IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 22, NO. 1, JANUARY 2007
`
`21
`
`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 mul-
`tilayer 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 over-
`laps 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.
`
`I. INTRODUCTION
`
`P LANAR contactless battery charging platform is an
`
`emerging technology that has the potential of unifying
`the charging protocols of portable consumer electronic prod-
`ucts 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, 2006. 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 publi-
`cation by Associate Editor J. A. Ferreira.
`The authors are with the Department of Electronic Engineering, City Univer-
`sity of Hong Kong, Hong Kong, China (e-mail: eeronhui@cityu.edu.hk).
`Color versions of one or more of the figures in this paper are available online
`at http://ieeexplore.ieee.org.
`Digital Object Identifier 10.1109/TPEL.2006.886655
`
`(a)
`
`(b)
`
`(a) Inductive battery charging platform (with magnetic flux lines flow
`Fig. 1.
`“horizontally” along the charging surfaces) proposed by Beart et al [1]. (b) Sec-
`ondary 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 sur-
`face. The lines of flux of this charging platform flow “perpen-
`dicularly” 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” sec-
`ondary energy-receiving device to be developed for the charging
`platform.
`
`0885-8993/$20.00 © 2006 IEEE
`
`Momentum Dynamics Corporation
`Exhibit 1020
`Page 001
`
`

`

`22
`
`IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 22, NO. 1, JANUARY 2007
`
`Inductive battery charging platform (with magnetic flux lines flowing
`Fig. 2.
`in and out perpendicularly of the charging surface) proposed by Hui [2].
`
`Fig. 3. Photograph of a universal planar charging platform [5].
`
`Fig. 5. Structure of the combination of different columns.
`
`TABLE I
`GEOMETRY OF EACH WINDING OF THE THREE PLATFORMS
`
`Fig. 4. Structure of three-layer hexagonal-spiral PCB winding arrays to gen-
`erate uniform MMF over the planar surface.
`
`For both planar charging platforms described above, it is nec-
`essary 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 electromag-
`netic 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 funda-
`mental 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 multi-
`layer 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 con-
`sidering 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
`
`Momentum Dynamics Corporation
`Exhibit 1020
`Page 002
`
`

`

`LIU AND HUI: EQUIVALENT CIRCUIT MODELING
`
`23
`
`TABLE II
`CALCULATED, SIMULATED AND MEASURED INDUCTANCE OF ONE WINDING OF THE THREE PLATFORMS
`
`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 sub-
`strate, 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 Green-
`house method offers sufficient accuracy and adequate speed,
`but cannot provide an inductor design directly from specifica-
`tions and is cumbersome for initial design. Reference [8] pro-
`vides 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 tech-
`niques, 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 rect-
`angular cross sections. Comparison between the calculated re-
`sults 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
`
`(1)
`
`where
`is the average diameter
`is the number of turns,
`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 mea-
`sured 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
`
`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 param-
`eter 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 appro-
`priate 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]
`
`Momentum Dynamics Corporation
`Exhibit 1020
`Page 003
`
`

`

`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-induc-
`tance 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 be-
`tween them,
`, must be included. The mutual coupling coef-
`ficient,
`which is equal to
`numerically, can be cal-
`culated 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 demon-
`strated 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 satis-
`fied in this design because a smaller spacing improves the inter-
`winding magnetic coupling and reduces the power loss [8]. For
`three adjacent windings as shown in Fig. 7(c), the mutual cou-
`pling coefficient between two nonoverlapped windings is rep-
`resented 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 con-
`sists 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 approx-
`imated by the use of (2)
`
`(2)
`
`is the number of windings connected in series to form
`where
`a column. Derivatoin of (2) is given in the Appendix.
`The simulated values of
`and
`by the use of the sim-
`plified 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 co-
`efficients 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 cal-
`culation, the calculated inductance of one winding as listed in
`Table II is used. The agreement between the calculated and mea-
`sured 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, re-
`spectively. In Fig. 7(e),
`represents the capacitance between
`two adjacent columns. The circuit model can be easily imple-
`mented 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 ad-
`jacent columns. The overall inductance, resistance, capacitance
`of
`(where
`2,3,
`8) series-connected columns are de-
`fined as
`,
`, and
`, respectively.
`Assuming that the parameters of each individual column are
`identical to those of the other column, the parameter equations
`
`Momentum Dynamics Corporation
`Exhibit 1020
`Page 004
`
`

