Applied Science, Faculty of
Engineering, School of (Okanagan)
DSpace
UBCO
Wenngren, Suzanne Elizabeth
Master of Applied Science - MASc
University of British Columbia
An analysis of a capacitive wireless power transfer system is presented to investigate the relationship between spatial impedance variation between the transmitter and receiver. The capacitively coupled link is modeled by an equivalent circuit that includes all coupling paths between the four conductors which make up the link. The final model is formed using an observational tuning approach, beginning with a simple model and iteratively adding complexities introduced by non-idealities until good agreement is shown between experimental and simulated results. Seven resonator configurations are designed to resonate at the ISM band frequencies of 13.56 MHz, 27.12 MHz, and 40.68 MHz, and at critical coupling distances of 10 cm, 20 cm and 30 cm to investigate the impact of each parameter on efficiency and spatial variation. The analysis shows that the behavior of the link over a spatial variation is generally independent of frequency and critical coupling distance, and only the peak efficiency at critical coupling is notably impacted. Measurement results from all seven configurations are shown to verify the analysis. The results indicate that for applications with high power transfer efficiency demands, a lower frequency and smaller critical coupling distances are preferred.
ANALYSIS AND DESIGN OF CAPACITIVELY COUPLEDRESONATORS FOR WIRELESS POWER TRANSFER SYSTEMAPPLICATIONSbySuzanne Elizabeth WenngrenB.ASc., The University of British Columbia (Okanagan), 2014A thesis submitted in partial fulfillment of the requirements forthe degree ofMaster of Applied ScienceinThe College of Graduate StudiesElectrical EngineeringThe University of British Columbia(Okanagan)February 2019c© Suzanne Elizabeth Wenngren, 2019 The following individuals certify that they have read, and recommend to the College of Graduate Studies for acceptance, a thesis/dissertation entitled: Analysis and design of capacitively coupled resonators for wireless power transfer system applications. submitted by Suzanne Elizabeth Wenngren in partial fulfillment of the requirements of the degree of Master of Applied Science. Dr. Thomas Johnson - School of Engineering Supervisor Dr. Stephen OâLeary - School of Engineering Supervisory Committee Member Dr. Liwei Wang - School of Engineering Supervisory Committee Member Dr. Yang Cao - School of Engineering University Examiner Additional Committee Members include: Dr. Wilson Eberle - School of Engineering Supervisory Committee Member iiAbstractAn analysis of a capacitive wireless power transfer system is presented to investigate therelationship between spatial impedance variation between the transmitter and receiver. Thecapacitively coupled link is modeled by an equivalent circuit that includes all coupling pathsbetween the four conductors which make up the link. The final model is formed using anobservational tuning approach, beginning with a simple model and iteratively adding com-plexities introduced by non-idealities until good agreement is shown between experimentaland simulated results. Seven resonator configurations are designed to resonate at the ISMband frequencies of 13.56 MHz, 27.12 MHz, and 40.68 MHz, and at critical coupling distancesof 10 cm, 20 cm and 30 cm to investigate the impact of each parameter on efficiency andspatial variation. The analysis shows that the behavior of the link over a spatial variation isgenerally independent of frequency and critical coupling distance, and only the peak efficiencyat critical coupling is notably impacted. Measurement results from all seven configurationsare shown to verify the analysis. The results indicate that for applications with high powertransfer efficiency demands, a lower frequency and smaller critical coupling distances arepreferred.iiiLay SummaryThe objective of this work is to determine any favorable conditions for transferring powerover a wireless power link. Typically, wireless power systems of this nature are reservedfor static applications. The investigations of this thesis reveal the usefulness in dynamicapplications. This work will contribute to the understanding of this sector of resonant powertransfer systems, offering explanations for some of the behavior of these systems. This workintroduces a method for modeling the system and verifying conclusions with experimentalresults. These results can provide insight to future scholars who may be seeking to makesimilar design decisions for their systems.ivPrefaceAll of the work presented in this thesis was completed under the supervision of Dr. ThomasJohnson, and was co-supervised by Dr. Stephen OâLeary, at the School of Engineering withinthe University of British Columbia.Part of the high-level system modeling in Chapter 4 is primarily based off of the workpublished by S. Wenngren, A. Clements, and T. Johnson, âCapacitively coupled resonatormodels for investigating spatial impedance variation in WPT systems,â presented at theWireless Power Transfer Conference, Montreal, QC, Canada, 2018. I worked collaborativelywith Dr. Johnson on this paper to derive the resonator model and the equation for thequality factor.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Inductive Power Transfer Systems . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Capacitively Coupled Power Transfer Systems . . . . . . . . . . . . . . . . . 31.3 Research Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Overview of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Modeling a Set of Resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1 The Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.1 The Capacitance Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 8vi2.2 Modeling the Inductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.1 Model 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2.2 Model 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2.3 Model 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.3 Modeling the Balun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4 Modeling the Resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.5 Coupling Two Resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 Resonator Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.1 Classes of Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2 Designing for Different Critical Coupling Distances and Frequencies . . . . . 363.2.1 Using the Transformer Turns Ratio to Set the Critical Coupling Distance 363.3 Resonator Designs For Different Frequency Bands . . . . . . . . . . . . . . . 463.3.1 Designs at 13.56 MHz . . . . . . . . . . . . . . . . . . . . . . . . . . 463.3.2 Designs for 27.12 MHz . . . . . . . . . . . . . . . . . . . . . . . . . . 513.3.3 Designs for 40.68 MHz . . . . . . . . . . . . . . . . . . . . . . . . . . 513.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.1 The Test Bench . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.2 Frequency Response for Critical Coupling . . . . . . . . . . . . . . . . . . . . 564.3 Results Over a Spatial Bandwidth . . . . . . . . . . . . . . . . . . . . . . . . 604.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71viiList of Tables2.1 Summary of element values for model 1. . . . . . . . . . . . . . . . . . . . . 142.2 Summary of element values for model 2. . . . . . . . . . . . . . . . . . . . . 182.3 Summary of element values for model 3. . . . . . . . . . . . . . . . . . . . . 223.1 Balun properties at 13.56 MHz. . . . . . . . . . . . . . . . . . . . . . . . . . 473.2 Inductor properties at 13.56 MHz . . . . . . . . . . . . . . . . . . . . . . . . 503.3 Balun properties at 27.12 MHz. . . . . . . . . . . . . . . . . . . . . . . . . . 513.4 Inductor properties at 27.12 MHz. . . . . . . . . . . . . . . . . . . . . . . . . 513.5 Balun properties at 40.68 MHz. . . . . . . . . . . . . . . . . . . . . . . . . . 513.6 Inductor properties at 40.68 MHz. . . . . . . . . . . . . . . . . . . . . . . . . 524.1 Comparison of Simulation and Experimental Results . . . . . . . . . . . . . 604.2 The relative bandwidth at which there is an attenuation of 3 dB for eachfrequency band. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61viiiList of Figures1.1 A typical IPT configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1 A single resonator: (a) the electrode side and (b) the matching circuit. . . . 72.2 A simplified overview of the resonator components. . . . . . . . . . . . . . . 72.3 Capacitively coupled conductors. . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Equivalent circuit model capacitances as a function of d. The online versionis available in color. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.5 Single inductor soldered to SMA connectors for a 2-Port measurement. . . . 112.6 Model 1 circuit layout for y-parameter analysis. . . . . . . . . . . . . . . . . 122.7 Schematic of model 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.8 Imaginary part of y-parameters for model 1. . . . . . . . . . . . . . . . . . . 152.9 Real part of y-parameters for model 2. . . . . . . . . . . . . . . . . . . . . . 162.10 Schematic of model 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.11 Imaginary part of y-parameters for model 2. . . . . . . . . . . . . . . . . . . 172.12 Real part of y11 and y22 for model 2. . . . . . . . . . . . . . . . . . . . . . . 182.13 Comparison of the imaginary parts of y11 and y22 for models 2 and 3. . . . . 192.14 Inductor model 3 versus measurement results: a comparison of the real partof the two port y-parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . 