# Effective Mid-Range Wireless Power Transfer with Compensated Radiation Loss

N. Ha-Van<EMAIL_ADDRESS>C. R. Simovski  F. S. Cuesta  P. Jayathurathnage
S. A. Tretyakov  Department of Electronics and Nanoengineering, Aalto
University, P.O. Box 15500, Aalto FI-00076, Finland

###### Abstract

In conventional inductive wireless power devices, the energy is transferred
via only reactive near fields, which is equivalent to non-radiative Förster
energy transfer in optics. Radiation from transmitting and receiving coils is
usually considered as a parasitic effect that reduces the power transfer
efficiency. As long as the distance between the two antennas is small as
compared to the antenna size, conventional WPT devices offer rather high power
transfer efficiency, of the order of 80-90%. However, for larger distances,
the transfer efficiency dramatically drops, making such devices not practical.
In this paper, we develop a dynamic theory of wireless power transfer between
two small loop antennas, clarify the role of far-field radiation, and find a
possibility to realize efficient wireless power transfer at large distances
utilizing the regime of radiation suppression due to optimized mutual dynamic
interactions between the transmitting and receiving antennas. The analytical
results have been validated by simulations and measurements, and they open a
possibility to greatly expand the range of distances of compact wireless power
transfer devices. The developed theory can be applied also to coupling between
antennas of different types and to energy transfer between nano-objects.

##  I Introduction

Wireless power transfer (WPT) technologies become more and more important for
diverse applications, such as charging of mobile telecommunication devices,
electric vehicles, implantable medical devices, robots, wearable electronics,
and in energy harvesting systems (see e.g. in [1, 2, 3]). During recent
decades, especially proliferation of wireless devices has motivated fast
developments of wireless powering and charging technologies.

All WPT systems can be classified into near-field and far-field ones (see e.g.
in [4]). When the distance between the transmitting and receiving antennas is
small compared to the wavelength, the reactive near fields at the receiver
position are much stronger than slower decaying fields of propagating waves
created by the transmitting antenna. In this short-range WPT regime, the power
is transferred predominantly by the near fields, whereas the radiation is
usually considered as a parasitic factor resulting in some radiation loss that
decreases efficiency (see e.g. in [5, 6, 7, 8, 9, 10, 11, 12]). Respectively,
it appears that in this regime it is desirable to suppress antenna radiation.
However, the radiative fields are present also in the near zone and they may
contribute to the received power.

If the distance between the transmitting and receiving antennas is
electrically large, the near fields of the transmitting antenna become
negligibly small as compared with the radiation fields. For electrically small
and weakly directive antennas, the transfer efficiency becomes very small. For
this reason, in long-range radiative WPT systems high-directivity antennas are
used in both receiving (Rx) and transmitting (Tx) devices. Obviously, in this
case, radiation is used as the main power-transfer mechanism, and radiation
losses can be reduced only by using higher-gain antennas of large sizes in
comparison with the wavelength.

In this paper, we consider the intermediate case of mid-range wireless power
transfer, where at the receiver position near fields of the transmitter
significantly decay and the slower-decaying radiative fields become comparably
strong or even stronger. In this situation, radiation of energy into far zone
is still unwanted, as this is one of the loss mechanisms. On the other hand,
radiative fields can significantly contribute to the power transfer to the
receiving antenna. In practice, this situation corresponds to scenarios where
the distance between the transmitting and receiving antennas is large as
compared to the sizes of the antennas, while the antennas themselves are
electrically small. Realization of such WPT systems would allow wireless power
transfer using compact devices, as the transfer distance can be large as
compared to both connected devices. It is expected that the power transfer
efficiency, in this case, will be lower than for near-field coupling devices
working at small transfer distances, but it may be possible to realize
reasonable efficiency also at large separations by exploiting coupling by both
near fields and radiative fields.

In this study, we develop the dynamic theory of wireless power transfer
between two magnetic dipole antennas and use it to study the role of radiation
fields (intermediate and far-zone) in transfer of power. We answer the
following questions: What is the dominant role of dynamic (far-zone) fields —
disadvantage of parasitic radiation loss or an advantage of higher
electromotive force induced in the receiving antenna? Is it possible to create
WPT devices where radiation into far zone is suppressed while the dynamic
fields between the antennas effectively deliver power to the load? How to
properly engineer the optimal regime of long-distance WPT between small
antennas? To our knowledge, these questions have not been answered in the
available literature. In spite of the great diversity of radio-frequency WPT
systems, the comparative role of the radiative and non-radiative regimes was
not properly elucidated. An important question is the suppression of radiation
in the short-range systems of non-radiative WPT. Dynamic coupling between two
loop antennas was recently considered in [13], where this question was
addressed, focusing on visualizations of the power flow in space. In that
paper, it has been found that it is possible to maximize the ratio of the
delivered power to the power available from the source and at the same time
minimize the radiated power by optimizing the load resistance for a fixed
value of the internal resistance of the source.

In this paper, we find a simple formula for the optimal load impedance and the
corresponding maximized power transfer efficiency that are valid for arbitrary
internal impedance of the source. We use these results to understand physical
mechanisms of power delivery accounting for dynamic interactions, clarify the
role of far fields, and find the optimal frequency range where the parasitic
radiation into the far zone is effectively suppressed while dynamic, far-zone
fields strongly contribute to the power delivered to the source.

The developed theory of the optimal load impedance and the maximized power
transfer efficiency can be used for optimizing coupling between arbitrary
antennas, including electric dipoles. However, we focus the study on loop
antennas, because electric dipoles are used in WPT devices quite rarely,
mainly in very short-range systems for high power transfer via capacitive
coupling [14, 15] and in WPT systems operating in lossy media, e.g., for
biomedical implants [16]. Most WPT systems operating in free space utilize MDs
implemented as coils [1, 2, 17, 18, 19].

This study is also relevant for understanding and use of non-radiating
antennas – those creating only near fields, such as anapole antennas [20, 21].
The evident advantage of anapole antennas is negligibly small parasitic
radiation for all arrangements of the two antennas where mutual coupling of
the antennas does not destroy the anapole properties. In paper [22], anapole
antennas were claimed to be more beneficial for the short-range WPT than MDs
due to suppressed radiation. The use of the theory developed in this work
allows us to understand full implications of radiation cancellation for power
coupling between antennas.

