Applied Science, Faculty of
Engineering, School of (Okanagan)
DSpace
UBCO
VanderMeulen, Tristan Michael
Master of Applied Science - MASc
University of British Columbia
The rapid growth of battery powered electronic devices continues to motivate research to improve charging methods that are compact and efficient. Wireless power is an attractive solution, as it eliminates the need to physically connect the battery powered device to a charger. The focus of this thesis is on the design of resonant inductive power transfer systems. The synthesis of the resonant coupling circuits is approached using canonical filter structures. In order to convert filters to inductive power transfer systems, impedance inverter theory is utilized to model the inductive power transfer coupling. The use of filter theory has not yet been used to derive shunt-shunt inductive power transfer systems; there are additional complications in the synthesis of the shunt-shunt topology related to higher order IPT networks. A solution is proposed, which allows for higher order filter networks to be converted to a shunt-shunt inductive power transfer system. The use of impedance inverter theory is examined as a method of compensation for variations in the spatial distance between resonators. Impedance inverters are used to represent the mutual coupling between the resonators, which shows how variations in the mutual coupling between resonators impacts the impedance level of the coupling link. Therefore, the input impedance to the system can be constant if the load resistance is varied to compensate for variations in spatial distance. As filter theory inherently has matched loads, it is shown that only the load resistance needs to be varied in the inductive power transfer system to compensate for spatial variation changes. This thesis concludes with the implementation of an end-to-end inductive power transfer system with a class F⁻¹ amplifier as a source. The class F⁻¹ source has not been explored extensively for IPT systems. Of specific interest in the system design was evaluating whether the shunt resonators in the coupling network could replace the output shunt resonator in the class F⁻¹ amplifier. When testing the configuration of the class F⁻¹ amplifier with only the shunt resonator of the IPT system, it was concluded further analysis into the configuration is necessary.
Using Canonical Filter Structures forDesigning Inductive Power TransferSystemsbyTristan Michael VanderMeulenB.A.Sc.,The University of British Columbia, 2020A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinThe College of Graduate Studies(Electrical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)December 2022© Tristan Michael VanderMeulen, 2022The following individuals certify that they have read, and recommend to the Collegeof Graduate Studies for acceptance, a thesis/dissertation entitled:Using Canonical Filter Structures for Designing Inductive Power Trans-fer Systemssubmitted by Tristan Michael VanderMeulen in partial fulfilment of the require-ments of the degree of Master of Applied ScienceThomas Johnson. School of EngineeringPrinciple SupervisorLo¨ıc Markley, School of EngineeringSupervisory Committee MemberDean Richert, School of EngineeringSupervisory Committee MemberKlaske Van Heusden, School of EngineeringUniversity ExamineriiAbstractThe rapid growth of battery powered electronic devices continues to motivate researchto improve charging methods that are compact and efficient. Wireless power is an attractivesolution, as it eliminates the need to physically connect the battery powered device to acharger.The focus of this thesis is on the design of resonant inductive power transfer systems.The synthesis of the resonant coupling circuits is approached using canonical filter struc-tures. In order to convert filters to inductive power transfer systems, impedance invertertheory is utilized to model the inductive power transfer coupling. The use of filter theoryhas not yet been used to derive shunt-shunt inductive power transfer systems; there areadditional complications in the synthesis of the shunt-shunt topology related to higher or-der IPT networks. A solution is proposed, which allows for higher order filter networks tobe converted to a shunt-shunt inductive power transfer system.The use of impedance inverter theory is examined as a method of compensation forvariations in the spatial distance between resonators. Impedance inverters are used torepresent the mutual coupling between the resonators, which shows how variations in themutual coupling between resonators impacts the impedance level of the coupling link.Therefore, the input impedance to the system can be constant if the load resistance isvaried to compensate for variations in spatial distance. As filter theory inherently hasmatched loads, it is shown that only the load resistance needs to be varied in the inductivepower transfer system to compensate for spatial variation changes.This thesis concludes with the implementation of an end-to-end inductive power trans-fer system with a class F−1 amplifier as a source. The class F−1 source has not beenexplored extensively for IPT systems. Of specific interest in the system design was eval-uating whether the shunt resonators in the coupling network could replace the outputshunt resonator in the class F−1 amplifier. When testing the configuration of the classF−1 amplifier with only the shunt resonator of the IPT system, it was concluded furtheranalysis into the configuration is necessary.iiiLay SummaryThe goal of this work is to explore the field of wireless power transfer as a means toreduce the dependence of electronic devices on wired connections to charge batteries. Amethod of designing inductive power transfer systems using filter theory is presented thatprovides a systematic way to design efficient wireless power transfer networks. Filter theoryoffers many benefits to designers, such as establishing a frequency response and impedancecontrol in the system. It is also shown in this work how changes in the separation distancebetween coils in the wireless power system can be fully compensated by modifying theload in the system. An end-to-end system is implemented with an amplifier and load toevaluate the performance of inductive wireless power transfer systems designed using thefilter theory methods presented in this work.ivPrefaceParts of Chapters 2 and 3 have been published in conference proceedings. My co-author for these publications was my colleague Mr. Masoud Ahmadi under the supervisionof Dr. Thomas Johnson and Dr. Lo¨ıc Markley. For all the works which are listed below,the contributions from both Mr. Ahmadi and I are considered equal. We worked closelytogether while researching these topics.The design of the IPT coils was completed in COMSOL multiphysics, with the licensingprovided through a partnership between CMC Microsystems and UBC. The final gerberfiles were generated in AWR through their educational license version. All of the circuitsimulations were completed in AWR. The figures were generated in MATLAB with a licenseobtained through UBC.Parts of Chapter 2 have been published in a conference proceeding. This includesderivations, figures and experimental results which were published in the proceedings.T. VanderMeulen, M. Ahmadi, L. Markley, and T. Johnson, “Comparison of shunt-shunt and series-series resonator topology for second-order inductive power transfer sys-tems,” in 2022 Wireless Power Week (WPW), pp. 532–535, 2022.Parts of Chapter 2 have been submitted for a journal publication. This includes deriva-tions, figures, and experimental results which were published in the proceedings.M. Ahmadi, T. VanderMeulen, L. Markley, and T. Johnson, “A filter theory approachto the synthesis of inductive power transfer systems with multiple resonators,” Progress InElectromagnetics Research C, 2022.Parts of Chapter 3 have been published in a conference proceeding. This includesderivations, figures, and experimental results which were published in the proceedings.M. Ahmadi, T. VanderMeulen, L. Markley, and T. Johnson, “Adaptive load impedancecompensation theory for inductive power transfer systems,” in 2022 Wireless Power Week(WPW), pp. 526–531, 2022.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiChapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 A System Overview of Inductive Power Transfer Networks . . . . . . . . . . 11.2 Review of Current Inductive Power Transfer Networks . . . . . . . . . . . . 31.3 Class F and F−1 Power Amplifiers . . . . . . . . . . . . . . . . . . . . . . . 51.4 Research Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Chapter 2: Filter Theory Techniques for Designing IPT Systems . . . . . . 82.1 Review of Microwave Filter Theory . . . . . . . . . . . . . . . . . . . . . . . 92.2 Derivation of the Series-Series IPT Network from Filter Theory . . . . . . . 112.3 Derivation of the Shunt-Shunt Topology IPT Network from Filter Theory . 172.4 Comparing Shunt-Shunt and Series-Series Topologies . . . . . . . . . . . . . 252.4.1 Analysis of the Simulation Results for Series-Series and Shunt-ShuntIPT Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.4.2 Comparing Component Values for Series-Series and Shunt-Shunt IPTNetworks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.5 Design and Testing of IPT Coils . . . . . . . . . . . . . . . . . . . . . . . . 322.6 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35viTABLE OF CONTENTS2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Chapter 3: An Inverter Model for Impedance Compensation Techniquesin IPT Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.1 Impedance Compensation Technique for Series-Series IPT networks . . . . . 453.1.1 Analysis of Series-Series IPT Networks using Impedance Inverters . 453.1.2 Impedance Compensation Technique for Series-Series IPT Networks 493.2 Impedance Compensation Technique for Shunt-Shunt IPT Networks . . . . 503.2.1 Impedance Compensation for Second Order Shunt-Shunt IPT Networks 503.2.2 Impedance Compensation for Higher Order Shunt-Shunt IPT Networks 523.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.3.1 Simulation Results for Third Order Series-Series Impedance Com-pensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.3.2 Simulation Results for Shunt-Shunt IPT Network Compensation . . 573.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Chapter 4: A Class F−1Amplifier for IPT Systems . . . . . . . . . . . . . . . 644.1 Class F−1Amplifier Design for Wireless Power Transfer . . . . . . . . . . . 654.1.1 Circuit Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.1.2 Circuit Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . 674.1.3 Circuit Board Layout and Design . . . . . . . . . . . . . . . . . . . . 684.1.4 Board Design Simulation Results . . . . . . . . . . . . . . . . . . . . 704.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724.2.1 Class F−1Amplifier Measurement Results . . . . . . . . . . . . . . . 744.2.2 Class F−1Amplifier Performance with Shunt-Shunt IPT Networks . 754.2.3 Impedance Variation with a Class F−1Amplifier . . . . . . . . . . . 784.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80Chapter 5: Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.2 Recommendations and Future Work . . . . . . . . . . . . . . . . . . . . . . 83Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85viiList of TablesTable 2.1 Resonator inductance expressions for IPT networks with shunt-shuntresonator port terminations. . . . . . . . . . . . . . . . . . . . . . . . 25Table 2.2 Resonator capacitance expressions for IPT networks with shunt-shuntresonator port terminations. . . . . . . . . . . . . . . . . . . . . . . . 26Table 2.3 Approximate expressions for inductances in an IPT network withshunt-shunt resonator port terminations when ∆2 → 0. . . . . . . . 26Table 2.4 Bandwidth and center frequency for fourth order IPT designs. . . . . 30Table 2.5 27.12 MHz IPT link design summary. . . . . . . . . . . . . . . . . . . 37Table 2.6 40.68 MHz IPT link design summary. . . . . . . . . . . . . . . . . . . 40Table 3.1 Summary of impedance compensation simulation for the third-orderIPT link. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Table 3.2 Summary of impedance compensation measurements for the second-order IPT link. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Table 4.1 Class F−1 amplifier measurement . . . . . . . . . . . . . . . . . . . . 75Table 4.2 Class F−1 amplifier measurements with IPT coupling links . . . . . . 77Table 4.3 Class F−1 amplifier measurement with second order IPT coupling linkand no output resonator . . . . . . . . . . . . . . . . . . . . . . . . . 78Table 4.4 Class F−1 amplifier measurements with spatial variation . . . . . . . 79viiiList of FiguresFigure 1.1 A block diagram of a general IPT system. . . . . . . . . . . . . . . . 2Figure 1.2 The idealized drain current and drain voltage waveforms for a ClassF power amplifier are shown. . . . . . . . . . . . . . . . . . . . . . . 5Figure 1.3 The circuit schematic for Class F power amplifiers is shown in (a)where N is any odd number. The circuit schematic for Class F−1 poweramplifiers is shown in (b) where M is any even number. . . . . . . . 6Figure 2.1 Canonical low-pass filter prototype with a series element first (a)and a shunt element first (b). . . . . . . . . . . . . . . . . . . . . . . 10Figure 2.2 A third-order filter prototype is shown in (a). Impedance invertersare used to transform the shunt resonator into a series resonator in(b). The elements of the third resonator are scaled with respect tothe capacitance and inductance of the first resonator in (c). . . . . . 13Figure 2.3 The impedance inverters are transformed to a T-network of lumpedinductors in (a). The T-network of inductors is modelled as aninductive coupling link in (b). . . . . . . . . . . . . . . . . . . . . . 14Figure 2.4 General band-pass filter prototype with N resonators is shown in(a). The transformed filter with series-series resonator coupling isshown in (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Figure 2.5 Transformation of a third-order filter network into a circuit withshunt-shunt resonator coupling is shown. Inverters are used to con-vert the series resonator to a shunt resonator in (a). In (b), a re-dundant inverter is added to the load. The impedance level of thethird resonator is scaled in (c). In (d), the third inverter is absorbedinto an equivalent load resistance and the inverters are replaced byan equivalent Π-network of inductors. . . . . . . . . . . . . . . . . . 19Figure 2.6 Shunt inductors are combined in (a). Π-networks of inductancesconverted to T-networks using ∆-Y transformation in (n). The se-ries inductances are partitioned to form mutually coupled inductorsin (c). The L-network is approximated by an equivalent series ca-pacitance at ωo in (d). The final transformed network is shown in(e). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20ixLIST OF FIGURESFigure 2.7 The resistance (a) and reactance (b) seen at the impedance planesZ2,P and Z3,P from Fig. 2.6(c) and Fig. 2.6(d). . . . . . . . . . . . . 22Figure 2.8 The canonical band-pass filter prototype with N resonators is shownin (a). The final transformed shunt-shunt IPT network is shown in(b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Figure 2.9 Comparison of simulated transmission coefficients (|S21|) for a filter,a series-series IPT and a shunt-shunt IPT. Comparisons are givenfor a third order network (a), a fourth order network (b), and a fifthorder network (c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Figure 2.10 Frequency response for modified series-series and shunt-shunt IPTnetworks, compared with an ideal filter response. . . . . . . . . . . . 29Figure 2.11 The transmitter inductance is shown vs. the normalized frequencybandwidth for both series (blue) and shunt (red) IPT networks atfour different ISM bands. . . . . . . . . . . . . . . . . . . . . . . . . 32Figure 2.12 Geometry of a spiral inductor is shown in (a). The inductor isdefined by an inner diameter d with n turns that have a turn-to-turn spacing S and trace width w. Geometry of an inductor with asingle circular loop is shown in (b). A multi-coil IPT link is shownin (c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Figure 2.13 Screenshots from the COMSOL simulation environment are shownfor the 2D axis symmetric simulation in (a) and the 3D full wavesimulation in (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Figure 2.14 Photographs of the experimental IPT systems. The second-orderlink in (a) has series-series resonator coupling and operates at 27.12 MHz.The IPT links in (b), (c), and (d) have shunt-shunt resonator cou-pling at 40.68 MHz with two, three and four resonators respectively. 36Figure 2.15 The simulated and measured transmission coefficient (|S21|) of thesecond order 27.12 MHz IPT links with series-series and shunt-shuntresonator coupling are shown in (a) and (b), respectively. Both linkswere designed for an impedance of 50 Ω, a normalized bandwidth of0.16, and an asymmetry factor of m = 1. . . . . . . . . . . . . . . . 38Figure 2.16 Simulated and measured transmission responses for shunt-shunt IPTlinks with (a) two resonators, (b) three resonators and (c) four res-onators. The simulated response of the final transformed networkis labelled as “Final IPT”. Simulation responses are also providedincluding cross-coupling in third and fourth order designs. . . . . . 41Figure 3.1 A block diagram of the impedance compensation method. . . . . . . 44Figure 3.2 Summary of steps to transform the equivalent network of two mu-tually coupled series coils into an impedance inverter at the centerfrequency ωo. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46xLIST OF FIGURESFigure 3.3 Summary of steps to transform the equivalent network of three mu-tually coupled series coils into a cascade of two impedance invertersat the center frequency ωo. Cross-coupling between non-adjacentcoils is assumed to be negligible. . . . . . . . . . . . . . . . . . . . 47Figure 3.4 Summary of steps to transform the equivalent network of two mu-tually coupled shunt coils into an impedance inverter at the centerfrequency ωo. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Figure 3.5 Summary of steps to transform an equivalent network of three mu-tually coupled shunt coils into a cascade of two impedance invertersat the center frequency ωo. Cross-coupling between non-adjacentcoils is assumed to be negligible. . . . . . . . . . . . . . . . . . . . . 53Figure 3.6 Simulated mutual inductance and input impedance (Zin) of a third-order IPT link with a fixed load resistance of 50 Ω. The plot in(a) shows how the mutual inductance and cross-coupling inductancevary with distance. The plot in (b) shows how the input resistanceRe(Zin) changes with distance, and the plot in (c) shows the inputreactance Im(Zin) changes with distance. . . . . . . . . . . . . . . . 56Figure 3.7 The simulated input impedance (Zin) of the second-order shunt-shunt IPT network with a fixed load resistance of 50 Ω is shownversus coil separation distance. The plot in (a) shows the inputresistance Re(Zin) and the plot in (b) shows the input reactanceIm(Zin). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Figure 3.8 The planar spiral inductor used for testing is shown in (a). Theexperimental setup is shown in (b) with the LC matching networkused to modify the load impedance. . . . . . . . . . . . . . . . . . . 59Figure 3.9 Characterization of mutual inductance M12 with respect to the sep-aration distance between two planar inductors is shown in (a). Ata separation distance of 6.9 cm and a frequency of 27.12 MHz, theinput resistance is 50 Ω for a 50 Ω load. Shown in (b) is the S21frequency response for the IPT system for the optimal distance of6.9 cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Figure 3.10 The simulated and measured input impedance (Zin) of the second-order IPT link with a fixed load resistance of 50 Ω is shown versusdifferent separation gaps. The plot in (a) shows the input resistanceRe(Zin), while the plot in (b) shows the input reactance Im(Zin). . 62Figure 4.1 The circuit schematic for the class F−1 power amplifier is shown in(a). Two modifications to the amplifier are tested. In step 1, theload RL is replaced with the second order shunt-shunt IPT networkshown in (b) and, in step 2, the shunt resonator in the dashed boxand the load are replaced by the IPT network in (b). . . . . . . . . 64xiLIST OF FIGURESFigure 4.2 AWR circuit simulation schematic for the class F−1 design. . . . . . 66Figure 4.3 Simulated drain current and voltage waveforms for the class F−1 amplifierdesign. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Figure 4.4 KiCad circuit schematic for the class F−1 amplifier design. . . . . . 69Figure 4.5 KiCad printed circuit board for the class F−1 amplifier design. . . . 70Figure 4.6 Simulated drain current and voltage waveforms for the class F−1 amplifierdesign with the addition of parasitic trace capacitance. . . . . . . . 71Figure 4.7 Simulated gate and probe voltage waveforms for the class F−1 design.The probe voltage is shown for simulated results with and withoutparasitic trace capacitance. . . . . . . . . . . . . . . . . . . . . . . . 72Figure 4.8 Assembled class F−1 amplifier. . . . . . . . . . . . . . . . . . . . . . 73Figure 4.9 Testing apparatus for the class F−1 amplifier. . . . . . . . . . . . . . 73Figure 4.10 Measured drain and gate voltage waveforms for class F−1 amplifier. 74Figure 4.11 Experimental results for the class F−1 amplifiers with shunt-shuntIPT links with (a) two resonators, (b) three resonators and (c) fourresonators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76Figure 4.12 Drain and gate voltage waveforms for class F−1 amplifier with nooutput resonator and a second order IPT network as the load. . . . 