1. technology and computing

Maximum Efficiency Control of Wireless Power
Transfer via Magnetic Resonant Coupling
Considering Dynamics of DC–DC Converter
for Moving Electric Vehicles
Katsuhiro Hata, Takehiro Imura, and Yoichi Hori
The University of Tokyo,
5–1–5, Kashiwanoha, Kashiwa, Chiba, 277–8561, Japan
Phone: +81-4-7136-3881, Fax: 81-4-7136-3881
Email: [email protected], [email protected], [email protected]
Abstract—Wireless charging for moving electric vehicles could
extend their cruising distance. Wireless power transfer via
magnetic resonant coupling is suitable for this application. The
transmitting efficiency can be maximized by using a DC–DC
converter on the secondary side. The control system, however,
must be designed properly to satisfy the response requirements
depending on motion of the vehicle. Previous controllers were
designed without considering the dynamics of the DC-DC converter for wireless power transfer via magnetic resonant coupling.
This paper proposes the design method of secondary voltage
control with a feedback controller using a novel DC–DC converter
model based on the analysis of wireless power transfer system.
Experiments show that the proposed model is effective and that
the secondary voltage control improves not only the transmitting
efficiency but also the charging power at any transmitting
distance.
I.
C1
i1
v1
R1
L1
Power source
Lm
R2
C2
L2
i2
v2
Transmitter and Receiver
RL
Load
(a) Equivalent circuit of magnetic resonant coupling.
i1
C1
v1
R1
L1-Lm
L2-Lm
Lm
R2
C2
v2
i2
RL
I NTRODUCTION
Electric vehicles (EVs) have gathered attention as a solution for environmental problems. Their electric motors have
the advantage of a faster response over internal combustion
engines and EVs can achieve a high performance in motion
control [1]. However, EVs need to be charged frequently due to
their limited mileage per charge. Making a charging network
for EVs will allow for the reduction in size of their energy
storage systems, but it is important to keep such a network
simple.
Wireless power transfer (WPT) can mitigate complicated
charging operations by eliminating the use of wiring. In recent
years, wireless charging for moving EVs have been receiving
focus and are expected to extend the cruising distance of EVs
[2]–[5]. In addition, an energy storage system of EVs could
be reduced in size by connecting EVs to electrical infrastructure because the required power for driving an EV could
be supplied by electricity infrastructure. When transmitting
coils of the wireless charging system are buried underground,
they should be installed at least several tens of centimeters
deep, considering road maintenance. Therefore, long distance
transmission is required.
WPT via magnetic resonant coupling is suitable for this application because of its highly efficient mid-range transmission
and its robustness to misalignment [6], [7]. The transmitting
Power source
Transmitter and Receiver
Load
(b) T–type equivalent circuit.
Fig. 1.
Equivalent circuit of wireless power transfer system.
efficiency can be maximized by using a DC-DC converter
on the secondary side [8]–[13], because it is determined not
only by the parameters of transmitting and receiving coils
but also the load condition [14]. Previous controllers were
designed without considering the dynamics of the DC-DC
converter for WPT via magnetic resonant coupling. In the
case of wireless charging for moving EVs, however, it is
important that the control system design satisfies the response
requirement depending on the motion of the vehicle.
This paper proposes the design method of a feedback
controller for secondary voltage control using a novel DC-DC
converter model, which is based on the analysis of WPT via
magnetic resonant coupling instead of the DC circuit model
of the inductive power transfer system [15], [16]. Experiments
show the validity of the proposed model and the effectiveness
of the maximum efficiency control by the secondary voltage
control.
Ground facility
DC-DC
converter
Receiver
M
Battery
Rectifier
Transmitter
Power source
Fig. 2.
Motor drive
Electric vehicle
System configuration for maximum efficiency control.
II.
W IRELESS POWER TRANSFER VIA MAGNETIC
RESONANT COUPLING
A. Input/output characteristics at resonance frequency
WPT via magnetic resonant coupling applies series or parallel resonance to a transmitter and a receiver. This paper uses
a series–series (SS) circuit topology and its equivalent circuit
is shown in Fig. 1. The transmitter and the receiver consist of
the inductance L1 , L2 , the series–resonance capacitance C1 ,
C2 , and the internal resistance R1 , R2 respectively. v1 , i1 , v2 ,
and i2 stand for root–mean–square voltages and currents. RL
is the load resistance and Lm is the mutual inductance between
L1 and L2 .
