PLANAR contactless battery charging platform is an

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1 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 22, NO. 1, JANUARY Equivalent Circuit Modeling of a Multilayer Planar Winding Array Structure for Use in a Universal Contactless Battery Charging Platform Xun Liu, Student Member, IEEE, and S. Y. Ron Hui, Fellow, IEEE Abstract In this paper, an equivalent circuit model of a multilayer planar winding array structure that can be used as a universal contactless battery charging platform is presented. This model includes the mutual-inductive effects of partial overlaps of planar windings in the multilayer structure. It has been successfully simulated with PSpice and practically verified with measurements obtained from three prototypes. This circuit model forms the basis of an overall system model of the planar charging platform. It is demonstrated that model parameters can be derived from the geometry of the winding structure. Errors between the calculated and the measured results are found to be within a tolerance of 5%. Index Terms Battery charger, equivalent circuit model, planar spiral inductance. (a) I. INTRODUCTION PLANAR contactless battery charging platform is an emerging technology that has the potential of unifying the charging protocols of portable consumer electronic products such as mobile phone, CD players, etc. Recently, two approaches have been proposed and are documented in several patent documents [1], [2], [10]. The first approach [1] adopts a horizontal flux approach in which the line of magnetic flux flows horizontally to the planar charging surface. This horizontal flux principle is in fact similar to that of the ac electromagnetic flux generated in a cylindrical motor, except that the cylindrical structure is compressed into a flat pancake shape. As the flux needs to flow horizontally along the upper and lower surfaces, two inherent limitations arise. First, an electromagnetic flux guide must be used to guide the flux along the bottom surface. This is usually a layer of soft magnetic material such as ferrite or amorphous alloy. In order to provide sufficient flux, this layer must be thick enough so that the flux can flow along the layer of soft magnetic material without magnetic saturation. Second, a similar problem applies to the secondary device that has to pick up the flux (and energy) on the upper surface of the charging platform. Fig. 1(b) shows the energy-receiving device required for the charging platform of Manuscript received January 24, 2006; revised March 29, This work was supported by the Hong Kong Research Grant Council under Project CityU 1223/03E and by the City University of Hong Kong. Recommended for publication by Associate Editor J. A. Ferreira. The authors are with the Department of Electronic Engineering, City University of Hong Kong, Hong Kong, China ( eeronhui@cityu.edu.hk). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TPEL (b) Fig. 1. (a) Inductive battery charging platform (with magnetic flux lines flow horizontally along the charging surfaces) proposed by Beart et al [1]. (b) Secondary device for use with the charging platform proposed by Beart et al [1]. The horizontal flux has to go through the shaded cross-sectional area. Fig. 1(a). It consists of a magnetic core and a winding. In order for the winding to sense the ac flux, the flux must flow into the cross-sectional area [shaded in Fig. 1(b)] that is vertical to the charging surface. Therefore, this cross-sectional area must be large enough (thick and wide enough) so that enough flux and energy can be picked up by the secondary device. It should be noted that this secondary device must be housed inside the electronic equipment to be charged on the charging platform. The thickness of the secondary device is crucial to the applicability and practicality of the device. If it is too thick, it simply cannot be housed inside the electronic equipment. Another planar inductive battery charging platform based on a perpendicular flux approach was proposed in [2]. Unlike the one described in [1], this charging platform generates an ac flux that has almost uniform magnitude over the entire charging surface. The lines of flux of this charging platform flow perpendicularly into and out of the charging surfaces (Fig. 2). This perpendicular flow of flux is very beneficial to the slim design of the energy-receiving element because it allows the energy transfer over the surface on which the electronic equipment (to be charged) is placed [3]. Particularly, it allows a thin secondary energy-receiving device to be developed for the charging platform /$ IEEE

