Implementation of the Neumann Formula for Calculating the Mutual Inductance between Planar PCB Inductors

Transcription

1 Proceedings of the 2008 Internationa Conference on Eectrica Machines Paper ID 876 Impementation of the Neumann Formua for Cacuating the Mutua Inductance between Panar PCB Inductors C. L. W. Sonntag, E. A. Lomonova, and J. L. Duarte Eectromechanics and Power Eectronics Group, Department of Eectrica Engineering, Eindhoven University of Technoogy, 5600 MB, Eindhoven, The Netherands E-mai: Abstract- In this paper, a method for estimating the mutua inductance between two panar printed circuit board inductors with straight tracs is presented. The thin copper tracs are first modeed as mutipe straight ine fiaments. The Neumann formua, derived from the magnetic vector potentia and Faraday s aw of induction, gives an anaytica soution of the mutua inductance. Unfortunatey, due to the compexity of this formua, it can ony be soved anayticay for reativey simpe geometries. However, by foowing the procedure presented in this paper, the mutua inductance between two arbitrary-positioned and orientated panar PCB inductors can be estimated with reduced computationa effort by soving the Neumann formua using numerica integration. The proposed method is demonstrated at the hand of two experiments; and the measured and estimated mutua inductances show exceent agreement with a maximum absoute error of ess than 5%. I. INTRODUCTION Panar inductors and transformers constructed as copper tracs on printed circuit boards (PCBs) are becoming increasingy popuar in the fied of contactess energy transfer (CET) and other high frequency eectronic systems [2]-[5], [9]. In the case of CET, eectronic devices embedded with PCB inductors of various shapes and sizes, transfer power through their shared magnetic fied and the inductive couping between them. When these inductors (often in the shape of spiras) are excited with reativey ow frequency current waveforms, the system is said to operate in a magneto-static mode where the magnetic fieds created by these inductors can be approximated by static magnetic fieds. With the ac of soft magnetic materias, these fieds can be accuratey described by the Maxwe equations [1]. The mutua inductance between two inductors is a measure of their magnetic couping and the amount of shared magnetic fied between them. In the CET systems, estimating the mutua inductance between the primary and secondary inductors is vita in determining the capabiity and efficiency of the power transfer between the devices. Unfortunatey, due to a the different shapes and forms these inductors may tae, no straightforward method for cacuating their mutua inductances exists. One existing method for estimating the mutua inductance between such inductors is the finite eement method (FEM). This numerica technique approximates the mutua inductance by modeing the inductors in a three-dimensiona environment and soving the eectromagnetic fied equations in and around the devices. Using this purey numerica method, can however be very time consuming especiay when compex structures are simuated. Some papers exist, describing anaytica and semi-anaytica methods for estimating the mutua inductance between circuar conductors. In these papers, the thin conductors are modeed as singe current fiaments, and thicer conductors as groups of fiaments, often caed the mesh-matrix method [6], [7], [10], [11]. Unfortunatey, in the case of the hexagona spira windings (Fig. 1), which are quite attractive in practica appications as used in [2] and [3], the tracs are not circuar but consist of mutipe straight ine segments. A different approach is thus needed for cacuating the mutua inductance between theseand other panar PCB inductors with straight tracs. The probem addressed in this paper is that of deveoping a method for estimating the mutua inductance between panar PCB inductors with straight tracs. Firsty, the copper tracs of the panar PCB inductors are modeed as mutipe straight ine fiaments. Secondy, various fiament parameters are extracted from the mode, e.g. the amount of fiaments, the amount of fiament vertices as we as their three-dimensiona coordinates. Fig. 1. A panar hexagon spira winding used in contactess energy transfer appications /08/$ IEEE 1 Authorized icensed use imited to: Eindhoven University of Technoogy. Downoaded on March 10,2010 at 10:12:08 EST from IEEE Xpore. Restrictions appy.

