Digital Coil: Transmitter Coil with Programmable Radius
for Wireless Powering Robust Against Distance Variation
Hao Qiu, Yoshiaki Narusue, Yoshihiro Kawahara, Takayasu Sakurai, and Makoto Takamiya
University of Tokyo, Japan
Email: hqiu@iis.u-tokyo.ac.jp

Abstract—A wireless powering system robust against distancevariation is required. In this paper, a digital transmitter (TX) coil,whose radius depends on the distance (d) between the TX and receiver (RX) coils, is proposed that maximizes the efficiency of wireless powering. Firstly, we analytically derived the optimum TX coilradius (rTX, OPT) to maximize the coil-to-coil efficiency (η) by cooptimizing the coupling coefficient and the quality factors of thecoils. It was found that rTX, OPT was approximately equal to d. Then,a practical implementation of the digital TX coil, whose radius waselectrically varied without mechanical motion, was proposed. It
was found by a measurement that, compared with a conventional
TX coil with a constant radius, the proposed digital TX coil increased η from 12% to 20% when d was four times the RX coilradius.
Index Terms—wireless power transfer, inductive coupling, magnetic resonance, coil-to-coil efficiency, distance variation, coil design, optimum coil radius, 13.56MHz.

rRX
Efficiency (η1)

Constant rTX(Conventional)
Efficiency(η2, CONV)978-1-5386-5159-9/18/$31.00 ©2018 IEEE

TX

rTX: TX coil radiusrRX: RX coil radius

Programmable rTX(Proposed)
RX

RX
Efficiency(η2, PROPOSED)

d2
rTX

RX

rRX

rRX

d2

rTX, PROG = d2

TX

TX

(a)

Efficiency (η)

I. INTRODUCTION
The efficiency (ηSYS) of a wireless power transfer (WPT) system rapidly decreases as the distance (d) between the transmitter (TX) and receiver (RX) coils increases. The most effective
way to realize wireless powering robust against d variation is todirectly maximize the coil-to-coil efficiency (η). Moreover, thetuning mechanism against d variation should be included on theTX side rather than the RX side, because the RX side should be
as simple as possible in, for example, the wireless charging ofcellphones. Several wireless powering systems robust against dvariations [1-5] have been proposed. In [1-4], impedance
matching on both the TX and RX sides of a WPT system wasfocused on, where ηSYS was limited to η. In addition, RX tuning– such as impedance matching between a load impedance andload [1-3] and RX coil layout optimization in [5] – has been
proposed but does not satisfy our design target of TX-only tuning. To solve these problems, we proposed a novel digital TX
coil with a programmable radius (rTX, PROG) depending on d,which makes it possible to directly maximize η and thus realizea system with ηSYS robust against d variation. Additionally, thedigital TX coil uses TX-only tuning. Throughout this work, wefocus on η.
Fig. 1 shows the key concept of this paper. Fig. 1 (a) showsa conventional TX coil with a constant radius and the proposed
TX coil with rTX, PROG, which depends on d (rTX, PROG=d). Fig. 1(b) shows a schematic diagram of the d dependence of η. Whend is increased from d1 to d2, the proposed TX coil has a higher
η for d2 than the conventional TX coil.
In Section II, an optimum TX coil radius is derived analytica-d1rTXη1η2, PROPOSED

Programmable rTX(Proposed)
Efficiency
increase

η2, CONV

Constant rTX(Conv.)
d1

d2

Distance between TX and RX coils (d)(b)
Fig. 1.
Key concept of this paper. (a) Conventional TX coil with constant radius and proposed TX coil with rTX, PROG. Here, single-turn coil isplotted for simplicity. (b) Schematic diagram of d dependence of η.

