented that the mobile phone gained relatively instant popularity.
One particularly point of interest is that initially there were two independent candidate networks to apply wireless techniques to: the information network and the energy supply network. The information should ideally be available everywhere to be in principle picked up by any mobile phone. The only reason that such concept works in practice is that the energy required to carry the information is relatively small and thus energy transfer efficiency was not a differentiating topic until recently [2]. On the other hand, supplying wireless energy to operate each mobile phone should be more efficient, because of the several orders of magnitude more power that is involved in this case. From Fig.
mobile phone call Safety
Flexibility
Innovation cord phone call Architecture
Fig. 1.6. Cost added as a third dimension to the classification model.
1.5 it is therefore not a surprise that historically the choice has been made to only transfer the information wirelessly, and to leave the operational energy supply for the mobile phones a task for batteries. Regarding batteries, note that their use and the fact that they require regular recharging is responsible for the difference between obtained and intended functionality. From Fig. 1.5 it is indeed clear that, although outperforming the cord-phone call, the current mobile phone system is still not optimal regarding the flexibility classifier. This explains the current research interest in wireless charging platforms for mobile devices [3].
Cost is included in the application area by adding a third dimension in Fig. 1.6, leading to a score volume. A little increase or decrease in the cost classifier has large impact on the volume, representing its major influence. However, the classifier can be easily removed, giving notice to the complexity of the cost aspect. This finalizes the introduction of the classification framework. In the following section the framework is applied to candidate residential contactless systems.
1.3 Residential contactless energy transfer
A general property of contactless energy transfer systems is that electrical source(s) and electrical load(s) are electrically isolated from each other. Contrarily, in a conventional electrical system source and load share the same current loop, which can be opened or closed by means of electrical contacts. Fig. 1.7 presents a schematic of conventional energy distribution in-house. Note
Convenience Cost Architecture Safety Flexibility Innovation
10 Residential contactless energy transfer
Fig. 1.7 Conventional residential energy transfer system.
Fig. 1.8 Impression of residential multiple-load contactless system.
indeed that a current loop is closed with an electric plug. Fig. 1.8 gives an impression of a contactless version of the conventional distribution system. Primary and secondary (wired) current loops remain isolated and energy is transferred by means of a wireless inductive or radiative (antenna) coupling across certain distance d. From the figure it is clear that the terms contactless
and wireless become potentially confusing if d approaches zero. Electrically contactless is the technically correct term, however, in the context of this thesis the meaning of contactless (alone) is considered sufficiently clear. Contactless systems with a relatively large distance d are furthermore denoted as wireless.
voltage control at power plant
utility grid
electrical current flow
meter cupboard electric plugs power conversion power conversion power conversion + V ri load A load load secondary current loop d energetic coupling (wireless) power conversion
primary current loop utility
grid
-Fig. 1.9 Impression of a contactless charging pad on a table.
Due to inherent electrical isolation a contactless technique is applied for one or both of the following reasons:
Increasing the safety classifier score Increasing the flexibility classifier score
The risk of electrocution and gas explosions due to electric sparks is naturally reduced compared to conventional systems. Indeed, a paper that addresses one of the first contactless applications in mines is about safety [4]. Whereas safety is a natural feature of any contactless system, a flexibility improvement is often less easily collected. This is due to its relation with contactless distance d. A larger d in general provides more flexibility, but at the cost of a substantial system efficiency penalty. Modern commercial contactless systems all use an inductive coupling to transfer the energy, of which the working principle is a weakly-coupled transformer [5]-[7].
Analyzing these systems, it is observed that their commercial success is based on the absence of a functionally equivalent conventional type of system. An example of a successful contactless application is the precise positioning of a waver that processes silicon microchips [5]. Energy supply cables attached to the waver disturb its positioning and therefore the minimum microchip size. Higher precision with contactless technology therefore extends the entire functionality of the waver, which clearly results in a much higher score for the (crucial) flexibility classifier. Score reductions of other classifiers (e.g., system efficiency) are then likely compensated for, the same as with the previous
energy supply
coils in a pad laptop y coils PDA mobile phone energy pickup coil(s)
12 Residential contactless energy transfer
Table 1.1 Classification summary questionnaire of charging pad (Fig. 1.9)
Classifier Question
Architecture How to supply the pad itself with energy?
