Coreless Planar MagneticWinding Structures for PowerConverters: Track-Width-RatiobySamuel Robert CoveM. Eng., Memorial University, 2011B. Eng., Memorial University, 2009A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Electrical & Computer Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)October 2016c© Samuel Robert Cove 2016AbstractWith the accelerated growth of slim consumer electronics has come the need to reduce theprofile of all electronic components. Planar magnetics provide an excellent solution to thisproblem, where copper strip conductors and flattened planar magnetic cores allow for theheight of the components to be severely decreased compared to traditional wire-wound com-ponents. Planar magnetics also provide more repeatable characteristics and easier manufac-turability.The major design goals for planar windings are low resistance, predictable inductance, andacceptable capacitance. This work investigates the application of a constant ratio betweenturn widths, called the Track-Width-Ratio (TWR) as a technique to attain these qualities inplanar spiral windings.This work introduces the generalized racetrack planar spiral winding, whose low-frequencyanalysis can be applied to a variety of common winding shapes while accommodating chang-ing track widths. The accompanying dimensional system provides the specification of thenovel winding arrangements, including predicting their inductance and resistance. A designexample demonstrates an 18% increase in low-frequency performance.The second part investigates the AC resistance from TWR. The proposed technique pro-vides a correction factor based on the most recent models for ac resistance. A windingtechnique which combines hollow windings with TWR is proposed to increase the qualityfactor of planar spiral windings at high frequency operation. A design example highlights achange in efficiency from 70% to 90% within a 5W Wireless Power Transfer system.iiAbstractFinally TWR is employed to reduce planar spiral capacitance. Through an inverse TWRwinding structure, a significant decrease in capacitance is observed with a moderate reductionin resistance and inductance. A quasi-analytical approach with finite element analysis isemployed to determine the winding capacitance. These windings show a 50% decrease incapacitance and a 20% decrease in resistance compared to traditional windings.All results from this work have been confirmed experimentally and highlight the excep-tional flexibility which is provided when the turn widths are included in the design of planarspiral windings.iiiPrefaceThis work is based on research performed at the Electrical and Computer Engineering depart-ment of the University of British Columbia by Samuel Robert Cove, under the supervisionof Dr. Martin Ordonez. Some experimental validation work was done in collaboration withNavid Shafiei.Chapter 1 contains modified portions of text from all below-listed publications, as well assome modified text from my master’s thesis [1]:• S. R. Cove, “Design methodology for the control of planar transformer parasitics, Mas-ters thesis, Memorial University, 2011Portions of chapters 2 and 3 have been published at the IEEE Energy Conversion Congressand Expo (ECCE) and IEEE Transactions on Industry Applications [2, 3]:• S. R. Cove, M. Ordonez, “Practical inductance calculation for planar magnetics withtrack-width-ratio,” in Energy Conversion Congress and Exposition (ECCE), 2013, pp.3733–3737.• S. R. Cove, M. Ordonez, “Wireless-power-transfer planar spiral winding design applyingtrack width ratio,” IEEE Trans. Ind. Appl., vol. 51, no. 3, pp. 2423–2433, May 2015.A portion of chapter 3 has been published with the IEEE Transactions on Power Elec-tronics [4]:ivPreface• S. R. Cove, M. Ordonez, N. Shafiei, J. Zhu, “Improving Wireless Power Transfer Ef-ficiency Using Hollow Windings with Track-Width-Ratio,” IEEE Trans. Power Elec-tron., vol. 31, no. 9, pp. 6524–6533, Sept. 2016.A portion of chapter 4 has been accepted for publication at IEEE ECCE 2016 [5]:• S. R. Cove and M. Ordonez, “Low-Capacitance Planar Spiral Windings Employing In-verse Track-Width-Ratio”, accepted for publication in IEEE Energy Conversion Congressand Expo. (ECCE), 2016.As first author of the above-mentioned publications, the author of this thesis developedthe theoretical concepts and wrote the manuscripts, receiving advice and technical supportfrom Dr. Martin Ordonez, and developed simulation and experimental platforms, receivingcontributions from Dr. Ordonez’s research team, in particular from the Ph.D. student NavidShafiei who developed some specific experimental tasks.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.1 Planar Spiral Winding Modeling . . . . . . . . . . . . . . . . . . . . 41.2.2 Planar Spiral Design and Track-Width-Ratio . . . . . . . . . . . . . 51.2.3 Capacitance Minimization in Planar Spiral Windings . . . . . . . . . 61.2.4 Design of Experiments Methodology . . . . . . . . . . . . . . . . . . 71.3 Contribution of the Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Low-Frequency Resistance and Inductance Modeling of Planar Spiral Wind-viTable of Contentsings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1 Generalized Racetrack Planar Spiral Winding . . . . . . . . . . . . . . . . . 152.1.1 Unified Dimensional System . . . . . . . . . . . . . . . . . . . . . . . 152.1.2 Turn Lengths and Conductor Width . . . . . . . . . . . . . . . . . . 172.2 Inductance Modeling of the Generalized Racetrack Planar Spiral Winding . 182.2.1 Self Inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.2 Mutual Inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.3 Multiple Layer Windings . . . . . . . . . . . . . . . . . . . . . . . . 232.2.4 Finite Element Simulation Validation . . . . . . . . . . . . . . . . . 242.3 DC Resistance Modeling of the Generalized Racetrack Planar Spiral Winding 262.3.1 General Resistance Formula . . . . . . . . . . . . . . . . . . . . . . . 272.3.2 Resistance Coefficients for Special Cases . . . . . . . . . . . . . . . . 292.3.3 Optimal Track-Width-Ratio . . . . . . . . . . . . . . . . . . . . . . . 302.3.4 Finite Element Simulation Validation . . . . . . . . . . . . . . . . . 312.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.4.1 Magnetic Substrate . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.4.2 Design Example: Low Frequency Planar Spiral Winding . . . . . . . 392.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 AC Quality Factor Improvement in Planar Spiral Windings . . . . . . . 463.1 High Frequency Resistance Modeling for Planar Spiral Windings . . . . . . 493.1.1 High Frequency Effects in Planar Spiral Windings . . . . . . . . . . 493.1.2 Modeling Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.1.3 Finite Element Simulation Validation . . . . . . . . . . . . . . . . . 563.2 Meta-Models for Resistance, Inductance, and Quality Factor . . . . . . . . . 563.2.1 High Frequency Resistance Meta-Model . . . . . . . . . . . . . . . . 58viiTable of Contents3.2.2 Inductance Meta-Model . . . . . . . . . . . . . . . . . . . . . . . . . 593.2.3 Resultant Quality Factor Meta-Model . . . . . . . . . . . . . . . . . 613.2.4 Meta-model Accuracy Confirmation . . . . . . . . . . . . . . . . . . 633.3 Improving Quality Factor by Applying Hollow Track-Width-Ratio . . . . . . 643.3.1 Inductance and Quality Factor Trends . . . . . . . . . . . . . . . . . 653.4 Experimental Confirmation and Application Example: Low Power WPT Spi-ral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.4.1 Practical Winding Considerations for Circular Spirals . . . . . . . . 693.4.2 Experimental Prototype and Performance . . . . . . . . . . . . . . . 713.4.3 Low Power Wireless Power Transfer Application . . . . . . . . . . . 743.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784 Capacitance Mitigation in Planar Spiral Windings . . . . . . . . . . . . . 804.1 Capacitance in Planar Spiral Windings . . . . . . . . . . . . . . . . . . . . . 824.1.1 Planar Spiral Winding Capacitance . . . . . . . . . . . . . . . . . . 844.2 Inverse Track-Width-Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.2.1 Low-Frequency Resistance Modeling of the Inverse TWR Planar SpiralWinding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.2.2 Inductance Modeling of the Inverse TWR Planar Spiral Winding . . 894.2.3 Capacitance Simulation of the Inverse TWR Planar Spiral Winding . 904.3 Experimental Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.3.1 Experimental Performance Comparison . . . . . . . . . . . . . . . . 984.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065.1 Conclusions and Contributions . . . . . . . . . . . . . . . . . . . . . . . . . 1065.1.1 Low Frequency Resistance and Inductance . . . . . . . . . . . . . . . 106viiiTable of Contents5.1.2 High Frequency Resistance and Quality Factor . . . . . . . . . . . . 1075.1.3 Capacitance Minimization . . . . . . . . . . . . . . . . . . . . . . . . 1085.1.4 Specific Academic Contributions . . . . . . . . . . . . . . . . . . . . 1095.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113ixList of Tables2.1 Summary of Resistance Model Coefficients . . . . . . . . . . . . . . . . . . . 292.2 Dimensional Model and Optimized Resistances: Simulation vs. Calculation . 332.3 Design Winding Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . 403.1 Factors range of operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.2 Factors range of operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.3 WPT Experimental Specifications . . . . . . . . . . . . . . . . . . . . . . . . 744.1 Simulated Winding Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 92xList of Figures2.1 The generalized racetrack planar spiral winding and its specific subsets ofwindings: circular, rectangular, octagonal, and traditional racetrack alongwith their resultant high performance windings after applying Track-Width-Ratio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Proposed set of dimensions for a generalized racetrack planar spiral windingwith a Track-Width-Ratio (TWR = a) applied. . . . . . . . . . . . . . . . . 142.3 Cross-section of a generalized racetrack planar spiral winding with TWR ap-plied. Definition of T is the total width of copper in one layer which stays thesame for a given footprint and number of turns (for a constant xi, xo, and Cl). 162.4 Schematic highlighting the relationship between the length of a circular race-track corner (blue) and the radius of an octagonal approximation (red) for thenth turn of an N turn winding. . . . . . . . . . . . . . . . . . . . . . . . . . 192.5 Two important cases for mutual inductance calculations with important di-mensions: (a) two parallel conductors of uneven length and width centered ona vertical axis (b) two uneven length conductors of even width connected atone end subtending an angle of 135◦. . . . . . . . . . . . . . . . . . . . . . . 212.6 Cross-section of a multi-layer planar spiral winding, indicating the distancebetween layers (s), and the number of layers (NL) for the unified dimensionalsystem, and the distances between the centers of conductors on different layersfor the mutual inductance calculations. . . . . . . . . . . . . . . . . . . . . . 24xiList of Figures2.7 Predicted vs. simulated inductance values for 50 generalized racetrack planarspiral windings with varying dimensions, number of turns, and number of layers. 252.8 Mapping from the generalized racetrack planar spiral winding to each of thespecific cases: circular, rectangular, generalized octagonal, and traditional pla-nar spiral windings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.9 Example of manufactured generalized racetrack planar spiral windings withvarious turns and Track-Width-Ratios applied. The windings in the coloredboxes represent the 4, 6, and 8-turn windings from Fig. 2.10, respectively. . . 352.10 Experimental validation of inductance prediction method for generalized race-track planar spiral windings. Inductance values for 4, 6, and 8 turns per layernormalized to 567 nH, 1.27 µH, and 2.51 µH respectively. The experimentaldata points are numbered in correlation with their winding design highlightedin Fig. 2.9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.11 Experimental validation of DC resistance prediction method for a variety ofplanar spiral shapes and sizes. Sizes range from xi = 1 to 15 mm, xo = 5 to 30mm, and N = 2 to 10, and Sarc = circ , rect, and oct. Exceptional accuracyis attained over a variety of windings shapes, sizes, and Track-Width-Ratio. . 372.12 Normalized figure of merit (LR) trends and experimental validation for gener-alized racetrack planar spiral windings from Fig. 2.9. . . . . . . . . . . . . . 392.13 10-Turn planar spiral windings employed in the design example to highlightthe accuracy of the proposed inductance and resistance models and to confirman improvement in the LRof the winding. Applying a TWR of 0.85 increasedLRby 18%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43xiiList of Figures3.1 Circular planar spiral windings: (a) baseline full planar spiral winding (b) twoupgrade paths previously presented for the baseline case (Track-Width-Ratioand hollowing by removing turns) (c) proposed hollow spiral winding with anincreased inner radius and a non-unity Track-Width-Ratio between turns. . . 483.2 Important dimensions for the generalized racetrack planar spiral winding withTrack-Width-Ratio (TWR = a) applied. . . . . . . . . . . . . . . . . . . . . 493.3 Select idealized flux lines in a 3-turn planar spiral winding, demonstrating theflux which causes skin effect, proximity effect, and the additive flux in thecenter of the winding. The direction of the current is indicated with dot andcross notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.4 (a) Magnetic field strength, H, for a horizontal cross-section of a circularplanar spiral winding. (b) Magnetic field intensity vector, ~H, for a verticalcross-section of a circular planar spiral winding. . . . . . . . . . . . . . . . . 513.5 (a) Descriptive magnetic field strength, H, for a horizontal cross-section of acircular planar spiral winding with ten turns of equal width. (b) Descriptivemagnetic field strength, H for a horizontal cross-section of an improved 10-turnhollow planar spiral winding with TWR applied. Both the footprint and scalesare the same, and less copper is impinged with high magnetic field strength inthe proposed winding structure. . . . . . . . . . . . . . . . . . . . . . . . . . 523.6 Resistance trend of hollow planar spiral windings with Track-Width-Ratio ap-plied. Predicted values are compared to 16 finite element simulations, varyingnumber of turns and frequency of operation. . . . . . . . . . . . . . . . . . . 543.7 Finite element validation of the high frequency resistance models for the N =6 windings. The model accurately highlights the non-linearity of the resistancewith respect to frequency, and the frequency dependence of the optimal TWRvalue. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55xiiiList of Figures3.8 Normalized simulated vs. predicted data for L, R, and Q for circular planarspiral windings employing a variety of turns, inner radius, and Track-Width-Ratio. L is normalized to 40 µH, R was normalized to 250 mΩ, and Q wasnormalized to 25. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.9 Three distinct methods of hollowing the generalized racetrack planar spiralwinding: 1. Increasing yi, 2. Increasing xi, 3. Increasing ri. This workchooses the symmetric approach of increasing ri, such that xi and yi willalways increase by the same amount. . . . . . . . . . . . . . . . . . . . . . . 643.10 3-D representations of the inductance and quality factor for windings describedin Table 3.2 at 200 kHz (a) inductance of 2-layer windings in µH (b) qualityfactor of 2-layer windings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.11 3-D representations of the inductance and quality factor for windings describedin Table 3.2 at 200 kHz (a) inductance of 4-layer windings in µH (b) qualityfactor of 4 layer windings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.12 The practical implementation of a 10 turn circular winding with a = 1.0 and0.85, highlighting the method in which the turns of varying width are con-nected as well as the improvement in termination track width. . . . . . . . . 703.13 Subset of circular windings employed to test the accuracy of the proposedmodeling of the hollow planar spiral winding with TWR. . . . . . . . . . . . 71xivList of Figures3.14 (a) Normalized representation of the performance of hollow planar spiral wind-ings by removing turns. Q is normalized to 15, L is normalized to 25 µH, andR is normalized to 475 mΩ. A 15% reduction in L is observed in order toattain a Q which is 50% higher. (b) Normalized experimental confirmation forL, R, and Q of 2-layer proposed hollow circular planar spiral windings withTrack-Width-Ratio. Q is normalized to 15, L is normalized to 25 µH, and Ris normalized to 475 mΩ. It can be observed that by tuning the combinationof xi and a, the inductance stays almost constant while Q doubles. . . . . . . 733.15 Schematic of the experimental wireless power transfer circuit. The inductorsL1 and L2 are the spirals under test. . . . . . . . . . . . . . . . . . . . . . . 753.16 Wireless power transfer system with proposed hollow planar spiral windingswith Track-Width-Ratio. Several connections and alignment aids have beenremoved to add clarity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763.17 Input voltage, input current, output voltage, and output current of the WPTsystem employing the baseline full planar spiral windings with a unity Track-Width-Ratio at 170 kHz. Input power is 7.8W and output power is 5.5 W,demonstrating a 70.5% efficiency. . . . . . . . . . . . . . . . . . . . . . . . . 773.18 Input voltage, input current, output voltage, and output current of the WPTsystem employing hollow planar spiral windings with 4 turns removed at 170kHz. Input power is 6.9W and output power is 5.5 W, demonstrating a 79.7%efficiency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773.19 Input voltage, input current, output voltage, and output current of the WPTsystem employing the proposed hollow planar spiral windings with non-unityTrack-Width-Ratio at 170 kHz. Input power is 6.1W and output power is 5.5W, demonstrating a 90.2% efficiency, the highest performance of the experi-mental cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78xvList of Figures4.1 Planar spiral winding cross-sections and their impact on capacitance and re-sistance: Traditional winding techniques, removing overlapping windings, andthe proposed winding with Track-Width-Ratio (a) applied in opposite direc-tions on each layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.2 Important dimensions for the generalized racetrack planar spiral winding withTrack-Width-Ratio (TWR = a) applied. . . . . . . . . . . . . . . . . . . . . 834.3 Distributed capacitance and voltage differences in planar spiral windings whenvoltage V is applied. Each overlapping conductor pair has the same capaci-tance, but a very different voltage difference between them. . . . . . . . . . . 834.4 Top view of the two layers of the proposed planar spiral winding with (a)traditional TWR and (b) inverse TWR. . . . . . . . . . . . . . . . . . . . . . 