2052 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 23, NO. 4, JULY 2008

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1 2052 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 23, NO. 4, JULY 2008 Extended Theory on the Inductance Calculation of Planar Spiral Windings Including the Effect of Double-Layer Electromagnetic Shield Y. P. Su, Student Member, IEEE, Xun Liu, Member, IEEE, and S. Y. (Ron) Hui, Fellow, IEEE Abstract Recent progress on wireless planar battery charging platform highlights a requirement that the platform must be shielded underneath so that the electromagnetic (EM) flux will not leak through the bottom of the charging platform. The presence of the EM shield will inevitably alter the flux distribution and thus the inductance of the planar windings. In this paper, a theory of inductance calculation of spiral windings is extended to determine the inductance of planar spiral windings shielded by a double-layer planar EM shield which consists of a layer of soft magnetic material and a layer of conductive material. With the generalized equations, the impedance of the planar spiral windings on double-layer shielding substrate and the optimal thickness of shielding materials can be calculated accurately without using time-consuming finite-element method. Therefore, the influence of the double-layer electromagnetic shield on the inductance of the planar spiral windings can be analyzed. Simulations and measurements have been carried out for several shielding plates with different permeability, conductivity, and thickness. Both of the simulations and measurements of the winding inductance agree well with the extended theory. Index Terms Double-layer shield, impedance formulas, planar spiral inductance. NOMENCLATURE Internal radii of circular winding. External radii of circular winding. Height of filaments or winding centers above the substrate. Thickness of the winding track. Bessel functions of the first kind. Self-inductance of windings in air. Self-inductance with single-layer substrates. Self-inductance with double-layer substrates. Additional inductance of windings due to the presence of substrate. Mutual inductance between two windings or filaments in air. Manuscript received May 28, 2007; revised September 10, Published June 20, This work was supported by the Hong Kong Research Grant Council under the CERG Project (CityU ) and the City University of Hong Kong under the Strategic Research Grant Recommended for publication by Associate Editor C. Sullivan. Y. P. Su and S. Y. (Ron) Hui are with the Center for Power Electronics, City University of Hong Kong, Hong Kong, China ( eeronhui@cityu.edu.hk). X. Liu is with the ConvenientPower (HK) Ltd., Hong Kong, China. Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TPEL Defined in (4) and (5). Additional resistance of windings due to eddy loss in substrate. Substrates thickness. Mutual impedance between two windings or filaments. Additional mutual impedance due to presence of substrates. Angular frequency (rad/s). Electrical conductivity of the substrates. Relative permeability of the substrates. Permeability of free space ( H/m). Defined in (6). Defined in (7) and (8). Defined in (9) and (10). Defined in (11) and (12). I. INTRODUCTION I NDUCTANCE calculation methods for spiral planar windings [1], [2] have provided a useful tool for researchers to investigate various applications of planar windings. Planar magnetic components are attractive in portable electronic equipment applications in which high power density and slim designs are preferred. For example, coreless planar transformers for signal and power transfer have been studied [3], [4]. The power loss modeling of planar inductors for thermal management has also been investigated [5]. Planar spiral windings have been used for gate drive applications [6] and stray inductance cancellation in electromagnetic interference (EMI) filters [7] [9]. Recent industrial applications of planar windings include stand-alone battery chargers [10] and induction heating [11] [13]. An inductance calculation for spiral windings with variable widths is reported in [14]. So far, the inductance of planar spiral windings on a singlelayer substrate of finite thickness or in a sandwich structure has been analyzed [1], [2], [15]. However, it has been shown that a patented double-layer EM shield consisting of a soft-magnetic plate and a conductive plate can achieve a much higher shielding effectiveness (SE) [16], where SE is defined as the ratio between the field strength at a given distance from the source without the shield interposed and the field strength with the shield interposed [17]. In order to enhance the shielding effectiveness, a double-layer EM shield structure with very thin copper on the bottom of ferrite was presented in [16], as well as analyzed and measured in [17]. Fig. 1 shows the results when the ferrite plate /$ IEEE