202.15 Imaginary y11 comparison between models 2 and 3 and the correspondinginductor measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21ix2.16 Imaginary y22 comparison for models 2 and 3, and the corresponding inductormeasurement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.17 Comparison between an ideal and non-ideal modeled balun and the balunmeasurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.18 Full model of a single resonator. . . . . . . . . . . . . . . . . . . . . . . . . . 242.19 Simulation versus measurement of a single resonator. . . . . . . . . . . . . . 252.20 Comparison of real and imaginary S-parameters for a single resonator. . . . 252.21 Model of the complete WPT system. . . . . . . . . . . . . . . . . . . . . . . 262.22 Smith chart S21 measurement versus simulated model at 5 cm. . . . . . . . . 272.23 |S21| measurement versus simulated model at 5 cm. . . . . . . . . . . . . . . 272.24 Smith chart S21 measurement versus simulated model at 10 cm. . . . . . . . 282.25 |S21| measurement versus simulated model at 10 cm. . . . . . . . . . . . . . . 282.26 Smith chart S21 measurement versus simulated model at 35 cm. . . . . . . . 292.27 |S21| measurement versus simulated model at 35 cm. . . . . . . . . . . . . . . 293.1 An overcoupled case at 13.56 MHz. . . . . . . . . . . . . . . . . . . . . . . . 323.2 A critically coupled case at 13.56 MHz. . . . . . . . . . . . . . . . . . . . . . 333.3 An under coupled case at 13.56 MHz. . . . . . . . . . . . . . . . . . . . . . . 343.4 S-parameters of an over coupled case at 13.56 MHz. . . . . . . . . . . . . . . 353.5 S-parameters of a critically coupled case at 13.56 MHz. . . . . . . . . . . . . 353.6 S-parameters of an under coupled case at 13.56 MHz. . . . . . . . . . . . . . 363.8 Reducing C11, C22, and C12 to a single parallel capacitance. . . . . . . . . . . 373.7 Lumped element model of a single resonator. . . . . . . . . . . . . . . . . . . 373.9 Rearranging the single resonator model to a series equivalent form. . . . . . 383.10 General transformation from a series to parallel network. . . . . . . . . . . . 393.11 Final equivalent parallel network of the single resonator. . . . . . . . . . . . 403.12 |S11| for different balun turns ratios, n, inputs for critical coupling at 20 cm. 43x3.13 |S21| for different balun turns ratios, n, inputs for critical coupling at 20 cm. 443.14 The n values returned for various RL inputs. . . . . . . . . . . . . . . . . . 453.15 Visual representation of the open circuit (a) and short circuit (b) testing ofthe balun. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.1 The configuration of a pair of resonators mounted in foam guides: (a) viewedfrom the side, and (b) in profile view. . . . . . . . . . . . . . . . . . . . . . 554.2 Measured transmission characteristics for capacitively coupled resonator. fora frequency of 13.56 MHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.3 Measured transmission characteristics for capacitively coupled resonators fora frequency of 27.12 MHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.4 Measured transmission characteristics for capacitively coupled resonators fora frequency of 40.68 MHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.5 A high level model of the capacitively coupled resonators. . . . . . . . . . . . 594.6 A high level model of the capacitively coupled resonators with the completecapacitance matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.7 Experimental results of spatial sensitivity normalized to the critical couplingdistance at 13.56 MHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.8 Experimental results of spatial sensitivity normalized to the critical couplingdistance at 27.12 MHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.9 Experimental results of spatial sensitivity normalized to the critical couplingdistance at 40.68 MHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.10 Comparison between all frequencies at a critical coupling distance of 10 cm. 654.11 Comparison between all frequencies at a critical coupling distance of 20 cm. 65xiList of AbbreviationsAbbreviation DefinitionWPT Wireless power transferIPT Inductive power transferCPT Capacitive power transferEMI Electromagnetic interferenceAC Alternating currentDC Direct currentRF Radio frequencyISM Industrial, scientific, and medicalVNA Vector network analyserSMA SubMiniature version AxiiAcknowledgementsI would like to first extend my great thanks to my primary supervisor, Dr. Thomas Johnson,for taking me on as his student when knowing little about me, other than I had an interest inthe topic of wireless power. My conversations with Dr. Johnson served as great motivationand fueled my inspiration throughout my graduate degree, and continue to do so to this day.Thank you to my co-supervisor, Dr. Stephen OâLeary, for providing continuous feedback andsupport and for the role he played in aiding my application to graduate studies.I would like to thank my industry partner, Solace Power, for providing a set of theirlab resonators for me to test out all of my designs on, as well as for their ongoing supportthroughout my graduate program.Additionally, I would like to express my gratitude towards a particular two professorswho fostered my interest in engineering and graduate studies. Namely Dr. Yang Cao andDr. Kenneth Chau, thank you for your support.Lastly, I would like to thank my friends and family who have always supported me pur-suing my education throughout the years. In particular, my husband, Wilhelm, who alwayspushed me on in those moments of doubt and for constantly providing me with comic relief.xiiiChapter 1IntroductionIn the course of their professional practice, electrical engineers aim to create technologicalinnovations that will simplify and improve daily life. The emerging area of wireless powertransfer (WPT) systems is an example of an innovation that will contribute to the convenienceof deploying electrical systems by eliminating the need to power devices through an electri-cal power cord. Other benefits include the elimination of a mechanical connection, which issubject to wear and corrosion that eventually limits the reliability of the wired power con-nection. WPT technology also opens up new ways to power and recharge implanted medicaldevices [1â4], embedded sensors [5â7], and nanotechnology devices [8, 9].While WPT technologies may appear to be a recent phenomenon, the idea has beenaround for over a century, dating back to the 1890âs when Nikola Tesla first invented theTesla coil. Tesla used the principle of resonance to maximize the power transferred betweenthe coils. His design consisted of a coil of wire and a capacitive device, which, when excited,could exchange very high voltages between coils [10, 11]. Although Tesla may have beenahead of his time, his work provided a foundation for many scholars to build their workupon.In more recent times, a paper in 2007 published by several professors at MIT [12], restartedthe investigation into the feasibility of implementing wireless power transfer systems. In their1paper they showed that a 60 W light bulb could be powered wirelessly over a distance ofseveral meters. A key insight in their work was the concept of employing resonance to transferpower between two weakly coupled coils. Since 2007, there has been a surge in the pursuit ofapplications from powering consumer electronic devices to charging electric vehicles. Thereare also emerging standards to enable interoperability between different manufacturers.As wireless power research has progressed, near field power transfer systems have beeninvestigated using both inductive coupling and capacitive coupling. There are advantagesand disadvantages for each method and these are discussed in more detail to follow.1.1 Inductive Power Transfer SystemsInductive power transfer (IPT) systems have been the focus of significant research and thereare many examples in the literature of systems that can transfer power over a wide range ofdistances from centimeters to meters [13,14] and from low power systems in the watts rangeto high power systems transferring kilowatts of power [15â18]. In an IPT system, powertransfer is coupled through the magnetic field and the system typically consists of two ormore coils separated by air gaps. The coils are usually tuned to be resonant at a specificfrequency that is used to transfer the power. An example of an IPT system is shown inFig. 1.1. A voltage source, VS, provides DC voltage to an inverter, which converts the sourcevoltage to an AC radio frequency current that drives the transmitting coil. The receivingcoil is placed at some distance, d, away from the transmitting coil. Current induced in thereceiving coil is rectified, then passed to a load, usually a battery, for energy storage.Safety issues related to the intensity of field levels is a significant constraint that mustbe considered in the design of WPT systems [19]. Given safety regulations on field intensity,as well as compliance with electromagnetic interference (EMI) regulations, brings one toquestion what type of wireless power system would be best for a specific application. Forexample, if a high power kilowatt level system is required for electric vehicle charging, would2dInverter RectifierTx Coil Rx CoilVSLoadSourceFigure 1.1: A typical IPT configuration.IPT be a better choice compared to capacitively coupled power transfer (CPT) systems?There are many factors to consider including gap distance, power, and frequency and thesetrade-offs will influence the conclusions on which coupling method to choose.There are other distinctions between IPT and CPT which are important and may lead toa preferred selection. Since IPT systems use magnetic field coupling, any stray metal in theair gap, or surrounding the resonators, will contribute to eddy current losses and reduce thepower transfer efficiency [20â22]. Therefore, IPT systems are highly sensitive to metal objectsand power transfer though a metal enclosure is not easily implemented [23]. Shielding of themagnetic field to control the surrounding magnetic field intensity, or