Safety -
Flexibility How much increased compared to cables? Convenience How much increased compared to cables? Innovation -
mobile phone example where the (initial) power efficiency of the information transfer was very poor.
An obvious step thus is to formulate a residential energy distribution application which benefits from increased safety and flexibility. However, a key difference with commercial contactless applications is that energy transfer itself is the main function of in-house energy distribution. There is no additional overruling functionality that speaks for itself, such as precision in the previous microchip waver case. An essential aspect of energy transfer is that it should be efficient, and (unfortunately) the conventional cabled residential distribution network alternative is already highly efficient. The essence is that the inherent strong safety and flexibility aspects of contactless technology cannot compensate for a reduced efficiency in a residential application. This important observation justifies the relevance of the rather extensive classification framework approach. 1.3.1 Conventional residential contactless products
A modern commercially available residential contactless product is a charging pad for mobile electrical equipment [9]. Fig. 1.9 shows a table where such pad is integrated. The idea is to supply energy electrically contactless from the pad to mobile electrical devices that require battery charging [10]. Applying a classification to the charging pad, it follows that safety is not a differentiating issue. Flexibility is somewhat in favour of the contactless pad, compared to a conventional charger. This contactless advantage is maximized if the electronics in the pad is compatible with all types of secondary side electronics (inside the mobile device). It is not a coincidence that currently regulations are underway that standardize power electronic adapters.
Fig. 1.10 Classification of residential energy distribution application.
Another residential contactless product in the prototyping phase is being developed by Witricity Corp. [11]. This product differs functionally from the charging pad by its larger contactless transfer distance, which makes it a typical wireless contactless system. Applying a classification again, this product does only give a higher score for flexibility. However, this comes with the price of reduced energy transfer efficiency, as expected. It is uncertain if the Witricity Corp. contactless technology will eventually reach the market, because efficiency is not easily compensated for in a residential environment.
Table 1.1 summarizes the classification of the charging pad (and Witricity product) by means of asking questions. It is immediately recognized that for example the charging pad itself should be connected to an electric wall outlet, which limits its flexibility. It is proposed that their prospects of success would improve when the applications are extended to include contactless technology in the entire residential energy distribution network. Simply stated, fixed electric wall outlets are designated to be replaced by contactless technology. This result proves the importance of the classification procedure, in which essentially the conventional residential (contactless) applications have been redefined.
1.3.2 Classification of proposed residential application
An extended residential (contactless) application is proposed that considers the energy distribution network from the meter cupboard up to the electric devices. The idea is to add contactless technology in this network, in order to beforehand optimize the impact of its inherent strong safety and flexibility.
14 Residential contactless energy transfer
Fig. 1.10 presents the classification of the conventional residential energy distribution network with electric plugs and wall electric outlets. The core of the residential architecture consists of a fixed network of wiring inside walls and ceilings. Several wiring clusters (only a single one is presented in Fig. 1.7) depart in parallel from the meter cupboard, ending in wall or ceiling electric outlets.
Classifying the conventional architecture, it is noted that safety inherently is a relatively weak aspect. An additional measure to increase the safety is the use of fuses, however, the risk of electrocution is still realistically present. Regarding the classifier control, the energy flow in the conventional architecture is essentially uncontrolled. It is the task of the central power plants to regulate the power, which finally results in the common alternating voltage of 230 V, 50 Hz at our residential meter cupboards. The conventional network (see again Fig. 1.7) has a voltage-parallel structure, which implies that the electrical current is automatically distributed among the loads without (the need for) further power regulation. This elegant simplicity of the electrical architecture is perfect for electrical devices that naturally work with the provided 230 V alternating voltage. However, many external power regulators (adapters) are present in modern homes, because consumer electronic devices mostly require a relatively low, non-alternating (DC) voltage. It is assumed fair to consider these adapters as part of the energy distribution network, which explains the reduced score of the control classifier in Fig. 1.10. Continuing with the flexibility classifier, in a regular room there is typically a (very) limited number of fixed wall electric outlets. Additional wire extensions are often required to make a connection with an electrical device. Furthermore, it is observed that the architecture inside the walls is rather extensive compared to the number of outlets.
Fig. 1.10 presents also the classification of a contactless conceptual version (Inductive-coupled Cables) of the same energy distribution application. The key idea of this concept is to use the entire length(s) of the wiring in the residential network to tap energy from inductively. The principle working is the same as with the charging coils of the pad in Fig. 1.9: electrical current that flows through a primary wire magnetically induces a current in a nearby secondary wire.
