Zoran Blažević and Maja Škiljo
IMAGE LICENSED BY INGRAM PUBLISHING
Resonant
Near-Field
Power Transfer
Revisiting the frequency-splitting phenomenon
using the spherical mode theory antenna model.
n this article, the resonant coupling phenomenon is revisited using the antenna theory. The methods for extracting
the maximum power by the load (i.e., the frequency tracking, conjugate matching, and critical coupling adjustment)
are addressed. To obtain the best performances in terms of the
highest power and the greatest transmission range, the system’s
tuning needs to be very sharp regardless of the method applied.
Conjugate matching is shown to be the best method for reaching the highest possible antenna separations and the maximum
efficiency, but the critical coupling adjustment turns out to be
more practical in the sense of the bandwidth when the antenna
separations are not too large.
I
BACKGROUND
State-of-the-art technology is characterized by myriad smalland middle-size gadgets and electrocommunication devices,
such as sensors, smartphones and cell phones, and laptops and
tablets, all equipped with some rechargeable power source,
such as a battery or capacitor. Additionally, emerging new technologies, such as electric cars and the Internet of Things (IoT),
require charging methods that are much more convenient than
the usage of many cords that lead to sockets. For this reason,
wireless power transfer (WPT) becomes a promising solution to
deal with the challenge. As the aforementioned devices require
vastly different amounts of energy, ranging from very small (as
IoT sensors do) to quite large (as electric vehicles do), different
WPT principles can be adopted for this purpose.
Digital Object Identifier 10.1109/MAP.2019.2920102
Date of publication: 27 June 2019
IEEE ANTENNAS & PROPAGATION MAGAZINE
AUGUST 2019
For instance, in ultrahigh-frequency (UHF) radio-frequency (RF) identification systems, low-power passive tags
are supplied by radio waves emanating from the reader antenna in the far field. Another example of the same method is a
low-power IoT sensor charged by Wi-Fi router RF power. For
the purposes of larger PTs, such as for charging smartphones,
laptops, or electric cars, the best approach is the application
of electrically small antennas (ESAs), inductively or resonantly coupled in the near field. The definition of ESA was
first proposed by Wheeler [1] as an antenna whose maximum
dimension is less than a radianlength (m/2r), m being the
wavelength. Chu [2] gave an equivalent definition of ESA as
the antenna enclosed inside a Chu sphere (i.e., the minimum
sphere that encloses whole antenna, with a diameter less than
a radianlength).
The first exploration and application of ESAs for the purpose of WPT was done more than a century ago within the
scope of radio research by Nikola Tesla in Colorado Springs,
Colorado [3]. More details about the WPT development
through history can be found in [4] and [5]. One of the successful more-recent experiments on Tesla’s WPT technology
was done by Soljačić and his team from the Massachusetts
Institute of Technology [6], who proposed an HF version
of Tesla’s resonant four-coil system and analyzed the resonant magnetic coupling by coupled-mode theory (CMT) [7].
Another HF version of the magnetically coupled four-coil system is investigated in [8] using the equivalent-circuit scheme.
The authors provide the definitions of coupling zones based
on the degree of coupling between the resonators, analyzing the frequency-splitting phenomenon while proposing the
1045-9243/19©2019IEEE
39
method of critical-coupling operating-regime system setup
and control to obtain the highest WPT performances.
On the other hand, the problem of the resonant WPT at
midranges is considered from the standpoint of the antenna theory and Z-matrix in [9] and [10]. There, the conjugate-matched
near-field WPT system is analyzed, applying the approximation
of the spherical mode theory antenna model (SMT-AM) that
deduces the PT to the interaction of the lowest-order transverse
electric (TE) and transverse magnetic (TM) spherical modes.
In the literature, those two are often referred to as even mode
and odd mode. However, because they switch the position on
the frequency axis depending on the propagation geometry, this
nomenclature is omitted in this article. The approach is applied
to the design of very HF and UHF WPT systems with multiplearm folded ESAs in [11] and [12]. However, none of these papers
dealing with the AMs address the frequency-splitting problem.
However, the SMT-AM model is considered for the prediction of
split resonant frequencies in [13].