to control the shape ofthe coupled field, is usually implemented using ferrite shielding [24â27]. Ferrite shielding isheavy and bulky, which may be undesirable in certain applications.1.2 Capacitively Coupled Power Transfer SystemsMost recently, interest in CPT systems has grown as a way to address IPT limitations,particularly with regard to operating in the presence of metal objects. As an example of theimprovement in power transfer efficiency obtained with CPT compared to IPT, the authorsin [28] showed a power transfer efficiency increase from 4% with IPT to 70% with CPT fora case where the transmitter is enclosed in a metal box. CPT has demonstrated that it is3less prone to generating unacceptable levels of EMI [29â31], and that it is also capable oftransferring power in the kilowatt range [32â34]. The authors in [35] were able to achievea power transfer of 1.88 kW with 85.87% efficiency at a seperation distance of 15 cm usingtheir four plate design. An anaylysis in [36] suggests that CPT systems have the potentialto reach hundreds of kilowatts of power transfer.CPT systems use electric field coupling between capacitively coupled plates. The capaci-tance which links the transmitter and receiver is dependent on the surface area of the platesand the distance separating the plates, as in Eq. (1.1). The reactance of the coupling capaci-tance also depends on frequency and at very low frequencies, reactance is high, making powertransfer difficult. Therefore, CPT systems are typically operated at higher frequencies [37,38]than IPT in order to reduce the coupling reactance. The disadvantage of operation at higherfrequencies is a reduction in the power efficiency of the radio frequency (RF) circuits whichimplement the inverter and rectifier circuits. Therefore, there is a trade-off in the overallsystem efficiency that needs to balance the efficiency of the CPT resonators, as well as theefficiency of the RF power circuits.C =Ad(1.1)1.3 Research GoalsThe goal of this thesis is threefold. The primary objective is to study how the impedance of aCPT system changes with distance and to determine whether a CPT system can provide anyadvantage over IPT in terms of misalignment. Secondly, each set of designs was studied overthree frequencies within the industrial, scientific, and medical radio band (ISM band). Thethree frequencies which were selected are 13.56 MHz, 27.12 MHz, and 40.68 MHz. Withinthe realm of power electronics, it is a commonly known phenomenon that as the frequency4increases in transistor-based circuits, the switching losses experienced in these transistorsalso increases [39, 40], affecting the systems overall performance. Despite this relationshipbetween efficiency and frequency, this work investigates the effect of frequency on impedancemismatching and the resulting efficiency. Lastly, out of all the designs presented in thework, is there a single design which can provide the best performance over the greatestrange of separation distance and misalignment between transmitter and receiver. The termâspatial bandwidthâ is introduced in this work to describe the range of distances at whichthe performance of the link will maintain predetermined threshold levels. This is perhapsthe most provocative topic, given that this work was motivated by applications where thereceiving resonator is not fixed, but rather hovering in free space.1.4 Overview of ThesisThis thesis is organized into the following chapters. Chapter 2 is a detailed account ofhow the system of resonators was modeled. The model is then simulated and comparedwith experimental measurements of the resonators. Chapter 3 gives an overview of eachcase of the resonator designs and how each configuration was derived. In Chapter 4, thetest bench set up and the experimental environment are discussed. Then, the experimentalmeasurements for each configuration from Chapter 3 are presented. Finally, Chapter 5 willdiscuss the results of each configuration and what they mean, and opportunities for futurework are proposed.5Chapter 2Modeling a Set of ResonatorsIn this chapter, we work through the development of an equivalent circuit model for twocapacitively coupled resonators. Initial work in the modeling of the resonators was madeusing a design which was provided by an industry sponsor (Solace Power) for this researchproject. The resonators were designed to resonate at a frequency of 13.56 MHz. Eachresonator consists of two electrodes, which are etched onto a printed circuit board, anda matching network that consists of inductors and a balun. A picture of the resonators,and corresponding matching circuit, is shown in Fig. 2.1. The equivalent circuit model ismotivated by the interest in having a circuit model that can be used in circuit simulationtools, which are used to design the inverter and rectifier circuits. A circuit model of the link,which is parameterized in terms of a spatial variable that relates to the air gap between thetransmit and receive resonators, can then be used to analyze the end-to-end system efficiencyof a wireless power link as a function of spatial variation. Spatial variation changes the linkimpedance significantly, and this in turn significantly affects the efficiency of the inverter andrectifier circuits.6(a) (b)1 2L1L2T1Figure 2.1: A single resonator: (a) the electrode side and (b) the matching circuit.L1L2T1L4L3T21 32 4dTransmitter RecieverFigure 2.2: A simplified overview of the resonator components.2.1 The MethodThe physical design of the metal electrodes in the capacitive link are modeled in an electro-magnetic simulator (COMSOL) and used to extract an equivalent circuit capacitance matrix.Referring to Fig. 2.2, the capacitive link is made up of conductors 1, 2, 3, and 4. The ca-pacitances vary as a function of spatial separation and can be used to parameterize thecapacitance matrix values. Once the equivalent circuit capacitances for the electrodes are7found, models are extracted for the inductors, L1 through L4, and baluns, T1 and T2, used inthe matching circuits for the resonators. Is is expected that, due to symmetry in the design,L1 = L2 = L3 = L4, and, similarly, T1 = T2. The inductors and baluns are wound on ferritecores and high frequency inductor models are required to accurately model the character-istics of the resonators. Different model levels of the matching circuit are constructed andcompared. After developing models for the resonators, simulation results, using the circuitmodels, are compared with experimental results.2.1.1 The Capacitance MatrixIn order to determine how the net capacitance of the resonators changes as a function ofseparation distance, d, a lumped element model was derived, based on the resonators. Thepair of resonators contain a total of four conductors, two on the transmitter, and two onthe receiver. All four conductors are identical rectangular conductors with a width of 5 cmand a height of 25 cm, similar to the conductor shape depicted in Fig. 2.3. Each pair ofconductors, 1 and 2 for the transmitter, and 3 and 4 for the receiver, are separated by adistance, s, equal to 10 cm. The conductors are aligned along the y-axis and the distance, d,along the x-axis is varied. The distance between the resonators is referred to as the capacitivelink. The capacitively coupled link shown in Fig. 2.3 consists of the four conductors and thecorresponding equivalent circuit capacitance matrix is a four by four matrix:C =ï£®ï£¯ï£¯ï£¯ï£¯ï£¯ï£¯ï£¯ï£°C11 C12 C13 C14C21 C22 C23 C24C31 C32 C33 C34C41 C42 C43 C44ï£¹ï£ºï£ºï£ºï£ºï£ºï£ºï£ºï£». (2.1)The capacitive network is reciprocal, where C = CT, and Cij = Cji, for all i 6= j.8C22C24C121234C13C34C11 C33C44C23C14xyzd1234Earth ground(a) (b)Tx RxsFigure 2.3: Capacitively coupled conductors.The capacitances between the conductors in Fig. 2.3(a) can be represented by an equiv-alent lumped element circuit model, shown in Fig. 2.3(b). The capacitances in the circuitmodel depend on the physical structure of the conductors as well as the spatial relationshipbetween the conductors, and a field solver can be used to determine the circuit capacitances.Symmetry relations can significantly reduce the number of variables in the capacitancematrix. If all four conductors are identical, and, if the conductors are all the same distancefrom a common earth potential, as shown in Fig. 2.3(a), then there is physical symmetry inboth the x and y directions. Under these conditions,C11 = C22 = C33 = C44 (2.2)C12 = C34, (2.3)(2.4)andC13 = C24. (2.5)Assuming the symmetry conditions, the equivalent capacitance circuit model in Fig. 2.3(b)has the form of a balanced symmetric lattice, where the link capacitances, C13 and C24, areidentical, and the cross-link capacitances, C14 and C23, are identical.90 5 10 15 20 25 30 35 400246810Separation Distance (cm)Capacitance (pF) C11C12C13C14Figure 2.4: Equivalent circuit model capacitances as a function of d. The online version isavailable in color.COMSOL was used to simulate the physical structure and extract equivalent circuit ca-pacitances. The corresponding capacitances are shown in Fig. 2.4. Note that the capacitancessatisfy the symmetry conditions in Eqs. (2.2)-(2.5).Since it is not obvious how all the capacitances in the circuit model affect the overallimpedance of the capacitively coupled link, a series of simplified models, of increasing com-plexity, are evaluated to gain insight into design of capacitively coupled link.2.2 Modeling the InductorAs shown earlier in Fig. 2.2, the matching circuit for the CPT system as two series inductorsto resonate with the link capacitance. Forming a high frequency model for the inductorswas the next step. To perform this analysis, an inductor model, derived from experimentalmeasurements, was used. A single inductor was separated from the resonator, and an SMAconnector was soldered onto each end of the coil, with a wire grounding the pins betweenthem, as shown in Fig. 2.5. The inductor position was oriented such that it was suspended infree space, as far off the test bench as the VNA cables would allow, to reduce any magnetic10Figure 2.5: Single inductor soldered to SMA connectors for a 2-Port measurement.coupling to any surrounding metal in order to minimize stray capacitance. Next, the VNAwas calibrated to make a two port measurement over a frequency range of 10-100 MHz. Oncethe measurements were acquired, the data was uploaded into a software tool called AWR forreview and analysis.The first model under consideration (Fig. 2.6) was based off a general lumped elementmodel of a non-ideal inductor. The model includes interwinding capacitance and resistancedue to core losses and high frequency resistance. Two-port measurements are made over awide frequency range and element values in the model are determined from the measureddata. An extraction technique, based on admittance, or y-parameters, was used to find the11C2C1C3RpLI1 I2V1 V2Figure 2.6: Model 1 circuit layout for y-parameter analysis.element values. The method is described next. These parameters are very useful to describethe electrical behavior of networks. The y-parameter matrix for a two-port network, as inour case, is described in Eq. (2.6). Performing a short-circuit test on the output, V2, Eq. (2.7)is realized.