Classifying the contactless architecture, safety is obviously a strong aspect. All system elements can be touched without risk, because the wires are electrically isolated. Regarding the classifier control, the energy flow requires control due to the naturally current-series architecture in Fig 1.8. Current circulates in the primary system side, while the loads share this same electrical (induction) current. If not, each load would require its own private energy source. As a result, there is no degree of freedom to provide unequal powers to each load device without active power regulation. Compared to the conventional network with power adapters there is thus hardly a score difference. Continuing with the flexibility classifier, the whole application has been shaped to be beneficial for applying contactless technology. Wiring inside walls can be safely placed inside the room (e.g. in a baseboard), resulting in a virtually unlimited number of inductive energy tapping points.
The result of the classification is a beforehand promising application of contactless technology, which is represented by the significantly larger score area in Fig. 1.10. It is emphasized that the main challenge for the contactless alternative is to have energy efficient energy transfer and power regulation, because additional flexibility and safety is not likely expected to compensate a reduced efficiency.
1.4 Problem definition
Summarizing results from the previous sections, a difference between residential and industrial environments is that in the last there is often a secondary overruling system functionality that compensates for a reduced contactless system efficiency in the actual energy transfer. In the residential environment the conventional energy distribution network is highly power efficient and compensation is not likely. However, the classification framework showed a way out by considering the entire residential energy distribution network, e.g. including adapters. The resulting contactless Inductive Cables concept is, given the further positive classifier scores, designated for research and development according to the following problem definition:
Investigate and design a residential energy distribution network system for multiple electrical devices, which makes use of contactless technology. The
16 Design objectives
network must be able to compete in efficiency with the extended conventional cabled network. Benefits must include a highly improved end-user flexibility and safety.
1.5 Design objectives
This section focuses on individual design objectives that follow from the problem definition. Besides a special focus on system efficiency, further objectives are to provide design values for typical contactless systems, to aim for system simplicity, and to apply a relative optimization procedure.
1.5.1 Efficiency and relative contactless distance
Clearly, a design wish is to achieve high system efficiency. Such wish however only makes sense if the context of the following engineering boundaries:
Restriction to the system dimensions (see (1.5)) Required power rating
Regulations (e.g., electromagnetic environment) These boundaries lead to the following design objective:
To calculate the relationship between relative contactless distance (1.5) and optimal system efficiency as a function of system dimensions and power rating.
Chapters 2 and 3 relate system dimensions, power rating and system efficiency for a generalized contactless energy transfer system. Regulations is a subject of Chapter 4, in which a specific residential system is designed. Chapters 5, 6 and 7 treat which treat a specific contactless system design.
1.5.2 System simplicity
Simplicity of the final contactless system implementation is a second objective. For system cost holds the same as for efficiency: compensation with other contactless features is not likely. Residential energy distribution essentially is a commodity, which should be efficient and cost effective.
1.5.3 Relative optimization
In the previous design objectives a type of optimization is involved. It is therefore relevant to formalize what can be expected from these optimizations. An optimization problem is defined by variables and non-variables that are individually related to each other. This thesis considers only the following 2 types of relative optimization:
Optimization of the relations between variables Optimization by turning non-variables into variables
Stated differently, the mathematical evaluation procedure to find the actual optimum is not a main interest. The idea behind relative optimization is that it automatically leads to a new design guide, where the designers should make their own specific optimization calculations by choosing certain materials and system components. Furthermore, relative optimization explicitly provides building blocks for future relative optimizations. To summarize, the third objective is to relatively optimize efficient contactless technology.
1.6 Thesis layout
Chapters in this thesis are all the results outcome of specific research topics that follow from the research objectives. A layout of these topics is provided in Fig. 1.11.
This chapter introduced the classification framework, leading to a residential application where the impact potential of contactless technology beforehand is relatively optimized.
Chapters 2 and 3 are in-depth treatments of (obtainable) contactless efficiency in relation with design boundaries 1 and 2 in section 1.5.1. The chapters bring the fundamental principles of radiation and induction together in a single example of two opposing dipoles. It is shown that step-wise building upon conventional contactless technology inevitably leads to a reduced efficiency as a function of relative contactless distance. Therefore, in chapter 4 a residential contactless system is proposed that is essentially designed without air gap (i.e., contactless distance ≈ 0).