864.5 Cross-section of a planar spiral winding with the inverse TWR structure. Eachlayer has its turns, ntrad and ninv extend from the interior to the exterior of thewinding, each ranging from 1 to N , while p is defined from the turn of highestto lowest voltage, ranging from 1 to the product of the number of layers bythe number of turns per layer, NLN . . . . . . . . . . . . . . . . . . . . . . . 884.6 FEA capacitive energy cross-section of (a) a traditional planar spiral windingand (b) the proposed inverse TWR planar spiral winding. The traditionalwinding exhibits more overlapping conductor and an area of extremely highcapacitive energy at the winding input terminals. The proposed winding pro-vides double areas without overlap and much less energy trapped at the inputterminal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.7 Normalized simulated winding capacitance in the proposed planar spiral wind-ing with inverse TWR. Baseline parameters are included in Table 4.1. . . . . 924.8 Analytical models of the DC resistance of the proposed planar spiral windingwith inverse TWR based on (4.10). Baseline parameters indicated in Table 4.1. 93xviList of Figures4.9 Normalized calculated inductance results of the proposed planar spiral windingwith inverse TWR based on the analysis in Chapter 2. Baseline parametersindicated in Table 4.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944.10 Subset of windings employed to test the accuracy of the proposed resistanceand capacitance modeling of the inverse TWR planar spiral winding. . . . . 954.11 Experimental (discrete) vs. predicted (continuous) capacitance values for theinverse TWR planar spiral winding structure. . . . . . . . . . . . . . . . . . 964.12 Experimental (discrete) vs. predicted (continuous) DC resistance values forthe inverse TWR planar spiral winding structure. . . . . . . . . . . . . . . . 974.13 Normalized experimental vs. predicted inductance values for the inverse TWRplanar spiral winding structure. . . . . . . . . . . . . . . . . . . . . . . . . . 984.14 Impedance magnitude and angle vs. frequency of the standard planar spiralwinding structure. The inductance is measured at low frequency by the rateof change of the impedance, while the capacitance is measured by the rate ofchange of impedance at high frequency. The SRF confirms the measurements. 994.15 AC resistance of the standard planar spiral winding structure. . . . . . . . . 1004.16 Impedance magnitude and angle vs. frequency of the planar spiral windingstructure without overlapping conductors. The inductance is measured at lowfrequency by the rate of change of the impedance, while the capacitance ismeasured by the rate of change of impedance at high frequency. The SRFconfirms the measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014.17 AC resistance for the planar spiral winding structure without overlapping con-ductors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102xviiList of Figures4.18 Impedance magnitude and angle vs. frequency of the proposed inverse TWRplanar winding structure with a = 0.925. The inductance is measured at lowfrequency by the rate of change of the impedance, while the capacitance ismeasured by the rate of change of impedance at high frequency. The SRFconfirms the measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034.19 AC resistance of the proposed inverse TWR planar winding structure with a= 0.925. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104xviiiAcknowledgementsI would like to sincerely thank my supervisor Dr. Martin Ordonez for accepting me as partof his research team. His support, dedication and technical advice during my program aregreatly appreciated.I would also like to acknowledge our research team, for graciously sharing their experienceand knowledge with me, and for making these years a series of enjoyable experiences. Specialthanks go to Navid Shafiei and Federico Luchino for developing the experimental platformsthat were used to confirm the validity of this work.I must thank the University of British Columbia and the Natural Sciences and EngineeringResearch Council of Canada (NSERC) for funding this research.A special thank-you is extended to Delta-Q Technologies for their collaboration with UBC,which allowed me to further develop my research ideas while experiencing the practical sideof electronics development.Finally, it is my pleasure to be able to thank my fiance´e, Deanne Drover, for her incrediblepatience, constant understanding, and unwavering support throughout the entirety of thisprocess.xixChapter 1Introduction11.1 MotivationWith the research surge in slim portable electronics and and viability of high-frequencyelectronic power conversion, there has been a pressing need to decrease the profile and increasethe power density of magnetic components, which have traditionally been the bulkiest andheaviest components in electronic systems. Increasing research in the area has demonstratedthat planar magnetic components are viable in a variety of applications, including WirelessPower Transfer (WPT), isolated power converters, resonant power converters, radio-frequencycircuits, and micro-film inductors and transformers.Planar winding layout optimization is a challenging task due to the variety of possiblewinding techniques and geometries which can be employed. The ideal planar spiral wind-ing shape is application-specific and depends on the trade-off between efficiency and powerdensity as well the physical constraints such as maximum footprint, design complexity, andmanufacturability. Even the most basic planar spiral winding requires an exorbitant number1Portions of this chapter have been modified from the following publications, including my master’s thesisfrom Memorial University: [S. R. Cove, “Design methodology for the control of planar transformer parasitics,Masters thesis, Memorial University, 2011.], [S. R. Cove, M. Ordonez, F. Luchino, J. Quaicoe, “ApplyingResponse Surface Methodology to Small Planar Transformer Winding Design,” IEEE Trans. Ind. Electron.,vol. 60, no. 2, pp. 483–493, Feb. 2013.], [S. R. Cove, M. Ordonez, “Practical inductance calculation forplanar magnetics with track-width-ratio,” in Energy Conversion Congress and Exposition (ECCE), 2013,pp. 3733–3737.], [S. R. Cove, M. Ordonez, “Wireless-power-transfer planar spiral winding design applyingtrack width ratio,” IEEE Trans. Ind. Appl., vol. 51, no. 3, pp. 2423–2433, May 2015.], [S. R. Cove, M.Ordonez, N. Shafiei, J. Zhu, “Improving Wireless Power Transfer Efficiency Using Hollow Windings withTrack-Width-Ratio,” IEEE Trans. Power Electron., vol. 31, no. 9, pp. 6524–6533, Sept. 2016.], and [S. R.Cove and M. Ordonez, “Low-Capacitance Planar Spiral Windings Employing Inverse Track-Width-Ratio”,accepted for publication in IEEE Energy Conversion Congress and Expo. (ECCE), 2016.].11.1. Motivationof decisions before it can be manufactured. Some of the decisions that must be made, include:• winding shape• inner dimensions• outer dimensions• number of turns• spacing between conductors• number of layers• conductor thickness• conductor widthwhere each choice affects the spirals inductive, resistive, and capacitive behaviour. Thissituation is compounded with high-frequency operation where closed-form analytical solutionsdo not exist, or when some of the factors are not kept constant throughout each turn of thespiral, such as the conductor width.In order to push the power density of planar magnetic technology to the limit, novelwinding techniques need to be developed to reduce resistance, control inductance, and reducewinding capacitance. The final point is particularly important since it is a major drawbackof planar technology when compared to wire-wound alternatives. To analyze these noveltechniques and facilitate the proliferation of planar magnetic components in future electronicsystems, a consolidated approach to planar spiral winding specification and analysis is re-quired. One which can account for the majority of winding design choices and can adapt tofuture winding techniques and developments.This work introduces the technique of changing track widths of each turn of planar spiralwindings with a constant ratio, the Track-Width-Ratio (TWR). This technique will be proven21.2. Literature Reviewto be beneficial in the design of planar spiral windings in that it can decrease resistance,tune inductance, and when strategically arranged it can reduce capacitance. This work isaccompanied with the introduction of the generalized racetrack planar spiral winding whichis the parent winding shape to multiple of the most popular winding shape. This windingshape includes TWR and can be analyzed through the use of a proposed unified dimensionalsystem which is flexible enough to represent circular, rectangular, octagonal, and traditionalracetrack winding shapes.1.2 Literature ReviewThere has been an vast amount of research into improving the performance of planar magneticdevices, and it is an extremely active topic in power electronics research today. Commonresearch goals include the reduction of resistance, the maximization of quality factor, andthe minimization of capacitance. When high-frequency excitations are considered, the mostrecent research in the field must rely on finite element simulation as a design tool, becausethe analytical models do not have any closed form solutions. In addition, each work presentstheir results for a limited number of winding shapes, mostly just one. Each work defines itsown dimensional system, which is often a subset of a larger and more general dimensionalsystem.Important applications for planar magnetics include resonant and traditional power con-version [6–9], power supply on chip (PwrSoC) [10–12], and wireless power transfer [13–18].The low profile, manufacturing repeatability, and ease of integration have established planarmagnetics as a viable option to traditional wire-wound magnetics. As electronics get smaller,and frequencies increase, planar spiral windings will be the preferred choice for inductors andtransformers, as they have been with radio frequency applications [19–22].31.2. Literature Review1.2.1 Planar Spiral Winding ModelingThe practical modeling of the inductance of planar spiral windings has been approached invarious forms [23–34]. The fundamental theories on the self and mutual inductances of avariety of winding shapes were summarized with various experimentally-fit equations andtables [23]. Curve-fitting solutions for circular [24, 25], square [26], and rectangular [27,28] were later developed. Greenhouse later developed a technique for rectangular spiralwindings which derives from the work of Grover [29] which provided exceptional accuracyfor single-layer windings. Mohan later employed curve-fitting to establish quick equations fora variety of winding shapes which was a compromise between the accuracy of Greenhouse’stechnique and the quickness of previous curve-fitting results [30]. When two planar windingsare considered to be sandwiched between two layers of magnetic material, the self and mutualinductances were also modeled [31–34]. These exceptional advances in inductance predictionof planar spirals are extremely elegant because they employ various dimensional systems,and do not consider advanced winding design techniques such as varied track widths for eachturn of the spiral.Previous work has aimed to model the effect of high frequency winding resistance andlosses in planar spirals [35–38]. A 1-D model provided an approximation for wire-woundmagnetics [35], which was later improved to include edge effects with a series of 2-D modelsfor rectangular [36] and arbitrary [37] cross section conductors. This work was later expandedto include concentric planar spiral windings by employing a model based on Finite ElementAnalysis [38]. These works have expanded the ability to predict the behavior of traditionalplanar and wire-wound windings, but the tools are not readily available to observe the effect ofadvanced winding techniques such as employing a variety of track widths within one winding.This work develops models for the inductance, low-frequency resistance, and high-frequencyresistance for the generalized racetrack planar spiral winding. This winding is analyzed as the41.2. Literature Reviewparent of the circular, rectangular, octagonal, and traditional racetrack planar spiral wind-ings. The proposed unified dimensional system that accompanies it can include the specifi-cation of turn widths which change with a constant ratio, the Track-Width-Ratio (TWR).These models are confirmed to be accurate with the use of finite element simulations andexperimental measurements. The TWR technique is confirmed to decrease inductance asit decreases from unity, while the resistance decreases to a minimum and then increases asthe TWR is decreased further. An experimental design example confirms that the ratio ofthe inductance to the resistance, LRactually increases to a peak and then decreases as TWRdecreases further from unity.1.2.2 Planar Spiral Design and Track-Width-RatioThere has been interesting research into methods in order to improve the design of planarspiral windings [39–45]. A method of improving the performance of circular planar spi-rals through removing the inner turns has been proposed [39, 40]. A resistance calculationmethod for circular and rectangular spiral windings with varying track widths has also beenpresented [41]. Preliminary work has been completed on characterizing parasitic effects forspecific spiral geometries for interlocking-square [42] and circular [43] spiral windings forplanar transformers. Track width variations have also been demonstrated to improve theQ of square integrated inductors for radio frequency (RF) applications [44, 45] for squarespirals. These important contributions use various dimensioning systems, resistance models,and width-varying techniques to improve particular cases of a more general planar spiralwinding structure.These improvements to the modeling and performance of planar spiral windings is par-ticularly important in resonant inductive Wireless Power Transfer (WPT) systems, wherehigh-quality windings are required to optimize the power transfer. Planar spiral windingshave been the choice for many recent WPT systems without any spiral optimization tech-51.2. Literature Reviewniques [13–18]. In these systems, the value of the inductance is important in order to designthe resonant network, while the quality factor (Q = ωL/R) has a significant impact on theefficiency of power transfer [46–51]. By increasing the Q factor of the inductor, lower lossescan be achieved in transmission, improving the overall efficiency of the WPT system.This work presents the hollow planar spiral winding with TWR as a technique of improv-ing Q in planar spiral windings while allowing for a stable inductance value. An accurate highfrequency resistance technique is developed and confirmed to be accurate through simulationand experimental measurements. Design of Experiments (DoE) methodology is employedto develop 3-D plots of R, L, and Q for a subset of planar spiral windings to highlight thetrends for each. These models are employed to design a planar spiral winding for use in a5 W WPT setup, for which the transfer efficiency is measured to be much higher than thetraditional winding, or the case of removing internal turns.1.2.3 Capacitance Minimization in Planar Spiral WindingsOne of the documented drawbacks to planar transformer design is the additional capacitancebetween turns, layers, and windings [52]. The combination of large planar conductors over-lapping with a high voltage differential between layers creates high capacitive energy storagein the PCB material, which can cause waveform distortion, reduction in self-resonance fre-quency, and high shoot-through currents in the presence of high dVdt. Previous attempts havebeen made to minimize capacitance in planar magnetics [53–58]. Preliminary work has beenperformed on predicting capacitance in planar transformers for 2-layer coreless planar spiralwindings [53] and for multi-layer planar transformers when an equal-voltage-drop model isemployed between each turn [54]. Improved shielding techniques have been proposed to mit-igate the impact of planar transformer capacitance [55] and a method of alternating PCBlayers has shown promise for extending PCB spiral winding operating frequency [56]. Re-cently an approach which removes the overlapping copper from the winding window has been61.2. Literature Reviewproposed [57] and was later improved to include a connection to ground in the open space toreduce the capacitance and serve as an EMI filter [58].This work presents the planar spiral winding structure with inverse TWR, which alter-nates between a TWR less than unity with the next layer being greater than unity to decreasethe amount of overlapping copper between layers. In addition to decreasing overlapping con-ductors, this technique also reduces the overall voltage gradient between layers, successfullyreducing the overall capacitive energy that the spiral winding can store. The voltage distri-bution across each turn is established and employed in finite element simulations to confirma significant reduction in winding capacitance, with the potential to reduce low-frequencyresistance. A prototype planar spiral winding is designed and its R, L, and C characteristicsare compared to both the traditional planar spiral winding and the case of no overlappingconductors and the proposed technique demonstrated a similar reduction in capacitance buta significant reduction in resistance.1.2.4 Design of Experiments MethodologyWhen modeling the effects of systems that either have many input factors or have highlynonlinear behaviour, it can be very beneficial to develop statistical meta-models in order toobserve trends and investigate areas where optimization can occur. Design of Experiments(DoE) methodology analyzes specific datasets using a combination of hypothesis testing andANOVA regression to identify relationships between system inputs and outputs. It hasbeen applied to a variety of nonlinear systems where accurate and insightful models arerequired [59–61]. Response Surface Methodology (RSM) is a powerful technique within DoEthat has been applied to the design of a C-core actuator [59], magnetic levitation systems [60],high temperature superconducting transformers [61], and planar transformer parasitics [43].The methodology is general as it can be applied to a variety of applications and designs, andpowerful as it can provide accurate results for optimization applications [62]. The existing71.3. Contribution of the Workliterature provides a basis for the use of DoE techniques to analyze the effect of nonlinearsystems. This work applies RSM to eliminate the need for finite element simulations whenpredicting the high-frequency resistance in planar spiral windings that apply TWR, which isan applications which has not previously been investigated.1.3 Contribution of the WorkThe goal of this work is to introduce the concept of Track-Width-Ratio in the design of planarspiral windings and to prove that it can be an effective tool in improving the performance ofplanar spiral windings. The important contributions of the work include:• First, the generalized racetrack planar spiral winding and the unified dimensional sys-tem are proposed as a technique for the analysis of planar spiral windings of a varietyof winding shapes and designs. These concepts allow for the analysis of planar spiralwindings with varied track widths for circular, rectangular, octagonal, and traditionalracetrack windings, which is not currently possible with any other winding shape ordimensional system. Models for the DC resistance and inductance are derived andconfirmed with finite element analysis and experimental measurements. The resistancemodel is derived in order to find the optimal Track-Width-Ratio (TWR) which mini-mizes the overall winding resistance. This technique is employed on many previouslypresented windings to highlight that this technique is capable of improving the perfor-mance of windings in a variety of applications. Finally, a design procedure is presentedin order to design a generalized racetrack planar spiral winding for its highest perfor-mance, when the ratio of LRis maximized. The successful design employed a TWR of0.85 and improved LRby 18%.