ï£®ï£¯ï£° I1I2ï£¹ï£ºï£» =ï£®ï£¯ï£° y11 y12y21 y22ï£¹ï£ºï£»ï£®ï£¯ï£° V1V2ï£¹ï£ºï£» (2.6)andy21 =I2V1â£â£â£â£V2=0. (2.7)Using the model shown in Fig. 2.6, Eq. (2.8) is evaluated to give:y21 = â(jÏC1 +1jÏL+1Rp)= âj(Ï2C1Lâ 1ÏL)â 1Rp. (2.8)Solving for the real components of the equation first, an expression using Re {y21} was12evaluated to solve for the value Rp, as in Eq. (2.9). Im {y21} was used to evaluate the valuesof L and C1, as in Eq. (2.10).Re {y21} = â 1Rp(2.9)Im {y21} = âÏC1 + 1ÏL(2.10)Noting that there are two unknowns in Eq. (2.10), one can take a look at the results forIm {y21} in Fig. 2.8. It can be seen that the inductor has a self resonant point at approxi-mately f=103 MHz. Using this frequency, Eqs. (2.11)-(2.13) can be derived based on the localslope at that point in order to solve for the inter-winding capacitance of the inductor. Theslope of the line can be determined by taking the derivative of Im {y21} as seen in Eq. (2.11).Since the slope is found local to the resonant point at Ïo, Eq. (2.12), a substitution for Ï canbe made, resulting in Eq. (2.13), where Ï = 2pif .d (Im {y21})dÏâ£â£â£â£Ï=Ïo=d(âÏC1 + 1ÏL)dÏâ£â£â£â£â£Ï=Ïo,= âC1 â 1LÏ2o, (2.11)Ïo =1âLC, (2.12)and13d (Im {y21})dÏâ£â£â£â£Ï=2pif= âC1 â 1L 1LC1= â4piC1. (2.13)A similar analysis was used to find the values for C2 and C3, using Im {y11} and Im {y22}respectively, as derived for Im {y11} below in Eqs. (2.14) and (2.15). All values derived formodel 1 can be seen in Table 2.1.y11 =I1V1â£â£â£â£V2=0= jÏ (C1 + C2) +1jÏL+1Rp(2.14)andIm {y11} = Ï (C1 + C2)â 1ÏL(2.15)Table 2.1: Summary of element values for model 1.Element ValueL 14.8 µHC1 0.16 pFC2 1.8 pFC3 3.5 pFRp 167 kâ¦14C2C1C3RpLFigure 2.7: Schematic of model 1.Figure 2.8: Imaginary part of y-parameters for model 1.2.2.1 Model 1Using the previously derived values, the model was simulated in AWR (Fig. 2.7). Comparingthe measured results for both imaginary (Fig. 2.8) and real (Fig. 2.9) y-parameters with thesimulations of model 1, it was apparent that although the values are very close, somethingwas still missing from the lumped element model.As seen in Fig. 2.8, the simulation traces of the imaginary components are almost anexact match with the measured values. This is encouraging, however when looking at theresults for the real portions in Fig. 2.9, it can be seen that the simulated results do notfollow the same overall characteristics as the measured values. Model 1 must be modified insome way that will only affect the characteristics of the real y-parameters, while leaving the15Figure 2.9: Real part of y-parameters for model 2.imaginary parameters unchanged. The shunt capacitors are the only elements that do notcontribute to the values of the imaginary components. Therefore it stands to reason that theaddition should be made there. Note that Fig. 2.9 shows that the real part is negative insome regions, which is not possible. The noisy appearance of the measured traces is due tothe VNA attempting to show a zero measurement. This trace indicates that the uncertaintyof the y-parameter values is approximately ±2Ã 10â5.2.2.2 Model 2Model 2, shown in Fig. 2.10, replaces the ideal shunt capacitors in model 1 with lossy capac-itors. Adding a resistor in series with each shunt capacitor improved the correlation betweenexperimental and simulated measurements. With some tuning of the resistor values, thetrace now resembles the dominant characteristics of the measured values seen in Fig. 2.11.As expected, this modification to the model has a negligible affect on the characteristics ofthe imaginary traces seen in Fig. 2.12. An updated summary of values for each element inmodel 2 can be seen in Table 2.2.16C2C1C3RpLRC2RC3Figure 2.10: Schematic of model 2.Figure 2.11: Imaginary part of y-parameters for model 2.17Figure 2.12: Real part of y11 and y22 for model 2.Table 2.2: Summary of element values for model 2.Element ValueL 14.8 µHC1 0.16 pFC2 1.8 pFC3 3.5 pFRp 167 kâ¦RC2 20â¦RC3 5â¦2.2.3 Model 3While the second model provides a good fit with the experimental measurements of a singleinductor from the resonator, the discrepancy in the values for C2 and C3 was a cause forfurther consideration. Based on the physical inductor design, one would expect that thestray capacitances would be symmetric, which is not supported by the results of model2. Recalling the measurement set up from the previous section, it was realized that theadditional port extension was not included in the initial calibration of the VNA, which islikely the cause for the additional capacitance observed on port 2. In order to test this18Figure 2.13: Comparison of the imaginary parts of y11 and y22 for models 2 and 3.conjecture, the inductor was removed from the SMA connectors and the measurements wererepeated for just the port extension. Once the results were uploaded into AWR, it becameclear that the SMA connectors certainly had some fixture capacitance, which would be inaddition to the stray capacitance of the inductor. The value of each fixture capacitancewas extracted from the y-parameter data of each SMA connector. The capacitance of port1 and port 2 was found to be 0.80 pF and 2.36 pF, respectively. Note that port 2 has asignificantly higher capacitance than port 1 due to the additional adapter. These resultssupport the reasoning that the discrepancy between C2 and C3 in model 2 is due to thispost-calibration port extension. Fig. 2.13 shows the results for the imaginary y-parametercomponents of model 3. The red traces represent the reflection measurements at each end ofthe inductor, prior to the de-embedding. The green trace represents the contribution of theport 2 extension that contributes to the increase experienced by y22. The blue traces showthe values of the imaginary components at each end of the inductor, once the effects of theSMA connectors and port extension have been eliminated. These results are consistent withwhat is expected for an inductor.In adjusting the values of the shunt capacitors, it is seen in Fig. 2.14 that the values19Figure 2.14: Inductor model 3 versus measurement results: a comparison of the real part ofthe two port y-parameters.for RC2 and RC3 have increased relative to model 2. The green trace shows that the SMAconnectors contribute virtually no real part to the overall values measured. Therefore theeffects of any resistance seen in the measurement are due to the losses in the inductor. Usinga tuner, the resistance is found to be around 50 â¦ at each end.Fig. 2.15 compares the reflection measurement of the inductor and the simulations. Model2 may align more closely with the measurements, however. after de-embedding it is clearthat the measurement of the inductance, independent of the SMA connector contribution,will behave more closely to the cyan trace.As a final verification of model 3, a comparison of y21 for models 2 and 3 and the actualmeasurements is shown in Fig. 2.16. The transmission measurement of model 3 is a nearlyperfect match to the measurement results. A summary of element values for model 3 is shownin Table 2.3.20Figure 2.15: Imaginary y11 comparison between models 2 and 3 and the corresponding in-ductor measurements.Figure 2.16: Imaginary y22 comparison for models 2 and 3, and the corresponding inductormeasurement.21Table 2.3: Summary of element values for model 3.Element ValueL 14.8 µHC1 0.16 pFC2 1.0 pFC3 1.12 pFRp 167 kâ¦RC2 55â¦RC3 50â¦2.3 Modeling the BalunModeling work for the balun first began by considering an ideal transformer model. Therequired turns ratio for an ideal balun is determined by the source, RS, and the load, RL,resistances. In this design, the source and load resistances are 50 â¦ and the correspondingturns ratio is 0.5, as see through Eq. (2.16). A comparison of the ideal balun model comparedto the measurement results is shown in Fig. 2.17.n =12âRSRL(2.16)A comparison of the ideal balun model compared to the measurement results is shown inFig. 