The paper is organized as follows. In Section II, we theoretically analyze the
dynamic coupling between two small antennas WPT and elaborate the model of
power transfer efficiency ($PTE$) applicable for a maximally broad frequency
range and maximal range of distances. In Section III, we study two key
arrangements of Tx and Rx loops; coaxial and coplanar. Finally, the developed
analysis is verified by full-wave simulations and experiments in Section IV.

Figure 1: Equivalent circuit model of the WPT system.

##  II WPT System Analysis

Figure 1 shows the equivalent circuit of WPT systems formed by two coupled
antennas. The two antennas are represented by their individual input
impedances $Z_{1}$ and $Z_{2}$. The transmitting antenna is fed by a voltage
source $V_{\rm G}$ with internal impedance $Z_{\rm G}$, and the receiving
antenna is loaded by the load impedance $Z_{\rm L}=R_{\rm L}+jX_{\rm L}$,
whose real part $R_{\rm L}$ is the useful load.

Similarly to [13], we find it convenient to express the electromotive forces
(EMFs) induced by antennas in one another through the complex coefficient
$M=M_{\rm R}+jM_{\rm I}$ instead of the commonly used mutual impedance,
generalizing the notion of mutual inductance to dynamic regimes, applicable to
arbitrary distances between the antennas:

$\displaystyle V_{\rm emf,\it i}=j\omega MI_{i}.$ (1)

Here, $i=1,2$ mark the first and second antennas. For inductive wireless power
transfer devices and in the quasi-static limit, $M$ is real-valued and equals
to the usual mutual inductance between the two coils. The value $Z_{m}=j\omega
M$ is the mutual impedance, not to be mixed with the induced impedances
$Z_{i1}=V_{\rm emf,1}/I_{1}=Z_{m}I_{2}/I_{1}$ and $Z_{i2}=V_{\rm
emf,2}/I_{2}=Z_{m}I_{1}/I_{2}$). For voltages $V_{1,2}$ we have

$\displaystyle V_{1}=Z_{1}I_{1}+j\omega MI_{2}=V_{\rm G}-I_{1}Z_{\rm G}$ (2)
$\displaystyle V_{2}=I_{2}Z_{\rm L}=-I_{2}Z_{2}+j\omega MI_{1},$ (3)

where $Z_{1,2}=R_{1,2}+jX_{1,2}$ are complex-valued antenna input impedances.
The impedances seen from the source and the load (see Fig. 1) can be expressed
in terms of the complex-valued mutual inductance $M$ in the usual form:

$\displaystyle Z_{\rm
in}=\frac{V_{1}}{I_{1}}=Z_{1}+\frac{\omega^{2}M^{2}}{Z_{2}+Z_{\rm L}}$ (4)
$\displaystyle Z_{\rm out}=\left.\frac{V_{2}}{I_{2}}\right|_{V_{\rm
G}=0}=Z_{2}+\frac{\omega^{2}M^{2}}{Z_{1}+Z_{\rm G}}.$ (5)

Aiming to the maximal power coupling between antennas, we use the conventional
definition of the power transfer efficiency $PTE$ as the ratio of the power
delivered to the load $P_{\rm L}$ and the power accepted by the input port of
the transmitting antenna $P_{\rm in}$ (see e.g. in [2]):

$\displaystyle{PTE}\equiv\frac{P_{\rm L}}{P_{\rm in}}=\frac{|I_{2}|^{2}R_{\rm
L}}{\mathfrak{Re}\\{V_{1}I_{1}^{*}\\}}.$ (6)

In this definition, the transfer efficiency does not depend on the internal
impedance of the source $Z_{\rm G}$, because the received power is normalized
to the power accepted by the transmitting antenna. Using (3) and (4), we write
Eq. (6) in the form

$\displaystyle{PTE}=\frac{|I_{2}|^{2}R_{\rm
L}}{|I_{1}|^{2}\mathfrak{Re}\\{Z_{\rm in}\\}}=\left|\frac{\omega
M}{Z_{2}+Z_{\rm L}}\right|^{2}\frac{R_{\rm L}}{\mathfrak{Re}\\{Z_{\rm
in}\\}}.$ (7)

The optimal load $Z_{\rm L,opt}$ maximizing the $PTE$ can be found by
nullifying the derivatives [23]:

$\frac{\partial{PTE}}{\partial X_{\rm
L}}=0,\quad\frac{\left.\partial{PTE}\right|_{X_{\rm L,opt}}}{\partial R_{\rm
L}}=0,$ (8)

from which we obtain the reactance and resistance of the optimal load that
depends on the operational frequency and complex-valued mutual inductance:

$\displaystyle X_{\rm L,opt}=-X_{2}-\frac{\omega^{2}M_{\rm R}M_{\rm
I}}{R_{1}},$ (9) $\displaystyle R_{\rm
L,opt}=\frac{\sqrt{R_{1}R_{2}-\omega^{2}M_{\rm
I}^{2}}\sqrt{R_{1}R_{2}+\omega^{2}M_{\rm R}^{2}}}{R_{1}}.$ (10)

This result generalizes expressions in [23] taking into account dynamic
interactions between Tx and Rx antennas. Substituting (9) and (10) into (7),
the $PTE$ of devices loaded by optimal loads after some algebra can be
expressed in a very simple form:

$\displaystyle PTE=1-\dfrac{2}{1+\sqrt{\dfrac{R_{1}R_{2}+\omega^{2}M_{\rm
R}^{2}}{R_{1}R_{2}-\omega^{2}M_{\rm I}^{2}}}}.$ (11)

Parameters $M_{\rm R},\,M_{\rm I},\,R_{1},\,R_{2}$ can be found analytically,
numerically, or experimentally.

The same expression for $PTE$ can also be written in form of the classical
definition of impedance parameters, where $R_{1}=r_{11}$, $R_{2}=r_{22}$,
$\omega M_{\rm I}=r_{12}$, and $\omega M_{\rm R}=x_{12}$ are the real and
imaginary parts of the corresponding components of the impedance matrix of the
wireless link:

$\displaystyle
PTE=1-\dfrac{2}{1+\sqrt{\dfrac{r_{11}r_{22}+x_{12}^{2}}{r_{11}r_{22}-r_{12}^{2}}}}.$
(12)

In this form, this result can be used for optimization of power transfer
between arbitrary emitters and receptors, not necessarily loop antennas.