2 SU et al.: EXTENDED THEORY ON THE INDUCTANCE CALCULATION OF PLANAR SPIRAL WINDINGS 2053 Fig D view of the model of planar windings on double-layer substrate. Fig. 1. Calculated and measured shielding effectiveness in decibels [17]. is made of 4F1 with a thickness of 0.4 mm, which clearly illustrate that the EM shielding effectiveness has been enhanced by the addition of the copper layer. The presence of the copper layer prominently enhances the shielding effectiveness by about 20 to 40 db when the frequency is higher than a few hundred kilohertz. In this example, the distance between the measuring point and the source is 5 mm. This double-layer EM shield structure is particularly important for designing planar wireless charging platform [18], [19] because the platforms must be shielded underneath so that the magnetic flux will not leak through the bottom of the platform. However, the presence of the double-layer EM shield will affect the inductance of the planar windings which is an important parameter in the system design. In this paper, an extended theory based on the equations in [1] and [2] for calculating the inductance of spiral windings is presented. A new set of formulas have been developed to calculate the inductance of planar spiral windings with a double-layer planar EM shield. This extended theory can be used to optimize the spiral windings and EM shield design. In this project, it is used to choose proper shielding parameters for the universal wireless battery charging platform. With the generalized equations presented in this paper, the mutual impedance of the planar spiral windings on a double-layer shielding substrate structure and the optimal thickness of shielding materials can be calculated directly without using time-consuming finite-element (FE) software. The frequency-dependent losses, particularly due to the eddy currents in the magnetic substrate, have also been considered in the calculation. Theoretical results agree favorably with the FE simulation results and practical measurements for several ferrite plates with different values of permeability and thickness. This extended theory also leads to some interesting discoveries that are useful to the optimal design of the shielding structure, such as the optimal thickness of the ferrite plate for a given application. Fig. 3. Cross-sectional view in R Z plane of the model of planar windings on double-layer substrate. equations. In this paper, the formula of the mutual inductance between two filaments in [1] is extended to cover the mutual inductance calculation in the presence of a double-layer EM shield. The mutual inductance between two planar windings is obtained by integrating the previous formulas over the cross section of the conductor, taking the current density distribution into account and assuming that the variation over the height of the cross section is negligible [1]. By extending the results in [1] and [2], a new set of impedance formulas is set up for planar spiral windings placed on a double-layer, planar EM shield constructed with a ferrite plate and a copper sheet. Figs. 2 and 3 show the 3-D view and cross-sectional view of two turns of concentric circular windings on a double-layer substrate having relative permeability of, electrical conductivity of, and thickness of, respectively. The mutual impedance between the two windings is given by where is the mutual inductance which would exist in the absence of the substrate and can be calculated by (2); is the additional impedance due to the presence of the double-layer substrate and can be calculated by (3). The real part of, the resistive component, represents the losses due to the eddy currents in the substrate, and the imaginary part of, the inductive reactance, enhances the inductance in the air. The specificdefinitions of nomenclature are given as (1) II. IMPEDANCE FORMULA A set of frequency-dependent impedance formulas for planar spiral windings on a magnetic substrate of finite thickness or a planar coils sandwiched between two magnetic substrates are reported in [1], [2]. Such theory considers the nonuniform current distribution in planar coils and the power loss in magnetic media. The impedance formulas are derived from Maxwell s (2) (3)

