Analytical Design for Resonant Inductive Coupling
Wireless Power Transfer System With Class-E
Inverter and Class-DE Rectifier
Tomoharu Nagashima1 †, Xiuqin Wei2 , Elisenda Bou3 , Eduard Alarcón3 , and Hiroo Sekiya1
1
2
Graduate School of Advanced Integration Science, Chiba University, Chiba, 263–8522 Japan
Department of Electronics Engineering and Computer Science, Fukuoka University, Fukuoka 814–0180, Japan
3
Electrical Engineering Department, UPC BarcelonaTech, Barcelona 08034, Spain
E-mail: †nagashima@chiba-u.jp
Abstract—This paper presents a RIC-WPT system with
class-E inverter and class-DE rectifier along with its analytical
design procedure. By using the class-E inverter as a transmitter
and the class-DE rectifier as a receiver, the designed WPT system
can achieve a high power-delivery efficiency because of the class-E
ZVS/ZDS conditions in both the inverter and the rectifier. In
the experimental results, the system achieved 73.0 % overall
efficiency at 9.87 W (50 Ω) output power, coil distance 10 cm,
and 1 MHz operating frequency. The experimental results showed
good agreement with the design specifications, which indicates the
validity of the design procedure.
Keywords—Wireless power transfer, resonant inductive
coupling, class-E inverter, class-DE rectifier, class-E ZVS/ZDS
conditions, high overall efficiency.
I. Introduction
Recently, there are many researches and development
about the wireless power transfer (WPT) systems [1]-[5].
Resonant inductive coupling (RIC) is one of the coupling
methods for WPT systems and it has been widely used for
a variety of applications, such as wireless battery charging
for electric vehicles [1], bio-medical implants [2], and so
on. For achieving a high power-delivery efficiency, it is
important to design not only low power-loss RIC, but also
high power-conversion-efficiency inverter and rectifier.
In [3]-[5], the class-E2 and class-DE-E WPT systems
were proposed for the power delivery efficiency enhancement.
By satisfying the class-E zero-voltage switching and
zero-derivative switching (ZVS/ZDS) conditions in both the
inverter and the rectifier, these systems can achieve high
power-delivery efficiency. However, in the class-E rectifier, the
diode-reverse voltage is several times higher than the output
voltage. Additionally, the large inductance for the low-pass
filter is needed in the class-E rectifier. These disadvantages
may lead a high cost implementation and a large circuit size.
The class-DE rectifier is also one of the high power
conversion efficiency rectifiers because of the class-E
ZVS/ZDS conditions [6]. Compared with the class-E rectifier,
the diode voltage can be suppressed to its output voltage.
Additionally, an inductance for the low-pass filter is
unnecessary in the circuit. By applying the class-DE rectifier
to WPT systems, it is possible to fabricate high output voltage
systems at low cost and small size.
This paper presents a RIC-WPT system with class-E
inverter and class-DE rectifier along with its analytical design
978-1-4799-8391-9/15/$31.00 ©2015 IEEE
686
procedure. By using the class-E inverter as a transmitter
and the class-DE rectifier as a receiver, the designed WPT
system can achieve a high power-delivery efficiency because
of the class-E ZVS/ZDS conditions in both the inverter and
the rectifier. In the experimental results, the system achieved
73.0 % overall efficiency at 9.87 W (50 Ω) output power,
coil distance 10 cm, and 1 MHz operating frequency. The
experimental results showed good agreement with the design
specifications, which indicates the validity of the design
procedure.
II.
ClassE-DE WPT System
Figure 1(a) shows a topology of the WPT system with the
class-E inverter [3], [4], [7], [8] and the class-DE converter [6].
In this system, it is regarded as the class-E inverter is connected
with the class-DE rectifier by the resonant inductive coupling
part. Figure 2 shows example waveforms of the WPT system,
where θ = ωt = 2π f t is the angular time and f is the operating
frequency. By applying the class-E switching technique to both
the inverter and the rectifier, the WPT system can achieve a
high power-delivery efficiency.
A. Class-E Inverter
The class-E inverter consists of dc-supply voltage VI ,
dc-feed inductance LC , MOSFET S , which works as a
switching device, shunt capacitance CS , and impedance
transformation component X p as shown in Fig. 1(a). The
impedance transformation component is used for transforming
the output network into the optimal load against VI . The switch
of the inverter is driven by the driving signal vg . During
the switch-off interval, the difference of currents through the
dc-feed inductance and the resonant filter flows through the
shunt capacitance. The current through the shunt capacitance
produces the switch voltage vS . The most important operation
of the class-E inverter is that the class-E ZVS/ZDS conditions
are satisfied at turn-on instant as shown in Fig. 2. The class-E
ZVS/ZDS conditions are expressed as
dvS vS (2π) = 0 and
= 0.
(1)
dθ θ=2π
Because of the class-E ZVS/ZDS conditions, the class-E
inverter achieves high power conversion efficiency under high
frequency operations.
Inductive coupling
Class-E inverter
LC
VI
II S
iS
vg
Cs
iinv
vs
i
i
C
RL 
Xp
L
Class-DE rectifier
vD
CD
C
vg
D
RL
D
L
CD
vD
0
k
Cs
RL (
-k)L
vs
Xp
VI iS S