18 Thesis layout
Fig. 1.11 Thesis layout
This simple measure gratuitously optimizes the inherent contactless efficiency, however, efficient power regulation by power electronic converters is also crucial. Chapters 5 and 6 are devoted to this non-trivial subject.
Parallel to the latter development, the results from chapters 2 and 3 are used to formulate tailored contactless base technology. As a result, a control concept is proposed in Chapter 7 that is specifically intended for efficient wireless contactless energy transfer. Chapter 8 (not presented in Fig. 1.12) concludes the work.
Inductive-coupled system design
(wireless distance ≈ 0)
(residential application)
Wireless system design
Chapter 7 Wireless energy transfer concept Chapter 4 Inductive design Chapters 5 and 6 Control design Engineering (reality) Application (desire)
Physics of wireless energy transfer
Chapter 3 Physics of induction Chapter 2 Physics of radiation Chapter 1 Application classification
1.7 Contributions summary
Contributions of the work are summarized in the following.
Chapter 1 contributes with a classification procedure, which helps to formulate a promising contactless residential application.
Chapter 2 contributes by merging the worlds of radiative and inductive energy transfer.
Chapter 3 contributes with a relationship between efficiency and relative contactless transfer distance for a harmonically excited system.
Chapter 4 contributes with the E-Snake system concept for residential energy distribution.
Chapter 5 contributes with a power regulating controller design for the E-Snake system. However, the resulting system efficiency is suboptimal. Chapter 6 contributes with an efficient control design for the E-Snake system.
Chapter 7 contributes with an unconventional control method for weakly magnetic coupled contactless systems that trades power density for increased power efficiency.
20 Contributions summary
Bibliography
[1] Tesla Memorial Society of New York, http://www.teslasociety.com [2] Energy Aware Radio and network tecHnologies, https://www.ict-earth.eu [3] On-line Electric Vehicle, http://olev.kaist.ac.kr
[4] K. W. Klontz, D. M. Divan, D. W. Novotny, and R. D. Lorenz, “Contactless power delivery system for mining applications,” IEEE Trans. Ind. Appl., vol. 31, pp. 27-35, Jan. 1995.
[5] J. de Boeij, E. Lomonova, and J. Duarte, “Contactless Planar Actuator With Manipulator: A Motion System Without Cables and Physical Contact Between the Mover and the Fixed World,” IEEE Trans. Ind. Appl., vol. 45, pp. 1930-1938, Nov. 2009.
[6] J. Sallan, J. L. Villa, A. Llombart, and J. F. Sanz, ”Optimal Design of ICPT Systems Applied to Electric Vehicle Battery Charge,” IEEE Trans. Ind. Elec., vol. 56, pp. 2140-2149, June 2009.
[7] Xun Liu and S. Y. Hui, “Optimal Design of a Hybrid Winding Structure for Planar Contactless Battery Charging Platform,” IEEE Trans. Pow. Elec., vol. 23, pp. 455-463, Jan. 2008.
[8] M. P. Theodoridis and S. V. Mollov, “Distant energy transfer for artificial human implants,” IEEE Trans. Biomed. Eng., vol. 52, pp. 1931-1938, Nov. 2005.
[9] POWERMAT Technologies, http://www.powermat.com
[10] C. L. W. Sonntag, “Design of a variable-phase contactless energy transfer platform using air-cored planar inductor technology”, Dissertation TU/e 2010.
Chapter 2
ELECTROMAGNETIC
THEORY
Linking induction and radiation
The principles of inductive energy transfer and radiative energy transfer in electrical engineering are almost completely separated worlds. Power electronics engineers mainly use the quasi-static low-frequency inductive approach, while telecommunication engineers are mainly interested in the radiative high-frequency properties of antennas.
This chapter introduces an example situation where both inductive and radiative phenomena are present, in order to get a clear understanding of the possibilities and limitations of contactless energy transfer with regard to efficiency. The example considers the energetic coupling of two dipoles and as a result the concepts of induction and radiation are linked together. However, due to the point-source nature of the dipole model there is not sufficient information to conclude the general energy transfer characteristics. In order to complete the picture, the dipoles get actual sizes in Chapter 3.
