Download Field Computational Aspects of Wireless Power Transfer

Field Computational Aspects of
Wireless Power Transfer
Szabolcs Gyimóthy1 , Sándor Bilicz1 , József Pávó1 , László Tóth1 ,
Péter Kis2 , Gábor Varga1 and László Szűcs1
1
Budapest University of Technology and Economics, Egry J. u. 18, H-1111 Budapest, Hungary
2
Furukawa Electric Institute of Technology, Késmárk u. 28/A, H-1158 Budapest, Hungary
E-mail: [email protected]
Abstract—The aim of this paper is, on one hand, to give an overview of the fast developing and diversified field of wireless
power transfer (WPT) from the point of view of the inevitable numerical modeling of electromagnetic fields. On the other
hand, we propose a fast method by which the full WPT system based on inductive resonant coupling (IRC) can be analyzed
or even optimized very efficiently. The proposed method has been validated by alternative methods and measurements.
Index Terms—wireless power transfer, inductive resonant coupling, method of moments
I. I NTRODUCTION
No doubt, the revolution of wireless power transfer is
facing us. Electric vehicles, mobile devices and several
industrial and medical applications are raring to use
this new feature. Among various techniques making
such energy-exchange possible, the inductive resonant
coupling [1] is preferred for the time being. Although the
capabilities of alternative techniques are far from being
exhaustively researched [2], yet in some cases the “big
idea” is waited for. Apparently Tesla’s original method
(1894) for wireless energy transfer based on inductive
resonant coupling was kind of reinvented around 2007,
in fact it had never been really forgotten (cf. Fig. 1).
Although the technique of IRC WPT has become more
or less mature for the mass application, there is still a lot
of work to do in order to improve factors like efficiency,
working distance, transmitted power and safety. Moreover, standardization, regulations and guidelines of the
field are still incomplete.
As a matter of fact, the numerical simulation of electromagnetic (EM) fields has currently growing interests
in this development. Besides its use for component (e.g.
coil) design [4], issues like adaptive coupling [5], foreign
object detection [6] and human exposure [7] have to
be dealt with, too. Hot topics are among others multifrequency coupling [8], meta-materials for enhancing
mutual coupling of coils [9], repeaters for relaying power
to larger distances [10] and resonant reactive shield-
Figure 1. Illustration of an experiment for inductive resonant wireless
power transfer as can be found in a Hungarian schoolbook [3] published
in 1954.
ing [11]. Some novel applications requiring sophisticated
EM simulation are “charge while driving” systems [12]
and board-to-board level interconnection in highly integrated packages [13].
In this paper we first review the most prevalent modeling approaches of resonant WPT and the role of EM simulation in each of them. Then we propose a lightweight
computational method based on integral equations, by
which the full IRC WPT system can be simulated and
optimized very efficiently. On the one part, the results
computed for a simple configuration are verified with
those obtained by the finite element method (FEM) and
compared with measured data as well. On the other part,
it is shown that full-wave modeling of the whole WPT
system, which is achievable by the proposed method, can
better predict the conditions of optimal power transfer
than other methods based on component analysis and
equivalent circuits.
II. T RANSFER
MODELS AND FIELD SIMULATION
The technology of wireless non-radiative mid-range energy transfer is based on the so-called long-lived oscillatory resonant electromagnetic modes with localized
slowly-evanescent field patterns in the regime of “strong
coupling”. This is something peculiar from the point of
view of computational electromagnetism in the sense
that quite different approaches —e.g. static analysis,
wave propagation or mode analysis— can provide useful
results. At the same time one or two methods may neglect
or ignore important features.
For example, static and quasi-static analyses certainly
do not account for the radiated losses. Indeed, when using
inductive coupling, these losses are almost negligible
from the point of view of transfer efficiency. On the
other hand, this radiation phenomenon may contribute
to interferences and human exposure, which can be
important factors too.
Yet in practice the preferred type of EM analysis is
primarily determined by the mathematical model used
for describing the energy transfer (see below).
Zs
Vs
Figure 2.
I1
V1
I2
s+1
s-1
Two-port
Sik or Zik
s+2
V
s-2 2
Zl
Simplified two-port representation of the WPT chain.
A. Two-port representation
From an electrical engineer’s point of view the most
general description of power transfer can be given by a
two-port (see Fig. 2), which is characterized either with
impedance parameters Zik or with scattering parameters
Sik , all complex and frequency dependent.