i
kL
iinv
i
vD
iS
0
0
Io
Zsec Zrec
(c)
C
RL
II
iinv
i
Req
Xp
Leq
vs
VI
II
C
Linv
L LX
vs
iinv
Rinv
i
}
LC
LC
Cs
0
Ri
Vind
(b)
VI iS S
0
Cr
L RL C Ci
}
II
C
vD
vS
(a)
LC
i 0
II
VO RL
Cf
OFF
ON
0
IO
Cs
iS S
inv
Fig. 2.
Zinv
Zinv
(d)
0
0
(e)
Fig. 1. Class-E-DE WPT system. (a) System overview. (b) Equivalent circuit
model including the coupling coils and the rectifier. (c) Equivalent circuit of
the rectifier. (d) and (e) Equivalent circuits boiled down to the class-E inverter.
Waveforms of class-E-DE WPT system.
The class-DE rectifier can be replaced by the input
capacitance Ci and the input resistance Ri , which are connected
in series as shown in Fig. 1(c). The input capacitance Ci and
the input resistance Ri are expressed as
Ci =
(3)
RL [1 − cos(2πDd )]2
2π2
(4)
and
B. Class-DE Rectifier
The class-DE rectifier consists of diodes D1 , D2 , shunt
capacitors C D1 , C D2 , a filter capacitance C f , and load resistance
RL . The waveforms of the class-DE rectifier are a reversed
version of those of the class-DE inverter. The diodes work
as half-wave voltage rectifiers and the rectified voltages are
converted into a dc voltage through the filter capacitance. At
the turn-off transition of the diodes, both the diode voltages
vD1 , vD2 and their slopes of dvD1 /dθ, dvD2 /dθ are zero as shown
in Fig. 2, which are also the class-E ZVS/ZDS conditions.
Therefore, the class-DE rectifier can also achieve the high
power conversion efficiency at high frequencies. Compared
with the class-E rectifier, the class-DE rectifier can suppress
the diode voltages to the output voltage Vo and does not require
the low-pass inductance.
Ri =
For the resonance in the rectifier part, C2 should resonate
with L2 and Ci . Therefore, C2 can be obtained from
Ci
C2 = 2
.
(5)
ω L2 C i − 1
The equivalent capacitance of C2 and Ci connected in serial
Cr is expressed as
C2Ci
.
(6)
Cr =
C2 + Ci
The amplitude of the rectifier input current I2 is
I2 =
III. Analytical Design Procedure
In this section, the analytical design procedure of the
class-E-DE RIC-WPT system is given. First, the rectifier part
is designed. After that, the rectifier part is expressed as the
reflected impedance in the inverter side. Finally, the inverter
part is designed to satisfy the class-E switching conditions.
A. Class-DE Rectifier Part
The rectifier part is designed using the model shown in
Figs. 1(a) and (c). The detailed analysis of the class-DE
rectifier was carried out in [6]. This subsection shows a
summary of the results in [6].
The diode shunt capacitances C D1 , C D2 are expressed as
C D1 + C D2
2π(C D1 + C D2 )
.
sin(4πDd ) + 2π(1 − 2Dd )
2π[1 + cos(2πDd )]
=
.
ωRL [1 − cos(2πDd )]
2πIo
,
1 − cos(2πDd )
where Io and Dd are the output current and the diode-on duty
ratio, respectively.
B. Coupling Part
The impedance Z sec seen by the induced voltage source
Vind is expressed as
(
1
1 )
Z sec = RL2 + Ri + j ωL2 −
−
.
(8)
ωC2 ωCi
At resonance, the imaginary part of Z sec equals zero. Therefore,
Vind can be obtained as
Vind = (RL2 + Ri )I2 .
(2)
687
(7)
(9)
The equivalent resistance Req and inductance Leq including
the coupling coils and rectifier, which is denoted in Fig. 1(d),
are
k2 ω2 L1 L2 (RL2 + Ri )
Req =
,
(10)
(
1 )2
(RL2 + Ri )2 + ωL2 −
ωCr
and
[
L2 ( 1 ) 2 ]
k2 L1 (RL2 + Ri )2 −
+
Cr
ωCr
2
Leq =
(
)2 + L1 (1 − k ), (11)
1
(RL2 + Ri )2 + ωL2 −
ωCr
respectively, where k is the coupling coefficient of the coupling
coils.
The current through the transmitting coil I1 , which is the
effective value of i1 , can be calculated from
Vind
I1 =
.
(12)
√
ωk L1 L2
It is known that the resonant filter of the inverter should
be inductive for achieving the class-E ZVS/ZDS conditions
in the inverter. In this analysis, Linv is divided into L0 and
L x virtually, where L0 and C1 realize resonant filter for ω as
shown in Fig. 1(e). In this case, L x , which is used for current
phase shift, for achieving the class-E ZVS/ZDS conditions can
be obtained from [8] as
ωL x {
= 2(1 − DS )2 π2 − 1 + 2 cos ϕinv cos(2πDS + ϕinv )
Rinv
}
− cos 2(πDS + ϕinv )[cos(2πDS ) − π(1 − DS ) sin(2πDS )] /
{
4 sin(πDS ) cos(πDS + ϕinv ) sin(πDS + ϕinv )
}
[(1 − DS )π cos(πDS ) + sin(πDS )] .
(19)
On the other hand, we can have another expression for Rinv
from the class-E inverter design viewpoint. According to the
power relationship between Figs. 1(d) and (e), we have
2
I12 (Req + RL1 ) = Iinv
Rinv .
From (16) and (20), Rinv can be obtained as
Rinv =
C. Class-E Inverter Part
The output impedance of the class-E inverter Req + RL1 +
jωLeq is transformed into Zinv by the impedance transformation
component X p as shown in Fig. 1(d). Zinv is
1
Zinv =
1
Req + RL1 + jωLeq
+ jωC p
= Rinv + jωLinv ,
(13)
(Req + RL1 )