2 22 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 22, NO. 1, JANUARY 2007 Fig. 2. Inductive battery charging platform (with magnetic flux lines flowing in and out perpendicularly of the charging surface) proposed by Hui [2]. Fig. 5. Structure of the combination of different columns. Fig. 3. Photograph of a universal planar charging platform [5]. TABLE I GEOMETRY OF EACH WINDING OF THE THREE PLATFORMS Fig. 4. Structure of three-layer hexagonal-spiral PCB winding arrays to generate uniform MMF over the planar surface. For both planar charging platforms described above, it is necessary to use an electromagnetic shield on the bottom surface. In case the charging platform is placed on a metallic desk, the ac flux generated in the charging platform may induce currents in the metallic desk, resulting in incorrect energy transfer and even heating effects in the metallic desk. A patented electromagnetic shield [4] has been shown to be effective for this type of planar charging platform. The electromagnetic shield in patent [4] simply consists of a thin layer of soft magnetic material (such as ferrite) and a thin layer of conductive material (such as copper). Based on the perpendicular flux approach, we focus on a contactless battery charging platform for portable consumer electronic equipment (Fig. 3). In [2], [3], and [5], the fundamental principles of the uniform magnetomotive force (MMF) generation based on multilayer printed circuit board (PCB) winding arrays were illustrated and verified by experiments. By representing a spiral hexagonal planar winding as a single hexagon, the structure of the three-layer hexagonal-spiral PCB winding arrays is shown in Fig. 4. Each layer of winding array is represented by a different color (red, blue and green). Fig. 5 shows a few combinations of the overlapped multilayer structures in different columns. In order to simplify the modeling, six hexagonal windings connected in series [shown in Fig. 5(a)] are considered as one column of windings. Each hexagonal winding spirals inwardly. The winding ends in the center of the hexagon and then moves into another layer as the beginning of another hexagonal winding. It can be seen that there is some overlap of the hexagonal windings even within each column. Fig. 5(b) shows the structure of two adjacent columns. Since there is overlap between them, they are considered to be overlapped columns. Fig. 5(c) shows that there is no overlap between the first and third columns. So they are considered to be nonoverlapped columns. In this paper, the equivalent circuit model is developed by considering the inductance of one column of six series-connected hexagonal windings as one unit (or column) of the multilayer winding structure. Attention is paid to the negative coupling effects between the overlapped and non-overlapped windings and columns. To verify the circuit model, the measured and simulated input impedance are compared. This equivalent circuit modeling of the multilayer planar winding arrays forms the basis of the overall modeling of the charging platform. If the parameters of the circuit model could be calculated based on the dimensions and geometry of the winding array structure without any measurement, the equivalent circuit model can be used to design and optimize the planar winding array structure

3 LIU AND HUI: EQUIVALENT CIRCUIT MODELING 23 TABLE II CALCULATED, SIMULATED AND MEASURED INDUCTANCE OF ONE WINDING OF THE THREE PLATFORMS Fig. 6. Configuration of the hexagonal winding [8]. in the early design stage. From previous research [6] [8], the inductance of each hexagonal spiral winding can be calculated from the geometry of the spiral winding. The mutual effects between the windings can be simulated out quickly by the use of the simplified one-turn model. Based on the calculated and simulated results, inductive parameters of the circuit model could be estimated, and the optimal operating frequency of the charging platform can be decided. The resistive parameters of the circuit model can also be calculated using the skin effect equations so that the power loss of the planar winding arrays can be predicted. The equivalent circuit model with the parameter estimation provides a useful tool not only for performance prediction but also for initial design of the charging platform. II. CIRCUIT MODELING OF THE MULTILAYER PLANAR WINDING ARRAYS In this study, three planar three-layer PCB winding array structures (or platforms) are used to evaluate the validity of the equivalent circuit. The three prototypes have eight, eight, and six columns, respectively. Structural details of each hexagonal winding unit of the three prototypes are given in Table I. The meaning of the parameters in Table I,,, and is illustrated in Fig. 6. A. Inductance of one Column The configuration of one hexagonal winding is shown in Fig. 6. The hexagonal winding and the winding array can not be revolved around an axis of symmetry, so that their inductance can not be computed directly by the use of the relatively easier and quicker 2-D finite element simulator in Ansoft [9]. The 3-D field simulator in Ansoft is an available replacement. But such computation is very time-consuming and is more appropriate for design verification than the design of an inductor, especially when the structure gets more and more complicated, like the multiturn winding arrays in this paper. FastHenry [11] is another alternative, which uses multipole-acceleration to reduce both required memory and computation time so that the complexity grows more slowly with problem size. It will be used as a reference in the later part. On the other hand, the more efficient way for self-inductance