Note that the full WPT chain is normally set up
with sophisticated electronics on both the transmitter
(Tx) and the receiver (Rx) side, e.g. converters, filters,
matching circuits, as well as elements for control and
communication. In our model the “two-port box” contains merely the “field part” i.e. the two resonators (most
often capacitively loaded coils) together with some coupling elements. At the same time the circuit environment
is greatly simplified for facilitating field analysis: the
system is fed by a generator of source voltage Vs and
intrinsic impedance Zs = Rs + jXs , whereas the load is
modeled as an adjustable impedance Zl = Rl + jXl .
Adequate computation of the frequency characteristics
of the two-port in a broad frequency range requires
full-wave EM analysis, which is expensive as a rule.
Some attempts have already been made for reducing the
computational burden of FEM simulation, by means of
e.g. homogenization [4][14], domain decomposition[15]
and “multi-slicing” [16].
B. Coupled mode theory (CMT)
CMT describes the behavior of two (or more) resonant
objects having (about) the same resonance frequency and
being in weak coupling with each other. It is based
on perturbation theory and is thoroughly described in
e.g. [17]. The governing equations of a two-resonator
system are as follows:
da1
= jω1 a1 (t) − (Γext,1 + Γ1 )a1 (t) + jκa2 (t)
dt
p
+ 2Γext,1 s+1 (t)
da2
= jω2 a2 (t) − (Γext,2 + Γ2 )a2 (t) + jκa1 (t)
dt
p
s−1 (t) = 2Γext,1 a1 (t) − s+1 (t)
p
s−2 (t) = 2Γext,2 a2 (t)
(1)
in which a1,2 are so-called complex mode amplitudes,
the squared magnitude of which gives the total energy
stored in the resonator; s±1,2 are incident and reflected
wave amplitudes, the square of which give the respective
powers. The system is characterized by the self-resonant
frequencies ω1,2 , the coupling coefficient κ, as well as the
internal and external losses Γ1,2 and Γext,1,2 , respectively.
As referring to the scheme of Fig. 2, s+1 and Γext,1
together represent the generator, while Γext,2 represents
the load.
Equations (1) are not confined to electromagnetics —
they can also describe coupled pendulums, for example—
thus CMT is very general in this respect. On the other
hand, its validity is limited to weak coupling. Moreover,
while CMT explains the transient flow of energy between
resonators very plausibly, in the steady state it provides
merely a transfer function, which is valid only in the
vicinity of the resonance frequency indeed.
CMT parameters of a coupled resonant electromagnetic system can be best obtained by means of mode
analysis. The coupling coefficient κ is simply equal to
the difference between the eigenfrequencies of the two
fundamental modes of the coupled system. In turn, the
attenuation coefficients Γ can be determined from the
respective quality factors of the resonators, as Q = ω/2Γ
holds (for more details see [18]).
C. Lumped circuit representation
For a designer, lumped circuit is undoubtedly the
readiest model of WPT. Moreover, an electrical engineer
believes he/she can intuitively assign circuit elements
to EM field phenomena in most cases. Note that —
unlike when using CMT model— we can have different
circuit models depending on the coupling phenomena and
the actual resonator configuration (cf. circuit model in
Section V).
Lumped parameters are commonly extracted from
static or quasi-static analysis, and sometimes from analytical formulas [19]. Self and mutual inductances L, M
are usually obtained from static magnetic analysis. Wire
resistance Rwire at high frequency can be approximated
analytically by considering the skin effect (but ignoring
the proximity effect). Radiation losses are represented
by Rrad and may be approximated analytically, too,
by modeling the coil as a current dipole. Finally, self
capacitance of the coil C is usually negligible to that of
the external capacitor Cext connected to the coil, but if
not, still an approximation can be given by assuming a
kind of sinusoidal current distribution along the coil [1].
Although widespread, this practice can be criticized
in many respects. Above all, once having obtained the
(approximate) lumped parameters, one is tempted to
treat the equivalent circuit as the primary model. For
example, parameters of the CMT model are usually
derived from one such circuit model — which method
is indeed adequate, provided the lumped parameters had
been calculated at the resonance frequency.
Far less acceptable however, if one derives the
impedance or scattering parameters of the circuit as a
two-port, and applies it in a broader frequency range
(see Section V-B for a comparison). Even if the frequency dependence of lumped parameters is taken into
account (e.g. R = R(ω)), it is the network structure
that cannot reflect all possible couplings, some of which
might be significant under unforeseen or unexamined
circumstances.