The former (CMT) approach assumes that the coupling
between two resonators is a consequence of interlapping and
interleaving of their reactive fields, without considering a
small radiation field contribution. As a result, WPT systems
are, on many occasions, operated at the antenna (first) selfresonant frequency [8], [14], [15], applying techniques that
avoid frequency splitting. However, it is shown in the latter
(SMT-AM) approach that, due to its interaction with the reactive coupling component, the near-field radiative component
can play a significant role in conjugate-matched WPT systems
at midranges.
In this article, the frequency-splitting phenomenon is investigated by adopting the SMT-AM model from [9]. We propose a
definition of the coupling zones (an alternative to the one given
z
θ0
Tx
y
G ~
θ1
z′
d
φ0
x
Rx
x′
φ1
L
FIGURE 1. The radio propagation geometry.
ZA – ZM
RG = RL
l1
U1
ZA – ZM
l2
ZM = Re[ZA]T U2
y′
in [8]) in the sense of the antenna theory approach, and we
reexamine both conjugate-matched and critically coupled WPT
systems. For that purpose, the validity of the applied SMT-AM
approximation is tested by comparison with the results of a fullwave model.
WPT MODELING
Let us consider the PT on some angular frequency ~ = 2rf,
f = c/m (c being the velocity of the electromagnetic waves in
the considered propagation medium) between two equal arbitrarily positioned ESAs separated by a center-to-center distance
d, as shown in Figure 1. The ESAs’ free-space input impedance
is Z A = R A + jX A, the frequency of the first self-resonance is
~ 0, and the radiation efficiency is h RAD = R RAD /R A, where
R RAD is the radiation resistance. The case of different antennas
can be approached in a similar manner to that shown here.
When the two Chu spheres satisfying the condition
b a 1 0.5 (where b = 2r/m and a is the radius of the minimum
sphere that encloses whole antenna) did not intersect (d 2 2a),
the problem was analyzed using SMT-AM in [9]. It is shown
that, by supposing a uniform current phase distribution across
the antenna, the coupling between two reciprocal ESAs can be
approached by applying the addition and translation theorem on
the TE10 and TM10 modes (the lowest-order spherical modes).
Those two are represented by their radiation resistances R TE
and R TM, respectively, R RAD = R TE + R TM, and the mode ratio
is a = R TM /R TE . The assumption that there are no higher-mode
interactions is shown not to be valid when two ESAs are very
close to each other, but then the WPT efficiency is high even
without proper receiver matching. The equivalent two-port network of two coupled ESAs is drawn accordingly, as in Figure 2.
For simplicity, in the antenna reactance X A, we include an ideal
(lossless) tuning reactance.
ESAs come close to the concept of the minimum-scattering
antenna (MSA) [16] (sometimes referred to as the equal-scattering antenna [17]). Such antennas have the property that, for
the specific reactive termination at the antenna port, they do
not scatter at all and thus become invisible for the transmitter.
Whether or not the termination is open circuit as in the case
of directly fed short dipoles, such an ESA is referred to as the
canonical MSA. Hence, the two-antenna WPT system can be
represented by a Z-matrix (i.e., by the scheme in Figure 2),
and the ESAs’ free-space impedances can be applied [18]. In
the case of four-coil systems, the inductively fed ESA can be
approximated by an MSA that vanishes if terminated by a short
circuit. In that case, a Y-matrix can be applied, resulting in dual
relations to that of the Z-matrix with impedances exchanged
with corresponding admittances.
The power transmission coefficient S 21 is given by
S 21 = 2 U 2
UG
RL
UG
FIGURE 2. An equivalent circuit for WPT between two
equal ESAs.
40
RG
,
RL
(1)
where U G and U 2 are the generator and the output voltage,
respectively. R L is load terminating the receiver port, and
RG = RL.
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IEEE ANTENNAS & PROPAGATION MAGAZINE
The reflection coefficient at the transmitter input port is
given as
S 11 =
Z in - R L
,
Z in + R L
(2)
where Z in is the input impedance.
Applying (1) to the scheme in Figure 2, one obtains
S 21 =
2Z M R L
2 .