78xiiAcknowledgementsDuring this stage of my educational career I have had the pleasure of working withmany outstanding colleagues with whom helped guide me through my research objectives.Without their support and direction, this thesis would not have been possible.My first expression of gratitude is to my supervisor Prof. Thomas Johnson who sup-ported me with his knowledge, expertise and instruction. Prof. Johnson was an amazingmentor who has helped me hone my academic skills and knowledge. His guidance andpatience throughout my research career was invaluable.I am extremely grateful for all of the members of my research team who supportedme with their time and knowledge. Masoud Ahmadi was an amazing mentor who assistedme throughout my research journey. Ziad Abdelsamad and Simon Faber were part ofour research team throughout my research journey and were always available for technicaldiscussions and friendly conversations. I would also like to thank Aaron Clements for histime and direction in the lab.My gratitude also extends to Prof. Lo¨ıc Markley for his time and expertise throughoutmy research. I would like to thank Prof. Mohammad Zarifi for sharing their lab equipmentthroughout my research. I also would like to thank Prof. Dean Richert for being a memberof my examining committee.I owe my greatest thanks to my wonderful family who have always supported me inany endeavour. I would not have been able to accomplish this achievement or any ofmy achievements without the love and support of my amazing fiance´e Mary Cooke. Myparents and siblings have always provided me with support and companionship for whichI am extremely grateful. Thank you Mom for always being there for me and for alwaysencouraging and nurturing me. Thank you Dad for your support and belief in me withwhich always drove me to achieve more. Thank you Mark for guiding me throughout mylife with your wisdom and knowledge. My siblings Abbigail, Maddison and Marcus havealways been there for me and, I know that in any endeavour they pursue, they will doamazing. Finally, thank you to all of my family and friends who have shared with me theirknowledge, support and wisdom.xiiiChapter 1IntroductionWith the growing demand for compact and mobile electronics, there is a need to reducethe reliance of mobile electronics on batteries. Wireless power transfer (WPT) presents asustainable solution to meet this demand by removing the components needed for wiredcharging and reducing battery sizes. To achieve this, methods to simplify the design ofefficient coupling networks need to become more accessible to designers. There are currentlymany wireless power techniques that are used to wirelessly transfer power at a range ofpower levels and distances. However, these systems can be complex to both design andimplement. Therfore, a need exists for a strong theoretical framework from which designerscan rely on to quickly and easily design WPT systems for a variety of applications. Thisthesis will explore the use of microwave filter theory as one such framework to quickly andreliably synthesize WPT networks.Wireless power transfer systems can be classified into three distinct categories: induc-tive power transfer (IPT), capacitive power transfer (CPT) and radiative power transfer(RPT). IPT systems use the magnetic field to transfer power from one coil structure toanother, where the coil structure can be resonant or non-resonant. Similarly, CPT systemsuse the electric field to transfer power from one plate to another and can also be resonantor non-resonant. Generally, for both IPT and CPT systems, resonant power transfer struc-tures improve the efficiency and coupling distance of the system [4]. RPT systems use farfield electromagnetic radiation to send power from an antenna to a receiver and will notbe explored in this work.1.1 A System Overview of Inductive Power TransferNetworksAn IPT network utilizes the phenomena of mutual inductance to couple inductors andtransfer power wirelessly. However, at a system level, there are additional components thatare needed to implement an IPT network. A block diagram of an IPT system is shown inFig. 1.1.11.1. A SYSTEM OVERVIEW OF INDUCTIVE POWER TRANSFER NETWORKSAmplifier/InverterMatchingNetworkMatchingNetworkRectifier Load/DeviceMagnetic CouplingRepeater CoilFigure 1.1: A block diagram of a general IPT system.In Fig. 1.1, the magnetically coupled multi-resonator system is implemented with anamplifier and matching network on the transmitter side, and a rectifier and matchingnetwork on the receiver side. The matching networks are necessary to ensure a highoperating efficiency for both the amplifier and the coupling link [4–6]. The coupling linkswith the matching network will be referred to as the IPT network. Included in the matchingnetworks is a shunt or series capacitor which resonates with the coupling inductor. Usingresonance in IPT networks greatly improves the efficiency of energy transfer from thetransmitter to the receiver at longer spatial distances between coils [4, 5] The matchingnetwork may also include reactive elements which are used to tune the impedance seenby the amplifier/rectifier to improve the efficiency of the amplifier/rectifier circuits [7, 8].Repeater coils can be used in the IPT network to extend the transmission distance withouta loss of efficiency in the coupling link [9].To ensure the IPT system has a high overall efficiency, a high efficiency amplifier mustbe designed to operate at the desired resonant frequency of the IPT network. Whilebroadband amplifier designs exist, for an IPT network the coils are designed to resonate ata single frequency and therefore the amplifier only needs to operate at a single frequency.A high efficiency amplifier class can be chosen for this operation. The rectifier efficiency isalso important in the overall efficiency of the network. There are many rectifier topologieswhich can be used to achieve high efficiency rectification. Rectifier topologies includesynchronous rectifiers which use time-reversal duality to convert the amplifier design toa rectification circuit [10, 11], or more common rectifier circuits which utilize diodes toachieve rectification [6].The amplifier, rectifier, and IPT link with the matching networks will all have a specificfrequency bandwidth over which a high efficiency can be achieved. The amplifier will ideallyoutput a signal with a single frequency component, which then transfers through the systemand is converted to a DC signal at the load [12]. The bandwidth of the IPT network can behigher than the amplifier/rectifier, with the frequency response of IPT networks mimickinga band-pass filter response [13]. The bandwidth of the IPT network depends on the loadedand unloaded quality factors of the coils as well as the ratio of the inductor size to thecapacitor size [14]. This will be explained in further detail when filter theory is directlyrelated to the design of IPT networks. The bandwidth over which the rectifier can convertthe RF signal to a DC signal is generally much smaller than the bandwidth of the IPTnetwork.21.2. REVIEW OF CURRENT INDUCTIVE POWER TRANSFER NETWORKSIn this thesis, the focus of the analysis is on the design of the IPT network whichincludes the coupling link and the matching networks. It is discussed how the couplinglink between the coils can impact the impedance seen through the IPT network and how theload impedance can be used to compensate for the variations in the coupling link. A highefficiency amplifier is designed and utilized with the IPT network to test the performanceof the IPT network at a system level.1.2 Review of Current Inductive Power Transfer NetworksHigh efficiency WPT systems offer a charging solution for mobile devices without theneed for wiring harnesses and power cords. Without the need for external power connec-tions, wireless devices can be fully isolated from dust and water which cause damage tothe device. The benefits of wireless devices are useful for a number of applications, suchas wearable electronics [15], implantable medical devices [5, 16], sensor networks [17], au-tonomous devices [18] and electric vehicle (EV) charging systems [7, 8, 19–21]. A popularmethod of WPT is through the magnetic field using using resonant coils to improve theefficiency of the power transfer link. This was first demonstrated by a research team atMIT where they demonstrated resonant IPT at a distance of 2 m [4]. Since then, researchinto IPT systems has grown, with many different circuit topologies and coil structuresbeing introduced. There are two resonant structures which form the basis for every IPTtopology which are shunt and series resonator coils [6]. The resonator coils can then befurther modified to achieve desired circuit characteristics which include impedance level,spatial bandwidth and frequency bandwidth. Series or shunt lumped element inductors andcapacitors can be inserted into the network to modify the characteristics of the network[6–8].A common physical implementation of the IPT coils which has been explored in theliterature is the planar spiral inductor [22–26], although other structures such as helicalinductors [27], wound loop inductors [28] and loop-gap resonators [29] have been explored.The benefit of using planar spiral inductors is that they are easy to design and fabricate,with analytical expressions derived for the inductance, self capacitance and resistive lossesin the coil [22, 30, 31].The most common method for designing IPT systems is through direct circuit analysiswhere KVL and KCL are applied to a specific network topology [32–42]. By using directcircuit analysis, the designer is required to find the voltage and current in each coil, whichis represented by a set of differential equations equal to the number of resonators in thesystem. Solving these differential equations can be difficult, especially in systems with alarge number of resonators.One method of reducing the complexity of the IPT system design is to utilize theconcept of impedance and admittance inverters to model to mutual coupling between theresonators [3, 43–45]. By using inverters, the voltage and currents in each resonator do not31.2. REVIEW OF CURRENT INDUCTIVE POWER TRANSFER NETWORKSneed to be calculated, which significantly simplifies the analysis. While the circuit analysisis reduced, the inverter method requires multiple design iterations to determine the desiredinverter impedance values for a fully matched IPT system. Another limitiation which boththe inverter method and the direct circuit analysis method share is the synthesis of the IPTnetwork is only considered for a single frequency of operation. Brute force optimization isrequired to shape the overall frequency response of the IPT network.To incorporate the frequency response in the synthesis of IPT networks, the concept ofusing band-pass filter theory for the design of the IPT network was introduced [14, 46–48].Filter theory relies on well established canonical filter structures which can be matchedfor specific termination impedances with a desired frequency response. Through the use offilter theory, circuit element values can be calculated without the need for complex circuitanalysis or iterative design processes. Instead, the component values are calculated basedon standardized tables of filter coefficients, each of which synthesize a different frequencyresponse. Filter theory also offers methods to determine the impact of finite resonatorquality factor on the insertion loss of the IPT network.The synthesis of critically-coupled IPT networks using a low-pass canonical filter struc-ture with a maximally flat (Butterworth) frequency response was first investigated in [14].In the work, the canonical filter network is transformed to a network of coupled resonatorsusing impedance inverters. The impedance inverters model the mutual coupling betweenthe resonators, which is the same as the inverter method for IPT design. The filter networkcan then be directly related to the physical geometry of the IPT network. Further investi-gations on this method were conducted where a second-order Butterworth band-pass filternetwork was transformed into an IPT system with two identical resonators [1, 49, 50]. Thistheory was further developed in [51] and [13] when they synthesized general expressionsfor series-series IPT networks with an arbitrary number of resonators. In [13], they estab-lishing expressions for identical and non-identical resonators, as well as expressions for thepower delivered to the load and efficiency of the IPT network based on the quality factorof the resonators.One potential downside of using the filter theory method is that the impedance level,frequency and normalized frequency bandwidth determine the corresponding resonatorvalues necessary for the IPT network. While this gives great insight into the design ofthe IPT network, there is less freedom to chose a specific coil geometry or size based onthe filter theory method. All of the works which explored the use of filter theory for thedesign of IPT networks only focused on the synthesis of series-series IPT networks. Thiswork expands the use of the filter theory approach to shunt-shunt IPT networks with anarbitrary number of resonators.41.3. CLASS F AND F−1 POWER AMPLIFIERS1.3 Class F and F−1Power AmplifiersHigh efficiency amplifiers and rectifiers are crucial in achieving a high end-to-end effi-ciency in WPT systems. The class E amplifier topology has been widely researched in theliterature for WPT systems due to its high efficiency and simple structure at the operatingfrequency [12, 52–55]. However, the class E amplifier is known to be sensitive to load varia-tion due to the zero voltage switching and zero voltage derivative switching characteristicsof the amplifier [12, 53]. This is undesirable for WPT systems as a change in coupling cancause the amplifier load to vary significantly.An alternative to the class E topology which has not been explored extensively inthe literature is the class F amplifier. Class F amplifiers are a continuous mode amplifierwhich utilize harmonic tuning in the load to shape the drain voltage and current waveforms[11, 56–62]. Due to the harmonic tuning characteristics of the class F amplifier, the draincurrent is shaped into a half sinusoidal waveform and the drain voltage is shaped into asquare wave as shown in Fig. 1.2 [56].0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Normalized Time per Cycle (s)0Current Waveform (A)Voltage Waveform (V)Figure 1.2: The idealized drain current and drain voltage waveforms for a Class F poweramplifier are shown.The dual of the class F amplifier is the class F−1 amplifier. The class F−1 amplifieroperates similarly to the class F amplifier in that harmonic tuning is used to shape the drainwaveforms, however, in the class F−1 amplifier, the current and voltage drain waveformsare switched. This is achieved by the harmonic resonators for the class F−1 being tunedto having an open circuit for even harmonics and a short circuit for odd harmonics. Bothcircuit topologies are shown in Fig. 1.3 with both amplifiers theoretically having the samenumber of components.Due to the reverse characteristics of the class F−1 amplifier, class F−1 has a higher maxi-mum drain voltage, which increases the risk of damaging the switch during load variations.However, the efficiency of the class F−1 amplifier is higher in practical implementations51.4. RESEARCH OBJECTIVESwhen the input DC power is constant [59]. Therefore, the class F−1 amplifier was thechosen over the class F amplifier due to the higher potential efficiency.f3 = 3 f0 f5 = 5 f0 fN = N f0RLf0VDDLDD(a)RLf2 = 2 f0 f4 = 4 f0 fM = M f0f0VDDLDD(b)Figure 1.3: The circuit schematic for Class F power amplifiers is shown in (a) where Nis any odd number. The circuit schematic for Class F−1 power amplifiers is shown in (b)where M is any even number.For practical WPT systems, a high efficiency rectifier is needed to achieve an efficientend-to-end system. High efficiency amplifiers such as class F and class F−1 can be con-figured in a rectification mode using the concept of time reversal duality [11]. However,these circuits require additional phase shifting elements or a phase locked loop to achievethe required switching at the input of the transistor. While this is achievable, it was notthe focus of this work as the implementation of the rectification circuit can be achieved avariety of ways. Therefore, a RF power meter with an attenuator was used as equivalentload for the rectifier in this work.1.4 Research ObjectivesAn objective of this thesis is to investigate how the filter theory method can be usedfor designing IPT systems to include shunt-shunt resonators topologies. The shunt-shunt61.5. SUMMARYtopology requires additional analysis which is not included in the synthesis of series-seriesIPT networks from filter networks, which has already been examined in the literature. Theresulting shunt-shunt IPT networks is compared with series-series IPT network to examinethe relative performance of the different topologies.Another objective of this research is to examine impedance techniques which can beused to compensate for spatial variations between resonators. In the synthesis of IPTnetworks from filter networks, the coupling link between mutually coupled resonators canbe represented by an ideal impedance inverter. Using this same concept, it can be shownthat for IPT systems with series resonators, the input impedance of the system is directlyrelated to the load resistance through the impedance inverter. In an ideal system, as themutual coupling between the coils varies due to spatial misalignment, the impedance ofthe inverter scales relative to the mutual coupling. Therefore, the load resistance can beused to compensate for variations in spatial misalignment between resonators. Adaptiveload compensation based on impedance inverter theory will be examined as a method forspatial compensation in IPT networks where the resonators are not critically coupled. Theadaptive load method will analyzed for both series-series and shunt-shunt IPT networks.The last objective of this research is to design and evaluate the IPT system using a classF−1 amplifier as an RF source. The class F−1 amplifier offers a high efficiency amplifier witha shunt output resonator. It will be examined whether the shunt resonator at the outputof the class F−1 amplifier can be eliminated and replaced with the input shunt resonatorin the shunt-shunt IPT networks.1.5 SummaryFor this work, the filter theory method will be explored and expanded for shunt-shuntIPT systems to create a strong theoretical framework for designing WPT systems. Throughthe use of impedance inverters, it will be shown that any filter network can be convertedto a wireless power transfer network. This has been done previously in the literaturefor capacitive power transfer networks and for series-series IPT networks. The resultingshunt-shunt IPT networks will be compared with the series-series IPT networks to analyzethe relative performance of the shunt-shunt topology. The coupling link between IPTresonators will then be analyzed and equated to an impedance change in the IPT system.Mutual coupling can be modelled as an impedance inverter which varies as a function ofthe mutual inductance between the resonators. It will be shown that variations in themutual coupling, which relate to variations in the spatial distance between coils, can befully compensated by modifying the load resistance of the system. Finally, to examine theperformance of the shunt-shunt IPT network, an end-to-end system will be implementedwith a class-F−1 amplifier, the shunt-shunt IPT system and an RF load.7Chapter 2Filter Theory Techniques forDesigning IPT SystemsIn this chapter, a detailed analysis of the design of IPT networks using a filter theoryapproach is presented. The insertion loss method of filter design is used for the design ofthe filters in this work. The insertion loss method begins with canonical low-pass filterprototypes that are normalized in both frequency and impedance scaling [63]. The low-pass filter can then be transformed into a band-pass filter through the addition of resonantelements with a specified frequency bandwidth. The insertion loss method is easy to use andcan be applied to many applications as the resultant filters can be scaled to any frequency orimpedance level without modifying the frequency response. The canonical filter prototypecan also be directly related to an IPT network through the use of impedance inverters.The idea of coupling has been discussed in [64] as an addition to filter theory. In [64],Zverev analyzes coupled resonators as either having parallel (shunt) or series resonators thatare coupled through reactive elements. This concept relates directly to WPT systems whichare classified based on the topology of the transmit and receive resonators [6]. The mutualcoupling between the coils can be directly related to T or Π lumped element networks withinductor values equal to the mutual inductance. It has also been shown that impedanceinverters can be represented as either T or Π lumped element networks [65]. Therefore,the mutual coupling between resonators can be represented as an impedance inverter inthe system.The use of impedance inverters in the framework of the insertion loss method of filtertheory creates a method of designing WPT systems with a specific frequency responsewith no additional circuit analysis. The concept of transforming a filter network to anIPT network has already been explored in detail for series-series IPT topologies and CPTtopolgies in [1, 3, 13, 43–45, 50]. However, shunt-shunt IPT networks have not yet beensynthesized using the filter theory approach. This is due to the transformation of a filternetwork to a shunt-shunt IPT network having additional complexities which are not presentin the transformation to a series-series IPT network. As a way of resolving the analyticdifficulties of using filter theory, an approximation is introduced to provide a closed formmethod of synthesizing shunt-shunt IPT resonator links. The details of the methodologyare described in this Chapter and the impact of the approximation in the synthesis