• Second, the concept of the hollow planar spiral winding with TWR is introduced toimprove the high-frequency performance of planar spiral windings. An ac resistance81.3. Contribution of the Workmodel for the proposed structure is developed based on a quasi-analytical techniquepreviously employed for windings with equal track widths. In order to demonstrate theeffectiveness of this technique, Design of Experiments (DoE) methodology is employedto develop parametric equations for a subset of winding dimensions in order to provethat for a desired inductance and footprint, the hollow planar spiral winding with TWRcan greatly improve the ac quality factor of a multi-layer planar spiral winding for ahigh-frequency 5W WPT setup. The power transfer efficiency is measured and com-pared to traditional winding design, and a design with turns removed. The proposedstructure improved the transfer efficiency from 70% to 90%, while the design with turnsremoved could only reach 80%.• Third, a planar spiral winding structure with inverse TWR is proposed in order toreduce inter-layer capacitance. Track widths for each turn will be reduced by a constantratio from the outside to the inside of one layer of turns, then the reverse is performedon the next layer. The amount of overlapping copper is greatly reduced, and thevoltage difference between the largest overlapping areas is greatly reduced. The storedelectrical energy will be investigated via Finite Element simulations, and meta-modelswill be developed based on the winding dimensions and layer separation. In a designexample it is demonstrated that capacitance can be reduced by 50% while reducing acresistance by upwards of 20%.In the area of planar spiral winding characterization, this work makes great strides throughthe proposed generalized racetrack planar spiral winding and associated unified dimensionalsystem through which the dc resistance and inductance are derived. The characterization isstrengthened through the ac resistance which is based off a quasi-analytical model, and finallythe capacitance which requires finite element simulation. With regard to the performanceof planar spiral windings, the solid windings with TWR, hollow windings with TWR, and91.4. Dissertation Outlinewindings with inverse TWR have proven to have particular niche uses, depending on thedesign criteria.1.4 Dissertation OutlineThis work is organized in the following manner:• In Chapter 2, the generalized racetrack planar spiral winding and the unified dimen-sional system are proposed as a technique for the analysis of planar spiral windings ofa variety of winding shapes and designs. The inductance of the structure is derivedfrom the work of Grover and Greenhouse while the low-frequency resistance is derivedfrom the relationship between the length to the width of each turn. These models areconfirmed through finite element simulations and experimental measurements, and adesign procedure is presented to illustrate how the performance at low frequency canbe optimized through increasing the ratio of inductance to resistance (LR).• In Chapter 3, the concept of the hollow planar spiral winding with TWR is introducedto improve the high-frequency performance of planar spiral windings. The effects whichcause high frequency resistance in planar spiral windings are investigated, and a modelis created to predict this resistance. In order to investigate the effect of increasing theinternal radius and employing TWR, high-accuracy meta-models employing Designof Experiments methodology are presented. The resultant models are employed todemonstrate that the quality factor of a planar spiral winding can be improved for aconstant inductance. The proposed winding structure is employed in a 5 W, 105-200kHz WPT setup and compared to the traditional winding case and the case of removinginner spiral turns.101.4. Dissertation Outline• In Chapter 4, a planar spiral winding structure with inverse TWR is proposed in orderto reduce inter-layer capacitance. The conditions which cause capacitance in planarspiral windings are investigated, and a model for the voltage profile of the planarspiral winding structure with inverse TWR is developed and employed to perform finiteelement simulations to determine spiral capacitance. The trends in capacitance andresistance are investigated with respect to changing the TWR employed in the proposedstructure and an experimental prototype is compared to the case of the traditionalplanar spiral winding, and a winding which removes overlapping copper.• Chapter 5 contains the relevant conclusions, contributions, and planned areas of futurework. The work contributed significantly to the modeling and design of planar spiralwinding, which is highlighted in eight relevant publications in international conferencesor IEEE Transactions journals.11Chapter 2Low-Frequency Resistance andInductance Modeling of Planar SpiralWindings2As discussed in the introduction, there are no tools available for a magnetics designer tospecify or analyze the resistance or inductance of planar spiral windings in which the windingturn widths change with a constant ratio - the Track-Width-Ratio (TWR). The change intrack widths provides a factor for which previous techniques cannot account for, and thereis no literature available which can predict the performance of these windings over a varietyof winding shapes.It is the aim of this chapter to define the generalized racetrack planar spiral windingand provide the necessary tools to analyze its inductance and low-frequency resistance. Theanalysis is provided using a proposed unified dimensional system which can transfer the di-mensions from the generalized racetrack winding to the specific cases of circular, rectangular,octagonal, and traditional racetrack windings. The analysis is applicable at DC and also anyfrequency at which the current density remains approximately constant over a cross-sectionof the conductor. While the current across a planar spiral is not 100% constant at DC, it2Portions of this chapter have been published in [S. R. Cove, M. Ordonez, “Practical inductance calculationfor planar magnetics with track-width-ratio,” in Energy Conversion Congress and Exposition (ECCE), 2013,pp. 3733–3737.] and [S. R. Cove, M. Ordonez, “Wireless-power-transfer planar spiral winding design applyingtrack width ratio,” IEEE Trans. Ind. Appl., vol. 51, no. 3, pp. 2423–2433, May 2015.]12Chapter 2. Low-Frequency Resistance and Inductance Modeling of Planar Spiral WindingsFigure 2.1: The generalized racetrack planar spiral winding and its specific subsets of wind-ings: circular, rectangular, octagonal, and traditional racetrack along with their resultanthigh performance windings after applying Track-Width-Ratio.132.1. Generalized Racetrack Planar Spiral WindingFigure 2.2: Proposed set of dimensions for a generalized racetrack planar spiral winding witha Track-Width-Ratio (TWR = a) applied.is close enough for this analysis to provide a practical and otherwise completely accuraterepresentation of the spiral inductance and resistance.The resistance and inductance models are confirmed using finite element analysis andexperimental validation, and then a design example is performed in which a winding shapeis chosen and designed such that the maximum ratio of inductance to resistance can beattained. This ratio (LR) is analogous to the quality factor of the winding at a frequency of 1rads, which is a suitably low frequency for the models derived in this chapter. The conclusionfrom this example is a winding with an 18% higher LRvalue than the case with equal turnwidths.Fig. 2.1 demonstrates the generalized racetrack planar spiral winding, all of it’s sub-winding shapes, and schematics for examples of high performance versions of each shapeafter employing TWR.142.1. Generalized Racetrack Planar Spiral Winding2.1 Generalized Racetrack Planar Spiral WindingA simplified generalized racetrack structure with high accuracy is shown in Fig. 2.2 withno interconnects between turns. The generalized racetrack winding is characterized by arectangular-shaped winding with 90◦ circular arcs at each corner, and a constant ratio (a)between each turn width referred to as the Track-Width-Ratio (TWR). This shape is theparent shape of many sub-winding shapes when the physical dimensions change, most notablycircular (when the straight edges reduce to zero length), rectangular (when the corners containzero radius), octagonal (when the corners are centered at an infinite distance from the corner)and the traditional racetrack spiral (when only two opposite straight edges have zero length).This shape will be analyzed employing a novel unified dimensional system in order to attainthe DC inductance and resistance values. An important constraint on the shape is that itmust be symmetric in both x and y axes, so meander-type windings cannot be characterizedin this fashion.2.1.1 Unified Dimensional SystemAs will be seen, the proposed unified dimensional system will successfully characterize thegeneralized racetrack winding. The dimensions, highlighted in Figs. 2.2 and 2.3, represent:the distance from the center of the spiral to the inner edge of the inner turn in the x and ydirections (xi and yi); the distances from the center of the spiral to the outer edge of the outerturn (xo and yo); the distances from the center of the spiral to the center of the concentriccircular arcs (xc and yc); the number of turns (N); the clearance distance between conductors(Cl); the thickness of the conductor (t); the corner arc shape (Sarc = circ for circular arcs, octfor octagonal arcs, and rect for rectangular); and the Track-Width-Ratio (a) which representsthe ratio of the width of one turn to its adjacent outer turn. It is interesting to note that thedimension W , highlighted in Fig. 2.2 is defined once the winding is fully designed. Using the152.1. Generalized Racetrack Planar Spiral WindingFigure 2.3: Cross-section of a generalized racetrack planar spiral winding with TWR applied.Definition of T is the total width of copper in one layer which stays the same for a givenfootprint and number of turns (for a constant xi, xo, and Cl).required dimensions, the generalized racetrack shape can be reduced to the following specificwinding shapes:• Circular: set xc and yc to 0, Sarc = circ.• Traditional racetrack: either xc or yc are 0, Sarc = circ.• Rectangular: set (xc,yc) = (xi,yi), Sarc = rect.• Generalized octagonal: observe Fig. 2.8 for (xc,yc) placement, Sarc = oct.Since the winding cross section is equal in the x and y directions, xo − xi is equal to yo − yi,so only three of the four of those dimensions are required. In this work, yo is consideredredundant. Similarly xi − xc is equal to yi − yc such that yi is considered to be redundant.Therefore, the proposed dimensional system can successfully cover the general and specificgeometries and will serve as the foundation for the analysis.The TWR is the proposed method to improve performance and is indicated by the variablea in Fig. 2.2. The outer conductor will always be referred to as having a width given by W162.1. Generalized Racetrack Planar Spiral Windingand the inner conductor widths are given by the TWR (a) multiplied by the previous turn’swidth.2.1.2 Turn Lengths and Conductor WidthThe unified dimensional system provides a foundation to effectively derive physical and elec-trical characteristics. Most importantly, the length and width of each turn (ln and wn) arerequired for the calculation of the inductance and resistance values. The dotted line in Fig. 2.2highlights one quarter of the length of one turn. This line is divided into three sections: onehorizontal, one 90◦ arc, and one vertical section. In terms of the unified dimensional system,this gives:ln = 4xc + 4yc + 4(π2rn) (2.1)where n is defined in Fig. 2.3 as the turn number, referenced to the inner turn. The radiusfrom (xc,yc) to the center of the conductor of the nth turn (rn) is defined as:rn = xi − xc + (n− 1)Cl + aN−n2W +n−1∑k=1aN−kW (2.2)The individual turn lengths can be calculated from substituting (2.2) into (2.1) under theassumption thatW is known. In many situations, this is not the case, and instead the overallfootprint of the winding is known (xi,xo,yi, yo, and Cl). To calculate the length without W ,the total copper cross-section of the layer is defined as T , which is highlighted in Fig. 2.3 andis given by:T =N∑n=1aN−nW =1− aN1− a W = xo − xi − (N − 1)Cl (2.3)172.2. Inductance Modeling of the Generalized Racetrack Planar Spiral WindingRearranging (2.3) to solve for W and substituting into (2.2), a more applicable equation forthe corner radius of the nth turn is then given by:rn = xi − xc + (n− 1)Cl + (xo − xi − (N − 1)Cl)[aN−n2+n−1∑k=1aN−k](2.4)Substituting (2.4) into (2.1) defines the length of the nth turn of the generalized racetrackplanar spiral winding in the condition where W is unknown. This is the case analyzed bythis work. The individual track-widths (wn) are defined using (2.3):wn = aN−nW = (xo − xi − (N − 1)Cl)aN−n(1− a)1− aN (2.5)It can be concluded that the unified dimensional system for the racetrack planar spiralwinding provides a strong basis for calculating winding parameters such as length and trackwidth, which are essential for winding manufacturing and the analysis of electrical parameterssuch as inductance and low frequency resistance.2.2 Inductance Modeling of the GeneralizedRacetrack Planar Spiral WindingNow that the fundamental characteristics of the generalized racetrack planar spiral windinghave been developed, the unified dimensional system will be employed to find equations tocalculate its inductance. This will be attained by approximating the circular arcs with threespecific straight planar conductors as described in Fig. 2.4, which provides an accurate rep-resentation of the self inductance of the circular corners as well as their mutual inductanceswith the other winding components. The lengths of the three straight conductors are de-termined in such a way that the overall conductor length of the corner is the same as the182.2. Inductance Modeling of the Generalized Racetrack Planar Spiral WindingFigure 2.4: Schematic highlighting the relationship between the length of a circular racetrackcorner (blue) and the radius of an octagonal approximation (red) for the nth turn of an Nturn winding.circular arc (adding to pi2rn) for the nth corner of an N turn spiral as in Fig. 2.4. All self andmutual inductance calculations will consider this efficient approach.2.2.1 Self InductanceThe self inductance (LsN ) of the generalized racetrack planar spiral winding with a non-unityTrack-Width-Ratio is the sum of the self inductances of all straight planar conductors of theoctagonal approximated winding. Applying the unified dimensional system to Grover’s selfinductance formula [23], it is defined byLsN (µH) =N∑n=10.002ln[loge(2lnt+ aN−nW)+ 0.50049 +(t+ aN−nW3ln)](2.6)192.2. Inductance Modeling of the Generalized Racetrack Planar Spiral Windingwhere ln is defined in (2.1), W is defined intrinsically within (2.3), and all dimensions arein cm. This interesting equation (2.6) shows the versatility of the generalized dimensionalsystem to obtain self inductance.2.2.2 Mutual InductanceThe total inductance of an N turn generalized racetrack planar spiral winding (LTN ) is theaddition of its self and mutual inductances (LsN and MN):LTN = LsN +MN (2.7)where:MN = 2∑j 6=kMjk (2.8)and Mjk is the mutual inductance between conductors j and k. Two particular cases ofmutual inductance calculations are used to calculate the mutual inductances of a generalizedracetrack planar spiral winding as observed in Fig. 2.5: two uneven length planar conductorsoriented (a) parallel to each other with uneven width and (b) connected on one end forming a135◦ angle. All other cases of mutual inductance within the generalized racetrack planar spiralwinding are considered to be negligible and can be disregarded without adding significanterror.The mutual inductance of two equal-length parallel planar conductors is [29]:Ml(µH) = 2l[loge{lGMD+(1 +l2GMD2) 12}−(1 +GMD2l2) 12+GMDl](2.9)where l is the length of the conductors and GMD is the geometric mean distance between thetwo conductors. The GMD for our case is assumed to be the distance between the centers202.2. Inductance Modeling of the Generalized Racetrack Planar Spiral WindingFigure 2.5: Two important cases for mutual inductance calculations with important dimen-sions: (a) two parallel conductors of uneven length and width centered on a vertical axis (b)two uneven length conductors of even width connected at one end subtending an angle of135◦.of the two parallel conductors without significant error. The mutual inductance between twouneven parallel planar conductors as presented in Fig. 2.5 is given by [29]:Mjk =Mk+p −Mp (2.10)where Mk+p is the mutual inductance calculated in (2.9) for l = lk + lp and Mp is the samewith l = lp. Both cases use GMD = rj − rk which is easily calculated using (2.2) given212.2. Inductance Modeling of the Generalized Racetrack Planar Spiral Windingthat it is known which turn each component belongs to. Mutual inductances are positive forparallel conductors whose currents are traveling in the same direction, and are negative forconductors whose currents are traveling in opposite directions.The mutual inductance of the case presented in Fig. 2.5 is calculated using an equationpresented by Grover simplified by the fact that the angle is always 135◦ [23]:Mjk(µH) = 0.001(√22)ljloge 1 +lklj+lj+klj1− lklj+lj+klj+lkljlogelklj+lj+klj+ 1lklj+lj+klj− 1 (2.11)where lj is represented by either (2.12) or (2.13) for the case of the generalized racetrackplanar spiral winding:lj = xc +16π94(2.12)lj = yc +16π94(2.13)depending upon whether it is a vertical (2.12) or horizontal 2.13 conductor. Equation (2.2) isused to calculate rn, given that the turn number (n) is known. The distance lj+k is calculatedusing the law of cosines:l2j+k = l2j + l2k −√2ljlk (2.14)By summing up all components of the mutual inductance, MN , in the planar spiral andcombining it with the self-inductance, Ls within (2.7), the total inductance of the generalizedracetrack planar spiral winding is calculated for a single layer winding. When multiple layersare involved, further analysis of the GMD is required.222.2. Inductance Modeling of the Generalized Racetrack Planar Spiral Winding2.2.3 Multiple Layer WindingsAll of the preceding discussion assumed a single layer winding, but in practice a single-layer spiral winding poses problems for the return path. A much more common scenariowould be a two-layer winding which spirals