The found simple analytical expression for the power transfer efficiency at
the optimal load allows us to perform a broadband analysis of the efficiency,
find the optimal operational frequency and clarify the role of radiation in
short- and mid-range WPT systems. Before going to the analysis of systems with
specific antennas, let us make some general observations. First, it is obvious
that in order to increase $PTE$ we should increase the value of the square
root in (11) or (12). Assuming that the two resistances are fixed, it is
clearly beneficial to increase $M_{\rm R}$, which, in the case of electrically
small distances, means simply that the transfer efficiency is larger at
smaller transfer distances. More interestingly, we see that increasing the
imaginary part of $M$ also leads to higher efficiency. The denominator under
square root cannot reach zero (which would mean 100% efficiency), because the
mutual impedance cannot be higher than the impedance of both coupled circuits,
but it can become rather small under some conditions. In the following, we
will discuss these two factors at an example of two coupled electrically small
loops.

Since the power transfer efficiency (6) and the optimal load impedance
(9)–(10) do not depend on the internal impedance of the source $Z_{\rm G}$,
that can be chosen based on application requirements. If the WPT device should
deliver the maximal power to the load, the source impedance can be conjugate
matched to $Z_{\rm in}$. In most applications, however, it is preferable to
maximize the end-to-end power transfer efficiency, in which case $Z_{\rm G}$
is made as small as possible, to minimize dissipative losses. Importantly, our
conclusions regarding optimization of $PTE$ remain valid for any value of
$Z_{\rm G}$ as long as the load impedance $Z_{\rm L}$ is equal to the optimal
load given by (9) and (10). If the load resistance is different, a matching
circuit can be used at the receiver side. This approach and the ultra-broad
range of possible operation frequencies is qualitatively different from that
of [13], where the optimization was done for the case when $R_{\rm
G}\equiv{\rm Re}(Z_{\rm G})=50~{}\Omega$, and the operation frequency range
was specified in advance.

Figure 2: Two magnetic dipoles in two case studies: a) coaxial position; b)
coplanar position.

##  III Analysis of a System Based on Loop Antennas

Our goal is to clarify the role of intermediate and far fields in WPT systems
in the maximally broad range of frequencies and for both short and middle
ranges of distances, that is, when the distance between antennas can be
comparable with the wavelength or electrically small. With this goal in mind,
we restrict the size of both antennas so that in the whole possible
operational frequency range the antennas remain electrically small. Otherwise,
the comparison of the two regimes would be veiled by the size resonances of
antennas.

In order to clarify the role of radiative fields, we consider small loop
antennas and study WPT for coaxial and coplanar arrangements of the Tx and Rx
antennas, as shown in Fig. 2. Because the loops are electrically small, we
model them as magnetic dipoles (MD). In the coaxial arrangement, the loops are
coupled solely by near fields, since a magnetic dipole does not radiate along
its magnetic moment. In this case, there is no radiative WPT, and the
radiation resistances of the loops are purely parasitic contributions to
resistances $R_{1,2}$. Thus, in order to increase $PTE$ it is desirable to
realize a regime of suppressed radiation. In the coplanar arrangement, the
situation is nearly opposite. In this case, the quasi-static mutual coupling
is weak because the mutual inductance is small, and the radiative WPT can
dominate already at mid-range distances.

###  III.1 Analytical model

####  III.1.1 Coaxial Arrangement

Let us consider two loops, Tx and Rx, with the radii $a$ and $b$,
respectively, positioned in free space at distance $d$ between them, as it is
shown in Fig. 2(a). In the spherical coordinate system, the magnetic field
components are as follows (see e.g. in [24]):

$\displaystyle
H_{r}=j\frac{ka^{2}I_{1}\cos{\theta}}{2d^{2}}\left[1+\frac{1}{jkd}\right]e^{-jkd},$
(13) $\displaystyle
H_{\theta}=-\frac{(ka)^{2}I_{1}\sin{\theta}}{4d}\left[1+\frac{1}{jkd}-\frac{1}{(kd)^{2}}\right]e^{-jkd}.$
(14)

Here $k=\omega/c$ is the wave number in free space. When $a,b\ll d$, we can
approximate $H_{z}\approx\rm const\it(x,y)$ in the area
$z=d,x^{2}+y^{2}<b^{2}$, i.e., the $z$-component of the magnetic field created
by Tx in the area of Rx loop is nearly equal to its $r$-component at the
receiving loop center:

$\displaystyle H_{z}\approx H_{r}=j\frac{kA_{\rm T}I_{1}}{2\pi
d^{2}}\left[1+\frac{1}{jkd}\right]e^{-jkd},$ (15)

where $A_{\rm T}=\pi a^{2}$. From (15) and Faraday’s law $V_{\rm
emf,2}=-j\omega\mu_{0}\pi b^{2}H_{z}$, we find the complex mutual inductance
$M$ defined by Eq. (1) as

$\displaystyle M\equiv M_{\rm R}+jM_{\rm I}=\frac{j\mu_{0}kA_{\rm T}A_{\rm
R}}{2\pi d^{2}}\left[1+\frac{1}{jkd}\right]e^{-jkd},$ (16)

where $A_{\rm R}=\pi b^{2}$ is the area of the Rx loop. In the quasi-static
limit $kd\ll 1$, and formula (16) transits to the known mutual inductance of
two coaxial loops, applicable when $d\gg\max(a,b)$ [25].

####  III.1.2 Coplanar Arrangement

Now we consider two loops positioned at the same $xy$-plane, as shown in Fig.
2(b). If $a,b\ll d$, we can write

$\displaystyle H_{z}\approx-\frac{k^{2}A_{\rm T}I_{1}}{4\pi
d}\left[1+\frac{1}{jkd}-\frac{1}{(kd)^{2}}\right]e^{-jkd}.$ (17)

Then we have for $M$ the expression:

$\displaystyle M\equiv M_{\rm R}+jM_{\rm I}=-\frac{\mu_{0}k^{2}A_{\rm T}A_{\rm
R}}{4\pi d}\left[1+\frac{1}{jkd}-\frac{1}{(kd)^{2}}\right]e^{-jkd}.$ (18)

In the low-frequency limit, this expression also transits to the known quasi-
static mutual inductance, when $d\gg\max(a,b)$ [25]. When $a=b$, the quasi-
static approximation is applicable with high accuracy when $d>4a$.