isevaluated.82.1. REVIEW OF MICROWAVE FILTER THEORY2.1 Review of Microwave Filter TheoryA filter is a two-port network which controls and shapes the frequency response ofsignals that pass through the network [63]. This shaping can be characterized by thepass-band and stop-band of the filter, where the pass-band is defined as having 100%transmission and the stop-band has 100% attenuation [65]. Filters can be classified intofour categories based on the characteristics of the pass-band and stop-band, those being:low-pass, high-pass, band-pass and band-reject [63]. A low-pass filter will allow all ofthe frequency components to pass through up to some cut-off frequency ωc at which allfrequencies higher will be attenuated. A high-pass filter is the opposite of a low-pass filterand will attenuate all of the frequency components up to some frequency ωc at which anyfrequency higher than the cut-off frequency will be transmitted. A band-pass filter canbe seen as a combination of a high-pass and a low-pass filter, where a range of frequencycomponents between ω1 and ω2 pass through with minimal attenuation, while frequencyoutside the bandwidth of the filter are attenuated. A band-reject filter is the inverse of theband-pass filter where only the frequency components between ω1 and ω2 are attenuatedand all other frequency components are transmitted.The development of filter theory began in the years preceding 1930, in which the imageparameter method of filter theory was introduced and was used for low-frequency radio andtelephone applications [63]. Since then, the insertion loss method for designing filters hasbeen developed as a more robust and useful tool for designing filter networks at a varietyof frequencies [63].The insertion loss of the filter is defined asInsertion Loss (IL) = 10 logPLR (2.1)where PLR is the power loss ratio which is defined asPLR =PsourcePload=11− |Γ(ω)|2 , (2.2)for a lossless system, where Γ(ω) is the reflection coefficient of the network. In [63], it isshown in Section 4.1 that |Γ(ω)|2 is an even function of ω. Therefore, the power loss ratioof the network can be defined asPLR = 1 +M(ω2)N(ω2)(2.3)where M and N are real polynomials in ω2 [63, 65]. This is where the versatility of theinsertion loss method becomes apparent as different frequency responses can be achieveddepending on how the polynomials M and N are chosen. After a specific frequency re-sponse is selected, the power loss ratio can be used to determine element values in a network92.1. REVIEW OF MICROWAVE FILTER THEORYthrough the input impedance. The input impedance can be used to find the reflection coef-ficient, which is directly related to the power loss ratio for the specific frequency response.A canonical filter prototype is a cascade of series inductors and shunt capacitors knownas a ladder network as shown in Fig. 2.1. The reactive elements in the ladder net-work are dependent on the power loss ratio and are defined by the filter coefficientsg0, g1, g2, ..., gN , gN+1. The filter coefficients are derived for an impedance level of 1 Ω,where N is the order of the filter. The filter coefficients g0 and gN+1 are used to determinethe resistance of the source and load. The reactive elements in the filter are alternatingseries inductors and shunt capacitors or shunt capacitors and series inductors for a low-pass frequency response. The resulting network is known as the low-pass canonical filterprototype, shown in Fig. 2.1.g1 gN−1g2 gNg0VS gN+1(a)g1 gN−1g2 gNg0VS gN+1(b)Figure 2.1: Canonical low-pass filter prototype with a series element first (a) and a shuntelement first (b).Tables can be found ([63, 65, 66]) which tabulate the filter coefficients for differentfrequency responses. The impedance of the filter can easily be scaled using the impedancescaling factor kz and the cutoff frequency of the filter can be chosen as ωc. The equationsfor calculating the required inductances and capacitances for the filter are given as102.2. DERIVATION OF THE SERIES-SERIES IPT NETWORK FROM FILTER THEORYR0 =kzg0RN = kz gNLk =kz g1ωcCk =g1kz ωc(2.4)In order to convert the low-pass filter network to a band-pass filter, a normalizedfrequency bandwidth of ∆ = ω2−ω1ω0 is defined where ω0 is chosen to be the arithmeticmean of ω1 and ω2. The series inductances in the low-pass filter are then transformed toseries LC resonatorsL′k =Lkω0∆, C ′k =∆ω0 Lk, (2.5)and the shunt capacitance in the low-pass filter are converted to shunt LC resonatorsL′k =∆ω0CkC ′k =Ckω0∆. (2.6)With this transformations complete, the resulting network is a band-pass filter with afrequency response which can be shaped depending on the filter coefficients, impedancelevel of the system and desired frequency pass-band. This is ideal for designers and offersgreat flexibility in the design process of band-pass filters.2.2 Derivation of the Series-Series IPT Network fromFilter TheoryThe derivation for a series-series IPT network from filter theory will be reviewed asit has already been derived in previous works [13, 14, 43–45]. It is useful to go throughthe derivation in order to fully compare the series-series IPT network to the shunt-shuntIPT networks. To reduce the complexity of the derivation, a third order filter network willbe transformed to a series-series IPT link with three resonators. This transformation willthen be expanded to series-series IPT networks with an arbitrary number of resonators.The third-order band pass filter prototype is shown in Fig. 2.2(a). The filter elementsareR0,S =kzg0R4,S = kz g4L1,S =kz g1ωbC1,S =1ω2o L1,SC2,S =g2kz ωbL2,S =1ω2o C2,SL3,S =kz g3ωbC3,S =1ω2o L3,S,(2.7)112.2. DERIVATION OF THE SERIES-SERIES IPT NETWORK FROM FILTER THEORYwhere ωo is the center frequency, kz is the impedance scaling factor, and ωb is the frequencybandwidth. The normalized frequency bandwidth can be defined as ∆ = ωb/ωo. Thesubscript “S” in (2.7) denotes that the filter prototype begins with a series resonator.For the first conversion step in Fig. 2.2(a), two impedance inverters are used to trans-form the middle shunt resonator L2,SC2,S into a series resonator. The resulting seriesresonator has an inductor value m21L1,S and a capacitor value C1,S/m21, where m21 is aresonator scaling factor, allowing the elements of the second resonator to be scaled withrespect to the element values of the first resonator. In the generalized expressions, the el-ements of an N th order system are scaled by mj1, where j = 2, 3, 4, ..., N . The subscriptnotation “j1” means that the scaling of the jth resonator is with reference to the elementsin the first resonator. These scaling factors do not change the frequency response or theresonant frequency of the resonators; rather they are used as a tool to modify the rela-tive size of the resonators in the system. The corresponding expression for the impedanceinverters in Fig. 2.2(a) isK1,S = K2,S =√m21L1,SC2,S= kz√m21g1g2. (2.8)122.2. DERIVATION OF THE SERIES-SERIES IPT NETWORK FROM FILTER THEORYL1,SC2,SR4,SR0,Sm21L1,SC1,Sm21(a)R0,S(b)L3,SC3,S(c)R4,Sm21L1,sC1,Sm21K1,S K2,SRedundant InverterL1,SC1,SL1,SC1,S L3,SC3,SK1,S K2,S K3,Sm21L1,SC1,Sm21L1,SC1,S m31L1,SC1,Sm31R0,S RL,S R4,SK1,S K′2,S K′3,SFigure 2.2: A third-order filter prototype is shown in (a). Impedance inverters are usedto transform the shunt resonator into a series resonator in (b). The elements of the thirdresonator are scaled with respect to the capacitance and inductance of the first resonatorin (c).132.2. DERIVATION OF THE SERIES-SERIES IPT NETWORK FROM FILTER THEORY(b)k12C1,Sm21m21L1,SC1,Sm31m31L1,Sk23RLR0,S RL,S(a)−M1,S −M1,SM1,S−M2,S −M2,SM2,Sm21L1,SC1,Sm21L1,SC1,S m31L1,SC1,Sm31K1,S K′2,SC1,SL1,SR0,SFigure 2.3: The impedance inverters are transformed to a T-network of lumped inductorsin (a). The T-network of inductors is modelled as an inductive coupling link in (b).For the next transformation shown in Fig. 2.2(b), a redundant inverter K3,S is addedto the output filter. This redundant inverter allows the third resonator to be scaled withrespect to the first resonator. The corresponding impedance of the redundant inverter isequal toK3,S = R4,S = kz g4. (2.9)With the redundant inverter in the circuit, the third resonator elements can be con-verted to elements scaled by m31 with respect to the element values of the first resonator.The newly scaled third resonator is shown in Fig. 2.2(c) where the second and third in-verters impedances areK ′2,S = K2,S√m31 L1,SL3,S= kz g1√m31m21g2g3(2.10a)K ′3,S = K3,S√m31L1,SL3,S= kz g4√m31g1g3(2.10b)142.2. DERIVATION OF THE SERIES-SERIES IPT NETWORK FROM FILTER THEORYThe third impedance inverter K ′3,S can then be absorbed into the load resistance by re-placing R4,S withRL,S =(K ′3,S)2R4,S=m31 kz g1 g4g3. (2.11)The network with the readjusted load resistance is shown in Fig. 2.3.(a).In Fig. 2.3(a), the impedance inverters are replaced by a lumped-element equivalentcircuit consisting of a T-network of inductors. The equivalence of the ideal impedanceinverter with the lumped element circuit is exact at the resonance frequency. However, theconversion introduces some deviations in the frequency response as the frequency movesaway from the center frequency. The inductances M1,S and M2,S in the T-networks arerelated to the impedance of the inverters asM1,S =K1,Sωo(2.12a)M2,S =K ′2,Sωo. (2.12b)The T-networks of inductors can be modelled as mutually coupled inductors and thecircuit is simplified to a network of coupled resonators as shown in Fig. 2.3(b). This ispossible due to the current contributions from both coupling links passing through them21 resonator, allowing the resonator to simultaneously couple to both the transmit andreceive resonators. Expressions for the resonator coupling coefficients are derived from theT-model ask12 =M1,SL1,S√m21=∆√g1g2(2.13a)k23 =M2,SL1,S√m31=∆√g2g3. (2.13b)The synthesis of the series-series IPT network from a third-order filter prototype is nowcomplete. The circuit model in Fig. 2.3(b) neglects cross coupling between the transmitterinductance L1,S and the receiver inductance L3,S . Neglecting cross-coupling terms at theinitial design phase for higher-order WPT systems is common and significantly reducesthe complexity of deriving general expressions [13, 43, 44]. Cross-coupling can also beaccounted for in the physical design of the coils, with the opportunity to design the coilssuch that cross-coupling has little effect on the circuit [9].The methodology to synthesize series-series IPT links from filter theory can now begeneralized as shown in Fig. 2.4. For an N th-order filter prototype the design equationsare as follows:152.2. DERIVATION OF THE SERIES-SERIES IPT NETWORK FROM FILTER THEORYR0,SL2,SC2,SL3,SC3,SL1,SC1,SR(N+1),SLN,SCN,S(a)L1,Sk12C1,Sm2,1m21L1,SC1,SmN1mN1L1,SRL,SC1,SC1,Sm(N−1)1m(N−1)1L1,Sk(N−1)(N)Transmitter ReceiverRepeater ResonatorsR0,S(b)Figure 2.4: General band-pass filter prototype with N resonators is shown in (a). Thetransformed filter with series-series resonator coupling is shown in (b).Li,S =kz giωb, if i odd1ω2o Ci,S, if i even(2.14a)Ci,S =1ω2o Li,S, if i oddgikz ωb, if i even(2.14b)R0,S =kzg0(2.14c)R(N+1),S = kz gN+1, (2.14d)162.3. DERIVATION OF THE SHUNT-SHUNT TOPOLOGY IPT NETWORK FROM FILTER THEORYRL,S =kzmN1 g1 gN+1gN, if N oddkzmN1 g1gN gN+1, if N even(2.14e)k(j−1)(j) =∆√gj−1 gj. (2.14f)For these equations, i and j are indices that range from 1 through N and i ̸= j. Inthe transformed network shown in Fig. 2.4(b), the elements of the repeater and receiverresonators are scaled with respect to the transmitter coil by scaling factorsmj1. This allowsfor additional flexibility in the design of the coils as the resonators may have different sizeswithout effecting the operation of the IPT network.2.3 Derivation of the Shunt-Shunt Topology IPT Networkfrom Filter TheoryIn the previous section, the synthesis of a series-series IPT network from a filter net-work was reviewed. In this section, the methodology is expanded to include the synthesis ofshunt-shunt resonant IPT networks from filter theory. Shunt-shunt refers to a shunt capaci-tance being used for matching the IPT coils at the transmitter and receiver. The subscript‘P’ is used throughout this work to differentiate the shunt (parallel) resonator networkconfiguration from the series resonator configuration that is identified by the subscript ‘S’.The transformation of a third-order filter to a shunt-shunt IPT network will be com-pleted first. The complexities of converting the filter network to a shunt-shunt IPT networkare highlighted by the third-order transformation. Due to the complexity of the conversionto a shunt-shunt IPT network, an approximation is necessary to complete in the derivation.Therefore the frequency response of the shunt-shunt IPT network will be distorted from thefilter network. The third order example will then be generalized to Nth-order shunt-shuntIPT networks.The third-order filter prototype is shown in Fig.2.5(a) and the circuit element valuesare:R0,P = kz g0 R4,P = kz g4C1,P =g1kz ωbL1,P =1ω2o C1,PL2,P =kz g2ωbC2,P =1ω2o L2,PC3,P =g3kz ωbL3,P =1ω2o C3,P.(2.15)172.3. DERIVATION OF THE SHUNT-SHUNT TOPOLOGY IPT NETWORK FROM FILTER THEORYFor the first conversion in Fig. 2.5(a), impedance inverters K1,P and K2,P are used totransform the series resonator into a shunt resonator. For this transformation step, theinverter impedances areK1,P = K2,P =√m21L2,PC1,P= kz√m21g2g1. (2.16)In Fig. 2.5(b), a redundant inverter (K3,P) is added to the output of the filter. Theredundant inverter is necessary to allow for the circuit elements in the third resonator tobe scaled. The impedance of the redundant inverter isK3,P = R4,P = kzg4. (2.17)The impedance level of the third resonator can now be changed to m31√L1,P/C1,P byapplying an impedance transformation to the second and third impedance inverters. Thetransformed impedance inverters areK ′2,P = K2,P√m31C3,PC1,P=kzg1√m31m21g2g3 (2.18a)K ′3,P = K3,P√m31C3,PC1,P= kzg4√m31g3g1. (2.18b)The resultant network with all resonators scaled with respect to the first resonator isshown in Fig. 2.5(c). Next, the redundant inverter K ′3,P is absorbed into an equivalentload resistance RL,P as shown in Fig. 2.5(d). The equivalent resistance isRL,P =(K ′3,P)2R4,P=m31 kz g3 g4g1. (2.19)182.3. DERIVATION OF THE SHUNT-SHUNT TOPOLOGY IPT NETWORK FROM FILTER THEORYL1,PC1,P(a)R4,PR0,PL3,P C3,PL2,P C2,P(b) Redundant Inverter(c)K1,P K2,Pm21L1,PC1,Pm21C1,PL1,PK1,Pm21L1,PC1,Pm21K2,PL3,PC3,P K3,PR4,PC1,PL1,Pm21L1,PC1,Pm21K1,P K ′2,P m31L1,PC1,Pm31K ′3,PR4,PR0,PR0,PM1,P−M1,P −M1,PC1,P L1,P 2m21L1,P 2m21L1,PC1,Pm21K1,P K ′2,P m31L1,PC1,Pm31RL,pR0,PM2,P−M2,P −M2,P(d)RL,PFigure 2.5: Transformation of a third-order filter network into a circuit with shunt-shuntresonator coupling is shown. Inverters are used to convert the series resonator to a shuntresonator in (a). In (b), a redundant inverter is added to the load. The impedance level ofthe third resonator is scaled in (c). In (d), the third inverter is absorbed into an equivalentload resistance and the inverters are replaced by an equivalent Π-network of inductors.192.3. DERIVATION OF THE SHUNT-SHUNT TOPOLOGY IPT NETWORK FROM FILTER THEORYk1,2 k2,3(e)(c)C1,PL′1,P(d)L-NetworkEquivalent atC ′2,pZ2,pLT4LT3LT1 LT2(a)M1,P M2,PC1,PC1,Pm31C1,Pm21R0,PRL,pL′2,P−M ′1,P −M ′1,PM ′1,PR0,PL′′2,PC1,Pm21L′3,P−M ′2,PC1,Pm31−M ′2,PM ′1,PRL,PZ1,PL′1,P L′2,P−M ′1,P −M ′1,PM ′1,PL′3,P−M ′2,P −M ′2,PM ′1,PC1,Pm31RL,PR0,P Z3,pC1,PC1,P L′1,P C ′2,pL′2,PL′3,PC1,Pm31RL,PR0,P(b)C1,PL1 L2M ′1,PR0,PC1,Pm21L3C1,Pm31L4M ′1,PRL,PωoFigure 2.6: Shunt inductors are combined in (a). Π-networks of inductances converted toT-networks using ∆-Y transformation in (n). The series inductances are partitioned toform mutually coupled inductors in (c). The L-network is approximated by an equivalentseries capacitance at ωo in (d). The final transformed network is shown in (e).202.3. DERIVATION OF THE SHUNT-SHUNT TOPOLOGY IPT NETWORK FROM FILTER THEORYThe filter circuit shown in Fig. 2.5(c) requires a transformation to directly map theimpedance inverters to inductively coupled resonators. In Fig. 2.5(d), the inductancem21L1,P is split into two shunt inductors, each with a value of 2 m21L1,P. The impedanceinverters are then replaced with an equivalent Π-network of inductors. The inductancesM1,P and M2,P have a reactance equal to the impedance of the inverters; thereforeM1,P =K1,Pωo(2.20a)M2,P =K ′2,Pωo(2.20b)The shunt inductances on each side of the Π-networks of M1,P and M2,P in Fig. 2.5(d) areconsolidated into equivalent inductances whereLT1 =M1,P L1,PM1,P − L1,P (2.21a)LT2 =2m21M1,P L1,PM1,P − 2m21 L1,P (2.21b)LT3 =2m21M2,P L1,PM2,P − 2m21 L1,P (2.21c)LT4 =2m31M2,P L1,PM2,P − 2m31 L1,P , (2.21d)After using these equations, the circuit is transformed to the circuit shown in Fig. 2.6(a).In the next step, the Π-networks of inductors are replaced by an equivalent T-networkusing a ∆-Y transformation as shown in circuit Fig. 2.6(b). The inductance’s L1, L2, L3,and L4 in Fig. 2.6(b) are then separated into two series inductances, either M′1,P or M′2,P ,as shown in Fig. 2.6(c). The final inductances in the network are given byL′1,P =∆ kz g2ωo(g1 g2 − 2∆2) (2.22a)L′2,P =2∆ kz g2m21ωo(g1 g2 − 2∆2) (2.22b)L′′2,P =2∆ kz g2 g3m21g1ωo(g2 g3 − 2∆2) (2.22c)L′3,P =∆ kz g2 g3m31ωo(g1 g2 − 2∆2) . (2.22d)The next step in the synthesis of the network is to transform the T-networks of M ′1,Pand M ′2,P in Fig. 2.6(c) into inductively coupled resonators. Observe that in the middle of212.3. DERIVATION OF THE SHUNT-SHUNT TOPOLOGY IPT NETWORK FROM FILTER THEORYthe circuit network, the inductances L′2,P and L′′2,P are separated by a shunt capacitanceC1,P /m21. With this configuration, there is no coupling between L′2,P and L′′2,P and thecircuit in Fig. 2.6(c) cannot be related to a physical transmission link with three resonators.However, if the L-network marked in circuit Fig. 2.6(c) is capacitive at the center frequency,it can then be replaced by an equivalent series capacitance C ′2,P as shown in Fig. 2.6(d).The inductance L′2,P can then be simultaneously coupled to both the transmit and receiveresonators through the two T-networks of inductances. The transformation of the networkfrom Fig. 2.6(c) to (d) requires the impedances seen through the L-network Z2,P andthe equivalent series capacitance Z3,P to be equal at the operating frequency. We solveZ2,P = Z3,P at ωo and findC ′2,P =g1(g2 g3 − 2∆2) (4∆2 g42 + g22)2∆ g2 kzm21 ωo (g22 g3 − 4∆2(g2 + g3 g42)) . (2.23)30 32 34 36 38 40 42 44 46 48 50Frequency [MHz]051015Resistance (+) N=3"=0.16Z2,PZ3,P(a)30 32 34 36 38 40 42 44 46 48 50Frequency [MHz]-150-100-500Reactance (+)N=3"=0.16Z2,PZ3,P(b)Figure 2.7: The resistance (a) and reactance (b) seen at the impedance planes Z2,P andZ3,P from Fig. 2.6(c) and Fig. 2.6(d).222.3. DERIVATION OF THE SHUNT-SHUNT TOPOLOGY IPT NETWORK FROM FILTER THEORYFrom (2.22) and (2.23), in order for the system to be realizable, the condition ∆ <min[√g22 g3/4(g3 g24 + g2),√g2 g3/2]must be met. For the third order network (2.23)requires ∆ < 0.5773 which is is easily met for practical applications. Fig. 2.7 comparesthe impedance seen at the planes Z2,P and Z3,P from Fig. 2.6(c) and Fig. 2.6(d), for acenter frequency of 40.68 MHz and a normalized frequency bandwidth of 0.16. As (2.23)is only valid at the center frequency ωo, there is a clear distinction in the impedance of theplanes over the bandwidth of the filter response. This causes a distortion in the frequencyresponse which will be explored through circuit simulations.With this approximation, the transformation of the filter network to a shunt-shunt IPTnetwork is complete and shown in Fig. 2.6(e). The coupling coefficients in Fig. 2.6(e) aredefined ask12 =M ′1,P√L′1,P L′2,P=√2∆√g1 g2(2.24a)k23 =M ′2,P√L′2,P L′3,P=√2∆√g2 g3(2.24b)Note that the shunt-shunt IPT network coupling coefficients are scaled by a factor of√2when compared with the coupling coefficients for the series-series IPT network in (2.13).The increase in the coupling coefficient relates to an increase in the frequency bandwidthof the final IPT network when compared to the filter network, which will be shown ina following section. With the steps shown in Fig. 2.5 and Fig. 2.6, the synthesis is nowcomplete.The methodology to synthesize third-order IPT systems can be extended to generalizedexpressions for higher order systems. A general N th-order filter prototype is shown inFig. 