lly, by o-optimization of the TX-to-RX coupling coefficient (k)and the quality factors of the coils, and it is found to be approximately equal to d. This is then confirmed by electromagnetic
simulation and coil-to-coil measurement. Then, as reported inSection III, the proposed digital TX coil with rTX, PROG is fabricated and measured.
II. OPTIMUM TX COIL RADIUS FOR MAXIMUM EFFICIENCY
A. Derivation of Equation
We first analytically derived the optimum TX coil radius forthe maximum η under the condition that the RX coil radius (rRX)
is much smaller than that of the TX coil (rTX). In WPT systems,
η depends on the quality factors of the TX (QTX) and RX (QRX)
coils and k, a combination of which (k2QTXQRX) must be maximized at the resonance frequency (f0) to maximize η [6]:k 2 QTX QRX = (σ Af 0 ) 21 M2.
mn rTX rRX(1)Here, σ is the conductivity of the coil metal, A is the cross-sectional area of the current flow in the metal considering skin effects, and M is the mutual inductance between the TX and RXcoils. m and n are the numbers of turns of the TX and RX coils,
respectively. Here, the increase in resistance due to proximityeffects between different turns is neglected.
Using Maxwell’s equations, M can be obtained as [7]
 2
2 
M = μ0 mn rTX rRX  − β  K − E ββ(2)2 rTX rRX(3)β=d + ( rTX + rRX )22,where μ0 is the permeability and the complete elliptic integrals
π /2π /2
are K =  dθ / 1 − k 2 sin 2 θ and E =  dθ 1 − k 2 sin 2 θ . Here,00

the physical thicknesses of both the TX and RX coils are assumed to be zero.
For rRX<<rTX, M in (2) can be approximated asμ0πrTX 2 rRX 2(4)M0 =2mn(d + r )2 3/ 22.

B. Validation by Electromagnetic Simulation
To confirm that (6) is a good guideline for TX coil design in
WPT systems, electromagnetic simulation software (EMPro,
Keysight Technologies) was used. Table I summarizes the design specifications of both the TX and RX coils. f0 was finely
tuned to 13.56MHz for both coils using compensation capacitors. Considering the physical thickness of the coils, d is definedas the distance from the geometrical center point of the TX coilto that of the RX coil.
Fig. 3 shows the simulated relationship of η versus rTX fordifferent d. As expected, η is maximized at rTX, OPT, although it
deviates from (6). The deviation is ascribed to the simplified
assumptions made in the analysis. Considering that the electromagnetic simulation gives more accurate results than equation
analysis, in this work, we set rTX, OPT as 35mm, 45mm, 70mm,and 80mm for d of 35mm, 50mm, 65mm, and 80mm, respectively. However, it is emphasized that (6) still acts as a good
design guideline with a tolerable percentage error (<12%).
TABLE IDESIGN SPECIFICATIONS OF TX AND RX COILSTXThen, k2QTXQRX in (1) can be simplified as
μ0πσ Af 0 223k QTX QRX = (error decays rapidly as d increases and thus (6) gives a reasonable approximation with error<15% from (7) for d>30mm.

2) mnrRX

1(r + d / r )
2TX

3.

(5)

TX

Note that when all other parameters (f0, m, n, rRX, d) are constants, k2QTXQRX in (5) reaches at its maximum value, whichalso corresponds to the maximum η, if and only if rTX is at its
optimum value (rTX, OPT) of(6)rTX, OPT = d.

Design specifications
Metal materialCu
1Radius (mm)rTX

Number of turns

520
3
2100

rTX, OPT=35mm
280

Efficiency (η) (%)

rTX, OPT /dMetal wire pitch (mm)3RX Coil
13.56

Metal wire diameter (mm)Approx. solution (Eq. (6))
Exact solution (Eq. (7))

4TX Coilf0 (MHz)

Error <15%, when d >30mm.
1

rRX=20mm
00153045607590105d (mm)Fig. 2.
d dependence of rTX, OPT normalized by d showing approximate
and exact solutions.

In addition, the error introduced in (4) is estimated. Insteadof using (4) for the mutual inductance, (2) is substituted into (1)and rTX, OPT can be obtained by solving∂ 2k QTX QRX = 0 .
∂rTX(7)Here, we set f0=13.56MHz, m=5, n=3, and rRX=20mm. Fig. 2
shows a comparison between rTX, OPT in its approximate form
[rTX, OPT(approx.)] and its exact solution [rTX, OPT(exact)]. According to Fig. 2, rTX, OPT(approx.) deviates from rTX, OPT(exact)
at a small d, while they converge at a large d. The percentage

d=35mmrTX, OPT=45mm

60rTX, OPT=70mm

d=50mm40d=65mm

200Simulated
rRX=20mm
01530d=80mm
rTX, OPT=80mm
45607590105

rTX (mm)Fig. 3.