####  III.1.3 Input Resistances of Tx and Rx loops

The input resistances $R_{1}$ and $R_{2}$ of the Tx and Rx antennas are the
sums of the Ohmic resistances $R_{\rm O1,O2}$ and radiation resistances
$R_{\rm r1,r2}$, i.e., $R_{1,2}=R_{\rm r1,r2}+R_{\rm O1,O2}$. For electrically
small loops made of thin round wires in the regime of strong skin effect, we
have (e.g., [24]):

$\displaystyle R_{\rm
O1}=\frac{a}{r_{0}}\sqrt{\frac{\omega\mu_{0}}{2\sigma_{w}}},\quad R_{\rm
O2}=\frac{b}{r_{0}}\sqrt{\frac{\omega\mu_{0}}{2\sigma_{w}}}.$ (19)
$\displaystyle R_{\rm r1}=\eta\frac{\pi}{6}(ka)^{4},\quad R_{\rm
r2}=\eta\frac{\pi}{6}(kb)^{4}.$ (20)

Here, $r_{0}$ is the radius of the antenna wire, $\sigma_{w}$ is the
conductivity of the wire, and $\eta=\sqrt{\mu_{0}/\varepsilon_{0}}$ is the
free-space impedance. In numerical examples, we assume copper wires with
$\sigma_{w}=58.7\times 10^{6}$ S/m, $a=b=36$ mm, and $r_{0}=2$ mm.

Using formulas (16), (18), and (20), it is easy to verify that
$\omega^{2}M_{\rm I}^{2}\leq R_{\rm r1}R_{\rm r2}$ for $d\geq\max(a,b)$ in
both coaxial and coplanar cases, confirming that in Eq. (11) the expression
under the square root is always positive.

Figure 3: Ratio of the mutual resistance to the radiation resistance for
coaxial and coplanar arrangements of the Tx and Rx loops versus the normalized
distance $d/\lambda$ between the loop centers.  Figure 4: Mid-range coupling
between two lossless loops: a) phase difference between the loop currents; b)
magnitude of the currents ratio $I_{2}/I_{1}$; c) $PTE$ for different values
of the load resistance $R_{\rm L}$. The dashed curves correspond to the
optimal resistance, maximizing $PTE$.

###  III.2 Suppression of Radiation of Power into Space

As discussed in Sec. I, it is important to clarify the role of radiation
fields, since on one side they create parasitic radiation of power into space,
but on the other hand, far fields can carry power to the receiver. In this
part, we consider this question for coaxial loops.

Let us first discuss the case when the dissipation of energy in both
transmitting and receiving coils can be neglected. Then, the quasi-static
theory of wireless power transfer between two loops tells that $PTE=1$
identically, because the power accepted by the transmitting antenna can go
only to the load in the receiver. However, it is expected that the fully
dynamic model will show finite transfer efficiency even in the limit of zero
distance between antennas, because the antennas will radiate some power into
space. Thus, in [13] it was claimed that suppression of radiation into far
zone is a useful mean to increase power transfer efficiency. Let us analyse
this issue using the analytical formulas for the complex-valued mutual
inductance and power transfer efficiency. First, we note that in the absence
of losses in coils, the resistances $R_{1,2}$ in (11) contain only radiation
resistances of the two loops (20). On the other hand, the term $\omega M_{\rm
I}$ in (11) is the mutual resistance of the two antennas. It is obvious that
in the limit $d\rightarrow 0$, the mutual resistance becomes equal to the
radiation resistance of the loops (for simplicity we consider two identical
loops), and the $PTE$ (11) tends to unity, although the radiation of energy
from both loops is fully accounted for. Physically, this means that at small
transfer distances, selection of the optimal load value given by (9)–(10)
leads to a WPT system with suppressed radiation into far zone.

Perfect radiation suppression takes place only in the limit of zero distance,
because the mutual resistance decays with increasing transfer distance. For
the case of two coupled loops, this decay is illustrated in Fig. 3. We see
that for $d<0.2\lambda$ the mutual resistance exceeds 80% of the radiation
resistance. This means that if the currents in the loops have opposite phases,
the dynamic interaction of the two loops will cancel at least 80% of radiation
into space, enhancing the received power. To show that the mutual coupling
namely decreases the radiation, we derive the ratio between $I_{2}$ and
$I_{1}$ for the case when $R_{\rm L}=R_{\rm L,opt}$ and $X_{\rm L}=X_{\rm
L,opt}$, which reads

$\displaystyle\dfrac{I_{2}}{I_{1}}=\frac{-R_{1}\omega(M_{\rm I}+jM_{\rm
R})}{R_{1}R_{2}+j\omega^{2}M_{\rm R}M_{\rm
I}+\sqrt{(R_{1}R_{2}-\omega^{2}M_{\rm I}^{2})(R_{1}R_{2}+\omega^{2}M_{\rm
R}^{2})}}.$ (21)

It can be observed from (21) that when $R_{1}=R_{2}=\omega M_{\rm I}$, the
ratio between the currents indeed becomes equal to $-1$, which indicates that
the currents have equal amplitudes and opposite phases ($180^{\circ}$).
Obviously, the magnetic dipole radiation of the set of two antennas is
suppressed in this case. Next, we plot the ratio of the currents in the two
loops, $I_{2}/I_{1}$, both the amplitude and phase, see Figs. 4(a) and (b). We
take as an example the case when the distance between the loops is five times
larger than the loop radius, $d=5a$. We see that at the optimal load (dashed
curves) the regime of radiation suppression holds in the low frequency range,
since the currents in the two loops are approximately equal in the amplitude
and opposite in phase.

Thus, such set of two antennas radiates into far zone only via its higher-
order multipoles. An estimation of the ratio between the radiated power from a
single magnetic dipole and the power radiated from a quadrupole formed by two
counter-directed dipoles of the same amplitude (both in the coaxial and
coplanar arrangements) gives

$\frac{P_{q}}{P_{d}}=\frac{1}{48\pi}(kd)^{2},$ (22)

which is approximately $6\cdot 10^{-4}$ for $kd=0.5$. Thus, for the case of
mid-range WPT distances, this higher-order multipole radiation can be
neglected.