2.8(a), with the elements in the filter network given byCi,P =gikz ωb, if i odd1ω2o Li,P, if i even(2.25a)Li,P =1ω2o Ci,P, if i oddkz giωb, if i even(2.25b)R0,P = kz g0 (2.25c)R(N+1),P = kz gN+1, (2.25d)232.3. DERIVATION OF THE SHUNT-SHUNT TOPOLOGY IPT NETWORK FROM FILTER THEORYC1,P L1,PC2,P L2,PC3,P L3,P CN,P LN,PR0,P R(N+1),P(a)l1,2C ′1,PlN−1,NTransmitter ReceiverRepeater ResonatorsL′1,PL′2,PC ′2,PL′(N−1),PC ′(N−1),PL′N,P C′N,PR0,PR′L,P(b)Figure 2.8: The canonical band-pass filter prototype with N resonators is shown in (a).The final transformed shunt-shunt IPT network is shown in (b).The transformed shunt-shunt IPT network is shown in Fig. 2.8(b). The values of thereactive elements are shown in Tables 2.1 and 2.2 for orders up to five. The couplingbetween adjacent resonators is given byk(j−1)(j) =√2∆√gj−1 gj. (2.26)Note that (2.26) is only valid for N > 2. For N = 2, there are no repeater coils and thecoupling coefficient is equal to ∆/√g1 g2, which is the same as the coupling coefficient inthe series-series IPT network in (2.14f). The equivalent load resistance for even and oddorder configurations areRL,P =kzmN,1 gN gN+1g1, if N oddkzmN,1 gNg1 gN+1, if N even(2.27)242.4. COMPARING SHUNT-SHUNT AND SERIES-SERIES TOPOLOGIESTable 2.1: Resonator inductance expressions for IPT networks with shunt-shunt resonatorport terminations.N L′1,P L′2,P L′3,P L′4,P L′5p2g2 kz∆ωo (g1 g2 −∆2)g2 kz∆m21ωo (g1 g2 −∆2)3g2 kz∆ωo (g1 g2 − 2∆2)2 g2 kz∆m21ωo (g1 g2 − 2∆2)g2 g3 kz∆m31g1 ωo (g2 g3 − 2∆2)4g2 kz∆ωo (g1 g2 − 2∆2)2 g2 kz∆m21ωo (g1 g2 − 2∆2)2 g2 g3 kz∆m31g1 ωo (g2 g3 − 4∆2)g3 g4 kz∆m4,1g1 ωo (g3 g4 − 2∆2)5g2 kz∆ωo (g1 g2 − 2∆2)2 g2 kz∆m21ωo (g1 g2 − 2∆2)2 g2 g3 kz∆m31g1 ωo (g2 g3 − 4∆2)2 g3 g4 kz∆m4,1g1 ωo (g3 g4 − 4∆2)g4 g5 kz∆m5,1g1 ωo (g4 g5 − 2∆2)The expressions given in Tables 2.1 and 2.2 are accurate for the shunt-shunt IPT net-works and have zero mismatch loss at the center frequency. However, there are usefulapproximations that can be made to simplify the expressions in Tables 2.1 and 2.2. Ingeneral, WPT links are narrowband (∆2 ≪ 1); therefore the denominator term for the in-ductor values shown in Table 2.1 can be simplified by neglecting the ∆2 term. The resultingsimplified expressions for inductances are given in Table 2.3. The corresponding resonatorcapacitances using the simplified inductance expressions are given by C ′N,P = (ω2o L′N,P)−1.The scaling of the repeater and receiver resonators with respect to the transmitter resonatoris simplified to a factor of 2mj1 for the repeater resonators and 2mj1 for the receiver res-onator. As further optimization will be needed in the design process, these simplifiedexpressions establish a starting point for the shunt-shunt IPT design.2.4 Comparing Shunt-Shunt and Series-Series TopologiesWith the derivation of both IPT network topologies from filter theory complete, theperformance of the topologies can be compared. The series-series IPT network has afrequency response which is ideally the same as the canonical filter prototype, with afrequency shift due to the lumped element model of the impedance inverter having limitedbandwidth. On the other hand, for the shunt-shunt topology, there is a clear distinctionbetween the canonical filter network response and the final IPT network response where theapproximation of an interstage L-matching network is made using a single capacitor (see2.6 (c) and (d)). The approximation of the L-network has a larger impact on the frequencyresponse than the approximation of the ideal impedance inverter in the coupling link. Thiscauses the frequency response of the final shunt-shunt IPT network to differ significantlyfrom the ideal filter response. The final IPT network also does not have the same elementvalues as the filter network it was derived from, due to the ∆-WYE conversion in 2.6 (a)to (b). Consequently,the shunt-shunt IPT network and the original filter prototype have252.4. COMPARING SHUNT-SHUNT AND SERIES-SERIES TOPOLOGIESTable 2.2: Resonator capacitance expressions for IPT networks with shunt-shunt resonatorport terminations.N C′1,P C′2,P C′3,P2g1kz ωbg1kz ωbm213g1kz ωbg1(g2 g3 − 2∆2) (4 g42∆2 + g22)2 g2 kz ωbm21 (g22 g3 − 4∆2(g2 + g3 g42))g1kzm31 ωb4g1kz ωbg1(g2 g3 − 4∆2) (g22 + 4 g52∆2)2 g2 kzωbm21 (g22 g3 − 4∆2 (2 g2 + g3 g52))g1(g3 g4 − 2∆2) (4∆2 + g32 g52)2 g3 kz ωbm31 (g32 g4 g52 − 4∆2(g4 + g3 g52))5g1kz ωbg1(g2 g3 − 4∆2) (g22 + 4 g62∆2)2g2 kzωbm21 (g22 g3 − 4∆2(2 g2 + g3 g62))g1(g3 g4 − 2∆2) (4∆2 + g32 g62)2 g3 kz ωbm31 (g32 g4 g62 − 4∆2(g4 + g3 g62))N C′4,P C′5p4g1kzm4,1 ωb5g1(g4 g5 − 2∆2) (g42 + 4 g62∆2)2 g4 kz ωbm41 (g42 g5 − 4∆2(g4 + g5g62))g1kzm5,1 ωbTable 2.3: Approximate expressions for inductances in an IPT network with shunt-shuntresonator port terminations when ∆2 → 0.N L′1,P L′2,P L′3,P L′4,P L′5p2kz∆ωo g1kz∆m21ωo g13kz∆ωo g12kz∆m21ωo g1kz∆m31ωo g14kz∆ωo g12kz∆m21ωo g12kz∆m31ωo g1kz∆m4,1ωo g15kz∆ωo g12kz∆m21ωo g12kz∆m31ωo g12kz∆m4,1ωo g1kz∆m5,1ωo g1Note that the corresponding capacitance of each resonators is C ′N,P =1ω2o L′N,P.262.4. COMPARING SHUNT-SHUNT AND SERIES-SERIES TOPOLOGIESdifferent frequency responses.In the following sections, examples are given to evaluate the differences in frequencyresponse for the synthesis of series-series and shunt-shunt IPT coupling networks. A com-parison is also made in terms of the size of the inductors (coils) for a specific IPT couplingbandwidth.2.4.1 Analysis of the Simulation Results for Series-Series andShunt-Shunt IPT NetworksTo compare series-series and shunt-shunt networks, simulations were made for the idealfilter prototype, the series-series IPT network, and the shunt-shunt IPT network in CadenceAWR Microwave Office. The networks are assumed to be lossless, with ideal lumpedelement components used in the simulation. To compare the networks, the magnitude of thetransmission coefficient frequency response is plotted for all the systems. The transmissioncoefficient offers insight into the impact of the transformation from a filter network to anIPT network, as the pass-band characteristics can be examined. The frequency response ofthe filter network, series-series IPT network and shunt-shunt IPT networks are comparedin Fig. 2.9 for a third order network in (a), a fourth order network in (b), and a fifth ordernetwork in (c).Fig. 2.9(a) shows the frequency response for the third order filter and the IPT networksderived from the filter. The series-series IPT system matches the filter response very closely,with only a slight shift in the band edges due to the non-ideal impedance inverter. Theshunt-shunt frequency response is clearly different from the filter response, which is dueto the simplification of the middle resonator in the derivation of the IPT network. Thiscauses a -3 dB bandwidth expansion from 16% to 25% with the frequency response nowcentered at 40.18 MHz. At the center frequency of 40.68 MHz, |S21| is still equal to 0dB, corresponding to 100% transfer efficiency. When the approximated equations fromTable 2.3 are used for the third order shunt-shunt IPT network, the band edges of thefrequency response are shifted to the right and the in-band ripple overlaps with the designfrequency of 40.68 MHz. This causes a loss of 0.11 dB at 40.68 MHz.Fig. 2.9(b) shows the frequency response for the fourth order filter and the IPT net-works derived from the filter. Similar to the third order networks, the series-series IPTsystem matches the filter response very closely, with only a slight shift in the band edgesdue to the non-ideal impedance inverter. The fourth order shunt-shunt frequency responsehas a larger bandwidth expansion than the third order system due to the added approx-imation necessary in adding an additional repeater coil to the network. This causes a -3dB bandwidth expansion from 16% to 29.7% with the frequency response now centeredat 39.9 MHz. At the center frequency of 40.68 MHz, |S21| is still equal to 0 dB. Whenthe approximated equations from Table 2.3 are used for the fourth order shunt-shunt IPTnetwork, the band edges of the frequency response are shifted to the right. However, incontrast to the third order network, the in-band ripple no longer overlaps the design fre-272.4. COMPARING SHUNT-SHUNT AND SERIES-SERIES TOPOLOGIES30 32 34 36 38 40 42 44 46 48 50 52Frequency [MHz]-15-12-9-6-30Transmission Coefficient [dB] N=3"=0.16Filter ResponseSeries-Series IPTShunt-Shunt IPTShunt-Shunt IPT ("2 ! 0)(a)30 32 34 36 38 40 42 44 46 48 50 52Frequency [MHz]-15-12-9-6-30Transmission Coefficient [dB] N=4"=0.16Filter ResponseSeries-Series IPTShunt-Shunt IPTShunt-Shunt IPT ("2 ! 0)(b)30 32 34 36 38 40 42 44 46 48 50 52Frequency [MHz]-15-12-9-6-30Transmission Coefficient [dB] N=5"=0.16Filter ResponseSeries-Series IPTShunt-Shunt IPTShunt-Shunt IPT ("2 ! 0)(c)Figure 2.9: Comparison of simulated transmission coefficients (|S21|) for a filter, a series-series IPT and a shunt-shunt IPT. Comparisons are given for a third order network (a), afourth order network (b), and a fifth order network (c).282.4. COMPARING SHUNT-SHUNT AND SERIES-SERIES TOPOLOGIESquency of 40.68 MHz. Therefore, there is ideally no loss in the approximated fourth ordershunt-shunt IPT network.Fig. 2.9(c) shows the frequency response for the fifth order filter and the IPT networksderived from the filter. Similarly to the previous results, the series-series IPT systemmatches the filter response very closely. The fifth order shunt-shunt frequency responseshows a larger bandwidth expansion than both the third and fourth order systems due tothe added approximations necessary in adding more repeater coils to the network. Thiscauses a -3 dB bandwidth expansion from 16% to 34.3% with the frequency response nowcentered at 39.8 MHz. At the center frequency of 40.68 MHz, |S21| is still equal to 0 dB.When the approximated equations from Table 2.3 are used for the fifth order shunt-shuntIPT network, the band edges of the frequency response are shifted to the right and thein-band ripple overlaps with the design frequency of 40.68 MHz. This causes a loss of0.29 dB at 40.68 MHz.34 36 38 40 42 44 46 48Frequency [MHz]-15-12-9-6-30Transmission Coefficient [dB] N=4Shunt "=8.75%Filter Response ("=0.16)Series-Series IPT (f0=40.30)Shunt-Shunt IPT (f0=41.02)Figure 2.10: Frequency response for modified series-series and shunt-shunt IPT networks,compared with an ideal filter response.A design example is shown in Fig. 2.10 where the fourth order shunt-shunt and series-series IPT networks were designed such that their frequency response matches the idealfilter response. A summary of the design changes can be found in Table 2.4. Table 2.4clearly shows that the series-series IPT network has a slight shift in center frequency due tothe approximation of the ideal impedance inverter. However, the impact on the bandwidthof the frequency response is negligible for the series-series IPT network. To match theseries-series IPT network to the filter response, the designed center frequency was loweredto 40.30 MHz. With the design parameters changed, the frequency response of the series-series IPT network is the same as the filter response. For the shunt-shunt IPT network,there is a frequency shift and a bandwidth expansion due to the approximation of the seriescapacitance in the repeater coils. To match the ideal filter response, the bandwidth of the292.4. COMPARING SHUNT-SHUNT AND SERIES-SERIES TOPOLOGIESTable 2.4: Bandwidth and center frequency for fourth order IPT designs.Design TransmissionParameters CharacteristicsOrder Topology ∆ f0 ∆ f0(N) (%) (MHz) (%) (MHz)4 Series 16.0 40.68 16.2 41.24 Shunt 16.0 40.68 29.7 39.94 Series 16.0 40.30 16.2 40.84 Shunt 8.75 41.02 16.2 40.8shunt-shunt IPT network was lowered to 8.75% and the center frequency was increased to41.02 MHz. The frequency response of the shunt-shunt IPT network still includes the in-band ripple in the pass-band of the response, which cannot be eliminated by changing thedesign parameters. However, the insertion loss at 40.68 MHz is 0 dB for the shunt-shuntIPT network, even with the designed center frequency being different than 40.68 MHz.2.4.2 Comparing Component Values for Series-Series and Shunt-ShuntIPT NetworksAs both the series-series and shunt-shunt IPT networks derived in this work beganwith the canonical low-pass filter prototype, it is easy to compare the resulting componentvalues in the networks. Filter theory is ideal for comparing the two topologies as theanalytical expressions derived from filter theory allows for parametric sweeps over thedesign variables. One of the main interests in comparing the two topologies relates to thesize of the inductors for series-series and shunt-shunt IPT systems. Second-order systemswere chosen to compare the two topologies.When comparing the two topologies, constraints were made on some of the designparameters to allow for parametric sweeps of other parameters. For this analysis, thesource and load resistances are assumed to be equal, relating to an asymmetry factor forboth series-series and shunt-shunt of m21 = 1. The impedance scaling factor kz is set to50 which relates to the source and load termination impedance being equal to 50 Ω. Thefilter coefficients for a maximally flat (Butterworth) response were chosen for the design,which for a second order system are g0 = g3 = 1 and g1 = g2 =√2. The frequencies atwhich the IPT networks were evaluated were chosen based on the ISM frequency bandsand are 6.78 MHz, 13.56 MHz, 27.12 MHz and 40.68 MHz.The values of the resonator inductance for both the series-series (blue) and shunt-shunt(red) are shown in Fig. 2.11 as a function of the normalized frequency bandwidth. As302.4. COMPARING SHUNT-SHUNT AND SERIES-SERIES TOPOLOGIESshown in Fig. 2.11, both the series and shunt resonator inductance decrease as frequencyincreases. For the shunt-shunt IPT network, the inductance levels are in the nH-levelrange, which is much smaller when compared to the series-series IPT network for whichthe inductance levels are in the µH-level range. Depending on the frequency bandwidth ofthe design, the shunt-shunt IPT network inductance can range from several nH to 100’sof nH. The series-series IPT network inductance ranges from 100’s of µH to several µHover the same frequency bandwidth. From this observation, as the frequency bandwidthdecreases, the shunt-shunt IPT inductance becomes smaller in contrast to the series-seriesIPT inductance which grows larger. This is true for higher order networks as the differencein inductance size is due to the scaling differences between shunt and series resonators infilter theory.The normalized frequency bandwidth is related to the coupling coefficient through(2.14f) for the series-series topology and (2.26) for the shunt-shunt topology. Note that(2.26) is only valid for filter networks with N > 2 and, for this analysis, the N = 2 shunt-shunt IPT network has the same coupling coefficient as the series-series IPT network. Forpractical applications, the resonators in an IPT link will be loosely coupled, where looselycoupled is defined as ∆ < 0.2 in terms of the normalized bandwidth. In Fig. 2.11, there isa clear separation in inductance values between the two topologies in this region.Due to practical considerations, deciding which topology to implement may dependon the IPT application. Applications where the coil design needs to be small and haveloose coupling, such as in biomedical fields, shunt-shunt coils may be preferred. Theseapplications include implantable medical devices, wearable electronics, and sensors [5, 16,22]. The transmission distance for these applications is short-range and the shunt-shuntIPT network to be preferred. For other applications such as electric vehicle (EV) chargingsystems, where the transmission distance is larger, the series-series IPT network is preferred[8, 20, 27, 28]. For a larger application such as EV charging, the coils can be designed tobe much larger which would allow for easier alignment of the coils.The analysis of the inductor size shows that each topology has merit for a number of dis-tinct applications. The physical design constraints of an application, such as transmissiondistance or available space, can influence the decision for which topology to chose.312.5. DESIGN AND TESTING OF IPT COILS0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5Normalized Frequency Bandwidth (")101102103104105106Transmitter-Side Inductance (nH)6.78 MHz 13.56 MHz 27.12 MHz 40.68 MHz6.78 MHz 13.56 MHz 27.12 MHz 40.68 MHzN=2SeriesShuntkz=50Figure 2.11: The transmitter inductance is shown vs. the normalized frequency bandwidthfor both series (blue) and shunt (red) IPT networks at four different ISM bands.2.5 Design and Testing of IPT CoilsTo verify the filter theory method for designing IPT networks, multiple coils weredesigned for N = 2, 3, 4 order networks. The coils were designed using the COMSOL FEMsimulation environment with help from their technical applications [67, 68]. The shunt-shunt coils have a low inductance so a simple planar coil with a single loop was chosen asthe coil geometry. The series-series coils required a larger inductance, so a planar spiralcoil was chosen for the coil geometry. Chip capacitors (size 0805) were soldered on eachcoil in either shunt or series with an SMA port to ensure the coils resonate.Analytical expressions were found from other works which were used to calculate aninitial coil design [5, 22, 25, 26, 30]. The coil design was then refined and tuned in COMSOLto ensure the geometry exhibited the proper inductance value. The 2D axisymmetricsimulator in COMSOL was chosen to design the shunt-shunt coils, due to the simplicityof the coil structure chosen. The 2D axisymmetric environment simplifes the analysis byassuming that any structures in the environment are rotated around the center axis a full360 degrees [68]. The magnetic field physics was used in the simulator which calculatesthe inductance and mutual inductance of the coils in the simulation environment [68]. The322.5. DESIGN AND TESTING OF IPT COILSmutual inductance can only be calculated in the stationary simulator, so to calculate themutual inductance at the desired frequency, the magnetic field flux through the coils wasused.The coil geometry for the planar coils is shown in Fig. 2.12 for series spiral coils (a)and shunt loop coils (b). The design parameters which was used to modify the inductanceof the coils was the inner diameter d, the number of turns n and the spacing betweenadjacent turns S. The trace width for all of the designs was chosen to be 2.5 mm. ForIPT designs with multiple coils, the spacing between adjacent coils was defined as ℓ(j−1)(j)where j = 2, 3, 4, ..., N . The coil to coil spacing parameter is illustrated in Fig. 2.12(c).For verification of the filter theory method, four shunt-shunt IPT networks were de-signed and one series-series IPT network was designed. The series-series network was asecond order network designed at 27.12 MHz and one of the shunt-shunt networks was alsoa second order network designed at 27.12 MHz. Three shunt-shunt networks were designedat 40.68 MHz: a second order network, a third order network and a fourth order network.The experimental results from the third and fourth order networks are used to verify theapproximation method used to synthesize higher order shunt-shunt IPT networks.For the shunt-shunt IPT designs, the analytical expressions from Tables 2.1, 2.2 wereused as an initial starting point to calculate the inductance’s and capacitance’s. As anexample, for the 40.68 MHz second order shunt IPT coils, a symmetric design was chosenwith both coils equal to L′1. Therefore, the calculated inductance and capacitance for thedesign were 22.4 nH and 692 pF. In COMSOL, the 2D simulator was used to find thediameter of the coil needed to achieve the required inductance (shown in Fig. 2.13(a)),which was 1.21 cm. For the second order design, the coupling coefficient is 0.1131 whichrelates to a mutual inductance between the coils of 2.536 nH. The ideal spacing betweenthe coils was found to be 0.85 cm which achieves the desired mutual inductance.The shunt coil designs from the 2D simulation were exported to the 3D full wavesimulation environment in COMSOL. The full wave environment simulates both the electricand magnetic fields in the coil geometry [67]. Lumped elements and ports can be defined inthe coil structure to estimate the performance of the IPT system. The simulation geometryis shown in Fig. 2.13(b), where the copper traces are simulated as flat perfect conductorsto reduce the simulation time and complexity. The S-parameters for the IPT network canbe calculated in the full wave environment to compare with the AWR simulations to ensurethe coil design is optimal. For the higher order shunt IPT coils, the design process is thesame with the additional coils added into the EM simulation, first in the 2D COMSOLenvironment, and then in the 3D COMSOL environment.332.5. DESIGN AND TESTING OF IPT COILSds w(a)dw1mm(b)`23Coil 1 Coil 2 Coil N-1 Coil N`(N−1)(N)Coil 3t `12(c)Figure 2.12: Geometry of a spiral inductor is shown in (a). The inductor is defined byan inner diameter d with n turns that have a turn-to-turn spacing S and trace width w.Geometry of an inductor with a single circular loop is shown in (b). A multi-coil IPT linkis shown in (c).342.6. EXPERIMENTAL RESULTS(a) (b)Figure 2.13: Screenshots from the COMSOL simulation environment are shown for the 2Daxis symmetric simulation in (a) and the 3D full wave simulation in (b).The design process for the series-series IPT coils had additional challenges due to thesize of the inductance needed for the coils. The 2D simulation environment in COMSOLwas configured with a magnetic field solver, which did not properly simulate the self-capacitance of the geometry. Therefore, the series coil design started in the 3D simulationenvironment, with the input impedance of the coil measured to estimate the equivalentinductance of the coil. The schematic for the series coil and the separation distance isshown in Fig. 2.12(a) and (c), respectively. Once the equivalent inductance was matchedto the analytically derived value of 2.59 µH, the matching capacitance of 13.3 pF was addedto ensure the coil was resonant at 27.12 MHz. In order to find the ideal separation distancebetween the coils, the separation distance was swept and the input impedance of the coilswas measured until the desired input impedance of 50 Ω was found. The ideal separationdistance was found to be 7.2 cm.2.6 Experimental ResultsWith the design of the coils completed in COMSOL, the coils were fabricated for test-ing. The frequency response of the IPT networks were measured using a PNA-X vectornetwork analyzer from Keysight Technologies. The coil design parameters are summarizedin Table 2.5 for the 27.12 MHz series-series and shunt-shunt IPT networks with a nor-malized bandwidth of 0.16 and port resistances of 50 Ω. The coil design parameters aresummarized in Table 2.6 for the 40.68 MHz shunt-shunt IPT networks with a normalizedbandwidth of 0.16 and port resistances of 50 Ω. The measurement setup for the second352.6. EXPERIMENTAL RESULTSorder IPT systems are shown in Fig. 2.14. All of the planar spiral coils were fabricated ona 1.57mm thick FR4 substrate with a copper trace thickness of 35 µm (1 oz. copper). Thetrace width used for all of the designs was 2.5 mm.