towards the middle, then enters an adjacent layerthrough vias, and spirals back to the outer turn. The self inductance of the new layer wouldbe calculated employing (2.6), and the mutual inductance components would have to berecalculated considering the addition of the new turns. It is assumed that the inductancecontribution from the via(s) is negligible compared to the total.The new dimensions added to the unified dimensional system from the addition of furtherlayers include the number of layers (NL) and the distance between the centers of conductorson adjacent layers (s). The choice of using the centers of the conductors becomes clear whenobserving the mutual inductance calculations. Fig. 2.6 highlights the new GMDs required tobe placed into (2.9) for the case of equal length conductors and (2.10) when the conductorsare of unequal length. For the case of directly overlapping conductors of the same length,the mutual inductance is defined byMl(µH) = 2l[loge{ls+(1 +l2s2) 12}−(1 +s2l2) 12+sl](2.15)where s is the GMD between the conductors. When the turn components do not overlap,the GMD is represented as:GMD =√s2 + (rn − rn−i) (2.16)where rn and rn−i are determined by (2.2) and i represents how many turns closer to thecenter the inner turn is, with respect to the outer turn being considered. This concept isrepresented in Fig. 2.6.232.2. Inductance Modeling of the Generalized Racetrack Planar Spiral WindingFigure 2.6: Cross-section of a multi-layer planar spiral winding, indicating the distance be-tween layers (s), and the number of layers (NL) for the unified dimensional system, and thedistances between the centers of conductors on different layers for the mutual inductancecalculations.2.2.4 Finite Element Simulation ValidationThe first step for confirming the proposed inductance modeling technique was to comparecalculated data to 3D finite element simulations. To simulate LTN , the inductive energy wasdocumented under a current excitation, I. The inductive energy (Wind) within the simulationis:Wind =14∫V~B · ~HdV (2.17)and this total energy is related to LTN by the relation:Wind =12LTN I2 (2.18)242.2. Inductance Modeling of the Generalized Racetrack Planar Spiral Winding0 2 4 6 8 10012345678910Predicted Winding Inductance (µH)Simulated Winding Inductance (µH) SimulationsIdealFigure 2.7: Predicted vs. simulated inductance values for 50 generalized racetrack planarspiral windings with varying dimensions, number of turns, and number of layers.Equating (2.17) and (2.18) results in:LTN =2I2∫V~B · ~HdV (2.19)When I2 = 2, the equation simplifies to:LTN =∫V~B · ~HdV (2.20)This inductance was simulated for 50 generalized racetrack planar spiral windings with252.3. DC Resistance Modeling of the Generalized Racetrack Planar Spiral Windingeither one or two layers to confirm the technique’s accuracy. The predicted and the simu-lated inductances were plotted against each other in Fig. 2.7, demonstrating a high accuracyfor the proposed model. The discrepancies can be attributed to the approximation of theGMD between windings and the approximation of the circular corners by their octagonalequivalents.2.3 DC Resistance Modeling of the GeneralizedRacetrack Planar Spiral WindingJust as the unified dimensional system for the generalized racetrack planar spiral windingprovides an effective approach to analyze its inductance, it also allows for the rapid calculationof DC resistance.The resistance of the winding (RN) is defined as the sum of the resistances of the individualturns:RN =ρtN∑n=1lnwn(2.21)where ρ represents the resistivity of the metal and t represents the metal thickness. Usingthe definitions of ln and wn defined in (2.1) and (2.5), the ratio of length to width of a singleturn is:lnwn= 4(xc + rn(pi2) + yc)aN−nW(2.22)where rn is defined as in (2.2).262.3. DC Resistance Modeling of the Generalized Racetrack Planar Spiral Winding2.3.1 General Resistance FormulaExpanding (2.22) considering (2.3) and simplifying, the general form of the resistance of thegeneralized racetrack planar spiral winding is:RN =4ρtT[NRN0 +N−1∑n=1(RN−an−N +RN+aN−n)] (2.23)where RN−, RN+ , and RN0 indicate the components of the metal resistance of an N -turngeneralized racetrack spiral with negative, positive, and zero powers of a, respectively, where:RN−6= RN+ 6= RN0 (2.24)This general form of the metal resistance highlights its polynomial nature, containing powersof TWR (a) ranging from −(N−1) to N−1. After applying (2.3) the resistance componentsare given by:RN0 =(1− π2)xc + yc +π2(xi + xo2)(2.25)RN−= n((1− π2)xc + yc +π2(xi +(n− 1)2Cl))(2.26)RN+ = n((1− π2)xc + yc +π2(xo + (2− n)Cl))(2.27)Equations (2.23) and (2.25-2.27) are exact solutions of winding resistance and can be easilyimplemented in a spreadsheet or technical computing software for a target footprint andnumber of turns. In the end, the solution is a polynomial of TWR (a) so even the mostsimple of calculating methods can be used to model the resistance of the resulting spiral. Itis important to note that this model is accurate in the absence of skin and proximity effects.272.3.DCResistanceModelingoftheGeneralizedRacetrackPlanarSpiralWindingFigure 2.8: Mapping from the generalized racetrack planar spiral winding to each of the specific cases: circular, rectangular,generalized octagonal, and traditional planar spiral windings.282.3. DC Resistance Modeling of the Generalized Racetrack Planar Spiral Winding2.3.2 Resistance Coefficients for Special CasesTable 2.1: Summary of Resistance Model CoefficientsSpiral Term ModelGeneralizedRN0(1− pi2)xc + yc +pi2(xi+xo2)RacetrackRN−n((1− pi2)xc + yc +pi2(xi +(n−1)2Cl))RN+ n((1− pi2)xc + yc +pi2(xo + (2− n)Cl))RN0(circ)pi4(xi + xo)Circular RN−(circ)npi2(xi +(n−1)2Cl)RN+(circ)npi2(xo + (2− n)Cl)RN0(rect) yi + xoRectangular RN−(rect) n (xi + yi + (n− 1)Cl)RN+(rect) n (−xi + yi + 2xo + (1− n)Cl)GeneralizedRN0(oct)(1− pi4)xi + yi +pi4xo − d(2− pi2) (√22+ 831)OctagonalRN−(oct) n(xi + yi +pi4(n− 1)Cl − d(2− pi2) (√22+ 831))RN+(oct) n((1− pi2)xi + yi +pi2(xo + (2− n)Cl)− d(2− pi2) (√22+ 831))TraditionalRN0(RT )(1− pi2)xc +pi2(xi+xo2)RacetrackRN−(RT ) n((1− pi2)xc +pi2(xi +(n−1)2Cl))RN+(RT ) n((1− pi2)xc +pi2(xo + (2− n)Cl))The previous derivation of winding resistance is universal to the generalized racetrack pla-nar spiral winding. The winding is also simplified to the specific cases of circular, rectangular,and octagonal winding shapes as indicated in Fig. 2.8. The specific resistance coefficientswere successfully extracted and are presented in Table 2.1. A pattern resulted, and each292.3. DC Resistance Modeling of the Generalized Racetrack Planar Spiral Windingresistance coefficient could be represented by one general form:RN0 = (1− k) xc + yc + k(xi + xo2)(2.28)RN−= n((1− k) xc + yc + k(xi +(n− 1)2Cl))(2.29)RN+ = n ((1− k) xc + yc + k (xo + (2− n)Cl)) (2.30)where k is a winding shape coefficient which is equal to pi2for circular corner shapes (general-ized and traditional racetrack, circular), 2 for rectangular, and√2 for generalized octagonalplanar spiral windings.2.3.3 Optimal Track-Width-RatioThe general method of finding the minimum of any function is applied to find the model forthe optimal TWR for a given footprint and number of turns. The derivative of the resistanceformula with respect to the TWR is found and set equal to zero:∂RN∂a= 0 (2.31)Considering the model of the resistance is a polynomial of a, the power rule is applied, inconjunction with the fact that the derivative of a sum of functions is the sum of the functions’derivatives, and the resultant equation for the optimal TWR (aopt) for a pre-determinedfootprint and number of terms is:0 =N−1∑n=1((n−N)RN−an−N−1opt + (N − n)RN+aN−n−1opt)(2.32)302.3. DC Resistance Modeling of the Generalized Racetrack Planar Spiral WindingMultiplying both sides of the equation by aopt will eliminate the newly added factors of a−1optintroduced by the derivative, leaving:0 =N−1∑n=1((n−N)RN−an−Nopt + (N − n)RN+aN−nopt)(2.33)Many methods are available to solve the roots of polynomials, such as the secant methodor Newton’s method. Many calculators and technical computing software contain built-infunctions for solving such problems. This is a very quick and effective method of determiningthe optimal TWR to apply to a given planar spiral inductor geometry. Substituting thesolution of (2.33) into (2.23) results in the theoretical minimum resistance.2.3.4 Finite Element Simulation ValidationIn order to determine the accuracy of the proposed resistance model, a variety of planarspiral windings were simulated using finite element analysis and their resistance extractedfor comparison. To determine the resistance of the spiral, the ohmic loss energy (WΩ) wassimulated using the relation:WΩ =12σ∫V~J · ~JdV (2.34)which is equivalent to the ohmic loss in a lumped sum resistor, given by:WΩ = I2RN (2.35)Equating (2.34) and (2.35) results in:RN =12∫V~J · ~JdV (2.36)A selection of spirals were simulated using 3D finite element analysis and compared to312.3. DC Resistance Modeling of the Generalized Racetrack Planar Spiral Windingthe predicted resistance at a = 1 and a = aopt values. The dimensions of the windings wereconverted from a number of specific systems into the proposed unified dimensional system,which provides a simple method to compare all of the specific cases of windings. Table 2.2summarizes the results which emphasize the benefits of applying TWR to planar spirals fora variety of shapes and sizes. References have been included for any windings that wereextracted from other published work. It can be observed from the results that the simulatedresistances match closely with the calculated values using (2.23) for a variety of footprintsand number of turns. It is also critical to note that applying TWR to the planar spiralsdecreased resistance significantly, even on top of the improvements provided by each citedwork. Both the resistance model and the TWR technique provide valuable tools for designof planar spiral windings.322.3.DCResistanceModelingoftheGeneralizedRacetrackPlanarSpiralWindingTable 2.2: Dimensional Model and Optimized Resistances: Simulation vs. CalculationUnified Dimensional System Traditional OptimizedSource Spiral Family (Proposed in this work) Rsim Rcalc aopt Rsim|a=aopt Rcalc|a=aoptxi = 7.5mm , xo = 17.5mm14.6mΩ 14.5mΩ 0.635 13.4mΩ 13.3mΩGeneralized yi = 4mm , Cl = 0.25mmRacetrack xc = 6.25mm , yc = 2.75mmN = 2 , t = 35µm[56] Circularxi = 8.9mm , xo = 21.4mm14.6mΩ 14.9mΩ 0.905 13.3mΩ 13.5mΩyi = 8.9mm , Cl = 0.3mmxc = 0 , yc = 0N = 8 , t = 70µm[63] Rectangularxi = 7.5 , xo = 17.5157mΩ 157mΩ 0.88 149mΩ 149mΩyi = 4mm , Cl = 1.04mmxc = N/A , yc = N/AN = 6 , t = 70µm[64] Squarexi = 2.4mm , xo = 7.5mm779mΩ 783mΩ 0.86 720mΩ 723mΩyi = 2.4mm , Cl = 0.15mmxc = N/A , yc = N/AN = 7 , t = 10µm[19] Octagonalxi = 37.5µm , xo = 83.5µm7.84Ω 7.80Ω 0.833 7.55Ω 7.49Ωyi = 37.5µm , Cl = 2µmxc = 0 , yc = 0N = 4 , t = 0.35µm[10]xi = 1.465mm , xo = 2.065mm182mΩ 180mΩ 0.92 177mΩ 176mΩTraditional yi = 315µm , Cl = 50µmRacetrack xc = 1.15mm , yc = 0N = 5 , t = 50µm332.4. Experimental Results2.4 Experimental ResultsMultiple panels of planar spiral windings with various shapes and sizes were manufactured inorder to test the accuracy of the developed models with the final objective of obtaining a highratio of inductance to resistance. Several of these windings are included in Fig. 2.9, reflectingthe change of trace width resulting from modifying TWR and the number of turns. Everywinding contained the same internal and external dimensions, just the distribution of thatcopper changed as the TWR was changed. The unified dimensional system representation ofthe windings is:• xc, yc = 2 mm, 13.5 mm• xi = 4.5 mm• xo = 14.5 mm• t = 35 µm• Cl = 254 µm• Sarc = circ• N = 2, 4, 6, 8, 10• a = 0.2 - 1.0• NL = 2• s = 0.5 mmRecall that TWR (Fig. 2.3) defines how much the inner trace widths are reduced as thewinding approaches the center. The inductance and dc resistance of all of the windings was342.4. Experimental ResultsLow QHigh QFigure 2.9: Example of manufactured generalized racetrack planar spiral windings with vari-ous turns and Track-Width-Ratios applied. The windings in the colored boxes represent the4, 6, and 8-turn windings from Fig. 2.10, respectively.352.4. Experimental ResultsFigure 2.10: Experimental validation of inductance prediction method for generalized race-track planar spiral windings. Inductance values for 4, 6, and 8 turns per layer normalized to567 nH, 1.27 µH, and 2.51 µH respectively. The experimental data points are numbered incorrelation with their winding design highlighted in Fig. 2.9.362.4. Experimental Results0 0.2 0.4 0.6 0.8 100.10.20.30.40.50.60.70.80.91Predicted Winding Resistance (Ω)Actual Winding Resistance (Ω) MeasurementsIdealFigure 2.11: Experimental validation of DC resistance prediction method for a variety ofplanar spiral shapes and sizes. Sizes range from xi = 1 to 15 mm, xo = 5 to 30 mm, and N= 2 to 10, and Sarc = circ , rect, and oct. Exceptional accuracy is attained over a variety ofwindings shapes, sizes, and Track-Width-Ratio.372.4. Experimental Resultsmeasured using a frequency response analyzer and their LRratios were calculated. By usingthe equations obtained in this work, Fig. 2.10 presents the resulting predictions with dashedlines. As well, the experimental measurements are presented in Fig. 2.10 to compare predictedversus estimated. Each curve is normalized to the inductance at unity TWR (a = 1) suchthat an even comparison can be made. The generality and accuracy of the DC resistancecalculation method is emphasized in Fig. 2.11. The resistance was calculated for over 100planar spiral windings and compared to measured values for racetrack, circular, rectangular,and octagonal spiral windings with various footprints and number of turns. The line forwhich the expected and measured values are exactly the same has been included on theplot in order to highlight the significant accuracy of the proposed DC resistance calculationmethod. Finally, the objective of obtaining a high ratio of L to R is fulfilled in Fig. 2.12, whichpresents the comparison of the predicted to the experimentally measured. The figure includesa subset of generalized racetrack planar spiral windings. By inspection, the best TWR forN=4,6,8 to improve LRis obtained. Lower number of turns (e.g., N=4) require a lower TWRto obtain the highest performance, thus departing significantly from the traditional windingwith TWR = 1.2.4.1 Magnetic SubstrateThese inductive and resistive models contain no magnetic material. A preliminary investiga-tion into the effect of a magnetic substrate with a relative permeability of 1000 was performedto observe the effect. FEA simulations were used to observe the change in inductance forthe windings measured in Fig. 2.10 with an infinite magnetic substrate placed below thewinding. As was discussed in [32], in each case the inductance increased by a factor of two,but continued with the same normalized curves represented in Fig. 2.10. This is a very inter-esting result, as the prediction models from this work can be used to predict the inductancewith a magnetic substrate through a simple multiplicative factor. As expected from [32], this382.4. Experimental ResultsFigure 2.12: Normalized figure of merit (LR) trends and experimental validation for generalizedracetrack planar spiral windings from Fig. 2.9.factor changes with the permeability of the material as well as the thickness of the material.This multiplier changes greatly when sandwiched between two infinite planes of magneticmaterial.2.4.2 Design Example: Low Frequency Planar Spiral WindingIn this section the previously presented models for inductance and resistance are employedin order to improve the LRfor a circular planar spiral winding. Low-frequency refers to a392.4. Experimental ResultsTable 2.3: Design Winding SpecificationsParameter Valuexo 15 mmxi 1 mmShape CircleN 10Cl 0.25 mmt 35 µmρcu 1.68 × 10−8 Ωma TBDfrequency at least two orders of magnitude below that which would exhibit the skin effect ina planar conductor of a particular width. It is important that the current density does notchange due to skin or proximity effect. The specifications of the baseline winding are includedin Table 2.3. The experimental planar PCB is shown in Fig. 2.13. In order to predict theinductance and resistance, and thus determine the TWR which will improve LRthe most, thefollowing procedure was followed:1. Apply the dimensions from Table 2.3 in (2.2) such that it is only a factor of a and n.In this casern = 0.75 + 0.25n+ 11.75[a10−n2+n−1∑k=1a10−k](2.37)2. Substitute (2.37) into (2.1) to determine each turn’s length as a factor of n. In thiscaseln = 2πrn (2.38)As an example, for the equal-turn-width case (a = 1), the turn lengths range from 9.96mm for the inner turn up to 90.6 mm for the 10th turn.3. Apply the unified winding dimensions and the results from (2.38) within (2.6) to cal-402.4. Experimental Resultsculate the self inductance of the windings. In this caseLs(µH) =10∑n=10.002ln[loge(2ln0.035 + a10−nW)+ 0.50049 +(0.035 + a10−nW3ln)] (2.39)whereW = 11.751− a1− a10 (2.40)the only variable which remains to solve this equation is a, the TWR.4. Calculate the lengths of each straight trace section of the octagonal approximate wind-ing. The horizontal component lengths are calculated applying (2.12), the verticalcomponent lengths are calculated applying (2.13), and the diagonal corner lengths aredetermined from the expression in Fig. 2.4:lj =31π94rn (2.41)where the radius of each turn, rn, was tallied in step 1. The final length that must becalculated is lj+k from (2.14).5. Calculate the mutual inductances using (2.8)-(2.11)6. Apply the method of Greenhouse to calculate the final inductance of the winding. Asimple script to change the TWR of the design and recalculate the inductances wasused to determine the inductance trends presented in Fig. 2.137. Calculate the resistance coefficients by substituting the dimensions into (2.25) - (2.27).412.4. Experimental ResultsIn this case:RN0 = 4π (2.42)RN−=πn2(1 +(n− 1)8)(2.43)RN+ =πn2(15.5− n4)(2.44)8. Calculate T from (2.3):T = 15− 1− 9 ∗ 0.25 = 11.75mm (2.45)9. Substitute (2.42) - (2.45) into (2.23) to determine the resistance as a function of a:RN =3200ρ329[40π +9∑n=1(πn2(1 +(n− 1)8)an−10 +πn2(15.5− n4)a10−n)](2.46)10. Now all models of LN and RN for the winding only contain a as a factor. Technicalcomputing software such as Microsoft Excel or MATLAB can be employed to plot LN ,RN , andLNRNfor a wide range of a.422.4.ExperimentalResultsFigure 2.13: 10-Turn planar spiral windings employed in the design example to highlight the accuracy of the proposedinductance and resistance models and to confirm an improvement in the LRof the winding. Applying a TWR of 0.85increased LRby 18%.432.5. SummaryWith the models from the preceding design procedure, LN , RN andLNRNwere plottedvs. TWR in Fig. 2.13, in dashed lines. What was observed is that both resistance andinductance decreased as the TWR decreased from unity, but at different rates. This changein rates caused the ratio LNRNto increase until a peak was observed and then decrease again,highlighting a peak at a TWR of 0.85. Also contained in Fig. 2.13 is a panel of planar spiralwindings employed in experimental confirmation of the models employed in this chapter,specifically five windings used to confirm the trends observed in this design example. LN andRN were measured with an impedance analyzer and the data included in Fig. 