If the power source angular frequency ω0 satisfies eq.
the voltage ratio Av and the transmitting efficiency η
expressed as eq. (2) and (3) [17].
1
1
ω0 = √
=√
L1 C1
L2 C2
v2
ω0 Lm RL
Av =
=
v1
R1 R2 + R1 RL + (ω0 Lm )2
(ω0 Lm )2 RL
η=
(R2 + RL ){R1 R2 + R1 RL + (ω0 Lm )2 }
(1),
are
(1)
(2)
The system configuration is shown in Fig. 2. Ground facilities
apply a DC source and an inverter to power source on the
primary side. An EV consists of a receiver, a rectifier, a DC–
DC converter, batteries, and a motor drive system. In this paper,
the motor drive system is neglected as this is a fundamental
study.
If the primary voltage is regulated, the load resistance can
be determined by the secondary voltage control. Then the
transmitting efficiency can be maximized by the secondary
voltage, which is expressed as follows [12]:
√
R2
ω0 Lm
√
v2 ∗ =
v1 .
(5)
√
R1 R1 R2 + (ω0 Lm )2 + R1 R2
This paper applies the secondary voltage control with
a feedback controller to a maximum efficiency control and
proposes the novel DC–DC converter model. Then, the plant
model includes a characteristic of the secondary current in
WPT via magnetic resonance coupling and the feedback controller is designed by the pole placement method.
(3)
B. Maximization of transmitting efficiency
The transmitting efficiency is determined by the angular
frequency of the power source, the internal resistance of the
coils, the mutual inductance between the two coils, and the
load resistance. The angular frequency of the power source and
the parameters of the coils can be designed before installation.
However, the mutual inductance changes depending on the
motion of the vehicle and the load resistance is determined by
the state of charge and the output power of the vehicle. In order
to maximize the transmitting efficiency, the load resistance
should be controlled according to the change in the mutual
inductance. The optimal value of the load resistance RLηmax
for maximizing the transmitting efficiency is expressed as
follows [14]:
√ {
}
(ω0 Lm )2
+ R2 .
(4)
RLηmax = R2
R1
Previous research has controlled the load resistance by
using a DC–DC converter on the secondary side [8]–[10].
III.
M ODELING OF DC–DC CONVERTER
A. Circuit configuration
Fig. 3 shows the circuit configuration of the DC–DC
converter, where E is battery voltage, C is capacitance of
the DC–link capacitor, L is inductance of the reactor coil, r
is internal resistance of the reactor coil and batteries, vdc is
voltage of the DC–link capacitor, and iL is an average value of
the inflowing current to batteries. When the resonant frequency
of WPT is much higher than the switching frequency of the
DC–DC converter, idc is defined as the average current flowing
into the DC–link capacitor.
B. State space averaging method
The plant model of the DC–DC converter can be described
by using the state space averaging method [18]. In this paper,
the DC–DC converter is operated in a continuous conduction
mode because switches of the DC–DC converter are alternately
turned on and off. When d(t) is defined as the duty cycle of the
upper switch, the state space model of the DC–DC converter
is expressed as follows:
d
x(t)
dt
vdc (t)
[
=
A(d(t))x(t) + B
=
cx(t)
A
c
idc
]
(6)
(7)
[iL (t) vdc (t)]

d(t)
− Lr
]
L
B
 d(t)
:=  −
0
C
O
0
1
− L1
0
0
1
C

L
vdc

.
x(t) := X + ∆x(t), ∆x(t) := [∆iL (t)
SW2
∆vdc (t)]
u(t) := U + ∆u(t), ∆u(t) := [∆d(t) ∆idc (t)]
T
Vdc ] , U := [D
Idc ]
iL
T
T
T
Then, the equilibrium point satisfies following equations.