Section 2.1 introduces the concepts of induction and radiation. For an example situation of two opposing dipoles in section 2.2 the wireless contactless energy
22 Introduction
Fig. 2.1. The force exerted by a transmitting charge on a receiving charge depends both on their mutual distance and acceleration (~frequency) of the transmitting charge.
transfer characteristics are calculated as a function of dipole excitation frequency. Implications from this exercise are discussed in section 2.3.
2.1 Introduction
This chapter focuses on the analysis of a simple energy transfer scheme consisting of two electric dipoles.
The objective is to observe the energy transfer efficiency as a function of electrical frequency and relative transfer distance. This section provides the mathematical tools to support and evaluate the last scheme. Each individual tool is conventional, but the added value is the connection between radiative and inductive energy transfer.
Fig. 2.1 presents an important aspect of James Maxwell’s famous set of equations that encompass the basic laws of classical electromagnetic theory [21]. This aspect considers that the magnitude of the force between two electrical charges (or charge distributions) does not only reversely depend on
100 MHz
= direction of force arrow size = force magnitude transmitting electric charge q transmitting electric charge q receiving electric charges receiving electric charges 1 kHz
Fig. 2.2 Centre-fed current element, which is a basic model of an antenna.
the distance between them, but also on the acceleration of the charges themselves. It leads directly to the concept of electromagnetic wave radiation. As input, Maxwell’s equations require the exact initial geometry of all charges and materials, together with their initial acceleration and speed. Solving the latter dynamic equations, the result is the new position of all charges at subsequent time instants.
One way of representing Fig. 2.1 in a mathematical equation is an expression for the electric field due to a moving charge q t( ) as a function of relative position, time, acceleration and distance [22]:
' ' 2 ' 2 2 2 2 0 ' 1 , , , 4 ' ' r r r e e d e q r d E x y z t c dt r r c dt
(2.1)Symbols denote the speed of light c (i.e.3 10 8m s/ ), the electric charge strength in Coulomb q, the retarded distance r’ between the point of observation (i.e., at location xyz) and charge q, and the time retarded unit vector er'. This vector is directed from the location of the charge (x0,y0,z0) to the
point of observation (x,y,z). Retardation takes into account the time that the electric field requires to reach the point of observation. Furthermore, time derivatives are associated with the electrical frequency of the charge.
Interpreting (2.1), the time-dependence of q t( )is included in er'. This unit vector changes according to the movements of the charge. The left-most term between brackets shows that the electric field decreases with distance squared.
l
0
( )
0j t
24 Introduction
Fig. 2.3. Field components r, and
, for a point p(x,y,z) on a sphere.The middle-term compensates for the retardation in the first term in case of low frequencies. The right-most term proves the existence of electromagnetic radiation, because it depends only on the acceleration of the charge and not on the distance between the point of observation and the charge. Note that the electric charge acceleration should be in the order of the speed of light c to give the term some weight compared to the other terms.
If (2.1) is applied to a particular charge distribution, for example to the short electrical current-carrying centre-fed element in Fig. 2.2, the result is written as (2.2)-(2.4) [23].
0 2 3 0 cos 1 2 j t r r q le j E cr r
(2.2)
2 0 2 2 3 0 sin 1 4 j t r q le j E c r cr r
(2.3)
0 2 sin 1 4 j t r I le j H cr r
(2.4)The current I0 is equal to the time derivative of the charge q0 divided by the
length l of the element. The exponential powers incorporate the
time-z r x y p ϕ θ
retardation of both current and charge. The E- and H-field subscripts r, and
as shown in Fig. 2.3 denote respectively radial, inclination (z-axis-angle) and azimuth (x-axis-angle) components of the electric and magnetic field strengths. The constant
(i.e.,2
) is the wave number, where λ (i.e.,2 c
) is thewavelength of the electromagnetic field, due to a current with angular frequency
ω. A relation between energy transfer and (2.2)-(2.4) is provided by the Poynting vector [24]:
, , ,
, , ,
, , ,
S x y z t E x y z t H x y z t (2.5)
It is possible to directly substitute (2.2)-(2.4) in (2.5), however, it is considered useful first to clarify the terms inductive and radiative energy transfer. The quasi-static inductive approximation assumes a relatively low angular frequency (i.e.,
/
c
1
) and is typically applied in power electronics engineering. The approximation implies that all terms containing c are omitted:0 3 0 0 3 0 0 2 cos( ) 2 -sin( ) 4 ( 1 ) cos( ) 2 r q l E r near field q l inductive approximation E r r c I l H r
(2.6)Note from (2.6) that both E and H field strengths rapidly fall off as a function of transfer distance r, which motivates the use of the term near-field inductive approximation. It is observed that Eθ and Hφ are both perpendicular and
anti-perpendicular to each other (dependent on time), which means that the Poynting vector is directed partially towards the source and outwards. Furthermore, the radial component of the electric field Er introduces a rotational
component in the Poynting vector. This rotational component in Fig. 2.4 explains the fact that energy can be temporarily stored in a magnetic or electric field.