Corresponding calculations for non-optimal values of the load resistance
(solid curves in Fig. 4) show that radiation is compensated in a certain
frequency range but not in the low-frequency regime. However, the analysis of
the $PTE$ dependence [Fig. 4(c)] reveals that the efficiency remains
practically unity in a broad frequency range, also at low frequencies even for
non-optimal loads, which is due to the fact that radiation is in any case weak
at low frequencies. The frequency of the sharp drop of efficiency at high
frequencies corresponds approximately to the frequencies above the first null
of the dependence of the mutual resistance on the frequency (Fig. 3). The
value $d/\lambda=0.5$ corresponds to approximately $8\cdot 10^{8}$ Hz for the
chosen example value of $d$. We conclude that if the dissipative losses in the
loop antennas can be neglected, the radiation into far zone is effectively
suppressed due to dynamic antenna interactions up to the transfer distances of
the order of one wavelength. Obviously, there is no need to use any special
means to suppress parasitic radiation losses in this regime.

Figure 5: Mid-range coupling between two lossy loops: a) phase difference
between the loop currents; b) magnitude of the currents ratio $I_{2}/I_{1}$;
c) $PTE$ for different values of the load resistance $R_{\rm L}$. The dashed
curves correspond to the optimal resistance, maximizing $PTE$.

Let us next consider mid-range interactions of loop antennas taking onto
account dissipation in the loops. Figure 5 shows the same plots as Fig. 4, but
accounting for dissipative losses in both loops, as defined by Eq. (19). We
see that in this case selecting the optimal load impedance we realize the
regime of effective suppression of radiation, but only in a certain frequency
range, approximately between $10^{8}$ and $10^{9}$ Hz. In this range, the
currents in the two loops are approximately of the same amplitude and of the
opposite phase. At low frequencies, the phase difference between the currents
is rather close to $90^{\circ}$, for a broad range of the load resistances (in
all cases, the load reactance is the optimal one). This conclusion follows
from (21) by setting the imaginary part of the mutual inductance $M_{\rm I}$
to zero.

To understand these effects, we need to consider the relative values of the
Ohmic resistance and the radiation resistance, because the significance of
radiation loss suppression is determined by the relative level of losses due
to dissipation in the antennas and parasitic radiation into far zone. To
estimate this ratio, we plot in Fig. 6 all relevant resistances as functions
of the frequency for the considered example of two loops.

Figure 6: Comparison of Ohmic resistance (19) and radiation resistances of one
loop antenna (20), together with the mutual resistance between two antennas
$\omega M_{\rm I}$. The loop radius $a=36$ mm, and $d=5a$.

We see that at low frequencies the loss resistance strongly dominates over the
radiation resistance. For this reason, radiation suppression has negligible
effect on performance in the quasi-static regime. The power transfer
efficiency is low because the quasi-static coupling between loops at such
large distance is weak. On the other hand, we observe that exactly in the
frequency range of the radiation loss suppression for each considered value of
the loss resistance, the power transfer efficiency $PTE$ has a strong peak.

For electrically small transfer distances, the imaginary part of the mutual
inductance $M_{\rm I}$ is negligibly small, and the general formula (11) tells
that higher efficiency corresponds to smaller distances $d$ between the
antennas (to larger values of the mutual inductance $M_{\rm R}$). It is the
case of conventional inductive WPT devices. Now we see that at electrically
larger distances between the two antennas there is another possibility to
realize high-efficiency wireless power transfer. This regime takes place when
$M_{\rm I}$ is large and the far-field radiation is suppressed. In this
regime, we do not need high values of the real part of the mutual inductance.
These results open a possibility to realize high-efficiency wireless power
transfer to distances that are large compared to the antenna sizes.

Figure 7: Coaxial arrangement: a) $PTE$ versus $kd$; b) $\kappa_{\rm R}$
versus $kd$; c) $\kappa_{\rm I}$ versus $kd$ for different values $d/a$ from
$5$ to $50$.  Figure 8: Coplanar arrangement: a) $PTE$ versus $kd$; b)
$\kappa_{\rm R}$ versus $kd$; c) $\kappa_{\rm I}$ versus $kd$ for different
values $d/a$ from $5$ to $50$.

###  III.3 Analytical Results for Optimized Mid-range WPT

Next, we study high-frequency mid-range power transfer in more detail,
considering various transfer distances and both coaxial and coplanar
orientations of the loops. In our analysis, we focus on the almost radiation
suppression regime in the mid-range WPT, where the corresponding distance $d$
is not very small compared to the wavelength but large as compared to the loop
radii $a,b$. In this case $M_{\rm I}$ is large in a rather broad range of
frequencies, which helps to achieve high power transfer efficiency.

In order to better clarify the roles of $M_{\rm R}$ and $M_{\rm I}$ in
attaining the maximal $PTE$, we introduce two normalized coupling parameters
$\kappa_{\rm R}=\omega^{2}M_{\rm R}^{2}/R_{1}R_{2}$ and $\kappa_{\rm
I}=\omega^{2}M_{\rm I}^{2}/R_{1}R_{2}$, so that Eq. (11) takes the form

$\displaystyle PTE=1-\dfrac{2}{1+\sqrt{\dfrac{1+\kappa_{\rm R}}{1-\kappa_{\rm
I}}}}.$ (23)

We fix the radii of the loops $a=b=36$ mm and plot their values versus the
normalized distance $kd$ for many values of $d$ from $d=5a$ to $d=50a$ and
compare these coupling parameters with the similar plots of $PTE$. For each
curve, the value of $d$ is fixed, that is, the curves effectively show
dependence on $k$ (the frequency). The three corresponding plots are presented
in Fig. 7 (coaxial arrangement) and Fig. 8 (coplanar arrangement).

We see that at $kd<1$, at all mid-range distances $PTE$ is low because the
real part of the mutual inductance is large only in the near-field zone, when
$d<a$, and also the imaginary part $M_{\rm I}$ is small compared to the Ohmic
resistance. In other words, when $d\gg a$ and $kd<1$, the loops are weakly
coupled, leading to small values of $PTE$. In the conventional quasi-static
regime of WPT, when $kd\ll 1$ and the antenna size $a$ is comparable to the
distance $d$, the mutual reactance $\omega M\approx\omega M_{\rm R}$ is
significantly larger than the Ohmic resistance. Therefore, in this
conventional case of small distances and large loops we can achieve a high
$PTE$ according to (11) and (23). Since the dipole-moment model of loops is
not applicable in the quasi-static case, we have studied the quasi-static
regime numerically and calculated the $PTE$ in both coaxial and coplanar
arrangements. The results are shown as dash-dot lines in Fig. 7(a) and Fig.
8(a), respectively. For very large electromagnetic distances where $kd\gg 1$,
both coupling parameters and, respectively, $PTE$ again becomes low. This is
because the mutual impedance decreases in the far zone, whereas the radiation
resistance of antennas rapidly grows versus frequency, i.e., versus $kd$.
Thus, we can conclude that at the mid-range distances, where $kd\sim 1$, for
any $d\gg a$ there is a range of optimal electrical distances $kd$
corresponding to the maximal total coupling and, therefore, to the maximized
$PTE$.