(Z A + R L)2 - Z M
3
T = - 4 6(R A + R L)2 + Re (Z 2M)@ - 27 Im 2 (Z 2M) (R A + R L)2 . (6)
(3)
Z M is the mutual impedance between the antennas that,
following the SMT-AM and applying the MSA concept, can be
calculated as [9], [18]
Z M = R A T,
to the frequency tracking. Strictly speaking, there is no possibility that the passive WPT system can operate in resonance
(Im (Z in) = 0) while the antennas are in self-resonance (X A = 0)
unless the antenna separation is infinite (far field). The number
of antenna reactance values that bring the system in resonance
(i.e., the number of the system resonant frequencies) is determined by the discriminant T:
(4)
where T is the transmission coefficient of the spherical modes
and it is a function of the electrical distance d/m between the
antennas, their radiation efficiency, and the space orientation of
the receiver relative to the transmitter. The complete relations
for the transmission coefficient of the lowest-order spherical
modes are given in “The Lowest-Order Spherical Modes Transmission Coefficient T.”
MAXIMUM WPT PERFORMANCES
To obtain the maximum WPT performances for a given scenario
(i.e., to ensure the absorption of the maximum power by the
load), ;S 21 ; must be maximized. With the load resistance fixed,
the calculation of the maximum |S 21 | by (3) with respect to the
antenna reactance X A leads to a determinantal equation in the
form of a depressed cubic equation for the optimum antenna
reactance given by
2
2
X 3A + 6(R A + R L)2 + Re (Z M
)@ X A - Im (Z M
) (R A + R L) = 0. (5)
It can be shown that the whole system is tuned (i.e., brought
to resonance at the selected frequency) by adjusting the antenna
reactance to one of the real solutions of (5). This is analogous
When T 1 0, there is one single real solution to (5), and the
resonant WPT system operates in an undercoupled regime. On
the other hand, when R L is such that T 2 0, there are three
real solutions. Then, the frequency splitting occurs, which
characterizes the overcoupled zone of the receiver. In the overcoupled zone, there is one local minimum near X A = 0 (i.e., not
far from ~ = ~ 0). Also, there are two maximums—the global
one and the local one—each representing one PT mode. They
are positioned on the left and right of the frequency axis relative
to ~ = ~ 0 .
The boundary between the two regions of the receiver
(T = 0) is called the critical coupling, and it is characterized by
;S 21(~); with the global maximum and with one double extremum in the form of inflection point. At this point, the frequency
splitting is still suppressed. When the system is tuned to the frecrit
quency of the global maximum, then ;S 21 ; = ;S 21
;, and the fixed
load absorbs maximum power.
The solutions to (5) are derived in “Tuning Reactances as
Solutions to the Depressed Cubic Equation.” However, as R L
can also be adjustable, there is an infinite number of (R L, X L)
combinations that bring the system in resonance and provide
peak WPT performances. Therefore, in the text that follows,
two approaches for selecting the best possible (R L, X L) pairs
that provide the maximum power are addressed.
CONJUGATE MATCHING
OPT
The best combination (R OPT
L , X L ) of the antenna reactance
and the load resistance among all of the possible (R L, X L) pairs
is the one that provides the maximum PT efficiency (PTE).
PTE is the ratio of the power absorbed by the load at the
THE LOWEST-ORDER SPHERICAL MODES TRANSMISSION COEFFICIENT T
The transmission coefficient T of the lowest-order spherical modes
between two equal minimum-scattering antennas (as in Figure 1)
is derived in [9] as
T = h RAD c Al10,10 +
a
1 + a2
Bl10, 10 m .
(S1)
Al10, 10 and Bl10, 10 are translated and rotated spherical-mode
coefficients between the lowest-order spherical transverse electric
or transverse magnetic modes given by
A10, 10
A10, 10e
A10, 10o
Al10, 10
;
E = cos i 1;
E + sin i 1' cos z 1;
E + sin z 1;
E1,
B 10, 10
B 10, 10e
B 10, 10o
Bl10, 10
(S2)
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where
A 10, 10 = P0 (cos i 0) h (02) ( bd ) + P 20 (cos i 0) h 2(2) ( bd ),
(S3)
B 10, 10 = 0,
(S4)
cos
(2)
A 11, 10eo = - 1 P 21 (cos i 0)
z 0 h 2 ( bd ),
2
sin
(S5)
sin
(2)
B 11, 10eo = ! 3 P 11(cos i 0)
z 0 h 1 ( bd ).
2
cos
(S6)
In (S3)–(S6) h (n2) are n-order spherical Hankel functions of the
second kind, and P nm are the spherical Legendre polynomials.