III. I NTEGRAL FORMULATION
Computing “frequency scan” within a wide band, performing parameter studies, or optimizing WPT systems,
all of these call for an efficient numerical method. A fastto-evaluate integral formalism can be proposed, in case
the following conditions hold: 1) The system is made
up of air-cored coils, optionally loaded with elements
that can be considered lumped. 2) Each coil is made of
a homogeneous and thin wire (compared to the radius
of the turns) with circular cross-section. 3) The coil
configuration is surrounded by air. Note that the extension
of the method for handling conducting extraneous objects
in the proximity is straightforward, but is not dealt with
hereafter.
The numerical simulation is based on the discretization
of the wire, resulting in a much smaller number of
degrees of freedom (DoF) than using FEM. The method
is similar to the classical scheme of wire-antenna calculation based on the method of moments (MoM). However,
it goes beyond that by exploiting the special geometry
of coils. In addition to this, a quasi-static approximation
—to be used for the extraction of lumped parameters, for
instance— can be given in an elegant way. The equations
are summarized below for only one single coil in order
to keep simplicity, but they can be easily generalized for
an arbitrary number of coils (for more details see [20]).
In the proposed model, the wire of each coil is
represented by a 1-dimensional curve, along which we
define the coordinate ζ, so that ζ = 0 corresponds to the
beginning, and ζ = l to the end of the wire, respectively.
Sinusoidal time dependence with angular frequency ω
is assumed for all quantities (the exp(jω) convention is
used). The magnetic vector potential A and the electric
scalar potential Φ along the wire are expressed with the
retarded source quantities, i.e. current I and charge per
unit length q, respectively, as follows:
Z
µ0 l I(ζ ′ )dζ ′ −jβχ(ζ,ζ ′ )
Aζ (ζ) = êζ
e
(2)
4π 0 χ(ζ, ζ ′ )
Z l
1
q(ζ ′ )dζ ′ −jβχ(ζ,ζ ′ )
Φ(ζ) =
e
(3)
4πε0 0 χ(ζ, ζ ′ )
where β = ω/c, and êζ denotes the unit vector tangential to the wire (note that we address only the ζcomponent of A). Then we apply the differential Ohm’s
law, Eζ = rI(ζ) to express the tangential electric field
on the surface of the conductor, where r is the resistance
per unit length of the wire. Hence using the potentials
one can write
dΦ
− jωAζ
(4)
rI(ζ) = −
dζ
Figure 4.
The studied helical coil.
additional relationships have to be given for the terminals
of each coil. In our simulations, simply the current is
prescribed, i.e. I(0) = I(l) = Icoil , but this is also
the point where more complex lumped port boundary
conditions can be entered.
Once the system of equations is solved (cf. [20] for the
details of discretization and numerical implementation)
one can compute the complex amplitude of voltage
between the coil terminals —which makes sense only
if their separation is small— as Vcoil = Φ(l) − Φ(0),
and then find the impedance. Alternatively, the complex power flowing through the coil terminals can be
expressed. A notable advantage of the formulation from
the point of view of EMC studies is that the radiated
part of complex power can be easily expressed by the
Poynting’s theorem:
Z l
1
[Aζ (ζ)I ∗ (ζ) − q(ζ)Φ∗ (ζ)] dζ (6)
Smedium = jω
2
0
in which superscript (*) stands for the complex conjugate.
IV. A NALYSIS
OF A SINGLE COIL
For evaluating the above introduced numerical method,
we have examined the impedance of a helical coil made
of copper, in the frequency range 10 . . . 100 MHz. The
radius of the coil is 110 mm, its height is 24.5 mm, the
wire diameter is 1.5 mm and the number of turns is 6
(see Fig. 4).
We used an Agilent/HP 4191A RF impedance analyzer
for the measurement. The simulation of the impedance
characteristics was carried out both by the integral formulation and by FEM — for the latter the RF Module
of Comsol Multiphysics was used [21]. The performance
of the two solutions are compared in Table I.