3 2054 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 23, NO. 4, JULY 2008 where (4) (5) is a Bessel function of the first kind. Due to the use of the double-layer substrate, the parameter function in (3) can be expressed as are defined in (7) (10), respec- where tively (6) (7) (8) (9) (10) (11) (12) The derivations of these equations are given in the Appendix. It is important to note that the above equations are an extended form of the calculation method presented in [1], [2]. When the two layers are made of the same material, (i.e., 1 ), the new parameter function of (6) can be simplified to (13) as follows: (13) It can be seen that (13) is identical to the equation for the spiral windings placed on a single-layer substrate of finite thickness as reported in [1]. Hence, the extended theory can be reduced to the original theory if a single-layer substrate is used. III. VERIFICATION AND ANALYSIS The formula is proposed for two turns of concentric circular windings on a double-layer substrate made of any materials. For an -turn spiral coil, the total impedance is the summation of each mutual impedance pairs between two concentric circular windings where is substituted into in (1). (14) Fig. 4. Cross-sectional view of the prototype in R 0 Z plane. A. Verification To evaluate the validity of the formula presented, a prototype shown in Fig. 4 has been analyzed as an example. The spiral planar winding has 10 turns. The geometric parameters of the planar spiral winding and the double-layer shielding structure are illustrated in Fig. 4. It can be seen in (6) that the parameter function depends on seven variables: namely the thickness, the relative permeability values, the conductivity values, and the frequency, where subscripts 1 and 2 refer to the first layer and the second layer of the substrate, respectively. In this project, copper is used for the second layer but two different types of materials are used for the first layer so as to check the validity of the proposed theory. In the first study, a piece of dielectric material is used as the first layer of EM shield. In the second study, ferrite (4F1) material with a relative permeability value 80 is used. The self-inductance of the spiral winding is obtained by calculated results (MATLAB), simulated results (Ansoft) and measured results in order to evaluate the accuracy of the formula. For comparison, all the inductance results are normalized with the inductance of windings in the air (without substrate), H as the base value. 1) Study 1: Using Dielectric and Copper Sheets: In the first case study, a double-layer substrate consisting of a piece of dielectric material (PCB lamination material FR4: 4.4) and copper is considered. This dielectric-conductive structure is not recommended for use in the wireless charging platform, but is studied to evaluate the extended theory. Fig. 5(a) shows the inductance curve of this structure as a function of frequency with the thickness of the dielectric substrate 0.3 mm and that of copper sheet 0.03 mm. Fig. 5(b) and (c) show the inductance curves as a function of frequency with the thickness of the copper layer changed to 0.06 mm and then to 0.3 mm, respectively, while other parameters remain unchanged. It can be observed that, without using soft ferrite in the first layer, the winding inductance reduces dramatically with increasing operating frequency. Without ferrite shield, the copper sheet behaves like a secondary conducting plate and the induced eddy current in the copper sheet dissipates some of the energy generated in the winding. It can also be found that the increase of the thickness of the copper sheet intensifies the negative effect on inductance at low frequency. When the frequency is high enough (above 100 khz), the thickness change has little influence, due to the skin effect. Fig. 6 shows the inductance value when is changed from 0 mm to 0.5 m, the operation frequency is kept at 500 khz and the

4 SU et al.: EXTENDED THEORY ON THE INDUCTANCE CALCULATION OF PLANAR SPIRAL WINDINGS 2055 Fig. 6. Inductance with double-layer substrate (dielectric and copper) as a function of t ; f = 500 khz; t = 0.03 mm; L = H; (one layer: dielectric; two layers: dielectric and copper). the validity of the formula. Without using a first layer of soft ferromagnetic material, the copper layer has a detrimental effect on the inductance of the spiral winding, especially when the relative permeability of the first layer is low. 2) Study 2: Using Ferrite (4F1) and Copper Sheets: Another double-layer substrate consisting of ferrite (4F1: S/m) and copper sheets (i.e., a magnetic-conductive double-layer structure) is then analyzed. Fig. 7(a) illustrates the inductance curve as a function of frequency while the thicknesses of the substrates are 0.5 mm and 0.03 mm. Fig. 7(b) shows the inductance curve as a