2 Proceedings of the 2008 Internationa Conference on Eectrica Machines Finay, the mutua inductance is approximated by impementing a numerica representation of the Neumann formua as an agorithm in a computer program. The presented method is demonstrated by means of two experiments. In the first experiment, the mutua inductance between two panar hexagon spira windings used in [2], is estimated. The mutua inductance is first cacuated for different reative winding pacements, and afterwards measured using the actua PCB windings. In the second experiment, a high frequency panar PCB current transformer is designed. For a certain primary current and required secondary no-oad votage, the desired mutua inductance is first estimated. With the primary winding as a straight trac and the secondary winding as a panar rectanguar spira winding, the transformer is reaized. In both experiments the measured and estimated mutua inductances show exceent agreement with a maximum absoute error of ess than 5%. II. PCB FILAMENT MODEL The copper tracs of the PCB inductors are modeed as current fiaments, where each straight ine segment is modeed by a fiament, paced at the center of the trac, as show in Fig. 2. is the infinitesimay sma integration eement on the contour path, and I 1 is the tota current through the structure. Fig. 3. A muti-fiament structure, and the observationa point P, where the vector potentia is evauated. The magnetic fied reates to the magnetic vector potentia though the cur operator, and can be expressed by B = A. (2) The magnetic fux ining two fiamentary structures can be cacuated by integrating the magnetic fied produced by the current in the primary structure defined by its contour, through the surface of the secondary structure bounded by its contour. The fux inage between the two is given as ( ) λ, = Bi da, (3) (a) Fig. 2. (a) A square PCB spira inductor with two turns, (b) with the copper tracs modeed as current fiaments. III. ANALYTICAL SOLUTION The anaytica soution for cacuating the mutua inductance between two muti-fiament structures is derived using the magnetic vector potentia. With the ac of soft magnetic materia, and assuming a magneto-static regime, the magnetic vector potentia produced by a ine current in a fiamentary structure is expressed as (b) where B represents the magnetic fux density created by the primary structure on da, a surface eement on the secondary structure. Using Stoes theorem [1], the surface integra in (3) is converted into a contour integra so that it becomes λ ( ), = Aid. (4) Combining (1) and (4), the foowing expression for the fux inage between the two fiamentary structures is formuated as A r µ I = 4π d r - r' 0 1 ( ), 1 (1) λ µ I did,. 4π r - r' 0 1 ( ) = (5) where, as iustrated in Fig. 3, P is the observation point, r is the vector pointing from the axis origin to the observation point, is the primary fiamentary structure contour, r is the vector pointing from the axis origin to the structure contour, d Here, as shown in Fig. 4, and are the contours of the primary and secondary fiament structures, respectivey, d and d are infinitesimay sma integration eements. 2 Authorized icensed use imited to: Eindhoven University of Technoogy. Downoaded on March 10,2010 at 10:12:08 EST from IEEE Xpore. Restrictions appy.

3 Proceedings of the 2008 Internationa Conference on Eectrica Machines q qm n =, m [1... a ]. (7) m m+ 1 cm Here, n represents the normaized position vectors for the primary structure fiaments. Simiary, the normaized position vectors for the secondary fiaments can be cacuated and is represented as n. The subscript m denotes the appropriate fiament. 6. The fux inage between the primary and secondary structures is then estimated as: Fig. 4. Primary and secondary fiamentary structures, with their respective contours, integration eements, and vectors connecting them to the axis origin. µ I λ = 4π n i n a a Χ Ε 0 1 α β (, ), K K α= 1 β= 1 χ= 0 ε= 0 χ ε (8) The mutua inductance between the two fiamentary structures is then cacuated as: ( ) 1 M (, ) = λ, I. (6) where ( )( ) K = q + χ n, χ α α ( )( ) K = q + ε n, ε β β Formua (5) is aso nown as the Neumann formua [8]. The integra can unfortunatey ony be competed anayticay for reativey few configurations. IV. NUMERICAL SOLUTION Since an anaytica soution of (5) is in genera quite impractica, numerica integration can be used to approximate it by evauating the integrant in very sma increments. The advantage of using the numerica integration as proposed in this section is that the definite integra in (5) can be impemented with ow computationa effort to evauate arbitrary positioned and orientated fiamentary structures. The procedure for impementing the numerica integration is shown in the next seven steps: 1. The tota amount of straight ine segments on the primary fiament contour,, and the secondary fiament contour,, is determined. They are stored as a and a, respectivey. 2. The tota amount of vertices (incuding the start and stop positions) on the primary and secondary structures are determined. They are stored as b, where b = a + 1, and b, where b = a + 1, respectivey. 3. The three-dimensiona coordinates of the primary- and secondary vertices are determined and stored in vectors q and q, respectivey. 4. The Eucidean engths of a the primary and secondary fiaments are then cacuated and stored in the arrays c, and c, respectivey. 5. The normaized fiament vectors are cacuated as: Χ =, Ε =. c α Here, the engths and c β are sma numerica integration eements. 7. Finay, the mutua inductance between the two fiaments can then be estimated from (6) as: M (, ) = λ (, ) I. (9) Caution must be taen when choosing the vaues for and. On the one hand, choosing these vaues too sma wi increase the accuracy of the mode but aso increase the processing time needed to sove the numerica integra. On the other hand, choosing these vaues to big reduces the mode accuracy. By iterating (8) with decreasing vaues of and the convergence of the resutant mutua inductance can be used as a gauge in determining the accuracy of the mode. Generay, these vaues shoud be about the same order of magnitude (preferaby smaer) than the PCB trac thicness and trac spacing. V. EXPERIMENTAL VALIDATION The method for estimating the mutua inductance proposed in this paper is demonstrated at the hand of two experiments. In these experiments no soft magnetic materia is present, a magneto-static regime is assumed, and a current and votage cacuations and measurements are sinusoida steady state waveforms. In both cases the mutua inductances are measured by exciting the primary winding with a current waveform and measuring the no-oad secondary induced votage. Assuming no (or negigibe) secondary current, the mutua inductance is 1 3 Authorized icensed use imited to: Eindhoven University of Technoogy. Downoaded on March 10,2010 at 10:12:08 EST from IEEE Xpore. Restrictions appy.