2.17. The ideal balun operated as expected, placing S11 as a single point in the centerof the Smith chart, as can be seen by the blue trace seen in Fig. 2.17.Based on the measurement data, the model for the balun must include some non-idealinductance in addition to the ideal balun. This is a logical conclusion, given that each turnon the core of the balun creates some inductance based on the properties of the chosencore material. A second look at the VNA confirms that the balun exhibits an equivalentinductance, 2Lb, of 0.49 µH. Using the tuning feature in AWR, the inductance correspondingto a result that lined up best with the measured data, in terms of overall shape and arc length22Figure 2.17: Comparison between an ideal and non-ideal modeled balun and the balun mea-surements.of the trace, is a total inductance of 2Lb = 0.42 µH. AddingLb2in series at each output ofthe balun yields the blue trace seen above in Fig. 2.17.2.4 Modeling the ResonatorsWith an accurate model of a single inductor and balun realized, the next step is to incorporatethis into a model of a complete resonator. The final step in modeling the resonator is to putall of the individually modeled components together as they are configured on the resonator.The equivalent circuit can be seen in Fig. 2.18. From left to right, we have the balun model,the non-ideal inductance of the balun, the two full inductor models in parallel, leading to thecapacitance matrix at the end, representing the electrodes.The results from this simulation are found to be a very good match with the measuredresults. Very minor tweaking was required in order to align the traces. The only change23C22C11LbLbC2C1C3RpLRC2 RC3C2C1C3RpLRC2RC3C12Figure 2.18: Full model of a single resonator.made from the previously found values is an increase in the second shunt resistance, foundat the output of each inductor in series with the stray capacitance, from a previous 55 â¦ to118 â¦. The corresponding results can be seen in Fig. 2.19 and 2.20.24Figure 2.19: Simulation versus measurement of a single resonator.Figure 2.20: Comparison of real and imaginary S-parameters for a single resonator.25LbLbC2C1C3RpLRC2 RC3C2C1C3RpLRC2 RC3LbLbC2C1C3RpLRC2RC3C2C1C3RpLRC2RC3C22 C24C121234C13C34C11 C33C44C23C14Figure 2.21: Model of the complete WPT system.2.5 Coupling Two ResonatorsTo complete the modeling for this system, the last step is to capacitively couple two resonatorsso as to ultimately realize the system as a WPT link. At this point, all of the componentsof the system are modeled, with the exception of the capacitive link between the resonators.The complete model of two resonators coupled with this link can be seen in Fig. 2.21.Implementing the lumped element equivalent of the above capacitor network providedsome promising results. A comparison of the resonators at three distances is studied forexperimental confirmation. Results are shown for distances of 5 cm (Fig. 2.22 and 2.23),10 cm (Fig. 2.24 and 2.25), and 35 cm (Fig. 2.26 and 2.27).As observed in the comparisons, the simulated results show good agreement with themeasured results. It is also noted that as the distance between the resonators increases,the measurements experience some asymmetry. This was later discovered to be due to theresonators not being properly matched when measurements were taken for this analysis,which leads to the asymmetry. When the separation distance increases, the stray capacitancesbegin to dominate over the inter-electrode capacitance, thus making any asymmetries thatare present more apparent.26Figure 2.22: Smith chart S21 measurement versus simulated model at 5 cm.Figure 2.23: |S21| measurement versus simulated model at 5 cm.27Figure 2.24: Smith chart S21 measurement versus simulated model at 10 cm.Figure 2.25: |S21| measurement versus simulated model at 10 cm.28Figure 2.26: Smith chart S21 measurement versus simulated model at 35 cm.Figure 2.27: |S21| measurement versus simulated model at 35 cm.292.6 Chapter SummaryIn this chapter, we iteratively derived a complete model for a pair of capacitively coupledresonators. We began with an analysis of the capacitive network between the resonators. Aseparate model for each component was formed using both analysis and a visual inspectionof simulated results. Finally, all of the components were put together to complete a modelfor the full system.A low level high frequency model for an inductor was derived and verified with simu-lation results. It was determined that the measurement fixture can contribute additionalcapacitance to a network, which is important to take into consideration for high frequencymodels.30Chapter 3Resonator DesignsIn their original state, as provided by Solace Power, the resonators were observed to have acritical coupling distance at approximately 29 cm at 13.56 MHz, and they were designed tomatch at an impedance of 20 â¦. Given the equipment available in the RF Lab, it would bemore useful to have the resonators matched to a 50 â¦ impedance. To achieve this, the mostcommon solution is to add an LC matching network. However, adding a matching networkwould result in the addition of more components to the otherwise very simple resonatordesign. Bearing simplicity in mind, it was realized that by changing the turns ratio of thebalun, the resonant point of the trace could be shifted along the real axis of the Smith chart,changing the matching impedance as a result. The method of obtaining the optimal turnsratio will be covered in depth in the upcoming sections.3.1 Classes of CouplingThroughout the matching design formulation, we were able to identify the three cases of cou-pling, depending on the distance between the resonators. When the resonators are relativelyclose together, the S21 trace on the Smith chart developed a loop within the outer circle, asseen in Fig. 3.1.31This type of result is known as over coupling. Separating the resonators further, theloop becomes smaller until it is non-existent, leaving a perfectly formed carotid, as seen inFig. 3.2. The cusp of the carotid is centered exactly at the r = 1 contour of the Smith chart,with no imaginary impedance, along the x = 0 line, which lays horizontally in the centerof the chart. This result indicates that at this point the resonators are critically coupledat the desired output impedance, which is the ideal situation for maximum power transferefficiency. Increasing the distance between the resonators to any point past critical couplingresults in a loss of definition in the cusp of the carotid, shown in Fig. 3.3, eventually resultingin a perfect circle. This region is known as the under coupled region.Figure 3.1: An overcoupled case at 13.56 MHz.32Figure 3.2: A critically coupled case at 13.56 MHz.33Figure 3.3: An under coupled case at 13.56 MHz.In terms of the log magnitude results for the case of S11, the over coupled cases have twodistinct peaks, separated by a flattened region, as seen in Fig. 3.4. The distance between thepeaks becomes greater as the resonators become increasingly over coupled. The frequenciesat which these peaks occur are synonymous with the frequencies where, on the Smith chart,the trace crosses the purely real axis.As the resonators come closer together, they reach the point of critical coupling. At criticalcoupling, the two peaks from the over coupled case merge together to form one defined peak,as seen in Fig. 3.5. The ideal design will place the desired resonant frequency at the tip ofthe peak.Under coupled cases result in both a decrease in the amplitude, and a softening of thesharp peak of the critically coupled case, as seen in Fig. 3.6.Measurements for S21 follow a similar pattern, where over coupled results show a cleftin an otherwise flat response, which would be observed at critical coupling, and the under34coupled result shifts the whole trace down in decibels, as seen in Figs. (3.4 through to 3.6),inclusive.Figure 3.4: S-parameters of an over coupled case at 13.56 MHz.Figure 3.5: S-parameters of a critically coupled case at 13.56 MHz.35Figure 3.6: S-parameters of an under coupled case at 13.56 MHz.3.2 Designing for Different Critical Coupling Distancesand FrequenciesIn this project, resonators were designed to be matched for critical coupling at distancesof 10 cm, 20 cm, and 30 cm. In addition to the multiple critical coupling distances, threedifferent frequencies of interest, 13.56, 27.12, and 40.68 MHz, will also be examined for eachcase.3.2.1 Using the Transformer Turns Ratio to Set the Critical Cou-pling DistanceAs mentioned earlier in Chapter 3, changing the turns ratio of the balun can serve as amatching network at any given distance. By the same logic, one can change the criticalcoupling distance by manipulating the turns ratio as well, for the same impedance, without36C22C12C11(a)C12 +C112(b)Figure 3.8: Reducing C11, C22, and C12 to a single parallel capacitance.dramatically affecting the resonant frequency. Given that the equivalent capacitance of thelink changes with distance, the resulting change in inductance required can be made upfor by changing the turns ratio. For example, moving from a critical coupling distance of27 cm to 20 cm, will result in an increase in the capacitance of the link. From the relationshippresented in Eq. (2.12), it is known that to hold the same resonant frequency when increasingthe capacitance, the inductance must decrease in order to compensate. This can be done byincreasing the turns ratio of the transformer. The analysis begins with the model of a singleresonator, which was developed in Chapter 2. Using a single resonator, one can omit theinfluence of the cross capacitance experienced between each pair of electrodes when the tworesonators are facing one another. This allows the capacitor matrix to be simplified down toone reactance in parallel with a load resistance, RL. For the purposes of this analysis, theinductor model has been simplified to exclude the inter-winding capacitance, wire resistance,and parallel resistance to ground, as shown in Fig. 3.7.LC11C22C12C1C2RLFigure 3.7: Lumped element model of a single resonator.As seen in the following diagrams, the inter-electrode capacitance, C12, can be considered37L C1C2RLC12 +C112L C12RLC12 +C112(a)(b)Figure 3.9: Rearranging the single resonator model to a series equivalent form.combined in parallel with the individual self-capacitances for each electrode, C11 and C22,has to ground. Given that C11 and C22 are intrinsically connected to the same node throughearth ground, they can be considered in parallel with one another. Additionally, the case ofsymmetry makes it so that C11 = C22, therefore making their parallel equivalent equal toC112, as shown in Fig. 3.8.C12 â C112= C12 +C112(3.1)Having simplified the inter electrode capacitance and the electrode capacitance to ground,shown in Eq. (3.1), the circuit can be redrawn, as shown in Fig. 3.9. To further simplify,the parallel inductors, L, and electrode capacitance to RL, C1 and C2 can be moved to oneside, thus creating a series network with XS and RL as shown in Fig. 3.9. Using Eqs. (3.2)through to (3.8), a parallel equivalent circuit can be formed, as seen in Fig. 3.11.38L XsRsC12 +C112LXp RpC12 +C112(a)(b)Figure 3.10: General transformation from a series to parallel network.Xp =Xs(Q2 + 1)Q2(3.2)whereQ =RpXp=XsRs(3.3)andRp = Rs(Q2 + 1) (3.4)39LRpC12 +C112 + CpFigure 3.11: Final equivalent parallel network of the single resonator.Xs =1jÏC12= â 2ÏC1(3.5)RS = RL (3.6)Cp = â 2C1R2LÏ2 + 4C1(3.7)andRp =4RLC21R2LÏ2+RL (3.8)Now that the circuit is in this parallel configuration, Cp can be combined with the pre-viously derived Eq. (3.1) by Eq. (3.9). This creates a simplified parallel circuit, as shown inFig. 