calculation is to consider some analytical methods. The Hurley method [6] takes full account of the current density distribution in the coil cross section and the eddy current losses in the substrate, which makes it adequately precise. But the formulas are based on the equation of the mutual inductance between two circular filaments. So when it is used to calculate the hexagonal inductors, some approximation must be made in advance. With the increase of the number of turns, such approximation may accumulate to a potential error. Another technique is to use the Greenhouse method [7] to calculate the inductance. The Greenhouse method offers sufficient accuracy and adequate speed, but cannot provide an inductor design directly from specifications and is cumbersome for initial design. Reference [8] provides three new approximation expressions for the inductance of square, hexagonal, octagonal, and circular planar inductors. The third expression in [8] is obtained by using data-fitting techniques, so that it is only applicable to inductors in the author s database. By comparing the first two expressions, the second one considers the concepts of geometric mean distance (GMD), which makes it more suitable for the PCB tracks with rectangular cross sections. Comparison between the calculated results with the simulated and measured results also shows that the second expression can achieve an accuracy high enough. The self-inductance of one winding is expressed by the second method in [8] as where is the number of turns, is the average diameter and equals 0.5, is the fill ratio and defined as, and are illustrated in Fig. 6, and are 1.09, 2.23, 0.00, and 0.17, respectively, for hexagonal winding. With the use of (1), the inductance values of one winding of the three platforms are calculated and compared with the measured results, as listed in Table II. The simulated results by the use of FastHenry are also presented. The errors are not higher than 2%. As shown in Fig. 5(a), six windings are connected in series to form a column, with some overlaps. Similarly, as explained in the latter part, there exists a negative mutual inductance between the partial overlapped windings so that the self-inductance of (1)

4 24 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 22, NO. 1, JANUARY 2007 Fig. 7 Equivalent circuit of (a) one winding, (b) two adjacent windings, (c) three adjacent windings, (d) one single column, and (e) two adjacent columns. one column is definitely a portion of the sum of the self-inductance of the six windings. Fig. 7(a) shows the circuit model of one winding. It consists of a resistor and an inductor connected in series. When two partly-overlapped windings are considered, as shown in Fig. 7(b), the mutual inductance between them,, must be included. The mutual coupling coefficient, which is equal to numerically, can be calculated out by fully modeling it in finite element software. But even with the accelerated method, such as using FastHenry, the required memory and computation time still grow faster than, where is the number of volume-elements [11]. This problem becomes much more serious when the coupling effects between columns are considered later in the paper. In [12], it is demonstrated that the mutual coupling coefficient between windings can be approximated out by simplifying each winding into one turn with the same dimension, as shown in Fig. 6. From [12], a conclusion can be extended that such approximation is accurate enough only if the spacing between tracks is smaller than the track width,. Such condition is always desired and satisfied in this design because a smaller spacing improves the interwinding magnetic coupling and reduces the power loss [8]. For three adjacent windings as shown in Fig. 7(c), the mutual coupling coefficient between two nonoverlapped windings is represented by which equals. Its value can also be simulated out quickly with the one-turn winding structure. models the capacitive coupling between the overlapped areas. It is in the order of only a few tens of pico-farads. Fig. 7(d) shows the detailed equivalent circuit of one single column which consists of six windings or more connected in series. For simplicity, and are represented by two PSpice K_Linear parts in the figure. The inductance of one single column,, is approximated by the use of (2) (2) where is the number of windings connected in series to form a column. Derivatoin of (2) is given in the Appendix. The simulated values of and by the use of the simplified one-turn winding structure are listed in Table III. It can be found that the coupling coefficients between the windings of Platform A and B are the same because they have the same dimensions. In addition, the simulated value of the coupling coefficients between far-apart windings (,, etc.) shows that they are small enough to be neglected. The self-inductance of one column of the three platforms are calculated with (2) and compared with the measured results in Table III. In the calculation, the calculated inductance of one winding as listed in Table II is used. The agreement between the calculated and measured results proves the accuracy of such one