Simulated η vs rTX for different d.

Fig. 4 shows the simulated relationship of η versus d for conventional and proposed programmable TX coils with the same
RX coil. As can be seen, compared with the conventional TX
coil, the proposed coil with rTX, OPT has a maximum η for different d. For example, for d=80mm, η is 10% using the TX coilwith rTX=35mm but is increased to 26% (more than doubled)
when the TX coil with rTX, OPT=80mm is used. Similarly, ford=35mm, η is 63% using the TX coil with rTX=80mm but isincreased to 78% when the TX coil with rTX, OPT=35mm is used.
100

Efficiency (η) (%)

8060

rTX=35mm
rTX=45mm
Conv.
rTX=70mm
rTX=80mm
Programmable rTX(Proposed)

78%

63%

4026%
200Simulated
rRX=20mm3040the same RX coil. Compared with the simulation results in Fig.
4, the absolute value of η is slightly lower, which is reasonableconsidering the mismatch between the simulated and fabricatedcoils. Note that a similar conclusion can be arrived at from thefact that the proposed programmable TX coil with rTX, OPT has amaximum η for different d. Taking d=80mm as an example, ηis 9.8% using the TX coil with rTX=35mm but is increased to22% (still more than doubled) when the TX coil withrTX, OPT=80mm is used.
L1C1 = L2 C 2 = L3C3 = L4C 4

10%50607080RLRX coil
1= LRXC RX =( 2π f 0 )290d (mm)Fig. 4.
Simulated η vs d using TX coil with conventional constant rTX
and proposed rTX, PROG.

LRXCRXdSelector
SW2SW4SW3RS C

4C3 C2TX4
TX3
TX2
TX1SW1C1L1L2 L3L4VSf0

Digital TX coilFig. 7.
Schematic of the proposed digital TX coil, in which one of four
sub-coils with different radii is selectively turned on depending on d. Here,single-turn coil is plotted for simplicity.

Fig. 5.
Fabricated TX coils with rTX of 35mm, 45mm, 70mm, and80mm. rRX is 20mm.
100

Efficiency (η) (%)

80

rTX=35mm
rTX=45mm
Conv.
rTX=70mm
rTX=80mm
Programmable rTX(Proposed)

76%604020051%

22%

MeasuredrRX=20mm30409.8%5060708090d (mm)Fig. 6.

Measured η vs d using TX coils in Fig. 5.

C. Experimental Validation
In addition, measurements were performed for comparison
with the simulation results. As shown in Fig. 5, four TX coils(TX1: rTX=35mm; TX2: rTX=45mm; TX3: rTX=70mm; TX4:rTX=80mm) and an RX coil (rRX=20mm) were fabricated according to the design specifications in Table I. f0 was finely
tuned to 13.60MHz for all coils using compensation capacitors.
A network analyzer (E5061B, Keysight Technologies) wasused to measure the S parameter for each pair of the TX and RXcoils, and η was calculated as |S21|2/(1-|S11|2). Fig. 6 shows themeasured relationship of η versus d for different TX coils with

Fig. 8.

Fabricated digital TX and RX coils.

III. PROPOSED DIGITAL TX COIL
On the basis of the above analysis, the TX coil with rTX, PROG
designed according to (6) can alleviate η degradation with dvariation. However, its practical implementation is still an issue. Rather than using motors to mechanically change rTX for
the TX coil, we propose the use of switches to electronicallyprogram rTX. As shown in Fig. 7, the proposed digital TX coil
is composed of four concentric sub-coils (TX1, TX2, TX3, andTX4), four compensation capacitors, and four switches (SWs).