function of frequency with the thickness of the copper layer changed from 0.03 mm to 0.06 mm while keeping other parameters unchanged. These results show that the magnetic-conductive double-layer shield does not reduce the winding inductance significantly. However, this kind of EM shield will provide a significant shielding effectiveness. Fig. 5. Inductance with double-layer substrate as a function of frequency; L = 1.312H (one-layer: dielectric; two-layers: dielectric and copper). (a) t = 0.3 mm, t = 0.03 mm (b) t = 0.3 mm. t = 0.06 mm. (c) t = 0.3 mm, t = 0.3 mm. thickness of copper sheet is kept at 0.03 mm. The theoretical results and measurements are highly consistent, confirming B. Analysis For the purpose of enhancing the inductance and providing an effective EM shield, soft ferromagnetic material (ferrite) is usually chosen for the first layer substrate. In this section, calculation and simulation on the inductance values and SE of the system are carried out for the same prototype shown in Fig. 4. The effects of the variation of the thickness of the ferrite plate,, or the frequency,, are examined. The results for magnetic-conductive double-layer shield are compared with those for single-layer shield (i.e., using ferrite sheet only), in order to highlight the effect of the copper sheet on the inductance of planar spiral windings and the significant SE improvement due to the presence of copper sheet. 1) Effects of the Thickness of the First Layer, : First, only the thickness of the ferrite plate is changed and the thickness of copper is fixed at 0.07 mm, while other parameters are unchanged and the frequency is fixed at 500 khz. The calculated inductance results are compared with finite-element analysis (FEA) results in Fig. 8. All the results are normalized as

5 2056 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 23, NO. 4, JULY 2008 Fig. 7. Inductance with double-layer substrate (4F1 and copper) as a function of frequency; L = H; (one layer: 4F1; two layers: 4F1 and copper): (a) t = 0.5 mm, t = 0.03 mm and (b) t = 0.5 mm, t = 0.06 mm. a ratio to the inductance of windings in the air (without substrate), H. Fig. 8(a) and (b) show the results when the material of the ferrite plate is 4F1 80 and 3F3 900, respectively. Using the SE analytical method proposed in [17]. the SE for each shielding structure is plotted in Fig. 9. Results in Fig. 8(a) and (b) indicate that the winding inductance can be enhanced when a single-layer substrate of ferrite is used. The addition of copper sheet could cause a significant decrease of the winding inductance when the ferrite plate is thin. However, if the thickness of ferrite plate is increased to a certain critical value, the reduction of the winding inductance due to the copper sheet can be minimized. By using this additional copper sheet (only 0.07 mm) the shielding effectiveness of the magnetic-conductive double-layer EM shield can increase by at least 25 db as shown in Fig. 9. So the extended theory allows the designer to determine the optimal thickness of the ferrite plate in order to avoid significant reduction in the winding inductance caused by the addition of Fig. 8. Inductance with double-layer substrate (ferrite and copper) as a function of ferrite plate thickness; L = H: (a) Ferrite plate 4F1 ( = 80; = S/m); (one layer: 4F1; two layers: 4F1 and copper) and (b) Ferrite plate 3F3 ( = 900; = 0:1 S/m); (one layer: 3F3; two layers: 3F3 and copper). the copper sheet. If is larger than the critical value, the negative effect of the copper layer on the winding inductance becomes negligible. For example, if one sets to be greater than 0.97 (i.e., is almost the same as ), the thickness of the ferrite plate can be chosen to be 0.4 mm for ferrite 4F1 in Fig. 8(a) and 0.03 mm for ferrite 3F3 in Fig. 8(b). (Note that ferrite plates of 0.5 mm are available in the market.) This critical thickness value decreases with increasing relative permeability of the first layer of ferrite material. Fig. 10 shows a plot of the lines of magnetic flux generated by a planar spiral winding on a printed-circuit board shielded with a magnetic-conductive double-layer EM shield [20]. It can be seen that the double-layer EM shield has the effect of guiding the vertical flux leaving the plane into the horizontal direction. The thickness of this ferrite layer must be great enough for the lines of flux to curve horizontally. Based on the theoretical results from the extended theory