(a) (b)(c) (d)Figure 2.14: Photographs of the experimental IPT systems. The second-order link in (a)has series-series resonator coupling and operates at 27.12 MHz. The IPT links in (b),(c), and (d) have shunt-shunt resonator coupling at 40.68 MHz with two, three and fourresonators respectively.362.6. EXPERIMENTAL RESULTSThe second order 27.12 MHz series-series IPT system was designed to have identicalresonators, each with an inductance of 2.59 µH and a matching capacitance of 13.30 pF.The IPT coils were implemented with a planar spiral coil with four turns (n), an innerdiameter (d) of 6.27 cm, and a spacing between adjacent turns (S) of 6 mm. To completethe coil structure, an additional copper trace was added to the backside of the board witha chip capacitance. The copper trace on the back of the board was connected to the topof the board through a via in the copper trace. A photo of the experimental setup for theIPT system is shown in Fig. 2.14(a).Table 2.5: 27.12 MHz IPT link design summary.Topology Parameter Value Parameter ValueNSeries fo 27.12 MHz n1(n2) 42 L1,S 2.59 µH d1(d2) 6.27 cmC1,S 13.30 pF ℓ12 7.2 cmk12 0.1131 S 6.00 mmm21 1Shunt fo 27.12 MHz n1(n2) 12 L′1,P 33.6 nH d1(d2) 1.69 cmC1,P 1.03 n F ℓ12 1.0 cmk12 0.113m21 1The second order 27.12 MHz shunt-shunt IPT system was designed to have identicalresonators, each with an inductance of 33.6 nH and a matching capacitance of 1.03 pF.The IPT coils were implemented with a circular loop coil with a single turn (n) and aninner diameter (d) of 1.69 cm. Trace extensions are added on either side of the loop toallow for chip capacitors to be soldered to the coil to achieve resonance. A photo of theexperimental setup for the IPT system is shown in Fig. 2.14(b).The comparison between the second order series-series IPT network and the secondorder shunt-shunt IPT network is shown in Fig. 2.15. The simulation results shown in thefigures refers to the ideal IPT network from the filter transformation which was simulated inAWR. The series-series frequency response shown in Fig. 2.15(a) shows a slight bandwidthexpansion when compared with the ideal simulation. The difference is due to the inductanceof the series coil being slightly higher than what was designed for. The reason for thisdiscrepancy may be due to the COMSOL simulation using flat perfect conductors, causingthe self-capacitance of the coils to not be fully accounted for in the simulation. It may also372.6. EXPERIMENTAL RESULTSbe due to the mesh size between the turns of the coils not being fine enough. The self-capacitance of the fabricated coils was higher which led to the inductance of the coils beinghigher as well. To compensate for the shift in the center frequency, the series capacitancewas lowered from 13.3 pF to 10 pF in the final design. The series-series coil had a loss of0.7 dB at 27.12 MHz. The measured shunt-shunt IPT frequency response in Fig. 2.16(a)matches the theoretical simulated IPT frequency response, with the exception of losses inthe measured data. The measured results show a loss of 0.9 dB for the the shunt-shuntIPT network which is slightly higher than the series-series IPT network.20 21 22 23 24 25 26 27 28 29 30 31 32 33 34Frequency [MHz]-15-12-9-6-30Transmission Coefficient [dB]N=2, "=0.16Series-SeriesSimulation ResultsExperimental Results(a)20 21 22 23 24 25 26 27 28 29 30 31 32 33 34Frequency [MHz]-15-12-9-6-30Transmission Coefficient [dB] N=2, "=0.16Shunt-ShuntSimulation ResultsExperimental Results(b)Figure 2.15: The simulated and measured transmission coefficient (|S21|) of the second or-der 27.12 MHz IPT links with series-series and shunt-shunt resonator coupling are shown in(a) and (b), respectively. Both links were designed for an impedance of 50 Ω, a normalizedbandwidth of 0.16, and an asymmetry factor of m = 1.382.6. EXPERIMENTAL RESULTSThe second order 40.68 MHz shunt-shunt IPT system was designed to have identicalresonators, each with an inductance of 22.40 nH and a matching capacitance of 692.0 pF.The IPT coils were implemented with a circular loop coil with a single turn (n) and aninner diameter (d) of 1.21 cm. Trace extensions are added on either side of the loop toallow for chip capacitors to be soldered to the coil to achieve resonance. A photo of theexperimental setup for the IPT system is shown in Fig. 2.14(b).The third order 40.68 MHz shunt-shunt IPT system was designed to have non-identicalresonators. The transmit and receive coils were designed to have an inductance of 32.2 nHand a matching capacitance of 489 pF. The transmit and recieve coils were implementedwith a circular loop coil with a single turn (n) and an inner diameter (d) of 1.63 cm. Therepeater coil was chosen to have a inductance double (m21 = 1)) that of the transmit andreceive coils of 64.4 nH, with a matching capacitance of 250.8 pF. The repeater coil wasimplemented with a circular loop coil with a single turn (n) and an inner diameter (d) of2.90 cm. Trace extensions are added on either side of the loops to allow for chip capacitorsto be soldered to the coil to achieve resonance. A photo of the experimental setup for theIPT system is shown in Fig. 2.14(c).The fourth order 40.68 MHz shunt-shunt IPT system was designed to have non-identicalresonators. The transmit and receive coils were designed to have an inductance of 42.34 nHand a matching capacitance of 374 pF. The transmit and recieve coils were implementedwith a circular loop coil with a single turn (n) and an inner diameter (d) of 2.05 cm. Therepeater coils were chosen to have a inductance slightly smaller (m21 = m31 = 0.4) thatof the transmit and receive coils of 33.33 nH, with a matching capacitance of 513 pF forC ′2,P and 520 pF for C′3,P . The repeater coils were implemented with a circular loop coilwith a single turn (n) and an inner diameter (d) of 1.69 cm. Trace extensions are addedon either side of the loops to allow for chip capacitors to be soldered to the coil to achieveresonance. A photo of the experimental setup for the IPT system is shown in Fig. 2.14(d).The measurement results for the 40.68 MHz shunt-shunt IPT networks are shown inFig. 2.16. A frequency of 40.68 MHz was chosen for the shunt coils as the design of theseries coils became quite challenging at higher frequencies due to the self-resonance of thecoils. Therefore, the shunt coils are preferred at higher frequencies due to the simplicity ofthe coils and the design process.As can be seen in Fig. 2.16(a), the second order shunt IPT experimental result showsgood agreement with the theoretical response which is expected, as the shunt-shunt IPTsystem at 27.12 MHz also followed the theoretical response. The coils for the second order40.68 MHz IPT design have a separation distance of 0.8 cm. The insertion loss for thesecond order 40.68 MHz design is 0.78 dB due to the copper losses in the coil.392.6. EXPERIMENTAL RESULTSTable 2.6: 40.68 MHz IPT link design summary.Topology Parameter Value Parameter ValueNShunt fo 40.68 MHz n1(n2) 12 L′1,P 22.40 nH d1(d2) 1.21 cmC1,P 692.0 pF ℓ12 0.8 cmk12 0.113m21 1Shunt fo 40.68 MHz n1(n2) 13 L′1,P(L′3,P) 32.20 nH n3 1L′2,P 64.40 nH d1(d3) 1.63 cmC1,P(C′3,P) 489.0 pF d2 2.90 cmC ′2,P 250.8 pF ℓ12 0.8 cmk12(k23) 0.16 ℓ23 0.8 cmm21(m31) 1Shunt fo 40.68 MHz n1(n4) 14 L′1,P(L′4,P) 42.34 nH n2(n3) 1L′2,P(L′3,P) 33.33 nH d1(d4) 2.05 cmC1,P(C′4,P) 374.0 pF d2(d3) 1.69 cmC ′2,P 513.0 pF ℓ12 0.8 cmC ′3,P 520.0 pF ℓ23 0.8 cmk12(k34) 0.1902 ℓ34 0.8 cmk23 0.1224m21(m31) 0.4m4,1 1402.6. EXPERIMENTAL RESULTS30 32 34 36 38 40 42 44 46 48 50Frequency [MHz]-15-12-9-6-30Transmission Coefficient [dB] N=2, "=0.16Shunt-ShuntFinal IPT (Ideal)Experimental Results(a)30 32 34 36 38 40 42 44 46 48 50Frequency [MHz]-15-12-9-6-30Transmission Coefficient [dB] N=3, "=0.16Shunt-ShuntFinal IPT (Ideal)Final IPT (Cross Coupling)Experimental Results(b)30 32 34 36 38 40 42 44 46 48 50Frequency [MHz]-15-12-9-6-30Transmission Coefficient [dB] N=4, "=0.16Shunt-ShuntFinal IPT (Ideal)Final IPT (Cross Coupling)Experimental Results(c)Figure 2.16: Simulated and measured transmission responses for shunt-shunt IPT links with(a) two resonators, (b) three resonators and (c) four resonators. The simulated responseof the final transformed network is labelled as “Final IPT”. Simulation responses are alsoprovided including cross-coupling in third and fourth order designs.412.7. SUMMARYThe measurement results for the third order 40.68 MHz shunt-shunt IPT networks isshown in Fig. 2.16(b). Two simulation results are shown for the third and fourth order IPTsystems, one without cross-coupling (the ideal case) and the other with cross-coupling. Ascan be seen in Fig. 2.16(b), the third order shunt IPT system should not be affected bycross-coupling when comparing the ideal IPT system with the simulation result with cross-coupling. The repeater coil for the third order IPT network is larger than the transmitand receive coil and therefore helps the IPT system reject the impact of cross-coupling.The experimental result shows good agreement with the theoretical response, with a slightfrequency shift in the response and losses due to the finite Q of the coils. The separationdistance between adjacent coils for the third order design is 0.8 cm with the total separationdistance from the transmit coil to the receive coil being 1.6 cm. The insertion loss for thesecond order 40.68 MHz design is 0.87 dB due to the copper losses in the coils. The additionof the repeater coil which only contributed to an additional insertion loss of 0.09 dB whileincreasing the separation distance from 0.8 cm to 1.6 cm.The measurement results for the fourth order 40.68 MHz shunt-shunt IPT networks isshown in Fig. 2.16(c). As can be seen in Fig. 2.16(c), the fourth order shunt IPT system isaffected by cross-coupling when comparing the ideal IPT system with the simulation resultwith cross-coupling. The repeater coils for the fourth order IPT network are smaller thanthe transmit and receive coil which increases the impact of cross coupling. The experimentalresult shows good agreement with the simulated response with cross-coupling with a slightlylarger dip at the higher frequency end of the pass-band. The separation distance betweenadjacent coils for the fourth order design is 0.8 cm with the total separation distance fromthe transmit coil to the receive coil being 2.4 cm. The insertion loss for the second order40.68 MHz design is 0.89 dB due to the copper losses in the coils. When compared withthe third order IPT system, the addition of another repeater coil had minimal effects onthe insertion loss or frequency response of the system while increasing the total separationdistance to 2.4 cm.2.7 SummaryThe filter theory method was explored for both the series-series and shunt-shunt IPTtoplogies. The series-series topology was re-synthesized using filter theory and generalizedequations were presented for designers. The derivation of shunt-shunt IPT networks fromfilter theory is more difficult and impedance approximations are used to synthesize higherorder shunt-shunt IPT networks from filter theory. The approximation led to some vari-ation in the overall frequency response but the transmission loss was still ideally zero atthe center frequency. The frequency responses of higher order IPT networks with bothseries-series and shunt-shunt topologies were compared and, it is clear that for higher or-der networks, that the series-series IPT network matches the filter response, while theshunt-shunt IPT network only approximates the filter response. The inductance levels for422.7. SUMMARYboth topologies were compared and it was determined that both topologies offer designersdifferent applications for which either topology may be considered to be best. Finally, thecoils were designed and simulated in COMSOL to verify the filter theory approach. Thecoils were fabricated and tested in the lab using a vector network analyzer to measure thefrequency response. The frequency response for both the series-series and the shunt-shunttopologies were consistent with what was expected from theory. Other effects such as cross-coupling and losses due to finite quality factor were observed, and it was discussed howto mitigate the effects of cross-coupling by using the asymmetry factors in the generalizedequations for the IPT network.43Chapter 3An Inverter Model for ImpedanceCompensation Techniques in IPTSystemsIn Chapter 2 it was shown that impedance inverters can be used to transform fil-ter networks into IPT systems. In this chapter, the benefit of representing IPT links asimpedance inverters will be discussed as a method to achieve impedance compensation forspatial variations between the transmitter and receiver coils.There are currently many methods for impedance matching in the design of WPT sys-tems which involve the use of lumped LC circuits for impedance matching [6–8]. However,in order to achieve the desired impedance with these methods, circuit simulation is requiredto optimized the lumped components. The use of impedance inverters to compensate forvariations in gap spacing has been used in [43] with variable capacitors at the input of thenetwork to modify the input impedance.Amplifier/InverterMatchingNetworkMatchingNetworkRectifier Load/DeviceMagnetic CouplingZin(k) =K(k)2ZLZLkK(k)Figure 3.1: A block diagram of the impedance compensation method.In this chapter, an impedance inverter model for adapting the load impedance to com-pensate for changes in gap spacing will be derived. A block diagram of the system setup isshown in Fig. 3.1, where the input impedance of the IPT network can directly be relatedto the load impedance. Using this model, it will be shown that only the load impedanceneeds to be modified in order to compensate for variations in the gap spacing between thecoils. This will be compared for both series-series and shunt-shunt IPT networks.443.1. IMPEDANCE COMPENSATION TECHNIQUE FOR SERIES-SERIES IPT NETWORKS3.1 Impedance Compensation Technique for Series-SeriesIPT networksIn this section, it will be shown how variations in the separation distance betweenseries-series IPT coils can be fully compensated by varying the load resistance. FromChapter 2, it was shown how impedance inverters can be used to transform a filter networkto a series-series IPT network by modelling impedance inverters in the filter network asmutually coupled coils. If the synthesis is reversed, any series-series IPT network can berepresented as a cascade of series resonators and impedance inverters. A second ordersystem is analyzed first, then the theory is expanded to include higher order networks.3.1.1 Analysis of Series-Series IPT Networks using Impedance InvertersThe circuit model for a second order series-series IPT network is shown in Fig. 3.2(a).The inductors L1 and L2 model the self inductance of the coils, and the capacitors C1 andC2 are the matching capacitance which resonate with the self inductances. The transmitresonator is L1C1 and the receive resonator is L2C2. The coupling coefficient K12 modelsthe mutual coupling between the coils. RS and RL are the termination resistance for thesource and load respectively.The coupling between the transmit and receive coil can be represented by an equivalentT-network of inductors, shown in Fig. 3.2(b), whereM12 is the mutual inductance betweenthe transmit and receive coils. The coupling coefficient k12 and the mutual inductance arerelated by M12/√L1L2. An asymmetry factor m = L2/L1 is established to relate the sizeof the transmit coil to the receive coil. Both resonators, L1C1 and L2C2, are resonant atωo; therefore C1 and C2 are given byC1 =1ω2oL1C2 =1ω2oL2=1ω2omL1. (3.1)As was seen in Chapter 2, the T-network of inductors inside the dashed box in Fig. 3.2(b)is an impedance inverter at ωo. The ideal impedance inverter can be used as an approx-imation of the T-network impedance inverter, as shown in Fig. 3.2(c). The impedance ofthe inverter is determined from the mutual inductance asK1 = ωoM12. (3.2)Since the resonators, L1C1 and L2C2, are series resonators, they become short circuitsat the resonant frequency ωo. Therefore, at resonance, the circuit in Fig. 3.2(c) can besimplified to the circuit in Fig. 3.2(d).With reference to the equivalent circuit at resonance in Fig. 3.2(d), the input impedanceisZin =K21RL=(ωoM12)2RL. (3.3)453.1. IMPEDANCE COMPENSATION TECHNIQUE FOR SERIES-SERIES IPT NETWORKSL1k12L2C1 C2(c)RS RLL2(b)L1C1 −M12 −M12M12InverterC2RS RLL2L1C1 C2K1Z1 Z2RS RLK1Zin(d)(a)RS RLFigure 3.2: Summary of steps to transform the equivalent network of two mutually coupledseries coils into an impedance inverter at the center frequency ωo.This equation shows that the input impedance seen through the IPT link is directly relatedto the mutual coupling from the transmit to the receive coil. Further, the input impedanceis purely resistive at ωo if the load impedance is resistive. Therefore, if the system isoperating at the resonant frequency, the input impedance is determined only by the loadimpedance and the mutual coupling between the two coils. If there are any changes in themutual coupling between the coils, the load resistance can be varied to compensate for thechange in the mutual coupling thereby maintaining a constant input impedance.The analysis can be expanded to higher order IPT networks and a third order series-series IPT network shown in Fig. 3.3(a) is analyzed next. There are three resonator struc-tures, L1C1, L2C2 and L3C3, that are all resonant at ωo. Asymmetry factors m21 = L2/L1andm31 = L3/L1 are used to relate the relative size of the resonators to each other. For theinitial analysis, cross-coupling between coils L1 and L3 is neglected. The effects of cross-coupling on the impedance compensation method will be evaluated later using simulationtools. It is common during the initial design phase to neglect the effects of cross-couplingto reduce the complexity of the analytical derivation for the general expressions [13, 43, 44].As further justification for neglecting cross-coupling, it was shown in Chapter 2, that withcareful design of the geometry of the coils, the asymmetry factors can also be used toreduce the impact of cross-coupling.In the next step, similar to the second order network, the coupling between coils L1L2and L2L3 is replaced by an equivalent T-network of inductors shown in Fig. 3.3(b). Themutual inductancesM12 andM23 represent the mutual inductance between coils L1L2 andL2L3, respectively. The coupling coefficient is related to the mutual inductance betweencoils L1L2 by k12 =M12/√L1L2; similarly, k23 =M23/√L2L3. The matching capacitance463.1. IMPEDANCE COMPENSATION TECHNIQUE FOR SERIES-SERIES IPT NETWORKSL1k12L2C1C2(c)RS RLL2(b)L1C1 −M12 −M12M12InverterC2RS RLL2L1C1 C2K1Z1 Z2RS RLK1Zin(d)(a)RSL3C3RLk23−M23 −M23M23L3 C3InverterK2L3 C3Z3K2Figure 3.3: Summary of steps to transform the equivalent network of three mutually cou-pled series coils into a cascade of two impedance inverters at the center frequency ωo.Cross-coupling between non-adjacent coils is assumed to be negligible.473.1. IMPEDANCE COMPENSATION TECHNIQUE FOR SERIES-SERIES IPT NETWORKSof each resonator is related to the inductance through the resonant frequency ωo asC1 =1ω2oL1C2 =1ω2om21 L1C3 =1ω2om31 L1. (3.4)The T-networks of inductors shown in the dashed boxes in Fig. 3.3(b) are then re-placed with ideal impedance inverters in Fig. 3.3(c). The impedance inverters are relatedthrough the mutal inductance of the coils as K1 = ωoM12 and K2 = ωoM23. Since all ofthe resonators L1C1, L2C2 and L3C3 are series resonators, the circuit can be simplified toFig. 3.3(d) at the resonant frequency ωo. The input impedance of the IPT network canthen be expressed as a cascade of impedance invertersK1 andK2 with the input impedancegiven byZin =K21 RLK22=(M12M23)2RL. (3.5)Note that the expression for the input impedance of the third-order network is differentthan the input impedance for the second order network. For the third order network, theinput impedance of the network is no longer dependent on the resonant frequency of thecoils. Also, the relationship between the load impedance RL is linear for the third ordernetwork and is inversely proportional to RL for the second-order network.To simplify the impedance analysis for IPT networks with an arbitrary number of res-onators, cross-coupling between non-adjacent coils is considered to be negligible. An IPTlink with N resonators can be converted to an impedance inverter network with N − 1inverters with impedances Ki = ωoM(i)(i+1) for i = 1, 2, 3, ..., N − 1. The mutual induc-tance between adjacent resonators is represented as M(i)(i+1) and the resonant frequencyof the resonators is ωo. If Li is known, then the corresponding resonator capacitance’sare Ci = 1/(ω2oLi). The method for synthesis the input impedance from the second andthird order networks can then be expanded for higher order networks and the generalizedexpression for the input impedance is given byZin =(ωoM12)2RL, if N = 2RL[(N−1)/2∏i=1(M(2i−1)(2i)M(2i)(2i+1))2], if N odd(ωoM12)2RL[N/2−1∏i=1(M(2i+1)(2i+2)M(2i)(2i+1))2], otherwise.(3.6)As can be seen from (3.6), the input impedance is inversely proportional to RL for evenorder IPT networks and is linearly proportional to RL for odd order networks.483.1. IMPEDANCE COMPENSATION TECHNIQUE FOR SERIES-SERIES IPT NETWORKS3.1.2 Impedance Compensation Technique for Series-Series IPTNetworksFor an IPT design, the frequency of operation, load resistance, and source resistance areimportant characteristics which determine the performance of the system. The variablesare linked to the size and spacing of the IPT resonators in the system. For IPT networksto operate at a high efficiency, the spacing between resonators is important to ensurethe proper coupling between the resonators. Variations in the spacing between adjacentresonators can have drastic effects on the efficiency of the IPT system based on the changesin coupling coefficient which effect the input impedance. The simplified inverter modelwhich corresponds to the IPT network at resonance, can be used to show how the variationin the spacing between adjacent coils can be compensated if the load resistance is changed.As shown in (3.6), the input impedance of the IPT networks is directly related to themutual inductance between the coils. As coil spacing changes, the mutual inductances be-tween coils changes, and consequently the input impedance changes. As the self-inductanceof the coils is not determined from the spacing between adjacent coils, there should be noeffect on the resonant frequency of the inductors and all series resonators will still act asshort circuits at ωo. Therefore (3.6) is still valid for any separation distance between coilsand the IPT network can be simplified into a cascade of impedance inverters which modelthe mutual coupling between adjacent coils.If the load resistance is considered to be constant, then it is clear from (3.6) that theinput impedance will vary as a function of the separation distance between coils. However,if the load resistance is assumed to be variable, the input impedance can be set to a constantZ0. The optimal load resistance for a given spatial distance between adjacent coils isRopt =(ωoM12)2Z0, if N = 2Z0[(N−1)/2∏i=1(M(2i−1)(2i)M(2i)(2i+1))2]−1, if N odd(ωoM12)2Z0[N/2−1∏i=1(M(2i+1)(2i+2)M(2i)(2i+1))2], otherwise.