For the coaxial case, when there is no contribution of far fields to the power
transfer, the decrease of the $PTE$ at high frequencies is more rapid than its
increase at low frequencies. For the coplanar case, the $PTE$ curves are more
symmetric, because at high frequencies not only the radiation loss increases,
but also the far-field coupling becomes stronger.

For the coaxial arrangement, the maxima of $\kappa_{\rm R}$ and $\kappa_{\rm
I}$ as functions of $kd$ occur at different values of $kd$. For example, for
$d=5a$ these maxima take place at $kd\approx 0.3$ and $kd\approx 0.5$,
respectively. For $kd=0.5$, we have from (16) and (20)

$\omega M_{\rm I}={\pi\eta k^{2}a^{4}\over 2d^{2}}\left({\sin kd\over kd}-\cos
kd\right)\approx{\pi\eta k^{4}a^{4}\over 6.15},$ (24)

which is close to ${\pi\eta k^{4}a^{4}\over 6}=R_{\rm r1,r2}$. This shows that
the radiation resistance of each of the two loops is almost compensated by the
mutual resistance, and this case corresponds to the optimal non-radiative
power transfer. Due to some residual radiation and the presence of losses
$PTE|_{kd=0.5}\approx 0.89$, i.e., about 11% of power is dissipated in
resistors $R_{\rm O1,O2}$ and radiated. However, the maximum of $PTE$ (also
about $0.9$) holds not at $kd=0.5$ but at $kd=0.4$. At this point the sum
$\kappa_{\rm R}+\kappa_{\rm I}=\omega^{2}|M|^{2}/R_{1}R_{2}$ is maximal. This
regime can be still called the radiation-suppression regime because the
maximum of the function $\kappa_{\rm I}(kd)$, as we can see in Fig. 7, is wide
and overlaps with the maximum of $\kappa_{\rm R}(kd)$, which is also wide in
accordance to Fig. 8. We note that for our specific example $kd=0.4$ in the
case $d=5a$ corresponds to the frequency of 110 MHz. Below we will see that
this is the optimal WPT frequency for the considered example of two loops.

In the coplanar arrangement, maxima of $\kappa_{\rm R}(kd)$ and $\kappa_{\rm
I}(kd)$ occur at $kd=0.4$ and $kd=0.8$, respectively, whereas $\kappa_{\rm
R}|_{kd=0.4}\approx 190$ and $\kappa_{\rm I}|_{kd=0.8}\approx 0.7$. In this
case the $PTE$ also attains its maximum close to $0.7$ at $kd=0.6$ in the
middle between the maxima of $\kappa_{\rm R}(kd)$ and $\kappa_{\rm I}(kd)$.

In the coaxial case, the far field plays a parasitic role, because the Tx loop
does not radiate into the Rx direction. However, the important coupling
parameter $\kappa_{\rm I}$ attains the maximum at a substantial
electromagnetic distance ($kd>1$) because the expression for $M_{\rm I}$
comprises two terms that compete with one another when $kd$ is not very small
and not very large. At still larger distances the efficiency quickly decays,
since far-field coupling becomes small. For example, the maximal $PTE$ at
$d=50a$ is as small as $0.001$.

In the coplanar mid-range ($1<kd<10$) case, the far field brings a
contribution into both $M_{\rm R}$ and $M_{\rm I}$ but also brings radiation
loss measured by $R_{\rm r1,r2}$. The optimal WPT is achieved for a
substantial electromagnetic distance $kd\approx 5$ where $\kappa_{\rm I}$ is
maximal and $\kappa_{\rm R}$ is close to the middle-range maximum. The far
field plays an advantageous role at this distance, and $PTE$ in this optimal
case is equal to $0.007$. In other words, due to the contribution of radiation
the maximal $PTE$ in the mid-range region turns out to be 7 times higher than
that we have obtained for the same large physical distance in the coaxial
arrangement. In terms of the electromagnetic distance $kd$, the advantage
granted by the far field is even higher because the optimal $kd$ for the
coplanar case ($kd=5$) is larger than that for the coaxial case ($kd=4$).

To conclude the discussion of Figs. 7 and 8, it is worth noting that in the
far zone ($kd>\pi$) there are periodic maxima and minima of $\kappa_{\rm
R,I}(kd)$, whereas maxima of $\kappa_{\rm R}(kd)$ coincide with the minima of
$\kappa_{\rm I}(kd)$ and vice versa (that results in a smooth decrease of
$PTE$ with the normalized distance $kd$). Two adjacent maxima (and minima) of
$\kappa_{\rm R}$ (and $\kappa_{\rm I}$) are distanced by $2\pi$ because the
phase shift between the Tx excitation and the field at the Rx antenna position
oscillates with the period $\lambda$. At such large distances, we approach the
far-field WPT regime, where one needs to use directive antennas instead of
small dipole antennas.

Figure 9: $PTE$ in a broad frequency range for different normalized transfer
distances $d/a$.
a) coaxial arrangement; b) coplanar arrangement.  Figure 10: $PTE$ in a broad
frequency range for different antenna sizes. The distance between the two
antennas $d=0.5$ m. a) Coaxial arrangement; b) coplanar arrangement.

Figure 9(a) shows ${PTE}$ for the coaxial arrangement at frequencies from 500
kHz to 5 GHz with the normalized distance $d/a$ changing from 5 ($d=18$ cm) to
50 ($d=1.8$ m). These results allow us to find the optimal operation frequency
for practical long-distance (transfer distance much larger than the antenna
sizes) WPT devices. Therefore, we performed a numerical validation of the
model using a full-wave electromagnetic solver CST Studio together with the
ADS simulation tool that modeled lossless matching circuits for the receiving
loop at any frequency.