Idc
(10)
IL =
D
ED − rIdc
Vdc =
(11)
D2
C. Secondary current on wireless power transfer
Fig. 4 shows the WPT system for maximum efficiency
control using the DC-DC converter. In this paper, a rectangular
voltage is used as a power source on the primary side. When
the SS circuit topology is applied to the WPT system, the
transfer function from the primary voltage to the secondary
current has a high gain at the resonant frequency in comparison
with other frequencies [19]. Therefore, the waveform of the
secondary current i2 becomes a nearly sinusoidal wave with
the resonant frequency regardless of the waveform of the
primary voltage. Additionally, if the variation of the DC–link
voltage can be ignored, the waveform of the secondary voltage
v2 becomes a rectangular wave, which has the same amplitude
as the DC–link voltage vdc and the resonant frequency due
to the conduction of the diodes. This paper focuses on fundamental waves with the resonant frequency and analyzes the
equivalent circuit of the WPT system at the resonant frequency.
If the power factor of the fundamental waves on the rectifier
is equal to 1 and losses of the rectifier can be ignored, the
whole of the rectifier and load are equated with a pure electrical
resistance [10]. An equivalent resistance RL is expressed as
follows:
v20
,
(12)
RL =
i2
Fig. 3.
r
C
O
To design a feedback controller by using linear control theory, the state space model is linearized around an equilibrium
point because the DC–DC converter is a non–linear system.
When IL , Vdc , Idc , and D are defined as the equilibrium point
of iL (t), vdc (t), idc (t), and d(t), the linearized model of the
DC–DC converter is expressed as:
d
∆x(t) = ∆A∆x(t) + ∆B∆u(t)
(8)
dt
∆vdc (t) = ∆c∆x(t)
(9)


Vdc
D
r
−L L
0
[
]
L
∆A ∆B


IL
:=  − D
0 − C C1  .
C
∆c
O
0
1
O
X := [IL
SW1
T
x(t) :=
[
E
idc (t)
E
Circuit configuration of DC–DC converter.
where v20 is defined as the root–mean–square voltage of the
fundamental secondary voltage. On the other hand, the voltage
ratio of the fundamental secondary voltage v20 to fundamental
primary voltage v10 is described as follows:
v20
.
(13)
Av =
v10
From eq. (2), (12), and (13), the root–mean–square value of
the secondary current i2 is expressed as follows:
ω0 Lm v10 − R1 v20
i2 =
,
(14)
R1 R2 + (ω0 Lm )2
where v10 and v20 can be expressed by the Fourier series
expansion of the primary voltage v1 and the secondary voltage
v2 . As a result, i2 can be described as follows:
√
2 2 ω0 Lm v1 − R1 v2
i2 =
.
(15)
π R1 R2 + (ω0 Lm )2
Then, the average current flowing into the DC–link capacitor
idc is expressed as follows:
8 ω0 Lm v1 − R1 vdc
idc = 2
,
(16)
π R1 R2 + (ω0 Lm )2
where vdc is equal to v2 . When Eq. (16) is linearized around
an equilibrium point, the microscopic fluctuation in Idc is
expressed as follows:
8
R1
∆idc = − 2
∆vdc .
(17)
π R1 R2 + (ω0 Lm )2
Applying eq. (17) to eq. (8), the linearized model of the
DC–DC converter is expressed as follows:
d
∆x(t) = ∆A∆x(t) + ∆B∆u(t)
(18)
dt
∆vdc (t) = ∆c∆x(t)
(19)
[
]
∆A ∆B
∆c
O


Vdc
D
− Lr
L
L


R1
:=  − D
− π82 C(R1 R2 +(ω
− ICL 
2
C
0 Lm ) )
0
1
0
x(t) := X + ∆x(t), ∆x(t) := [∆iL (t) ∆vdc (t)]
u(t) := D + ∆d(t), ∆u(t) := ∆d(t)
X := [IL
T
Vdc ] , U := D
T
i1
C1
Lm
R1
R2
C2
RL
i2
idc
d
SW1
L
v1
L1
L2
v2
iL
r
C
vdc
SW2
Power source
Fig. 4.