3.11.40(C12 +C112)â Cp =(C12 +C112)+ Cp (3.9)The final step required in order to obtain the pieces necessary to find the ideal turns ratiois to do one last conversion of the most recent parallel circuit to a series RLC circuit. Thisis derived through Eqs. (3.10) through (3.14).RS2 =RpX2pR2p +X2p(3.10)andXS2 =R2pXpR2p +X2p(3.11)whereXp = â 1Ï(C12 +C112+ Cp) (3.12)and Rp is given by Eq. (3.8)XS2 =XpQ2Q2 + 1(3.13)and41RS2 =RpQ2 + 1(3.14)We now have an expression for RS2 in terms of the equivalent capacitive matrix of theelectrode for a single resonator. When the two resonators are coupled, the turns ratio, n,required for critical coupling between them, can be determined from Eq. (3.15).n =âRs2Rs=turns on secondaryturns on primary(3.15)Lastly, the series inductance needed for critical couping is given by Eq. (3.16).L =12Ï2oCs(3.16)4210 15 20 25 30 35 40Separation Distance (cm)-40-35-30-25-20-15-10-505|S 11| (dB)n = 0.435n = 0.687n = 0.970n = 1.364n = 1.575n = 1.886n = 2.700Figure 3.12: |S11| for different balun turns ratios, n, inputs for critical coupling at 20 cm.The method described above will yield a unique turns ratio for any RL. Given that RLis not a physical element in the resonator design, there is freedom to adjust it to provideoptimal results. Now that there are two resonators involved in the system, the effects of crosscapacitance, which was neglected in the early stages of this analysis, are apparent. However,by increasing the free variable, RL, the system can be desensitized to the impact of the crosscapacitance. For example, in the case of critical coupling for 20 cm at 13.56 MHz, if a valueof 200 â¦ is input for RL, the series of equations will output a turns ratio of 0.4346, whereasif 4 kâ¦ is input for RL, the turns ratio is 1.886. Fig. 3.14 graphically depicts the relationshipbetween various RL inputs and the resulting n values returned.Although it is possible to achieve critical coupling at the desired frequency for all valuesof RL, both S11 and S21 vary greatly over spatial changes. Fig. 3.13 and 3.12 show that4310 15 20 25 30 35 40Separation Distance (cm)-40-35-30-25-20-15-10-505|S 21| (dB)n = 0.435n = 0.687n = 0.970n = 1.364n = 1.575n = 1.886n = 2.700Figure 3.13: |S21| for different balun turns ratios, n, inputs for critical coupling at 20 cm.440 2000 4000 6000 8000 10000 12000 14000RL (ohms)00.511.522.53nFigure 3.14: The n values returned for various RL inputs.45there is clearly an optimal n value that will provide the greatest spatial bandwidth, whichin the case of critical coupling for 20 cm at 13.56 MHz is n = 1.575. Fig. 3.12 shows thatwhen n = 1.575, the spatial bandwidth is maximized while S11 maintains a value of lessthan -30 dB. Fig. 3.13 also shows the greatest spatial bandwidth when n = 1.575, whilemaintaining excellent efficiency.3.3 Resonator Designs For Different Frequency Bands3.3.1 Designs at 13.56 MHzUsing the method outlined in the previous section, the turns ratio needed for the first designunder test, the case of a 20 cm critical coupling distance, was found to be n = 1.4. Similarly,the inductance needed to maintain 13.56 MHz over this separation distance was found tobe 14.4 µH. With the simulations providing satisfactory results, it is time to move to thebench for experimental verification. All inductors and baluns involved in the design seriesare made in house by the researcher. First, an appropriate core material was selected forthe given frequency and then the toroid size was determined. As an example, material 67from Fair-Rite Products Corp. was selected for the designs at 13.56 MHz due to the stablelinearity of the complex permeability about the frequency. The size of the core depends onthe inductance per turn of wire, known as the Al value, for the material and number of turnsneeded to acquire either the desired turns ratio, in the case of the baluns, or inductance, inthe case of the inductors. For example, if a 20 µH inductor is needed, it would take manywindings on an Al = 1 nH/turn core, so the size of the core must accommodate the numberof turns. In the example above, the core would need to be very large, which is rarely practicalin hardware designs, especially at high frequencies.Building the baluns for each design is relatively straightforward. Once the ideal turnsratio and appropriate core material is determined, the number of windings on the primary and46secondary side are chosen to achieve the desired turns ratio. Since it is possible to achievea specific turns ratio with multiple combinations of turns on the primary and secondarywindings, the most convenient combination is selected. For the case of n = 1.4, the amountof windings chosen is 14 on the secondary and 10 on the primary, thus satisfying Eq. (3.15).A summary of the turns ratio required for each design at 13.56 MHz are summarized inTable 3.1.Table 3.1: Balun properties at 13.56 MHz.Critical Coupling Distance Core Type No. Turns n Wire Gauge(cm) (p:s) (AWG)10 140-67 6:22 2.6 2020 140-67 9:16 1.4 2030 140-67 14:12 0.9 20In Table 3.1, it can be seen that the actual number of turns in the third column donot agree exactly with the expectations given by Eq. (3.15). For example, in the case forn = 1.4, the balun is designed to have 14 turns on the secondary and 10 on the primary,but this was adjusted to 16 on the secondary and 9 on the primary. Based on those turns,the expected turns ratio would be approximately 1.8, which is significantly higher than thedesign. This discrepancy is due to the coupling coefficient between the primary and secondarycoils. An ideal transformer, as in the simulated design, has a coupling coefficient, k = 1. Thiswould mean that one hundred percent of the load on the primary side is transformed to thesecondary side of the transformer. Based on the expected and actual number of windings,the k value for the 20 cm design at 13.56 MHz is approximated to be 0.72.An analysis was performed on the baluns from the 13.56 MHz design critically coupledfor 10 cm. First, the primary terminals of the balun were connected to a port on the VNAand the secondary terminals were left as an open circuit, as in Fig. 3.15(a). A single portmeasurement was taken and the open-circuit inductance of the balun, L1, was measured tobe approximately 3.16 µH at 13.56 MHz. Performing the short-circuit test, as in Fig. 3.15(b),47the short circuit inductance, L2, was measured to be approximately 0.623 µH.Using the relationship described in Eq. (3.17), the final equation of the series inductancecan be derived for when the impedance across the secondary winding of the balun is equalto zero. This is done by short-circuiting the secondary impedance, seen by the primary, sothat ZS = 0.ZP = jÏL1 +(ÏM)2Z2 + jÏL2= jÏL1 +ÏM2jL2= jÏ[L1 â M2L2](3.17)In this case, the actual inductance on the primary coil, Lâ²1, can be found by subtractingL2 from L1, resulting in Lâ²1 = 2.52 µH. The mutual inductance between the primary andsecondary coils can then be found to be 9.80 µH. Finally, the coupling coefficient, k = 0.89,can be found through Eq. (3.18).k =MâLâ²1L2(3.18)While the k value derived is greater than the expected value of 0.72, the analysis verifiesthat the flux linkage between the primary and secondary turns of the balun is not ideal,justifying the greater number of turns required to achieve the desired turns ratio. Thediscrepancy between the calculated and expected value indicates that the analysis requiresaccounting for any second order affects on the balun such as the inter-winding capacitance.However, an analysis of this depth falls beyond the scope of this thesis.48P S P S(a) (b)Figure 3.15: Visual representation of the open circuit (a) and short circuit (b) testing of thebalun.L = N2Al (3.19)N =âLAl(3.20)The inductance found to match the first design was found to be 14.4 µH. A ferrite coremade of material 67 was selected, due to its properties at the desired frequency. The numberof turns can be found using Eq. (3.19), where L is the desired inductance, Al is the value ofnH/turn specific to the core material, and N is the number of turns.Referring to the datasheet, the Al value for core material 125-67 is 39 nH/turn. Byrearranging to solve for N , as shown in Eq. (3.20), it is found that 19 turns will providean inductance of 14.4 µH. A summary of the inductors made for each design is shown inTable 3.2.49Table 3.2: Inductor properties at 13.56 MHzCritical Coupling Distance Core Type No. Turns Measured Inductance Wire Gauge(cm) µH (AWG)10 125-67 16 11.75 1820 125-67 18 14.27 2030 125-67 18 14.80 20Although the simulation results suggest that the inductance can remain fixed, while stillmaintaining the resonant frequency, the experimental results do differ. The discrepancy isdue to the non-ideal inductance contributed by the windings of the transformer to the overallinductance of the resonator. Referring to Table 3.1, a