4 Proceedings of the 2008 Internationa Conference on Eectrica Machines estimated using: V = jω I M. (10) Here I 1 is the current ampitude and ω is the radia frequency of the primary current; V 2 is the secondary induced votage and M 12 is the mutua inductance between the primary and secondary winding. A. Hexagon spira winding mutua inductance In [2], a contactess energy transfer (CET) patform is presented. In this system, power is transferred from a primary CET patform embedded with a matrix hexagon spira windings to secondary devices (e.g. mobie phones, PDAs and aptops) embedded with simiar receiving windings, through inductive couping. Accurate estimation of the mutua inductance between the primary and secondary windings are vita in deveoping the power eectronics circuits required to power the system, and determining the systems capabiities as a whoe. The physica dimensions of the primary and secondary hexagon spira windings are specified in Tabe I. Fig. 6, shows part of the setup used to measure the mutua inductance. Here the secondary hexagon spira winding is paced on top of a matrix of hexagon spira windings. The activated primary winding is hidden underneath the secondary winding. White pastic spacers are used to separate the windings. The current probe, measuring the primary current, and votage probe, measuring the secondary induced votage, can aso be seen. TABLE I PHYSICAL DIMENSIONS OF THE HEXAGON SPIRAL WINDINGS Parameter Primary Winding Vaue Secondary Winding Vaue Radius 12 mm 20 mm Turns Thicness 105 µm 105 µm Trac width 0.5 mm 0.5 mm Trac spacing 0.25 mm 0.25 mm Using the techniques described in this paper, the mutua inductance between a singe primary winding and a singe secondary winding is first cacuated and then measured. With, = = 0.5 mm, the primary winding is ept at the axis origin, the secondary winding is shifted parae to the primary at different x-, y-, and z-coordinates, as shown in Fig. 5. Fig. 6. The setup used to measure the mutua inductance between the hexagon spira windings. The resuts of the cacuated and measured mutua inductances are shown in Tabe II. TABLE II RESULTS OF THE MUTUAL INDUCTANCE CALCULATIONS AND MEASUREMENTS. Secondary winding position (u, v, w) [mm] Cacuated Mutua Inductance Measured Mutua Inductance Error 0, 0, µh 3.16 µh % 0, 0, µh 2.64 µh % 0, 0, µh 2.35 µh % 0, 0, µh 2.01 µh % 10.4, 0, µh 1.43 µh 2.10 % 10.4, 0, µh 1.29 µh 3.10 % 10.4, 0, µh 1.15 µh 3.48 % 10.4, 0, µh 1.03 µh 2.91 % 12, 0, nh 916 nh 1.75 % 12, 0, nh 802 nh % 12, 0, nh 751 nh % 12, 0, nh 716 nh 2.23 % Fig. 5. The position of the primary- and secondary hexagon spira windings during mutua inductance cacuation and measurements. B. High frequency panar current transformer In order to contro the power transfer in the CET system presented in [2], the controer needs to measure the current through each activated primary hexagon spira winding. To do this, a high frequency current transformer is impemented as copper traces on a PCB. The primary winding is a straight 4 Authorized icensed use imited to: Eindhoven University of Technoogy. Downoaded on March 10,2010 at 10:12:08 EST from IEEE Xpore. Restrictions appy.