For verification we have also computed the selfresonance frequencies of the coil by Comsol RF, using
Finally, charge conservation law can be written as
jωq(ζ) +
dI(ζ)
= 0.
dζ
(5)
The two integral and two differential equations (2)-(5)
form a coupled system for the unknown current, charge,
and potential distributions. To provide a unique solution,
Table I
P ERFORMANCE OF NUMERICAL SIMULATIONS
Degrees of freedom
CPU time for 100 pts. [sec]
FEM
3055664
18273
Integral formulation
1200
13
6
10
integral formulation
FEM sweep
FEM eigenfrequency
measurement
5
impedance magnitude (kΩ)
10
4
10
X: 52
Y: 2704
3
10
2
10
1
10
10
30
frequency (MHz)
its eigenfrequency application mode. Note that when
considering open ended coil one obtains the resonance
frequencies at which the impedance has a local maximum. In turn, with short circuited coil terminals one
gets the so-called anti-resonance frequencies where the
impedance tends to zero. All the simulated and measured
results are plotted in Fig. 3.
As the figure shows, the two simulated impedance
curves agree well, and the computed resonance frequencies coincide with their extrema, too. However, the
measured curve shows a systematic frequency shift from
the computed ones in terms of the peaks. This may
be originated either from calibration — although the
measurement was carried out carefully enough— or from
the glue material used to fasten the windings.
More interestingly, the measured impedance curve
shows a double peak at about 50 MHz, but this does not
appear at all in either of the simulated curves. At the
same time, FEM eigenfrequency analysis predicts both
resonance and anti-resonance at about 58 MHz — a good
example of how different types of analysis can catch or
dismiss certain features. This resonance corresponds to a
specific mode in which both the current and the voltage at
the terminals are zero. We suspected that for a symmetric
coil this singular behavior is confined to a such narrow
frequency range, that with the chosen frequency steps we
simply could not catch it.
On the other hand, we believed this range is widened
0.02
0.02
0.015
0.015
z (m)
z (m)
40
50
60
70
80
90
100
Comparison of the simulated and measured impedance curves, as well as the computed resonance frequencies of the helical coil.
0.01
0.01
and the effect becomes stronger, if the perfect symmetry
of the coil is broken, as it can always happen with handmade coils. Therefore we modeled a coil having the same
overall dimensions but a variable pitch of winding (see
Fig. 5). The comparison of its impedance characteristics
with that of the original coil (having uniform winding)
clearly shows the difference: the expected double peak
appears around 58 MHz (see Fig 6), while the remaining
parts of the characteristics are not affected.
V. A NALYSIS OF A WPT SYSTEM
In order to point out the necessity of the full-wave
field analysis of a WPT system, instead of resorting
to its lumped circuit model, we have compared the
efficiency predicted by the two models w.r.t. frequency
and resonator distance. The configuration was adopted
from [1]. It contains two identical coils of 30 cm radius
and 20 cm height. There is a planar loop of 25 cm radius
on the side of each resonator for the inductive coupling
of the source and the load, respectively. All wires are
made of copper having 6 mm diameter. The axes of the
components are aligned (see Fig. 7). The distance range
[0.5, 2.5]m and the frequency range [8, 11]MHz were
examined in the simulations.
graded winding
regular winding
impedance magnitude (kΩ)
Figure 3.
20
100
10
X: 58.55
Y: 0.7432
1
0.1
0.005
0.005
0.01
0
−0.1
−0.05
0
x (m)
0.05
0.1
0
−0.1
−0.05
0
x (m)
0.05
0.1
Figure 5. Coil models of uniform and “graded” winding. Note that
the two axes have different scaling for better visualization.
10
Figure 6.
20
30
40
frequency (MHz)
50
60 70
100
Impedance characteristics of the coils shown in Fig 5.
1
z (m)
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0.2
y (m)
0.8
0
−0.2
efficiency
0.7
−0.2
0
Figure 7.
0.2
x (m)
fullwave 1m
fullwave 2m
lumped 1m
lumped 2m
0.9
0.6
0.5
0.4
0.3
Configuration of the two-resonator WPT system.
0.2
0.1
For the full-wave frequency-domain modeling of the
system we have used the method introduced in Section III. The lumped circuit model of the system is
depicted in Fig. 8. Inductance (self and mutual) and
capacitance parameters of the circuit were computed by
the quasi-static approximation of the above mentioned
method (cf. [20]). The resistance values representing
joule loss and radiation, respectively, were calculated
with analytical formulas as described in Section II-C.
Both the full-wave model and the circuit model are
represented by the impedance matrix of the two-port
(symmetric, in this case) the entries of which, Z11 =
R11 + jX11 and Z12 = R12 + jX12 , are frequency
dependent. Note however, that this dependence differs
for the two models.