Transmitter and Receiver
Rectifier
E
DC-DC converter
Battery
Wireless power transfer system for maximum efficiency control.
D
Therefore, the transfer function from ∆d(s) to ∆vdc (s) is
described as follows:
b1 s + b0
∆Pv (s) = 2
.
(20)
s + a1 s + a0
8
R1
r
a1 := + 2
L π C{R1 R2 + (ω0 Lm )2 }
{
}
1
8
rR1
a0 :=
D2 + 2
LC
π R1 R2 + (ω0 Lm )2
IL
rIL − DVdc
b1 :=
, b0 :=
C
LC
IV.
C ONTROL SYSTEM DESIGN
A. Feedback controller for secondary voltage control
A feedback controller for the secondary voltage control is
designed by eq. (20). To apply the pole placement method, we
use a PID controller CP ID (s), which is expressed as follows:
CP ID (s) = KP +
KI
KD s
+
.
s
τD s + 1
(21)
A discretized controller of CP ID (s) is obtained by Tustin
transform, which is defined as follows:
s=
2(z − 1)
,
T (z + 1)
(22)
where T is a control period and equates the switching frequency of the DC–DC converter in this paper.
v
*
dc
+
Fig. 5.
∆d
CPID (z)
+
-
vdc
+
DC-DC Converter
Block diagram of secondary voltage control.
rectangular wave, Vdc can be obtained by eq. (5) and Idc can be
calculated by eq. (16). D and IL are obtained by eq. (10) and
(11). Therefore, the equilibrium point is expressed as follows:
√
R2
ω0 Lm
√
Vdc =
v1 , (23)
√
R1 R1 R2 + (ω0 Lm )2 + R1 R2
8 ω0 Lm v1 − R1 Vdc
Idc =
,
(24)
π 2 R1 R2 + (ω0 Lm )2
√
E + E 2 − 4rVdc Idc
D =
,
(25)
2Vdc
Idc
IL =
.
(26)
D
V.
E XPERIMENT
A. Experimental setup
B. Defining the equilibrium point
The experimental equipment is shown in Fig. 6. The
characteristics of the transmitter and the receiver are expressed
in Table 1 and the specification of the DC–DC converter is
shown in Table. 2. The transmitting distances were gradually
increased in 50 mm increments from 200 mm to 450 mm
during the experiment. The mutual inductance between the
transmitter and the receiver for each transmitting distance is
indicated by Table 3. These values were measured by a LCR
meter (NF Corporation ZM2371).
Vdc is determined as a reference of the secondary voltage which maximizes the transmitting efficiency. When the
waveform of the secondary voltage can be equated with a
The primary voltage v1 was generated by a function
generator (Tektronix AFG3021B) and a high speed bipolar
amplifier (NF Corporation HSA4014). The voltage amplitude
The secondary voltage control is implemented by using
the discretized controller CP ID (z) and its block diagram is
shown in Fig. 5. Then an equilibrium point has to be defined
adequately because the plant model is a small signal model
around the equilibrium point.
Secondary current i2 [A]
ϭ͘ϴ
ϭ͘ϲ
Calculated
ϭ͘ϰ
Measured
ϭ͘Ϯ
ϭ
Ϭ͘ϴ
Ϭ͘ϲ
Ϭ͘ϰ
Ϭ͘Ϯ
Ϭ
(a) Transmitter and receiver.
Experimental equipment.
TABLE I.
ϮϬϬ
ϮϱϬ
(b) DC–DC converter.
ϯϬϬ
ϯϱϬ
ϰϬϬ
ϰϱϬ
ϱϬϬ
ϰϬϬ
ϰϱϬ
ϱϬϬ
Transmitting distance [mm]
Fig. 7.
Experimental result of secondary current.
C HARACTERISTICS OF TRANSMITTER AND RECEIVER .
Primary coil
Secondary coil
1.28 Ω
638 µH
3990 pF
99.8 kHz
1.24 Ω
642 µH
3990 pF
99.4 kHz
Resistance R1 , R2
Inductance L1 , L2
Capacitance C1 , C2
Resonant frequency f1 , f2
Outer diameter
448 mm
Number of turns
56 turns
TABLE II.