6 SU et al.: EXTENDED THEORY ON THE INDUCTANCE CALCULATION OF PLANAR SPIRAL WINDINGS 2057 Fig. 10. FE simulation of the field plot for double-layer substrate (ferrite and copper) [20]. Fig. 9. SE performance of double-layer substrate (ferrite and copper) as a function of ferrite plate thickness: (a) Ferrite plate 4F1 ( = 80; = S/m); (one layer: 4F1; two layer: 4F1 and copper) and (b) Ferrite plate 3F3 ( = 900; = 0.1 S/m); (one layer: 3F3; two layer: 3F3 and copper). and the measurements, it can be seen that the extended theory can predict the optimal thickness of the ferrite plate for a given permeability. 2) Variation of Operating Frequency: A test has been performed to evaluate the frequency effects of the magnetic-conductive double-layer substrate on the inductance of planar spiral windings. The 4F1 ferrite plate is used in the test for a frequency range from 1 to 10 MHz. Fig. 11(a) and (b) show the results when the thickness of the ferrite plate,, is 0.05 mm and 0.4 mm, respectively. The thickness of copper is fixed at 0.07 mm. If the thickness of the ferrite plate is less than the critical value, e.g., 0.05 mm ( 0.4 mm), the induced current in the copper sheet will reduce the winding inductance as the frequency increases, as shown in Fig. 11(a). However, once the critical thickness of the ferrite is reached ( 0.4 mm), the adverse effect of the copper sheet on the winding inductance diminishes as confirmed in Fig. 11(b). Fig. 11. Frequency effect of the inductance with different ferrite plate thickness; (one layer: 4F1; two layers: 4F1 and copper): (a) t = 0.05 mm (less than critical thickness) and (b) t = 0.4 mm.

7 2058 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 23, NO. 4, JULY 2008 Fig. 12. Shielding effectiveness as a function of frequency; (one layer: 4F1; two layer: 4F1 and copper). Fig. 13. Cross-sectional view of the tested device in R Z plane. The SE of the EM shield structures tested in this part is plotted in Fig. 12. The SE of single-layer shield (ferrite only) is almost constant (i.e., independent on frequency). When a magnetic-conductive double-layer shield is used, the SE can be enhanced significantly, especially in the high-frequency region. 3) Variation of the Distance Between Windings and Shielding Substrate: Since this research is related to an inductive charging platform, the effect of the double-layer shield on coupling between transmitter winding and receiver winding is an important issue in charging platform design. In this section, the electromagnetic coupling between planar transmitter and receiver windings for a range of distance from the windings to the shielding substrate,, is considered. The cross-sectional view of the structure is shown in Fig. 13. The magnetic-conductive double-layer EM shield is composed of 4F1 ferrite plate ( 0.4 mm) and copper sheet ( 0.07 mm). The distance between planar transmitter and receiver windings is fixed at 1 mm, and 500 khz is chosen as the operation frequency. The self-inductance of the transmitter winding and receiver winding, and the mutual-inductance between them are calculated and compared with simulated results in Fig. 14(a) and (b), respectively. With increasing, the enhancement effect of the substrate on winding inductance is reduced. If the distance between winding and substrate reaches 10 mm, this kind of enhancement effect can be negligible. Fig. 14. Effect of the double-layer shield on the inductances as a function of d; (one layer: 4F1; two layers: 4F1 and copper): (a) self-inductance of transmitter winding and receiver winding and (b) mutual-inductance between transmitter winding and receiver winding. 4) Power Loss in the Copper Layer: As discussed previously, the second layer of copper sheet can enhance the SE performance prominently. For the first layer of ferrite, its thickness should be beyond the critical value in order to minimize the negative effect on the inductance. One important issue needs to be considered is the increased power loss due to the added copper layer. In this part of discussion, 3F3 and 4F1 are considered separately as the material of first layer substrate and the thickness is fixed at 0.4 mm. The thickness of the copper sheet is 0.07 mm. The prototype in Fig. 4 is used as an example in this FEA here. Simulations have been carried out for three situations, with 1) a planar winding without any shielding substrate, 2) a planar winding with only one layer of 3F3 or 4F1 ferrite plate, and 3) a planar winding with a double-layer shield consisting of a ferrite plate and copper sheet. The planar winding is excited with an ac current of a peak value of 1 A, so the RMS value of the excitation current is. The total power losses at operating frequency of 10 khz and 10 MHz are obtained from the FEA so that the equivalent resistance of the system, which is proportional to the power loss can be derived by (15).