41
TUNING REACTANCES AS SOLUTIONS TO THE DEPRESSED CUBIC EQUATION
The determinantal (5) in the form X 3A + pX A + q = 0, where
p = (R A + R L)2 + Re (Z M2 ) and q = -Im (Z 2M)(R A + R L) can be solved by
deducing it to the quadratic equation, applying the substitution
X A = x - (p/3x) . By solving the obtained quadratic equation
in x followed by calculating X A(x) = X extr
A , the first solution is
obtained as
3
X extr
A =
q
+
2
q
- T +3 - 108
2
- T ,
108
(19)
receiver port and the power that enters the system through the
transmitter antenna port. It can be written using (1) and (2) as
PTE =
|S 21 |2
.
1 - |S 11 |2
(7)
The optimum antenna reactance (tuned antenna) and
the optimum resistance for the maximum PT between two
identical antennas separated by a distance can be derived by
2PTE/2X A = 0 as
X OPT
A
Im (Z 2M)
=
.
2R A
(8)
The optimum resistance can be found by inserting (8) into
(7), and then 2PTE/ 2R L |X A = XOPT
= 0. It is given by
A
2
2
R OPT
= Re 6(Z OPT
L
A ) @ - Re (Z M) ,
(9)
where Z OPT
= R A + jX OPT
A
A .
When the system is conjugate matched at the selected frequency, S11 = 0. And, according to (7), the maximum PTE is
equal to the maximum |S 21 |2. Also, the system radiation loss is
then at the minimum, as is the influence of the WPT system on
living beings in its surroundings [19].
Inserting (9) into the Vieta’s formulas for (5), the other two
solutions for the extremum |S 21 | in the case of R L = R OPT
are
L
derived as
X EXT
=
A
2
)
-Im (Z M
)1 !
4R A
1 - 32R 3A
R A + R OPT
L
3,
2
Im 2 (Z M
)
(10)
where the negative sign in front of the square root in (10) corresponds to the local minimum, whereas the positive sign corresponds to the local maximum of |S 21 |. For the load resistance
determined by (9), the frequency of transmission can be set to each
of these points (the maximum PTE, the local maximum |S 21 |, or
the local minimum near the antenna resonance) by setting the
antenna reactance at values given by (8) and (10), respectively.
CRITICAL COUPLING
The critical coupling method of adjustment must provide the
crit
critical mutual impedance Z M = Z M
for the maximum |S 21 |, at
which frequency splitting ceases to exist. Considering (5) and (6)
42
where T is given by (6).
The other two solutions are obtained by using Vieta’s formulas,
or by multiplying each term on the right side of (19) with a
different complex solution to 3 -1 , one with - (1/2)(1 - j 3 ) and
Y 0, depending
the other with - (1/2)(1 + j 3 ). As in our case q =
on the sign of T, the depressed cubic (5) can have either three
single real solutions (T 2 0), one single and one double real
solution (T = 0), or else one real and two conjugate complex
solutions (T 1 0).
at a given antenna separation, the adjustment is accomplished
by satisfying the condition T = 0 for one single real solution
of (5) representing the maximum |S 21 | settled at the operating frequency: first by the proper choice of the load resistance
R L = R crit
L that fulfills it and then by selecting the tuning reactance X A = X crit
A that emerges as one single solution of (5). After
some mathematical manipulations, one obtains
2
3
3
R crit
Re 2 (Z M) @ - R A,
L = 6 Im (Z M) -
(11)
X crit
A
(12)
3
=
3
4 Im (Z 2M) (R A
+
R crit
L ).
Additionally, there also exists one double real solution of (5)
representing the inflection point of the |S 21 | characteristics, and
crit
the adjustment of the antenna reactance to X inf
A = -(1/2) X A
settles the operating frequency at the inflection point.
The conditions for critical coupling can be established efficiently in the four-coil systems by controlling the couplings
between the driving loop and the antenna body on both the
transmitter and the receiver side, as shown in [8], followed by
a fine tuning with the aid of (12). As a matter of fact, the analysis
of the four-coil system would lead to a dual equation for critical susceptance Bcrit [i.e., with Z M exchanged with the mutual
admittance YM in (12), and with R A and R crit
L exchanged with
conductivities G A = Re (1/ Z A) and G Lcrit, respectively.]