2.13, confirmingthe modeled trends, with a peak performance at a TWR of 0.85. The optimized windingshowed an improvement in LNRNof 18%.One of the most important insights from this design example is that the TWR whichprovides the minimum resistance is not the same value that provides the maximum ratio ofinductance to resistance. This is due to the decrease in inductance with TWR, that shiftsthe maximum towards a higher TWR value. As observed in 2.13 the minimum resistancepoint actually occurs at a TWR of 0.765.2.5 SummaryIn this chapter, the generalized racetrack planar spiral winding and the unified dimensionalsystem were proposed as a technique for the analysis of planar spiral windings of a varietyof winding shapes and designs. The most novel aspect of the planar winding structure isthe capability of modeling the effects when track widths are changed with a constant ratio,defined the Track-Width-Ratio (TWR).The self and mutual inductance for this complex structure were modeled based on thework of Greenhouse and Grover, with the application of the proposed unified dimensionalsystem. Single and multiple layer windings were considered and the results were compared442.5. Summaryto simulation and experimental measurements in order to confirm their accuracy. The induc-tance trends observed as a part of this work suggest that the inductance of a planar spiralwinding decreases exponentially as TWR decreases from unity, and that this decay is moredistinct in windings with more turns. This decay was then investigated with an infinitely-thick magnetic substrate below the winding and the same normalized trend was observed,with an absolute multiplier of 2.0 of the coreless inductance value.Also, models for the low-frequency resistance of the generalized racetrack planar spiralwinding were developed analytically as a summation of the length to width of each turnof the winding. One general resistance model was presented which encompassed all subsetwinding families: circular, rectangular, octagonal, and traditional racetrack. This modelwas compared to simulation and experimental measurements of a series of windings fromthis work and previous literature in order to demonstrate the capabilities of TWR to reduceresistance. The resistance trends observed with the use of TWR involved a parabolic decreasein resistance until a minimum is reached, followed by a parabolic increase in resistance asTWR is decreased further. A model to predict at which TWR this resistance minimumoccurs was then derived.A design example was then performed in order to find the best TWR to be employedto maximize the performance of a planar spiral winding. This performance was measuredby the ratio of the inductance to the resistance, LR. The design example highlighted thatthere is a maximum for this ratio, and that it diverges from the minimum in resistance. Theoptimized design winding had a TWR of 0.85 and improved LRby 18%.45Chapter 3AC Quality Factor Improvement inPlanar Spiral Windings3In the previous chapter it was established that changing the turn track widths with a constantratio can improve the lRperformance of planar spiral windings under low-frequency operation.This improvement is an excellent first-step towards a better design for spiral windings, but israrely encountered in common applications. In practical scenarios, the designer must accountfor high-frequency magnetic field effects (skin effect and proximity effect) which can disturbthe current density in the cross-section of the planar spiral. In this situation, the previousanalysis cannot adequately predict the winding losses, and new techniques must be applied.This chapter aims to present a modeling approach to high frequency resistance for thegeneralized racetrack planar spiral winding. This technique will be derived from the state-of-the art in high-frequency resistance modeling [38], which marries analysis with normalizedfinite element simulations. The prediction method in this work will apply a parabolic ap-proximation based on the previous work when TWR is considered.Once this prediction method is established, a technique is presented in order to mitigatehigh-frequency resistive losses in planar spiral windings. This technique uses a combinedapproach of applying TWR and increasing the internal radius to find the optimal design toreduce the effects of high-frequency resistance. When this approach is combined with the3Portions of this chapter have been published in [S. R. Cove, M. Ordonez, N. Shafiei, J. Zhu, “ImprovingWireless Power Transfer Efficiency Using Hollow Windings with Track-Width-Ratio,” IEEE Trans. PowerElectron., vol. 31, no. 9, pp. 6524–6533, Sept. 2016.]46Chapter 3. AC Quality Factor Improvement in Planar Spiral Windingsinductance prediction modeling presented in the previous chapter, an optimal quality factorcan be determined for a desired winding inductance.Fig. 3.1 highlights the two steps in the design process and how they come together to formthe final hollow winding with TWR. Circular spirals are presented to increase the clarity ofthe image, but this technique is applicable to the generalized racetrack planar spiral windingand its subset winding families. This chapter will establish that the proposed design techniquecan increase the Q of the winding while allowing for an adjustable inductance, which is afeature that is not exhibited in any other technique. Inductance tuning is achievable becauseincreasing the internal radius increases the inductance, while applying TWR reduces it. Thiscontrol over the inductance is favorable for any application which is sensitive to the designedinductance value.In order to exhibit the benefit of the proposed techniques, Design of Experiments (DoE)methodology is employed to develop meta-models of the R, L, and Q of a subset of planarspiral windings, to observe the trends with a variety of winding design parameters. Themodels will be experimentally confirmed to be highly accurate, and a design example willbe included in order to design a high performance planar spiral winding for a 5 W WirelessPower Transfer system, based on the Qi specification.47Chapter3.ACQualityFactorImprovementinPlanarSpiralWindingsFigure 3.1: Circular planar spiral windings: (a) baseline full planar spiral winding (b) two upgrade paths previouslypresented for the baseline case (Track-Width-Ratio and hollowing by removing turns) (c) proposed hollow spiral windingwith an increased inner radius and a non-unity Track-Width-Ratio between turns.483.1. High Frequency Resistance Modeling for Planar Spiral WindingsFigure 3.2: Important dimensions for the generalized racetrack planar spiral winding withTrack-Width-Ratio (TWR = a) applied.3.1 High Frequency Resistance Modeling for PlanarSpiral WindingsThe previous chapter introduced the generalized racetrack planar spiral winding and theunified dimensional system which continues to be employed for analysis in this chapter. Abrief reminder of the shape and the important dimensions has been included in Fig. 3.2.Other important dimensions are the number of layers, NL, and the layer spacing, s.3.1.1 High Frequency Effects in Planar Spiral WindingsPlanar spiral windings suffer from AC losses due to high magnetic field strength normal totheir surface, creating high frequency eddy-current losses. These fields can be generatedfrom high-frequency currents within the conductor of interest (skin effect) or from externalconductors in close proximity (proximity effect). Fig. 3.3 presents select flux lines in a cross-section of a typical 3-turn planar spiral winding, demonstrating the flux which causes the493.1. High Frequency Resistance Modeling for Planar Spiral WindingsFigure 3.3: Select idealized flux lines in a 3-turn planar spiral winding, demonstrating theflux which causes skin effect, proximity effect, and the additive flux in the center of thewinding. The direction of the current is indicated with dot and cross notation.skin and proximity effect, as well as an important phenomenon in the center of the windingwhere the flux lines are all directed in the same direction. This additive flux presents a poorenvironment for planar conductors, as they will be susceptible to higher eddy-current losses.Finite element simulation was used to highlight the impact of this issue in a qualitativesetting. Fig. 3.4 (a) shows a top-down view of the magnetic field strength simulation resultsof a circular planar spiral winding. The field is strongest in the center, which is inducinghigh-frequency losses on the inner turns. Fig. 3.4 (b) displays a cross-sectional look at the fluxvectors whose magnitudes were represented in Fig. 3.4 (a). A distinct pattern of increasedmagnetic flux in the center of the winding is displayed, with very little extension vertically.The strength of the magnetic flux vectors decrease radially until they are at their weakestoutside the winding, where the majority of the flux cancels itself.This situation poses a critical problem when the internal turns in a traditional planarspiral winding contribute the least to the overall inductance, and are so wide that they con-tribute the highest resistive losses. The traditional approach to this problem was to removeinner turns, which provides a trade-off of lower inductance but much lower resistance [39, 40].503.1. High Frequency Resistance Modeling for Planar Spiral Windings(a) (b)Figure 3.4: (a) Magnetic field strength, H, for a horizontal cross-section of a circular planarspiral winding. (b) Magnetic field intensity vector, ~H, for a vertical cross-section of a circularplanar spiral winding.The problem is that the inductance can decrease a significant amount, which is unacceptableif this is for a resonant application, such as wireless power transfer. Even a 5 or 10% changein inductance will alter the resonant frequency of the system significantly.The proposed technique builds on two different techniques to combat the high resistivelosses in the internal turns: increasing the internal radius, ri = xi − xc and applying a non-unity TWR to the turns. In this way, the internal turns avoid more of the flux, and haveless surface area for the flux to make contact with the winding. This scenario is qualitativelydescribed in Fig. 3.5, where the magnitude of the magnetic flux, H, is plotted from an FEAsimulation for the case with even widths and a proposed winding with increased internalradius and reduced TWR. The scales in both cases are the same, and demonstrate how lesscopper is impacted with strong magnetic fields. The work in this chapter will demonstratethat this represents a reduction in high-frequency resistance and that these two factors canbe used to tune the inductance. It will be found that increasing the internal radius of thewinding will increase the inductance of the spiral, while the application of TWR decreases513.1. High Frequency Resistance Modeling for Planar Spiral WindingsFigure 3.5: (a) Descriptive magnetic field strength, H, for a horizontal cross-section of acircular planar spiral winding with ten turns of equal width. (b) Descriptive magnetic fieldstrength, H for a horizontal cross-section of an improved 10-turn hollow planar spiral windingwith TWR applied. Both the footprint and scales are the same, and less copper is impingedwith high magnetic field strength in the proposed winding structure.it. In this way the two parameters can be tuned to provide the maximum quality factor fora given footprint, inductance, and operating frequency.3.1.2 Modeling ResistanceThe AC resistance model of the hollow spiral winding with TWR employed is based onprevious work on a quasi-analytical approach to high frequency resistance estimation inplanar spiral windings [38]. Finite element analysis (FEA) was inserted into the analyticalmodel in a normalized fashion based on the conductor width and thickness in terms of theskin depth. The losses were broken down into conduction losses and proximity losses. Sincethe proximity losses are induced by magnetic fields which impinge the conductors at variousangles, there is an x and z component to the proximity loss term (where z is vertically523.1. High Frequency Resistance Modeling for Planar Spiral Windingsupward). The models are duplicated here for clarity [38]:Rac = Rcond +Rprox,x +Rprox,z (3.1)whereRcond =1wntσΦcond,rec(wnδ,tδ)(3.2)Rprox,x =1wntσΦprox,rec(wnδ,tδ)|Ho,x|2 (3.3)Rprox,z =1wntσΦprox,rec(wnδ,tδ)|Ho,z|2 (3.4)Each resistance value is per unit of length, and the Φ values are normalized frequency andgeometry dependent values which are determined from FEA and are visually representedin [38]. The magnitudes of the orthogonal x and z components of the magnetic field fromthe proximity effect are estimated by FEA as well.These models were established for windings with a constant conductor width. In this work,further FEA was performed in order to establish the effect of a non-unity TWR for hollowplanar spiral windings. The result is a quadratic function of resistance based on frequencyand the previously defined physical characteristics of the winding, which is normalized to theresistance value, Rac from (3.1). The resultant model of ac resistance of the hollow planarspiral winding with TWR is given by:RRac=(NNLxiClfxoa2 +NLfxoNxiCla+x2o√fClNNLxi)(3.5)which is a parabola with no roots and a minimum that falls between a = (0,1). That minimum533.1. High Frequency Resistance Modeling for Planar Spiral Windings0.5 0.6 0.7 0.8 0.9 10.60.650.70.750.80.850.90.951Track Width RatioNormalized Winding Resistance N = 2, f = 50 kHzN = 2, f = 250 kHzN = 8, f = 50 kHzN = 8, f = 250 kHzFigure 3.6: Resistance trend of hollow planar spiral windings with Track-Width-Ratio ap-plied. Predicted values are compared to 16 finite element simulations, varying number ofturns and frequency of operation.can be determined by differentiating (3.5) and setting the result equal to zero:0 =(2NNLxiClfxoamin +NLfxoNxiCl)(3.6)The model is accurate over the interval a = [2amin − 1,1], at which point the resistanceof the winding exceeds Rac and there is no benefit derived from employing TWR at that543.1. High Frequency Resistance Modeling for Planar Spiral Windings0 0.2 0.4 0.6 0.8 10.080.090.10.110.120.130.14Frequency (MHz)Winding Resistance (Ω) a = 1.0a = 0.85a = 0.7Figure 3.7: Finite element validation of the high frequency resistance models for the N = 6windings. The model accurately highlights the non-linearity of the resistance with respect tofrequency, and the frequency dependence of the optimal TWR value.553.2. Meta-Models for Resistance, Inductance, and Quality Factorfrequency.3.1.3 Finite Element Simulation ValidationFigs. 3.6 and 3.7 present resistance curves for various combinations of turns, frequency, andTWR with comparisons with the models provided in this chapter. Beyond indicating the highaccuracy of the model, some insights from Fig. 3.6 are that amin increases as the number ofturns increase, and that the impact of a on resistance increases with the number of turnsand the frequency of operation. Fig. 3.7 takes a different approach and keeps the numberof turns constant while varying the TWR and frequency. This plot demonstrates that asthe frequency changes, the preferred TWR changes, but at no point is the traditional caseever the optimal one. At low frequencies a conservative value of TWR provides a preferredresponse, which eventually is usurped by a TWR that departs further from unity.3.2 Meta-Models for Resistance, Inductance, andQuality FactorWhile the previously derived models for high-frequency resistance and inductance can ac-curately predict the performance of planar spiral windings with TWR applied, their use indesign is limited by reliance on finite element simulations or summations of summations. Inorder to quickly observe the effect of TWR with frequency, or any other physical parameter,statistical Design of Experiments (DoE) will be employed in conjunction with the unifieddimensional system to extract meta-models of the inductance, high frequency resistance, andthen quality factor. DoE [62] is a technique to deal with complex multivariate non-linearsystems. In particular, the variables for the analysis will be the parameters of the unifieddimensional system. Modeling the effects of every combination of the unified dimensional563.2. Meta-Models for Resistance, Inductance, and Quality FactorTable 3.1: Factors range of operationFactor Low Mid High Unitsxc 0 20 40 mmyc 0 15 30 mmxi − xc 0 2 4 mmxo − xc 5 15 25 mmN 2 6 10 turnsFrequency 0 0.5 1.0 Mhza 0.80 0.90 1.0system coordinates and frequency one at a time would be computationally impractical andrequire significant resources without providing any insight into the interactions between fac-tors. To produce accurate parametric models, a face-centered Central Composite Design(CCD) is used. The methodology employs ANOVA regression analysis to provide parametricmodels of the formF (Y ) = a0 +m∑i=1aiui +m∑i=1m∑j=1bijuiuj (3.7)where Y is the response being measured (inductance, resistance, quality factor), F representsa functional transform of Y (such as natural log or square root), a0 is the overall average ofmeasurements, ai are linear regression coefficients, bij are quadratic and interaction regressioncoefficients, m represents the number of factors while ui and uj represent the factors beingvaried. These equations are valid over the entire factor ranges with an accuracy proportionalto the adjusted R2 value of the model.The unified dimensional system combined with frequency provide the factors of the ex-periment, allowing for a thorough investigation based on geometry and non-linear currentdistribution. The factor ranges were chosen in order to encompass all of the winding designscurrently used by the Qi standard for WPT [65], in preparation for the design example atthe end of this chapter, and are summarized in Table 3.1. A quarter-fractional face-centered573.2. Meta-Models for Resistance, Inductance, and Quality FactorCCD with 7 factors requires 25 = 32 calculations with factors at high and low values, 2× 7= 14 calculations with all but one factor at its mid point, and 1 calculation where all factorsare at their mid point, totalling 47 calculations [62].One important note is that the number of layers is a non-continuous variable, meaningthat it can only be employed in integer values. In this case, different meta-models will berequired for different numbers of layers. This work covers 2 and 4 layer winding models.3.2.1 High Frequency Resistance Meta-ModelAll 47 factor combinations were calculated through the application of the high frequencyresistance model proposed in this chapter, within the ranges specified. The resultant para-metric model of high frequency resistance of generalized racetrack planar spiral windings with2 layers (in mΩ) is:ln(Rac2l)× 103 = 3190 + 30.3xc + 37.6yc + 28.4(xi − xc)− 194(xo − xc) + 814N + 459f − 946a− 0.429xcyc − 0.571xc(xo − xc) + 0.668xcN− 0.694yc(xi − xc)− 4.28N(xo − xc)+ 19.8f(xo − xc) + 93.7a(xo − xc)− 36.2N2 − 356f 2 + 7.967a2 (3.8)583.2. Meta-Models for Resistance, Inductance, and Quality Factorand the model for 4 layers is:ln(Rac4l)× 103 = 6409 + 46.8xc + 70.2yc + 49.6(xi − xc)− 276(xo − xc) + 892N + 651f − 680a− 0.872xcyc − 0.693xc(xo − xc) + 0.888xcN− 0.977yc(xi − xc)− 7.34N(xo − xc)+ 34.0f(xo − xc) + 140.1a(xo − xc)− 59.2N2 − 563f 2 + 11.4a2 (3.9)These high frequency resistance meta-models highlights the highly nonlinear and interac-tive nature of resistance. They also support each other in that all of