`

`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)
`
`(5)
`
`From (3),
`columns is
`
`the mutual
`
`inductance between two adjacent
`
`When the inductance of three adjacent columns as shown in
`Fig. 5(d) is taken into account, the mutual inductance
`be-
`tween 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,
`rep-
`resents 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
`
`(6)
`
`The mutual inductance,
`
`is
`
`(7)
`
`(8)
`
`can be measured with an
`, and
`,
`,
`,
`,
`impedance analyzer (HP4194A), or be determined from the di-
`mensions 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 calcu-
`lated 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 calcula-
`tion 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
`
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`
`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 oppo-
`site, 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-over-
`lapped columns (i.e., they are not the “two nearest” non-over-
`lapped 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.
`
`C. Resistance
`The resistance of a conductor is
`
`(9)
`
`is its conductivity and
`where is the length of the conductor,
`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
`
`Fig. 9. Calculated resistance of one winding of platform B, compared with the
`measured results.
`
`(10)
`
`Fig. 10. Calculated resistance of two columns of platform B, compared with
`two types of measured results.
`
`is the mag-
`is the frequency (in Hertz) of the current,
`where
`netic 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,
`0.093 mm at 500 kHz, 0.066 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)
`
`is the number of
`is the number of windings included,
`where
`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.
`
`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 as-
`terisks
`are obtained from the measurement by an impedance
`analyzer (HP4194A), and the circles
`are acquired with the
`
`Momentum Dynamics Corporation
`Exhibit 1020
`Page 006
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`

`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 ac-
`quired 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 con-
`ducted 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 interconnec-
`tions across the PCB layers.
`
`D. Capacitance
`
`Fig. 11. Circuit model of the arrays consisted of eight columns.
`
`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
`
`inductance of the “two nearest”
`does include the mutual
`non-overlapped columns but not those “far-apart” nonover-
`lapped 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.
`
`(14)
`
`A. Platform A
`
`is the distance between
`is the overlapped area and
`where
`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 fre-
`quency 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 param-
`eters 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
`
`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 cir-
`cuit 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
`0.1829
`and
`0.0719, 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 in-
`cludes mutual inductances between “far-apart” non-overlapped
`columns.
`
`Momentum Dynamics Corporation
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`Page 007
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`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” nonover-
`lapped 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
`0.1829 and
`0.0719, 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 con-
`figuration 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.
`the simulated and measured input
`Fig. 14(a)
`shows
`impedance of six columns of the first simulation without
`considering the mutual inductance of the “far-apart” nonover-
`lapped columns. They are in fairly good agreement. When
`10% of
`and 5% of
`are added as compensation so that
`0.1833 and
`0.0709, their agreement is even
`better as shown in Fig. 14(b).
`The three sets of comparison based on three different proto-
`types have confirmed that the equivalent circuit model is suffi-
`ciently accurate as a tool for performance prediction. In the sim-
`ulation, 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” nonover-
`lapped columns.
`
`Fig. 14. Simulated and measured input impedance of six columns of platform
`C: (a) without and (b) with mutual inductances between “far-apart” nonover-
`lapped columns.
`
`Momentum Dynamics Corporation
`Exhibit 1020
`Page 008
`
`

`

`LIU AND HUI: EQUIVALENT CIRCUIT MODELING
`
`29
`
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`
`Xun Liu (S’04) was born in China in 1978. He
`received the B.S. and M.S. degrees in electrical en-
`gineering 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 con-
`tactless power transfer systems, and current source
`inverters.
`
`S. Y. Ron Hui (F’00) was born in Hong Kong in
`1961. 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 Im-
`perial College of Science and Technology, University
`of London, London, U.K., in 1987.
`He was a Lecturer in power electronics at the Uni-
`versity of Nottingham, Nottingham, U.K. from 1987
`to 1990. In 1990, he took up a lectureship at the Uni-
`versity of Technology, Sydney, Australia, where he
`became a Senior Lecturer in 1991. He joined the Uni-
`versity of Sydney in 1993 and was promoted to Reader of Electrical Engineering
`in 1996. 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 1997. He has been an At-Large member of the IEEE
`PELS AdCom since October 2002. He was appointed as an IEEE Distinguished
`Lecturer by IEEE PELS in 2004.
`
`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 multi-
`layer PCB winding array structure that can b