Obviously, a theoretical possibility for the critical coupling exists within the near-field zone determined by
R crit
L $ 0. Following (4), this range can be expressed by
3
Im 2 ^T h - 3 Re 2 (T) $ 1. Note that the maximum electrical distance limit of the critical coupling method depends
on the antennas’ radiation efficiency, and it varies with the
receiver position and orientation relative to the transmitter.
Inside this area, any terminating resistance that falls below
the value of the critical resistance causes the overcoupling of
the resonant WPT system. One may also note that, because
Y ~ 0, the coupling zone definition proposed
of tuning at ~ =
herein diverges from those given in [14] and [15].
Furthermore, by inserting (11), (12), and (4) into (3), the
critically coupled |S 21 | can be expressed in the form |S 21 |crit =
1/h RAD {[Im 2/3 (T) - Re 2/3 ^T h]3/2 - 1} G (bd), where G is not
a function of the radiation efficiency. The critical transmission
efficiency at some near-field distance from the transmitter
is inversely proportional to -h RAD . The radiation efficiency
of the knee point 2|S 21 |crit / 2h RAD = 1 rises with the antenna
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separation and frequency as h knee
RAD = G (bd). Thus, to reach
high critical coupling performances (above the knee value) at a
given electrical distance, the WPT antenna design should aim
for the radiation efficiency h RAD 2 h knee
RAD . For example, in the
case of the antenna separation of 0.1 m, this value for the coaxially arranged nonrotated antennas is h knee
RAD = 50%. However,
for a given radius of the Chu sphere a 1 m /4r, as R TM + f 2,
R TE + f 4 and the antenna loss resistance R LOSS + f , the
increment of the radiation efficiency is directly opposed to lowering the frequency of transmission. The necessary tradeoff is
investigated in [20], whereas some other aspects of WPT antenna characterization regarding matching network are analyzed in
[21] and [22].
very low at 0.05 m). Moreover, it is found that this discrepancy
is due to the error in the imaginary part of the mutual impedance exclusively. The calculation of the real part gives accurate results regardless of the size of the gap (d 2 2a) between
the loops.
From the results in Figure 4, although the real component
Re(Z M) has a small influence at low antenna separations, it has
a crucial role at midranges. When a precise tuning is executed,
the critical |S crit
21 | decays much slower with distance until the
knee point than in the case of ~ = ~ 0 (X A = 0). Even at the
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S21
NUMERICAL EXAMPLE AND DISCUSSION
0.2
Critical Coupling Versus Conjugate Matching
1
Coaxial PEC
0.9 Loops D = 0.1 λ
0.8 at 13.56 MHz
0.7
S21
Take, for example, the scenario where a conjugate-matched
mobile receiver is entering the overcritical zone, determined by the condition that the term under the square
root in (10) is lower than zero, which is analogous to T 1 0
(R L = R OPT
1 R crit
L
L ). Let the antennas be equal perfectly conducting (PEC) loops of 20-cm diameter, each connected serially to the capacitor C = 206.6 pF that settles the antennas’
self-resonance near an industry, science, medicine frequency
of 13.56 MHz. The operating frequency is selected to be exactly
13.56 MHz. The magnitude of S 21 [calculated by SMT-AM
using (3), (4) and (13)] for the case of the nonrotated loops in
a coaxial position is given in Figure 3. These results are compared to the ones obtained by a full-wave model using FEKO,
where the antennas are concluded with the Linville impedance
Z Linville in the case of the conjugate matching.
crit
When the receiver arrives on the boundary Z M = Z M
OPT
crit
(T = 0), then R L = R L . Besides the existing maximum
PTE point on the operating frequency f = 13.56 MHz tuned
EXT
= X crit
by X A = X OPT
=
A
A , the inflection point X A = X A
OPT
inf
-(1/2) X A = X A appears at the opposite side on the frequency
axis relative to ~ = ~ 0 . This case is illustrated in Figure 3(a).
When the receiver finds itself inside the overcoupled zone, the
WPT system can be maintained at optimum performance by
continuing to adjust X A and R L to the conjugate-matched condition with (8) and (9). Alternatively, it can be turned back to the
critical conditions by adjusting R L and X A with (11) and (12).
This is illustrated in Figure 3(b). In the former case, the frequency splitting exists while the peaks of the |S 21 | are sharp. The latter choice ensures a larger fractional bandwidth and suppressed
frequency splitting but at the expense of a somewhat smaller
PTE than in the former case. The maximum WPT performances obtained by the two methods are compared in Figure 4.