trend can be observed between thebalun turns ratio and the inductance at a given critical coupling distance. As the turnsratio increases, the number of turns on the secondary coil also increases. The increase insecondary turns increases the self inductance of the secondary winding. Since there is agreater inductance contributed by the transformer, the inductor in the matching networkmust decrease to compensate.When building the 30 cm design for 13.56 MHz, it was observed that due to link ca-pacitance decreasing so much that the self-capacitance between each capacitor and groundbecomes dominant, the matching network becomes very difficult to practically implement.The matching network was so sensitive to changes in inductance, that even a slight movementof either the positioning of the inductors on the resonator, or even moving an individual turnon the coil of wire on the inductor, would cause a large mis-match in the results. As well asbeing difficult to match, when a match was achieved, the results at this point, which will becovered in a later chapter, were so inefficient that it became an overall impractical design forour particular application. Multiple attempts were made to construct a 30 cm critically cou-pled experimental design at 27.12 MHz, however, the design proved to be even more sensitivethan the 13.56 MHz design. Therefore a design for 30 cm was not implemented.503.3.2 Designs for 27.12 MHzThe components for the 27.12 MHz designs were built using the same process outlined in theprevious sections. The summary of the components used for the baluns and inductors foreach case can be seen in Tables 3.3 and 3.4.Table 3.3: Balun properties at 27.12 MHz.Critical Coupling Distance Core Type No. Turns n Wire Gauge(cm) (p:s) (AWG)10 140-67 7:18 1.8 2020 140-67 9:10 1 20Table 3.4: Inductor properties at 27.12 MHz.Critical Coupling Distance Core Type No. Turns Measured Inductance Wire Gauge(cm) µH (AWG)10 87-68 14 1.85 2020 87-68 17 3.38 203.3.3 Designs for 40.68 MHzThe balun and inductors used in the 40.68 MHz designs are summarized in Tables 3.5 and3.6.Table 3.5: Balun properties at 40.68 MHz.Critical Coupling Distance Core Type No. Turns n Wire Gauge(cm) (p:s) (AWG)10 87-68 5:18 1.5 2220 87-68 9:10 0.8 2251Table 3.6: Inductor properties at 40.68 MHz.Critical Coupling Distance Core Type No. Turns Measured Inductance Wire Gauge(cm) µH (AWG)10 87-68 14 1.85 2020 87-68 17 3.38 20523.4 Chapter SummaryThis chapter describes the methodology used to match each design to 50 â¦ for critical couplingat various distances. Each coupling class is discussed, and examples for each case are shownon both the Smith chart and in log magnitude of the S-parameters. The balun turns ratiowas determined to be the most critical design element, and the method of determining theexact ratio is analytically derived. Finally, a summary for each design is presented in tableform.53Chapter 4ResultsIn this chapter, experimental results are shown to characterize the spatial sensitivity of acapacitively coupled wireless power transfer system. The resonators used to construct thelink were described in Chapter 3, and a test bench was constructed to evaluate the efficiencyof the wireless power link for three difference frequencies of 13.56 MHz, 27.12 MHZ, and40.68 MHz. Details of the test bench and the experimental methodology are described, andthen, the spatial sensitivity measurements are shown for the different frequency bands.4.1 The Test BenchThe experimental test bench used to measure the performance of the capacitively coupledwireless power transfer link is shown in Fig. 4.1. The capacitive link consists of two identicalresonators, separated by a distance, d. The resonators are mounted on a high density foamcut from foam insulation. The relative permittivity of the foam is approximately 1, and thepresence of the foam does not significantly affect the characteristics of the power transfersystem. Slots are cut into the foam in one centimeter increments in order to adjust thespacing between the transmitter and receiver resonators. The resonators are arranged facingeach other and the spatial variation measured in this work consists of a one dimensional54change along the x-axis, as shown in Fig. 4.1(a).(a)(b)xyzFigure 4.1: The configuration of a pair of resonators mounted in foam guides: (a) viewedfrom the side, and (b) in profile view.A 2-port calibration was performed on the VNA to the ends of the connecting cables.Twelve inch semi rigid cables were added to the connecting cables to provide extra lengthto accommodate varying the separation distance, which were not included in the calibration.The resonators were used as a link in a larger system, which included a class E inverter andrectifier. A cable was necessary to connect the transmitting resonator to the inverter and55the receiving resonator to the rectifier. To simplify the modeling required for both set ups,the twelve inch cable was later included in the model of the resonators, so that a 50 â¦ matchwas made with the cables connected.4.2 Frequency Response for Critical CouplingOne goal of this thesis was to determine if the value of the critical coupling distance wouldimpact the spatial bandwidth. To investigate this, the frequency response was comparedfor different critical coupling values for each frequency band. The experimental results areshown in Figs. 4.2-4.4.Figure 4.2: Measured transmission characteristics for capacitively coupled resonator. for afrequency of 13.56 MHz.56Figure 4.3: Measured transmission characteristics for capacitively coupled resonators for afrequency of 27.12 MHz.Figure 4.4: Measured transmission characteristics for capacitively coupled resonators for afrequency of 40.68 MHz.57Beginning with the resonator designs at a frequency of 13.56 MHz, Fig. 4.2 shows that thequality factor, Q, of the matched resonators increases significantly when the critical couplingdistance increases. For the 10 cm results, the absolute bandwidth is more than ten times thatof the 30 cm results. This pattern is observed across all three of the resonant frequencies.To look further into why this is the case, a model of the coupled resonators can beformed. For simplicity, this model does not include the full model of the inductors or the crosscapacitance terms. In a capacitively coupled link with a small gap between the transmit andreceive resonators, the conductors can be arranged to provide strong direct coupling betweennodes 1 and 3 and nodes 2 and 4. Assuming C13 and C24 are the dominant capacitances inthe model, a capacitively coupled link can be modeled with the circuit shown in Fig. 4.5. Inthis circuit, series inductors, Lm, are selected to resonate with the gap capacitances, C13 andC24, and input and output baluns transform the source and load resistances. The equivalentcircuit for the link can be reduced to a series resonant circuit, as shown Fig. 4.5(b). Theresonant frequency for the wireless power link is Ïo = 1/âLmC13, and the Q of the resonatoris adjusted by the turns ratio of the baluns. Assuming that the source and load resistancesare equal (RS = RL), the loaded resonator Q is given by Eq. (4.1).Q =1Ïo (2N2RL)C13(4.1)If we want to design a capacitively coupled link that reduces impedance sensitivity tospatial variation in C13, then a low Q circuit is desirable. In the frequency domain, thereactance function of a low Q circuit is less sensitive to frequency change than in a highQ circuit. In a WPT system operating at a constant frequency, the reactance function isdependent on spatial variation rather than frequency variation, and similar to the frequencydomain analogy, a reactance function that is less sensitive to spatial variation is desirable.58Eq. (4.1) shows that by transforming the source and load resistances up (N > 1), the Q ofthe series resonator decreases. The equation also shows why there is interest in increasingthe frequency of the wireless power link. Since the physical capacitances are dependent onlyon the geometry of the link, as the frequency is increased, Q decreases, which in turn reducesthe sensitivity to a spatially varying capacitance.C241234C13Lm/2k = 1N : 1RLN2RLk = 11 : NVSRSTransmitter (Tx) Link (Gap d) Receiver (Rx)N2RSLm/2Lm/2 Lm/2C13/22LmN2RLN2RS(C24 = C13)(a)(b)Figure 4.5: A high level model of the capacitively coupled resonators.The model from Fig. 4.5 is modified to include the full capacitance matrix discussed inChapter 2, as shown in Fig. 4.6, while keeping the high level balun and inductors. This modelwas simulated, and it was found that there was good agreement between the simulated andexperimental results. A comparison summary between all the experiments and accompanyingsimulations is given in Table 4.1.59Table 4.1: Comparison of Simulation and Experimental ResultsFrequency d N Lm/2 (µH) BW (MHz) N Lm/2 (µH) BW (MHz)(MHz) (cm) Exp. Exp. Exp. Sim. Sim. Sim.13.56 10 2.6 11.75 2.65 2.75 16.1 2.6513.56 20 1.4 14.27 0.68 1.4 16.1 0.6813.56 30 0.9 14.8 0.22 0.79 16.2 0.2127.12 10 1.8 1.85 5.02 1.95 4 5.3127.12 20 1.0 3.38 1.35 0.95 4 1.3840.68 10 1.5 0.43 8.02 1.6 1.8 7.9140.68 20 0.8 1.46 1.99 0.8 1.8 2.06Lm/2Lm/2Lm/2Lm/2C22 C24C121234C13C34C11 C33C44C23C14Figure 4.6: A high level model of the capacitively coupled resonators with the completecapacitance matrix.In terms of the relative bandwidth, whereby the bandwidth is divided by the resonantfrequency, 13.56 MHz has the greatest advantage. The results for the relative bandwidthbetween frequencies, at which there is an attenuation of 3 dB at each frequency band, isshown in Table 4.2. The results indicate that as the frequency and critical coupling distanceincreases, the relative bandwidth decreases.4.3 Results Over a Spatial BandwidthBased on the results of the fixed distance experiments, a hypothesis was posed that the spatialsensitivity of the resonators would decrease as the Q of the system decreased. In other words,60Table 4.2: The relative bandwidth at which there is an attenuation of 3 dB for each frequencyband.Frequency d BWRelative(MHz) (cm) %13.56 10 19.413.56 20 10.313.56 30 6.627.12 10 1.827.12 20 3.740.68 10 3.740.68 20 0.2the more closely that the resonators were coupled, the greater the spatial bandwidth. Toexplore this, the transmission characteristics at the resonant frequency were compared at1 cm intervals from 1 to 40 cm. The results are shown in Figs. 4.7 through 4.9.