the same factors weredeemed statistically significant in each model, and only the coefficients changed. The 2-layermodel has an adjusted R2 value of 0.9942 indicating an exceptional fit with the calculateddata (the ideal value is 1.0), while the 4-layer model has an adjusted R2 value of 0.9931.The proposed models provide deep insight into the impact of each factor of the experimentas well as highlight the interactions that exist between many of the factors. It is an efficientway to analyze the resistance of a wide variety of generalized racetrack planar spiral windingsand the accuracy is at its highest within the range of factors presented in Table 3.1. Designsoutside of this space can be investigated using the proposed equation as an initial designcheck.3.2.2 Inductance Meta-ModelThe inductance was calculated for the same 47 factor combinations as the resistance was.The inductance was determined by the models from Chapter 2, repeated here for clarity. The593.2. Meta-Models for Resistance, Inductance, and Quality Factorself inductance was calculated from:LsN (µH) =N∑n=10.002ln[loge(2lnt+ aN−nW)+ 0.50049 +(t+ aN−nW3ln)](3.10)and the mutual inductances were calculated from either the relation:Ml(µH) = 2l[loge{lGMD+(1 +l2GMD2) 12}−(1 +GMD2l2) 12+GMDl](3.11)orMjk(µH) = 0.001(√22)ljloge 1 +lklj+lj+klj1− lklj+lj+klj+lkljlogelklj+lj+klj+ 1lklj+lj+klj− 1 (3.12)where the GMD was assumed to be the distance between the center of each conductor. Theresultant inductance was calculated as a sum of the self and mutual inductances as per thework of Greenhouse. The 47 datapoints were analyzed statistically for the 2 and 4-layer casesand the resultant meta-models are given as:L2Layer = 12 + 2.6xc + 4.75yc + 28.4(xi − xc)− 194(xo − xc) + 814N − 946a− 0.429xcyc − 0.571xc(xo − xc) + 0.668xcN+ 93.7a(xo − xc)− 36.2N2 − 1.64a2 (3.13)603.2. Meta-Models for Resistance, Inductance, and Quality Factorfor the 2-layer case and:L4Layer = 23 + 5.0xc + 7.9yc + 11.2(xi − xc)+ 13.6(xo − xc) + 814N − 946a− 0.429xcyc − 0.571xc(xo − xc) + 0.668xcN+ 15.7a(xo − xc) + 36.2N2 − 3.23a2 (3.14)for the 4-layer case. L is calculated in µH, and the adjusted-R2 of each equation is 0.966 and0.974 respectively. This is considered to be an excellent fit to the simulated data. It is alsonoticed that in each case the significant factors remain the same and only the coefficientschange. This is one more sign of consistency in the results. It is also important to noticethat frequency was not deemed to be a significant factor in the inductance, since it is alwaysmeasured at DC.3.2.3 Resultant Quality Factor Meta-ModelHaving the meta-models for L and R, the data for the quality factor meta-model were derivedfrom the equation:Q =ωLR(3.15)613.2. Meta-Models for Resistance, Inductance, and Quality Factorwhere L and R were the same data that was used for (3.8), 3.9, (3.13), and (3.14). Theresultant model for 2-layers isQ2Layer = 25 + 1.8xc + 4.3yc + 12.7(xi − xc)− 1.98(xo − xc) + 9.34N + 2.23f − 1.93a− 0.112xcyc − 0.445xc(xo − xc) + 0.215xcN− 0.872yc(xi − xc)− 5.23N(xo − xc)+ 13.2f(xo − xc) + 16.8a(xo − xc)− 24.5N2 − 113f 2 − 4.22a2 (3.16)and the model for 4-layers is:Q4Layer = 40 + 3.2xc + 7.65yc + 22.3(xi − xc)− 3.62(xo − xc) + 18.72N + 6.31f − 2.49a− 0.343xcyc − 0.788xc(xo − xc) + 0.451xcN− 1.38yc(xi − xc)− 7.33N(xo − xc)+ 16.2f(xo − xc) + 20.2a(xo − xc)− 30.8N2 − 148f 2 − 7.2a2 (3.17)where Q is a dimensionless quantity. The adjusted R2 of the Q models are 0.988 and 0.967respectively. Each model contains the same terms of different weighting, and it is a functionof every dimension of the investigation in a highly nonlinear manner.623.2. Meta-Models for Resistance, Inductance, and Quality Factor0 0.5 1 1.5 200.20.40.60.811.21.41.61.82Normalized Predicted Winding L, R, QNormalized Actual Winding L, R, Q L (Measured)R (Measured)Q (Measured)IdealFigure 3.8: Normalized simulated vs. predicted data for L, R, and Q for circular planar spiralwindings employing a variety of turns, inner radius, and Track-Width-Ratio. L is normalizedto 40 µH, R was normalized to 250 mΩ, and Q was normalized to 25.3.2.4 Meta-model Accuracy ConfirmationFifty planar spiral windings that fit within the dimensions from Table 3.1 were simulatedusing finite element analysis in order to evaluate the accuracy of the proposed meta-models.The normalized results for R, L, and Q at various layers are presented in Fig. 3.8. The meta-models fit the data exceptionally well, within 5% of the simulated value in all occurrences.With these powerful meta-models on hand, the proposed technique of the hollow planar spiralwinding with TWR can be evaluated in a design example, and then in a dynamic example633.3. Improving Quality Factor by Applying Hollow Track-Width-RatioFigure 3.9: Three distinct methods of hollowing the generalized racetrack planar spiral wind-ing: 1. Increasing yi, 2. Increasing xi, 3. Increasing ri. This work chooses the symmetricapproach of increasing ri, such that xi and yi will always increase by the same amount.in a 5 W WPT system.3.3 Improving Quality Factor by Applying HollowTrack-Width-RatioNow that the high-frequency resistance of the generalized racetrack planar spiral has beenestablished, this work presents the hollow generalized racetrack planar spiral winding as amethod to reduce the high frequency losses. The combination of hollowing the winding withthe application of TWR allows for designs which can reduce the amount of copper affected bythe high frequency magnetic field generated by the winding. There are three ways in whichthe internal dimensions of the unified dimensional system could be changed to indicate ahollowing of the winding, which are highlighted in Fig. 3.9:1. Increasing yi, while maintaining xi constant.643.3. Improving Quality Factor by Applying Hollow Track-Width-RatioTable 3.2: Factors range of operationFactor Value Unitsxc, yc 0, 0 mmNL 2, 4f 150 kHzN 10 turnsxo 15 mmCl 0.25 mmt 0.105 mms 1.0 mm2. Increasing xi, while maintaining yi constant.3. Increasing ri, which increases both yi and xi uniformly.This work focuses on only option 3, in which the internal corner radius, ri, is increasedsuch that xi and yi increase uniformly. In this way, the increase matches closely with that ofa purely circular winding, and the assumptions from the previous chapter can still apply, inthat:xo − xi = yo − yi (3.18)andxi − xc = yi − yc (3.19)and also, it removes xi − xc and xo − xi from the analysis and replaced with ri and ro.3.3.1 Inductance and Quality Factor TrendsWhile the meta-models were presented earlier, it is not readily apparent how L and Q changewhen isolated to just the changes of the internal radius and the TWR. In this section, a subsetof the windings from the meta-model domain are investigated in order to determine the trends653.4. Experimental Confirmation and Application Example: Low Power WPT Spiralwith the change of ri and a. The static dimensions of the investigated windings are includedin Table 3.2. The results of the investigation can be observed in the 3-D plots in Fig. 3.10for the 2-layer spirals and Fig. 3.11 for the 4-layer spirals. They highlight that there is amaximum Q for a given footprint and frequency, for this case. The quality factor improveswith an increase in a and ri until a peak is reached, and then reduces. The peak Q can besignificantly higher than the baseline winding, in this case an increase of 400% is observedin the 2-layer case. This is natural because of the way TWR reduces the resistance of thewinding. If this plot is observed in slices of constant inductance, from Fig. 3.10 it still exhibitsa maximum Q for each slice. Fig. 3.11 demonstrates the changes in the surface when morelayers are added. In this case, the peak Q is attained from higher values of xi and a, at avalue higher than the 400% observed in Fig. 3.10. This makes sense since the field is strongerin the middle with the added layers, requiring a higher internal radius and stronger TWRto reduce the high frequency resistance. It is important to notice that the inductance doesnot reduce to provide this exceptional behavior. The following section confirms the modelsexperimentally, while investigating the performance of the proposed windings compared totraditional hollow windings.3.4 Experimental Confirmation and ApplicationExample: Low Power WPT SpiralIn this section, many aspects of experimental testing of the previous meta-models are dis-cussed. First, some practical winding considerations are presented in regards to designingcircular windings. This spirals are the only ones which require some approximation withmanufacturing, in order to keep clearances constant. Then the resistance, inductance, andquality factor meta-models will be confirmed with experimental measurements. Following663.4. Experimental Confirmation and Application Example: Low Power WPT SpiralFigure 3.10: 3-D representations of the inductance and quality factor for windings describedin Table 3.2 at 200 kHz (a) inductance of 2-layer windings in µH (b) quality factor of 2-layerwindingsthat, an application example of a 5W Wireless Power Transfer system will be presented, inwhich the proposed hollow planar spiral winding is compared to the traditional planar spiral673.4. Experimental Confirmation and Application Example: Low Power WPT SpiralFigure 3.11: 3-D representations of the inductance and quality factor for windings describedin Table 3.2 at 200 kHz (a) inductance of 4-layer windings in µH (b) quality factor of 4 layerwindings.683.4. Experimental Confirmation and Application Example: Low Power WPT Spiralwinding, and the planar spiral winding which has been hollowed by removing turns.3.4.1 Practical Winding Considerations for Circular SpiralsThere are various practical implementation considerations when employing windings with anon-unity TWR. One is the way in which the turns of various width are connected. Theapproach that was taken for the physical experimentation in this work has minimized theeffect of turn connections while still maintaining a constant clearance so that the voltagerating of the spiral is not impacted. For the rectangular spirals, the width changes at the 90degree angles between turns such that no additional connection is required. Circular windingsare comprised of N concentric circular arcs of approximately 300◦ and the outer turn isconnected to the inner turn by a straight trace, as observed in Fig. 3.12. The connectionsare as seamless as possible as to not deviate far from the ideal concentric-turn analysis.The overall conductor length was compared between the ideal construction and the practicalcircular winding for all windings tested in this work and they all portrayed less than 2%difference when compared.693.4.ExperimentalConfirmationandApplicationExample:LowPowerWPTSpiralTWR = 1.0 TWR = 0.70Concentric Circular ArcsTurn ConnectionsTurn InterfaceTerminationWidthTerminationWidthFigure 3.12: The practical implementation of a 10 turn circular winding with a = 1.0 and 0.85, highlighting the methodin which the turns of varying width are connected as well as the improvement in termination track width.703.4. Experimental Confirmation and Application Example: Low Power WPT SpiralFigure 3.13: Subset of circular windings employed to test the accuracy of the proposedmodeling of the hollow planar spiral winding with TWR.The other consideration is connecting the layers, and then the inductor to the externalcircuit. In order to connect the winding layers, vias are used at the center of the windingto allow for external connection on a lower PCB layer. A conservative estimate of less than1 A per via is used when determining the number of vias. All tests performed in this workcontained the exact same connection to the winding from the bottom layer of a 2 layer board.When connecting the winding to the external circuit, one of the benefits of applying a non-unity TWR is acknowledged in Fig. 3.12, which is that the input termination track widthcan be much wider than the unity TWR case which can reduce connection losses. In thisparticular case the windings with a = 0.85 has an input connection which is 3 times widerthan the unity TWR case.3.4.2 Experimental Prototype and PerformanceFig. 3.13 shows a series of circular planar spiral windings that were manufactured in orderto test the models developed in the previous section and their L, R, and Q were measured713.4. Experimental Confirmation and Application Example: Low Power WPT Spiralby employing a high-accuracy impedance analyzer. Excellent correlation was found betweenthe measurements and the calculated results from the models employed in this work, as canbe observed in Fig. 3.8. Tight tolerances, less than 5%, are noticed between predicted andactual values for L, R, and Q. In this situation, they are normalized parameters, where Lwas normalized to 40 µH, R was normalized to 250 mΩ, and Q was normalized to 25.723.4.ExperimentalConfirmationandApplicationExample:LowPowerWPTSpiralFigure 3.14: (a) Normalized representation of the performance of hollow planar spiral windings by removing turns. Qis normalized to 15, L is normalized to 25 µH, and R is normalized to 475 mΩ. A 15% reduction in L is observed inorder to attain a Q which is 50% higher. (b) Normalized experimental confirmation for L, R, and Q of 2-layer proposedhollow circular planar spiral windings with Track-Width-Ratio. Q is normalized to 15, L is normalized to 25 µH, and Ris normalized to 475 mΩ. It can be observed that by tuning the combination of xi and a, the inductance stays almostconstant while Q doubles.733.4. Experimental Confirmation and Application Example: Low Power WPT SpiralTable 3.3: WPT Experimental SpecificationsFactor Value UnitsL 25 µHNL 2fr 100 kHzf 105-200 kHzl 10 mmVin 19 VPout 5 W3.4.3 Low Power Wireless Power Transfer ApplicationThe proposed hollow planar spiral windings with TWR have the benefits of greatly increasedquality factor compared to traditional approaches and have the added benefit of being ableto tune the inductance up or down without changing the footprint. This makes the tech-nique much more suitable for low power, resonant WPT systems such as those covered by theWireless Power Consortium’s Qi standard. In this section, the trends observed in Figs. 3.10and 3.11 are confirmed experimentally, then compared to experimental data from the ap-proach of removing inner turns, and finally three windings are chosen for testing in a 5 WWPT system and their transmission efficiencies are documented: the baseline case, the bestcase winding with turns removed, and the best case proposed hollow planar spiral windingswith TWR applied. The specifications of the WPT system are contained in Table 3.3.The maximum frequency of operation is 200 kHz, which is where the ac losses will be attheir worst. This is the frequency the windings will be designed for. At 25 µH inductance,a 2 layer solution is chosen due to inspection of Fig. 3.10, whose surface was used to definea locus of designs for which the inductance is predicted to stay at 25 µH. A series of spiralswere manufactured to confirm this behavior, and were compared to the trends provided fromFigs. 3.10 and 3.11. The results are presented in Fig. 3.14(c). The data points follow 25 µH743.4. Experimental Confirmation and Application Example: Low Power WPT SpiralFigure 3.15: Schematic of the experimental wireless power transfer circuit. The inductors L1and L2 are the spirals under test.inductance line and plot the resistance and Q of the combination, up to the point calculatedhere as the maximum. The inductance stays approximately constant, while the resistancedecreases greatly, culminating in a Q which is over twice the initial value. These resultsare then compared with those achieved from hollowing the planar spiral by removing innerturns. In this case, the procedure was straightforward: measure L, R, and Q of a seriesof spirals in which one turn is removed at a time and observe the results. The data forthis particular case is displayed in Fig. 3.14(b). The Q only reaches 50% higher than thebaseline case, and at that point the inductance is reduced by 20%. This will also require acapacitor which is 25% larger in order to achieve the same resonant frequency for wirelesspower transmission. The proposed technique of increasing the internal radius and applyingTWR has more flexibility and provides a nearly constant value of inductance. This is intuitiveas the resistance decreases when the turns are shifted from the high field in the center, andthe shorter turn lengths get matched with smaller winding widths. The inductance increasesfrom the effect of hollowing, then decreases from the application of TWR, which providesthe tunable inductance.Three windings were placed in a WPT system, whose simplified circuit diagram is included753.4. Experimental Confirmation and Application Example: Low Power WPT SpiralFigure 3.16: Wireless power transfer system with proposed hollow planar spiral windingswith Track-Width-Ratio. Several connections and alignment aids have been removed to addclarity.in Fig. 3.15: the completely filled winding (baseline), the highest-Q option when turns wereremoved, and the highest-Q option when the inner radius was increased and TWR wasapplied. The series capacitors were chosen in order for resonance to occur at 100 kHz, andoperation was swept from 105 kHz to 200 kHz. A deconstructed view of the proposed windingsetup in the WPT transmission system is displayed in Fig. 3.16 with no alignment aids orconnections such that the spiral can be observed.Efficiency was compared at full load (5 W) condition. The best efficiency in each case wasat around 170 kHz, in which the baseline case had a 70% transmission efficiency (Fig. 3.17),the winding with turns removed had 80% (Fig. 3.18), and the proposed winding had 90%(Fig. 3.19). As predicted by the modeling, the proposed design method pushes planar spiral763.4. Experimental Confirmation and Application Example: Low Power WPT SpiralFigure 3.17: Input voltage, input current, output voltage, and output current of the WPTsystem employing the baseline full planar spiral windings with a unity Track-Width-Ratio at170 kHz. Input power is 7.8W and output power is 5.5 W, demonstrating a 70.5% efficiency.Figure 3.18: Input voltage, input current, output voltage, and output current of the WPTsystem employing hollow planar spiral windings with 4 turns removed at 170 kHz. Inputpower is 6.9W and output power is 5.5 W, demonstrating a 79.7% efficiency.773.5. SummaryFigure 3.19: Input voltage, input current, output voltage, and output current of the WPTsystem employing the proposed hollow planar spiral windings with non-unity Track-Width-Ratio at 170 kHz. Input power is 6.1W and output power is 5.5 W, demonstrating a 90.2%efficiency, the highest performance of the experimental cases.windings to exceptional efficiencies. The resulting low profile, high efficiency winding can beused for WPT in slim consumer electronics for battery charging purposes.3.5 SummaryThis chapter introduced the hollow spiral winding with Track-Width-Ratio (TWR) as ameans of improving the Quality Factor (Q) of planar spiral windings. The proposed windingstructure involved a two-factor interaction between extending the internal radius of the planarspiral winding in order to make it hollow, and then to apply TWR in order to decrease thewidth of internal turns of the winding. In doing so, the area of copper which can be impingedby the magnetic field is reduced, decreasing the