5 Proceedings of the 2008 Internationa Conference on Eectrica Machines trac (to minimize the introduced inductance) paced in the current path, and the induced votage over the secondary winding is samped and measured by the controer using a high speed anaog-to-digita converter (ADC). The ADC has an extremey arge input resistance (5-20 Ω) and the secondary current is assumed to be zero. For a current frequency of 2.7 MHz, a secondary induced votage of 1 V (rms) per 1 A (rms) of primary current is required. The necessary mutua inductance is estimated as nh. The genera shape of the transformer is shown in Fig. 7, and the physica dimensions are determined through a parametric search through the variabes shown in the figure. Vaues of and equa to 1 mm are used. Fig. 7. The panar PCB transformer mode. The physica dimensions of the deveoped PCB transformer are presented in Tabe III. TABLE III PHYSICAL DIMENSIONS OF THE PCB CURRENT TRANSFORMER Parameter H 1 H 2 W 1 W 2 W 3 Vaue 5 mm 40 mm 3 mm 42 mm 5 mm The impemented PCB transformer has a trac thicness of 1 mm and copper thicness of 35 µm, and is shown in Fig. 8. Fig. 8. The actua panar PCB current transformer. The mutua inductance is measured as nh, which gives an error of approximatey -2.5 %. VI. CONCLUSIONS In this paper, a method for estimating the mutua inductance between PCB inductors with straight tracs is presented. The copper tracs of the PCB inductors are first modeed as mutipe straight ine fiaments of which various parameters are then extracted (e.g. the amount of fiaments, the amount of fiament vertices as we as their three-dimensiona coordinates). The mutua inductance is then approximated by impementing a numerica representation of the Neumann formua as an agorithm in a computer program. The presented method is demonstrated by means of two experiments. In the first experiment, the mutua inductance between two panar hexagon spira winding used in a CET system [2], is estimated. The mutua inductance is first cacuated and afterwards measured using the actua PCB windings, for different reative winding pacements. In the second experiment, a high frequency panar PCB current transformer is designed. For a certain primary current and required secondary no-oad votage, the desired mutua inductance is first estimated. With the primary winding as a straight trac and the secondary winding as a panar rectanguar spira winding, the transformer is reaized. In both experiments the measured and estimated mutua inductances show exceent agreement with a maximum absoute error of ess than 5%. ACKNOWLEDGEMENTS The authors woud ie to than F. B. M. v. Horc, M. A. M. Hendrix, A. J. M. Pemen and M. J. P. C. Uyt de Wiigen for their vauabe discussions and hep regarding this wor. REFERENCES [1] Herman A. Hous, and James R. Mecher, Eectromagnetic Fieds and Energy, Massachusetts Institute of Technoogy, pp , [2] C. L. W. Sonntag, E. A. Lomonova, J. L. Duarte and A. J. A. Vandenput, Speciaized Receiver for Three-Phase Contactess Energy Transfer Destop Appications, The 12th European Conference on Power Eectronics and Appications, EPE 2007, Aaborg, pp. 1-11, September [3] S. Y. R. Hui and W. C. Ho, A New Generation of Universa Contactess Battery Charging Patform for Portabe Consumer Eectronic Equipment, The 35 th Annua IEEE Power Eectronics Speciaists Conference, PESC 2004, Germany, pp , June [4] Byungcho Choi, Jaehyun Nho, Honnyong Cha, Taeyoung Ahn, and Seungwon Choi, Design and Impementation of Low-Profie Contactess Battery Charger Using Panar Printed Circuit Board Windings as Energy Transfer Device, IEEE Transactions on Industria Eectronics, Vo. 51, No. 1, pp , February [5] S.C. Tang, S. Y. Hui, and Henry Shu-Hung Chung, Coreess Panar Printed-Circuit-Board (PCB) Transformers - A Fundamenta Concept for Signa and Energy Transfer, IEEE Transactions on Power Eectronics, Vo. 15, No. 5, pp , September [6] Ki-Bing Kim, Enrico Levi, Zivan Zabar, and Leo Birenbaum, Mutua Inductance of Noncoaxia circuar Cois with Constant Current Density, 5 Authorized icensed use imited to: Eindhoven University of Technoogy. Downoaded on March 10,2010 at 10:12:08 EST from IEEE Xpore. Restrictions appy.

6 Proceedings of the 2008 Internationa Conference on Eectrica Machines IEEE Transactions on Magnetics, Vo. 33, No. 5, pp , September [7] C. Fernandez, R. Prieto, O. Garcia, and J. A. Cobos, Coreess Magnetic Transformer Design Procedure, The 36 th IEEE Power Eectronics Speciaists Conference, PESC 2005, Brazi, pp , June [8] H. L. Chan, K. W. E. Cheng, and D. Sutanto, A simpified Neumann s Formua for Cacuation of Inductance of Spira Coi, The 8 th Internationa Conference on Power Eectronics and Variabe Speed Drives, London, pp , September [9] J. Achterberg, E. Lomonova, J. de Boeij, Coi Array Structures Compared for Contactess Battery Charging Patform, IEEE Transactions on Magnetics, Vo. 44, No. 5, pp , May [10] Y. P. Su, Xun Liu, S. Y. R. Hui, Mutua Inductance Cacuation of Movabe Panar Cois on Parae Surfaces, The 39 th IEEE Annua Power Eectronics Speciaists Conference, PESC 2008, Greece, pp , June [11] Sobodan I. Babic, and Cevdet Aye, New Anaytica-Numerica Soutions for the Mutua Inductance of Two Coaxia Circuar Cois with Rectanguar Cross Section in Air, IEEE Transactions on Magnetics, Vo. 42, No. 6, pp , June Authorized icensed use imited to: Eindhoven University of Technoogy. Downoaded on March 10,2010 at 10:12:08 EST from IEEE Xpore. Restrictions appy.