In Fig. 9(a) besides of the analytical results (solid curves) we show the
simulated $PTE$ for $d=5a$ (dash-dotted curve). Qualitatively, this curve
confirms the analytical model. Quantitative differences are as follows: the
optimal operation frequency 160 MHz instead of 110 MHz, and $PTE_{\rm
max}=0.7$ instead of $0.9$. The reasons of the disagreement are clear. In the
analytical model, we replace the Tx loop by a magnetic dipole, whereas in the
simulations it is a split loop fed by a generator connected to a finite gap of
width $g$. Therefore, besides of a magnetic current mode, there is an electric
dipole mode induced in the Tx loop. The electric coupling of two loops results
in the electric dipole mode also in the Rx loop. Parasitic impact of electric
coupling leads to a shift of the optimal frequency and makes the cancellation
of radiation less effective. At a first glance, this parasitic effect can be
made negligible decreasing $g$, however, simulations show that $g\ll 1$ mm
correspond to a so large split capacitance that it completely shunts the loop
preventing its excitation.

In principle, it is possible to analytically optimize the loop even if the
split is large. For such optimization it is necessary to take into account the
electric coupling of loops together with the magnetic one. However, it is not
the purpose of this study. We aim to developing a theory clearly explaining
the role of near and far fields in mid-range WPT, which calls for a simplest
possible analytical model to be used. Therefore, in simulations we simply
numerically optimized the gap width $g$. The best correspondence of
simulations and analytical theory holds when $g=1$ mm and when the wire of the
loop is slightly enlarged in the near vicinity of the split edges. Note that
the problem of a finite gap exists only for the Tx loop and does not arise for
the Rx loop. The electromagnetic solver allows us to load it by a lumped
resistance whose value is determined by formula (10).

The corresponding numerical curve for the coaxial arrangement and $d=5a$ is
depicted in Fig. 9(a). In Fig. 9(b), we present a similar set of theoretical
curves and the validation of the upper curve ($d=5a$) for the coplanar
arrangement.

Besides of comparison of the analytical results with full-wave simulations,
Figs. 9(a) and (b) lead to the following observations. First, we see that the
optimal frequency of 110 MHz is the same for the coaxial arrangement at any
considered $d/a$. The coincidence of these frequencies for the case $d=5a$ and
$d=50a$ has been already discussed above. Now, we observe it in the whole
range $d/a$. Second, for the coplanar arrangement, as we can see in Fig. 9(b),
the absolute maximum of $PTE$ shifts versus $d/a$. In accordance to the
theory, it shifts from about 80 MHz ($d=5a$, $PTE_{\rm max}=0.5$) to nearly
180 MHz ($d=50a$, $PTE_{\rm max}=0.007$). Third, for both arrangements the
optimal frequency range is about 100–200 MHz, that is significantly lower than
the self-resonance frequency of a wire loop with the radius $a=36$ mm. This
frequency range results from the formulas of the analytical model, confirmed
by numerical simulations, and it is general: if a WPT device is based on two
MDs, the maximal efficiency is achieved at a frequency much lower than the
self-resonance of the individual antenna. However, this frequency is higher
than those corresponding to the quasi-static regime – the regime in which
$\omega M_{\rm I}\ll R_{\rm r1,r2}$.

In [26], numerical calculations similar to those presented in Fig. 9 were done
for a different type of WPT. In [26], a spherical muscle phantom with a very
small receiving antenna in the center was excited by a uniformly distributed
source over the sphere surface. That ideal Tx was creating a fixed incident
electromagnetic power at all frequencies (reflections from the sphere boundary
were assumed to be eliminated by an ideal antirefelecting coating). The Rx
antenna was assumed to be perfectly matched with the load and $R_{\rm L}$ was
optimized for the maximal power transfer at every frequency. Full-wave
simulations were performed in [26] with the aim to find the optimal frequency
of WPT for both electric and magnetic Rx antennas. Interestingly, the set of
curves plotted in Fig. 3 of [26] for different $d/a$ qualitatively resembles
our plots in both Figs. 9(a) and (b). The main qualitative difference is a
negative shift of the optimal frequency versus $d$ in [26]. This shift
definitely results from electromagnetic energy dissipation in the medium. In
the muscle tissue, the near-field coupling exponentially decays when $d$
increases, and this decay factor is proportional to the frequency. Meanwhile,
in the considered case two MDs are located in free space, and their near field
at a given distance weakly depends on the frequency. However, the main
qualitative result of the study [26] is the optimal frequency range for WPT
located below the self-resonance of Rx but higher than the low-frequency
region in which the quasi-static approximation for the whole system is
applicable. This result fits our one. In work [27], the coupling of two MD-
based antennas in a lossy medium is considered with the purpose to optimize
the antenna sizes for the given operation frequency. Again, the operation
frequency turns out to be located between the low-frequency region and the
band of the antenna self-resonance. And the same refers to the range of
optimal operation frequencies when the Tx and Rx antennas are located in free
space.

In Figs. 10(a) and (b), we present $PTE$ versus frequency for both cases of
coaxial and coplanar arrangements for a comparatively large fixed transfer
distance $d=0.5$ m and different loop radii $a$. We see that the optimal
frequency decreases versus $a$. This decrease, evidently, results from the
increase in the coupling for given frequency when the loop gets larger. The
top theoretical curves in Fig. 10(a) and (b), i.e., the case $a=4$ cm, is
validated by numerical simulations. Again, we note a qualitative agreement,
but the simulated $PTE_{\rm max}$ is triply lower than the theoretical
prediction for the coaxial arrangement and twice lower for the coplanar one.
In other words, the numerical disagreement between the theory and simulations
in the case $d=12.5a$ is much larger than that in the case $d=5a$, when the
radiation suppression regime is achieved. This result confirms our insight
that the main inaccuracy of the magnetic dipole model is not in the
approximate formulas for the mutual inductance, but mostly in negligence of
the electric dipole mode. For the loop $a=4$ cm this mode is more noticeable
than for the loop $a=3.6$ cm operating at the same frequency.

To conclude this section, let us formulate the most important theoretical
observations:

  * •

In the mid-range coupling regime between two electrically small loops there is
a possibility to reach reasonably high power transfer efficiency due to
suppression of radiation into far zone (compensation of the radiation
resistances of both loops);

  * •

This regime is realized by tuning the load impedance to the optimal value for
the maximal transfer efficiency;

  * •

The optimal frequency for WPT between two small loops to distances much larger
than the loop sizes in both cases of the arrangement of Tx and Rx loops lies
above the low frequency region (in which the coupling of two antennas is
quasi-static and strong) but below the resonance band of the individual loop
antennas;

  * •

To realize effective wireless power delivery at the maximally possible
distance between small antennas, the mid-range regime with the coplanar
arrangement of loops is suitable. In this case the far-field coupling grants a
significant improvement of $PTE$ compared to the case when WPT holds solely
due to near fields, which makes the coplanar arrangement of loops advantageous
compared to the coaxial one.