overall ac resistance of the winding.This work applied a quadratic factor to the previous work to account for the applicationof TWR in order to predict the behavior at any frequency of operation. Since this technique783.5. Summaryis still reliant on finite element simulation data, a statistical Design of Experiments approachwas employed to create meta-models of the resistance values for a subset of windings in orderto observe the resistance trends with frequency, and later as part of a design example.Similar meta-models were created for the inductance, employing the models from Chap-ter 2. This investigation confirmed that the increase in internal radius can increase theoverall inductance of the winding, while TWR can decrease it. This allowed for combina-tions of these factors to be employed to tune the inductance, rather than the decrease ininductance observed from previous techniques.A specific design example was performed in order to maximize the Q of a circular planarspiral winding for application in a Wireless Power Transfer (WPT) system. Studying wind-ings based on the Qi specification, the proposed technique was able to improve Q by 100%while keeping the inductance constant. This was compared to the technique of removingturns, which experienced a 20% reduction in inductance and only 50% increase in Q. Remov-ing turns fails to have a second degree of freedom to tune the inductance without changingthe footprint of the winding.The proposed winding was then compared to the traditional planar spiral winding andthe winding with turns removed in a 5 W, 110-200 kHz WPT system. At the rated load, thetraditional planar spiral winding had an efficiency of 70%, the winding with removed turnshad an efficiency of 80%, while the proposed hollow planar spiral winding with TWR had anefficiency of 90%.79Chapter 4Capacitance Mitigation in PlanarSpiral Windings4One of the biggest drawbacks of employing planar spiral windings in power electronic ap-plications is the excessive parasitic capacitance between layers and windings. The wideoverlapping conductors act like capacitive plates which will reduce the operating frequencyof the coil and can distort current and voltage waveforms. In the worst-case scenario, inplanar transformers or wireless power transfer windings, high dVdtcan cause shoot-throughcurrents which can trip protections or damage devices.Previous attempts have been made to minimize capacitance in planar magnetics [53–58].Preliminary work has been performed on predicting capacitance in planar transformers for2-layer coreless planar spiral windings [53] and for multi-layer planar transformers when anequal-voltage-drop model is employed between each turn [54]. Improved shielding techniqueshave been proposed to mitigate the impact of planar transformer capacitance [55] and amethod of reducing voltage differential between overlapping conductors has shown promisefor extending PCB spiral winding operating frequency with an extensive number of layerconnections [56]. Recently an approach which removes the overlapping copper from thewinding window has been proposed [57] and was later improved to include a connection toground in the open space to reduce the capacitance and serve as an EMI filter [58].4Portions of this chapter have been accepted for publication in [S. R. Cove and M. Ordonez, “Low-Capacitance Planar Spiral Windings Employing Inverse Track-Width-Ratio”, accepted for publication inIEEE Energy Conversion Congress and Expo. (ECCE), 2016.]80Chapter 4. Capacitance Mitigation in Planar Spiral WindingsFigure 4.1: Planar spiral winding cross-sections and their impact on capacitance and re-sistance: Traditional winding techniques, removing overlapping windings, and the proposedwinding with Track-Width-Ratio (a) applied in opposite directions on each layer.In these previous investigations, either copper was removed from the winding or shieldswere added in order to lower capacitance. This means leakage inductances were increased, andat many frequencies the resistance was increased in order to reduce capacitance. The proposedtechnique in this work builds upon the techniques presented in the previous chapters. Insteadof applying the same TWR to each layer of a multi-layer winding, every layer changes thereference of the TWR from the outside turn, to the inside turn, or vice-versa. Applyingthis technique, named Inverse TWR, the amount of overlapping conductor is reduced whichresults in a significant reduction in stored capacitive energy. Since the outer dimension,inner dimension, and clearance values are not changed. Compared to other techniques, theproposed method maintains better resistive and inductive properties.814.1. Capacitance in Planar Spiral WindingsThis proposed inverse TWR technique is displayed in Fig. 4.1 with a qualitative descrip-tion of the claims within this chapter when compared to traditional winding designs anddesigns which reduce the copper within the winding window. The proposed technique re-duces the overlapping copper in the multi-layer winding while reducing the overall voltagegradients between overlapping turns, resulting in a significant reduction in overall capac-itance. This chapter provides a revised low-frequency resistance model for the proposedinverse TWR structure, a specific case of the windings in Chapter 2. This model is employedto find the voltages for each turn in order to simulate the capacitance employing 2D finiteelement analysis. The inductance model in Chapter 2 is employed to calculate the inductanceof the proposed structure.The analytical results for low frequency highlight the capability for improvement over thetraditional case. A comparative example is included, in which the optimized planar spiralwinding employing TWR was able to reduce the capacitance by 50%, reduce ac resistanceup to 20%, and keep the inductance within 4% of the nominal. The winding performancewas measured with an impedance analyzer to confirm the improved operation.4.1 Capacitance in Planar Spiral WindingsThe previous chapter introduced the generalized racetrack planar spiral winding and theunified dimensional system which continues to be employed for analysis in this chapter. Abrief reminder of the shape and the important dimensions has been included in Fig. 4.2.Other important dimensions are the number of layers, NL, and the layer spacing, s.824.1. Capacitance in Planar Spiral WindingsFigure 4.2: Important dimensions for the generalized racetrack planar spiral winding withTrack-Width-Ratio (TWR = a) applied.Figure 4.3: Distributed capacitance and voltage differences in planar spiral windings whenvoltage V is applied. Each overlapping conductor pair has the same capacitance, but a verydifferent voltage difference between them.834.1. Capacitance in Planar Spiral Windings4.1.1 Planar Spiral Winding CapacitanceTraditionally, the capacitance between overlapping layers in planar inductors can be approx-imated by considering them as parallel plate capacitors:C =ǫAs(4.1)where ǫ represents the permittivity of the material between the winding layers, A representsthe overlapping area between conductors, and s represents the vertical distance betweenthe two turns, as per the unified dimensional system presented in Chapter 2. This is acrude approximation due to the effects of fringing fields within the winding window, but itcan be used to approximate the individual capacitive effects between turns and layers thatmust be summed to achieve the total capacitance of the winding, as illustrated in Fig. 4.3.This highlights that while each overlapping conductor pair has the same capacitance, C0,the voltage differential between then is starkly different, decreasing exponentially as theconductors approach the center of the winding. This greatly affects the capacitive energythat is stored between each conductor pair (E), from the relationship:E =12CV 2 (4.2)This demonstrates that the outer turns are the most important for determining the overallcapacitive energy stored in the winding, and suggests that there are a variety of methodswhich can improve the capacitance of planar spiral windings:• Reduce the overlapping area (A) in adjacent layers.• Increase the layer separation (s) between layers.• Decrease the voltage differential between overlapping conductors.844.2. Inverse Track-Width-RatioConsidering the fact that s is generally chosen based on the required inductance and powerdensity of the application at hand, the degrees of freedom for reducing the capacitance inplanar spiral windings involves techniques to either reduce A or reduce the voltage differen-tial. At their extremes, techniques that remove all overlapping area are prone to increasedresistive losses, while techniques that completely reduce the voltage differential are prone torestrictive construction techniques and increased losses in connections between layers. Thiswork combines both techniques but takes neither of them to the extreme, and the results area significantly reduced capacitance, and a reduction in winding resistance.4.2 Inverse Track-Width-RatioThe proposed planar winding structure with inverse TWR is a way of designing multi-layerplanar spiral windings in which the width of each track changes as it approaches the centerof the winding. In one layer the widest conductor is on the outside of the spiral and in thenext layer the widest conductor is in the center of the winding as illustrated in Fig. 4.4. Forclarity, the spiral with the smaller inner turns will be referred to as the traditional layer, andthe spiral with the wider inner turns will be referred as the inverted layer. The TWR (a) is aratio of track widths which stays constant throughout the entire winding. In the traditionallayer of Fig. 4.4 it is referenced to the outer turn, which is of width W , thus reducing thewidth of inner turns. This is different from the inverted layer, in which the inner turn is ofwidth W and the outer turns reduce in width by the same ratio. The clearance (Cl) staysconstant between each turn.854.2. Inverse Track-Width-RatioFigure 4.4: Top view of the two layers of the proposed planar spiral winding with (a) tradi-tional TWR and (b) inverse TWR.864.2. Inverse Track-Width-Ratio4.2.1 Low-Frequency Resistance Modeling of the Inverse TWRPlanar Spiral WindingThe planar spiral winding structure with inverse TWR has some slight changes when com-pared to the generalized racetrack planar spiral winding, so an improved resistance modelhas been derived. This model will be employed to model the voltage gradient in the spiralwinding and used as an input to the finite element simulations in order to model the windingcapacitance. This is achieved from calculating the resistance of each turn as a ratio of thetotal resistance of the winding. The resistance of each turn (Rn) is calculated by employing:Rn =ρlntwn(4.3)where wn is a function of W , the widest turn width, which is defined in Chapter 2 as:W =1− a1− aN (xo − xi − (N − 1)Cl) (4.4)Then each winding width of the traditional layer follows as:wn(trad) = aN−nW (4.5)and the inverted layers as:wn(inv) = an−1W (4.6)where n always starts from 1 at the innermost turn and extends to N at the outermost turnof the winding, whether it is a traditional or inverted layer. The length of each individualturn, ln, is defined for any layer as:ln = 4(xc + rn(π2) + yc) (4.7)874.2. Inverse Track-Width-RatioFigure 4.5: Cross-section of a planar spiral winding with the inverse TWR structure. Eachlayer has its turns, ntrad and ninv extend from the interior to the exterior of the winding, eachranging from 1 to N , while p is defined from the turn of highest to lowest voltage, rangingfrom 1 to the product of the number of layers by the number of turns per layer, NLN .which is a function of the radius of each turn’s corner, rn, which now is different dependingon which layer is being discussed. The radii of the traditionally defined layer turns (rn(trad))is given by:rn(trad) = xi + (n− 1)Cl + (xo − xi − (N − 1)Cl)[aN−n2+n−1∑k=1aN−k](4.8)while the radii of the inverted layers (rn(inv)) is described as:rn(inv) = xi + (n− 1)Cl + (xo − xi − (N − 1)Cl)[an−12+n−1∑k=1ak](4.9)From that, the total resistance (Rtot) is:Rtot =NL∑m=1N∑n=1Rn (4.10)where NL is the number of layers.With this data, the turns can be iterated through in the order in which they are seenfrom the source (using p as the index as described in Fig. 4.5) and the voltage for each turn884.2. Inverse Track-Width-Ratio(Vp) can be derived as:Vp =VRtotNLN∑i=pRi (4.11)where V is the input voltage. With this data, Vp for every turn can be used in 2-D FiniteElement Analysis to find the overall capacitance of the proposed planar spiral winding withinverse TWR.4.2.2 Inductance Modeling of the Inverse TWR Planar SpiralWindingThe inductance calculation for the inverse TWR planar spiral structure will follow the calcu-lations from Chapter 2, which are duplicated here for clarity. The self inductance is calculatedfrom:LsN (µH) =N∑n=10.002ln[loge(2lnt+ aN−nW)+ 0.50049 +(t+ aN−nW3ln)](4.12)while the mutual inductances between straight section conductors is calculated through:Ml(µH) = 2l[loge{lGMD+(1 +l2GMD2) 12}−(1 +GMD2l2) 12+GMDl](4.13)orMjk(µH) = 0.001(√22)ljloge 1 +lklj+lj+klj1− lklj+lj+klj+lkljlogelklj+lj+klj+ 1lklj+lj+klj− 1 (4.14)based on their orientation. The self and mutual inductance components are summed throughthe use of Grover’s algorithm as per the relation:LTN = LsN +MN (4.15)894.2. Inverse Track-Width-Ratiowhere:MN = 2∑j 6=kMjk (4.16)The important dimensions in these calculation is the GMD which is approximated as thedistance between the center of conductors. With these models the inductance of the inverseTWR planar spiral winding can be investigated.4.2.3 Capacitance Simulation of the Inverse TWR Planar SpiralWindingCapacitance is measured in FEA by applying a DC voltage to the winding and measuringthe total capacitive energy developed (Etot), through the relation presented in (4.2), and theenergy density relationship:Etot =12∫A~D · ~EdA (4.17)where V is the lumped voltage applied to the winding and A is the area where capacitiveenergy exists. Setting (4.2) equal to (4.17) and setting V to 1 V, the resultant capacitanceis calculated from the FEA data by the equation:C =∫A~D · ~EdA (4.18)In order to simulate the correct amount of capacitive energy in a 2-D FEA model, thevoltage of each turn is required, as it is distributed across the winding. These voltages weredetermined through the application of (4.10) and (4.11). Fig. 4.6 displays the capacitive en-ergy density between a traditional planar spiral winding and the proposed winding structure.It can be seen that the proposed winding structure has a more even energy distribution dueto the reduced surface area between turns with the highest voltage differential between them.904.2. Inverse Track-Width-RatioFigure 4.6: FEA capacitive energy cross-section of (a) a traditional planar spiral windingand (b) the proposed inverse TWR planar spiral winding. The traditional winding exhibitsmore overlapping conductor and an area of extremely high capacitive energy at the windinginput terminals. The proposed winding provides double areas without overlap and much lessenergy trapped at the input terminal.While the same total cross-sectional width of conductor is used in each case, a lot of the fieldis dissipated through the reduced overlap provided by the staggered clearances.A series of planar spiral windings were simulated and calculated in order to observe theeffect of changing TWR on the overall inductance, resistance, and capacitance of the spirals.914.2. Inverse Track-Width-Ratio0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.950.10.20.30.40.50.60.70.80.91Track Width Ratio (a)Normalized Winding Capacitance 2 Turns4 Turns6 Turns8 TurnsFigure 4.7: Normalized simulated winding capacitance in the proposed planar spiral windingwith inverse TWR. Baseline parameters are included in Table 4.1.Table 4.1: Simulated Winding ParametersFactor Value Unitsxc, yc 5,2 mmri 1 mmNL 2N 2, 4, 6, 8 turnsro 15 mmCl 0.25 mmt 0.105 mms 1.1 mm924.2. Inverse Track-Width-Ratio0.5 0.6 0.7 0.8 0.9 10.850.90.951Track Width RatioNormalized Winding Resistance 2 Turns4 Turns6 Turns8 TurnsFigure 4.8: Analytical models of the DC resistance of the proposed planar spiral windingwith inverse TWR based on (4.10). Baseline parameters indicated in Table 4.1.The dimensions of the windings are contained in Table 4.1. Figs. 4.7, 4.8, and 4.9 displaynormalized capacitive, resistive and inductive results for 5 different 2-layer winding casesof changing number of turns. It can be observed that as TWR decreases, the capacitancerapidly falls and then stabilizes at a much lower value than the initial capacitance, while theresistance decreases to a minimum and then rebounds back as the TWR is further decreased.The inductance stays constant for the area of resistance decrease and then falls off rapidlyonce the capacitance settled at its expected value. In each case there is an optimal regionwhere the capacitance is within 5% of its minimum value, the resistance is still below the934.2. Inverse Track-Width-Ratio0.5 0.6 0.7 0.8 0.9 10.850.90.951Track Width RatioNormalized Winding Inductance N = 2N = 4N = 6N = 8Figure 4.9: Normalized calculated inductance results of the proposed planar spiral windingwith inverse TWR based on the analysis in Chapter 2. Baseline parameters indicated inTable 4.1.initial case, and the inductance is within 1% of its nominal value. This is considered to bethe optimal design point for applications in which the current density is roughly constantacross the conductor surfaces.944.2.InverseTrack-Width-RatioFigure 4.10: Subset of windings employed to test the accuracy of the proposed resistance and capacitance modeling ofthe inverse TWR planar spiral winding.954.3. Experimental Validation0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.950.10.20.30.40.50.60.70.80.91Track Width Ratio (a)Normalized Winding Capacitance 2 Turns4 Turns6 Turns8 TurnsFigure 4.11: Experimental (discrete) vs. predicted (continuous) capacitance values for theinverse TWR planar spiral winding structure.4.3 Experimental ValidationMultiple panels of planar spiral windings were manufactured in order to confirm the capac-itive, resistive, and inductive behaviour and to observe the effect of the proposed techniquewhen the frequency is changed. Fig. 4.10 presents a subset of the investigated windings.The capacitance and resistance of the planar spiral windings with dimensions presentedin Table 4.1 were measured with an impedance analyzer, and the results are presented inFigs. 