(3.7)The variations in separation distance in (3.7) are represented by the mutual inductancebetween adjacent coils. For coil structures where the mutual inductance can be solvedanalytically, this method allows the designer to find the optimal load resistance necessaryto fully compensate for variations in the spatial distance between coils. For the largemajority of coil structures, the complexity of the geometry will not allow the designer todetermine the mutual inductance analytically. Therefore, electromagnetic simulations arerequired to characterize the mutual inductance for this case. The inverter based methodfor impedance compensation requires the physical IPT network to be equivalent to thecircuit structures in Fig. 3.2(a) or Fig. 3.3(a). If the IPT network can be equated to these493.2. IMPEDANCE COMPENSATION TECHNIQUE FOR SHUNT-SHUNT IPT NETWORKSfilter structures, then this method provides the designer a way to theoretically calculatethe optimal load resistance based on the separation distance between coils.3.2 Impedance Compensation Technique for Shunt-ShuntIPT NetworksIn this section, the conversion of shunt-shunt IPT networks to a cascade of impedanceinverters will be shown. It is interesting to note the differences between the series-series andshunt-shunt topologies, as the shunt-shunt topology requires additional steps to convert thenetwork to cascade of impedance inverters. For series-series IPT networks, the resonantfrequency of each resonator is constant and equal to ωo. For shunt-shunt IPT networks,the filter theory approach from Chapter 2 shows that for a maximally flat response, theresonant frequency of the transmitter and receiver coils is not equal to the center frequencyof the frequency response.3.2.1 Impedance Compensation for Second Order Shunt-Shunt IPTNetworksThe circuit model for a second order shunt-shunt IPT network is shown in Fig. 3.4(a).The inductors L1 and L2 model the self inductance of the coils and the capacitors C1 andC2 are the matching capacitances which do not resonate with the self inductances for theshunt topology. This is because the capacitors C1 and C2 will resonate with L′1 and L′2for the circuit in Fig. 3.4(e). The transmit resonator is L1C1 and the receive resonator isL2C2. The coupling coefficient km models the mutual coupling between the coils. RS andRL are the termination resistance for the source and load, respectively.The mutual coupling between the transmit and receive coils is represented by anequivalent T-network of inductors where M1 is the mutual inductance between the coils,shown in Fig. 3.4(b). The mutual inductance is related to the coupling coefficient whereKm = M12/√L1L2. Unlike in the case of the series-series IPT network, the T-network ofinductors cannot be equated into an impedance inverter as the resulting network would nothave a shunt-resonator on either side of the inverter. Therefore, an additional ∆-Y conver-sion is necessary to convert the T-network of inductors to a Π-network of inductors. Thereis also an additional conversion between Fig. 3.4(b) and (c), as the shunt combination ofL′1 and −M ′1 need to be separated. The equations for the circuit elements in Fig. 3.4(c)areM ′1 =L1 L2 −M12M1, L′1 =L1 L2 −M12L2, L′2 =L1 L2 −M12L1. (3.8)503.2. IMPEDANCE COMPENSATION TECHNIQUE FOR SHUNT-SHUNT IPT NETWORKSRS RLK1Zin(e)(a)RLK1RSRLRSL′2C2C1L′1(d)(c)RLRSL2 C2C1 L1km−M ′1 −M ′1M ′1InverterL2(b)L1 −M1 −M1M1RSC1RLC2L′2C2C1L′1Figure 3.4: Summary of steps to transform the equivalent network of two mutually coupledshunt coils into an impedance inverter at the center frequency ωo.With the ∆-Y conversion complete, the Π-network of M ′1 inductors can be representedas an ideal impedance inverter K1 with a value of K1 = ωoM′1 where ωo is the centrefrequency of the IPT network. The equivalent network of the circuit is shown in Fig. 3.4(d).When the IPT network is critically coupled, resonators L′1C1 and L′2C2 will resonate at ωo.Therefore, both resonators will become open circuits and the IPT network can be simplifiedto the circuit network in Fig. 3.4(e). The circuit configuration in Fig. 3.4(e) clearly showsthe input impedance is related to the load impedance through the impedance inverter asZin ≈ K21RL=(ωoM′1)2RL=ωo2 (L1 L2 −M12)2M21 RL. (3.9)However, it should be noted that this equation is only exact when the system is criticallycoupled. If the mutual coupling M1 is varied from the critical coupling value, the equationbecomes an approximation for the input impedance because L′1 and L′2 are dependent on themutual coupling in the system. Consequently, the resonators can no longer be representedas open circuits. However, equation (3.9) can still approximate the input impedance for theIPT network at various separation distances. IPT networks are generally loosely coupled,which means M1 is much less than L1 and L2. Therefore, variations in M1 should not513.2. IMPEDANCE COMPENSATION TECHNIQUE FOR SHUNT-SHUNT IPT NETWORKSsignificantly impact the calculated value for L′1 or L′2 as the M12 term is much smallerthan L1 L2 in (3.8).With (3.9), the approximate input impedance seen by the shunt-shunt IPT networkcan be used for impedance compensation in the network. If the required input impedanceis Zo, equation (3.9) can be solved for the optimal load impedance Ropt:Ropt ≈ L2 (L1 L2 −M12)M21 Zo. (3.10)It is clear that impedance compensation is more complicated for the shunt-shunt net-work due to the additional ∆-Y conversion compared to series-series networks. The impactof the ∆-Y conversion is that the resonant frequency of L′1C1 will shift as the mutual induc-tance varies. This leads to additional reactance in the network which impacts the accuracyof equation (3.9).3.2.2 Impedance Compensation for Higher Order Shunt-Shunt IPTNetworksWhen considering shunt-shunt IPT networks for higher order impedance matching, itis important to consider the topology of the repeater coil in shunt-shunt networks. Themutual coupling between two inductors can be modelled as current-dependent differentialvoltage sources in the repeater coil. Therefore, the repeater coil would have only one loopwith two voltage sources, an inductor, and a capacitor which are in series. This is shownin Fig. 3.5(a) where IPT coupling links are replaced with the equivalent T-network ofinductors in Fig. 3.5(b). In Fig. 3.5(b), current contributions from the mutual couplingfrom L1 and L3 both link the middle resonator L2C2 which makes further simplification ofthe circuit difficult.If the transformation of the network is continued in a similar manner as was done forthe second order network, then the T-networks of inductors are converted to equivalentΠ-networks of inductors as shown in Fig. 3.5(c). This creates shunt resonators for thetransmitter and receiver coils; however the repeater coil now has a topology which is nolonger a shunt or series resonator. In this circuit topology, if the Π-network of inductorsare replaced with impedance inverters (shown in Fig. 3.5(d)), then the transmitter andreceiver resonators will be open circuits at the resonant frequency. However, due to thecomplex structure of the middle resonator it is clear that the impedance of the middleresonator will not be a short or open circuit.Based on this analysis of the transformation of the third order shunt-shunt IPT network,it is clear that the impedance compensation method is not suitable for higher order shunt-shunt IPT systems. All higher order shunt-shunt IPT networks have the same repeatercoil circuit structure; which makes impedance compensation through the load resistanceimpossible. The issue of analyzing higher order shunt-shunt IPT networks is similiar to thefilter theory method of designing higher order shunt-shunt IPT networks in Chapter 2. In523.2. IMPEDANCE COMPENSATION TECHNIQUE FOR SHUNT-SHUNT IPT NETWORKSL2C2L1RLRS(a)L3 C3C1k12k23L2(b)L1 −M12 −M12M12RSC1RLC3L3−M23 −M23M23C2RLRSC1L′1(c)InverterC2L′2 L′′2M12 M23−M12 −M12 −M23 −M23 L′3C3InverterRLRSC1L′1(d)C2L′2 L′′2 L′3C3K12 K23Figure 3.5: Summary of steps to transform an equivalent network of three mutually coupledshunt coils into a cascade of two impedance inverters at the center frequency ωo. Cross-coupling between non-adjacent coils is assumed to be negligible.533.3. SIMULATION RESULTSthat chapter, an approximation was found which allowed the filter network to be convertedinto a shunt-shunt IPT network with only minimal effects on the frequency response. Itmay be possible to address the issue of higher order shunt-shunt impedance compensationwith a similar approximation, however that was not explored further in this work.3.3 Simulation ResultsTo verify the inverter based method of impedance compensation, magnetic field simu-lations were completed in COMSOL multiphysics to model the mutual inductance betweenthe coils. This was done for both third order series-series IPT networks and second ordershunt-shunt IPT networks.3.3.1 Simulation Results for Third Order Series-Series ImpedanceCompensationTo verify the application of the inverter based compensation theory, a third order IPTlink was designed at 27.12 MHz with identical resonantors. The IPT link was synthesizedto have a maximally flat (Butterworth) response using the filter theory approach whichwas described in Chapter 2. The IPT link has an impedance of 50 Ω for both the sourceand load impedances and a normalized frequency bandwidth of 0.16. To design the IPTlink, the 2D magnetic field physics in COMSOL multiphysics was used with the same coildesign which was used in section 2.5 in Fig. 2.12 [68]. The final planar spiral inductordesign has three turns of 35 µm (1 oz) copper trace with an inner diameter (d) of 100 mm.The copper traces have a width (w) of 2.5 mm and a turn to turn spacing (S) of 5 mm.The planar spiral inductor has an inductance of 1.86 µH which requires a capacitance of18.5 pF to achieve resonance at 27.12 MHz. To achieve the desired mutual coupling, thecoils were spaced at separation distances of ℓ1 and ℓ2 (see Fig. 2.12) equal to 6.7 cm whichcorresponds to mutual inductances (M12 and M23) of 0.208 µH. The coils are assumed tobe lossless for the simulation analysis.To verify the impedance compensation method, ℓ2 was chosen as the variable to changeseparation distance. An example of a practical application for this design would be anEV-charging system where the transmitter and repeater coils are fixed to a docking stationor stand and the receiver coil on the body of the EV is free to change position. Themagnetic field solver was used to characterize the mutual inductance between the coils asℓ2 was varied. The variation of the mutual inductances, M23 and M13, with respect toℓ2 is plotted in Fig 3.6(a). Note that M13 is the cross-coupling inductance between thetransmitter and repeater coil, andM12 is fixed at 0.208 µH due to ℓ1 being fixed at 6.7 cm.The results in Fig 3.6 shows that the cross coupling inductance has no impact on the in-put resistance of the IPT network. However, the cross-coupling term introduces a negativeinput reactance in the IPT network which increases as the separation distance increases.When comparing the mutual inductance to the cross coupling inductance in Fig 3.6(a) the543.3. SIMULATION RESULTSTable 3.1: Summary of impedance compensation simulation for the third-order IPT link.No Cross Coupling With Cross CouplingDistance Ropt Rin Xin Rin Xin(cm) (Ω) (Ω) (Ω) (Ω) (Ω)3.0 345.3 49.99 -0.070 50.12 -13.934.0 194.7 49.99 -0.060 50.13 -15.275.0 114.5 49.99 -0.046 50.14 -16.566.0 69.58 50.00 -0.023 50.15 -17.897.0 43.46 50.00 0.010 50.16 -19.068.0 27.82 50.00 0.060 50.17 -20.539.0 18.20 50.00 0.14 50.19 -21.9010.0 12.16 50.00 0.25 50.20 -23.19mutual coupling decreases at a faster rate than the cross coupling as the separation distanceincreases. Therefore the ratio of the cross coupling to the mutual coupling is increased,causing the input reactance to the network to increase as the separation distance increases.As the separation distance increases, the input resistance will increase in the network.The separation distance ℓ2 was swept from 3 cm to 10 cm to test the verification of theimpedance compensation method. For each distance, the load resistance was calculatedaccording to (3.7), with the resulting input impedance measured in the simulation. Theresults for the compensated input impedances are shown in Table 3.1 for the IPT networkwith and without cross-coupling. The results in Table 3.1 confirm the input impedanceof the network is maintained at 50 Ω for all distance when cross-coupling is neglected.When cross-coupling is included in the network, the input resistance is nearly maintainedat 50 Ω, for all distances. There is an additional input reactance in the network whencross-coupling is included which cannot be compensated for using the inverter impedancecompensation method. Despite the fact the cross-coupling terms were neglected in thederivation of the optimal resistance, the efficiency of the link is maintained above 95% atall distances. Therefore, the inverter impedance compensation method was found to beeffective for the compensation of spatial variation in third order series-series IPT networks.Additional impedance compensation could be obtained by adding a reactive element to theinput of the filter network to compensate for the negative reactance caused by the mutualcoupling.553.3. SIMULATION RESULTS3 4 5 6 7 8 9 10Separation Distance (cm)00.20.40.6Mutual Inductance(7H)Mutual Inductance (M23)Cross Coupling Inductance (M13)Optimal Spacing(a)3 4 5 6 7 8 9 10Separation Distance (cm)050100150200250Resistance (+)Rin No Cross CouplingRin Cross Coupling(b)3 4 5 6 7 8 9 10Separation Distance (cm)-25-20-15-10-505Reactance (+)Xin No Cross CouplingXin Cross Coupling(c)Figure 3.6: Simulated mutual inductance and input impedance (Zin) of a third-order IPTlink with a fixed load resistance of 50 Ω. The plot in (a) shows how the mutual inductanceand cross-coupling inductance vary with distance. The plot in (b) shows how the inputresistance Re(Zin) changes with distance, and the plot in (c) shows the input reactanceIm(Zin) changes with distance.563.3. SIMULATION RESULTS3.3.2 Simulation Results for Shunt-Shunt IPT Network CompensationTo test the compensation method shown in section 3.2, the 40.68 MHz second ordershunt-shunt IPT network was used to simulate the input impedance as a function of theseparation distance between coils. The mutual inductance for the IPT network was firstsimulated in the magnetic field solver in COMSOL multiphysics 2D axis symetric environ-ment described in section 2.5. The 2D simulation environment is accurate at determiningthe mutual inductance as shown from the experimental results from Chapter 2. The IPTnetwork was designed using the filter theory method and has a maximally flat (Butter-worth) response with an impedance of 50 Ω for both the source and load impedances. Thenormalized frequency bandwidth for the design is 0.16.The coils were designed in COMSOL as single loop indcutors with an inner diameterof 1.21 cm. The coils were printed on 35 µm (1 oz) copper with a trace width of 2.5 mm.The inductance for the coils was measured as 22.4 nH. The mutual inductance betweenthe coils was measured as the separation distance between the coils was swept from 0.4 cmto 1.12 cm. The mutual inductance measurements were then exported to AWR where themeasurements were used to calculate the resulting coupling coefficient in the IPT network.The comparison was made for the input impedance of the IPT network with a constantload impedance of 50 Ω and a compensated load impedance calculated from equation(3.9). Fig. 3.7 shows how the input resistance and input reactance vary as a function ofthe mutual inductance between the coils. The optimal spacing distance of 0.85 cm relatesto a mutual inductance of 2.536 nH.When the separation distance is decreased from 0.8 cm to 0.5 cm, the impedancecompensation method can reliably predict the load resistance necessary to compensatefor the separation distance variation. However, due to the mutual inductance varyingthe resonant frequency of the L′1C1 resonator, there is a residual input reactance termthat is not compensated for. When the separation distance is increased from 0.8 cm to1.2 cm, the input reactance of the network increases. The input resistance can still be fullycompensated for separation distances up to 1.2 cm. Beyond, 1.2 cm, the input reactanceincreases at a fast rate and the load compensation method is compromised. However, theimpedance compensation method is still useful as the insertion loss of the network shows.When impedance compensation method is used, there is only 0.4 dB loss for a separationdistance of 1.2 cm which equates to a transmission efficiency of 91.2%. On the other hand,without load compensation, the insertion loss is significantly higher and equal to 3.6 dB,or an equivalent transmission efficiency of 43.3%. It is clear from simulation results that,while the impedance compensation method is not exact for shunt-shunt IPT networks, itstill offers significant improvements in system efficiency.573.3. SIMULATION RESULTS0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2Separation Distance (cm)0255075100125150175Resistance (+) Input ResistanceCompensated Input Resitstance(a)0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2Separation Distance (cm)-40-30-20-1001020Reactance (+)Input ReactanceCompensated Input Reactance(b)Figure 3.7: The simulated input impedance (Zin) of the second-order shunt-shunt IPTnetwork with a fixed load resistance of 50 Ω is shown versus coil separation distance. Theplot in (a) shows the input resistance Re(Zin) and the plot in (b) shows the input reactanceIm(Zin).583.4. EXPERIMENTAL RESULTS(a) (b)Figure 3.8: The planar spiral inductor used for testing is shown in (a). The experimentalsetup is shown in (b) with the LC matching network used to modify the load impedance.3.4 Experimental ResultsFor the experimental verification of the inverter impedance compensation theory, the27.12 MHz series coils from Chapter 2 were used. The design specification for the coilswere four turns of 35 µm (1 oz) copper with a trace width of 2.5 mm, an inner diameter of62.7 mm and a turn-to-turn spacing distance of 6 mm. The inductor is shown in Fig. 3.8(a)with the experimental setup shown in Fig. 3.8(b) used to modify the load impedance ofthe IPT network.The coils were designed in Chapter 2 to have a maximally flat (Butterworth) frequencyresponse with a center frequency of 27.12 MHz and a normalized frequency bandwidth of0.16. The IPT network was designed for equal termination impedances of 50 Ω for boththe source and load terminations.The full wave FEM solver in COMSOL multiphysics [67] was used to characterize theIPT link and find the self-inductances of the coils as well as the mutual inductance betweenthe coils. The full wave simulation was chosen for the characterization of the coils to includethe effects of interwinding capacitance and mutual capacitance between the coils. In thesimulation, the separation distance between the transmitter and receiver coil (ℓ) was variedto observe the changes in the mutual inductance. The simulation results show that the coilinductance is equal to 2.59 µH. A series capacitance of 13.3 pF was added to the simulationto achieve resonance at 27.12 MHz for both coils. A plot of the simulation results for themutual inductance as a function of separation distance is shown in Fig. 3.9(a). The optimal593.4. EXPERIMENTAL RESULTSseparation distance was found to be 6.9 cm from the simulation results, which correspondsto an input impedance of 50 Ω. The mutual inductance is equal to 0.29 µ at 6.9 cm.The frequency response (S21) of the IPT link was measured with a PNA-X vectornetwork analyzer (VNA) from Keysight Technologies. The frequency response is shownin Fig. 3.9(b) for both the simulated and experimental IPT networks. Due to the self-capacitance of the inductors, the matching capacitance in the experimental network wasmodified from 13.3 pF to 10.4 pF to resonate at 27.12 MHz. The frequency bandwidthof the experimental IPT network is within 3% of the theoretical bandwidth. The largestdiscrepancy between the measured and simulated results is the insertion loss of the filters.The minimum insertion loss is approximately 0.67 dB, which is attributed to the finitequality factor of the inductors.The objective of the experimental work was to verify the load compensation theory asa method of maintaining a constant input impedance over a range of separation distancesbetween the IPT coils. To achieve this, a variable load resistance was implemented withan LC matching circuit which is connected to a 50 Ω load. The variable load networkis shown in Fig. 3.8(b). For load impedances (Ropt) less than 50 Ω, a series inductor incascade with a shunt capacitor was used. For load impedances greater than 50 Ω, a shuntcapacitor in cascade with a series inductor was used.The separation distance was varied over a range from 2.4 cm to 10.4 cm and the inputimpedance of the IPT network was measured with a fixed 50 Ω load. The measured andsimulated results are compared in Fig. 3.10. The input reactance is approximately zero forboth simulated and experimental results when the separation distance is increased fromthe optimal spacing distance. When the separation distance is decreased from the optimalspacing distance, the input reactance becomes capacitive