0
8
8.5
9
9.5
10
frequency (MHz)
10.5
11
Figure 9. Maximum efficiency w.r.t. operating frequency at two fixed
distances, as predicted by the two models. (The full-wave model suffers
from numerical instabilities at the certain frequencies; some outlying
data points were skipped.)
decrease efficiency). Hence the three real parameters Rl ,
Xs and Xl are to be chosen such that η is maximal (cf.
Fig. 2). Their “matched” values depend on the frequency
and the coil distance as well. It can be shown that the
maximum efficiency does not depend on Xs ; it can be
used to compensate the input reactance at the primary
port so as to the maximal power can enter the twoport at a given source voltage Vs . Thus, the optimization
problem reduces to
A. Impedance matching for optimal efficiency
The conditions of optimal efficiency of WPT are
usually determined with the assumption that the output
power of the generator is maximized. This leads to the
so-called double conjugate matching conditions [22]. At
the same time, the internal losses of the generator are
not dealt with. Since in our opinion these losses should
be considered in the evaluation of the whole system,
we define efficiency in a slightly different manner. We
consider the net power coming from the source,
1
Re{Vs I1∗ }
2
and the power consumed by the load,
Ps =
(7)
1
1
(8)
Pl = − Re{V2 I2∗ } = Rl |I2 |2 .
2
2
The efficiency of power transfer is defined as η = Pl /Ps .
For the optimization of this efficiency we assume that
the resistance of the source Rs is given (a lower bound
exists in practice, and an artificial increase of Rs can only
Ml-c
Lloop
V1 Rloop
I1
Figure 8.
Mc-c
Lcoil
Rcoil
Ccoil
Ml-c
Lcoil
Rcoil
Ccoil
Lloop
Rloop V2
I2
Lumped circuit model of the WPT system shown in Fig. 7
(R̂l , X̂l ) = arg max η(Rl , Xl |Rs , Z11 , Z12 )
(9)
The optimal choice of Rl and Xl is as follows (deduced
simply by calculating the partial derivatives; details are
omitted herein)
p
2 − R2 + R R )(R2 + X 2 + R R )
(R11
s 11
s 11
12
11
12
R̂l =
R11 + Rs
R12 X12 − (R11 + Rs )X11
X̂l =
.
(10)
R11 + Rs
B. Results and comparison
Figure 9 shows how much the maximum achievable
efficiency depends on the frequency. Note that the two
models suggest different operating frequencies (9.0 MHz
by lumped, and 10.3 MHz by full-wave). At the same
time, the distance does not have considerable effect on
the location of the best operating frequency.
In Fig. 10 the dependence of the maximum available
efficiency on the coil distance is presented, provided that
the best operating frequency has been chosen for each
model (cf. Fig 9). In this figure, the measured data of [1]
are also indicated, although the authors defined efficiency
in the other way (see above). The predictions of the two
models are different, again.
Finally, the wide variation of the optimal load resistance is presented in Figure 11 as function of the distance
(9.0 MHz frequency is set for the lumped model, and
10.3 MHz for the full-wave model). This is definitely an
1
R EFERENCES
0.9
0.8
efficiency
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.5
fullwave
lumped
measured*
1
1.5
distance (m)
2
2.5
Figure 10. Maximum efficiency w.r.t. coil distance predicted by the
two models, and compared with measured data borrowed from [1].
1
optimal load resistance (kΩ)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
fullwave
lumped
0.2
0.1
0
0.5
1
1.5
distance (m)
2
2.5
Figure 11. Optimal load resistance w.r.t. coil distance, predicted by
the two models
issue for the design of the matching electronic circuit,
which connects the receiver loop and the device to be
supplied, in order that the loop is loaded with the optimal impedance. However, model predictions are slightly
different.
VI. C ONCLUSIONS
We believe that different transfer models and field analysis methods can be applied complementary to have a better insight to the mechanism and characteristics of WPT.
At the same time, we underline the superiority of the fullwave field computational model over the lumped circuit
model. In this paper, at least the remarkable difference
in their behavior and predictions was demonstrated.
We have also proposed a fast numerical field simulation tool based on the classical method of moments by
which the whole inductive resonant WPT system can be
efficiently analyzed. The method has been verified by
alternative methods and measurements; its extension to
handling extraneous objects is underway.
ACKNOWLEDGMENT
This work was supported by the Hungarian Scientific
Research Fund under grants K-105996 and K-111987.
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