-30 -20 -10 0 10 20 30Relative Distance (cm)-30-25-20-15-10-50|S 21| (dB)10 cm20 cm30 cmFigure 4.7: Experimental results of spatial sensitivity normalized to the critical couplingdistance at 13.56 MHz.61-20 -15 -10 -5 0 5 10 15 20 25 30Relative Distance (cm)-30-25-20-15-10-50|S 21| (dB)10 cm20 cmFigure 4.8: Experimental results of spatial sensitivity normalized to the critical couplingdistance at 27.12 MHz.62-20 -15 -10 -5 0 5 10 15 20 25 30Relative Distance (cm)-25-20-15-10-50|S 21| (dB)10 cm20 cmFigure 4.9: Experimental results of spatial sensitivity normalized to the critical couplingdistance at 40.68 MHz.The results indicate that there is very little change in the absolute spatial bandwidth ofthe resonators as a function of the critical coupling distance. At each frequency, the resultsconsistently show that there is an allowable deviation of approximately ±6 cm to stay abovea -3 dB gain (or efficiency greater than 50%) for any of the three critical coupling distancesconsidered. While the spatial bandwidth at each critical coupling distance remains relativelyunchanged, the results reveal that the maximum efficiency achieved by each design at criticalcoupling decreases as the critical coupling distance increases. This is thought to be due todielectric losses as the air gap between the resonators increases.These results deviate from any expectation set by the fixed distance experiments, however,they can be explained by referring to Fig. 2.4. Although Q increases as the critical couplingdistance between the resonators increases, the spatial sensitivity is relatively constant for thethree designs because the increase in Q is offset by a decrease in the spatial sensitivity to thelink capacitances. C13 may be decreasing, however, C11 is increasing by a similar rate from 1063to 30 cm, with some small contributions from C12 and C14. As the equivalent link capacitancebecomes increasingly dominated by the capacitance of each electrode to ground, the indirectground connection allows for greater distances between the resonators while maintaining arelatively constant spatial bandwidth.The question was posed at the beginning of this work as to whether or not there wouldbe any benefit to increasing the resonant frequency of the capacitive link to achieve greaterspatial bandwidth. To investigate this, the transmission characteristics of all frequencies foreach critical coupling distance were compared. The results can be seen in Figs. 4.10 and 4.11.The comparisons show that there is no advantage to moving up in resonant frequency. At10 cm, it is seen that there is very little difference among the frequency responses in terms ofefficiency nor spatial bandwidth. The 20 cm results indicate that the efficiency decreases atthe frequency increases, suggesting that increasing the frequency is a disadvantage in termsof the efficiency of the WPT link. This result can also be explained by Eq. (4.1). While theQ of the link will decrease as the frequency increases, this increase in frequency is offset bya decrease in N . Referring to Table 4.1, it is shown that for the case of 10 cm, N = 2.75 at13.56 MHz, N = 1.95 at 27.12 MHz, and N = 1.6 at 40.68 MHz. Now consider that goingfrom 13.56 MHz to 27.12 MHz is an increase by a factor of two for frequency, thus changingQ by a factor of one half. However, since the N term is squared in the expression, goingfrom N = 2.75 to N = 1.95 corresponds to a decrease in Q by a factor of one half. Thisrelationship holds for moving from 27.12 MHz to 40.68 MHz as well. Therefore, any increasein frequency is nearly completely canceled out by the decrease in N , meaning that the Q ofthe system is independent of frequency.640 5 10 15 20 25 30 35 40Separation Distance (cm)-30-25-20-15-10-50|S 21| (dB)13.56 MHz27.12 MHz40.68 MHzFigure 4.10: Comparison between all frequencies at a critical coupling distance of 10 cm.0 5 10 15 20 25 30 35 40Separation Distance (cm)-20-18-16-14-12-10-8-6-4-20|S 21| (dB)13.56 MHz27.12 MHz40.68 MHzFigure 4.11: Comparison between all frequencies at a critical coupling distance of 20 cm.65However, consider the relative spatial bandwidth shown by Eq. (4.2) where âd is thedistance range on either side of critical coupling and dcc is the critical coupling distance. Therelative spatial bandwidth for the 10 cm designs is approximately 57% and 30% for the 20 cmdesigns. This indicates that the relative spatial bandwidth decreases as the critical couplingdistance increases.BWRelativeSpatial =âddcc(4.2)Revisiting our research objective regarding determining a relationship between frequencyand spatial bandwidth, this analysis shows that there is indeed a correlation between fre-quency and relative spatial bandwidth. As previously discovered, as the frequency and crit-ical coupling distance increases, the relative bandwidth decreases. Similarly, as the criticalcoupling distance increases, the relative spatial bandwidth decreases. To achieve the best pos-sible relative spatial bandwidth and relative bandwidth, lower frequency, and closely coupleddesigns are desirable.4.4 Chapter SummaryThis chapter began by comparing the frequency response for each frequency band at eachcritical coupling distance. The results indicate that while the absolute bandwidth increasesas the frequency increases, the relative bandwidth decreases. This makes a case for designsat lower frequencies if a high relative bandwidth is a design objective.Next, the results for each frequency band over a spatial bandwidth are compared. Theresults show that the absolute spatial bandwidth for efficiencies greater than 50% remainsrelatively constant over all designs. However, the relative spatial bandwidth decreases asthe critical coupling distance increases. It is also observed that the critically coupled peak66efficiency decreases as both frequency and the critical coupling distance increases.67Chapter 5Conclusions and Future WorkThis work was motivated by the idea that there could be a best configuration, based ona combination of resonant frequency and critical coupling distance, which would providethe greatest spatial bandwidth for dynamic applications in a resonant capacitively coupledWPT system. Different levels of models were extracted from a physical system in an effortto analytically predict how the efficiency of the link would respond to changing a set ofvariables. A test bench was constructed and seven different resonator designs were built andtested experimentally to verify the analytical and simulated results.The results demonstrated, both analytically and experimentally, that while a low Q circuitprovides the greatest absolute and relative bandwidth at a fixed point, this does not translateto a greater efficiency over a spatial bandwidth. It was proven that for a resonant capacitivelycoupled WPT system, the Q, and associated absolute spatial bandwidth, is independent ofboth frequency and critical coupling distance. While spatial bandwidth is independent offrequency and the critical coupling distance, the efficiency at the critical distance sufferedwith both an increase in frequency and the critical coupling distance. If high efficiency isthe primary objective for a given application, this work suggests that designing for a shorttransmitting distance at a lower frequency would be a preferable option, such as the case ofthe 13.56 MHz design, which provided up to 95% efficiency at a critical coupling distance68of 10 cm. For applications requiring smaller devices, or devices that require any amount oflow power over a greater distance, increasing the frequency has the advantage of allowing forsmaller inductors and transformers.This work revealed that the relative bandwidth decreases as both frequency and couplingdistance increases, and similarly, the relative spatial bandwidth decreases as the couplingdistance increases. This relationship indicates that the best configuration for the greatestrelative bandwidth and relative spatial bandwidth would be a lower frequency, closely coupleddesign.5.1 Future WorkThe derivation of the lumped element model of the resonators included some analytical work,but was supplemented with tuning components until agreement between the simulated andexperimental results was found by inspection. Further analytic work of the capacitive link andhigh frequency inductor model would be very valuable in providing insight into capacitivelycoupled WPT systems and to confirm the findings of this work.For wireless power systems to be successful in dynamic applications, further research intoadaptive impedance matching is an important step. As seen in this work, the efficiencycan become less than 50% within a ±6 cm range of motion. Dynamic impedance matchinghas the potential to maintain high efficiencies as the source and load impedances experiencechanges due to the motion.In addition to adaptive impedance matching, there would be great value in investigatingways to make a capacitive WPT link smaller. It is well known that as the surface area ofa parallel plate capacitorâs electrodes decreases, so does the capacitance. This can be anadvantage for applications at higher frequencies, however, there is the disadvantage of largerinductors with smaller capacitances to maintain resonance. 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Thesis/Dissertation
eng
Electrical Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Analysis and design of capacitively coupled resonators for wireless power transfer system applications
Text