turn simplification for coupling calculation. B. Mutual Effects Between Columns The equivalent circuit of one single column and two adjacent columns are shown in Fig. 7(d) and (e)., and represent the inductance, capacitance, and resistance of one column, respectively. In Fig. 7(e), represents the capacitance between two adjacent columns. The circuit model can be easily implemented in PSpice. The PSpice K_Linear part, represents the mutual inductance between two adjacent columns [e.g., columns 1 and 2, or columns 2 and 3, or columns 3 and 4, etc. as shown in Fig. 5(b)]. Numerically, equals. Fig. 7(e) also shows the simplified equivalent circuit of two adjacent columns. The overall inductance, resistance, capacitance of (where 2,3, 8) series-connected columns are defined as,, and, respectively. Assuming that the parameters of each individual column are identical to those of the other column, the parameter equations

5 LIU AND HUI: EQUIVALENT CIRCUIT MODELING 25 TABLE III CALCULATED AND MEASURED INDUCTANCE OF ONE COLUMN OF THE THREE PLATFORMS TABLE IV CALCULATED AND MEASURED INDUCTANCE OF TWO COLUMNS OF THE THREE PLATFORMS TABLE V CALCULATED AND MEASURED INDUCTANCE OF THREE COLUMNS OF THE THREE PLATFORMS of two series-connected adjacent columns (such as columns 1 and 2) can be expressed as (3) (4) From (3), the mutual inductance between two adjacent columns is (5) When the inductance of three adjacent columns as shown in Fig. 5(d) is taken into account, the mutual inductance between the two nearest nonoverlapped columns [columns 1 and 3, or columns 2 and 4, or columns 3 and 5, etc. as shown in Fig. 5(c)] should be considered. Another K_Linear part, represents the mutual inductance between the two nearest non-overlapped columns. It is equal to numerically. The inductance of columns 1, 2 and 3 connected in series, can be expressed as (7) (6) The mutual inductance, is,,,,, and can be measured with an impedance analyzer (HP4194A), or be determined from the dimensions and geometry of one winding column. To determine the value of, the one-turn simplification method is also used. In the simulation, each column is simplified to six one-turn windings connected in series so that the required memory and computation time decrease obviously. The simulated of the three platforms is listed in Table IV. It is found interestingly that is almost a fixed value for the three platforms. The calculated inductance of two adjacent columns of the three platforms by the use of (3), as well as the measured results is also given out. In the calculation, the calculated inductance of one column as listed in Table III is used. Similarly, the simply simulated is listed in Table V. It can be found that the difference between the three platforms is very small too. The calculated inductance of three columns and the measured results are also given out. The agreement between the calculated and measured results in Table IV and V proves that such one-turn simplification is also applicable for the calculation of coupling effects between columns. The negative sign of mutual inductance can be explained by the definition of mutual inductance. Let us use two adjacent columns as an example. Columns 1 and 2 have some overlap (but they are not exactly on top of each other). In the overlap (8)

6 26 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 22, NO. 1, JANUARY 2007 Fig. 8. Current distribution on the cross section of the conductor. area, the current directions of the two overlap layers are opposite, resulting in some flux cancellation. Based on the above approach, mutual inductance terms of far-apart non-overlapped columns (,,, etc.) can also be derived. One may expect that such far-apart non-overlapped columns (i.e., they are not the two nearest non-overlapped columns) have negligible mutual inductance. Although such assumption could simplify the equivalent circuit greatly, it will be shown later in the paper that such omission may bring a little error to the simulation and some compensation can bring out a perfect result. Fig. 9. Calculated resistance of one winding of platform B, compared with the measured results. C. Resistance The resistance of a conductor is where is the length of the conductor, is its conductivity and is the area of its cross section. At low frequency, is the whole area of the cross-section. With the increase of the frequency, due to the skin effect, decreases and increases correspondingly. The skin effect depth could be calculated with (9) (10) where is the frequency (in Hertz) of the current, is the magnetic permeability of the conductor, and is the conductivity of the conductor. With the use of (10), the skin effect depth of copper could be calculated out as 0.66 mm at 10 khz, 0.21 mm at 100 khz, mm at 500 khz, mm at 1 MHz, and 0.02 mm at 10 MHz. Because the thickness of the conductor (0.14 mm) is smaller than the double of the skin effect depth even nearly 1 MHz, only the width of the cross-section of the conductor is needed to consider in the operating frequency range from 100 khz to 500 khz, as shown in Fig. 8. So at high frequency, when 2, the resistance of the conductor is a function of the frequency if (11) In (9) and (11), the length of the conductor could be calculated from the geometry of one winding (12) where is the number of windings included, is the number of turns of one winding, is the outer diameter, is the track width and is the spacing between the tracks, as shown in Figs. 6 and 8. Fig. 10. Calculated resistance of two columns of platform B, compared with two types of measured results. Summarized, because PCB is very thin but very wide, skin effect is more important than proximity effect [12]. So in the frequency range below 1 MHz, the resistance of the winding array structure could be calculated out with this function below, neglecting the influence of proximity effect if if (13) With (13), the resistance of the platform working at different frequencies could be calculated. The calculated and measured resistance of one winding of platform B is shown in Fig. 9. With the same method, the calculation is carried out for two columns of platform B. The results are shown in Fig. 10 and compared with two types of measured results. In Fig. 10, the asterisks are obtained from the measurement by an impedance analyzer (HP4194A), and the circles are acquired with the

7 LIU AND HUI: EQUIVALENT CIRCUIT MODELING 27 TABLE VI CALCULATED PARAMETERS IN THE EQUIVALENT CIRCUIT OF THREE PLATFORMS concept of energy conservation. In the second measurement, the active power inputted into the platform is totally consumed by the resistance of the conductors as power loss, and could be acquired by the measurement of the voltage and current across the platform. At a certain frequency, the resistance value equals to the division of the measured active power by the square of the rms current value. With the frequency limitation of the PWM control IC for the inverter, the second measurement is only conducted in the frequency range from 144 to 424 khz. The small discrepancy between the calculation and the measurement is thought to be due to the unknown resistance in the interconnections across the PCB layers. Fig. 11. Circuit model of the arrays consisted of eight columns. D. Capacitance As shown in Fig. 5, there are some overlaps inside a single column and between two adjacent columns. The electric field between the overlapped areas is represented by,, and in Fig. 7, respectively. The capacitance could be calculated by the use of (14) where is the overlapped area and is the distance between two layers. It should be noted that is only of the enveloped area because of the spacing between the tracks. In general, prediction of and is more important in this equivalent circuit modeling than and. Together with an externally connected capacitor (that is much greater than and ), the overall inductive value of columns forms a resonant tank that affects the suitable operating frequency range. As it is necessary to include an external capacitor of several tens of nano-farads in the charging platform [5] for its charging operation, the determination of the capacitive parameters in the equivalent circuit is not essential because the model capacitance is usually in the order of a few tens of pico-farads. III. VERIFICATION OF THE CIRCUIT MODEL Fig. 11 shows the detailed circuit of a prototype with eight columns. To verify the circuit model, three different planar winding platforms are tested. The input impedance of each platform is simulated and compared with the measurements obtained from an HP4194A impedance analyzer. In the PSpice simulation, the circuit parameters in Fig. 11 come from the calculated results. Two sets of simulation are carried out for each platform. In the first simulation, the equivalent circuit does include the mutual inductance of the two nearest non-overlapped columns but not those far-apart nonoverlapped columns. In the second simulation, compensation is included in the circuit model so that the mutual inductance of far-apart nonoverlapped columns is taken into consideration. A. Platform A This platform consists of eight columns altogether. Each column consists of six hexagonal windings connected in series. The geometrical information of each winding is listed in the first row of Table I. Table VI gives out the calculated results of the inductance, resistance and capacitance of one single column and the capacitance between two adjacent columns. The resistance is a function of frequency as expressed by (13). The simulated coupling coefficients between the columns, and are also listed. Two simulations have been carried out based on the circuit model (i) without and (ii) with the inclusion of mutual inductance of far-apart nonoverlapped columns. Fig. 12(a) shows the simulated and measured input impedance of the arrays without including the mutual inductance of far-apart nonoverlapped columns. The measured and simulated results are close to each other (within 10% tolerance). It is found that by adding about 10% of and 5% of so that and , the mutual inductance between far-apart non-overlapped columns can be reflected in the equivalent circuit. Fig. 12(b) shows the good agreement of the measured and theoretical values based on the compensated model that includes mutual inductances between far-apart non-overlapped columns.