While deriving the results, we separately tested the discrepancies of mutual impedance produced by neglecting highermodes interactions in the SMT-AM approximation by the
full-wave model. Note that the agreement between the SMTAM approximative model and the results of the full-wave model
in Figures 3 and 4 are quite satisfactory. The conclusion from [9]
about the rise of error when the antenna separation becomes
less than 0.1 m is confirmed here as well. However, unless the
separations are very small, this error is acceptable (e.g., it is still
Conjugate Matching at Critical Distance
1
0.9 Coaxial PEC Loops
D = 0.14 λ at 13.56 MHz
0.8
0.7
SMT-AM
0.6
ω = ω0
f = 13.56 MHz
0.5
FEKO
0.4
0.3
0.2
0.1
0
–0.6 –0.5 –0.4 –0.3 –0.2 –0.1
0
0.1
Frequency Offset (kHz)
(a)
0.6
0.5
0.4
0.3
0.2
0.1
0
–0.6 –0.5 –0.4 –0.3 –0.2 –0.1
0
Frequency Offset (kHz)
0.1
0.2
SMT-AM, Conjugate Matching at 13.56 MHz
ω = ω0, Conjugate Matching
f = 13.56 MHz, Conjugate Matching
FEKO, Conjugate Matching at 13.56 MHz
SMT-AM, Critical Coupling at 13.56 MHz
ω = ω0, Critical Coupling
f = 13.56 MHz, Critical Coupling
FEKO, Critical Coupling
(b)
FIGURE 3. The transmission coefficient ; S 21 ; versus frequency
f for WPT between PEC loops separated by d and conjugate
matched for f = 13.56 MHz. (a) The receiver is very near the
critical coupling border, and the differences between the
two characteristics are within the thickness of the curve, and
(b) the conjugate-matched WPT system is overcoupled.
43
separations larger than the knee one, the tuned system reaches
higher values of |S crit
21 | than the detuned one as the knee of the
former is settled at a significantly larger antenna separation than
the one of the latter.
Although, according to (5) and (6), the real component is not
the cause of the frequency splitting [it is caused solely by the
existence of Im (Z M)], it has a profound influence on the S 21(~)
characteristic, displacing the position of the minimum from the
antenna’s self-resonant frequency ~ 0 in the overcritical zone
and producing the inflection point in terms of the critical coupling. Moreover, the real component is responsible for the frequency splitting in the conjugate-matched condition. Without its
influence, the conjugate-matched WPT system would be locked
in an undercoupled regime regardless of the antenna separation,
as previously noted in [15]. This can be deduced from the comparison of (9) and (11) by neglecting Re (Z M).
Maximum WPT Performances
1
AN EXPERIMENTAL OBSERVATION OF FREQUENCYSPLITTING PHENOMENON
0.9
Max (S21)
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.02 0.05
0.1
0.15
0.2
Antenna Separation (λ)
0.25
SMT-AM; RL = RLOPT, XA = X AOPT
SMT-AM; RL = RLOPT, XA = X AEXT
FEKO; ZL = Z Linville
SMT-AM; RL = RLCRIT, XA = XACRIT
FEKO, Critical Impedance
SMT-AM; RL = RLCRIT, XA = 0
FIGURE 4. The maximum WPT performances between two
coaxial PEC loops tuned at 13.56 MHz versus distance.
d = 2.5 cm
VNA
ω = ω0
0.9
d = 5 cm
VNA
ω = ω0
0.9
d = 7.5 cm
VNA
ω = ω0
0.9
0.8
0.8
0.8
0.7
0.7
0.7
0.6
0.6
0.6
0.5
0.4
1
S21
1
S21
S21
1
0.5
0.4
0.5
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0.1
0
10 15 20
Frequency (MHz)
0
10 15 20
Frequency (MHz)
0
10 15 20
Frequency (MHz)
(a)
(b)
(c)
FIGURE 5. The measured magnitude of S 21 between two
coaxial electrically small two-turn HF cylindrical coils
terminated with 50 X: (a) Taken at 2.5-cm separation, (b)
measured very close to the critical distance, and (c) typical
for the system operating in terms of undercritical coupling.