Figure 11: Experimental setup and results: a) Two loop antennas in coaxial and
coplanar arrangements; b) Phase difference of the loop currents; c) Magnitude
ratio $I_{2}/I_{1}$; d) $\kappa_{\rm R}$; e) $\kappa_{\rm I}$; f) $PTE$
comparison between calculated, simulated, and measured results for coaxial and
coplanar antenna arrangements.

##  IV Experimental Verification

In this section, we present an experimental validation of the theoretical
results. The experimental setup consists of two loop antennas designed for the
frequency range of interest. In order to minimize the effect of the electric-
dipole mode, we use shielded loop antennas made from coaxial cables. The
center conductor of the cables forms transmitting and receiving loops, and the
cable screen serves as a shield, minimizing the electric-dipole mode current.
The two terminals of the inner copper wire of the loop are welded to the
center pins and the ground parts of SMA connectors. The shield is cut close to
the connection points. Both Tx and Rx loops have the same radius of 3.6 cm and
are placed at a center-to-center distance of 18 cm ($d=5a$) for coaxial and
coplanar arrangements, as shown in Fig. 11(a). For each arrangement, the
antennas are connected to two ports of the ZND Vector Network Analyzer (VNA)
to measure $Z$-parameters of the system in a wide frequency range, which
allows evaluating the optimal frequency and $PTE$ for the optimal load. The
measured $Z$-parameters are used for analytical calculations of the optimal
load impedance and power transfer efficiency using the formulas presented
above. The experimental results are shown in Fig. 11.

We see that the phase difference and magnitude ratio $I_{2}/I_{1}$ of the
currents in the transmitting and receiving loops show a qualitative agreement
with numerical calculations (see blue dashed lines in Fig. 5). At the optimal
frequency of 59.38 MHz for the coaxial and coplanar cases, the currents in the
two loops have approximately opposite phases
($-177.451^{\circ}$/$-175.341^{\circ}$) and similar amplitudes (0.67/0.607),
as is seen in Figs. 11(b)-(c). Here, the power transfer efficiency ($PTE$) has
a peak (0.35/0.353) corresponding to the peak of $\kappa_{\rm I}$, although
$\kappa_{\rm R}$ is small, as is seen in Figs. 11(d)-(f).

Let us also compare $PTE$ in the dynamic regime with the system working in the
conventional quasi-static coupling regime. In this case, the loop coupling is
defined by the mutual inductance, conventionally determined by the geometry
and positions of the loops. To do that, we measured the mutual inductance and
the $Z$-parameters of the two coupled loops at distance $d=5a$ at the
frequency of 4 MHz. The measured mutual inductance is approximately 0.56 nH
for the coaxial case (that is around 0.508 nH in numerical calculations) and
0.384 nH for the coplanar case (0.313 nH in numerical calculations).
Substituting the measured mutual inductances and measured resistances
$R_{1,2}={\rm Re}(Z_{11,22})$ into formula (11), we find an estimation for the
maximal achievable $PTE$ at this distance $PTE\approx 0.0065$ for the coaxial
arrangement and $PTE\approx 0.0035$ for the coplanar arrangement. These values
are dramatically smaller than those corresponding to the theoretical,
simulated, and experimental curves shown in Fig. 11(f), confirming a
possibility to significantly enhance power transfer efficiency using properly
tuned dynamic coupling.

Overall, we observe qualitative agreement between the theoretical and
experimental results. Quantitative differences are a small shift in the
optimal frequency range and an order of smaller magnitude of the $PTE$ in the
experiment compared to numerical calculations. This big difference can be
explained by losses in the dielectric support and in the galvanic contacts.

##  V Conclusion

In this paper, we have developed the dynamic theory of wireless power transfer
between two antennas, specializing it for identical loop antennas. The theory
covers a very broad range of frequencies and refers to both short-range and
mid-range distances between the loops. Assuming that the Rx antenna reactance
is properly matched at every possible operational frequency and its useful
load resistance is similarly optimized, we aimed to find the optimal frequency
range for a given loop size and track how and why the power transfer
efficiency changes with the distance. We studied two most important mutual
arrangements of the loops: the coaxial and the coplanar ones. In the first
arrangement, the coupling between the Tx and Rx is solely due to the near
fields, in the second case the near-field coupling is low and the radiative,
far-zone coupling exists.

We have shown that when the distance $d$ is very small as compared to the
antenna size, and dissipation is negligible, the power transfer efficiency can
approach unity at whatever (enough low) operation frequency, and the only
issue is proper matching of antennas. However, if $d$ is larger than the loop
diameter $2a$, finding the optimal operation frequency becomes crucial for WPT
system design. For the coaxial arrangement, we have found the optimal
frequency which turned out to be unique for a broad range of distances
($5a\leq d\leq 50a$). For the coplanar arrangement, the optimal frequency
depends on the distance, but this dependence is quite weak. In both cases, the
optimal frequency lies below the self-resonance of the loop but above the band
in which the quasi-static model is applicable for the WPT system. In this
frequency range the interaction between the loops is dynamic, measured with
both inductive and resistive parts of the mutual impedance.

The maximum value of the $PTE$ turns out to be quite high in spite of the
geometrically and electromagnetically substantial distance between the two
antennas. In this optimal regime, the mutual coupling of two loops suppresses
the radiation from the system, and almost all the power, which is not lost due
to parasitic dissipation in the Tx antenna, is transferred to the Rx. We have
thoroughly studied the prerequisites and peculiarities of this regime, that
holds for both coaxial and coplanar arrangements, considering contributions
and advantages of the near-field and far-field couplings for mid-range
distances. The developed analytical model has been verified by full-wave
simulations and partially validated experimentally. We believe that the regime
of the radiation suppression is very important not only because it grants high
$PTE$ for substantial transfer distances but also because it prevents
parasitic heating of the ambient around the WPT system.

## Acknowledgements

This work was partially supported by the Academy of Finland, project 338786
and the Academy of Finland postdoctoral researcher grant 333479.

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