8 28 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 22, NO. 1, JANUARY 2007 Fig. 12. Simulated and measured input impedance of eight columns of platform A: (a) without and (b) with mutual inductances between far-apart nonoverlapped columns. B. Platform B There are eight columns altogether in Platform B. Each column consists of six hexagonal windings connected in series. The configuration of each winding is listed in the second row of Table I. The parameters in the simulation are listed in the second row of Table VI. Without considering the mutual inductance of the far-apart non-overlapped columns, Fig. 13(a) shows the simulated and measured input impedance of Platform B. It can be seen that the simulated and measured results are fairly accurate. When 10% of and 5% of are added to include the mutual inductance of the far-apart non-overlapped columns so that and , the compensation yields more accurate results as shown in Fig. 13(b). C. Platform C There are six columns altogether in Platform C. Each column consists of six hexagonal windings connected in series. The configuration of each winding is listed in the third row of Table I. The third row of Table VI contains the calculated parameters in the equivalent circuit. Fig. 14(a) shows the simulated and measured input impedance of six columns of the first simulation without considering the mutual inductance of the far-apart nonoverlapped columns. They are in fairly good agreement. When 10% of and 5% of are added as compensation so that and , their agreement is even better as shown in Fig. 14(b). The three sets of comparison based on three different prototypes have confirmed that the equivalent circuit model is sufficiently accurate as a tool for performance prediction. In the simulation, the value of the proportion added to and (10% Fig. 13. Simulated and measured input impedance of eight columns of platform B: (a) without and (b) with mutual inductances between far-apart nonoverlapped columns. Fig. 14. Simulated and measured input impedance of six columns of platform C: (a) without and (b) with mutual inductances between far-apart nonoverlapped columns.