due to the effects of mutualcapacitance between the coils. This is most apparent at the closest separation distance of2.4 cm where the measured reactance is -j97 Ω. If the mutual capacitance is significant,then the circuit model used for the derivation of the impedance compensation method isno longer valid and the method cannot be used to accurately predict the input impedance.Therefore, there is a practical range over which the impedance compensation method canbe used which is dependent on the geometry of the IPT coils.To compensate for the variation in the separation gap, the load impedance was changeduntil an input impedance (Zin) of 50 Ω was observed at the input of the IPT network. Theinput impedance into the matching network (ZMN) was then measured to compare withthe analytical value of the load impedance (Ropt) predicted through the load compensationtheory using equation (3.7). The results from the experiment are tabulated in Table 3.2.Table 3.2 shows the IPT network can be fully compensated over a separation dis-tance from 4.4 cm to 10.4 cm. The load impedance seen through the matching networkis approximately equal to the optimal load resistance which is predicted through the the-oretical method. This is especially apparent for gap distances larger than 5.6 cm. Atcloser separation distances, the load impedance starts to become capacitive because ofmutual capacitance between the coils. Based on these experimental results, it shows that603.4. EXPERIMENTAL RESULTS2 4 6 8 10 12 14 16Separation Distance (cm)00.20.40.60.81Mutual Inductance(7H)Optimal Spacing(a)20 21 22 23 24 25 26 27 28 29 30 31 32 33 34Frequency [MHz]-15-12-9-6-30Transmission Coefficient [dB]N=2, "=0.16Series-SeriesFinal IPT (Ideal)Experimental Resultsf0=27.12 MHzS21= -0.67 dB(b)Figure 3.9: Characterization of mutual inductance M12 with respect to the separationdistance between two planar inductors is shown in (a). At a separation distance of 6.9 cmand a frequency of 27.12 MHz, the input resistance is 50 Ω for a 50 Ω load. Shown in (b)is the S21 frequency response for the IPT system for the optimal distance of 6.9 cm.613.4. EXPERIMENTAL RESULTS2 3 4 5 6 7 8 9 10 11Separation Distance (cm)0200400600Resistance (+) Simulated RinMeasured Rin(a)2 3 4 5 6 7 8 9 10 11Separation Distance (cm)-80-60-40-20020Reactance (+)Simulated XinMeasured Xin(b)Figure 3.10: The simulated and measured input impedance (Zin) of the second-order IPTlink with a fixed load resistance of 50 Ω is shown versus different separation gaps. The plotin (a) shows the input resistance Re(Zin), while the plot in (b) shows the input reactanceIm(Zin).623.5. SUMMARYTable 3.2: Summary of impedance compensation measurements for the second-order IPTlink.Gap Distance Ropt RMN XMN Rin Xin(cm) (Ω) (Ω) (Ω) (Ω) (Ω)3.6 290.1 286.8 -12.5 51.5 6.74.4 183.9 143.0 -4.4 52.0 05.6 96.2 92.0 -5.5 50.1 06.6 57.8 57.8 -5.0 48.5 07.6 35.7 40.3 -0.7 50.0 08.4 24.8 22.5 0.3 50.0 09.4 16.0 14.0 2.0 50.1 010.4 10.6 9.0 3.75 50.5 -2.0the theoretical circuit model using an equvivalent impedance inverter to model couplingcan be used to predict the required load resistance that is required to maintain a con-stant input impedance. The model incorporates changes in mutual coupling by calculatingcorresponding changes in the inverter impedance.3.5 SummaryA theoretical framework to compensate for variations in the separation distance betweencoils for IPT networks was described. This was done by relating the mutual coupling inan IPT network to an impedance inverter at the design frequency. It was shown that thismethod can accurately predict the optimal load impedance for series-series IPT networksof any order. For shunt-shunt IPT networks, it was shown that this method can alsobe used to predict the optimal load impedance for second order networks, but is unableto predict the optimal load impedance for higher order networks. Simulated results wereshown to confirm the load compensation method for a third order series-series IPT networkand a second order shunt-shunt IPT network. Experimental results were then shown fora second order series-series IPT network, which showed the practical limitations of thetheory. At closer separation distances, the experimental results shows the coupling beingthe transmitter and receiver coils cannot be modelled as purely inductive, due to the mutualcapacitance between the coils. Overall, the experimental results demonstrated substantialcompensation for a range of distances between the transmitter and receiver and validatedthe theoretical predictions from the inverter compensation model.63Chapter 4A Class F−1Amplifier for IPTSystemsA high efficiency amplifier is required when considering IPT systems for commercialuse to ensure the end-to-end efficiency of the system is maximized. For this work, the ClassF−1 amplifier topology is selected for its high efficiency. In the circuit design of the classF−1 amplifier there is an output shunt resonator tuned to the fundamental frequency of thedesired RF signal. The shunt-shunt IPT networks which were designed in Chapter 2 allrequire a shunt resonator at the input of the IPT network which is tuned to approximatelythe fundamental frequency of the RF signal. Tests are conducted to determine whetherthe output resonator of the class F−1 amplifier can be replaced by the shunt-resonator atthe input of the IPT network, removing redundant shunt-resonators in the design process.The first test will be to replace the load of the amplifier with the shunt-shunt IPT network,shown in Fig. 4.1 step 1. The second test will be to replace the output resonator with theshunt-shunt IPT network, shown in Fig. 4.1 step 2.RLf2 = 2 f0 f4 = 4 f0f0VDDLDDL′2 C2C1 L′1kmRLStep 1Step 2(a) (b)Figure 4.1: The circuit schematic for the class F−1 power amplifier is shown in (a). Twomodifications to the amplifier are tested. In step 1, the load RL is replaced with thesecond order shunt-shunt IPT network shown in (b) and, in step 2, the shunt resonator inthe dashed box and the load are replaced by the IPT network in (b).The amplifier was chosen to operate at a frequency of 40.68 MHz which allowed fortesting of the amplifier with the higher order shunt-shunt IPT coils which were designed inChapter 2. Finally, the amplifier was tested with the second order shunt-shunt IPT coilswith a varying distance to determine the impact of load variation on the class F−1 amplifier.644.1. CLASS F−1AMPLIFIER DESIGN FOR WIRELESS POWER TRANSFER4.1 Class F−1Amplifier Design for Wireless Power TransferThe initial stage of the design process for the class F−1 amplifier is to choose a transistorwhich is suitable for the required power levels of the amplifier. For the design, a 10 WBLP27M810 power LDMOS transistor from Ampleon was chosen due to its rugged package,high efficiency, and high power gain [69]. The transistor has a high maximum drain voltageof 65 V which is required for the class F−1 operation [69]. A 10 W device was chosenfor testing the IPT coils as the shunt-shunt coils are much smaller than the series-seriescoils. The size and separation distance of the shunt-shunt coils is ideal for lower powerapplications, such as wireless charging for phones or other small devices.The class F−1 amplifier topology requires a switch mode input to ensure proper har-monic components in the drain voltage and current waveforms. A Texas InstrumentsLMG1020 gate driver was chosen to generate a square wave signal at the gate of the ampli-fier. The LMG1020 has a 1 ns pulse width with an input high threshold between 1.7-2.6 Vand and input low threshold between 1.1-1.8 V [70]. The switching characteristics of thegate driver are suitable for a switching frequency of 40.68 MHz or equivalently a switchingperiod of 24.6 ns. The output of the LMG1020 is a 5 V switched signal which is sufficientlyhigh to switch the BLP27M810 transistor which has a gate-source threshold voltage of1.5-2.3 V.4.1.1 Circuit DesignThe class F−1 amplifier circuit was simulated in AWR Microwave Office using the tran-sistor model for the BLP27M810 device. In place of the LMG1020 gate driver, which wasunable to be imported into AWR, an ideal current source with two resistors was used tosimulate the gate voltage signal. To model potential losses in the DC choke inductor, theinductor was modelled with a series resistance have an equivalent quality factor of 300.The inductance of the DC choke inductor was chosen to be 952 nH which allows for a highimpedance of 243 Ω while minimizing potential losses in the physical implementation ofthe design. An AC bypass capacitor was added to the input of the DC source to mitigatethe potential AC signals damaging the DC source. A DC blocking capacitor was addedbetween the drain of the transistor and the harmonic tuning network and the capacitorhas negligible impact on the circuit as the impedance of the DC blocking capacitor is 3 Ω.The final simulation circuit is shown in Fig. 4.2. The circuit in Fig. 4.2 includes induc-tance from the source node to ground to model the physical vias that connect the sourceto the ground plane. The circuit also includes capacitance models for the testing probesat the gate and the drain. At the drain, a 2.2 pF capacitance was added in series to thetesting probe to minimize the impact of the testing probe on the operation of the circuit.At the gate, the capacitance of the test probe is in shunt with the gate-source capacitancewhich minimizes the impact of the test probe on the operation of the amplifier.654.1. CLASS F−1AMPLIFIER DESIGN FOR WIRELESS POWER TRANSFERCAPID=C1C=Cdc pFCAPID=C2C=Cbypass pFDC_VID=V2Sweep=NoneV=Vdd VM_PROBEID=VP1M_PROBEID=VP3M_PROBEID=VP4PLCID=LC1L=L2*nH nHC=C2*pF pFPLCID=LC2L=L4*nH nHC=C4*pF pFPLCID=LC3L=Lo*nH nHC=Co*pF pFRESID=R1R=RL OhmINDID=L2L=2.5 nHM_PROBEID=VP2INDID=L3L=2.5 nHM_PROBEID=VP5M_PROBEID=VP6RESID=R4R=rloss OhmINDID=L1L=Ldc nHCAPID=C5C=2.2 pFM_PROBEID=VP7CAPID=C3C=Ctest pFRESID=R3R=1 OhmRESID=R2R=1 OhmAC_IID=I1Signal=SquareSpecType=Specify freqSpecBW=Min time resSweep=NoneTone=1Freq=40.68 MHzTimeRes=0.5 nsHI=5000 mALO=0 mATR=1 nsTF=1 nsTD=0 nsWINDOW=DEFAULTDCVal=0 mAACMag=5000 mAACAng=0 DegCAPID=C4C=Ctest pFGDS123SUBCKTID=S1NET="AMP_BLP27M810"SELF_HEATING=0.6Co: 3.912e-10 Lo: 3.912e-08RL=50f2=2*40.68e6Lo=1/(w^2*Co)L4=1/(w4^2*C4)nH=1e9w2=2*_PI*f2L2: 1.956e-08Q=5Vgpk=5 w4=2*_PI*f4C4: 9.781e-11w=2*_PI*fCo=Q/(w*RL)f=40.68e6Vdd=20C2: 1.956e-10L4: 9.781e-09L2=1/(w2^2*C2)f4=4*40.68e6pF=1e12C2=Q/(w2*RL)C4=Q/(w4*RL)rloss=w*Ldc*1e-9/QindQind=300Cdc=1320Cbypass=750Ldc=952rloss: 0.8111Ctest=11.5tw=8Figure 4.2: AWR circuit simulation schematic for the class F−1 design.For the harmonic tuning resonators, a loaded quality factor of 5 is used to ensure thebandwidth of the harmonic resonators is large enough to compensate for additional traceinductance and capacitance in the practical circuit. The resistive load at the output of theclass F−1 amplifier is assumed to be shunting the harmonic tuning resonators at 2 fo and4 fo respectively. When the transistor is on, there is a clear path from either resonator toground which allows for the load to shunt either resonator. When the transistor is turnedoff, there is only high impedance paths from the left side of the resonators to ground.Therefore, only the on mode for the transistor was considered and the load resistance wasused for the quality factor calculation for the harmonic tuning resonators. The outputresonator is in shunt with the load resistance and parallel resonator expressions were usedfor calculations.664.1. CLASS F−1AMPLIFIER DESIGN FOR WIRELESS POWER TRANSFER4.1.2 Circuit Simulation ResultsWith the circuit design of the class F−1 amplifier complete, a harmonic balance analysiswas completed in AWR to test the characteristics of the circuit. The circuit simulationincludes losses due to the transistor and the DC choke inductor, with the LMG1020 gatedriver simulated by a square wave AC current source.The simulated drain voltage and drain current waveforms for the BLP27M810 amplifierin Fig. 4.2 are shown in Fig. 4.3. The voltage waveforms are shown for a duration of 25 nswhich includes a full period of the 40.68 MHz signal, which is 24.6 ns. When comparing thesimulated results in Fig. 4.3 to the ideal class F waveforms in Fig. 1.2, it is clear that thecircuit does not achieve the ideal class F waveforms. The drain current in the simulatedcircuit has significant ripples and only approximates a square wave. The drain currentis also not equal to zero when the transistor is turned off, with the current flow into thetransistor being both positive and negative. The drain voltage waveform approximates ahalf-sinusoidal wave with the inclusion of an additional slope in the sinsoidal waveshapedue to the non-ideal characteristics of the amplifier. There is also slight ringing in thedrain voltage waveform when the transistor is turned off, likely due to the large ringing inthe drain current waveform.0 5 10 15 20 25Time (ns)-0.5-0.2500.250.50.7511.251.5Current (A)-10010203040506070Voltage (V)Drain Current (A)Drain Voltage (V)Figure 4.3: Simulated drain current and voltage waveforms for the class F−1 amplifierdesign.As shown in Fig. 4.3, there are clearly regions where both the drain current and drainvoltage are not equal to zero. This leads to power losses in the transistor as 100% powerefficiency is obtained only if the current and voltage waveforms are orthogonal. The currentthrough the DC choke inductor is also a source of power loss as there is a finite qualityfactor associated the the drain inductor. The circuit in Fig. 4.2 has a dc input power674.1. CLASS F−1AMPLIFIER DESIGN FOR WIRELESS POWER TRANSFERof 9.07 W, an output RF power of 8.34 W resulting in a simulated power efficiency of92%. While the ideal class F−1 design would have 100% efficiency, the efficiency of thesimulated circuit is a more practical estimation due to the non-ideal characteristics of theBLP27M810 transistor and the losses from the drain choke inductor.4.1.3 Circuit Board Layout and DesignThe class F−1 amplifier circuit board layout was completed in KiCAD PCB designsoftware. The circuit design from the previous section was converted to the KiCAD circuitboard schematic shown in Fig. 4.4. The circuit board schematic in Fig. 4.4 includes SMA,BNC and 0.5” header connectors as well as capacitor banks, which were not included inthe previous circuit design. The LMG1020 bias circuit is also included in the board layoutschematic.The 0.5” header connectors were used for the two DC power supplies, one forthe LMG1020 gate driver (J1) and another for the DC drain supply for the BLP27M810power transistor (J2). For the input waveform, a right-angle BNC connector (J3) wasused to supply the switching waveform from the waveform generator. At the output of theamplifier, a SMA connector (J4) was used which was connected to the RF power meter. Forthe probing of the gate (J5) and drain (J6) waveforms, standard upright BNC connectorswere used which where connected to oscilloscope probes.One of the objectives of using the class F−1 amplifier was to test the viability of replacingthe output resonator in the class F−1 design with the resonator in the shunt-shunt IPTnetwork. In order to test this setup without having to remove all of the circuit componentsin the output resonator, a 0 Ω resistor was used to connect the output resonator to theoutput of the amplifier. Therefore, to remove the output resonator from the circuit, onlythe 0 Ω resistor is required to be desoldered from the circuit.684.1. CLASS F−1AMPLIFIER DESIGN FOR WIRELESS POWER TRANSFERFigure 4.4: KiCad circuit schematic for the class F−1 amplifier design.The inductors in the circuit are hand wound inductors, with L2, L3 and L4 being air coreinductors to reduce losses in the circuit. The inductance of L1 needed to be large to providea large impedance to block the RF frequency components. Therefore, a ferromagnetic corewas needed to boost the inductance of the coil. For L1, the toroidal core which was usedis a “67” material core from Fair-Rite Products Corp. with a permeability of 40 [71]. Thiscore is suitable for applications up to 50 MHz making it an ideal choice for this application.Custom through-hole layouts were used to allow for easy interchangeability of the inductorsin the circuit.694.1. CLASS F−1AMPLIFIER DESIGN FOR WIRELESS POWER TRANSFERFigure 4.5: KiCad printed circuit board for the class F−1 amplifier design.4.1.4 Board Design Simulation ResultsWith the board design for the class F−1 amplifier complete, the AWR amplifier circuitsimulation was updated to evaluate the impact of the parasitic trace capacitances on thecircuit design. The parasitic capacitances for the large copper traces were measured as0.722 pF, 1.551 pF, 2.315 pF and 3.527 pF from left to right between the BLP27M810transistor and the output resonator (C17-C21 and C30) shown in Fig. 4.4. The impact ofthe parasitic trace capacitance on the drain voltage and current waveforms is not significanton the overall shape of the waveforms. However, it is clear that the addition of the parasiticcapacitance has a large impact on the ringing in the drain current waveform as shown inFig. 4.6. The ringing in Fig. 4.6 is far more pronounced than the ringing in Fig. 4.3.The impact of the parasitic capacitance is less impactful on the drain voltage waveform;however, the slope of the drain voltage sinusoidal wave in Fig. 4.6 is slightly different thanthe slope of the drain voltage waveform in Fig. 4.3.704.1. CLASS F−1AMPLIFIER DESIGN FOR WIRELESS POWER TRANSFER0 5 10 15 20 25Time (ns)-0.5-0.2500.250.50.7511.251.5Current (A)-10010203040506070Voltage (V)Drain Current (A)Drain Voltage (V)Figure 4.6: Simulated drain current and voltage waveforms for the class F−1 amplifierdesign with the addition of parasitic trace capacitance.The impact of the parasitic trace capacitance also has implications for the measurementof the drain waveform in the lab. The test probe is connected in series with a 2.2 pF capac-itor to minimize the impact of the test probe of the amplifier’s operation. A comparisonof the drain voltage waveform with and without trace capacitance is shown in Fig. 4.7.From Fig. 4.7 it is clear that the inclusion of the parasitic trace capacitance will causethe DC coupling of the circuit to increase. When observing the safe operating level forthe transistor, the DC coupling needs to be taken into account when measuring the peakdrain voltage through the oscilloscope. It is observed that even with the increased DCcoupling, the peak probe voltage with the trace capacitance is similiar than the peak probevoltage without the trace capacitance. Therfore, the scaling ratio from the probe voltageto drain voltage is only slightly impacted by the inclusion of the trace capacitance. Theprobe voltage to drain voltage ratio was found to be 8.81 V/V with the inclusion of thetrace capacitance.714.2. EXPERIMENTAL RESULTS0 5 10 15 20 25Time (ns)-4-202468Voltage (V)Gate VoltageProbe VoltageProbe Voltage with Trace CapacitanceFigure 4.7: Simulated gate and probe voltage waveforms for the class F−1 design. Theprobe voltage is shown for simulated results with and without parasitic trace capacitance.The addition of the trace capacitance has an impact on the efficiency of the amplifier aswell. The simulated circuit with the trace capacitance has a dc input power of 9.02 W, anoutput power of 8.20 W and an efficiency of 90.9%. Therefore, the impact of including thetrace capacitance in the design of the amplifier adds 1.1% of efficiency loss in the amplifier.4.2 Experimental ResultsThe circuit board design was fabricated by OSH park and assembled in the lab. Thecircuit components were placed on the board with solder paste and baked in a re-flowoven. All of the active components on the board were tested for continuity, using anAgilent 34401A digital multimeter, to ensure there were no shorts on the board from there-flow process and all of the passive components were checked to ensure they had a goodelectrical connection. The final step in the fabrication process was to solder the SMA, BNCand 0.5” header connectors onto the board as well as the hand wound inductors. The finalassembled circuit board is shown in Fig. 4.2.After fabricating the circuit, the experimental verification of the board could be com-pleted. For testing purposes, one Agilent E3632A DC power supply was used as a 5 Vvoltage source for the gate driver circuit and one Agilent DC E3634A power supply wasused as a variable voltage source for the BLP27M810 transistor. A 50 Ω, 40 dB attenuatorwas used with a Agilent E4417A EPM-P series power meter as the load for the amplifier.724.2. EXPERIMENTAL RESULTSThe gate and drain voltage waveforms were observed using an Agilent DSO-X-3034A digi-tal storage oscilloscope. The square wave input for gate driver circuit was a Rigol DG4162arbitrary waveform generator. In order to achieve a 50 % duty cycle at the gate of theBLP27M810 transistor, the 40.68 MHz square wave amplitude was set to 3.8 V. The testingapparatus is shown in Fig. 4.9 with the amplifier connected to the attenuator using a 50 Ωtransmission line. The transmission line was replaced with the shunt-shunt IPT networksduring experimental testing.Figure 4.8: Assembled class F−1 amplifier.Figure 4.9: Testing apparatus for the class F−1 amplifier.734.2. EXPERIMENTAL RESULTS4.2.1 Class F−1Amplifier Measurement ResultsThe class F−1 amplifier was first tested using a 50 Ω attenuator and power meter as theload. This was done to ensure that the harmonic resonant tanks and the output resonatorwere tuned properly for maximum power efficiency. One of the consequences of testingthe prototype in the lab was the realization that the copper trace distance between thecapacitors and the inductors for the harmonic tanks was causing the harmonic tanks to bemistuned in the prototype. To compensate for this, the circuit board mask was removeddirectly under the inductors and the capacitors were soldered directly across the inductors.This improved the efficiency of the prototype and the drain voltage waveforms.The measured drain and gate voltage waveforms for the BLP27M810 transistor areshown in Fig. 4.10. The gate voltage waveform was supplied by the LMG1020 gate driverand has a duty cycle of 50 % to ensure proper operations of the harmonic tanks in the classF−1 amplifier. The duty cycle was tuned by adjusting the voltage amplitude of the 40.68MHz square wave supply signal into the LMG 1020 gate driver. There is some overshootin the gate waveform due to the inductance of the copper traces between the LMG1020gate driver and the BLP27M810 transistor. The drain voltage waveform is not ideal andincludes harmonic elements in the waveform which prevent it from achieving the ideal half-sine wave characteristic. The waveform is also AC coupled, causing the voltage waveformto be offset by approximately -2.5 V relative to the waveform at the drain of the transistor.