8 SU et al.: EXTENDED THEORY ON THE INDUCTANCE CALCULATION OF PLANAR SPIRAL WINDINGS 2059 TABLE I SIMULATED SYSTEM RESISTANCE The equivalent resistance of cases 1), 2), and 3) are denoted as and, respectively (15) The simulated system resistance values of the three situations are displayed in Table I. The last column displays the difference of the equivalent resistance between cases 2) and 3) (i.e., extra loss caused by the addition of the copper sheet). It can be seen that the equivalent resistance (and hence the power loss) at low-frequency (10 khz) operation remains almost the same in all cases. At high-frequency (10 MHz) operation, the increase in power losses in the ferrite plate and the ohmic loss in the windings becomes obvious. But the loss introduced by the copper sheet, as indicated by in the last column of Table I, remains relatively small. Details of winding loss analysis can be found in [21]. In summary, when exceeds the critical value (e.g., 0.4 mm for 4F1 ferrite plate), the copper sheet has limited detrimental effect on the inductance of the windings and the power loss, even if the frequency is increased to 10 MHz. But the copper sheet can improve the shielding effectiveness by more than 40 db with the same structure parameters [17]. This finding indicates a very good property of shielding effectiveness improvement without significantly reducing winding inductance or increasing power loss if the thickness of the ferrite plate is beyond a critical value. This extended theory can be used to optimize the thickness of the ferrite layer in the practical design. IV. CONCLUSION In this paper, an extended theory on the inductance calculation of planar spiral windings on a double-layer substrate is presented and verified with FEA and practical measurements. This formula is based on the physical dimensions of the spiral planar windings and the electrical and the magnetic properties of the substrates. An analysis has been carried out for a range of the thickness of the ferrite plate and also operating frequency. With the help of this extended theory, it is found that a critical value of the thickness of the ferrite layer exists for a given ferrite material. The critical thickness value decreases with increasing relative permeability of the first layer of ferrite material. This new information is important for the optimal design of the ferrite layer in the magnetic-conductive double-layer EM shield structure. The extended theory has been confirmed with FEA Fig. 15. Filamentary turns above a double-layer substrate. and practical measurements. It can also be reduced to the original theory if the double-layer EM shield structure is reverted to a single-layer structure. APPENDIX The following procedure provides the derivation of (1) (3) in detail, which is similar to the method described in [1]. In order to obtain the impedance formula of planar windings, the case of filamentary turns must primarily be considered as shown in Fig. 15. The filamentary turn at carries a sinusoidal current. The solution of Maxwell s equations must be considered in five distinct regions and [1] has presented the solution of each region about this structure. The solution of the electric field intensity in each region is: Region 1: Region 2: Region 3: Region 4: (A1) (A2) (A3) (A4)

9 2060 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 23, NO. 4, JULY 2008 Region 5: (A5) There are six constants to be established on the basis of the boundary conditions. The electric field is continuous at the boundaries of and (A6) (A7) (A8) (A9) The boundary condition on the radial component of the magnetic field intensity is given by (A10) where is the unit vector normal to the plane at the boundary and is the surface current density at the boundary. The radial component the magnetic field intensity is given by Maxwell s equations (A11) At and, there is no surface current and equating at either side of the boundary gives (A12) (A13) (A14) At, and in terms of available in [1], the final boundary condition equation could