4.11, 4.12, and 4.13. The experimental data closely matched the predicted data, except964.3. Experimental Validation0.5 0.6 0.7 0.8 0.9 10.850.90.951Track Width RatioNormalized Winding Resistance 2 Turns4 Turns6 Turns8 TurnsFigure 4.12: Experimental (discrete) vs. predicted (continuous) DC resistance values for theinverse TWR planar spiral winding structure.for the small amount of capacitance in each case which comes from the overlap induced fromconnecting the turns, and any stray capacitance from the connections. The trends that canbe observed are similar to the proposed models, where the capacitance tends towards anasymptotic value below the initial capacitance while the low frequency resistance drops asmall amount and then increases rapidly after a minimum value is reached. The inductanceremains constant and then drops off.974.3. Experimental Validation0 0.2 0.4 0.6 0.8 100.10.20.30.40.50.60.70.80.91Normalized Predicted InductanceNormalized Actual Inductance N = 2N = 4N = 6N = 8Figure 4.13: Normalized experimental vs. predicted inductance values for the inverse TWRplanar spiral winding structure.4.3.1 Experimental Performance ComparisonA comparative study was performed for a set of 8-turn windings in order to clarify the benefitsof employing the proposed planar spiral winding structure with inverse TWR. Three windingcases are considered: the traditional equal-turn-width planar spiral winding, the planar spiralwith no turns overlapping, and the proposed planar spiral winding with inverse TWR. Animpedance analyzer is employed to measure the impedance (Z) from 500 kHz to 50 MHz inorder to observe their inductance (L), capacitance (C), and self-resonant frequency (SRF ).984.3. Experimental ValidationFigure 4.14: Impedance magnitude and angle vs. frequency of the standard planar spiralwinding structure. The inductance is measured at low frequency by the rate of change of theimpedance, while the capacitance is measured by the rate of change of impedance at highfrequency. The SRF confirms the measurements.The inductance is measured by observing the slope of the impedance at low frequency by therelationship:Z = jωL (4.19)and the capacitance is measured by observing the slope of the impedance at high frequencyby the relationship:Z =−1jωC(4.20)The values are confirmed through the measurement of the SRF , which is the point at994.3. Experimental ValidationFigure 4.15: AC resistance of the standard planar spiral winding structure.which the impedance angle is 0◦ by the relationship:SRF =12π√LC(4.21)After the capacitance was determined, the resistance of the windings were measured withinan operating range from 20 Hz to 1 MHz. The resistances at 20 Hz, 200 kHz, and 1 MHzwere noted as comparison points between the three windings.The impedance profile for the traditional planar spiral winding is presented in Fig. 4.14.In this case all turns are 1.5 mm wide and overlap fully, separated by 0.254 mm spacing. Asis expected, this case is the worst case scenario. For a 3.36µH winding, it has an SRF ofonly 22.4 MHz, representing a capacitance of 15 pF. This capacitance reduces the workingfrequency range of the winding, and can cause current spikes when met with rapid changes in1004.3. Experimental ValidationFigure 4.16: Impedance magnitude and angle vs. frequency of the planar spiral windingstructure without overlapping conductors. The inductance is measured at low frequency bythe rate of change of the impedance, while the capacitance is measured by the rate of changeof impedance at high frequency. The SRF confirms the measurements.voltage, which are characteristic of switch-mode power supplies. The resistive characteristicsof the winding are presented in Fig. 4.15. This resistance profile is the standard by whichthe two options are compared, with 230 mΩ at 20 Hz, 527 mΩ at 200 kHz, and 1.3 Ω at 1MHz.Employing the same footprint as the traditional case, an 8-turn planar spiral winding withno overlap between turns was manufactured and tested. These turns were only 0.875 mmwide, with 0.875 mm spacing, so the low frequency resistance is predicted to be much higher,while the capacitance should be much lower. The impedance plot for this winding is presentedin Fig. 4.16. The SRF has increased, as expected, but only to 27.4 MHz due to a 30%1014.3. Experimental ValidationFigure 4.17: AC resistance for the planar spiral winding structure without overlapping con-ductors.increase in inductance. The inductive performance of this technique is problematic in manyapplications, especially resonant techniques in which it will heavily impact the operatingfrequency. The capacitive performance, though, is exceptional, with a 48% decrease. Some ofthe remnant capacitance can be attributed to connections between turns, and stray electricfield between layers. The resistive profile of this winding is contained in Fig. 4.17. Asexpected, the low frequency resistive performance is much worse than the traditional case,with an increase of 275% near DC. This performance gets better as the frequency increasesthough, with only a 40% increase in resistance at 200 kHz and a 16% decrease in resistanceat 1 MHz. This improved response at higher frequency can be attributed to the decrease inproximity effect losses due to spacing between conductors. If the increase in inductance canfit within the design specifications, this technique is applicable in some frequencies where the1024.3. Experimental ValidationFigure 4.18: Impedance magnitude and angle vs. frequency of the proposed inverse TWRplanar winding structure with a = 0.925. The inductance is measured at low frequency bythe rate of change of the impedance, while the capacitance is measured by the rate of changeof impedance at high frequency. The SRF confirms the measurements.proximity effect is a dominant issue.The final option for capacitance reduction is the proposed planar spiral winding structurewith inverse TWR. The design of this winding is based upon the performance observed inFigs. 4.12 and 4.11. The chosen TWR is a = 0.925 as that is where the resistance is thelowest, and is at an adequate reduction in capacitance to compete with the non-overlappingcase, roughly 50% of the traditional capacitance. The resulting impedance plot for thiswinding is included in Fig. 4.18, demonstrating an increase in the SRF to 32.9 MHz, thehighest frequency of the 3 tested windings. In addition, the inductance of the winding isonly 3.13 µH, within 4% of the traditional planar spiral winding. The capacitance reduced1034.3. Experimental ValidationFigure 4.19: AC resistance of the proposed inverse TWR planar winding structure with a =0.925.by over 50%, down to 7.48 pF. While the capacitance reduction is similar to the windingwith removed turns, the applications of this technique are more varied, due to its abilityto maintain inductance. The resistive trend for this winding is included in Fig. 4.19. At20Hz the resistance decreased by 7%, at 200 kHz the decrease is also 7%, but by 1 MHz,the decrease as spread to a 20% reduction in resistance. This performance is better thanthe case of removed turns at every frequency within the predicted operating range. Theadded benefit of this technique is that the TWR can be tuned to optimize behaviour at anyoperating frequency based on the technique proposed in Chapter 3. There are no degreesof freedom, beyond changing the footprint which can improve the technique of removingoverlap.1044.4. Summary4.4 SummaryThis chapter introduced the planar spiral winding with inverse Track-Width-Ratio as a meansof improving the capacitance in planar spiral windings. By changing track widths such thatthe traditional layer had wider turns on the outside of the spiral and the inverted layer hadwider turns on the inside of the spiral, the inverse TWR structure increased the amount ofnon-overlapping copper and reduced the area in the outside of the winding where the highestamount capacitive energy is stored. The analytical model for low frequency resistance fromChapter 2 was modified and included in a model for the voltage of every turn. The inductancemodel from Chapter 2 confirmed the inductance does not change substantially within thedesired design space of the inverse TWR structure. The resistance model was used in finiteelement simulations to confirm the decrease in capacitance and resistance experienced byemploying this technique. It was found that there is an optimal region where TWR can bechosen to decrease resistance to a minimum, the capacitance will be significantly reduced,and the inductance will be largely unchanged.The capacitance models were then confirmed through experimental measurements, andthen a comparative example was performed to highlight the benefits of the proposed tech-nique. Three windings were compared: the traditional planar spiral winding with equal turnwidths, the planar spiral winding with no overlapping copper, and a planar spiral windingwith inverse TWR that had been optimized for low frequency operation. It was found thatthe non-overlapping winding decreased the capacitance by 48% but increased the inductanceby 30% and the low frequency resistance by 275%. This became better at 1 MHz when areduction in resistance by 16% was observed. The proposed inverse TWR winding structurereduced the capacitance by over 50% while lowering the resistance by 6% at low frequencyup to 20% at 1 MHz. Finally, the inductance was kept within 4% of the traditional case.105Chapter 5Conclusions5.1 Conclusions and ContributionsThis dissertation explored three important aspects of the use of a Track-Width-Ratio (TWR)within concentric planar spiral winding design: low frequency resistance reduction and in-ductance prediction, high frequency quality factor improvement, and capacitance reduction.The generalized racetrack planar spiral winding was introduced and was accompanied by aunified dimensional system which could specify the dimensions of a variety of popular windingshapes:• Circular• Rectangular• Octagonal• Traditional Racetrack5.1.1 Low Frequency Resistance and InductanceIn Chapter 2, the self and mutual inductances for the generalized racetrack planar spiralwinding were modeled based on the work of Greenhouse and Grover. Many panels of planarspiral windings of varying shapes and sizes were employed in simulations and experimentalmeasurements in order to confirm their accuracy. In all cases, the inductance trends suggested1065.1. Conclusions and Contributionsthat the inductance of a planar spiral winding decreases exponentially as TWR decreasesfrom unity, and that this decay is more distinct in windings with more turns. In addition,a model for the low-frequency resistance of the generalized racetrack planar spiral windingwas developed analytically as a summation of the length-to-width ratio of each turn of thewinding. Applying TWR to spiral windings from previous work emphasized its capabilitiesto reduce low frequency resistance. The resistance trends observed with the use of TWRinvolved a parabolic decrease in resistance until a minimum is reached, followed by a parabolicincrease in resistance as TWR is decreased further. The derivative of the model was foundin order to predict the TWR of the lowest resistance.A design example followed, which demonstrated how to find the best TWR to be employedto maximize the performance of a planar spiral winding. This performance was measured bythe ratio of the inductance to the resistance, LR. The design example highlighted that thereis a maximum for this ratio, and that it diverges from the case of minimum resistance. Theoptimized design winding had a TWR of 0.85 and improved LRby 18%.5.1.2 High Frequency Resistance and Quality FactorChapter 3 introduced the hollow spiral winding with Track-Width-Ratio (TWR) as a tech-nique to improve the Quality Factor (Q) of planar spiral windings at high frequencies. Theproposed winding structure involved a two-factor interaction between extending the internalradius of the planar spiral winding in order to make it hollow, and then to apply TWR inorder to decrease the width of internal turns of the winding. By doing so, the effects of skinand proximity effect were significantly reduced, improving the overall performance of theplanar spiral windings.This work applied a quadratic approximation of the effect of TWR on high frequencyresistance based on normalized finite element simulation data. In order to remove finiteelement simulations from the model, a design area was chosen and a statistical Design of1075.1. Conclusions and ContributionsExperiments approach was employed to create meta-models of the resistance values. Similarmeta-models were created for the inductance, employing the models from Chapter 2. Thisinvestigation confirmed that the increase in internal radius can increase the overall inductanceof the winding, while TWR can decrease it. This allowed for combinations of these factors tobe employed to tune the inductance, rather than decrease inductance as was observed whenturns were removed.From the investigated windings, a specific design example was performed in order tomaximize the Q of a circular planar spiral winding for application in a Wireless PowerTransfer (WPT) system. The proposed technique was able to improve Q by 100% whiletracking a constant inductance. The same winding footprint was employed to study theeffect of removing inner turns, which resulted in a 20% reduction in inductance and onlya 50% increase in Q. Removing turns fails to have a second degree of freedom to tune theinductance without changing the footprint of the winding. The proposed hollow planar spiralwinding with TWR was tested dynamically and compared to the traditional planar spiralwinding and the winding with turns removed in a 5 W, 110-200 kHz WPT system. At therated load, the traditional planar spiral winding had a transmission efficiency of 70%, thewinding with removed turns had a transmission efficiency of 80%, while the proposed hollowplanar spiral winding with TWR had a transmission efficiency of 90%.5.1.3 Capacitance MinimizationChapter 4 introduced the planar spiral winding with inverse TWR as a technique to reducethe parasitic capacitance in planar spiral windings. By changing track widths such that thetraditional layer had wider turns on the outside of the spiral and the inverted layer hadwider turns on the inside of the spiral, the inverse TWR structure increased the amountof non-overlapping copper and reduced the area of the highest voltage differential betweenlayers, where the highest amount of capacitive energy is stored. A low-frequency voltage1085.1. Conclusions and Contributionsmodel for each turn was developed in order to simulate the capacitive energy distributionin the proposed structure. This model was used in finite element simulations to confirm thedecrease in capacitance and resistance experienced by employing this technique. It was foundthat there is a TWR-range where the resistance reduces to a minimum and the capacitancewill be significantly reduced.The capacitance models were then confirmed through experimental measurements, andthen a comparative example was performed to highlight the benefits of the proposed tech-nique. Three windings were compared: the traditional planar spiral winding with equal turnwidths, the planar spiral winding with no overlapping copper, and a planar spiral windingwith inverse TWR that had been optimized for low frequency operation. It was found thatthe non-overlapping winding decreased the capacitance by 48% but increased the inductanceby 30% and the low frequency resistance by 275%. This became better at 1 MHz when areduction in resistance by 16% was observed. The proposed winding structure reduced thecapacitance by over 50% while lowering the resistance by 6% at low frequency up to 20% at1 MHz. Finally, the inductance was kept within 4% of the traditional case.The concepts introduced in all chapters of this work were thoroughly tested with finiteelement simulations and experimental measurements employing a high-frequency LCR meter,a high-precision Frequency Response Analyzer, and a high-resolution oscilloscope.5.1.4 Specific Academic ContributionsThe work on low-frequency inductance and resistance modeling culminated in the followingpublications:1095.1. Conclusions and Contributions• S. R. Cove, M. Ordonez, “Practical inductance calculation for planar magnetics withtrack-width-ratio,” in Energy Conversion Congress and Exposition (ECCE), 2013, pp.3733–3737.• S. R. Cove, M. Ordonez, “Wireless-power-transfer planar spiral winding design applyingtrack width ratio,” IEEE Trans. Ind. Appl., vol. 51, no. 3, pp. 2423–2433, May 2015.These works cover the inductance and resistance prediction, and include a design examplewhich employs the meta-model for high-frequency resistance presented in Chapter 3.The following publication introduced the hollow planar spiral winding technique andcovered the high frequency resistance model, presented the spiral winding design technique,and included the WPT design example:• S. R. Cove, M. Ordonez, N. Shafiei, J. Zhu, “Improving Wireless Power Transfer Ef-ficiency Using Hollow Windings with Track-Width-Ratio,” IEEE Trans. Power Elec-tron., vol. 31, no. 9, pp. 6524–6533, Sept. 2016.The following publication introduced the inverse TWR structure for capacitance reductionin planar spiral windings:• S. R. Cove and M. Ordonez, “Low-Capacitance Planar Spiral Windings Employing In-verse Track-Width-Ratio”, accepted for publication in IEEE Energy Conversion Congressand Expo. (ECCE), 2016.In addition to the body of work described above, foundational and complementary workwas published which investigated Track-Width-Ratio and Design of Experiments methodol-ogy for planar transformers in LLC resonant converters:• S. R. Cove, M. Ordonez, F. Luchino, J. Quaicoe, “Applying Response Surface Method-ology to Small Planar Transformer Winding Design,” IEEE Trans. Ind. Electron., vol.60, no. 2, pp. 483–493, Feb. 2013.1105.2. Future Work• S. R. Cove, M. Ordonez, F. Luchino, J. Quaicoe, “Integrated magnetic design of smallplanar transformers for LLC resonant converters,” in Energy Conversion Congress andExposition (ECCE), 2011, pp. 1839–1844.• S. R. Cove, M. Ordonez, F. Luchino, J. Quaicoe, “Applying Response Surface Method-ology to planar transformer winding design,” in Energy Conversion Congress and Ex-position (ECCE), 2010 IEEE, 2010, pp. 2182–2187.Some publications resulted from collaborative research around magnetic design for planarinductors and transformers:• N. Shafiei, M. Ordonez, S. R. Cove, M. Craciun, C. Botting, “Accurate modeling anddesign of LLC resonant converter with planar transformers,” in Energy ConversionCongress and Exposition (ECCE), 2015, pp. 5468–5473.• J. M. Galvez, M. Ordonez, S. R. Cove, J. Quaicoe, “Accurate modeling and design ofLLC resonant converter with planar transformers,” in Canadian Conference on Elec-trical and Computer Engineering, 2012, pp. 1–4.5.2 Future WorkThis work opens up many full areas of research which can build upon its findings. Some ofthe main areas in which multiple Masters and PhD students could be supervised include:• Investigating the effects of a non-constant ratio of turns: While a constant ratio iseasy and convenient for fabrication and design, there is no guarantee that it is theoptimal design for every application. The effects of ratios which change quadratically,or exponentially will be investigated to observe the effects on parasitic elements.1115.2. Future Work• Application of the work in Wireless Power Transfer: While the windings investigatedin this work have been proven to be beneficial for low-power Wireless Power Transferapplications, a large investment will be needed to design windings which can work forapplications such as electric vehicle battery charging. A 3-D Wireless Power Transfertestbed is currently being developed to provide the necessary conditions to thoroughlytest a high-power WPT setup with planar spiral windings employing TWR.• Investigating the effect of adding a magnetic core: This work investigated corelesswindings and focused on the effect of winding design exclusively. The next step in thisregard is to add a magnetic core and investigate TWR along with core geometry, corethickness, airgap length, and see if it has a significant impact on parasitic elements andperformance. 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