-20 -15 -10 -5 0 5 10 15 20 25Time (ns)-4-20246810Voltage (V)Amplifier DrainAmplifier GateFigure 4.10: Measured drain and gate voltage waveforms for class F−1 amplifier.It is clear from the drain voltage waveform that there are points where the voltagecharacteristics are not ideal, which leads to efficiency losses. When the transistor is on,there is a dip in the drain voltage which leads to power losses as the drain current wouldbe a maximum at this stage. When the transistor is off, the voltage waveform has a largenumber of harmonics which indicates the circuit is not performing ideally at this stage.744.2. EXPERIMENTAL RESULTSAdditional losses in the circuit come from resistive losses through the DC choke inductorand the other passive components on the circuit board.Even with the voltage waveform being non-ideal and the additional losses in the circuit,the amplifier still performs well. The measurements for the amplifier are shown in Table4.1 with the efficiency of the amplifier measured at 85.9 % at a load power of 7.75 W.Under these conditions, the amplifier is within a safe operating range and the drain voltagewaveform peak is below 65 V, the breakdown voltage for the device.Table 4.1: Class F−1 amplifier measurementSupply Supply Input Output End-to-End Max DrainVoltage Current Power Power Efficiency Voltage(V) (A) (W) (W) (%) (V)20 0.451 9.02 7.75 85.9 59.34.2.2 Class F−1Amplifier Performance with Shunt-Shunt IPT NetworksWith the amplifier confirmed to be functioning at a high efficiency, the shunt-shunt IPTnetworks were combined with the amplifier to test the effectiveness of the combined stages.The IPT links were inserted into the circuit between the output of the amplifier and the50 Ω attenuator and power meter. The drain and gate voltage waveforms are shown inFig. 4.11 for the amplifier with a second order shunt-shunt IPT link in (a), a third ordershunt-shunt IPT link in (b), and a fourth order IPT link in (c). The shunt-shunt IPT linksare the same IPT links which were designed and tested for confirmation of the the filtertheory approach for the design of shunt-shunt IPT links in Chapter 2. In that Chapteritwas found that the IPT links had copper losses that translated to a power transfer efficiencyof 83.6 %, 81.8 % and 81.5 % for the second, third and fourth order networks respectively.Therefore with the amplifier efficiency of 85.9 % it is expected the amplifier network wouldhave an efficiency of 71.8 %, 70.3 % and 70.0 % with the second, third and fourth orderIPT networks respectively.754.2. EXPERIMENTAL RESULTS-20 -15 -10 -5 0 5 10 15 20 25Time (ns)-4-20246810Voltage (V)Amplifier DrainAmplifier Gate(a)-20 -15 -10 -5 0 5 10 15 20 25Time (ns)-4-20246810Voltage (V)Amplifier DrainAmplifier Gate(b)-20 -15 -10 -5 0 5 10 15 20 25Time (ns)-4-20246810Voltage (V)Amplifier DrainAmplifier Gate(c)Figure 4.11: Experimental results for the class F−1 amplifiers with shunt-shunt IPT linkswith (a) two resonators, (b) three resonators and (c) four resonators.764.2. EXPERIMENTAL RESULTSThe waveforms in Fig. 4.11 have similiar characteristics to the waveforms in Fig. 4.10which is expected as ideally the IPT network will present a 50 Ω load to the amplifier.For all three IPT networks, the amplifier consistently has four peaks in the drain voltagewaveform when the gate is off and ripples in the drain voltage waveform when the gate isturned on. The third order network has slightly larger peaks in the drain voltage wave-form, which may be indicative of some slight mistuning of the separation distance betweencoils. However, as is shown in Table 4.2, the efficiency of the amplifier with the IPT linksapproaches the expected efficiency of the two systems together. A sign that all the IPTnetworks appear to be presenting the desired input impedance to the amplifier is that thesupply current is approximately 0.45 A for all the measurements in Table 4.2. Overallthe results in table 4.2 show the viability of using the class F−1 amplifier with the IPTnetworks.Table 4.2: Class F−1 amplifier measurements with IPT coupling linksOrder Supply Supply Input Output End-to-End Max DrainVoltage Current Power Power Efficiency Voltage(N) (V) (A) (W) (W) (%) (V)2 20 0.479 9.58 6.76 70.5 57.23 20 0.440 8.80 6.14 69.7 66.94 20 0.412 8.24 5.70 69.1 60.8One of the main reasons for choosing the class F−1 topology with the shunt-shunt IPTnetworks was to test if the output shunt resonator in the class F−1 amplifier could bereplaced with the shunt IPT resonator. This was tested in the lab by desoldering the 0Ω resistors which connect the output resonator with the load. The simulated results withthis configuration were promising, however the finite quality factor of the IPT coils was notincluded in the simulation. As the output resonator on the circuit board was implementedwith an air coil helical inductor, the output resonator will have a much higher qualityfactor than the IPT coil which is implemented as a spiral inductor printed on an FR4circuit board. Therefore, the testing results were not ideal, with the drain and gate voltagewaveforms shown in Fig. 4.12, and the amplifier measurements shown in table 4.3.The drain waveform for the amplifier using the IPT network as the output resonatoris shown in Fig. 4.12. When compared to the waveforms in Fig. 4.11, the ripples in thedrain voltage are increased significantly compared to when the output resonator on theclass F−1 amplifier is used. The increased harmonic ripples in the drain voltage waveformappear to correspond to a small increase in the drain current through the transistor, seenin Table 4.3 as an increase in the supply current. The increased current through the deviceappears to correspond to a loss of efficiency due to the non-zero resistive losses in the774.2. EXPERIMENTAL RESULTSdevice. The efficiency of the prototype amplifier with no output resonator combined withthe IPT network is 67.9 % ,which is a loss of efficiency of 2.6 %. This, combined withreducing the number of capacitors on the board by five, and reducing the need for the 0Ω jumper resistor and the need for the inductor L4 in Fig. 4.5, shows that the design maybe feasible if the transistor in the circuit can be protected by high currents in the device.Table 4.3: Class F−1 amplifier measurement with second order IPT coupling link and nooutput resonatorSupply Supply Input Output End-to-End Max DrainVoltage Current Power Power Efficiency Voltage(V) (A) (W) (W) (%) (V)20 0.498 9.96 6.77 67.9 59.7-20 -15 -10 -5 0 5 10 15 20 25Time (ns)-6-4-20246810Voltage (V)Amplifier DrainAmplifier GateFigure 4.12: Drain and gate voltage waveforms for class F−1 amplifier with no outputresonator and a second order IPT network as the load.4.2.3 Impedance Variation with a Class F−1AmplifierThe last experiment to be tested with the class F−1 amplifier and the IPT networks wasthe impact of separation distance for the IPT network on the performance of the amplifier.It was shown in Chapter 3 that the impedance compensation method is not exact forshunt-shunt IPT networks due to the ∆-Y conversion for the impedance inverter. However,the simulation measurements show that the impact of the conversion only adds a slight784.2. EXPERIMENTAL RESULTSreactive component to the input impedance. Therefore, the shunt-shunt IPT network canbe used to approximate an impedance change seen by the amplifier and judge how theamplifier performs under different loading conditions.Table 4.4 shows measurement results for the class F−1 amplifier when the second ordershunt-shunt IPT network has spatial gap changes from 0.4 cm to 1.2 cm. For the measure-ment, the supply voltage had to be lowered as at closer distances the impedance mismatchcauses an increase in the supply current. This would cause damage to the transistor, sothe supply voltage was lowered to protect the device. This is evident from table 4.4; whenthe separation distance is only 0.4 cm, the supply voltage was lowered to only 5 V and thesupply current is still higher than the supply current under ideal loading conditions whenthe supply voltage is set to 20 V in table 4.1.As is expected in the measurement results, the end-to-end efficiency reduces as theseparation distance moves further and further away from the ideal separation distance of0.8 cm. It is interesting to note that when the coils are moved closer to each other, thereappears to be a greater impact on the efficiency than when the coils are moved furtheraway from each other. This is due to the fact that the input impedance from the IPTnetwork varies non-linearly with distance as the mutual inductance change is not linearwith distance. The input and output power both increase when the coils are moved closertogether, and conversely power decreases when the coils are moved further apart due tothe impedance change seen by the amplifier.Table 4.4: Class F−1 amplifier measurements with spatial variationSeparation Supply Supply Input Output End-to-End Max DrainDistance Voltage Current Power Power Efficiency Voltage(cm) (V) (A) (W) (W) (%) (V)0.4 5 0.64 3.2 1.28 39.8 18.60.6 10 0.66 6.6 4.11 62.3 28.60.8 10 0.39 3.9 2.82 72.1 27.21.0 10 0.22 2.2 1.53 70.8 25.01.2 10 0.11 1.1 0.566 50.9 23.6These results show that the amplifier can perform reasonably well if the separation dis-tance is increased with minimal risk of damaging the transistor, as both the supply currentand maximum drain voltage decrease as the separation distance is increased. However,when the separation distance is decreased, the supply voltage must be lowered to protectthe transistor in the amplifier.794.3. SUMMARY4.3 SummaryA class F−1 amplifier was designed for the purpose of testing the previously designedIPT coils in an end-to-end system. The class F−1 topology was chosen with the shunt-shuntIPT networks to test the potential of replacing the output resonator of the class F−1 designwith the shunt resonator in the IPT network. The prototype amplifier was tested firstwith a 50 Ω load and then with various IPT configurations. An efficiency of 85.9 % wasmeasured for the class F−1 amplifier through a 50 Ω attenuator, with the output powermeasured by a power meter. With the IPT networks added to the prototype amplifier,the efficiency for the end-to-end system was measured as 70.5%, 69.7% and 69.1% for IPTnetworks with two, three and four coils respectively. The input and output power forthe end-to-end system with the IPT resonators was lower than the power levels for theamplifier without the IPT networks. When the class F−1 prototype was tested with nooutput resonator and the second order IPT network, it was observed that the input andoutput power was similar to the power levels of the configuration of the network with theoutput resonator. The end-to-end efficiency of the network with the class F−1 amplifierhaving no output resonator was 67.9% which is 2.6% lower than the end-to-end efficiencyof the network with the output resonator. However, the reduction in the complexity of theamplifier prototype offers a promising outlook that this configuration can be possible if theprotection of the device can be ensured.The class F−1 amplifier prototype was then tested with the spatial distance in thesecond order IPT link varied to test the spatial bandwidth of the amplifier combined withthe IPT network. It was observed that the efficiency of the combined network decreasedat a higher rate as the coils were brought closer together, compared to when the coils werespaced further apart. This is due to the mutual coupling between the coils increasing at agreater rate when the coils are brought together than the rate of decrease when they arespaced further apart. These results are also consistent with results shown in Chapter 3.It is concluded that a load compensation network would be needed to compensate for thechanges in the spatial variation of the IPT network to protect the device in the amplifierfrom damage and to maintain high power efficiency.80Chapter 5ConclusionThis thesis describes a method for designing IPT networks using canonical filter struc-tures. The canonical filter structure gives designers control over key parameters such asimpedance, frequency response, and frequency bandwidth. Impedance inverters are usedto convert canonical filter prototype networks, that are based on ladder networks, intophysically realizable IPT networks that are implemented with coupled resonators. Theimpedance inverters model the IPT coupling link and the impedance of the inverters is re-lated to the mutual inductance between coupled coils. Mutual inductance depends on thephysical geometry of the inductors and the spatial distance between inductors. Therefore,spatial changes, such as a change in the distance between two inductors, can be modelledas a change in the impedance of the inverter. The impedance inverter model provides away to capture the impact of spatial changes in IPT systems, and this was used to inves-tigate load compensation techniques that can maintain constant input impedance at thetransmitter. By maintaining constant impedance, the efficiency and power transferred tothe load can be held constant when there are spatial changes.5.1 ContributionsThe transformation from a filter network into a series-series IPT network was re-derivedwith generalized equations for the components in the IPT network based on filter coeffi-cients. The ability to adapt the frequency response to a specific application is why thefilter theory approach offers such utility in the design of WPT systems. It was shown thatthe only approximation needed for the conversion from a filter network to a series-seriesIPT network was the conversion from an ideal impedance inverter to an equivalent lumpedelement model. The transformation of the filter network to a shunt-shunt IPT networkwas explored, and it was shown how additional approximations are needed to transformfilter networks with more than three resonators. The approximation is needed to simplifythe analysis of repeater coils (coils in the mid section of the IPT link). Repeater coils havemore complex coupling with contributions from adjacent coils on both sides of the repeatercoil. A solution was presented where a L-matching network in the circuit was replaced witha series capacitor. This substitution is only valid for cases where the L-match network iscapacitive, which is when the normalized bandwidth of the system is small. Since manyIPT systems have narrow bandwidth, this approximation is justified in many pratical ap-plications. Using the approximation, generalized expressions were derived for shunt-shunt815.1. CONTRIBUTIONSIPT networks with up to five resonators. It was shown that by assuming the normalizedbandwidth is much less than one, the expressions for the shunt-shunt IPT network can besimplified into a form which is similar to the series-series IPT networks.Series-series and shunt-shunt IPT networks were compared using both simulated andexperimental results. It was shown in simulation, that as the number of resonators in-creases, series-series IPT networks match the ideal filter response. The shunt-shunt IPTnetworks on the other hand include deviations in terms of bandwidth expansion and pass-band ripple that are introduced by the L-network approximation that was used in theanalysis. Theoretically both topologies have zero losses at the center frequency. The ex-perimental results match well with the simulation results for higher order shunt-shunt IPTnetworks as well as for second order IPT networks for both topologies. It was shown,that for a specific operating frequency and bandwidth that a series-series IPT networkhas larger inductor values compared to shunt-shunt IPT networks. Therefore, the designmethod for shunt-shunt IPT networks may be useful in many applications, such as wirelessdevice charging, medical applications for implanted devices and wireless sensors where thephysical size of inductors needs to be small.By using impedance inverters in the conversion of series-series IPT networks, it wasobserved that these IPT networks could be simplified to a cascade of impedance invertersat the resonant frequency. As was already shown in the filter theory method of IPTnetwork design, impedance inverters are directly related to coupling links between IPTcoils. Therefore, changes in the coupling link (i.e. a spatial change in the distance betweenadjacent coils) is equivalent to an impedance change in the inverter. By representing thespatial variation in this way, any variation in the spatial distance can be fully compensatedfor by modifying the load resistance. Generalized expressions are given for the optimal loadresistance if the mutual inductance between adjacent resonators is known. The impedancecompensation technique can be used for both simple resonator structures where the mutualinductance can be solved analytically or in more complex structures where the mutualinductance can be found through electromagnetic simulation. Experimental and simulatedresults are shown which confirm the use of modifying the load resistance in compensatingfor spatial variations in the IPT networks.The shunt-shunt IPT coils were then implemented in an end-to-end system with aclass F−1 amplifier. The class F−1 topology was chosen with the shunt-shunt IPT coils totest whether the output resonator in the class F−1 amplifier topology could be replacedwith the shunt resonator in the IPT network. First the amplifier was tested without theIPT networks and it was observed that there was substantial ringing in the drain voltagewaveform which was not observed in the simulation data. However, the efficiency of theamplifier was still high and operated at the design power levels. When the amplifier wastested with the IPT networks it was first tested with the second, third and fourth orderIPT shunt-shunt IPT networks. It was observed that the end-to-end efficiency for thecombined networks was 70.5%, 69.7% and 69.1% for the second, third and fourth ordernetworks respectively. With the operation of the combined network confirmed to be as825.2. RECOMMENDATIONS AND FUTURE WORKexpected, the output resonator of the amplifier prototype was removed and the combinednetworks were tested again to assess the impact of the change. The ripples in the drainvoltage waveform increased substantially with the removal of the output resonator and theinput DC current to the amplifier increased. The end-to-end efficiency of the combinednetwork with no output resonator in the amplifier prototype was 67.9% which is 2.6%lower than the end-to-end efficiency with the output resonator. The loss of the end-to-end efficiency is weighed with the reduction of components necessary for the amplifierprototype which leads to a reduction in cost and complexity for the design of the amplifier.It was then observed how the class F−1 amplifier behaved when the IPT network wassubjected to spatial variation changes. It was found that to protect the transistor in theclass F−1 amplifier, some impedance management system would need to be implementedto compensate for the spatial variation changes.5.2 Recommendations and Future WorkThis work introduced a novel design technique for shunt-shunt IPT networks usingcanonical filter structures which had previously been used to design series-series IPT net-works. It was observed that due to the approximation of the repeater coils in higher ordershunt-shunt IPT networks, there is a bandwidth expansion and a shift in the center fre-quency for the frequency response. There was also a slight shift in the center frequencyof the series-series IPT network due to the approximation of the ideal impedance inverterwith a lumped element model. This should be taken into account when using the filtertheory approach to design IPT networks for a specific frequency response. An exampleof modifying the design parameters to ensure the frequency response of the IPT networksmatches the ideal filter response is shown in Section 2.4.1.An impedance compensation technique was also introduced using the concept of impedanceinverters, which model the coupling link between adjacent resonators in IPT networks. Forboth the impedance compensation technique and the filter theory design technique, furtherresearch is needed to assess the impact of cross-coupling between non-adjacent resonators.Cross-coupling is typically neglected in the design stages for IPT networks, but can havean impact on the physical implementation of the IPT network. By including cross-couplinginto the design of the IPT networks, in-band ripple and a shifting of the frequency responsemay be mitigated without the need for changing the physical parameters of the repeatercoils to reduce the impact of cross-coupling. Further, by including cross-coupling terms inthe impedance compensation method, the required load impedance for load compensationin higher order networks may be impacted by cross-coupling terms when the coils are overcoupled.The class F−1 amplifier was chosen primarily to test if the shunt output resonator in theamplifier topology could be replaced with the shunt-shunt IPT network. 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Thesis/Dissertation
eng
Electrical Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Johnson, Thomas
Using canonical filter structures for designing inductive power transfer systems
Text