be acquired giving (A15) There are now eight equations in eight unknowns which can be readily solved. In terms of mutual impedance we are particularly interested in Region 1 where the electric field is (A16) (A17) Applying the inverse transform of the Fourier Bessel integral described in [1], the mutual impedance between two filaments is obtained, in the expression of (A18), (A19), and (A20) (A18) (A19) (A20) The mutual impedance between two planar windings is obtained by integrating (A20) formulas over the cross section of the conductor using the method based on the Fourier Bessel integral transformation. The final result is shown as (1) (12). REFERENCES [1] W. G. Hurley and M. C. Duffy, Calculation of self and mutual impedances in planar sandwich inductors, IEEE Trans. Magn., vol. 33, no. 3, pp , May [2] W. G. Hurley, M. C. Duffy, S. O Reilly, and S. C. O Mathuna, Impedance formulas for planar magnetic structures with spiral windings, IEEE Trans. Ind. Electron., vol. 46, no. 2, pp , Apr [3] S. Y. R. Hui, S. C. Tang, and H. S. Chung, Coreless printed-circuit board transformer for signal and energy transfer, Electron. Lett., vol. 34, no. 11, pp , Nov [4] S. C. Tang, S. Y. R. Hui, and H. S. Chung, Coreless planar printedcircuit-board (PCB) transformers A fundamental concept for signal and energy transfer, IEEE Trans. Power Electron., vol. 15, no. 5, pp , Sep [5] T. G. Imre, W. A. Cronje, J. D. van Wyk, and J. A. Ferreira, Loss modeling and thermal measurement in planar inductors-a case study, IEEE Trans. Ind. Appl., vol. 38, no. 6, pp , Nov./Dec [6] S. Y. R. Hui, S. C. Tang, and H. S. Chung, Optimal operation of coreless PCB transformer-isolated gate drive circuits with wide switching frequency range, IEEE Trans. Power Electron., vol. 14, no. 3, pp , May [7] T. C. Neugebauer and D. J. Perreault, Filters with inductance cancellation using printed circuit board transformers, IEEE Trans. Power Electron., vol. 19, no. 3, pp , May [8] B. J. Pierquet, T. C. Neugebauer, and D. J. 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10 SU et al.: EXTENDED THEORY ON THE INDUCTANCE CALCULATION OF PLANAR SPIRAL WINDINGS 2061 Y. P. Su (S 07) was born in China in He received the B.S. degree in electrical engineering from Tsinghua University, Beijing, China, in 2005 and is currently pursuing the Mphil. degree at the City University of Hong Kong, China. His main research interests include planar magnetic components, circuit integration in power electronics, numerical calculation of the electromagnetic field, and EMC/EMI. Xun Liu (M 07) 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 the Ph.D. degree from the City University of Hong Kong, China, in He is a Technology Manager with Convenient- Power (HK), Ltd., Hong Kong, and responsible for the R&D of a new generation of universal wireless charging platform for a wide range of consumer electronic products His main research interests include planar integration in power electronics, applied superconductivity, and EMC/EMI. S. Y. (Ron) Hui (F 03) 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 went to Australia and took up a lectureship at the University of Technology, Sydney, 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 (CityU). From 1999 to 2004, he was an Associate Dean of the Faculty of Science and Engineering at CityU. He has published over 200 technical papers, including over 120 refereed journal publications. He holds over 20 patents. Dr. Hui received the Teaching Excellence Award in 1999 and the Grand Applied Research Excellence Award in 2001 from the City University of Hong Kong, and the Best Paper Award from the IEEE IAS Committee on Production and Applications of Light in He is a Fellow of the IEE and has been an Associate Editor of the IEEE TRANSACTIONS ON POWER ELECTRONICS since Since 2007, he has been an Associate Editor of the IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS. He has been an At-Large member of the IEEE PELS AdCom since October He has been appointed as an IEEE Distinguished Lecturer by IEEE PELS for