(
)2  ,


1
2
2
2

ω C p (Req + RL1 ) + ωLeq −
ωC p 
(14)
Leq (1 − ω2 LeqC p ) − C p (Req + RL1 )2

(
)2  .

1

2
2
2

ω C p (Req + RL1 ) + ωLeq −
ωC 
(15)
p
The class-E-DE WPT system is transformed into the typical
class-E inverter as shown in Fig. 1(d). Therefore, many design
equations for the class-E inverter [8] can be applied to the
WPT system.
From the relationship between the input voltage and the
inverter current of the class-E inverter [8], the amplitude of
the inverter current Iinv and the input current II for achieving
the class-E ZVS/ZDS conditions are obtained as
2 sin(πDS ) sin(πDS + ϕinv )
Iinv =
VI ,
(16)
π(1 − DS )Rinv
and
II =
cos(2πDS + ϕinv ) − cos ϕinv
Iinv ,
2π(1 − DS )
(17)
respectively, where DS and ϕinv are the switch-on duty ratio
and the phase shift between the inverter current iinv and the
driving signal vg , which is expressed as
tan ϕinv =
cos(2πD s ) − 1
.
2π(1 − D s ) + sin(2πD s )
π2 (1 − DS )2 I12 (Req + RL1 )
.
(21)
Now, we have two expressions of Rinv as given in (14) and
(21). By equating the right-had side of (14) and (21), X p
for achieving both the class-E ZVS/ZDS conditions and the
specified output power is obtained as
[
]
2
−Rinv (Req + RL1 )2 + ω2 Leq
√
[
].
2
ωLeq Rinv ± Rinv (Req +RL1 ) (Req +RL1 )(Req +RL1 −Rinv )+ω2 Leq
(22)
It is shown in (22) that there are two solutions. The solution
that satisfies L0 = Linv − L x > 0 should be selected because L0
resonates with C1 .
and
Linv =
2 sin2 (πDS ) sin2 (πDS + ϕinv )VI2
Xp =
where Rinv and Linv are
Rinv =
(20)
(18)
688
From [8], the shunt capacitance CS and the dc-feed
inductance LC are calculated from
1
CS =
{2 sin(πD s ) cos(πD s + ϕinv )
ωπ2 (1 − D s )Rinv
· sin(πD s + ϕinv )[(1 − D s )π cos(πD s ) + sin(πD s )]} ,
(23)
and
LC =
)
Rinv ( π2
+2 .
f 2
(24)
IV. Design Example
A design example is shown in this section. First, the design
specifications were given as follows: operating frequency f =
1 MHz, dc-supply voltage VI = 24 V, output power Po = 10 W,
output resistance RL = 50 Ω, switch and diode on-duty ratios
D s = 0.5 and Dd = 0.35, and distance between the primary coil
and the secondary one dcoils = 10 cm. The parameters of the
handmade coupling coils are: self inductances L1 = 23.1 µH
and L2 = 22.7 µH, equivalent series resistances RL1 = 0.891 Ω
and RL2 = 0.809 Ω, and coupling coefficient k = 0.0559
at dcoils = 10 cm. From the previous design procedure, the
component values were obtained as given in Table I. In the
LC
CS
C1
Cp
C2
C D1
C D2
Cf
RL
L1
L2
k
r L1
r L2
dcoils
f
VI
Po
ηall
Analytical
186 µH
1.09 nF
768 pF
432 pF
1.15 nF
2.60 nF
2.60 nF
47 µF
50.0 Ω
1 MHz
24.0 V
10.0 W
-
Design values of design example
Simulated
186 µH
1.09 nF
768 pF
432 pF
1.15 nF
2.60 nF
2.60 nF
47 µF
50.0 Ω
23.1 µH
22.7 µH
0.0559
0.891 Ω
0.829 Ω
1 MHz
24.0 V
9.84 W
76.0 %
Measured
196 µH
1.12 nF
766 pF
429 pF
1.16 nF
2.63 nF
2.60 nF
50.3 Ω
23.1 µH
22.7 µH
0.0559