9 LIU AND HUI: EQUIVALENT CIRCUIT MODELING 29 and 5%, respectively) is only used to illustrate the function of the mutual inductance of far-apart nonoverlapped columns. So exact values are not necessary to be established. IV. CONCLUSION This paper presents an equivalent circuit model of a multilayer PCB winding array structure that can be used as a universal battery charging platform. The model includes the mutual effects of partial overlaps and nonoverlap of planar windings in the multilayer structure. The equivalent circuits of three different platforms have been successfully simulated with PSpice and practically confirmed with measurements. The model parameters are important in facilitating the optimal design of the resonant operation and the power loss of the universal battery charging platform. This circuit model forms the basis of an overall system model of the planar charging platform. With this model as well as the parameter estimation, the operating frequency of the planar winding arrays and the power loss of the conductor can be determined so that energy can be transferred optimally to the secondary circuit via near field coupling. Most importantly, this paper proposes a general method to analyze the inductive parameters of a similar complicated structure which is composed of many units (windings or columns). The self-inductance of one unit can be calculated analytically and the coupling coefficient between the units can be simulated out quickly with the simplified winding structure. Incorporating the inductance of a unit and the coupling effect between each, the distributed circuit model accelerates the design process, compared to the conventional finite element fully-modeled method. APPENDIX The derivation of (2): If we use as the number of windings that are connected in series to form a column, then REFERENCES [1] P. Beart, L. Cheng, and J. Hay, Inductive Energy Transfer System Having a Horizontal Magnetic Field, U.K. Patent GB , [2] S. Y. R. Hui, Planar Inductive Battery Charger, U.K. Patent GB238972, [3], Rechargeable Battery Circuit and Structure for Compatibility With a Planar Inductive Charging Platform, U.S. Patent , Jul [4] S. Y. R. Hui and S. C. Tang, Planar Printed Circuit-Board Transformers With Effective Electromagnetic Interference (EMI) Shielding, U.S. Patent , May [5] S. Y. R. Hui and W. C. Ho, A new generation of universal contactless battery charging platform for portable consumer electronic equipment, IEEE Trans. Power Electron., vol. 20, no. 3, pp , May [6] W. G. Hurley and M. C. Duffy, Calculation of self and mutual impedances in planar magnetic structures, IEEE Trans. Magnetics, vol. 31, no. 4, pp , Jul [7] H. M. Greenhouse, Design of planar rectangular microelectronic inductors, IEEE Trans. Parts, Hybrids, Packag., vol. PHP-10, no. 2, pp , Jun [8] S. S. Mohan, M. M. Hershenson, S. P. Boyd, and T. H. Lee, Simple accurate expressions for planar spiral inductances, IEEE J. Solid-State Circuits, vol. 34, no. 10, pp , Oct [9] Getting Started: A 2D Parametric Problem, Ansoft Corporation, Maxwell 2D Field Simulator (v. 6.4), Jul [10] S. Y. R. Hui, Apparatus for Energy Transfer by Induction, U.K. Patent GB , [11] M. Kamon, M. J. Tsuk, and J. White, Fasthenry, a multiple-accelerated 3D inductance extraction program, IEEE Trans. Microw. Theory Tech., vol. 42, no. 9, pp , Sep [12] C. Fernandez, O. Garcia, R. Prieto, J. A. Cobos, S. Gabriels, and G. Van der Borght, Design issue of a core-less transformer of a contact-less application, in Proc. 17th Annu. IEEE Appl. Power Electron. Conf. Expo (APEC 02), vol. 1, pp Xun Liu (S 04) was born in China in He received the B.S. and M.S. degrees in electrical engineering from Tsinghua University, Beijing, China, in 2001 and 2003, respectively, and is currently pursuing the Ph.D. degree at the City University of Hong Kong. His main research interests include electrical and thermal modeling, analysis and design of planar contactless power transfer systems, and current source inverters. S. Y. Ron Hui (F 00) was born in Hong Kong in He received the B.Sc degree (with honors) from the University of Birmingham, Birmingham, U.K., in 1984 and the D.I.C. and Ph.D degrees from the Imperial College of Science and Technology, University of London, London, U.K., in He was a Lecturer in power electronics at the University of Nottingham, Nottingham, U.K. from 1987 to In 1990, he took up a lectureship at the University of Technology, Sydney, Australia, where he became a Senior Lecturer in He joined the University of Sydney in 1993 and was promoted to Reader of Electrical Engineering in Presently, he is a Chair Professor of electronic engineering at the City University of Hong Kong. He has published over 190 technical papers, including over 110 refereed journal publications. Dr. Hui received the Teaching Excellence Award in 1999, the Grand Applied Research Excellence Award in 2001 from the City University of Hong Kong, the Hong Kong Award for Industry, and the Technological Achievement Award and Consumer Design Award, in 2001 and 2004, respectively. He is a Fellow of the IEE and has been an Associate Editor of the IEEE IEEE TRANSACTIONS ON POWER ELECTRONICS since He has been an At-Large member of the IEEE PELS AdCom since October He was appointed as an IEEE Distinguished Lecturer by IEEE PELS in 2004.