26 Introduction
Fig. 2.4. Field explanation for energy storage in a magnetic field: the reactive energy flow that is generated by a dipole at the origin is subsequently radial outward, rotational and radial inward.
Opposite to the induction principle, telecommunication engineers use the principle of electromagnetic radiation. EM-waves are transmitted first into space by an antenna.
A receiving antenna then is supposed to capture passing waves. In telecommunication applications EM-waves are modulated in frequency or amplitude to contain information. EM-waves are characterized by a high-frequency carrier wave (i.e., /c1) and typical transmitting distances are many times the wavelength of the wave (i.e., r). Under these conditions the radial electric field of (2.2) vanishes, as well as two of the three terms in the tangential electric field (2.3). The same holds for the magnetic field intensity vector H in (2.4). This leads to the far-field radiative approximation [23]:
0 0 -30 sin sin 4 ( 1 ) j t r j t r far field I l radiative E j e r approximation I l H j e r r c
(2.5)In the far-field approximation the E and H field components are perpendicular to each other. The Poynting vector is therefore directed outward, thus with only a radial component. This implies that energy that leaves the source will not return again, which is contrary to the inductive approximation.
energy flow radial components
energy source
(e.g. the current element in Fig. 2.2)
Fig. 2.5. Adjacent electric dipoles.
Summarizing both principles, the idea of inductive energy transfer is to put a load in the radial and rotational energy flow from Fig. 2.4, while in radiative transfer the energy is released into space by a transmitting antenna and caught by a receiving antenna.
2.2 Energetic coupling between dipoles
The objective in this section is to obtain insight in the concept of wireless energy transfer related to the concepts of induction and radiation. For this reason a simple energy transfer system geometry is chosen, in other to enable analytical mathematical solutions. The selected geometry consists of two opposing dipoles and the problem space is assumed vacuum without the presence of materials. An electric dipole is a basic model of two opposing electrical point charges of equal magnitude and opposite polarity. The building block of an electric dipole is often used to characterize or synthesize a physical antenna (i.e., multiple dipoles). On the other hand, the dipole can be seen as a charged capacitor. To transfer energy to a dipole an electrical source should be connected to the dipoles by means of leads. The electrical frequency of this source (related to the distance in between the poles) determines the dominant behavior of the dipole as capacitor or as radiating antenna.
z
+
-+
dipole 1 dipole 2 x d28 Energetic coupling between dipoles
A dipole can be described by equations (2.2). Indeed, if the charges at the poles of the dipole change in time and the distance between the poles is negligible, then from the outside it is as if a (displacement) current flows. The following power flow calculations between the adjacent dipoles are a shortened version of the material as presented by Maeda and Diament in [24].
First, the Poynting vector that describes the time-averaged interaction between the dipoles 1 and 2 is given by:
1 1 1 2 2 1 2 2
1 Re 2
S E HE HE HE H (2.6)
Besides two self-terms (with equal subscripts) there are two terms that describe the interaction between the dipoles. To solve (2.6) equations (2.2)-(2.4) are transformed into the Cartesian xyz coordinate system by using:
1 1 1 1 1 1 2 2 2 2 2 2 2 2 r r x z d d E x E z E z E x E E d d x z x z (2.7)
The transformations hold for the plane y=0 and for dipole 1. Note the absence of an y-component in the electric field. The magnetic field intensity vector in the Cartesian coordinate system is obtained in a similar way. At the plane y=0 there is only a single H-component:
1y 1
H H (2.8)
For the second dipole the transformation is similar but with different orientation:
2 2 2 2 2 2 2 2 2 2 2 2 2 2 r r x z d d E x E z E z E x E E d d x z x z (2.9)