0.891 Ω
0.829 Ω
10 cm
1 MHz
24.0 V
9.87 W
73.0 %
Overall efficiency (%)
TABLE I.
Difference∗
5.4 %
2.8 %
−0.24 % −0.65 % 0.57 % 1.3 %
0.039 %
0.6 %
0.0 %
0.0 %
−1.1 %
-
Secondary coil
Class-E inverter Class-DE rectifier Load resistance
(a)
vo (V) vD, (V) i (A) i (A) vs (V) vg (V)
dcoils
10 ON
0
100
50
0
2
0
-2
2
0
-2
20
0
20
0
20
0
OFF
ON
60
40
20
Ddopt
00
Fig. 4.
0.1 0.2 0.3 0.4 0.5
Diode-on duty ratio Dd
Overall efficiency as a function of Dd .
V. CONCLUSION
In this paper, the RIC-WPT system with class-E inverter
and class-DE rectifier along with its analytical design
procedure has been proposed. By using the class-E inverter
as a transmitter and the class-DE rectifier as a receiver, the
designed WPT system can achieve a high power-delivery
efficiency because of the class-E ZVS/ZDS conditions in both
the inverter and the rectifier. In the experimental results, the
system achieved 73.0 % overall efficiency at 9.87 W (50 Ω)
output power, coil distance 10 cm, and 1 MHz operating
frequency. The experimental results showed good agreement
with the design specifications, which indicates the validity of
the design procedure. Additionally, we revealed that the overall
efficiency highly depends on the diode-on duty ratio, which
also means that the derivation of the optimal diode-on duty
ratio is quiet important for maximizing the overall efficiency.
* “Difference” is the difference between analytical and
experimental results.
Primary coil
80
OFF
Acknowledgments
0.5
1.0 1.5
Time (µs)
2.0
(b)
Fig. 3. Experimental set up and result. (a) System overview. (b) Experimental
waveforms for design example.
This research was partially supported by Grants-in-Aid for
Scientific Research (No. 13J08797, 26289115 and 25820112)
of JSPS and Telecommunications Advancement Foundation,
Japan.
References
[1]
experiment, the IRFS4410 MOSFET and the STPS5H100B
Schottky Barrier Diode were chosen as switching devices.
Figure 3(a) shows the system overview of the design example.
[2]
[3]
Figure 3(b) shows the experimental waveforms of the
designed system. All the switch voltages in this system, namely
vS , vD1 , and vD2 , satisfy the class-E ZVS/ZDS conditions
and the output power was Po = 9.87 W. From these
results, the validity of the design procedure can be confirmed.
Additionally, 73.0 % overall efficiency was achieved, where
the overall efficiency is calculated from η = Po /VI II .
Figure 4 shows the PSpice simulated overall efficiency as
a function of the diode-on duty ratio Dd , where the design
specifications are the same as the design example and the
design values were recalculated for each Dd . It is seen from
Fig. 4 that the overall efficiency highly depends on Dd and is
maximized at Dd = Ddopt . From this result, it is important to
derive the optimal Ddopt for maximizing the overall efficiency
in this system.
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