S PECIFICATION
Battery voltage E
OF
DC-DC
CONVERTER .
12 V
Internal resistance r
800 mΩ
Inductance L
511 µH
Capacitance C
3400 µF
Carrier frequency fc
10 kHz
ϭ
Ϭ͘ϵϱ
Ϭ͘ϵ
η
Transmitting efficiency
Fig. 6.
ϭϱϬ
Ϭ͘ϴϱ
Ϭ͘ϴ
Ϭ͘ϳϱ
w/o control
w/ control
Ϭ͘ϳ
Ϭ͘ϲϱ
ϭϱϬ
ϮϬϬ
ϮϱϬ
ϯϬϬ
ϯϱϬ
Transmitting distance [mm]
TABLE III.
M UTUAL INDUCTANCE BETWEEN
TRANSMITTER AND
RECEIVER IN EACH TRANSMITTING DISTANCE .
Transmitting distance [mm]
Mutual inductance Lm [µH]
200
86.0
250
59.2
300
42.2
350
30.5
400
23.2
450
18.0
of v1 was tuned to 20 V and the frequency of v1 was set to 99.0
kHz. To equate the flowing current into the DC–link capacitor
with the average value, the switching frequency was set to
10 kHz, which is much lower than the resonant frequency of
WPT. The feedback controller was designed to place closed
loop poles at -1000 rad/s (multiple root).
B. Secondary current of wireless power transfer
The secondary current of WPT was measured to verify the
validity of eq. (15) in each transmitting distance. Then, the
secondary voltage was regulated to the amplitude of 18 V by
using the DC–DC converter.
Fig. 7 shows a comparison between the value calculated
by eq. (5) and the measured value. The results show that eq.
(5) is acceptable, as these values closely matched. However,
the error of the secondary current was increased according to
Fig. 8.
Experimental result of transmitting efficiency.
the increase of the transmitting distance. This is because the
waveform of the secondary voltage could not be equated with
a rectangular wave due to the variation of the DC–link voltage.
Using a DC–link capacitor which has a large capacitance and
using higher frequencies are effective for the problem.
C. Transmitting efficiency
The effectiveness of the secondary voltage control for maximizing the transmitting efficiency is verified by experiments.
The secondary voltage was controlled to satisfy eq. (5) by the
secondary voltage control. Without control, the duty cycle d
was set to 1, therefore conducting the upper switch of the
DC–DC converter at all times.
The experimental result of the transmitting efficiency is
shown in Fig. 8. At long distance transmission, the secondary
voltage control can increase the transmitting efficiency. When
the transmitting distance was short, however, high transmitting
efficiency was achieved regardless of the secondary voltage
control. In the case of wireless charging for moving EVs, the
secondary voltage control is effective because long distance
transmission is required.
30
w/o control
25
Charging power PL [W]
w/ control
20
15
10
5
0
ϭϱϬ
ϮϬϬ
ϮϱϬ
ϯϬϬ
ϯϱϬ
ϰϬϬ
ϰϱϬ
ϱϬϬ
Transmitting distance [mm]
Fig. 9.
Experimental result of charging power.
D. Charging power
The experimental result of the charging power is shown
in Fig. 10. At any transmitting distance, the charging power
with the maximum efficiency control was increased. Therefore
the maximum efficiency control can achieve not only highly
efficient transmission but also high power charging. This result
indicates that the secondary voltage control for maximizing
the transmitting efficiency is suitable for EV applications
regardless of the transmitting distance.
VI.
C ONCLUSION
This paper proposed a novel DC-DC converter model based
on the analysis of WPT and designed a feedback controller for
the maximum efficiency control of WPT. Experimental results
demonstrated that the analysis of the secondary current of WPT
is acceptable, and that the maximum efficiency control through
secondary voltage control is suitable for EV applications at any
transmitting distance.
In future works, the response characteristics of the secondary voltage control and its stability will be discussed. To
implement a wireless charging system for EVs in motion, the
maximum efficiency control according to the change in the
mutual inductance will be designed and the motor drive system
will be considered.
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