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 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%.
(a)
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 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, 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 publication by Associate Editor J. A. Ferreira.
The authors are with the Department of Electronic Engineering, City University 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
(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.
0885-8993/$20.00 © 2006 IEEE
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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
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.
, , and is
The meaning of the parameters in Table I,
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
(1)
is the average diameter
where is the number of turns,
, is the fill ratio and defined as
and equals 0.5
,
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
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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
and an inductor
one winding. It consists of a resistor
connected in series. When two partly-overlapped windings are
considered, as shown in Fig. 7(b), the mutual inductance be, must be included. The mutual coupling coeftween them,
which is equal to
numerically, can be calficient,
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 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
. Such condition is always desired and satistrack width,
fied 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 repwhich equals
. Its value can also be
resented by
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.
and
by the use of the simThe simulated values of
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,
, etc.) shows
efficients between “far-apart” windings (
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
and
represent
columns are shown in Fig. 7(d) and (e). ,
the inductance, capacitance, and resistance of one column, rerepresents the capacitance between
spectively. In Fig. 7(e),
two adjacent columns. The circuit model can be easily implerepresents the
mented in PSpice. The PSpice K_Linear part,
between two adjacent columns [e.g.,
mutual inductance
columns 1 and 2, or columns 2 and 3, or columns 3 and 4,
equals
.
etc. as shown in Fig. 5(b)]. Numerically,
Fig. 7(e) also shows the simplified equivalent circuit of two adjacent columns. The overall inductance, resistance, capacitance
2,3, 8) series-connected columns are deof (where
,
, and
, respectively.
fined as
Assuming that the parameters of each individual column are
identical to those of the other column, the parameter equations
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), the mutual inductance between two adjacent
columns is
(6)
, ,
,
,
, and
can be measured with an
impedance analyzer (HP4194A), or be determined from the dimensions and geometry of one winding column. To determine
, the one-turn simplification method is also used.
the value of
In the simulation, each column is simplified to six one-turn
windings connected in series so that the required memory and
of the
computation time decrease obviously. The simulated
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.
When the inductance of three adjacent columns as shown in
beFig. 5(d) is taken into account, the mutual inductance
tween the “two nearest” nonoverlapped columns [columns 1 and
3, or columns 2 and 4, or columns 3 and 5, etc. as shown in
repFig. 5(c)] should be considered. Another K_Linear part,
between the “two nearest”
resents the mutual inductance
numerically. The
non-overlapped columns. It is equal to
inductance of columns 1, 2 and 3 connected in series,
can
be expressed as
(7)
The mutual inductance,
is
(8)
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
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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 opposite, resulting in some flux cancellation.
Based on the above approach, mutual inductance terms of
,
, etc.) can
“far-apart” non-overlapped columns ( ,
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
(9)
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
(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,
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.
2 , the resistance of the
So at high frequency, when
conductor is a function of the frequency
if
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
(11)
if
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
is the outer diameter, is the track
turns of one winding,
width and is the spacing between the tracks, as shown in Figs. 6
and 8.
(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 asare obtained from the measurement by an impedance
terisks
are acquired with the
analyzer (HP4194A), and the circles
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
of
two layers. It should be noted that is only
the enveloped area because of the spacing between the tracks.
and
is more important in this
In general, prediction of
and . Together with an
equivalent circuit modeling than
(that is much greater than
externally connected capacitor
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
between two adjacent
column and the capacitance
columns. The resistance is a function of frequency as expressed
by (13). The simulated coupling coefficients between the
and
are also listed.
columns,
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
0.1829
0.0719, the mutual inductance between “far-apart”
and
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.
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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
and 5% of
are added to include the mutual
10% of
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).
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.
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
and 5% of
are added as compensation so that
10% of
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 prototypes have confirmed that the equivalent circuit model is sufficiently accurate as a tool for performance prediction. In the simand
(10%
ulation, the value of the proportion added to
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.
LIU AND HUI: EQUIVALENT CIRCUIT MODELING
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
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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 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
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 Imperial College of Science and Technology, University
of London, London, U.K., in 1987.
He was a Lecturer in power electronics at the University of Nottingham, Nottingham, U.K. from 1987
to 1990. In 1990, he took up a lectureship at the University of Technology, Sydney, Australia, where he
became a Senior Lecturer in 1991. He joined the University 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.