44
To validate the observed results further, we selected the measurement setup similar to the one of the Tesla’s earliest experiments with resonant coupling [14], where WPT measurements
between two coaxially positioned electrically small coils were
conducted by a vector network analyzer (VNA). For that purpose, two turns of approximately equal lengths of 2-mm isolated
copper wire with a 10-cm radius and a 1-cm height were wound
for each coil. The coils were made to antiresonate on approximately equal frequencies near 34 MHz. The self-resonant
frequency of each coil was adjusted to 13.56 MHz by a variable
capacitor C (C . 55 pF). The WPT on 4,001 VNA frequency
points between 5 and 35 MHz was measured at various antenna
separations between 2.5 and 15 cm in steps of 2.5 cm. The VNA
ports’ impedance was set to R L = 50 X, and a 50-X 1:1 balun
was applied on each side of the WPT link. The total insertion
loss was estimated to be approximately 1 dB.
Typical examples of |S 21 | frequency characteristics are
depicted in Figure 5. Each is the average of 16 consecutive VNA
measurements at the distance. Figure 5(a) was taken at 2.5-cm
separation. Two resonant peaks are observed, and the system
operates in an overcoupled regime. Note that the position of the
local minimum is displaced from the antennas’ self-resonant frequency. Figure 5(b) was measured very close to the critical distance, where the critical resistance is approximately 50 Ω. The
resonant frequency (i.e., that of the maximum) is 14.42 MHz, at
which the received power is 0.5% higher relative to the one at
~ = ~ 0 . Figure 5(c) is typical for the system operating in terms
of undercritical coupling. One clear peak very near the antennas’ self-resonant frequency can be observed.
Results of the measurements are summarized in Figure 6.
Note that as the antenna separation becomes larger, all three
monitored frequencies converge to the antenna-self-resonant
frequency. However, immediately after the 5-cm separation,
only one resonant frequency exists, and the frequency-splitting
phenomenon ceases. This is followed by a rapid decrease of
the received power. Thus, in these simple measurements of
WPT between two real coils, one can notice all of the effects
that confirm the conclusions from the “Maximum WPT Performances” section.
CONCLUSIONS
Despite the fact that the SMT-AM approximative model is
not capable of making a completely accurate prediction of
the performances of the near-field antenna system at very
AUGUST 2019
IEEE ANTENNAS & PROPAGATION MAGAZINE
VNA Measurements
19
18
0.8
17
S21
0.7
16
15
0.6
14
0.5
13
0.4
0.3
1.5 2.5
12
5
7.5
10
12.5
Coil Separation (cm)
Frequency of Extremum (MHz)
0.9
11
15
Global Maximum S21
S21 (f = 13.56 MHz)
Local Maximum S21
Global Maximum Frequency
f = 13.56 MHz
Local Maximum Frequency
Local Minimum Frequency
FIGURE 6. The measured magnitudes of S 21 (blue) and the
extrema frequencies (red) versus coil separation.
low electrical distances, it nevertheless reveals the true
potential of resonant WPT technology. Furthermore, by
applying the antenna theory, the frequency-splitting phenomenon can be predicted and controlled. The two methods
for obtaining maximum WPT performances are presented
and compared. The method of critical coupling is more convenient to be used at low ranges because of a larger fractional
bandwidth. Nonetheless, as a result of the SMT-AM-based
analysis, it turns out that the critically coupled WPT system
is the most effective when it is tuned properly, which means
that the antennas (including the tuning reactance) cannot
be in self-resonance at the operating frequency. To reach
the largest distances of transmission possible (relative to the
given frequency), the conjugate-matching approach must be
applied, while the used antennas’ loss resistance should be as
low as possible.
ACKNOWLEDGMENT
This work has been supported in part by the Croatian Science
Foundation under the project “Internet of Things: Research and
Applications,” UIP-2017-05-4206.
AUTHOR INFORMATION
Zoran Blažević (zblaz@fesb.hr) is with the Department of
Electrical and Computer Engineering, University of Split, Croatia. His current research interests include resonant wireless
power transfer, radio-frequency identification systems, antennas,
and radio-channel modeling. He is a Member of the IEEE.
Maja Škiljo (msekelja@fesb.hr) is with the Department of
Electrical and Computer Engineering, University of Split,
Croatia. His current research interests include resonant wire-
IEEE ANTENNAS & PROPAGATION MAGAZINE
AUGUST 2019
less power transfer, ground-penetrating radars, radio-frequency identification systems, and antennas. He is a Member of
the IEEE.
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