TABLE II
COMPARISON WITH PREVIOUSLY REPORTED WPT SYSTEMS ROBUSTAGAINST DISTANCE VARIATIONEach sub-coil resonates at the same f0 as the RX coil. The selector is used to control these four SWs. For each d, only the subcoil with rTX, OPT that can achieve the maximum η is turned on by
the selector. For example, when d is 80mm, TX4 with
rTX=80mm is turned on by selecting SW4. Fig. 8 shows a photograph of the digital TX coil, which was fabricated from thefour TX coils in Fig. 5. The geometric structure of the digital
TX coil is also given. The compensation capacitors were also
reselected to tune f0 for each sub-coil to 13.60MHz. Here, a relay (Panasonic Electric Works TQ2-L2-4.5) with an ON resistance of less than 50mΩ was used for each SW.

Ref.

100

Efficiency (η) (%)

80

rTX=35mm
rTX=45mm
rTX=70mm
rTX=80mm
Digital TX(Proposed)

75%

60
40
20051%

1.5

2.0

Conv.

12%
2.5

3.0

3.5

4.0

4.5

d/rRX

Fig. 9.

[1]

No

No

[2]

No

No

[3]

No

No

[4]

No

Yes ✔

[5]

Yes ✔

NoThis work

Yes ✔

Yes ✔

IV. CONCLUSION

20%

MeasuredrRX=20mm

Coil-to-coil efficiency
TX-only tuningmaximization

Measured η vs d/rRX using digital TX coil in Fig. 8.

Fig. 9 shows the measured relationship of η versus d normalized by rRX (d/rRX). As highlighted in blue, the proposed digital
TX coil provides the maximum η by selectively turning on TX1,TX2, TX3, and TX4 when d equals 1.75, 2.5, 3.25, and 4 times
rRX, respectively. Taking d=rRX×4 as an example, η can be improved from 12% to 20% by selectively turning on TX4 ratherthan TX1. In addition, η can be improved from 51% to 75% at
d=rRX×1.75, when TX1 rather than TX4 is turned on.
Here, we further discuss the proposed digital TX coil. Firstly,
the coupling from the other three unselected open sub-coils in
the proposed digital TX coil may cause η degradation [6]. However, by comparing Figs. 6 and 9, η only shows a minor fluctuation of ±2%, which means that the effect of such coupling isnegligible. Secondly, one of the four sub-coils in the digital TX
coil was manually selected according to d in this paper. In ourfuture work, however, an RX localization subsystem, based onwhich rTX can be automatically programmed, will be implemented to establish an advanced WPT system. Finally, in thiswork, only the ideal scenario has been discussed, in which theRX coil is always concentrically aligned with the TX coil andonly d is changed. In real applications, however, it is not uncommon for the RX coil to be misaligned with the TX coil. Although the discussion of this is beyond the scope of this work,
the digital TX coil with rTX, PROG, whose optimum value will
deviate from (6), is still promising for alleviating η degradation.
Table II shows a comparison of the proposed digital TX coilwith previously reported WPT systems robust against d variation. Only the proposed system achieved both coil-to-coil efficiency maximization and TX-only tuning, making it practicalfor wireless powering robust against d variation in real applications (e.g., the wireless charging of cellphones).

In this paper, a digital TX coil, whose radius was optimized
according to d, was proposed to maximize system efficiency in
WPT systems when an RX coil is given. Firstly, by co-optimization of both the coupling coefficient and the quality factors of
the coils, we derived the relationship between the optimum TX
coil radius and d. These two values were approximately equal
for a maximum coil-to-coil efficiency. In addition, a practicalimplementation of the digital TX coil using relay switches toselectively turn on one of the four sub-coils was proposed.
Compared with a conventional system without the proposed
coil, experimental results showed an increase in coil-to-coil efficiency from 12% to 20% at d of four times the RX coil radius.
Also, in contrast to previously reported systems, the proposed
digital TX coil maximizes coil-to-coil efficiency and thus realizes a value of system efficiency robust against d variation.
Moreover, no tuning in RX is required.
ACKNOWLEDGMENT
This work was partially supported by JST ERATO Grant
Number JPMJER1501, Japan.
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